2021
116
Edua do Royo Amonda ain
Non-pe u ba i e physics
in la ice gauge heo ies
Di ec o /es
Azcoi i Pé ez, Vicen e
Follana Adín, Edua do
© Uni e sidad de Za agoza
Se icio de Publicaciones
ISSN 2254-7606
Edua do Royo Amonda ain
NON-PERTURBATIVE PHYSICS IN LATTICE
GAUGE THEORIES
Di ec o /es
Azcoi i Pé ez, Vicen e
Follana Adín, Edua do
Tesis Doc o al
Au o
2021
UNIVERSIDAD DE ZARAGOZA
Escuela de Doc o ado
P og ama de Doc o ado en Física
Reposi o io de la Uni e sidad de Za agoza – Zaguan h p://zaguan.uniza .es
Non-pe u ba i e physics
in la ice gauge heo ies
Doc o al disse a ion o
Edua do Royo Amonda ain
Supe ised by
Vicen e Azcoi i P´
e ez
Edua do Follana Ad´ın
UNIVERSIDAD DE ZARAGOZA
Depa amen o de F´
ısica Te´
o ica
&
Cen o de As opa ´
ıculas y F´
ısica de Al as Ene g´
ıas
2020
Reposi o io de la Uni e sidad de Za agoza – Zaguan h p://zaguan.uniza .es

Aknowledgemen s
Ag adecimien os
Lle a a buen pue o el abajo que aqu´ı se p esen a, y que ha sido o igen de
no pocos des elos, hab ´ıa sido imposible de no con a con el apoyo de muchas
pe sonas. Dada la di icul ad de nomb a a odas ellas en an pocas l´ıneas, que ´ıa
empeza po da las g acias de o ma gene al a quien, desde su p opio ´ambi o
y en mayo o meno medida, haya con ibuido en es a emp esa. Sin pe juicio
de lo an e io , y ogando se disculpe cualquie omisi´on que pueda p oduci se a
con inuaci´on, en lo que sigue quisie a des aca algunos nomb es que conside o
especialmen e ele an es.
En lo acad´emico, esul a ineludible econoce la labo del P o eso ado con el
que he enido la opo unidad de coincidi en las di e en es e apas del Sis ema
Educa i o. Buenos ejemplos se ´ıan mi p o eso a de Lengua en 1ode la E.S.O.
o mi u o du an e ese mismo cu so, de Dibujo, sin ol ida la p o eso a que, en
2ode la E.S.O., me in oluc ´o en la Olimpiada Ma em´a ica. Del mismo modo,
ya du an e la e apa uni e si a ia, pude dis u a de un g an elenco de docen es
que supie on ansmi i su pasi´on po es a disciplina, con menci´on especial pa a el
Depa amen o de F´ısica Te´o ica, cuyas asigna u as siemp e dejaban con ganas de
segui indagando en la ma e ia. El ag adecimien o al Depa amen o es doble, de
hecho, pues su acogida y acompa˜namien o a lo la go de es os a˜nos han sido dignas
de elogio. Ha sido pa a m´ı un au ´en ico hono habi a , siquie a empo almen e,
los pasillos de Te´o ica, en los que siemp e me han hecho sen i como en casa.
A caballo en e lo acad´emico y lo pe sonal, engo que da las g acias a mis
di ec o es, Vicen e y Edua do, cuya gu´ıa po los no-siemp e-cla os caminos del
doc o ado ha sido encomiable. De ellos he podido ap ende mucho, pe o po si es o
ue a poco, su con ibuci´on asciende ampliamen e lo p o esional. Adem´as de se
buenos colegas, en el sen ido m´as o mal del ´e mino, ambi´en han demos ado se
buenos amigos. En es e mismo sen ido, no pod ´e ag adece lo su icien e la excelen e
acogida de Giuseppe du an e mis es ancias en L’Aquila. Muy di ´ıcilmen e se puede
da en la ida con una pe sona de mayo calidad humana. No quie o ol ida me
ampoco de o os compa˜ne os ocasionales de iaje, como Alex o Ma eo, ni po
i
ii
supues o de mi compa˜ne o de a igas y despacho, Filibe o.
Ya en el e eno pe sonal, g acias a odas las amis ades que han es ado ah´ı en
an os momen os. A mis compa˜ne os del S adium Casablanca, con los que du an e
muchos a˜nos he podido dis u a del balonmano. A los del Ins i u o y a los de la
ca e a, con menci´on especial al eje e o mis a-populis a. A las Colonas, eino
animal incluido, que sol´ıamos se de Ca ´an—y que cada ez somos m´as. Y, po
supues o, g acias a mi amilia. G acias a Menchu, Mus a, No a y A i, que siemp e
me han hecho sen i como uno m´as. G acias a A a e, Jes´us, Alejand o y Ga o,
po que deci que siemp e me hab´eis apoyado incondicionalmen e se ´ıa queda se
co o. Hab´eis sido el mejo apoyo que un hijo o un he mano puede ene . G acias
ambi´en, Robin, po saca me a pasea odos los d´ıas, pues no hay mejo mane a
de da le una uel a a un p oblema ebelde.
Finalmen e y po encima de odo, g acias a Cecilia y a Yasmina, sob e cuyo
abajo de cuidados descansa el g ueso de es a esis. Ambas hab´eis sido condici´on
sine qua non pa a que es e abajo haya podido sali adelan e, an o como lo sois
pa a m´ı. Va po oso as.
iii
Es e abajo ha sido inanciado po el Minis e io de Econom´ıa y Compe i i idad
a a ´es de la ayuda p edoc o al pa a la o maci´on de doc o es BES-2013-063567.
De igual modo, cabe menciona los p oyec os del Minis e io de Econom´ıa y Com-
pe i i idad/Fondo Eu opeo de Desa ollo Regional FPA2012-35453 y FPA2015-
65745-P, y de la Dipu aci´on Gene al de A ag´on/Fondo Social Eu opeo 2015-E24/2.
4CONTENTS
unde lying in e ac ions o qua ks and gluons in he QCD amewo k, cons i u es a
e y ac i e ield o esea ch, ha includes a la ge a ie y o app oaches [24]. The
in e es ha his ma e gene a es is di ided be ween he beha io o he coupling
in he in a ed egion—whe e nowadays he e is no consensus abou αS(q2→0)
ending o ze o, eezing o e en di e ging—and a la ge momen a, whe e pe -
u ba i e QCD can be applied and bo h expe imen al and heo e ical me hods
y o p o ide he mos accu a e app oxima ion. In his con ex , la ice-based
s a egies ha e been capable o deli e ing esul s bo h in he in a ed egion and
in he high ene gy egime, whe e in ac hey p o ide he mos p ecise de e mina-
ion o αS(MZ) [2]. Ou wo k can be amed p ecisely in o hese app oaches ha
come om la ice QCD, and i elies upon a ghos -gluon e ex compu a ion, as
in [25–27].
This hesis is o gnized as ollows. In Chap e 1 we e iew he undamen als
o some o he opics p esen ed in his in oduc ion, including a b ie his o ical
e iew o he la ice app oach. A mo i a ion o he inclusion o a θ e m in he
QCD ac ion is de eloped in Chap e 2, oge he wi h a b ie e iew o he exis ing
app oaches o his p oblem. Chap e 3, based on he wo k p esen ed in [28], is
dedica ed o he Ising model wi hin an imagina y magne ic ield. He ea e we
add ess he s udy o he massi e Schwinge model wi h a θ e m. The 1- la o
esul s, elying on he wo k p esen ed in [29], a e epo ed in Chap e 4, whe eas
he 2- la o case is ea ed wi h a pseudo e mions app oach in Chap e 5. Finally,
Chap e 6 co e s he compu a ion o αs(q2) ia he ghos -gluon e ex. Las ly,
ou conclusions a e summa ized in he homonymous chap e and echnical de ails
conce ning he compu a ion o he cumulan expansion pe o med in Chap e 3
a e gi en in Appendix A.

In oducci´on
Va ias d´ecadas han pasado desde que la c omodin´amica cu´an ica (QCD, po sus
siglas en ingl´es) se es ableci´o como la eo ´ıa que desc ibe las in e acciones ue es.
Es ampliamen e acep ada como una de las eo ´ıas m´as exi osas de la ´ısica mode na
y ha sido pues a a p ueba de o ma exhaus i a, an o desde el pun o de is a e´o ico
como del expe imen al.
A al as ene g´ıas, QCD es asin ´o icamen e lib e, lo que signi ica que sus cons i-
uyen es undamen ales, qua ks ygluones, in e accionan con una in ensidad que
dec ece con o me la ene g´ıa alcanza escalas m´as al as. En es a si uaci´on, esul a
ac ible aplica la eo ´ıa de pe u baciones pa a esol e las in e acciones a co a
dis ancia. Al es udia secciones e icaces en p ocesos de al as ene g´ıas, es necesa io
ene en cuen a las in e acciones p oducidas an o a co a como a la ga dis ancia.
Sin emba go, con la ayuda de los eo emas de ac o izaci´on [1], es posible com-
bina QCD pe u ba i a con cie o inpu no pe u ba i o, p o enien e de uen es
emp´ı icas o e´o icas, y ob ene as´ı p edicciones que pueden se con on adas ex-
pe imen almen e. De hecho, es e p og ama ha sido lle ado a cabo po una amplia
comunidad de cien ´ı icos y cien ´ı icas, abajando en cien os de Uni e sidades, lab-
o a o ios y habi ualmen e o ganizados en g andes colabo aciones in e nacionales,
incluyendo a ios acele ado es de pa ´ıculas. La impo an e can idad de e idencias
ecogidas en es e p oceso ha pe mi ido a QCD con e i se en un componen e muy
iable del ac ual Modelo Es ´anda de ´ısica de pa ´ıculas. Adem´as, QCD pe u -
ba i a es hoy en d´ıa un campo de in es igaci´on muy ac i o, oda ez que cen os
como el G an Colisionado de Had ones p o een de nue os da os expe imen ales
cada a˜no [2], y debe in e i se un g an es ue zo e´o ico en el c´alculo de los p ime os
´e minos de las expansiones co espondien es.
Po o o lado, pa a escalas de ene g´ıa no an ele adas, la in e acci´on ue e
no puede se educida a una se ie con e gen e de diag amas de Feynman. De he-
cho, una de sus p opiedades ca ac e ´ıs icas es el llamado con inamien o de colo .
Es o signi ica que los qua ks se encuen an siemp e (excluyendo ´egimenes de al a
densidad) en es ados ligados, llamados had ones, que son neu ales espec o del
colo . En es a si uaci´on pu amen e no pe u ba i a, hay pocas ´ecnicas que puedan
analiza la eo ´ıa con ´exi o. P obablemen e la que mejo es ablecida es ´a es QCD
5
6CONTENTS
en el e ´ıculo, denominada comunmen e la ice QCD. Desde el abajo undacional
de Wilson en 1974 [3], el ´exi o del m´e odo ha ido c eciendo con el iempo. Si bien
du an e los p ime os a˜nos ealiza los c´alculos necesa ios pa a ex ae esul ados
signi ica i os de QCD pa ec´ıa muy lejano, el p og esi o e inamien o de los algo i -
mos jun o con el c ecimien o exponencial de la capacidad compu acional mundial
dio la uel a a la si uaci´on. Muchos hi os han sido ya alcanzados: simulaciones
p ecisas incluyendo los e ec os de loops de qua ks i uales [4], la de e minaci´on
del espec o de had ones lige os con e o es sis em´a icos o almen e con olados [5]
o, m´as ecien emen e, la compu aci´on de los spli ings de isosp´ın (es o es, de las
di e encias de masa en e neu ´on y p o ´on, u o os canales had ´onicos) con g an
acue do con los da os expe imen ales, incluso excediendo su p ecisi´on en algunos
casos [6]. Todos ellos son buenos ejemplos del ´exi o de es e en oque.
Po los mo i os ci ados, QCD es conside ada como la eo ´ıa que desc ibe co ec-
amen e la in e acci´on ue e, an o pa a al as como pa a bajas ene g´ıas, y la ice
QCD es econocido po la comunidad como un m´e odo ab ini io iable que iene
una in e acci´on ´u il con lo expe imen al, pa a aseando a Wilson [7]. Es en ado
pensa que, con la e oluci´on ac ual de la po encia de c´alculo, se ´ıa simplemen e
cues i´on de iempo que el en oque de la la ice en en ase y esol iese cada uno de
los p oblemas no pe u ba i os que oda ´ıa espe an una soluci´on. Po supues o, las
cosas no son an sencillas, e incluso cuando pa a un subconjun o de p oblemas bas-
a ´ıa con con a con equipos m´as po en es, exis en emas undamen ales que hoy
en d´ıa cons i uyen p egun as abie as. Al menos dos p oblemas compa en es e es-
a us: el compo amien o de la ma e ia a densidad ba i´onica ini a—incluyendo su
diag ama de ases de empe a u a-densidad—y los es udios que in oluc an e ec os
opol´ogicos en QCD. Aunque, como e emos, los in en os han sido nume osos, los
a ances en ambos campos han sido escasos. La p incipal di icul ad de ´as de es e
modes o p og eso en ambas ´a eas es la misma: la acci´on de la eo ´ıa es compleja,
y no exis e e o mulaci´on conocida que pueda e i a la apa ici´on de un p oblema
de signo se e o (SSP, po sus siglas en ingl´es).
El SSP puede se de inido como el p oblema de e alua num´e icamen e la in-
eg al de una unci´on muy oscila o ia, que adem´as depende de un g an n´ume o
de a iables. Pe enece a la clase de p oblemas NP-comple os [8], lo que signi ica
que encon a un algo i mo que supe ase el SSP en iempo polin´omico equi ald ´ıa
a p oba que P=NP, uno de los sie e P oblemas del milenio p opues os po el
Clay Ma hema ics Ins i u e. En o as palab as, si un algo i mo as´ı exis iese, e-
sol e ´ıa odos los p oblemas de clase NP en iempo polin´omico. Pe o, has a el
d´ıa de hoy, el dilema P s NP con in´ua sin espues a, y consecuen emen e no
exis e soluci´on gene al pa a el SSP que pe mi a la aplicaci´on de las ´ecnicas de
Mon eca lo usuales. As´ı pues, los es ue zos dedicados a in en a supe a es a di-
icul ad se concen an en la elabo aci´on de di e en es m´e odos, adap ados a las
CONTENTS 7
pa icula idades del p oblema en cues i´on. Es e es el caso de QCD con compo-
nen e imagina ia en la acci´on, que apa ece cuando es ´an p esen es los ´e minos
co espondien es al po encial qu´ımico (µ > 0) o a e ec os opol´ogicos (θ6= 0).
Con el obje i o de a anza en la comp ensi´on de QCD con acci´on compleja,
a ias p opues as se han desa ollado a lo la go de las ´ul imas d´ecadas. La
din´amica de Lange in compleja [9–11], los m´e odos desa ollados po Azcoi i e
al [12,13] y, m´as ecien emen e, los dedales de Le sche z [14–16] y el m´e odo de la
densidad de es ados [17–19], son ejemplos ele an es de algunos de es os in en os.
En gene al, es as es a egias han pe mi ido es udia un n´ume o conside able de
oy models y, en algunos casos, han ob enido esul ados in e esan es pa a QCD
con po encial qu´ımico ini o. Sin emba go, en el caso de θQCD no se ha podido
ob ene casi ning´un p og eso, debido bien a las di e en es limi aciones undamen-
ales de las que los m´e odos ci ados adolecen, bien po las di icul ades p ´ac icas
que sus espec i as implemen aciones implican.
En es e con ex o, la pa e p incipal de es a esis se ha dedicado al es udio
de modelos que su en de un SSP, como el modelo de Ising bidimiensional con
campo magn´e ico pu amen e imagina io, o el modelo de Schwinge masi o con un
´unico la o y un ´e mino θ. En el p ime caso, es udiamos el conocido modelo
po medio de ´ecnicas anal´ı icas, explo ando una egi´on del espacio de pa ´ame os
( empe a u a eal y campo magn´e ico imagina io) algo desa endida en la li e -
a u a, posiblemen e debido a la di icul ad de aplica ´ecnicas an o anal´ı icas como
num´e icas. De acue do con los pocos abajos que explo an es e sis ema [20,21],
se espe a una es uc u a de ases con cie a iqueza. Nues o abajo p e ende p o-
ba la alidez de uno de los m´e odos de Azcoi i e al [13,21] en es e escena io y,
dado que puede a oja algo de luz en una egi´on c ´ı ica que su e de un SSP (que
ha obs aculizado la consecuci´on de una soluci´on num´e ica) espe amos que pueda
se i como e e encia pa a o os m´e odos que aspi en a supe a el p oblema del
signo en cualquie eo ´ıa gauge en el e ´ıculo.
Con el obje i o de en en a sis emas simila es a QCD con un ´e mino θ, y de
es e modo desa olla los m´e odos que lidian con el SSP, hemos es udiado el mod-
elo de Schwinge masi o con un la o y ´e mino θ, que se co esponde con QED
en dimensi´on 1 + 1. Como e emos, compa e un g an n´ume o de p opiedades
con QCD, incluyendo el con inamien o y, en cie o modo, la libe ad asin ´o ica,
po lo que de hecho es ampliamen e usado como su oy model. Adem´as, de ini
la ca ga opol´ogica en es e modelo es casi i ial, en con as e con cualquie a de
las de iniciones usuales pa a es e obse able en QCD, que esul an mucho m´as
in incadas. Es e hecho nos ha pe mi ido explo a el modelo con un cos e com-
pu acional ac ible, ob eniendo esul ados compa ibles con la p edicci´on anal´ı ica
de Coleman [22] y, lo que es m´as impo an e, poninedo a p ueba el m´e odo de-
sa ollado en [13] en una eo ´ıa gauge con e miones, lo que cons i uye un paso
8CONTENTS
impo an e en el camino a su aplicaci´on en QCD.
Como de i ado de la l´ınea de abajo an e io , y empujados po la necesidad
de una mayo op imizaci´on de los algo i mos an e io es, ambi´en hemos analizado
la e si´on de 2 la o s del modelo de Schwinge . En es e caso, se ha e i ado el
c´alculo comple o del de e minan e e mi´onico siguiendo un en oque basado en la
´ecnica de los pseudo e miones [23].
M´as all´a del es udio de sis emas a ec ados po un p oblema de signo, o o ema,
incluido den o de la ice QCD, se ha a ado en es a esis: el unning coupling
αS. La dependencia de αS(q2) con el momen o ans e ido q, que codi ica las
in e acciones subyacen es de qua ks y gluones en el ma co de QCD, cons i uye un
campo de in es igaci´on muy ac i o, que incluye una g an a iedad de me odolog´ıas
[24]. El in e ´es que es a ma e ia gene a se di ide en e el compo amien o de
es e acoplamien o en la egi´on in a oja, donde hoy en d´ıa no exis e consenso
sob e si αS(q2→0) iende a ce o, se congela o incluso di e ge, y a momen os
al os, donde QCD pe u ba i a puede se aplicada y m´e odos an o e´o icos como
expe imen ales in en an p o ee la ap oximaci´on m´as p ecisa. En es e con ex o,
las es a egias basadas en el e ´ıculo han sido capaces de o ece esul ados pa a
ambos casos, consiguiendo de hecho la de e minaci´on m´as p ecisa pa a αS(MZ) [2].
Nues o abajo puede ubica se p ecisamen e en e los en oques que ienen de la
la ice, y se apoya en un c´alculo del ´e ice ghos -gluon, como en [25–27].
Es a esis se o ganiza como sigue. En el Cap´ı ulo 1 epasamos lo undamen al
de algunos de los emas p esen ados en es a in oducci´on, incluyendo un b e e
esumen his ´o ico sob e el o igen de la ice QCD. Una mo i aci´on pa a la inclusi´on
del ´e mino θen la acci´on de QCD se desa olla en el Cap´ı ulo 2, jun o con
un b e e epaso de de los en oques exis en es pa a es e p oblema. El Cap´ı ulo
3, basado en el abajo p esen ado en [28], se dedica al modelo de Ising en un
campo magn´e ico pu amen e imagina io. A con inuaci´on en en amos el es udio
del modelo de Schwinge masi o con ´e mino θ. Los esul ados pa a el caso de
un ´unico la o , p esen ados en [29], se o ecen en el Cap´ı ulo 4, mien as que
el caso de 2 la o s se a a con un m´e odo basado en pseudo e miones en el
Cap´ı ulo 5. Finalmen e, el Cap´ı ulo 6 cub e la compu aci´on de αs(q2) median e el
´e ice ghos -gluon. Pa a acaba , nues as conclusiones se esumen en el cap´ı ulo
hom´onomio, y algunos de alles ´ecnicos, que concie nen al c´ompu o de la expansi´on
de cumulan es ealizada en el Cap´ı ulo 3, se discu en en el Ap´endice A.
Chap e 1
The la ice app oach
In his chap e we co e he essen ial poin s o he la ice app oach o QCD, includ-
ing a b ie his o ical e iew o i s bi h and e olu ion o e he pas ew decades.
The main aspec s o he o malism a e explained, discussing he s enghs and lim-
i a ions o Mon e Ca lo me hods when s udying la ice gauge heo ies. Finally,
some conside a ions abou he ype o e o s associa ed wi h his me hodology a e
discussed, ecalling how we can con ol hem and, e en ually, in which way we can
p o ide a p ecise es ima ion o a gi en obse able.
1.1 The dawn o colo
The appea ance o qua ks and gluons as he undamen al cons i uen s o ba yons
and mesons is ela i ely ecen . In o de o gi e he necessa y con ex , i is desi -
able o go back o he middle o he las cen u y. The disco e y o he pion h ough
cosmic ay expe imen s in 1947 [30], which was soon ollowed by hose o he i s
s ange pa icles, he kaon and he Λ, ma ked he beginning o a endency ha
con inued du ing he 50s, by means o di e en expe imen s in ol ing pa icle col-
lide s. This expe imen al ac , i.e., he disco e y o a la ge numbe o new pa icles
and esonances ha we e somehow ela ed, claimed o an explana ion in e ms o
a educed se o deg ees o eedom. I was in 1961 when Mu ay Gell-Mann (who
had p e iously in oduced he s angeness as a quan um numbe , conse ed by
bo h elec omagne ic and s ong in e ac ions) p o ided a success ul explana ion o
he so-called pa icle zoo, in his amous The Eigh old Way [31]. By ex ending he
SU(2) isospin symme y, Gell-Mann p oposed a SU(3) la o symme y, b oken
by mass di e ences, ha was capable o o ganize all he obse ed had onic s a es.
Mo eo e , his symme y p edic ed he exis ence o he Ω−ba yon, a pa icle ha
was obse ed h ee yea s la e , wi h a mass ha ma ched accu a ely he alue
an icipa ed by he model [32]. P ecisely Gell-Mann in 1964, and independen ly
9

10 CHAPTER 1. THE LATTICE APPROACH
Zweig [33], p oposed ha bo h mesons and ba yons we e composed o mo e un-
damen al cons i uen s (called qua ks by he men o , and aces by his pupil) which
hold ac ional elec ic cha ge.
A his poin , some ques ions s ill emained o comple e he qua k puzzle. Pa -
icula ly ele an was he ac ha he Ω−pa icle, composed o h ee squa ks in
i s g ound s a e, should ha e a symme ic wa e unc ion. This was in con adic-
ion wi h Pauli exclusion p inciple, which equi ed an an isymme ic wa e unc ion
unde he exchange o wo qua ks, which we e assumed o be e mions o spin 1/2.
Ano he conce n in ol ed he decay ampli udes p edic ed by he qua k model,
which di e ed om he alues measu ed in elec on-posi on collide s. Bo h issues
we e add essed by Gell-Mann, F i zsch and Ba deen, who du ing 1971 and 1972
in oduced a new exac SU(3) symme y o he qua ks, named colo [34–36],
which was exac ly conse ed. Colo was soon in e p e ed as a gauge symme y
and a ield heo y was cons uc ed by Gell-Mann, F i zsch and Leu wyle [37] and
independen ly by G oss and Wilczek [38] in 1973, who emphasized (as Poli ze
in [39]) one o he cha ac e is ic p ope ies o he new heo e ical a i ac : asymp-
o ic eedom. La e , F i zsch and Gell-Mann would inally gi e he heo y i s
mode n name: Quan um Ch omodynamics.
In o de o comple e he pic u e o QCD, ano he phenomenon had o be ex-
plained: why expe imen s only measu e colo single s, i.e., ba yons and mesons
a e ee o any colo cha ge, and indi idual qua ks a e no p esen in na u e as
ee pa icles. To explain his con inemen o qua ks and gluons in o had ons,
Wilson demons a ed in 1974 how la ice gauge heo ies con ine cha ged s a es in
he s ong coupling limi [3]. This wo k se he basis o he quali a i e unde -
s anding o colo con inemen —and in his way closes he ea ly pe iod in which
QCD was cons uc ed—bu i s in luence o e he heo y would anscend by a
his objec i e, since he egula iza ion de eloped by Wilson opened a whole new
ield wi hin high ene gy physics. Wi h espec o con inemen i sel , i should be
no ed ha , al hough he e exis b oad nume ical e idence o i s alidi y, oday a
igo ous ma hema ical p oo is s ill lacking.
1.2 A disc e ized space ime
The o mula ion o Quan um Ch omodynamics as he dynamical heo y o he
s ong in e ac ion, oge he wi h he unde s anding o asymp o ic eedom and
con inemen , is wi hou doub one o he key miles ones eached du ing he pas
cen u y. E en so, and due o he s ongly coupled na u e o he heo y, he p e-
dic i e powe o QCD was se e ely limi ed du ing i s i s s eps. Specially a low
ene gies, pe u ba i e me hods—de eloped wi h g ea success o p ocesses in ol -
ing Quan um Elec odynamics—we e o no use, making obse ables such as he
1.2. A DISCRETIZED SPACETIME 11
had on spec um un eachable.
In his con ex , Wilson in oduced—in which is conside ed o be he ounda-
ional wo k o he la ice app oach [3]—a no el non-pe u ba i e egula iza ion o
QCD. The essen ial ideas o he p ocedu e, applicable o any gauge heo y, ha e
endu ed up o he p esen day, and a e ela i ely simple. S a ing om Feynman’s
pa h in eg al o malism, space ime is disc e ized in a ou -dimensional (Euclidean)
hype cube. All objec s a e now de ined in he si es o he hype cube, o he links
joining hem. This p ocedu e is done in such a way ha gauge symme y is ex-
ac ly p ese ed. Al hough Lo en z (o Euclidean) in a iance is los o any ini e
la ice spacing a, Wilson a gued ha his obs acle could be o e come by means o
a eno maliza ion-g oup app oach, possible i he e exis s a c i ical poin a some
alue o he gauge-coupling o he heo y.
A his poin , and e en i la ice QCD had elucida ed he con inemen phe-
nomenon, i was unclea how he app oach could be exploi ed in o de o calcula e
ele an physical obse ables. The u ning poin came by he end o he decade,
when a p ocedu e widely used a he ime in S a is ical Mechanics was in oduced
in o he ealm o Field Theo y. In 1978, Wilson [40] p oposed o apply Mon e
Ca lo me hods, namely a Me opolis algo i hm, o la ice gauge heo ies, in o de
o elucida e nume ically he con inemen phenomenon. Wi h a de ailed p esc ip-
ion o how obse ables should be measu ed in such a amewo k, he also poin ed
ou ha he i s calcula ions we e al eady ongoing o he SU(2) gauge heo y.
The i s nume ical esul s, ha demons a ed he po en ial o he new echnique,
came by C eu z, Jacobs and Rebbi in 1979 [41], who pe o med a simula ion o
he gauge Z(2) model in ou dimensions, obse ing a i s -o de ansi ion ha
suppo ed he con inemen hypo hesis by coun e ing a p e ious conjec u e due o
Migdal [42]. In he same yea , Wilson [43] p esen ed ano he eno maliza ion-
g oup app oach o he SU(2) gauge heo y, impo ing block-spin echniques om
S a is ical Mechanics. Al eady in 1980, C eu z [44] p o ided u he e idence o
bo h con inemen and asymp o ic eedom in he SU(2) gauge heo y, ensu ing
in his way he possibili y o aking he con inuum limi by holding cons an a
physical obse able like he s ing ension.
These no el ideas c ys allized in se e al wo ks o e he yea s o ollow, including
he pionee ing compu a ions o he ho meson mass o a disc e e app oxima ion o
he SU(2) heo y, by Weinga en [45], o he calcula ion o se e al had on masses
due o Hambe and Pa isi [46], pe o med al eady wi h SU(3) as he gauge g oup.
In bo h cases, he limi s imposed by compu a ional esou ces we e alle ia ed by
he use o he so-called quenched app oxima ion, which neglec s en i ely he e ec
o i ual qua k loops—wi h he associa ed addi ion o uncon olled e o s. The
inclusion o ull dynamical e mions, i.e., aking in o accoun bo h gluon and qua k
dynamics, was achie ed i s in 1983 by Azcoi i and Nakamu a [47], who, leaning
12 CHAPTER 1. THE LATTICE APPROACH
on he pseudo e mions me hod p oposed by Fuci o e al [23], compu ed he mass
spli ing o he ho and omega mesons o he icosahed al app oxima ion o SU(2)
as he gauge g oup. A simila compu a ion, also elying in he pseudo e mionic
app oach, was made by Hambe in 1985 [48], al eady using SU(3) as he gauge
symme y g oup.
These ea ly s udies, e en when hey we e made wi h e y modes compu-
a ional powe by oday s anda ds, and wi h a numbe o sou ces o uncon-
olled sys ema ic e o s—especially signi ican o hose made wi hin he quenched
app oxima ion—we e success ul in p o ing he easibili y o he la ice app oach.
Du ing he subsequen decades, compu ing esou ces kep g owing s eadily. In pa -
allel, new algo i hmic and heo e ical ad ances we e de eloped, which ul ima ely
made possible b eak h oughs such as he compu a ion o he ligh had on spec um
om i s p inciples [5] o , mo e ecen ly, he de e mina ion o he isospin mass
spli ings [6]. A p esen ime, he e exis many esea ch g oups ac i ely wo king
in his a ea; s a e-o - he-a esul s can be e iewed annually in he p oceedings
o he In e na ional Symposium on La ice Field Theo y.
Fo mo e de ails abou he his o ical de elopmen o he la ice app oach, in-
cluding a mo e echnical discussion on he issue, we e e he in e es ed eade o
he ex ensi e e iew o Fodo and Hoelbling [49].
1.3 The o malism
A his poin , and be o e discussing some o he ca ea s o he la ice app oach, i
seems desi able o ske ch a leas he basic p emises o he me hod. In any case,
o a mo e exhaus i e in oduc ion o he opic, we ecommend he in e es ed
eade he lec u e no es o Da ies [50] o he mo e ecen book by Ga inge and
Lang [51], which in ac has se ed as a e e ence o some o he opics p esen ed
in his sec ion.
1.3.1 A ini e pa h in eg al
The s a ing poin o he app oach is he pa h in eg al o malism [52,53], which, o
a gi en quan um ield heo y, allows o w i e i s pa i ion unc ion as a unc ional
in eg al
Z=ZDΦe−S[Φ],(1.1)
whe e Sis he euclidean ac ion o he heo y—e en being conside ed a imagina y
ime, physical in o ma ion can be eco e ed as long as he heo y ul ills a se o
axioms [54,55]—and he in eg a ion is mean o be pe o med o e all possible con-
igu a ions o he ields o he heo y, deno ed gene ically by Φ. I we pa icula ize
1.3. THE FORMALISM 13
o QCD, he ields o be conside ed a e qua ks ψ, an iqua ks ¯
ψand gluons Aµ,
and (1.1) can be o mula ed schema ically as
ZQCD =ZDψD¯
ψDAµe−SQCD[ψ, ¯
ψ,Aµ],(1.2)
wi h acuum expec a ion alues o ope a o s O(ψ, ¯
ψ, Aµ) being gi en by
hOi =1
ZQCD ZDψD¯
ψDAµO(ψ, ¯
ψ, Aµ)e−SQCD[ψ, ¯
ψ,Aµ].(1.3)
The p ecise meaning o he in eg a ion symbol p esen in (1.2) and (1.3) is a sub le
ma hema ical issue; a his poin he heo y would be ill-de ined and a egula iza-
ion is manda o y in o de o ex ac any physical in o ma ion and a oid di e -
gences. To his end, he e a e se e al possibili ies ha lean on pe u ba i e ex-
pansions in he coupling, such as Pauli-Villa s (o dimensional) egula iza ion [56],
which ha e been widely used bo h in QED and QCD. The p esc ip ion o Wilson
ha was in oduced ea lie [3], is howe e he only ully non-pe u ba i e egu-
la iza ion known, allowing o s udy quan um ield heo ies om i s p inciples,
especially QCD beyond sho -dis ance in e ac ions, whe e con inemen — oge he
wi h he highly non- i ial s uc u e o he QCD acuum—poses an insu moun -
able obs acle o pe u ba i e-based app oaches.
The eby, going ahead wi h QCD, we disc e ize he con inuum ou -dimensional
euclidean space- ime in an hype cubic g id wi h cons an la ice spacing a; his
pa ame e will be he egula o o he heo y, in which limi a→0 i s o iginal
e sion is eco e ed. Mo eo e , we es ic he ull space ime o a simple 4-d box
o ini e ex en , limi ing in his way he spa ial olume and he imagina y- ime
e olu ion o he sys em. By doing so, he numbe o a iables o conside be-
comes ini e and he o iginal p oblem can begin o be pic u ed as compu a ionally
ea able. I we e u n o (1.2), he ields ψ(x) and ¯
ψ(x) now ake alues only o
x=a(n1, n2, n3, n4), whe e nia e in ege s sa is ying 0 ≤nia < Li,Libeing he
spa ial ( empo al) ex en o he 4-d box in each dimension.
Wi h espec o he gluon ield Aµ(x), i is con enien o pos pone b ie ly i s
de ini ion on he la ice. Fi s , we should no e ha he ac ion o he heo y, SQCD
in (1.2), is he space ime in eg al o he ollowing euclidean1lag angian densi y,
LQCD =¯
ψ(iγµDµ+m)ψ+1
2g2T (FµνFµν),(1.4)
1The o dina y eal- ime e sion o LQCD, i.e., wi h Minkowski space ime me ic, is jus
¯
ψ(iγµDµ−m)ψ−T (Fµν Fµν)/2g2.No e ha we a e using a a he compac no a ion, whe e
only space ime indexes a e explici , while hose co esponding o colo , spin and la o a e kep
implici . A mo e de ailed no a ion could label qua k (and an iqua k) ield componen s by ψ
α,c,
since ψis a 3-colo ec o , a 4-Di ac spino and a n - la o ec o . In he same way, mis a
diagonal ma ix in la o space, con aining each one o he qua k masses.
20 CHAPTER 1. THE LATTICE APPROACH
p esc ip ions ha minimize u he he o de o he e o s, he o iginal p oposal o
Wilson p o ides a good balance be ween he p ecision achie ed and he complexi y
o i s de ini ion. In any case, imp o ed gauge ac ions a e buil by adding highe
o de e ms o (1.17), in ol ing loops la ge han he plaque e ha a e supp essed
by powe s o a. This ype o gluonic ac ions a e s a e-o - he-a in mode n la ice
compu a ions.
1.3.3 In e lude: chi al symme y in he con inuum
Jus be o e add essing he cons uc ion o he e mionic ma ix M, we ound nec-
essa y o make a leas a shallow e iew o he ole o chi al symme y in QCD, as
i s o mula ion on he la ice has deep and di ec implica ions in how dynamical
e mions can en e in o he la ice ac ion.
In i s con inuum o mula ion, he ac ion o QCD o N la o s is gi en by he
space ime in eg al o he lag angian (1.4). Focusing on he e mionic pa and
keeping a compac no a ion, whe e ψand ¯
ψ ields a e unde s ood o be N - ec o s
in la o space, we ha e
SQCD-F =Zd4x¯
ψ(iγµDµ+m)ψ. (1.18)
Fo he pa icula case o ze o e mion mass, i.e., i all he componen s o he
diagonal ma ix m anish, he e exis wo se s o ans o ma ions ha lea e he
abo e ac ion in a ian . The i s amily is composed by he so-called ec o ans-
o ma ions, gi en by
ψ→eiα1ψ¯
ψ→¯
ψe−iα1,(1.19)
ψ→eiαTiψ¯
ψ→¯
ψe−iαTi,(1.20)
whe e he Tima ices a e he gene a o s o he la o g oup, SU(N ), and so, he
index i uns om 1 o N2
−1. When conside ing al oge he (1.19) and (1.20), he
g oup ex ends o U(N ). The o he amily o ans o ma ions can be cons uc ed
by including a γ5ma ix in he p e ious ones, and hus hey a e gi en by
ψ→eiαγ51ψ¯
ψ→¯
ψe−iαγ51,(1.21)
ψ→eiαγ5Tiψ¯
ψ→¯
ψe−iαγ5Ti.(1.22)
This las se ecei es he name o axial ans o ma ions. When conside ed oge he
wi h ec o ans o ma ions, hey ecei e he name o chi al ans o ma ions; in he
same way, he massless limi —in which hey p ese e SQCD—is called chi al limi .
To di e en ia e he wo sec o s ha compose chi al symme y, i is cus oma y o
label hem wi h a Vo a Asubsc ip . In his way, he ull chi al g oup is gi en by
SU(N )V×SU(N )A×U(1)V×U(1)A.(1.23)

1.3. THE FORMALISM 21
I is impo an o no e ha axial ans o ma ions a e symme ies o he massless
ac ion, bu his symme y is explici ly b oken o ini e mass qua ks. On he
con a y, ec o ans o ma ions (1.20) p ese e he ac ion also in he degene a e
mass case—i.e., N species o equal non-ze o mass—con o ming he well-known
isospin symme y. Fu he mo e, (1.19) is always a symme y o he ac ion, which
implies ba yon numbe conse a ion.
Equally ele an is he so-called axial anomaly. Al hough U(1)Ais a symme y
o he massless ac ion, i is explici ly b oken in he ully quan ized heo y. This
educes he ull exp ession (1.23) o he ollowing symme y g oup:
SU(N )V×SU(N )A×U(1)V.(1.24)
This g oup is usually exp essed in sligh ly di e en e ms. To his aim, i su ices
o ecall ha chi al symme y spli s bo h he qua k ields and he ac ion in o wo
sepa a e pieces, commonly named le - and igh -handed. De ining he p ojec o s
PLand PRas
PL=1−γ5
2PR=1+γ5
2,(1.25)
i is possible o pa i ion he ull la o space, since in oducing he p ojec ed qua k
and an iqua k ields
ψL,R ≡PL,Rψ¯
ψL,R ≡¯
ψPR,L,(1.26)
allows o exp ess he qua k (an iqua k) ield as he sum o ψLand ψR, sepa a ing
in his way he ac ion in o se e al componen s:
SQCD-F =Zd4x¯
ψL(iγµDµ)ψL+¯
ψR(iγµDµ)ψR+¯
ψLmψR+¯
ψRmψL.
(1.27)
F om he abo e exp ession i i ially ollows ha , in he chi al limi , only he i s
wo componen s su i e. Rema kably, hey a e comple ely decoupled in his limi ,
in e ac ing only h ough he mass e ms in he abo e o mula. Fo his easons,
he chi al symme y g oup (1.24) is o en e o mula ed in e ms o i s le - and
igh -handed deg ees o eedom, i.e., as
SU(N )L×SU(N )R×U(1)V.(1.28)
In any case, when conside ing qua ks o ini e bu degene a e mass, he axial sec o
symme y is b oken explici ly—jus he SU(N ) pa , since he one co esponding
o U(1)Awas al eady b oken by he anomaly—and he abo e g oup ge s educed
o i s ec o sec o U(N )V, o ollowing he s uc u e o (1.24)
SU(N )V×U(1)V.(1.29)
22 CHAPTER 1. THE LATTICE APPROACH
E en ually, i one conside s N la o s wi h di e en masses, he abo e symme y
ge s educed o he enso p oduc o N copies o U(1)V.
Be o e closing his in e lude, we wan o s ess a couple o issues conce ning
chi al symme y. The i s one is ha chi al symme y—o mo e p ecisely, i s
axial sec o —is spon aneously b oken in QCD a ze o empe a u e. Le us
conside he ac ion wi h jus wo la o s: up and down. Since hese qua ks ha e
a e y small mass compa ed o he QCD scale, i s explici symme y b eaking,
co esponding o he shi be ween (1.28) and (1.29) o N = 2, should gi e ise
o expe imen al e ec s. In pa icula , some pa icles—as, e.g., pa i y pa ne s—
should ha e almos -degene a e masses. Howe e , his expec ed symme y does no
ma ch he obse a ions; mass di e ences due o he explici b eaking o he QCD
ac ion a e signi ican ly smalle han he expe imen al measu es. The o igin o
hese disc epancies is, p ecisely, ha he global symme y (1.28) is spon aneously
b oken, since he acuum o he heo y is no in a ian unde he co esponding
ans o ma ions. In o he wo ds, al hough he ac ion is almos p ese ing chi al
symme y, he g ound s a e o he sys em is no .
The second poin o discuss is jus a co olla y o he p e ious a gumen : since
he e is a con inuous symme y being spon aneously b oken, Golds one’s heo-
em p edic he appea ance o se e al massless bosons. This is in ac he case,
since i we conside jus qua ks uand d, we should expec h ee ligh bosons—
since he symme y is sligh ly explici ly b oken—which can be iden i ied wi h he
h ee pions π0,π±. Including also he s ange qua k s, would accoun o eigh
bosons, which can also be iden i ied wi h he ou kaons K0,¯
K0, K±and he η
meson, which sums up o he h ee pions. No e ha i we o e look he anomaly,
in he las case 9 ligh mesons should appea , and he η0pa icle should be also
conside ed wi hin his pic u e. I is p ecisely he anomaly wha allows o explain
he η0−π0mass di e ence, as was s essed in he p e ious subsec ion.
Wi h hese conside a ions, we can now p oceed o alk o e mions in he la ice,
wi h a las wo d o cau ion ha summa izes he p e ious a gumen . The ligh es
pa icles o he QCD spec um gain hei low masses hanks o he spon aneous
b eaking o chi al symme y. I we we e o weak he heo y and emo e he chi al
in a iance om he ac ion, he explici b eaking o his symme y would p e en
hese pa icles om being pseudo Golds one bosons, and consequen ly hei masses
would be expec ed o inc ease.
1.3.4 Dynamical e mions: doubling and chi al symme y
In o de o implemen ull QCD in he la ice, we need o choose an ac ion Sla
which, as in (1.12), can be decomposed in o wo componen s. Fi s , a pu ely glu-
onic e m Sg, o which a iable candida e has al eady been e iewed in subsec ion
1.3.2. In second place, a e m in ol ing qua k and an iqua k ields is needed. Now
1.3. THE FORMALISM 23
ha he basics o chi al symme y in con inuum QCD ha e been discussed, we can
ace he ask o cons uc ing his e mionic e m, comple ing in his way he la ice
ac ion. The eby, in his subsec ion we will analyze he di icul ies in p oposing a
p ope la ice e sion o he e mionic ma ix M, which de ines how qua ks and
an iqua ks in e ac .
While he cons uc ion o he Wilson gauge ac ion (1.17) was no e y ou-
blesome, he si ua ion changes subs an ially when ying o p oceed wi h he
e mionic pa o he ac ion in a simila way. The simples app oach o dis-
c e ize he con inuum e m consis s in eplacing he i s o de de i a i es in LQCD
by a s anda d symme ic di e ence. In his way, he con inuum e mionic e m
¯
ψ(iγµDµ+m)ψbecomes
1
2a¯
ψ(n)iγµUµ(n)ψ(n+ ˆµ)−U†
µ(n−ˆµ)ψ(n−ˆµ)+mψ(n),(1.30)
which is indeed gauge-in a ian , as bilocal p oduc s o qua k-an iqua k ields a e
connec ed by he co esponding link a iables. Un o una ely, his p ocedu e in-
oduces he so-called double s: o each o hese nai e e mions included in he
la ice ac ion, 16 copies appea in he con inuum limi , 15 o hem being unphys-
ical. In o de o ix his si ua ion, an al e na i e is o use Wilson e mions, which
add an ex a e m o he nai e o mula ion. I s aim is o assign a di e gen mass,
o o de O(a−1), o each o he unwan ed double s, decoupling hem om he he-
o y in he con inuum limi . This mechanism is su icien o ix he con inuum limi
o he ac ion, howe e , he axial sec o o chi al symme y ge s explici ly b oken
in he la ice by he ex a e m—no only he anomalous pa U(1)A, which in
ac should be b oken, bu he whole g oup U(N )A. In con inuum QCD, as i was
discussed in he p e ious subsec ion, chi al symme y plays a cen al ole, since i s
non-anomalous sec o is spon aneously b oken, p oducing app oxima e Golds one
bosons. Wi h an explici ly b oken chi al symme y, his mechanism is no possi-
ble anymo e: pa icles such as pions, kaons o he η, which con o m he so-called
meson oc e , acqui e masses well abo e i s o iginal alues. This beha io hinde s
ac ual simula ions om eaching he physical poin . In ac , du ing decades o
la ice nume ical wo ks, he achie ed mass o he pion—a e y ele an quan i y,
being he ligh es had on in he spec um—has been e y a om i s physical
alue, which makes necessa y o pe o m ex apola ions o he physical poin .
The p e ious discussion ises a na u al ques ion: is i possible o ind a e mion
o mula ion ha , whi hou in oducing unphysical mul iplici ies, p ese es chi al
symme y? Fo a long ime, i was belie ed ha , un o una ely, i was no . In
ac , a om being a echnical complica ion, he double s issue has i s oo in he
chi al anomaly o QCD. In he nai e la ice o mula ion o (1.30) he anomaly dis-
appea s, canceled exac ly by he ex a double s. Adding he Wilson e m, which
decouples he ex a pa icles, can be in e p e ed as in oducing a la ice e sion o
24 CHAPTER 1. THE LATTICE APPROACH
he anomaly by hand—wi h one undesi able e ec : i also b eaks he non-anomalus
pa o he symme y, spoiling he spon aneous b eaking mechanism. In ac , as
was p o en in 1981 by Ka s en and Smi [67], his is a gene al esul : ei he he
axial anomaly is canceled by he p esence o ex a e mions, o i is in oduced
in he la ice by a e m ha , necessa ily, explici ly b eaks chi al symme y, in-
cluding i s non-anomalous sec o . In he same way, a con empo a y esul due o
Nielsen and Ninomiya— he so-called no-go heo em—s a es ha i is no possible
o cons uc a la ice o mula ion o QCD ha has a he same ime absence o
double s, chi al symme y, and locali y [68, 69]. Fu he mo e, he e also exis s a
scheme-independen e sion o his esul , wi h sligh ly di e en condi ions [70],
ha makes he impossibili y o egula izing a heo y wi h chi al e mions a a he
p o ound ques ion, no exclusi e o he la ice app oach. As a consequence, a
choice mus be made be ween p ese ing chi al symme y in i s con inuum o m
o a oiding unwan ed deg ees o eedom; e e y e mion o mula ion on he la ice
su e s om one pa hology o he o he .
Bu , e en hough he abo e esul s a e co ec , a wo ka ound o una ely exis s.
Almos 40 yea s ago, Ginspa g and Wilson ealized ha a emnan o chi al symme-
y could be iden i ied wi hin he la ice o mula ion, in such a way ha he axial
anomaly is s ill p ope ly p ese ed and, a he same ime, no unphysical deg ees o
eedom a e in oduced [71]. While he con inuum o m o chi al symme y—i.e.,
he p ese a ion o he ans o ma ion se s (1.19) o (1.22)— equi es he massles
Di ac ope a o o an icommu e wi h γ5, he emnan symme y p oposed in [71]
e i ies a b oade es ic ion, namely
Dγ5+γ5D=aDγ5D, (1.31)
whe e D≡iγµDµs ands o he massless Di ac ope a o . This condi ion is in
ac di e en om i s con inuum coun e pa o e e y ini e la ice spacing a,
al hough hey con e ge in he a→0 limi . In his sense, (1.31) can be in e p e ed
as an ex ension o he con inuum de ini ion, wi h a singula ad an age: i is able
o e ade he Nielsen-Ninomiya no-go heo em, since only he con inuum o m o
chi al symme y—wi h a anishing igh -hand side in (1.31)—is a ec ed by i [72].
Mo eo e , a modi ied chi al o a ion can be de ined in he la ice, in such a way
ha he gauge ields ans o m essen ially as i s con inuum e sions, p ese ing
ele an esul s as, i.e., he index heo em [73]. Howe e , he solu ion gi en in [71]
was no cons uc i e, in he sense ha a pa icula o m o Dcould no be ound a
ha momen ; i would ook almos wo decades o ind a p ac ical implemen a ion
o hese ideas.
Wi hin he abo e scena io, o he al e na i es would be explo ed o e he yea s
o come. In ac , a numbe o hese s a egies we e de eloped and nowadays a e
pa o he cu en lo e o he ield. They can be classi ied in g oups acco ding o
1.3. THE FORMALISM 25
how hey deal wi h he doubling p oblem. Some app oaches choose o gi e up chi al
symme y; his is he case o Wilson e mions. In his ca ego y a e also included
clo e e mions, a Symanzik imp o ed e sion o Wilson e mions ha emo es
O(a) disc e iza ion e o s [74], and wis ed mass e mions, which conside pai s o
mass degene a e Wilson e mions oge he wi h an isospin mass spli ing e m [75].
An al e na i e app oach consis s in p ese ing chi al symme y, while assuming
some o he double s degene acy. He e he p ime example a e Kogu -Susskind o
s agge ed e mions [76]. This a ian is cons uc ed om he nai e o mula ion,
bu dis ibu es he usual Di ac 4-spino componen s o e neighbo ing si es, in such
a way ha he double s degene acy ge s educed om 16 o 4 species. Mo eo e ,
o some obse ables i is possible o emo e he esidual deg ees o eedom by
aking he ou h oo o he e mion de e minan —a p ocedu e commonly e e ed
o as oo ing, o iginally p oposed by Ma ina i, Pa isi and Rebbi, in he con ex o
he massi e Schwinge model [77]. Al hough i s alidi y was a i s con o e sial,
bo h nume ical e idence [78] and heo e ical a gumen s [79] suppo ha his
echnique leads o he co ec heo y as long as con inuum and chi al limi s a e
aken p ecisely in his o de . As wi h Wilson e mions, he e also exis Symanzik
imp o ed e sions o he s agge ed ac ion; wo o hem, widely used by he la ice
communi y, a e he Asq ad ac ion [80]— o a-squa ed adpole imp o ed—and he
HISQ ac ion [81]— o highly imp o ed s agge ed qua ks. In Chap e 4 we make
use o s agge ed qua ks, while on Chap e 6 he analyzed con igu a ions we e
gene a ed wi h he HISQ ac ion.
Fo he sake o comple eness—e en i i is less ela ed wi h he wo k de el-
oped in his hesis— he e is s ill ano he class o e mions ha dese es a leas
a men ion: he pa icula solu ions o he Ginspa g-Wilson equa ion (1.31). They
p ese e chi al symme y wi hou he double s pa hology, so in his sense hey a e
he bes possible e mions, and hey should be p e e ed when compa ed o o he
al e na i es. Howe e , all hei implemen a ions su e om he same illness— hey
a e by a he mos expensi e e mions in compu a ional e ms. As a consequence,
i s use is ese ed o si ua ions whe e chi al symme y is needed wi h conside able
p ecision. The e exis wo di e en solu ions ha a e b oadly used: hey a e called
o e lap [82–84] and domain-wall [85,86] e mions. In he i s case, he e mionic
ma ix is cons uc ed by ope a ing wi h a Wilson-like ma ix— he esul being
a cos ly non-spa se ma ix. Fo domain-wall e mions, an ex a i h dimension
o in ini e ex en is needed o p ese e chi al symme y. Since in p ac ice his is
implemen ed as an addi ional dimension in a ini e la ice, a mild iola ion o he
symme y s ill emains, al hough i can be con olled.

26 CHAPTER 1. THE LATTICE APPROACH
1.3.5 Mon e Ca lo: ensembles and obse ables
Summing up all he p e ious conside a ions, we ha e now a ully egula ized e -
sion o QCD. The eby, he o iginal exp essions o he pa i ion unc ion (1.2) and
he acuum expec a ion alue (1.3) o a gi en obse able acqui e now a well-de ined
ma hema ical meaning.
I we now wan o conside , e.g., a pu ely gluonic obse able O(U), we should
compu e he ollowing in eg al:
hOi =RDUO(U)e−Sla (U)
R la hcalDUe−Sla (U).(1.32)
The abo e in eg a ion measu e DUis composed by he p oduc o as many indi-
idual Haa measu es as links a e in he gi en la ice, which amoun o V≡QiLi
imes he dimension o he la ice. So, o almos any size ha we can hink
o , (1.32) con ains a highly mul idimensional in eg al, wi h a as con igu a ion
space ha , e en o small la ices, can no be ully explo ed by any compu a ional
means. The e exis howe e a class o nume ical app oaches, commonly e e ed
o as Mon e Ca lo me hods, ha a e capable o p ope ly sampling hese kind o
spaces— o apply hem o he la ice egula iza ion was, in ac , one o he key
p oposals o Wilson [40].
In o de o compu e he expec a ion alue o (1.32), a i s Mon e Ca lo ap-
p oach could consis in gene a ing a andom sample o link con igu a ions, i.e.,
a collec ion o ensemble o gauge ields Ui, each one con aining he in o ma ion
o each indi idual link Uµ(n) in he la ice. Then, hOi could be compu ed as a
weigh ed a e age o e he p e ious ensemble, wi h weigh e−Sla (U). In a simila
ashion, s a is ical e o s could be compu ed by s anda d echniques. Howe e ,
his app oach is oo nai e o he p oblem a hand, since he exponen ial ac o in
(1.32) highly supp esses a g ea majo i y o he con igu a ion space elemen s; in
o he wo ds, only a small subse o he whole space con ibu es signi ican ly, and a
nai e andom sampling o he space would miss he a ea o in e es . In o de o ix
his issue, impo ance sampling is o be applied: again, an ensemble o con igu a-
ions Uiis gene a ed, bu hey should be selec ed wi h p obabili y e−Sla (Ui). Then,
p o ided a well-dis ibu ed ensemble o Nelemen s is a ailable, an es ima o o
he expec a ion alue o a gi en obse able can be s aigh o wa dly compu ed as
¯
O=1
N
N
X
iO(Ui).(1.33)
The o iginal p oblem is now shi ed o he gene a ion o he Uicollec ion. This can
be achie ed by s a ing om a gi en ini ial con igu a ion, say U1, and ollowing a
Ma ko chain p ocess, in which he selec ion o he nex con igu a ion is egula ed
1.3. THE FORMALISM 27
by a p obabili y ha depends only on he immedia e p e ious s a e. The idea is
ha , e en i one s a s a om he meaning ul con igu a ions, he Ma ko p ocess
should d i e he sys em o an equilib ium s a e, in which he dis ibu ion e−Sla (Ui)
is ep oduced. To gua an ee ha , he ansi ion p obabili ies om each Ui o any
o he Ujneed o be adequa ely de ined. I is su icien (bu no necessa y) o
equi e he de ailed balance condi ion o be ul illed, so i we label he ansi ion
p obabili y o he chain as p(Ui|Uj),
p(Ui|Uj)e−Sla (Uj)=p(Uj|Ui)e−Sla (Ui)(1.34)
would be equi ed o e e y iand j. In his case, i he sys em is also e godic—
meaning e e y con igu a ion is accessible om any o he in a ini e numbe o
s eps—i is gua an eed ha , wi h independence o he ini ial con igu a ion chosen,
he Ma ko p ocess will each equilib ium and, consequen ly, a well-dis ibu ed
ensemble will be gene a ed.
Then, o elabo a e a pa icula algo i hm ha ollows he desc ibed Ma ko
p ocess consis s hen in speci ying how he nex elemen o he chain is selec ed,
and which ansi ion p obabili ies connec he con igu a ion space. Wi h e-
spec o he la e poin , mos o he algo i hms used by he la ice communi y
e i y he de ailed balance condi ion. Maybe he mos elemen a y o hem is
Me opolis algo i hm, which accep s a change be ween Uiand Ujwi h p obabili y
max {1, e−∆Sla }. This is exac ly he app oach ollowed in Chap e 3. Mo e in-
ol ed al e na i es include he hea ba h algo i hm [44], which o en includes some
o e elaxa ion s eps [87–89]. Fu he mo e, o he mo e gene al case in which
e mions a e included in o he ac ion, one o he mos widely used algo i hms is
he Hyb id Mon e Ca lo [90,91], which a each upda e combines a mic ocanonical
e olu ion wi h a inal me opolis s ep o accep o ejec he p oposed change.
In any case, he s ochas ic na u e o Mon e Ca lo me hods in oduce some un-
ce ain y in o he compu ed obse ables, in he o m o s a is ical e o s associa ed
o he co esponding es ima o s. Ne e heless, hese can be deal wi h wi hou
much di icul y, jus by aking in o accoun wo undamen als. Fi s , ha he
gene a ed ensemble has o be he malized, meaning ha su icien i e a ions need
o be spen o each he equilib ium dis ibu ion o he Ma ko chain. This can
be achie ed by moni o ing a se o obse ables ha allow o de e mine when he
Ma ko e olu ion is s a iona y. Secondly, and e en mo e impo an , is he inhe -
en co ela ion o he con igu a ions gene a ed by he upda e p ocess—gene ally a
local algo i hm ha needs a high numbe o i e a ions o p oduce a new con igu-
a ion a om he o iginal. In his case, he use o s anda d e o analysis ools,
such as jackni e binning, o e en he di ec compu a ion o au oco ela ion imes
o each obse able, allows o es ima e in a eliable way he s a is ical e o s o
any compu ed obse able.
28 CHAPTER 1. THE LATTICE APPROACH
1.4 Reaching he con inuum and sou ces o e o
Up o his poin , we ha e co e ed how con inuum QCD can be egula ized in o a
space ime la ice whe e, aking ad an age o Mon e Ca lo me hods, i is possible
o es ima e acuum expec a ion alues o gi en obse ables. The compu ed ob-
se ables will depend in gene al on he dimensions o he la ice and on he ba e
alues o he gauge coupling βand he N masses o he e mion species in ol ed.
Al hough knowing hese alues can be enough o some applica ions, in mos cal-
cula ions he objec i e is o ob ain a physically measu able quan i y ha can be
e en ually con as ed wi h expe imen . This is no he case o he p esen hesis,
since he wo k being p esen ed is no di ec ly conce ned abou he las pa o a
gene al la ice calcula ion; howe e , in o de o gi e a gene al pe spec i e o wha
a comple e compu a ion would equi e, and o he sake o comple eness, he ea e
we p oceed wi h a b ie o e iew o he issues in ol ing he las s eps o he la ice
app oach.
Ob aining a physical p edic ion om ba e la ice esul s equi es a somewha
in ol ed limi p ocedu e; no only he la ice spacing ashould anish— his would
be he con inuum limi —bu also he ba e couplings should go o hei physical
alues. In o he wo ds, he physical poin needs o be eached; i no , he simula ed
heo y would be desc ibing an al e na i e scena io, one wi h, e.g., di e en had on
masses. In ac , he couplings o he heo y a e no di ec ly obse able (since
con inemen p e en s qua k ields o mani es ou side a colo -single , he mass o
he qua k is no an obse able and depends on he eno maliza ion scheme). In
place, o he obse ables, such as had on masses a ios—which o cou se depend
on he a o emen ioned pa ame e s—a e o be compu ed in he la ice, so hey
can be compa ed wi h he expe imen al alues. Thus, in o de o compu e a gi en
quan i y, i is necessa y o keep ack o se e al addi ional obse ables, one o each
o he ee pa ame e s—o la o species— ha a e included in o he calcula ion.
Mo o e , i should be no ed ha he la ice spacing ais no a ee pa ame e o he
la ice heo y, bu ano he quan i y ha needs o be measu ed wi hin he la ice.
The abo e scena io can be summa ized as ollows: we need o ake he con in-
uum limi , a→0, while d i ing a se o physically measu able quan i ies o hei
expe imen ally de e mined alues. Mo eo e , since doing he compu a ions on a
la ice o in ini e ex en is ou o each, he p ocedu e is o be pe o med on a 4D
box o ini e physical size.5In o de o ake he a→0 limi , i is equi ed o e-
pea he compu a ions a di e en la ice sizes, keeping he physical size o he box
cons an . In his way, la ices wi h di e en alues o he la ice spacing a—and
consequen ly o he numbe o nodes N—a e compu ed, while he olume o he
5In ac , epea ing he calcula ions in se e al box sizes allows o ex apola e he esul s o
he he modynamic limi .
1.4. REACHING THE CONTINUUM AND SOURCES OF ERROR 29
box a4Nis kep cons an . I should be ecalled ha while Nis a ee pa ame e ,
ais an obse able ha needs o be measu ed (depending no only on he gauge
coupling bu also on he inclusion o e mionic species). So, as a i s consequence,
keeping he physical olume cons an is no a comple ely i ial issue.
The same p oblema ic ha appea s when ying o ix he olume o he 4D
box applies o e e y physical obse able ha is in ended o be kep cons an (o
d i en o a sui able alue, ob ained om expe imen al sou ces). To achie e his,
essen ially wo op ions a e possible: ei he he ee pa ame e s a e uned in an
i e a i e p ocess—one la ice a a ime up o he physical poin —o he compu a-
ions a e epea ed o se e al alues o hese pa ame e s, and hen in e pola ed o
he desi ed poin . A less ideal a ia ion o he la e is o each he physical poin
by means o an ex apola ion. This was in ac he ule du ing se e al decades o
la ice nume ical compu a ions, since he di e en ypes o e mion disc e iza ions
( e iewed in Sec ion 1.3.4) ha e many di icul ies in eaching a su icien ly low
mass o he pion.6
As a inal ema k, i is impo an o s ess ha any ull compu a ion on he
la ice should include a ca e ul analysis o all sou ces o e o s. Fi s , s a is ical
e o s a e in oduced by he Mon e Ca lo e alua ion p ocess. These a e howe e
ela i ely easy o deal wi h, since hey can be es ima ed by s anda d me hods and,
e en mo e impo an ly, can be sys ema ically educed by inc easing he compu a-
ion ime. A mo e challenging obs acle is in es ima ing he di e en issues leading
o sys ema ic e o s, specially when a high p ecision is desi ed—and p e iously
unno iced sys ema ic e o s can become big enough o be aken in o accoun . A
non-exhaus i e bu common eco d o he di e en sou ces o sys ema ics consid-
e ed would include he e o s associa ed wi h he uning o in e pola ion o he
physical poin and he con inuum limi p ocess. Usually less se e e a e hose e-
la ed wi h he he modynamic limi — ini e olume e ec s—o wi h he exclusion
om he ac ion o elec omagne ic e ec s, which, gi en ha he qua ks a e elec-
ically cha ged pa icles, should be accoun ed o . In gene al, any app oxima ion
assumed should accoun o i s co esponding e o —as, e.g., he case o some la -
ice e mion ypes which conside uand dqua k ba e masses as degene a e, and
hus a e equi ed o accoun o he co esponding isospin b eaking e ec s.
6Al hough Ginspa g-Wilson e mions p ese e chi al symme y, hey su e om a la ge com-
pu a ional o e head ha spoils hei ad an age wi h espec o o he e mion ypes, when each-
ing a physical pion mass is conside ed.
36 CHAPTER 2. TOPOLOGY IN QCD
pieces,
Lθ-QCD =LQCD +iθ g2
64π2µνρσFa
µνFa
ρσ,(2.8)
whe e LQCD s ands o he s anda d exp ession (1.4), which is bo h CP conse ing
and eal. In con as , he second piece in he igh hand side o (2.8) b eaks CP—
as was s essed in he p e ious sec ion—and, mo e impo an ly o he upcoming
discussion, is a pu ely imagina y numbe .
The opological na u e o he θ e m has di ec physical e ec s: he acuum
o he heo y canno be cons uc ed as a quan um luc ua ion a ound a classical
de ini e s a e [97]. The e o e, in o de o p ope ly analyze in which way QCD
depends on he θpa ame e , a non-pe u ba i e app oach is equi ed. Unde his
ci cums ances, i would seem adequa e o deal wi h he inclusion o he ex a e m
Lθwi hin he la ice egula iza ion o malism. In p inciple, i would su ice o
ind a p ope disc e iza ion o he opological cha ge Q, and include i s e ec s in
he co esponding Mon e Ca lo algo i hm. Incon enien ly, his is a om being
enough i non- anishing alues o he acuum angle a e o be explo ed, since he
θ e m amoun s o a complex phase in he ac ion ha gi es ise o a se e e sign
p oblem. Usual impo ance sampling me hods a e o no use in his si ua ion, and
wo ka ounds need o be ound. A p esen , much o he achie ed p og ess in his
opic in ol es compu a ions o opological quan i ies a θ= 0, whe e s anda d
MC me hods a e s ill applicable. Al hough he de ini ion o he opological cha ge
on he la ice is a sub le ques ion, i is possible o compu e quan i ies such as
he opological suscep ibili y χ, e en wi h some di icul ies [102]. Howe e , li le
p og ess has been achie ed in he las decades in wha conce ns he s udy o he
θ > 0 case.
I is wo h no ing ha he inclusion o he θ e m is no he only example o
a sign p oblem being induced by a complex componen in o he ac ion o QCD.
Ano he majo example is gi en by ini e densi y QCD, which akes place when a
non-ze o chemical po en ial e m is included in o he e mionic ma ix—a equi ed
addi ion when conside ing high ba yonic densi ies. This, in ac , is a ele an
scena io which a ec s se e al a eas. I is needed in he s udies o as ophysical
objec s such as neu on s a s. Mo eo e , in ea ly Uni e se in es iga ions, dealing
wi h ex emely high alues o bo h empe a u e and densi y is equi ed. And, las
bu no leas , pa icle collide acili ies ep oduce hese condi ions in hea y ion
expe imen s. Al hough, hanks o asymp o ic eedom, pe u ba i e expansions
can o e an insigh o some limi ing cases—namely high ene gy, which ansla es
in high To high µ—and he µ= 0 case is accessible o he s anda d impo ance
sampling echniques, almos he ull µ−Tphase diag am is beyond he each o
hese app oaches.
In bo h o he cases abo e, conside ing a complex ac ion esul s in he appea -
ance o a sign p oblem, which, as i was wa ned a he beginning o his hesis,

2.3. TRYING TO OVERCOME THE SIGN PROBLEM 37
cons i u es one o he Millenium p oblems—in pa icula one ha is no expec ed
o be sol ed wi h a posi i e ou come, which o ou in e es s would be P=NP.
The e o e, he communi y has di ec ed hei e o s owa ds he de elopmen o
di e en al e na i es ha y o e ade, o in some cases amelio a e, he sign p ob-
lem in he sys ems o in e es , i.e., in QCD o QCD-like models. In he nex
sec ion, we men ion some o he mo e popula s a egies, including a b ie e iew
o he me hods de eloped by Azcoi i e al [12, 13], which in ac a e applied in
Chap e 4, when econs uc ing he θdependence o he Schwinge model.
2.3 T ying o o e come he Sign P oblem
In o de o p og ess in he s udy o complex ac ion sys ems, di e en me hods ha e
been de eloped o e he yea s. In he cases whe e he sign p oblem is mild enough,
such as in ini e densi y QCD o small alues o he chemical po en ial µ, se e al
s a egies can be ollowed wi h success. Pe haps he mo e s aigh o wa d ap-
p oach is he echnique known as eweigh ing. I in oduces an auxilia y pa i ion
unc ion wi h non-nega i e densi y, which is used o e o mula e he expec a ion
alue o a gi en obse able—compu ed in he o iginal ensemble—in e ms o o he
expec ed alues ha a e o be de e mined wi hin he auxilia y ensemble. Indeed,
his change does no elimina e he p oblem, since he compu a ional cos o his
me hod escala es exponen ially wi h he olume o he sys em; howe e , i is use ul
when applied o small sys ems, o in ce ain egions o he pa ame e space, such
as QCD wi h a small chemical po en ial µ, whe e he e ec s o he sign p oblem
a e a om being se e e. In his scena io, al e na i e app oaches include Taylo
expansions a ound µ= 0 and analy ic con inua ions om pu ely imagina y chem-
ical po en ial, al hough i s scope is hea ily bounded by he p esence o a SSP; o
an ex ended discussion on his opic, we ecommend he eade [103,104].
Apa om he abo e me hods, ha p o ide some insigh o mild sign p oblem
egions—bu ail o deli e o he wise—o he p ocedu es ha ha e, in p inciple,
g ea e scope, ha e been de eloped o e he las decades. In ch onological o de ,
he i s is Complex Lange in. The o iginal wo ks o Pa isi and Klaude p oposed
o gene alize he Lange in equa ion o malism o he case o a complex- alued
dis ibu ion [9,10]. E en when igo ous p oo s we e lacking—e.g., he exis ence o
a s a iona y solu ion was a conjec u e— he echnique seemed o gi e co ec esul s
in some cases [105]. Howe e , in o he sys ems he algo i hm ailed o con e ge,
o i did bu o he w ong limi [106]. In e es in he me hod was ebounded when
Be ges and S ama escu ealized ha he ins abili ies o he Lange in e olu ion can
be deal wi h i he s epsize is educed enough [11]. In ac , using an adap a i e
s epsize elimina es his p oblem [107]. No wi hs anding i s ecen successes, he
app oach s ill has some ca ea s ha need o be add essed, since in some scena ios
38 CHAPTER 2. TOPOLOGY IN QCD
i con inues o con e ge o a w ong esul . Cu en e o s a e de o ed o iden i y
unde which condi ions he p ope solu ion can be eached, and wha mechanisms
can be used o moni o i he Lange in e olu ion is o be us ed; o a e iew on
his opic, we e e o [108].
Wi h a special ocus on sys ems wi h a opological e m in he ac ion, a di e en
s a egy was p oposed by Azcoi i e al in 2002 [12]. In summa y, he opological
cha ge dependence on θis econs uc ed om he p obabili y dis ibu ion unc ion
a θ= 0, which is o be compu ed om simula ions a pu ely imagina y alues
o θ, possible since in his case he ac ion becomes eal. These esul s a e o
be adjus ed o a sui able analy ical exp ession ha , once in eg a ed, allows o
ob ain q(θ) wi h he help o mul ip ecision algo i hms. This app oach p o ed o
wo k well in a numbe o sys ems, including he one dimensional Ising model and
he U(1) compac model in wo dimensions, gi ing p edic ions o CP3and, in a
la e wo k, o CP9—a model ha , as QCD, exhibi s con inemen and asymp o ic
eedom [109]. Ne e heless, he scope o he app oach ge s hampe ed by he ac
ha i canno econs uc a non-mono one o de pa ame e [21]. In o he wo ds, i
a gi en model b eaking CP ge s his symme y es o ed a θ=π, hen q(θ) needs
o dec ease a some poin ; in his si ua ion, he me hod ails o ep oduce his
beha io and a la ening is obse ed—which, on he o he hand, i is gene ally
a oided in models wi h (spon aneously) b oken symme y a θ=π.
Ini ially wi h he aim o c osschecking he abo e me hod, an al e na i e ap-
p oach, using he same inpu —i.e., Mon e Ca lo simula ions a imagina y alues
o he acuum angle θ—was p oposed in 2003 [13]. In his case, an addi ional
assump ion needs o be made, namely ha in o de o econs uc he ull θde-
pendence, no c i ical poin s a e allowed, excep a mos a θ=π. This poses
a se e e obs acle o i s applicabili y in ini e densi y QCD, whe e a ich phase
diag am is expec ed, bu is in p inciple well adap ed o θQCD, whe e only one
phase ansi ion—i any—is su mised, p ecisely a θ=π[102]. Besides he la e
assump ion, a pa icula ex apola ion is equi ed, ha needs ce ain obse ables
o a y as slowly as possible. Fo his eason, he me hod is expec ed o wo k well,
among o he s, in asymp o ically ee gauge heo ies. This app oach, which has
al eady ob ained good esul s in a a ie y o models [109–111], has been applied
in Chap e 4 o econs uc he θdependence o he massi e Schwinge model; o
a mo e ex ensi e e iew, co e ing he speci ics o he ac ual implemen a ion, we
e e he eade o Sec ion 4.3.
Finally, o he ecen app oaches o complex ac ion sys ems can be men ioned,
such as Le sche z himbles [14–16] o he densi y-o -s a es o LLR me hod [17–19].
Thei de ails a e beyond he scope o his hesis, since hei applica ion o QCD
wi h a opological e m seems, o now, emo e.
Chap e 3
Ising model wi h a θ e m
In his chap e we p esen ou wo k on he wo-dimensional an i e omagne ic Ising
model wi h a pu ely imagina y magne ic ield, which can be in e p e ed as a oy
model o he usual θphysics, and ha was published in [28]. Ou mo i a ion,
as i was an icipa ed a he beginning o his hesis, is wo old. Fi s , we p e end
o p o ide a benchma k calcula ion in a sys em which su e s om a s ong sign
p oblem, so ha ou esul s can be used o es Mon e Ca lo me hods de eloped
o ackle such p oblems. In second place, we wan o es he p edic ions o
he me hod de eloped by Azcoi i e al. [13] ega ding his sys em [21], since i s
pe o mance wi h he expec ed non- i ial phase diag am could en ail addi ional
obs acles o i s eliable applica ion in o he scena ios.
In he o ecoming sec ions, we jus i y he choice o model and e iew hei
undamen als. Then, we discuss he analy ical echniques applied, including he
exac compu a ion o he i s eigh cumulan s o he expansion o he e ec i e
Hamil onian in powe s o he in e se empe a u e, which allows o calcula e phys-
ical obse ables o a la ge numbe o deg ees o eedom wi h he help o s an-
da d mul i-p ecision algo i hms. Finally, we epo accu a e esul s o he ee
ene gy densi y, in e nal ene gy, s anda d and s agge ed magne iza ion, and he
posi ion and na u e o he c i ical line, which con i m he mean- ield quali a i e
pic u e o [21], and which should be quan i a i ely eliable, a leas in he high-
empe a u e egime, including he en i e c i ical line.
3.1 Why Ising?
As has been a gued along he i s chap e s o his wo k, nume ical simula ion o
sys ems wi h a se e e sign p oblem is one o he majo challenges o high-ene gy
heo is s—a s a emen which is also alid o hei solid-s a e colleagues. I we
deno e he mic oscopic s a es o a gi en physical sys em by s, and he he mody-
39
40 CHAPTER 3. ISING MODEL WITH A θTERM
namics o such sys em is desc ibed by a pa i ion unc ion o he o m
Z=X
s
P(s),(3.1)
we say ha he sys em in ques ion p esen s a sign p oblem i he “weigh s” P(s)
a e no eal and posi i e: This implies ha we canno in e p e P(s) as a p ope
p obabili y dis ibu ion, and he s anda d, e icien Mon e Ca lo algo i hms canno
be applied. No all sign p oblems a e equally se e e. Le us es ic ou sel es o
simplici y o he case whe e he P(s) a e eal bu no posi i e de ini e1. One
can easily de ise a eweigh ing algo i hm ha uses he absolu e alue |P(s)|as
he weigh o each s a e, and shi s he sign o P(s) in o he obse ables. Now a
s anda d Mon e Ca lo me hod is applicable, and in he limi o in ini e s a is ics
we should ob ain he co ec esul . Wi h ini e s a is ics, howe e , a key quan i y
is he he modynamic a e age o he sign o each con ibu ion o he pa i ion
unc ion, ha is, hsign(P(s))i. I his quan i y goes o ze o exponen ially wi h he
olume, hsigni ∝ e−αV , hen we would need an exponen ial amoun (in he olume
o he sys em V) o s a is ics o ge co ec esul s, which is o cou se impossible
in p ac ice. In his case we say ha he sign p oblem is se e e.
Beyond QCD a ini e ba yon densi y o QCD wi h a opological e m in he ac-
ion, he e exis o he physically ele an sys ems which su e om a SSP. Some o
he mos popula examples include chains o quan um spins wi h an i e omagne ic
in e ac ions, he wo-dimensional O(3) non linea sigma model wi h a opological
e m o he Hubba d model. The exis ence o a SSP is he main eason o he
li le p og ess made on he heo e ical unde s anding o hese physical sys ems
ou side o phenomenological models.
In o de o check no el Mon e Ca lo me hods designed o ackle such p ob-
lems, i is highly desi able o ha e a se o benchma k calcula ions as ex ensi e
as possible. Fo e y ew sys ems an analy ic solu ion is known, o example,
he one-dimensional an i e omagne ic Ising model wi h an imagina y magne ic
ield, he wo-dimensional compac U(1) model wi h opological e m, o he wo-
dimensional Ising model wi h an imagina y magne ic ield h=iπ/2. In a ew o he
cases he sign p oblem can be a oided by e o mula ing he physical sys em wi h
new deg ees o eedom, aking ad an age o he ac ha a good choice o hese de-
g ees o eedom p o ides an equi alen physical sys em ee om he sign p oblem,
which can he e o e be simula ed by s anda d me hods; see e.g. [112] o a ecen
discussion on his dualiza ion app oach. Un o una ely his idea wo ks only in a
ew cases which, un il now, a e no he mos in e es ing physical sys ems—indeed
none o he examples p e iously men ioned ha e been sol ed wi h his idea.
Wi hin he abo e scena io, ou in en ion in his wo k is o p o ide a benchma k
calcula ion o a sys em o which we do no ha e an analy ic solu ion a ailable,
1The discussion o complex weigh s does no add any undamen al di icul y.
3.1. WHY ISING? 41
no a e o mula ion ha a oids he sign p oblem. We s udy he wo-dimensional
an i e omagne ic Ising model wi h a pu ely imagina y magne ic ield, which can
be hough o as a oy model o he usual θphysics. Indeed he Euclidean pa i ion
unc ion o QCD wi h a non anishing θ e m can be w i en in he o m
ZV(θ) = X
n
pV(n)eiθn (3.2)
whe e n, he opological cha ge, is an in ege , and pV(n) is, up o a no maliza ion,
he p obabili y o he opological sec o na θ= 0. This has he same s uc u e as
he pa i ion unc ion o he an i e omagne ic Ising model in an ex e nal pu ely
imagina y magne ic ield, as we will see in de ail la e on, and we expec ha he
SSP in bo h sys ems should also be simila .
This sys em was s udied in [20] by loca ing he ze os o he pa i ion unc-
ion in he complex empe a u e-magne ic ield plane, and hey ound, o pu ely
imagina y magne ic ield, a ich phase s uc u e wi h wo phases cha ac e ized by
a anishing (non anishing) s agge ed magne iza ion, sepa a ed by a phase ansi-
ion line. We s udy his sys em by an exac cumulan expansion o eigh h o de ,
ollowed by he analy ic compu a ion o he pa i ion unc ion and o he physical
quan i ies o a la ge numbe o deg ees o eedom wi h he help o a s anda d
mul ip ecision algo i hm. This amoun s essen ially o he compu a ion o he e -
ec i e Hamil onian up o o de T−8, and he e o e is expec ed o wo k well in he
high- empe a u e egime, and we p o ide s ong e idence ha his is indeed he
case. Ou esul s a e consis en wi h [20], and ex end he esul s o [21], ob ained
h ough he applica ion o algo i hms de eloped in [12,13], and h ough a mean-
ield analysis. We a e able o ob ain a mo e p ecise quan i a i e de e mina ion o
he ansi ion line sepa a ing he pa amagne ic and an i e omagne ic phases o
he model.
Fo some sys ems wi h a SSP, we know a p io i ha he pa i ion unc ion
will be posi i e, o example sys ems in he mal equilib ium wi h a (He mi ian)
Hamil onian desc ip ion. Such is he case in a quan um ield heo y wi h a θ e m.
In he oy model we s udy he e, al hough we do no ha e a igo ous p oo in his
case,2we ha e e idence ha , a leas in he egion whe e he app oxima ion we
use is alid, he pa i ion unc ion is indeed posi i e (i is i ially always eal).
Such e idence is wo old. Fi s , we can p o e igo ously ha up o he i h
cumulan , he pa i ion unc ion is indeed posi i e. Un o una ely we ha e no
been able o ex end his p oo o highe cumulan s, bu in ou mul ip ecision
calcula ions wi h up o eigh cumulan s, we ha e ne e seen an ins ance whe e
2This would imply a non i ial es ic ion on he posi ion o he Lee-Yang ze os o he
an i e omagne ic Ising model. To he bes o ou knowledge, e y li le is igo ously known
abou such ze os.

42 CHAPTER 3. ISING MODEL WITH A θTERM
he pa i ion unc ion is nega i e o anishes. This is highly non i ial: I ins ead
o a cons an imagina y magne ic ield we y, o example, o pu a s agge ed
imagina y ield in ou la ice ( his is o cou se equi alen o he e omagne ic
model wi h a cons an imagina y ield), we immedia ely ge a luc ua ing sign o
he pa i ion unc ion.
Second, he e ha e been s udies loca ing he Lee-Yang ze os o he an i e o-
magne ic wo-dimensional Ising model up o 142la ices [113], and in 12 ×13
la ices [20]. Up o ha size he e is no sign o any ze os cu ing he imagina y
axis a any empe a u e.
Whe eas his by no means amoun s o a igo ous p oo , we belie e i p o ides
a s ong indica ion ha , a leas in he egion o in e es o ou wo k, his model
should ha e a posi i e pa i ion unc ion.
He ea e , Sec ion 3.2 is de o ed o o mula e he model and o ecall he main
ing edien s and esul s o he mean- ield app oxima ion de eloped in [21]. In Sec.
3.3 we in oduce he cumulan expansion, epo he analy ical esul s o he i s
eigh cumulan s in he wo-dimensional model, and w i e he analy ical exp essions
o he ee ene gy and mean alues o in e es ing physical quan i ies. The esul s
o he s agge ed magne iza ion, suscep ibili y, and phase diag am o he model
a e epo ed in Sec. 3.4, whe e we also compa e ou esul s a h= 0 and iπ/2 wi h
he analy ical solu ions o [114–116]. In Sec. 3.5 we epo ou conclusions. The
echnical de ails o he analy ical compu a ion o he cumulan expansion can be
ound in Appendix A.
3.2 Two-dimensional Ising model
The Ising model [20, 114–119] has been s udied o a long ime now, and i has
known analy ical solu ions in he one-dimensional case a any ex e nal magne ic
ield h[117], and in wo dimensions only o he case wi hou magne ic ield h[114]
and o h=iθ/2 = iπ/2 [115, 116]. The model wi h a pu e imagina y magne ic
ield su e s om a SSP in any numbe o dimensions. In addi ion o ha , he
expec ed phase diag am o d≥2 is non i ial [21], making he econs uc ion
o he θdependence o he obse ables e en mo e challenging. All his makes he
model a good heo e ical labo a o y o es new me hods designed o deal wi h
he SSP. I is he e o e wo hwhile o ca y ou a de ailed s udy o his model
a pu ely imagina y magne ic ield, pa icula ly because li le p og ess has been
achie ed on econs uc ing he θdependence o he obse ables, apa om he
analysis o [21] and he ecen s udy in [120].
3.2. TWO-DIMENSIONAL ISING MODEL 43
The pa i ion unc ion o he model, ollowing he con en ions o [21], is:
Z=X
{si}
exp FX
<ij>
sisj+iθ1
2X
i
si!.(3.3)
The hal magne iza ion M
2≡1
2X
i
si,(3.4)
is an in ege aking any alue be ween −N/2 and N/2, whe e Nis an e en numbe
deno ing he o al numbe o spins in he la ice. I is in his sense ha we iden i y
M/2 wi h a opological cha ge and ega d he imagina y magne ic ield e m in
he ac ion as a θ e m. I is impo an o men ion ha , om now on, we will
conside only he an i e omagne ic case F < 0, since he model wi h imagina y
ield does no de ine a uni a y heo y o a bi a y alues o he e omagne ic
coupling [115,121].
As we shall see in de ail in he nex sec ion, by di iding he ec angula la ice
in o wo subla ices, in oducing he espec i e magne iza ions M1and M2, making
a cumulan expansion and keeping only he i s cumulan , we a i e a he ollow-
ing app oxima ion o he pa i ion unc ion (whe e ddeno es he dimensionali y
o he la ice):
Z1c(F, θ) = X
{si}
exp iθM1+M2
2+ 4Fd
NM1M2.(3.5)
We ecall now he mean- ield analysis ca ied ou in [21]. The esul ing pa i ion
unc ion,
ZMF (F, θ) = X
{si}
exp iθM1+M2
2−Fd
N(M1−M2)2,(3.6)
is di e en om Eq. (3.5). Howe e , i can be seen o gi e he same quali a i e
esul s o he obse ables and he phase diag am. In his ega d, we will conside
he i s -cumulan expansion Z1cas a mean- ield app oxima ion o Z, and he
gene al expansion i sel as an imp o emen o i , a leas o small F, whe e he
expansion is expec ed o con e ge.
Applying s anda d saddle-poin echniques o he mean- ield pa i ion unc ion
[21], one ob ains he F−θphase diag am shown in Fig. 3.1. A second o de c i ical
line,
dFc=1
2cos2θc
2,(3.7)
sepa a es wo di e en phases: a s agge ed one, wi h hmsi 6= 0, o F > Fc(θ), and
a pa amagne ic one, wi h hmsi= 0, o F≤Fc(θ).
44 CHAPTER 3. ISING MODEL WITH A θTERM
0
0.1
0.2
0.3
0.4
0.5
0π/2π
hmsi= 0
hmsi 6= 0
d|F|
θ
dFc(θ)
Figu e 3.1: Phase diag am o he mean- ield app oach o [21] o he an i e omag-
ne ic Ising model in he F−θplane.
3.3 Cumulan expansion and obse ables
Ou in e es is ocused on he an i e omagne ic model, whe e he s agge ed mag-
ne iza ion is a good o de pa ame e . F om now on we will wo k wi h a ec angula
wo-dimensional la ice, al hough he me hod is easily gene alizable o any numbe
o dimensions. We di ide he la ice in o wo subla ices Ω1and Ω2in a chess-
boa d ashion. In he wo-dimensional la ice his means ha i iand jindex,
espec i ely, he ow and he column o a gi en spin, his spin will be in he i s
(second) subla ice i he sum i+jis e en (odd). Fo simplici y we will equi e
bo h leng hs o he la ice o be e en. Deno ing by N he o al numbe o poin s
in he la ice, we de ine he magne iza ion densi ies m1and m2as
mj≡Mj
N/2≡Pi∈Ωjsi
N/2j= 1,2,(3.8)
and he densi y o s agge ed magne iza ion is
ms≡m1−m2
2.(3.9)
Le us deno e by g(m1, m2) he numbe o mic os a es wi h magne iza ion
3.3. CUMULANT EXPANSION AND OBSERVABLES 45
densi ies m1and m2in subla ices Ω1and Ω2, espec i ely, ha is,
g(m1, m2) = X
{si}
δ X
i∈Ω1
si−M1!δ X
i∈Ω2
si−M2!.(3.10)
A i ial compu a ion gi es:
g(m1, m2) = N/2
N1+N/2
N2+,(3.11)
wi h Nj+≡N(1 + mj)/4 o j= 1,2. Now, by es ic ing ou sel es o he se o
con igu a ions wi h gi en magne iza ion densi ies m1and m2, i is s aigh o wa d
o de ine he expec a ion alue o a gene al obse able O({si}) wi hin his subse —
i.e., a ixed m1, m2—as:
hOim1,m2≡1
g(m1, m2)X
{si}
δ(X
i∈Ω1
si−M1)δ(X
i∈Ω2
si−M2)O({si}).(3.12)
Then, he sum o e all possible spin con igu a ions in he o iginal pa i ion unc ion
(3.3) can be pa ially summed up—a leas o mally—g ouping oge he sec o s
wi h equal magne iza ion densi ies m1, m2. By doing so, and aking in o accoun
he abo e de ini ions, he e o mula ed pa i ion unc ion akes he ollowing o m:
Z=X
m1,m2
g(m1, m2)*exp iθ
2X
i
si+FX
<ij>
sisj!+m1,m2
.(3.13)
The θ e m in Eq. (3.13) is jus iθ (m1+m2)N/4, and he e o e cons an a ixed
m1and m2; we can ake i ou o he expec a ion alue, a i ing a
Z=X
m1,m2
g(m1, m2)e1
4Niθ(m1+m2)*exp FX
<ij>
sisj!+m1,m2
.(3.14)
We canno e alua e exac ly he expec a ion alue in Eq. (3.14), as ha would be
equi alen o sol ing exac ly he model o a bi a y alues o he ex e nal ield.
Ins ead we pe o m a cumulan expansion and unca e a a gi en o de . Le us
ecall he de ini ion:
e X≡exp ∞
X
n=1
κn
n
n!!,(3.15)
whe e he n h cumulan κnis an n h deg ee polynomial in he i s nnoncen al
momen s o X, gi en by he ollowing ecu sion o mula:
κn=µ0
n−
n−1
X
m=1 n−1
m−1κmµ0
n−m, µ0
n≡ hXni.(3.16)
52 CHAPTER 3. ISING MODEL WITH A θTERM
0.5
0.6
0.7
0.8
0.9
0 0.1 0.2 0.3 0.4 0.5 0.6
e|ns (F)
|F|
k=1
k=4
k=8
Analy ic
Figu e 3.6: Nonsingula pa o he in e nal ene gy a θ=π, N = 2000.
−0.25
−0.2
−0.15
−0.1
−0.05
0.1 0.2 0.3 0.4 0.5 0.6
c (F)
|F|
k=1
k=4
k=8
Analy ic
Figu e 3.7: Speci ic hea a θ=π, N = 2000, plo ed agains he analy ical
exp ession.

3.5. CONCLUSIONS 53
0
0.5
1
1.5
2
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
c (F)
|F|
k=1, N=2000
k=4, N=2000
k=8, N=2000
k=8, N=6000
Analy ic
Figu e 3.8: Speci ic hea a θ= 0, plo ed agains he analy ical solu ion. A
θ= 0, Fc= log(1 + √2)/2≈0.4407.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36
hm2
si
|F|
N = 400
N = 800
N = 1600
N = 3200
Figu e 3.9: hm2
sicu es a θ= 2, k = 8. Solid lines a e jus a guide o he eye.
54 CHAPTER 3. ISING MODEL WITH A θTERM
0
0.5
1
1.5
2
2.5
3
3.5
4
0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4
dhm2
si/dθ
|F|
N = 100
N = 200
N = 400
N = 800
N = 1600
N = 3200
Figu e 3.10: Scaling o dhm2
si/dθ a θ= 2, k = 8. Solid lines a e a guide o he
eye.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
π/4π/2 3π/4π0
|F|
θ
k = 1
k = 4
k = 8
Ma ee and Sh ock, 2008
Figu e 3.11: The c i ical line Fc(θ), compu ed as he maximum o dhm2
si/dθ a
N= 2000. The maximal Fpoin s ob ained in [20] a e also shown.
3.5. CONCLUSIONS 55
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36
c (F)
|F|
N = 400
N = 800
N = 1600
N = 3200
Figu e 3.12: Speci ic hea c wi h k= 8 and θ= 2. Solid lines a e jus a guide o
he eye.
magne iza ion as an o de pa ame e . The ini e-size scaling sugges s ha he
wo phases a e sepa a ed by a con inuous phase ansi ion line. The posi ion
o he c i ical poin a θ= 0 is in e y good ag eemen wi h he exac esul
Fc= log(1 + √2)/2≈0.4407, and he ee and in e nal ene gy densi ies a θ=π
ag ee also well wi h he analy ical p edic ion, a leas in he high- empe a u e
egime, hus gi ing eliabili y o ou esul s in his egion. The e o e his model
could be a good labo a o y o check p oposals o simula e physical sys ems a lic ed
by a SSP. Mo eo e , we ha e con i med ha he sys em unde discussion has a
mo e in ol ed phase diag am han in he expec ed θQCD case, which would ha e,
a mos , a single c i ical poin in θ=π. In his sense, he econs uc ion me hod
o [13], which when applied in [21] was capable o de ec ing he p esence o a
c i ical egion, is s ongly suppo ed.
56 CHAPTER 3. ISING MODEL WITH A θTERM
Chap e 4
Massi e 1- la o Schwinge model
wi h a θ e m
We analyze he e he massi e 1- la o Schwinge model wi h a θ e m and a quan-
ized opological cha ge. Ou wo k, published in [29], elies on he app oach o
Azcoi i e al in [13]. We a e able o calcula e he ull dependence o he o de
pa ame e wi h θin a sys em ha includes dynamical e mions. Mo eo e , ou e-
sul s a θ=πa e compa ible wi h Coleman’s conjec u e [22] on he phase diag am
o his model.
This chap e is o ganized as ollows: a e mo i a ing he opic in he i s
sec ion, we summa ize some ele an ea u es o he Schwinge model wi h a opo-
logical e m in Sec. 4.2. Since he p oposal [13] o analyze physical sys ems wi h
a opological e m in he ac ion has been ound o be pa icula ly well sui ed o
bypass he sign p oblem in asymp o ically ee gauge heo ies, we decided o apply
i , and Sec. 4.3 con ains a b ie e iew o he me hod. In Sec. 4.4 we gi e some
echnical de ails conce ning he la ice se up and he compu e simula ions pe -
o med. Sec. 4.5 shows ou esul s o he opological cha ge densi y as a unc ion
o θa se e al e mion masses and gauge couplings, and inally we end his chap e
by epo ing ou conclusions.
4.1 Mo i a ion
The na u e and o igin o da k ma e cons i u e one o he mos wide open ques-
ions in mode n physics. To gain some insigh in his puzzling p oblem, i is highly
desi able o elucida e he exis ence o new low-mass, weakly in e ac ing pa icles
om a heo e ical, phenomenological and expe imen al poin o iew. As was ou -
lined in Chap e 2, he ligh pa icle ha has ga he ed he mos a en ion is he
axion, p edic ed by Weinbe g and Wilczek [99], and Wilczek [100] in he Peccei and
57

58 CHAPTER 4. SCHWINGER MODEL WITH A θTERM
Quinn mechanism [98] o explain he absence o pa i y and empo al in a iance
iola ions induced by he QCD acuum. The axion is one o he mo e in e es -
ing candida es o make he da k ma e o he uni e se, and he axion po en ial,
ha de e mines he dynamics o he axion ield, plays a undamen al ole in his
con ex .
The QCD axion model ela es he opological suscep ibili y χTwi h he axion
mass maand decay cons an a h ough he ela ion χT=m2
a 2
a. The axion
mass is, on he o he hand, an essen ial ing edien in he calcula ion o he axion
abundance in he Uni e se. The e o e, a p ecise compu a ion o he opological
p ope ies o QCD and o hei empe a u e dependence becomes o p imo dial
in e es in his con ex . Unde s anding he ole o he θpa ame e in QCD and
i s connec ion wi h he s ong CP p oblem is one o he majo challenges o high
ene gy heo is s [122].
The calcula ion o he opological suscep ibili y in QCD is al eady a challenge,
bu calcula ing he comple e po en ial equi es a s a egy o deal wi h he p esence
o a highly oscilla ing e m in he pa h in eg al; in o he wo ds, one needs o
ci cum en a se e e sign p oblem. In ac euclidean la ice gauge heo y, ou main
non-pe u ba i e ool o s udying QCD om i s p inciples, has no been able
o help us much because o he imagina y con ibu ion o he ac ion coming om
he θ e m, ha p e en s he applicabili y o he impo ance sampling me hod
[102]. This is he main eason why he only p og ess in he analysis o he ini e
empe a u e θdependence o he acuum ene gy densi y in pu e gauge QCD,
ou side o app oxima ions, educes o he compu a ion o he i s ew coe icien s
in he expansion o he ee ene gy densi y in powe s o θ[123], and he si ua ion
in ull QCD wi h dynamical e mions is, on he o he hand, e en wo se [124–129].
Much expe ience has been de eloped in he las yea s conce ning he s eng hs
and weaknesses o he app oaches [12,13], which aspi e o e en ually o e come he
sign p oblem in θQCD. As a ma e o ac , i has been applied success ully o
he compu a ion o he acuum ene gy densi y and he opological cha ge densi y
in a hand ul o in e es ing physical sys ems [21,109–111,130]. Ou pu pose in he
p esen chap e is o ake ad an age o his expe ience o pe o m a i s s ep in
he ambi ious p og am o compu ing he θdependence o he QCD acuum ene gy
densi y.
The eby, we analyze he θdependence o a oy model o QCD, he Schwinge
model, on he la ice. S ic ly speaking, he Schwinge model in he con inuum is
no asymp o ically ee, as QCD, since i is supe - eno malizable and he Callan-
Symanzik β- unc ion anishes. Howe e , in he la ice e sion, since he con inuum
coupling is dimension ul, he con inuum heo y is eached a in ini e in e se squa e
gauge coupling β= 1/e2a2, much in he same way as ou -dimensional asymp o -
ically ee gauge heo ies such as QCD. Fu he mo e he model is con ining [131],
4.2. THE MASSIVE SCHWINGER MODEL WITH A θTERM 59
exac ly sol able a ze o e mion mass, has non- i ial opology and shows explici ly
he UA(1) axial anomaly [132] h ough a non- anishing alue o he chi al conden-
sa e in he chi al limi , in he one- la o case. These a e basically he easons why
his model has been ex ensi ely used as a oy model o QCD.
I should be no ed ha , o wo dimensional sys ems such as he Schwinge
model wi h a θ e m, he e exis nume ical me hods such as Hamil onian me hods
[133–135] and he G assmann enso eno maliza ion g oup me hod [136] ha ha e
been applied success ully. Howe e , such app oaches a e cu en ly only applicable
o wo-dimensional sys ems, whe eas ou aim is o es a me hod ha should, in
p inciple, be applicable also o ou -dimensional heo ies such as QCD.
4.2 The massi e Schwinge model wi h a θ e m
The Schwinge model is Quan um Elec odynamics in 1+1-dimensions [137]. The
euclidean con inuum ac ion eads
S=Zd2x¯
ψ(x)γµ(∂µ+ieAµ(x)) ψ(x) + m¯
ψ(x)ψ(x) + 1
4F2
µν(x),(4.1)
whe e mis he e mion mass and eis he elec ic cha ge o gauge coupling, which
has he same dimension as m. A e a simple escaling o he ields he ac ion can
be w i en as
S=Zd2x¯
ψ(x)γµ(∂µ+iAµ(x)) ψ(x) + m¯
ψ(x)ψ(x) + 1
4e2F2
µν(x),(4.2)
whe e Fµν(x) = ∂µAν(x)−∂νAµ(x) and γµa e 2×2 ma ices sa is ying he algeb a
{γµ, γν}= 2gµν.(4.3)
whe e gµν s ands o he Euclidean me ic enso .
A he classical le el his ac ion is in a ian in he chi al limi unde he UA(1)
global ans o ma ions
ψ→eiαγ5ψ, (4.4)
¯
ψ→¯
ψeiαγ5,(4.5)
leading o he conse a ion o he axial cu en
JA
µ(x) = ¯
ψ(x)γµγ5ψ(x).(4.6)
60 CHAPTER 4. SCHWINGER MODEL WITH A θTERM
Howe e he axial symme y is b oken a he quan um le el because o he axial
anomaly, as was discussed in de ail in Sec ion 1.3.3. The di e gence o he axial
cu en is
∂µJA
µ(x) = 1
2πµνFµν(x),(4.7)
wi h µν he an isymme ic enso , and he e o e does no anish. The axial
anomaly induces a opological θ e m in he ac ion o he o m
S op =iθ
4πZd2xµνFµν(x),(4.8)
whe e he opological cha ge Q=1
4πRd2xµνFµν(x) is an in ege .
Ou pu pose is hen o analyze he θdependence o he model desc ibed by
he ac ion (4.2)+(4.8)
S=Zd2x¯
ψγµ(∂µ+iAµ)ψ+m¯
ψψ +1
4e2F2
µν +iθ
4πµνFµν.(4.9)
A simple analysis o his model on he la ice sugges s ha i should unde go
a phase ansi ion a some in e media e e mion mass mand θ=π, e en a ini e
la ice spacing. Indeed he la ice model is analy ically sol able in he in ini e
e mion mass limi (pu e gauge wo-dimensional elec odynamics wi h opological
e m) [138, 139], and i is well known ha he densi y o opological cha ge ap-
p oaches a non- anishing acuum expec a ion alue a θ=π o any alue o he
in e se squa e gauge coupling β, exhibi ing spon aneous symme y b eaking. On
he o he hand by expanding he acuum ene gy densi y in powe s o m, ea ing
he e mion mass as a pe u ba ion [140], one ge s o he acuum expec a ion
alue o he densi y o opological cha ge he ollowing θdependence:
h−iqi=mΣsinθ +1
2m2sin (2θ) (χP−χS) + ··· ,(4.10)
wi h Σ he acuum expec a ion alue o he chi al condensa e in he chi al limi
and a θ= 0 (Σ = eγee/2π3/2in he con inuum limi ), and χPand χS he pseu-
doscala and scala suscep ibili ies espec i ely. Equa ion (4.10) shows how he Z2
symme y a θ=πis ealized o de by o de in he pe u ba i e expansion o he
opological cha ge in powe s o he e mion mass m, and he e o e a c i ical poin
sepa a ing he la ge and small e mion mass phases is expec ed.
Indeed he model was analyzed in he con inuum by Coleman in [22], whe e he
conjec u ed he exis ence o a phase ansi ion a θ=π, and some in e media e
e mion mass msepa a ing a ”weak coupling” phase ( e
m<< 1), whe e he Z2
symme y o he model a θ=πis spon aneously b oken, om a ”s ong cou-
pling” phase ( e
m>> 1) whe e he Z2symme y is ealized in he acuum. This
4.3. COMPUTING THE ORDER PARAMETER AS A FUNCTION OF θ61
conjec u e was co obo a ed in [133, 134] using he la ice Hamil onian app oach
wi h s agge ed e mions, and mo e ecen ly in [136] using he G assmann enso
eno maliza ion g oup and Wilson e mions.
4.3 Compu ing he o de pa ame e as a unc-
ion o θ
To compu e he θdependence o he densi y o opological cha ge we use he
app oach p oposed in e e ence [13]. The only assump ion in his app oach is he
absence o phase ansi ions a eal alues o θexcep a mos a θ=π. The
me hod is based in ex apola ing a sui ably de ined unc ion o he o igin. This
unc ion u ns ou o be e y smoo h in all he cases conside ed up o now [21,
109–111], and his makes us con iden on he whole p ocedu e. He e we summa ize
he main s eps.
F om nume ical simula ions o ou physical sys em a imagina y alues o θ=
−ih ( eal alues o h), which a e ee om he se e e sign p oblem, we compu e he
densi y o opological cha ge q(−ih) as a unc ion o h, and in oduce he ollowing
unc ions:
z= cosh h
2,(4.11)
y(z) = q(−ih)
anh h
2
.(4.12)
The p ocedu e o ind ou he densi y o opological cha ge a eal alues o θ
elies on scaling ans o ma ions [13]. We de ine he unc ion yλ(z) as
yλ(z) = yeλ
2z.(4.13)
Fo nega i e alues o λ, he unc ion yλ(z) allows us o calcula e he o de pa-
ame e  anh h
2y(z)below he h eshold z= 1. I y(z) is non- anishing o
any posi i e z,1 hen we can plo yλ/y agains y. Fu he mo e, in he case ha
yλ/y is a smoo h unc ion o yclose o he o igin, hen we can ely on a simple
ex apola ion o y= 0. O cou se, a smoo h beha io o yλ/y canno be aken o
g an ed; howe e no iola ions o his ule ha e been ound in he exac ly sol able
models.
1E en hough he possibili y o a anishing y(z) o some alue z > 0 canno be comple ely
excluded, i does no happen o any o he analy ically sol able models we know.
68 CHAPTER 4. SCHWINGER MODEL WITH A θTERM
0
0.5
1
1.5
2
0 0.01 0.02 0.03 0.04 0.05 0.06
γ
yλ
β = 2
β = 3
β = 4
m = 0.0
Figu e 4.4: Exponen γ o m= 0.0 and a ious coupling cons an s. The shaded
a eas gi e an es ima ion o he ambigui y in he ex apola ion o yλ= 0. The
con inuous ed line is he analy ic esul in he pu e gauge heo y, co esponding
o in ini e e mion mass.
We plo in Fig. 4.6 he esul s o each o he independen analysis o an in e al
o θ. As can be seen, he e o s we would ob ain by a e aging he independen
poin s a e ully consis en wi h he syn he ic-da a es ima ion.
In Fig. 4.6 we p esen q(θ) a β= 3 o wo masses in he symme y es o ed
phase, as well as a β= 2 and m= 0.5, in he symme y b oken phase (and also
he co esponding analy ic esul s o he pu e gauge case a bo h alues o β o
compa ison).
In Fig. 4.7 we show he esul s o m= 0 and he h ee di e en alues o he
coupling cons an we ha e simula ed. We can clea ly see he es o a ion o he
symme y as we app oach θ=π. In Fig. 4.8 we show, o β= 3.0 and m= 0,
he o de pa ame e q(θ) in he icini y o θ=π. Fi ing q(θ) nea θ=πin
he symme y es o ed phase allows us o ex ac he exponen (π−θ), which is
ela ed o γby =γ−1.6We p esen in Table 4.1 ou esul s o .
6The nume ical p ocedu e used o ex ac he wo exponen s is di e en , and he e o e he

4.5. RESULTS 69
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
y
yλ
β = 3.0 , m = 0.0
Figu e 4.5: Fi o y e sus yλ.
Table 4.1:
β m 
2.0 0.0 0.67(4)
2.0 0.05 0.43(5)
3.0 0.0 0.92(7)
3.0 0.05 0.70(21)
4.0 0.0 0.94(19)
To inish his Sec. we wan o discuss a li le bi mo e on he esul s o he
massless Schwinge model epo ed in Fig. 4.7. I is well known ha he con inuum
o mula ion o he massless Schwinge model shows no θdependence, because he
θ e m in he ac ion can be canceled by an anomalous chi al ans o ma ion which
esul s, al hough compa ible wi hin e o s, will also be di e en .
70 CHAPTER 4. SCHWINGER MODEL WITH A θTERM
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0 0.5 1 1.5 2 2.5 3
q
θ
β = 3.0
β = 2.0
m = ∞
m = 0.5
m = 0.05
m = 0.0
Figu e 4.6: O de pa ame e as a unc ion o θ. The da a a m= 0.0 and m= 0.05
co espond o β= 3.0, whe eas he poin s a m= 0.5 co espond o β= 2. Blue
poin s, co esponding o he esul s o ou independen uns, a e also shown, o
p o ide a di e en es ima e o he e o . The con inuous line labeled m=∞is he
pu e gauge analy ic esul o β= 3.0, whe eas he do ed line is he co esponding
analy ic esul o β= 2.0.
does no change he e mion-gauge ac ion i he e mion mass anishes. Hence he
non- i ial θdependence o he densi y o opological cha ge shown in Fig. 4.7
may seem su p ising. Howe e , he massless s agge ed Di ac ope a o does no
ha e exac ze o-modes, and he e o e, o a gi en gauge con igu a ion, a nonze o
alue o he quan ized opological cha ge Qdoes no imply he exis ence o a
co esponding numbe o ze o-modes in he s agge ed Di ac ope a o , as would be
he case, o example, wi h he o e lap Di ac ope a o . Wha we should expec
ins ead is ha , as we app oach he con inuum limi , he opological cha ge densi y
anishes. This is indeed wha seems o happen, as is sugges ed by Fig. 4.9.
4.6. CONCLUSIONS AND OUTLOOK 71
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0 0.5 1 1.5 2 2.5 3
q
θ
m = 0.0
β = 2.0
β = 4.0
β = 3.0
Figu e 4.7: O de pa ame e as a unc ion o θ, a m= 0.0 and di e en coupling
cons an s.
4.6 Conclusions and ou look
All ou esul s a e compa ible wi h he s anda d lo e on his model, and in pa ic-
ula wi h Coleman’s conjec u e on he exis ence o wo dis inc phases a θ=π, a
symme y b eaking phase a la ge mass, and a symme y es o ed phase a small
mass.
Ou simula ions a e a p oo o concep , and a e no ex ensi e enough o de-
e mine p ecisely he posi ion o he c i ical mass a θ=πo i s p ope ies in
de ail. Bu he impo an poin is ha we ha e succeeded in calcula ing he ull
dependence o he o de pa ame e in θin a gauge heo y wi h e mions and a
quan ized opological cha ge, using a me hod ha should, in p inciple, wo k also
in highe dimensional heo ies.
72 CHAPTER 4. SCHWINGER MODEL WITH A θTERM
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
2.8 2.85 2.9 2.95 3 3.05 3.1
q
θ
β = 3.0 , m = 0.0
Figu e 4.8: O de pa ame e as a unc ion o θnea θ=π.
4.6. CONCLUSIONS AND OUTLOOK 73
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 0.5 1 1.5 2 2.5 3
β2q
θ
β = 2
β = 3
β = 4
m = 0.0
Figu e 4.9: Rescaled opological cha ge densi y a m= 0.0 and di e en coupling
cons an s.

74 CHAPTER 4. SCHWINGER MODEL WITH A θTERM
Chap e 5
P elimina y esul s on wo- la o
Schwinge —a pseudo e mionic
app oach
He ea e we p esen ou wo k wi h he massi e Schwinge model wi h a θ e m and
wo dis inc e mionic species. Al hough he s a ing poin o he s udy consis s in
he applica ion o he same Mon e Ca lo algo i hm ha was de eloped o Chap-
e 4—a s anda d implemen a ion o Kogu -Susskind e mions wi h a Me opolis
upda e—a sea ch o mo e e icien algo i hms p o es o be necessa y in o de o
ully es he capabili ies o he q(θ) econs uc ion app oach o [13].
5.1 Mo i a ion
The s udy o he Schwinge model wi h a single la o o massi e e mions and
he inclusion o a θ e m was ca ied ou in he p e ious chap e wi h conside able
success, since he dependence o he opological cha ge on θwas de e mined on he
la ice o he whole domain o he acuum angle, up o θ=π. This was achie ed
hanks o s anda d Mon e Ca lo simula ions pe o med a pu ely imagina y alues
o θ, which a e he basic inpu o he econs uc ion me hod o [13]. In hese
eal-ac ion compu a ions, s agge ed (Kogu -Susskind) e mions we e used, and
he de e minan o he e mionic ma ix was calcula ed a e e y Me opolis s ep
by ex ac ing nume ically all i s eigen alues—a echnique as eliable as ine icien .
In o de o adjus he compu a ion o he one- la o case, he squa e oo o his
de e minan needs o be aken; since he ac ual weigh employed by he algo i hm
depends on he loga i hm o he de e minan , his oo ing p ocedu e amoun s
o mul iplying by a ac o o 1/2. I ins ead we conside a ac o o N /2, he
discussion is alid o he Schwinge model wi h N e mion species—in ac , his
75
76 CHAPTER 5. PRELIMINARY RESULTS ON NF= 2 SCHWINGER
is he only change equi ed in he algo i hm.
Ha ing de eloped an algo i hm ha can be i ially ex ended o he mul i-
la o ed case, i seems na u al o apply i , a leas , o he cases ha a e closely
ela ed o one o ou main in e es s du ing his hesis: he s udy o opological
objec s on la ice gauge heo ies, and i s implica ions in QCD. In ac , his is he
case o he wo- la o ed e sion o he model: i s ac ion holds a U(2) symme y
in he chi al limi , o which i s axial U(1) subg oup is b oken by he anomaly,
much in he same way as QCD. The emaining SU(2) g oup cons i u es a ue
symme y o he heo y ha , con a y o wha occu s in low empe a u e QCD,
is exac ly p ese ed—as g an ed by a Theo em due o Coleman,1a con inuous
symme y canno be spon aneously b oken in a wo-dimensional sys em, as long
as in e ac ions a e kep su icien ly local. This exac ly p ese ed symme y esul s
o be an in e es ing p ope y, as long as i is sha ed by he high empe a u e
phase o QCD. In o he wo ds, QCD a high empe a u es has in he chi al limi
an exac ly p ese ed chi al symme y (con a y o he less exo ic low empe a u e
phase, as was discussed in Sec ion 1.3.3). This ac a o s he s udy he wo- la o
Schwinge model as a mechanism o gain insigh abou he opological p ope ies
o ini e empe a u e QCD.
Beyond i s in e es as a oy model o QCD, he N = 2 Schwinge model
p esen s a mo e in ol ed θbeha iou han i s single la o ed coun e pa . The
la e p esen s, a θ=π, wo dis inc phases depending on he coupling e/m. While
Psymme y is spon aneously b oken a weak coupling, in he s ong coupling (o
ligh mass) limi he acuum ene gy densi y can be expanded in e ms o he
e mion mass m, i s leading con ibu ion being
E(θ)∼me cos θ. (5.1)
As a consequence, he symme y is exac ly p ese ed and he opological suscep-
ibili y emains ini e. By he con a y, he wo- la o e sion o he model, which
has a simila beha io in he weak coupling egion, has a mo e in ol ed θdepen-
dence on he acuum angle. As i was shown by Coleman [22], a s ong coupling
app oxima ion allows o w i e he ene gy densi y θdependence as
E(θ)∼m4
3e2
3cos4
3θ
2,(5.2)
which e en ually leads o Pexac conse a ion a θ=π, bu wi h a di e gen opo-
logical suscep ibili y— he cha ac e is ic o a con inuous phase ansi ion. This ap-
p oxima ion, alid in p inciple when e/m >> 1, implies a alue o δ= 1/3 o he
1Al hough his esul is p o en by Coleman in he con ex o quan um ield heo ies [147], i
is commonly known as Me min-Wagne heo m, since hey a i ed o he same conclusions in
s a is ical physics [148].
5.2. THE MODEL 77
associa ed c i ical exponen , which desc ibes how he opological cha ge densi y
anishes as θapp oaches π. Addi ionally, he ligh es bosons o he spec um a e
p edic ed o be an iso iple and an isosingle , he quo ien o i s masses being
√3. I is wo h no ing ha p ecisely his mass a io has been ecen ly he subjec
o some con o e sy, since a ecen wo k by Azcoi i [149] ound a sub an ial dis-
c epancy wi h espec o he o iginal compu a ion o Coleman [22]. Fu he mo e,
Geo gi [150] has d own e en mo e a en ion o his model, by p oposing a solu ion
o he h ee ques ions posed by Coleman in [22] ha could en ail he exis ence
o a no el mechanism, capable o gene a ing he appea ance o ine- uning in low-
ene gy e ec i e heo ies and, consequen ly, wi h p omising po en ial conce ning
any o he hie a chy p oblems ha a lic he S anda d Model. In any case, since
a c i ical poin is expec ed in his model a θ=π, his sys em poses a ele an
challenge o he econs uc ion me hod ha was applied du ing Chap e 4; o his
e ec , i se es as an addi ional mo i a ion o his wo k—a pa icula ly p agma ic
one, a guably.
5.2 The model
The ac ion o he one- la o massi e Schwinge model wi h a θ e m can be easily
gene alized o i s mul i- la o e sion by adding an index , unning om 1 o N ,
o he o iginal exp ession (4.9). In his manne , he ac ion o N la o s o equal
cha ge eand mass myields
SN =Zd2x


N
X
1
¯
ψ [γµ(∂µ+iAµ) + m]ψ +1
4e2F2
µν +iθ
4πµνFµν


.(5.3)
Following he easoning o he p e ious chap e , i is possible o disc e ize he
abo e con inuum ac ion by using Kogu -Susskind e mions and he s anda d Wil-
son ac ion o he gauge pa , as in (4.15). A his poin we ecall ha a p ocedu e
commonly known as oo ing was needed o ge he one- la o heo y om he
co esponding la ice ac ion, since s agge ed e mions a e no comple ely ee o
he doubling p oblem— hey desc ibe wo degene a e species o e mions, in wo
dimensions. Bu , as long as we a e in e es ed in he wo- la o e sion, i su ices
o conside he exac same ac ion (4.15) and dismiss he oo ing s ep.
The nex s ep would imply pe o ming a Mon e Ca lo simula ion, much in
he same way as in Chap e 4. Howe e , while in ou one- la o s udy i was
enough o pe o m a p oo -o -concep calcula ion, ou aim wi h he N = 2 case
is o de e mine mo e in ol ed quan i ies, such as he c i ical exponen o he
expec ed θ=πphase ansi ion. E en i he b u e o ce app oach o he p e ious
chap e was able o deli e esul s in, oughly speaking, a ew mon hs o compu e
84 CHAPTER 5. PRELIMINARY RESULTS ON NF= 2 SCHWINGER
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
hqi
h
Me hod [29]
Pseudo e mions, n= 10
Figu e 5.5: Resul s o he wo- la o Schwinge model in a 16 ×16 la ice, wi h
β= 3 and m = 0.12. δmax = 0.25 and n= 10. The pseudo e mion app oach is
con on ed wi h he me hod o Chap e 4, wi h g ea ag eemen o small alues
o h, al hough sys ema ic de ia ions a e o he wise obse ed.

Chap e 6
Explo a o y ghos -gluon s udy on
αswi h HISQ e mions
In his las chap e we pu aside he s udy o opological e ec s o ace a non-
pe u ba i e quan i y o he u mos ele ance in QCD—i s unning coupling con-
s an αs. Ou in en ion is o explo e he dependence o he coupling on he mo-
men um ans e q, by means o a pu ely gluonic me hod, ollowing S e nbeck e
al [26]. In his way, i is possible o compu e gluon and ghos p opaga o s, G(q2)
and D(q2), in he la ice, e en ually leading o he de e mina ion o αs(q2).
In he ollowing sec ions we mo i a e he opic and e iew b ie ly he mos
ele an ma hema ical ela ions, desc ibing wi h some de ail he echnical issues
in ol ed in he p opaga o s de e mina ion. A e ha , ou esul s—ob ained om
la ge se s o con igu a ions gene a ed by he MILC collabo a ion—a e p esen ed.
The chap e is inished wi h some conside a ions conce ning how he p esen s udy
could be ex ended.
6.1 Mo i a ion
The de e mina ion o he αscoupling cons an cons i u es a e y ac i e ield o
esea ch. In ac , he Pa icle Da a G oup pe iodically p o ide a global a e age
o his quan i y, including bo h heo e ical calcula ions o la ice simula ions and
expe imen al de e mina ions, such as om had onic τdecays o e+e−annihila ion
p ocesses, ha allow o gi e an es ima e—a a gi en scale, usually ha o he Z
boson—o he s ong coupling αs[2]. Rema kably enough, he las decade has
been ma ked by a se back o he p ecision achie ed when calcula ing he global
a e age o his quan i y, in pa due o he exis ence o p e iously unde es ima ed
sou ces o sys ema ic e o ha we e p esen in la ice compu a ions.
In his si ua ion, he e cohabi se e al independen app oaches wi hin he la -
85
86 CHAPTER 6. EXPLORATORY GHOST-GLUON STUDY ON αS
ice o malism, de eloped by a numbe o in e na ional collabo a ions such as
HPQCD [4, 151, 152], PACS-CS [153], ETM [154], and o he esea ch g oups
[155, 156]. These app oaches di e in a numbe o echnicali ies, including how
e mions a e implemen ed on he la ice. Mo eo e , di e en obse ables can be
s udied in o de o ex ac αs, an example being hea y qua k co ela o s in [152],
o he de e mina ion o he s a ic po en ial in [156], which in ac gi es a alue o
αs(MZ) ha exhibi s some ension wi h he es o la ice p edic ions.
Ou in en ion in he cu en chap e is o explo e he po en ial o combining he
ghos -gluon e ex echnique o S e nbeck e al [26], which was al eady applied o
wis ed mass e mions in [154], wi h he highly imp o ed s agge ed ac ion (HISQ),
used by he HPQCD collabo a ion [78]. To his aim, we will analyze ghos and
gluon ields in a collec ion o ensembles gene a ed by he MILC collabo a ion wi h
he HISQ ac ion [157].
6.2 αsand he ghos -gluon e ex
Be o e going any u he , i should be no ed ha he unning coupling αsis no a
physically obse able quan i y. Ins ead, i acqui es a p ecise meaning only in he
con ex o pe u ba ion heo y. Then, in o de o compa e wo gi en esul s o he
coupling i is necessa y o ake in o accoun he eno maliza ion scheme. Typical
choices in he li e a u e a e momen um sub ac ion schemes, he mo e common
including MS, MS and MOM. While he s anda d compu ed alue o αs(MZ) is
ipically gi en in he li e a u e in he MS scheme [2], he p esen app oach is
de ined in a MOM scheme, in which eno maliza ion cons an s a e de ined by
equi ing wo- and h ee-poin unc ions o equal hei ee le el exp essions a a
gi en ene gy scale µ[158].
The e exis a numbe o ways ha allow o calcula e he alue o he unning
coupling αson he la ice. As i has been jus men ioned, he p esen wo k ollows
he app oach o S e nbeck e al [26], al hough wi h he ocus se in a di e en egion
(since we a e no pa icula ly in e es ed in he in a ed limi o αs). He ea e , we
e iew he essen ials o he me hod.
The compu a ion o αsin he la ice s a s by ealizing how ghos and gluon
p opaga o s can be exploi ed. I s d essing unc ions can be used o de e mine he
unning coupling as a eno maliza ion g oup in a ian , in a momen um sub ac ion
scheme [25], as
αs(q2) = g2
0
4πZD(q2)Z2
G(q2),(6.1)
whe e ZDand ZGa e espec i ely he ba e d essing unc ions o gluon and ghos
p opaga o s; we discuss how o compu e hese unc ions in he nex sec ions.
6.3. THE GLUON PROPAGATOR 87
Al hough in usual la ice compu a ions, as we e iewed in Chap e 1, ixing he
gauge is no necessa y, he ma hema ical exp essions o ghos and gluon p opa-
ga o s adop a simple o m in he Landau gauge—which in ac makes he whole
compu a ion easible. Consequen ly, in wha ollows all exp essions will be unde -
s ood o be alid wi hin he Landau gauge.
6.3 The gluon p opaga o
To begin wi h, he s a ing poin is he s anda d 4-dimensional la ice o Vsi es,
and 4Vlink a iables Ux,µ ∈SU(Nc= 3). The la ice gluon ields, which li e in
he mid-poin o each link, Ax,µ ≡Aµ(x+ ˆµ/2), a e de ined as
Ax,µ := 1
2i(Ux,µ −U†
x,µ)−1
6iT (Ux,µ −U†
x,µ).(6.2)
Addi ionally, we ecall ha he colo componen s Aa
x,µ o he gluon ield can be
compu ed as
Aa
x,µ := 2 T (TaAx,µ)=2·Im T (TaUx,µ),(6.3)
whe e he de ini ion (6.2) has been used. Wi h hese exp essions, we can compu e
he ba e gluon p opaga o on he la ice as
Dab
µν(k) = D˜
Aa
µ(k)˜
Ab
ν(−k)EU,(6.4)
whe e ˜
Aµ=˜
Aa
µTaa e he Fou ie ans o med gluon ields. In o he wo ds,
Dab
µν(q(k)) = 1
V*X
x,y
Aa
x,µAb
y,νeik·(x+ˆµ/2)e−ik·(y+ˆν/2)+U
,(6.5)
whe e he momen um q(k) is gi en by
qµ(kµ) = 2
asin πkµ
Lµ.(6.6)
I we assume now ha Dab
µν(q(k)) has he same enso s uc u e han i s con inuum
coun e pa ,
Dab
µν(q) = δab δµν −qµqν
q2D(q2),(6.7)
i su ices a bi o algeb a o ob ain an exp ession o he scala pa o he p op-
aga o D(q2),
D(q2) = 1
(D−1)(N2
c−1) X
aµ
Daa
µµ(q),(6.8)
88 CHAPTER 6. EXPLORATORY GHOST-GLUON STUDY ON αS
which is ela ed wi h he d essed p opaga o simply by
ZD(q2)≡q2D(q2).(6.9)
In p ac ical e ms, he key s ep in his compu a ion—in e ms o compu a ional
complexi y—is he Fou ie ans o ma ion o he gauge ields. Fo una ely, Fas
Fou ie T ans o m algo i hms allow o compu e e y e icien ly exp ession (6.5),
especially aking in o accoun ha a single applica ion o he algo i hm deli e s
he p opaga o o e e y la ice alue o he momen a qµa once.
6.4 The ghos p opaga o
Following [27], he ghos p opaga o in he la ice is de ined in he Landau gauge
as
Gab(k) = a2*X
xy M−1ab
xy eik·(x−y)+=δabG(q),(6.10)
whe e he eal symme ic ma ix Mis he Fadee -Popo ope a o , de ined by
Mab
xy =X
µAab
x,µδx,y −Bab
x,µδx+ˆµ,y −Cab
x,µδx−ˆµ,y(6.11)
wi h
Aab
x,µ =Re T {Ta, Tb}(Ux,µ +Ux−ˆµ,µ),(6.12)
Bab
x,µ = 2 ·Re T TbTaUx,µ,(6.13)
Cab
x,µ = 2 ·Re T TaTbUx−ˆµ,µ.(6.14)
In o de o compu e (6.10), he ollowing sys em o equa ions needs o be sol ed



Max,bycby
c=δac cos (k·x),
Max,bysby
c=δac sin (k·x).(6.15)
The 8V-componen ec o s cc,sca e compu ed wi h he conjuga e g adien me hod
and can be used o de e mine he in e se o M. Toge he wi h (6.10), and assuming
he enso s uc u e o he con inuum, Gab(qµ) = δabG(q2), we ha e
G(q2) = 1
(N2
c−1) X
ax
[cos (k·x)cax
a+ sin (k·x)sax
a],(6.16)
wi h he co esponding d essed p opaga o being gi en by
ZG(q2)≡q2G(q2).(6.17)
6.5. RESULTS 89
β m0
l/m0
sm0
sm0
cN3
s×N a( m) # o cn gs
6.00 1/5 0.0509 0.0635 243×64 0.1218(7) 1053
6.30 1/5 0.0370 0.0440 323×96 0.0879(5) 1008
6.72 1/5 0.0240 0.0286 483×144 0.0573(4) 1017
Table 6.1: De ails o he h ee ensembles s udied.
In con as wi h he gluon de e mina ion o he p e ious sec ion, he s eps he e
desc ibed a e much mo e expensi e in compu a ional e ms. In pa icula , nume -
ically sol ing he sys em o equa ions (6.15) is a e y demanding ask which, a
la ge la ice sizes—as he ones s udied in his chap e a e— equi es la ge esou ces,
bo h in e ms o memo y and p ocessing ime. Fu he mo e, la ice a i ac s a e
expec ed o be mo e in ense bo h a la ge and a o -diagonal momen a, due o
he lack o o a ional symme y on he la ice in he la e case [159]. Fo his
easons, he ghos p opaga o , and as a consequence also αs, ha e been compu ed
only o a hand ul o selec ed diagonal momen a.
6.5 Resul s
We ha e analyzed h ee se s o con igu a ions, made a ailable by he MILC collab-
o a ion [157]. The de ails o hei pa ame e s a e summa ized in Table 6.1. P io
o he p opaga o s de e mina ion, we ixed e e y con igu a ion o Landau gauge.
To his end, i is necessa y o make use o an i e a i e op imiza ion algo i hm. As
is well known, his ype o algo i hms could su e om a se e e c i ical slowing
down p oblem. In he case o la ge la ices, as some o he ones analyzed he e, his
obs acle can u n insu moun able. Howe e , we ha e e aded his di icul y by ap-
plying a Fou ie -accele a ed algo i hm, o iginally p oposed by Da ies e al [160],
which allows o alle ia e he compu a ional o e head, making he gauge ixing
p ocedu e easible. In his p ocess, he ixing p ocedu e was s opped only when
e e y local gauge ield e i ied he ans e sali y condi ion— he la ice e sion o
∂µAµ= 0—up o Θ <10−14, wi h he same de ini ion o [160]. Such le el o
p ecision was p o ed o be necessa y, since he p opaga o s a e qui e sensi i e o
he gauge condi ion.
Once he whole se s we e ixed o Landau gauge, we ha e compu ed bo h
p opaga o s, G(q2) and D(q2), o a o al o se en diagonal momen a,
k= (n, n, n, n) o n= 1,...,7.(6.18)
As we men ioned ea lie , he ghos compu a ion, and in a lesse way he gauge

90 CHAPTER 6. EXPLORATORY GHOST-GLUON STUDY ON αS
# o con igu a ions 1 1053
Landau gauge- ixing ime (co e-h) 7.1 7.1k
P opaga o s o 7 momen a (co e-h) 19.7 20.8k
To al ime (7 momen a) (co e-h) 26.928.3k
Each addi ional momen um (co e-h) +3.0k
Table 6.2: Dis ibu ion o compu ing imes in ol ed in he 243×64 ensemble.
ixing p ocedu e, a e e y demanding in e ms o compu ing esou ces. This being
he case, ac ual calcula ions ha e equi ed o be pe o med in la ge clus e acil-
i ies, which p o ide bo h he compu ing powe and he memo y needed o s o e
he la ges con igu a ions. To his end, he Uni e si y o Camb idge compu ing
se ices ha e been used, including he clus e Da win, and i s 2017 upda e CSD3.
Fo he i s se , o olume 243×64, 28k co e-hou s we e used in Da win; o exem-
pli y how hese a e dis ibu ed, see Table 6.2. Fo he 323×96 se , a o al o 135k
co e-hou s, also in Da win, we e used. Finally, he ensemble co esponding o he
bigge la ice size, 483×144, was analyzed wi h a o al cos o 452k co e-hou s o
he newe clus e CSD3.
Ou esul s o bo h ba e un eno malized p opaga o s a e shown in Figu e 6.1,
as a unc ion o he squa ed la ice momen a in physical uni s. By means o Eq.
6.1, he p e ious esul s can be applied o de e mine αs. Taking in o accoun ha ,
acco ding wi h he pa icula implemen a ion o he HISQ ac ion in he analysed
MILC con igu a ions,
g2
0≡5
3
2Nc
β=10
β,(6.19)
ou inal esul s o he un eno malized e sion o he unning coupling a e shown
in Fig. 6.2. As a p elimina y conclusion, hese esul s can be checked o be
quali a i ely in ag eemen wi h hose o he li e a u e, see e.g. [25,26,154]. In any
case, in o de o p o ide an es ima e o he αs(MZ) in he MS eno maliza ion
scheme, he co esponding β unc ion should be ca e ully s udied and in eg a ed,
oge he wi h a ho ough s udy o p esen olume e ec s and la ice a i ac s, since
he knowledge o i s dependence would allow o es ima e he sys ema ic e o s o
he me hod. Finally, he ela i ely low alue o he es ima ed s a is ical e o s in
he s udied ensembles d i e us o conclude ha comple ing he p esen analysis
would be wo hwhile, po en ially leading o a aluable con ibu ion o he wo ld
a e age o he s ong unning coupling.
6.5. RESULTS 91
0
1
2
3
4
1 10 0
1
2
3
4
5
6
1 10
q2[GeV 2]
G(q)−243×64
323×96
483×144
q2[GeV 2]
D(q)−243×64
323×96
483×144
Figu e 6.1: Ghos and gluon ba e p opaga o s, in physical uni s.
0
0.5
1
1.5
2
1 10
q2[GeV2]
243×64
323×96
483×144
Figu e 6.2: The unning coupling αs(q2) as de ined in Eq. 6.1. Shaded a eas
connec 1σin e als.
92 CHAPTER 6. EXPLORATORY GHOST-GLUON STUDY ON αS
Conclusions and ou look
In he beginning o his hesis we emphasized he key ole o la ice QCD as he
main ool capable o dealing wi h non-pe u ba i e phenomena, in QCD and be-
yond. Thei successes a e many, as i has been documen ed in Chap e 1. Up o he
p esen day, he e exis se e al in e na ional collabo a ions ha wo k in ensi ely
o p o ide heo e ical es ima es ha ma ch he e e -inc easing p ecision o he ex-
pe imen al measu es. Bu , no wi hs anding he achie emen s in phenomenology
g ounds, he e exis open ques ions ha seem o e ade any a emp o esolu ion.
Indeed, his is he case o QCD when he opological θ e m is included in o i s
ac ion, as i has been e iewed in Chap e 2.
Deeply connec ed wi h he s ong CP p oblem, and o g ea impo ance in
axion physics, he s udy o he θdependence o QCD has been he long-sigh ed
goal o his hesis. Ce ainly, his is an ambi ious objec i e, since he e exis many
di icul ies when conside ing θQCD on he la ice, one o he mos undamen al
being he p esence o a se e e sign p oblem ha makes he sys em una ainable
o s anda d Mon e Ca lo simula ions. In his scena io, ou e o s ha e been i s
di ec ed owa ds he es ing o a p omising me hod ha would in p inciple allow
o econs uc he ull θdependence o obse ables as he opological cha ge [13].
The eby, in Chap e 3, ollowing [28], we ha e c osschecked a p e ious wo k ha ,
making use o he econs uc ion me hod o Azcoi i e al, ound signs o a ich
phase s uc u e in he an i e omagne ic Ising model wi hin an imagina y magne ic
ield—which can be hough o as a θ e m [21]. By making use o a combina ion o
analy ical and nume ical—bu exac — echniques, ou esul s ha e suppo ed he
p e ious quali a i e pic u e, hus con i ming he po en ial o he econs uc ion
me hod in a heo y ha , when compa ed o θQCD, holds a mo e in ica e phase
diag am. In he same spi i , Chap e 4 has been de o ed o he s udy o he
massi e one- la o Schwinge model wi h a θ e m in he ac ion, which equen ly
se es as a oy model o QCD, as p esen ed in [29]. To apply he econs uc ion
me hod [13] in his sys em implies a s ingen exam conce ning i s possibili ies o
being success ully applied o ull QCD, as i sha es many o i s po en ial obs acles—
bu no all, since i holds a e y elemen a y de ini ion o he opological cha ge, in
con as wi h he di icul ies ha appea in he SU(3) 4d heo y. Ac ually, we ha e
93
100 CHAPTER 6. EXPLORATORY GHOST-GLUON STUDY ON αS

Appendix A
Compu a ion o he cumulan s κn
In o de o use exp essions (3.17) and (3.20), we need o compu e he cumulan s
κn. The n h cumulan can be calcula ed in e ms o he i s nnoncen al momen s
µ0
n,
µ0
n≡* X
<ij>
sisj!n+m1,m2
,(A.1)
by means o he ecu sion ela ion (3.16). The summa ion o e < ij > uns
o e each couple o neighbo ing spins, o in o he wo ds, o e each link. Two
neighbo ing spins always belong o di e en subla ices.
Be o e going u he , le us commen on wo in e media e esul s. Fi s , we
conside a la ice o Nspins, he magne iza ion o which is he sum m=Pisi,
and ask abou he expec ed alue o he p oduc o no hese spins a ixed m(o
ixed N+, he numbe o posi i e spins), ha is, hs1s2···snim. One can pe o m
his calcula ion by means o he mic ocanonical o malism, a i ing a
hs1s2···snim=1
N
N+
n
X
k=0
(−1)kn
k N−n
N+−n+k.(A.2)
In he abo e exp ession, kcan be ead as he numbe o nega i e spins in he
p oduc s1s2···sn. In his way, he i s summand, k= 0, coun s he numbe
o s a es wi h ze o nega i e spins in he p oduc s1s2···snand mul iplies i by
he expec a ion alue o he p oduc in his case, (−1)0= 1. The second one,
k= 1, does he same o one nega i e spin in s1···sn, and so on. Di iding he
sum by he o al numbe o con igu a ions wi h magne iza ion m= 2N+/N −1,
one ob ains he p e ious expec a ion alue a ixed m. Secondly, conside an
obse able O(m1, m2) in ou wo subla ice sys em, wi h a dependence on m1and
m2such as we can w i e i as O1(m1)O2(m2). In his case, om he de ini ion
101
102 APPENDIX A. COMPUTATION OF THE CUMULANTS κN
(3.12) o he expec a ion alue a ixed m1and m2, we ha e
hO1(m1)O2(m2)im1,m2=hO1(m1)im1hO2(m2)im2.(A.3)
This immedia ely applies o he spin p oduc s1s2···sn. We can always di ide i
in o wo p oduc s sa···sband sα···sβ, each one con aining he spins o one o
he subla ices, and hen
hs1s2···snim1,m2=hsa···sbim1hsα···sβim2.(A.4)
Wi h he p e ious couple o esul s, we come back o Eq. (A.1), and apply he
linea i y o he expec a ion alue, a i ing a
µ0
n=X
<ij>,<kl>,··· ,<pq> hsisjsksl···spsqim1,m2,(A.5)
which is he sum o he expec a ion alues o he p oduc o nlinks, unning o e
all pe mu a ions wi h epe i ions o hese links. Then, in e e y summand we ha e
he p oduc o 2nspins, in some cases wi h some o hem iden ical. Taking in o
accoun ha s2
i= 1 ∀i, each summand can be educed o he expec a ion alue
o he p oduc o n1+n2di e en spins, n1and n2being he numbe o spins
in each subla ice. Since by means o Eq. (A.2) we al eady ha e an exp ession
ha compu es hs1···snim, he p oblem is educed o coun how many summands
in Eq. (A.5) ha e (n1, n2) spins. We call hese numbe s geome ical ac o s, and
deno e hem by G(n1, n2). Following his con en ion, we can w i e he n h cen al
momen as
µ0
n=X
{n1,n2}G(n1, n2)hsa···sb
| {z }
n1spins im1hsα···sβ
| {z }
n2spins
im2,(A.6)
whe e he sum uns o e he couples o in ege s (n1, n2) he sum o which is e en
and less han o equal o n.
The compu a ion o he geome ical ac o s G(n1, n2) can be done by hand o
he i s ew cumulan s. As an example, o he second noncen al momen µ0
2we
ha e o compu e ou cases: he wo links being he same (sha ing bo h spins),
sha ing only one spin belonging o he i s o he second subla ice, and inally
no sha ing any spin a all. Tha is, in e ms o he p e ious no a ion,
{(n1, n2)}={(0,0),(2,0),(0,2),(2,2)}.(A.7)
The ac o s G(n1, n2) can be compu ed easily in his case, e en o an hype cubic
la ice o a bi a y dimension d, a i ing a he ollowing exp ession o he second
momen
µ0
2=Ndh1i+Nd(d−1)(hs1s2im1+hs1s2im2)
+Nd(Nd −2(d−1) −1)hs1s2im1hs1s2im2.
(A.8)
A.1. TRANSLATIONAL SYMMETRY 103
We can use his exp ession o calcula e he second cumulan κ2,
κ2=µ0
2−µ02
1
N→∞
−−−→ Nd(m2
1−1)(m2
2−1),(A.9)
whe e we ha e aken he he modynamic limi , keeping only he e ms o o de
O(N), which is he leading o de o all cumulan s. Subleading o de s can be
p ese ed i needed, bu hey a e no ele an o ou pape . The di icul y o
he p e ious compu a ion escala es quickly wi h he o de no he cumulan , and
i is qui e cumbe some o jus n≥4. In o de o ge beyond his limi a ion,
we ha e de eloped a p og am which compu es he geome ical ac o s G(n1, n2)
nume ically o a ini e L×Lbidimensional la ice. Since hese ac o s G(n1, n2)
a e polynomials in No o de ≤n(and wi h in ege coe icien s), we can un he
p og am o la ices o n+ 1 di e en sizes, ob aining a se o (N,G(N)) poin s,
which we can use o eco e he exac in ege coe icien s o each geome ical ac o ,
by means o he Lag ange in e pola ion o mula.
The basic idea o he p og am is e y simple. We jus cons uc a pe iodic
ec angula L×Mla ice, wi h L, M > n,nbeing he o de o he cumulan we
wan o compu e. Wi h his es ic ion we a oid p oduc s o links c ossing he
en i e la ice, ha would no appea in he he modynamic limi o any ini e
cumulan . Once we ha e his, we s a a loop unning o e all he pe mu a ions
wi h epe i ions o nlinks, and pe o m he ollowing s eps,
•We ha e a p oduc o nlinks, o equi alen ly 2nspins, s1···s2n.
•Recu si ely, we emo e couples o equal spins om his p oduc .
•We classi y he emaining p oduc by he numbe o spins in each subla ice,
(n1, n2).
•We add one o he geome ic ac o G(n1, n2) and p oceed o he nex i e a-
ion.
When he algo i hm inishes, we ob ain all he G(n1, n2) alues o a gi en N=
LM. The compu a ional cos is associa ed o he numbe o i e a ions o he main
loop, which g ows as (LM)n, ha is, exponen ially wi h he o de o he cumulan .
In p ac ice, we ha e only eached he compu a ion o he ou h cumulan wi h
his p og am. Howe e , a numbe o op imiza ions can be implemen ed in o de
o each highe o de cumulan s, which we summa ize in wha ollows.
A.1 T ansla ional symme y
Ou la ice is symme ic unde ansla ions, implying ha all geome ical ac o s
a e p opo ional o Nd, he numbe o links. Fixing, e.g., he i s link o he
104 APPENDIX A. COMPUTATION OF THE CUMULANTS κN
p oduc , one ob ains he same G(n1, n2), bu di ided by a common ac o Nd.
The same ac o is gained in he o e all speed o he p og am. In addi ion o
ha , he deg ee o he polynomials G(n1, n2) is also educed by one, and i su ices
wi h n(ins ead o n+ 1) di e en sizes in o de o eco e he Ndependence.
One can go e en u he by ealizing ha he geome ical ac o co esponding o
non-neighbo ing links, G(n, n), is he only one wi h maximum deg ee Nn−1. This
allows us o exp ess i in e ms o he emaining ac o s,
1
NdG(n, n) = (Nd)n−1
−1
Nd X
{(n1,n2)} (n,n)G(n1, n2),(A.10)
which a e only o o de n−2 o less. This means ha i is enough o un he
p og am o n−1 la ice sizes, compu e all he geome ical ac o s bu G(n, n)
ia he Lag ange in e pola o , and hen wi h he p e ious exp ession ind he N
dependence o his las ac o .
A.2 F om pe mu a ions o combina ions
The p oduc o links commu es, so i s con ibu ion o he geome ical ac o s is he
same ega dless o he o de . Then, we can change he main loop o e pe mu a ions
wi h epe i ion o a loop o e combina ions wi h epe i ion, by aking in o accoun
he mul iplici y o each combina ion. Schema ically, we pe o m
Pi,j,...,k con ib(lilj···lk)
→X
i≤j≤···≤k
mul ×con ib(lilj···lk),(A.11)
whe e con ib ep esen s a unc ion in ou p og am ha akes a p oduc o links
and e u ns he con ibu ion o he geome ical ac o s. I he e a e di e en
links, each one appea ing k1, . . . , k imes, he mul iplici y o he combina ion is
gi en by
mul = n!
k1!···k !.(A.12)
A.3 Blocks - G ouping links oge he
Many o he link p oduc s ha e ew, i any, epea ed spins, and hei con ibu ions
o he geome ical ac o s can be coun ed wi hou ha ing o analyze one by one
each o hem. This is possible by g ouping hem in se s o links ha we will call in
A.4. CLUSTERS OF BLOCKS 105
wha ollows blocks, and eplacing he loop o e link p oduc s by a loop o e block
p oduc s. When he blocks in a p oduc a e no neighbo s (i.e., hey do no ha e
any common spin), we do no need o pe o m he compu a ion link by link and
he con ibu ion can be summed up i ially. Le b1and b3be wo non-neighbo ing
blocks, each one composed by Nblinks, and le us deno e he con ibu ions o he
geome ical ac o s by λ(n1, n2), whe e λis an in ege coun ing how many p oduc s
o links ha e n1(n2) spins in he i s (second) subla ice. Then we ha e
con ib(b1b3) = N2
b(2,2),(A.13)
o in gene al, o he p oduc o knon-neighbo ing blocks, Nk
b(k, k). Following
his s a egy, we di ide ou la ice in o unidimensional blocks o 2Mlinks, in a
way ha he j h block, bj, con ains all links he i s spin o which belongs o he
j h column. As a consequence, bjis a neighbo o blocks j−1 and j+ 1, and,
aking in o accoun he bounda y condi ions, b0and bL−1a e neighbo s oo.
When we ha e a p oduc o neighbo ing blocks, we p oceed as be o e, analyzing
he link p oduc s one by one, and he e is no compu a ional sa ing. Bu when he
nblocks a e no neighbo s, we mo e om (Nd)ni e a ions o a single one.
A.4 Clus e s o blocks
The block me hod, as de ined abo e, ails o sa e any compu a ion ime i wo o
mo e blocks a e neighbo s in a gi en block p oduc . Howe e , we can ex end he
me hod by di iding each block p oduc in o se e al subp oduc s, which we will
deno e as clus e s. In each clus e , one can always connec one block o ano he
by he equi alence ela ion o being neighbo s (sha ing spins). And in he same
way, in each p oduc di e en clus e s ne e sha e any spin. This allows us o
compu e he con ibu ions o each clus e sepa a ely, and hen compose hem wi h
he ollowing law,
λ1(a, b)⊕λ2(c, d) = λ1λ2(a+c, b +d).(A.14)
I he con ibu ions o he clus e s in ol e mo e han one geome ical ac o , lin-
ea i y applies,
X
ab
λab(a, b)⊕X
cd
λcd(c, d) =
X
ab,cd
λabλcd(a+c, b +d).(A.15)
P ocessing one clus e wi h kblocks akes a compu ing ime p opo ional o (Nd)k.
So di iding he whole block p oduc in smalle clus e s implies o almos e e y

106 APPENDIX A. COMPUTATION OF THE CUMULANTS κN
block p oduc a signi ican amoun o ime sa ed. Only when all he blocks a e
pa o he same clus e he e is no speed up.
Ano he majo op imiza ion can be pe o med by ealizing ha ansla ional
in a iance can also be applied he e, since a gi en clus e , say b0b1b1, and any o i s
ansla ions, b0+ b1+ b1+ , ha e he same con ibu ion o he geome ical ac o s.
Then, when a clus e is going o be compu ed, we can exp ess i in e ms o i s
equi alence class, compu e i s con ibu ion, and s o e i in memo y. E e y ime
one o i s ansla ions appea s, we jus ake he alue om he memo y, sa ing
a lo o compu ing ime. In addi ion o ha , once we ha e compu ed he ac o s
G(n1, n2) o he i s size L×M, we know in ad ance all he clus e con ibu ions
o any L0×Mla ice ( he blocks keep i s size cons an ). Since almos all he
compu ing ime is spen in igu ing ou he clus e con ibu ions, we educe in
his way he ull p oblem o compu ing he geome ical ac o s in la ices o n−1
di e en sizes o only one size, he smalles one, M×M. In p ac ice, he ime
spen by he es o he sizes needed is ba ely he 1 −2% o ha o he i s size.
A.5 Compu a ion o a clus e
The las op imiza ion conce ns he compu a ion o he clus e s hemsel es. Un il
now i is done simply by pe o ming a loop o e each possible pe mu a ion o
links belonging o each o he blocks in he clus e . Howe e , one can go one s ep
u he and di ide he blocks composing he clus e in o smalle se s, ha we will
call si es. A si e is simply he se o wo links he i s spin o which lies in he si e
i, j, ha is,
si e(i, j)≡ {sijsi+1,j, sijsi,j+1}.(A.16)
Wi h his new subdi ision, we can apply in he same way he echniques desc ibed
abo e. In o de o compu e he clus e b1. . . bk, we s a a loop o e e e y pe -
mu a ion o si es s1. . . sk, wi h si∈bi. Each si e p oduc is di ided in o clus e s,
he con ibu ions o which can be summed wi h Eq. (A.15) and a e calcula ed by
pe o ming ano he loop o e each link p oduc (2ki e a ions o a si e p oduc
o kelemen s). Finally, by summing up each si e p oduc con ibu ion, we ob ain
he whole clus e con ibu ion.
All he desc ibed op imiza ions do no emo e he exponen ial dependence on
no he algo i hm. Howe e , hey allow us o each he eigh h cumulan , which
akes abou h ee days o compu ing ime in a mode n lap op.
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