Absence of high-temperature ballistic transport in the spin-1/2 XXX chain within the grand-canonical ensemble

Available online at www.sciencedirect.com ScienceDirect Nuclear Physics B 914 (2017) 62–98 www.elsevier.com/locate/nuclphysb Absence of high-temperature ballistic transport in the spin-1/2XXX chain within the grand-canonical ensemble J.M.P. Carmelo a,b,c,∗, T. Prosen d aDepartment of Physics, University of Minho, Campus Gualtar, P-4710-057 Braga, Portugal bCenter of Physics of University of Minho and University of Porto, P-4169-007 Oporto, Portugal cBeijing Computational Science Research Center, Beijing 100193, China dDepartment of Physics, FMF, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia Received 30 August 2016; accepted 30 October 2016 Available online 5 November 2016 Editor: Hubert Saleur Abstract Whether in the thermodynamic limit, vanishing magnetic field h →0, and nonzero temperature the spin stiffness of the spin-1/2XXX Heisenberg chain is finite or vanishes within the grand-canonical ensemble remains an unsolved and controversial issue, as different approaches yield contradictory results. Here we provide an upper bound on the stiffness and show that within that ensemble it vanishes for h →0in the thermodynamic limit of chain length L →∞, at high temperatures T→∞. Our approach uses a representation in terms of the Lphysical spins 1/2. For all configurations that generate the exact spin-Senergy and momentum eigenstates such a configuration involves a number 2Sof unpaired spins 1/2in multiplet configurations and L −2Sspins 1/2that are paired within Msp =L/2 −Sspin–singlet pairs. The Bethe-ansatz strings of length n =1and n >1 describe a single unbound spin–singlet pair and a configuration within which npairs are bound, respectively. In the case of n >1pairs this holds both for ideal and deformed strings associated with ncomplex rapidities with the same real part. The use of such a spin 1/2representation provides useful physical information on the problem under investigation in contrast to often less controllable numerical studies. Our results provide strong evidence for the absence of ballistic transport in the spin-1/2XXX Heisenberg chain in the thermodynamic limit, for high temperatures T→∞, vanishing magnetic field h →0and within the grand-canonical ensemble. *Corresponding author. E-mail address: [email protected] (J.M.P. Carmelo). http://dx.doi.org/10.1016/j.nuclphysb.2016.10.021 0550-3213/©2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 63 ©2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 1. Introduction The anisotropic spin-1/2XXZ Heisenberg chain [1] with anisotropy parameter  ≥0, exchange integral J, and Hamiltonian, JL j=1(ˆ Sx jˆ Sx j+1+ˆ Sy jˆ Sy j+1+ ˆ Sz jˆ Sz j+1), where ˆ Sx,y,z jare components of the spin-1/2 operators at site j=1, ..., L, is a paradigmatic example of an integrable strongly correlated quantum many-body system. However, the isotropic point at  =1( the spin-1/2XXX Heisenberg chain [2,3]) is the most experimentally relevant [4–6]. It is also the case that poses the most challenging technical problems for theory. For instance, the problem of clarifying the possibility of ballistic spin transport at nonzero temperatures in the spin-1/2XXX chain in a magnetic field his one of the most intensely debated unsettled fundamental questions in the theory of strongly correlated systems. Its Hamiltonian with periodic boundary conditions reads, ˆ H=J L  j=1 ˆ  Sj·ˆ  Sj+1−2μBh L  j=1ˆ Sz j,(1) where h ∈[−hc, hc], μBis the Bohr magneton and ±hc=±J/μBare the critical fields for fully polarized ferromagnetism. The model’s spin stiffness D(T ), also called spin Drude weight, defined via the singularity in the real part of the spin conductivity, σ(ω,T)=2π D(T ) δ(ω) +σreg(ω, T ) , (2) can be interpreted as a quantitative measure of ballistic spin transport. In the thermodynamic limit (TL), L →∞, the corresponding stiffness expressions given below in this paper involve the expectation values of the z-component spin current operator, ˆ Jz=−iJ L  j=1 (ˆ S+ jˆ S− j+1−ˆ S+ j+1ˆ S− j), (3) where ˆ S± j=ˆ Sx j±iˆ Sy j. Different approximate approaches [4,7–26], ranging from numerical simulations through effective field-theoretical descriptions to calculations partially based on the Bethe ansatz (BA) have yielded different, contradictory results, either showing that the model’s spin stiffness D(T ) converges as h →0in the TL to zero [4,8,9,20] or to a finite value [11,13,15,21,22]. For instance, the schemes used in the studies of Refs. [11,13,15,21,22] lead to a finite value for the spin stiffness at nonzero temperature. In contrast, the investigations of Ref. [4] indicate that transport at finite temperatures is dominated by a diffusive contribution, the spin stiffness being very small or zero. Such studies exclude the large spin stiffness found in Ref. [15] by a phenomenological method that relies on a spinon and anti-spinon basis for the thermodynamic Bethe ansatz (TBA) [3]. The results obtained by a completely different and more direct use of the TBA in Refs. [8,9] as well as the more recent results of Ref. [20] that rely on the combination of several techniques find a vanishing spin stiffness for zero spin density. 64 J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 The nature of the exotic spin transport properties at nonzero temperature of one-dimensional (1D) correlated lattice systems has been a problem of also experimental interest [5,6,27–31]. The spin stiffness is directly related to the long-time asymptotic current–current correlation function as D(T ) =1 2LT lim t→∞ˆ Jz(t) ˆ Jz(0).(4) (The angle brackets .denote here the thermal average.) In integrable models there is a lower bound for D(T ), which is encoded in an inequality due to Mazur [32], D(T ) ≥1 2L j ˆ Jzˆ Qj2 ˆ Q2 j.(5) Here the sum runs over a complete set of linearly extensive orthogonal commuting conserved quantities ˆ Qjfor which ˆ Q2 j ∝L, local and quasilocal [18,24,33–35]. In the case of the spin-1/2 XXZ chain, the sum over strictly local conserved quantities responsible for integrability gives at nonzero temperatures (i) a finite value and thus ballistic spin transport for h =0 and (ii) vanishes and is inconclusive at h =0. Two recent results provided some essential preliminary steps for the clarification of the problem studied in this paper. The first of these results is that the Mazur’s inequality sum over quasilocal conservation laws associated with deformed symmetries gives for the spin-1/2XXZ chain a stiffness lower bound at h =0, Dl(T ) ≤D(T ), which for T→∞reads [24] Dl(T ) =16J2 T sin2(πl/l) sin2(π/l)1−l 2πsin2π l.(6) It refers to a dense set of commensurate easy-plane anisotropies,  =cos(πl/l), where l, l∈ Z+and l≤l>0are such that 0 ≤ ≤1. Since this lower bound vanishes at the isotropic point,  =1, it does not discard the possibility that the spin stiffness of the spin-1/2XXX chain is also vanishing as h →0. The second recent result presented in Ref. [26] is a upper bound for the spin stiffness of the spin-1/2XXX chain, Du(T ) ≥D(T ), valid within the canonical ensemble for spin densities m ∈[0, 1]and the whole T>0 range, in the TL. Its limiting behaviors are Du(T ) =(J π)2 2Tm2L, for m1, =J2 2T(1−m)2L, for (1−m) 1.(7) That Du(T ) vanishes as m2Lin the m →0 limit ensures that within the canonical ensemble the stiffness vanishes as m →0 yet leaves out, marginally, the grand canonical ensemble as h →0 in which m2 =O(1/L). A schematic phase diagram of temperature Tversus spin density m of ballistic spin transport is shown in Fig. 1. In this paper we provide new insights on the above unsolved problem concerning the spin stiffness for the spin 1/2XXX chain in the TL. Specifically, we provide strong theoretical evidence that for high temperatures T→∞it also vanishes for h →0, within the grand-canonical ensemble. While for a canonical ensemble one considers that the spin density mis kept constant, in the case of a grand-canonical ensemble it is the magnetic field hthat is fixed. In general the canonical-ensemble and grand-canonical ensemble lead to the same results in the TL. This J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 65 Fig. 1. Phase diagram of ballistic spin transport of the spin-1/2XXX Heisenberg chain. Ballistic regions with positive spin Drude weight, D>0, namely temperature T=0or spin density m =0, are painted in cyan, whereas in the complementary region, T>0and m =0 (white), the spin stiffness vanishes, D=0, in the thermodynamic limit. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) is generally true except near a phase transition or a critical point. Hence this issue deserves a careful analysis in the m →0 and h →0 limits, respectively. The use of effective spinon representations [36–38] provides a suitable description of the model low-energy physics and excitations of the S=0 ground state. However, they do not apply to high-temperature problems at a magnetic field h ∈[−hc, hc]that involve all 2Lenergy eigenstates, as that studied in this paper. Our approach then rather uses the representation of Ref. [26] in terms of the spin-1/2XXX chain Lphysical spins 1/2. Within such a representation, all configurations that generate the exact energy and momentum eigenstates of spin Sinvolve a number 2Sof unpaired spins 1/2i n multiplet configurations and L −2Sspins 1/2 that are paired within Msp =L/2 −Sspin–singlet pairs. Within the TBA, the imaginary part of the complex rapidities simplify in the TL, which corresponds to the ideal strings of length n >1[3]. For large Lvalues there is in addition two types of deformed complex rapidities that deviate from such an ideal behavior [39–41]. Importantly, the general representation in terms of 2Sunpaired physical spins 1/2 plus Msp = L/2 −Sspin–singlet pairs of physical spins 1/2 used in the studies of this paper applies both to the TBA [3] and to BA schemes including three types of complex rapidities [39], respectively. On the one hand, both for an ideal string and a deformed string of length n >1 the corresponding set of ncomplex rapidities with the same real part refer to an independent configuration with a number nof spin–singlet pairs bound within it. On the other hand, the real rapidities correspond to single unbound spin–singlet pairs. Our derivation relies on the spin stiffness expression in terms of matrix elements of the z-component current operator, Eq. (3), and the operator algebra relating that operator to both the other two SU(2)symmetry operator components, ˆ J+=(ˆ J−)†=2iJ L  j=1 (ˆ S+ jˆ Sz j+1−ˆ S+ j+1ˆ Sz j), (8) and the three generators ˆ Sη=L j=1ˆ Sη j, η=±, zof that global symmetry. This includes the commutators, ˆ Jz,ˆ S±=ˆ Sz,ˆ J±=±ˆ J±;ˆ J±,ˆ S∓=±2ˆ Jz ˆ Jz,ˆ Sz=0;ˆ Jz,(ˆ  S)2=ˆ J+ˆ S−−ˆ S+ˆ J−,(9) which follow directly from the SU(2)algebra for the operators under consideration. 66 J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 There is a general consensus that the use of ideal strings of TBA for energy and momentum eigenstates described by groups of real and complex rapidities [3] leads in the TL to exact results as long as either the temperature or the magnetic field are nonzero [39,42]. Our studies involve the spin stiffness at very hight temperature, T→∞, so that concerning thermal effects they are not affected in the TL by the string deformations. On the one hand, concerning the case h =0, we use a method other than the BA or TBA to compute the exact current operator expectation values of the corresponding Sz=0 energy and momentum eigenstates [26]. On the other hand, in what the contributions to the spin stiffness for the model at finite magnetic field from the square of current operator expectation values of finite-Szenergy and momentum eigenstates is concerned, we rely on upper bounds. In contrast to those used in Ref. [26], the present upper bounds involve sums that run over a large, macroscopic number, of energy and momentum eigenstates. As justified below in Sec. 6, such upper bounds are in the TL insensitive to the use of ideal [3] or deformed [39] BA strings. Our representation in terms of configurations of the Lphysical spins 1/2 provides useful physical information on the problem under investigation, in contrast to the often less controllable numerical studies on the occurrence or lack of ballistic spin transport in the spin-1/2XXX chain as h →0in the TL. The remainder of the paper is organized as follows. In Sec. 2the finite-temperature spin stiffness and the representation in terms of configuration of the Lphysical spins 1/2used in the studies of this paper are introduced. The general expressions of the spin stiffness at high temperature T→∞is the issue addressed in Sec. 3. In Sec. 4a non-BA-related method used to compute the spin currents of the Sz=0 energy and momentum eigenstates for the strictly zero magnetic-field case is briefly reported and the physical consequences of the corresponding exact results are discussed. Useful and needed inequalities and corresponding current absolute values upper bounds are introduced in Sec. 5. The effects of the string deformations on the spin currents in the TL at finite magnetic field is the issue addressed in Sec. 6. In Sec. 7the high-temperature stiffness upper bounds within the TL used in our study are derived. Finally, the concluding remarks are presented in Sec. 8. Additional technical information useful for details of our analysis is provided in Appendices A and B. 2. The finite-temperature spin stiffness and Lphysical spins 1/2 configurations We denote the energy eigenstate’s spin and spin projection by Sand Sz=−(N↑−N↓)/2, respectively. Here N↑and N↓such that L =N↑+N↓are the numbers of spins 1/2 with up and down spin projection, respectively. For the so-called lowest-weight-states (LWSs) and highestweight-states (HWSs) of the SU(2)algebra we have S=−Szand S=Sz, respectively. The class of LWSs and the non-LWSs generated from those that are used in our analysis are energy and momentum eigenstates. They are as well eigenstates of (ˆ  S)2and ˆ Szwith eigenvalues S(S +1)and Sz, respectively. We thus label all 2Lenergy, momentum (as well as spin and spin projection) eigenstates by |lr, S, Sz. Here lrstands for all quantum numbers other than Sand Sz needed to specify an energy and momentum eigenstate, |lr, S, Sz. This is independent of using the general BA or the TBA for these states, always holding that lr=Nsinglet(S) for the model in each fixed-Ssubspace. Here Nsinglet(S) =L L/2−S−L L/2−S−1is that subspace number of independent spin–singlet configurations and thus N(S) =(2S+1) Nsinglet(S) is its dimension. Since the LWSs and non-LWSs generated from them considered in this paper are energy and momentum eigenstates, these designations are often used for the latter states. J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 67 Within the canonical-ensemble description at fixed value of Sz, the spin stiffness D(T ) expression involves the current operator expectation values, lr, S, Sz|ˆ J|lr, S, Sz, which in the TL and for nonzero temperatures are the current matrix elements that contribute to it [7,26,43]. As justified below in Sec. 4, for the non-LWSs, which are generated from the corresponding LWSs |lr, S, −Sas |lr, S, Sz =1 √C(ˆ S+)ns|lr, S, −Swhere C=(ns!) ns j=1( 2S+1 −j)and ns≡S+Sz=1, ..., 2S, such current operator expectation values can be expressed in terms of that of the corresponding LWS by suitable use of the spin SU(2)operator algebra. From such considerations one finds that in the TL the spin stiffness reads D(T ) =0for Sz=0 and for |Sz| ≥1/2it can be written as [26] D(T ) =(2Sz)2 2LT L/2  S=|Sz| lr plr,S,Sz| ˆ Jz(lr,S)|2 (2S)2.(10) Here ˆ Jzis the zcomponent of the spin current operator, Eq. (3), plr,S,Szare the Boltzmann weights, and ˆ Jz(lr, S) ≡lr, S, −S|ˆ J|lr, S, −Sare the LWSs spin currents. In this and all following expressions for the spin stiffness, the sums over Salways increase in steps of 1, whereas Szand Shave to be integers (half-odd integers) for even (odd) L. For each Svalue there are N(S) =(2S+1) Nsinglet(S) energy and momentum eigenstates. Our study accounts for all corresponding L 2S=0(integers)N(S) =2Lenergy and momentum eigenstates. For S>0 each such a state is populated by a set of 2Sspins 1/2 that participate in its multiplet configuration, which is one of the 2S+1 multiplet configurations, and a complementary set of even number L −2Sof spins 1/2 that form a tensor product of singlet states. Since all the N(S)states with the same Svalue have the same ˆ  S2eigenvalue, the energy and momentum eigenstates are superpositions of such configuration terms. Each such terms is characterized by a different partition of Lphysical spins 1/2into 2Ssuch spins that participate in a 2S+1spin multiplet and a product of singlets involving the remaining even number L −2Sof spins 1/2. As in Ref. [26], we call unpaired spins and paired spins the members of such sets of 2Sand L −2Sspins, respectively. In the TL this partition is common to the general BA solution and the TBA representation of its energy and momentum eigenstates. Both for large Land within the TL the L −2Spaired spins 1/2a re contained in a number Msp =1 2(L −2S) =L 2(1−mS), (11) of spin–singlet pairs. Hence each fixed-Ssubspace is spanned by energy and momentum eigenstates with exactly the same number Msp =L/2 −Sof such pairs. Moreover, Msp =L/2 −S also is the total number of BA rapidities that describe such states. And this is independent of such rapidities being all real or some being real and other complex. Consistently, within the present representation each BA rapidity describes a spin–singlet pair. The derivation of the spin stiffness upper bound of Ref. [26], whose limiting behaviors are given in Eq. (7), used a large overestimate of the current absolute values | ˆ Jz(lr, S)|. Specifically, for the whole set of energy and momentum eigenstates with the same Szvalue corresponding to the sums lrL/2 S=|Sz|in Eq. (10) it used the largest magnitude of the current expectation value among these states. Since the probability distribution plr,S,Szin each fixed-Szcanonical ensemble is normalized as L/2 S=|Sz|lrplr,S,Sz=1, this then allowed performing exactly such sums for all nonzero temperatures, T>0. The large overestimate of the currents used in deriving that spin stiffness upper bound is behind its m →0 behavior reported in Eq. (7) leaving out the grand canonical ensemble in 68 J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 which m2 =O(1/L). Our main goal is to derive an alternative spin stiffness upper bound whose estimate of the current absolute values | ˆ Jz(lr, S)|is closer to yet larger than those of the currents in Eq. (10). Here we perform such a program for high temperatures, T→∞. The Msp =L/2 −Sspin–singlet pairs of each energy and momentum eigenstate include Mpunbound pairs. The remaining MB sp =Msp −Mpspin–singlet pairs of energy and momentum eigenstates described by groups of both real and complex BA rapitities are bound within a well-defined number MB st <M B sp of independent configurations. (For energy and momentum eigenstates described only by groups of real BA rapitities such numbers read Mp=Msp and MB sp =0, respectively.) As discussed in the following, there is a one-to-one correspondence between such MB st independent configurations and the MB st strings of length larger than one, each of which is associated with a set of complex BA rapidities with the same real part. The unbound and bound spin–singlet pairs of the L −2Spaired spins are indeed described by groups of real and complex solutions, respectively, of the model BA equation [2,3], 2arctan(j)=qj+1 L α=j 2arctanj−α 2(mod 2π). (12) Here the α=1, ..., Mpsummation is over the subset of occupied qαquantum numbers out of the full set, qj=2π LIj,where j=1, ..., Mb,(13) Mb=Mp+Mh, and Mh=2S+2(MB sp −MB st ). The different occupancy configurations of the related quantum numbers Ij(defined modulo L) such that j=1, ..., Mbgenerate different energy and momentum eigenstates. The latter are successive integers or half-odd integers according to the boundary conditions, Ij=0,±1, ..., ±Mb−1 2for Mbodd, =±1/2,±3/2, ..., ±Mb−1 2for Mbeven.(14) The set of j=1, ..., Mbquantum numbers qjcan only have occupancy zero and one, respectively. Within our representation, the α=1, ..., Mpoccupied momentum values qαrefer to the center of mass translation degrees of freedom of Mpneutral composite pseudoparticles. The internal degrees of freedom of each of these Mppseudoparticles refer to one of the Mpunbound spin–singlet pairs. Our functional representation involves a qjdistribution function Mp(qj)that reads 0 and 1for the Mh=2S+2(MB sp −MB st )unoccupied and Mpoccupied qjvalues, respectively. Since the contribution to the momentum eigenvalues of the Mppseudoparticles reads π+ Mb j=1Mp(qj) qj, the set j=1, ..., Mbof quantum numbers qjsuch that qj+1−qj=2π/L may be associated with the discrete momentum values of a pseudoparticle spin band. For LWSs described only by groups of real rapidities, all Msp =L/2 −Sspin–singlet pairs are unbound, so that Mp=Msp =L/2 −S, Mh=2S, Mb=L/2 +S, MB sp =0, and MB ps =0. Consistently with the 0and 1 allowed occupancies of the spin-band momentum values, the LWS BA wave functions formally vanish when two rapidities jand jin Eq. (12) become equal. If one considered all the rapidities to be real, this property could suggest that simply choosing α=1, ..., Mpdistinct occupied momentum values qαamong the set of j=1, ..., Mballowed J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 69 spin-band discrete momentum values qj, which gives a dimension Mb Mp=L/2+S L/2−S, would allow the reconstruction of all 2Lenergy eigenstates that span the model Hilbert space. However, only some of the solutions to the model general BA equation involve only a group of Msp =Mpreal rapidities j. As mentioned above, there also exist solutions involving groups of real and complex rapidities [2,3]. There are Msp =Mp+MB sp BA rapidities that describe the Msp spin–singlet pairs of a general energy and momentum eigenstate. Within our representation in terms of L −2Spaired physical spins 1/2, the Mpreal rapidities and MB sp complex rapidities describe their Mpunbound spin–singlet pairs and their MB sp spin–singlet pairs bound within the state MB st independent configurations, respectively. The following general relations between the different numbers under consideration apply Mp=Msp −MB sp =L 2(1−mS)−MB sp , Mh=2S+2(Msp −Mst)=2S+2(MB sp −MB st ), Mb=Mp+Mh.(15) Here Mst =Mp+MB st gives the total number of both Mpunbound spin–singlet pairs and corresponding spin-band pseudoparticles and MB st independent n-pair configurations with n >1 spin–singlet pairs bound within them. The ncomplex rapidities with the same real part that describe each such a n-pair configuration is labeled by a quantum number l=1, ..., n. It also labels each of the spin–singlet pairs bound within such a configuration. These l=1, ..., nrapidities with the same real part have the general form [39] n,l j=n j+i(n +1−2l) +Dn,l jwhere l=1, ..., n . (16) The roots of Eq. (12) are here partitioned in a configuration of strings. A n-string is a group of nroots also called rapidities. Within our representation such a string describes an independent n-pair configuration. The number nis often called the string length. The real part of the set of nrapidities, n j, is called the string center [39]. Hence Mst =Mp+MB st is in Eq. (15) the number of strings. Mpand MB st refer to the number of strings of length n =1 and length n >1, respectively. Note that for n =1 one has that l=1 and the corresponding single rapidity 1,1 jis real. The quantity Dn,l j=Rn,l j+iδn,l jin Eq. (16), where Rn,l jand δn,l jare real numbers, is the fine-structure deviation from the TBA ideal strings for which Dn,l j=0[3]. Importantly, D1,1 j=0 for the Mpreal rapidities 1,1 jof all energy and momentum eigenstates. There is a one-to-one correspondence between an energy eigenstate MB st strings of length n >1 and the MB st independent n-pair configurations with n >1 spin–singlet pairs bound within them, respectively. The string length n >1is thus the number of spin–singlet pairs bound within the corresponding n-pair configuration. The present representation clarifies the physical meaning of the imaginary parts of the n >1 complex rapidities with the same real part that refer to a string of length n, Eq. (16): Such imaginary parts are associated with the binding within the corresponding n-pair configuration of n >1spin–singlet pairs. Consistently and as mentioned above, for n =1the rapidity 1,1 jis real and describes a single unbound pair. The maximum possible value of the number nof spin–singlet pairs bound within a n-pair configuration and corresponding string length is obviously given by the number of spin–singlet pairs, Msp =(L −2S)/2, Eq. (11). The set of energy and momentum eigenstates that span each fixed-S subspace have all the same number Msp =(L −2S)/2of such pairs. Provided that (1 −ms)is 70 J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 finite, that number is such that Msp →∞as L →∞. Hence in general we consider in the TL that nhas the range n =1, ..., ∞. For a given large L, the complex solutions of the spin-1/2XXX chain BA equation, Eq. (12), are found to belong to three classes [39]. The first class refers to the ideal strings for which Dn,l j= 0in Eq. (16). The second class was first identified by Essler, Korepin and Schoutens (EKS) for n =2 complex rapidities [40] yet also occurs for n >2. The corresponding strings deviate from the ideal behavior and are known as EKS-strings [39]. The imaginary part of their complex rapidities are smaller than 1/2. It decreases upon increasing L, vanishing at some Lvalue. The third class of solutions corresponds to another type of deformed strings usually called V-strings, which have been first found by Vladimirov (V) [41]. In the case of a system with a fixed large L, the number of energy eigenstates obtained by accounting for the three classes of BA equations groups of real and complex solutions is given by the correct Hilbert space dimension, 2L[39]. In Sec. 6it is justified why concerning the model at finite magnetic field our final results are independent from the use in the TL of ideal or deformed strings of length n >1for the |Sz| ≥1/2 energy and momentum states described by groups of real and complex rapidities. The unbinding of spin–singlet pairs by processes associated with the vanishing of the EKS-strings imaginary parts, usually called collapse of narrow pairs, is for a large system and finite magnetic field the aberration from the ideal strings that must be accounted for. The effects of the V-strings are unimportant in the TL for the physical quantities studied in this paper. For large finite systems they behave in a rather normal way, consistent with the predictions of the 1/L expansion methods [39]. The direct relation reported in the following of the TBA quantum numbers to our representation configurations of 2Sunpaired spins 1/2, L −2Spaired spins 1/2, corresponding Msp =L/2 −Sspin–singlet pairs, and Mpand MB sp unbound and bound such pairs, respectively, is useful and needed for the studies of Secs. 5and 7. Within the TBA, the l=1, ..., n complex rapidities of a string, Eq. (16), simplify in the TL to their ideal form [3] n,l j=n j+i(n +1−2l) where l=1, ..., n . (17) Such rapidities are solutions of the TBA coupled integral equations given below. The number 2L of energy eigenstates prevails under the use of the TBA in terms of only ideal strings, Eq. (17). We call Mnthe number of n-pair configurations and corresponding strings of length n. Within our representation the Mst =Mp+MB st BA strings correspond to Mst =Mp+MB st n-pair configurations involving for each spin-Senergy and momentum eigenstate its Msp =L/2 −S spin–singlet pairs, Eq. (11). Consistently, the TBA quantum numbers obey the following sum rule [3], msp =∞  n=1 nm n=1 2(1−mS);Msp =∞  n=1 nM n=L/2−S=msp L, (18) where msp is the density of spin–singlet pairs and mS=2S/L ≥m, m n=Mn/L . (19) Within the momentum-distribution functional notation used here and in Ref. [26], the TBA equations derived in Ref. [3] from the general BA equation, Eq. (12), by means of real and complex rapidities associated with ideal strings, Eq. (17), read qj=kn j−1 L (n,j)=(n,j) Mn(qj) nn (n j−n j). (20) J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 77 number of 2Sholes also cancel each other. (The remaining virtual current canceling processes are similar to those of the S=0 energy and momentum eigenstates.) Such a virtual current cancelation mechanism is encoded both in the general BA equation, Eq. (12), and in the n =1, ..., ∞TBA equations, Eq. (20), and corresponding general spincurrent expressions. However, for increasingly larger numbers of spin/n =1 band holes it is technically difficult to access from direct solution of these equations. The problem can be explicitly solved in terms of such equations for the simplest case of the class of S=0 energy eigenstates with two holes in the spin/n =1 band. Such states thus have one n =2-pair configuration described by one string of length two. (Within the TBA its two bound pairs refer to the internal degrees of freedom of one n =2 composite pseudoparticle.) This simplest case has been studied within the BA solution, as in Ref. [44] for the present model, by use of the method of Ref. [45] for the related large-on-site-repulsion half-filled 1D Hubbard model. (In this paper the spin current operator, Eq. (3), and its expectation values are given in units of 1/2, which justifies that extra factor within the notation of Ref. [44].) One then explicitly finds that, independently of the momentum values qjand qjof the two holes, their virtual spin currents exactly cancel each other. As confirmed in the ensuing section, the virtual current mechanism also occurs for |Sz| >0 energy and momentum eigenstates. For such states it corresponds though to a partial cancellation [26]. 5. Useful inequalities and upper bounds on current absolute values The inequalities and corresponding current absolute values upper bounds introduced in this section refer to the TBA. More general inequalities accounting for the effects of the string deformations on the spin currents at finite magnetic field are introduced below in Sec. 6. The spin-1/2XXX chain in a uniform vector potential /L whose Hamiltonian is given in Eq. (A2) of Ref. [26] remains solvable by the BA. Within the TBA the LWSs momentum eigenvalues, P=P(/L), have the general form P(/L)=P(0)+L−∞ n=12nM n L=P(0)+mS=P(0)+2S L.(41) Here the  =0 momentum eigenvalue P(0)is given in Eq. (26) and the sum rule ∞ n=12n Mn= L −2Sinvolving the number L −2Sof paired physical spins 1/2 has been used. (Such a sum rule follows from that of the corresponding Msp =L/2 −Sspin–singlet pairs, Eq. (18).) Importantly, for large Lexactly the same exact momentum eigenvalues expression, P(/L) = P(0) +2S(/L), is obtained by use of the BA accounting for deformed strings. On the one hand, the expectation values of the current operator in the  →0LWSs, Eq. (35), can be derived from the /L dependence of the energy eigenvalues E(/L) as ˆ Jz =dE(/L)/d(/L)|=0[26]. On the other hand, dP (/L)/d(/L)|=0gives the number of spin carriers that couple to the vector potential. The natural candidates are the model L physical spins 1/2. The form of the exact momentum eigenvalues, Eq. (41), reveals that only the 2Sunpaired spins 1/2 contributing to the multiplet configurations couple to the vector potential /L. Since the L −2Sphysical spins 1/2left over are those within the Msp =L/2 −Sneutral spin–singlet pairs, this exact result is physically appealing. A second exact result is consistent with only the 2Sunpaired physical spins 1/2 coupling to the vector potential also holding for non-LWSs. For simplicity, we consider that Lis even yet within the TL the same results are reached for Lodd. For a general LWS carrying a spin 78 J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 current ˆ Jz LWS(lr, S)all 2Sunpaired spins 1/2have up-spin projection. Let Sσbe the number of unpaired spins 1/2 with spin projection σ=↑, ↓of a non-LWS such that σ=↑,↓Sσ=2S. The exact relation, Eq. (40), can then be written simply as ˆ Jz(lr,S ↑,S ↓)=(S↑−S↓) 2Sˆ Jz LWS(lr,S) =S↑×j+1/2+S↓×j−1/2,(42) where j±1/2=±ˆ Jz LWS(lr,S) 2S=±1 2S ∞  n=1 Mb n  j=1 Mn(qj)j n(qj). (43) The exact relation, Eqs. (42) and (43), confirms that only the 2S=S↑+S↓unpaired spins 1/2 contribute to the spin currents. For each spin flip generated by application of the off-diagonal spin generator ˆ S+(and ˆ S−) onto an energy eigenstate with finite numbers S↑and S↓, the spin current exactly changes by a LWS current quantum 2j−1/2(and 2j+1/2). Hence each unpaired spin 1/2 with spin projection ±1/2 carries an elementary current j±1/2, Eq. (43). For a LWS one has that S↑=2Sand S↓=0, so that ˆ Jz LWS(lr, S) =2S×j+1/2. That only the 2S=mSLunpaired physical spins 1/2 couple to the vector potential justifies the validity of the result of Ref. [26] that all spin currents exactly vanish as mS→0. This exact result can be used to confirm that, as found in that reference, within the canonical-ensemble description at fixed value of Sz, in the TL, and for nonzero temperatures the spin stiffness D(T ), Eq. (10), vanishes as mS→0. The main goal of this paper is to extend that result to the grandcanonical-ensemble description for T→∞. Relying on the exact relation, Eq. (40), the spin stiffness expressions given in Eqs. (10), (30), and (31) involve only spin current expectation values ˆ Jz(lr, S)of LWSs. It is thus useful to consider here the LWS fixed-Ssubspace that is spanned by the Nsinglet(S) LWSs with a given spin S. It is a subspace of the larger fixed-Ssubspace spanned by all N(S) =(2S+1) Nsinglet(S) energy and momentum eigenstates with the same spin S. A LWS fixed-Ssubspace can be further divided into smaller LWS reduced subspaces for which both the number 2Sof unpaired physical spins 1/2 and that of pseudoparticles Mps are fixed. The Mh 1=2S+2(Msp −Mps) =2S, ..., 2S+2(Msp −1)value is thus also fixed. Hence such subspaces refer to fixed values of the densities mS∈[0, 1]and mh 1∈[mS, 1]. Each LWS fixed-Ssubspace contains one real-rapidity reduced subspace. It is spanned by real-rapidity LWSs for which mh 1=mS. All Msp =L/2 −Sspin–singlet pairs that populate such LWSs are unbound and thus Mps =M1=Msp =L/2 −Sand Mn=0for n >1. We denote its finite numbers by M1=Mp S, Mh 1=Mh S, and Mb 1=Mb Swhere Mp S=L/2−S;Mh S=2S, Mb S=Mp S+Mh S=L/2+S. (44) All remaining reduced subspaces of a LWS fixed-Ssubspace are called complex-rapidity reduced subspaces. Indeed those are spanned by complex-rapidity LWSs described by groups of both real and complex rapidities. Their mh 1>m Svalues belong to the range mh 1∈[mS, 1]. We denote by | ˆ Jz LWS|T(m S,mh 1)the largest current absolute value of each LWS reduced subspace of a given LWS fixed-Ssubspace. It is of the general form, | ˆ Jz LWS|L(mS,mh 1)=cT L2J2SM ps =cT2JLm Smps .(45) J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 79 The coefficient cTin this expression obeys the inequality cT≤π. It is a function of the densities mSand mh 1with the following limiting behaviors, cT=π 2for mS=mh 1→0, =1formS=mh 1→1,(46) and cT=sin(πmS) mS for mS∈[0,1/2]and mh 1→1, =1 mS for mS∈[1/2,1]and mh 1→1.(47) On the one hand, for mS→0 and mh 1∈[0, 1]it is an increasing function of mh 1given by cT= πc 1where c1=1/2for mh 1→0 and c1=1for mh 1→1. On the other hand, for mh 1=mSit is a decreasing function of mh 1whose limiting values are given in Eq. (46). The LWSs spin currents result from processes that are simpler to be described in terms of local spins 1/2 occupancy configurations in the spin-1/2XXX chain lattice. Within these processes, each 2n-site configuration of the Mps =∞ n=1Mnpseudoparticles that populate a LWS interchanges position under its motion along the lattice with such a state single-site 2Sunpaired physical spins 1/2. This justifies why the largest current absolute value of a LWS reduced subspace is proportional to 2S×Mps, as given in Eq. (45). Consistently, LWSs for which 2S=0 and/or Mps =0 carry no spin current. The degrees of freedom of the 2Sunpaired spins 1/2are distributed over different quantum numbers of the exact BA solution. They are the physical spins 1/2 whose spin is flipped by the spin SU(2)symmetry algebra off-diagonal generators. The spin degrees of freedom of the S↑and S↓unpaired spins 1/2 with up and down spin projection, respectively, determine the spin S=(S↑+S↓)/2 and spin projection Sz=−(S↑−S↓)/2of all energy eigenstates. Their translational degrees of freedom are described in each n-band by its Mh n= 2S+∞ n=n+12(n−n) Mnholes. Hence in terms of the exact solution quantum numbers the above local processes that generate the spin currents refer to the relative occupancy configurations of the Mnpseudoparticles and corresponding Mh nholes in each nband for which Mn>0. Consistently, the LWSs spin currents ˆ Jz LWS(lr, S)in the general spin current expression ˆ Jz(lr, S↑, S↓) =([S↑−S↓]/2S) ˆ Jz LWS(lr, S), Eq. (42), can alternatively be expressed in terms of pseudoparticles, as given in Eq. (35), or of n-band holes. Within the latter representation, they read ˆ Jz(lr, S) =∞ n=1Mb n j=1Mh n(qj) jh n(qj)where Mh n(qj) =1 −Mn(qj)and jh n(qj) =−jn(qj). For non-LWSs, one can consider that Mh n=Mh n,↑+Mh n,↓where Mh n,σ =Sσ+∞ n=n+1(n− n) Mnfor σ=↑, ↓. The role of the additional number ∞ n=n+12(n−n) Mnof holes in Mh n= 2S+∞ n=n+12(n−n) Mnis to ensure that in each fixed-Ssubspace dimension, N(S) = (2S+1) Nsinglet(S), the factor Nsinglet(S) =L Msp−L Msp−1where Msp =L/2 −Sis exactly given by Nsinglet(S) ={Mn}mS∞ n=1Mb n Mh n. On the one hand, for the S=0energy eigenstates considered in Sec. 4, the number Mh n reads Mh n=Mh,0 n=∞ n=n+12(n−n) Mn. In this case the elementary currents carried by ∞ n=n+1(n−n) Mnn-band holes exactly cancel those carried by the remaining ∞ n=n+1(n− 80 J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 n) Mnsuch holes. On the other hand, for S>0 energy eigenstates for which ∞ n=n+12(n− n) Mn>0 there is a corresponding partial elementary current cancellation. In that case out of the Mh n=2S+∞ n=n+12(n−n) Mnn-band holes there is in average number 2Sof such holes that describe the translational degrees of freedom of the 2Sunpaired spins 1/2. Hence their elementary currents contribute to the LWSs spin currents. The elementary currents carried by an average number ∞ n=n+12(n−n) Mnof n-band holes cancel each other. In the case of LWSs, such a partial canceling does not occur in n-bands for which Mh n=2S. We denote by | ˆ Jz LWS|A(mS,mh 1)the average current absolute value of each LWS reduced subspace. It is given by | ˆ Jz LWS|A(mS,mh 1)=lmS,mh 1| ˆ Jz(lmS,mh 1)| {Mn}mS,mh 1∞ n=1Mb n Mn.(48) Here the sum lmS,mh 1 runs over all n =1, ..., ∞band occupancy configurations that generate the {Mn}mS,mh 1∞ n=1Mb n MnLWSs with the same number 2Sof unpaired physical spins 1/2 and Mps of pseudoparticles. Hence the summation {Mn}mS,mh 1 is over all sets of n-band pseudoparticle numbers {Mn}that obey both the sum rules ∞ n=1n Mn=1 2(L −2S) =Msp, Eq. (18), and ∞ n=1Mn=1 2(L −Mh 1) =Mps, Eq. (27), respectively. That each fixed-Sreduced subspace is spanned by energy eigenstates with exactly the same number Mps of pseudoparticles simplifies the form of the average current absolute values, Eq. (48). In the TL they are related to the corresponding largest current absolute values, Eq. (45), as follows, | ˆ Jz LWS|A(mS,mh 1)=cA cT | ˆ Jz LWS|L(mS,mh 1) 2Mps =cA LJ2S2Mps ≈Jm S2Mps .(49) The coefficient cAreads here cA=1for (1 −mh 1) 1 and otherwise obeys the inequality cA≤1, being of the order of unity. The factor 1/2Mps that multiplies | ˆ Jz LWS|L(mS,mh 1)stems from the LWSs that span the reduced subspace being generated by all possible occupancy configurations of the Mps pseudoparticles. In the case of the reduced subspace for which Mps reaches its maximum value at fixed S, the average current absolute value general form, Eq. (49), follows from the calculations of Appendix B. Its generalization to the remaining reduced subspaces involves in the TL lengthy yet straightforward calculations. The precise value of the coefficient cAremains though an involved open problem. Fortunately, the only related information needed for our studies is that cAis of the order of the unity. At fixed spin Sthe number 2Sof unpaired physical spins 1/2 that couple to a vector potential is fixed. Hence the current absolute values are largest for LWSs for which these 2Sunpaired spins 1/2have a larger number Mps of n-band pseudoparticles to interchange position with. On the one hand, for a given LWS fixed-Ssubspace the average current absolute value is thus smallest for its Mps =1 reduced subspace. For it the Msp =L/2 −Sspin–singlet pairs are all bound within a single gigantic n =Msp =L/2 −Spair-configuration. The single pseudoparticle J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 81 of the LWSs that span such a LWS reduced subspace has one of the j=1, ..., 2S+1 momentum values qj=0, ±2π L, ..., ±2π L(S −1), ±2π LS. For such LWSs the Msp =L/2 −Sspin–singlet pairs involving the L −2Spaired spins 1/2 reach the smallest dilution relative to the 2Sunpaired spins 1/2. The spin current of these LWSs, ˆ Jz LWS(lr, S) =ˆ Jz LWS(qj, S) =−2Jsinqj, results from the motion of the single gigantic pseudoparticle relative to a number 2Sof n =L/2 −S band holes. Those describe the translational degrees of freedom of the 2Sunpaired physical spins 1/2. On the other hand, both the largest current absolute | ˆ Jz LWS|T(m S,mh 1), Eq. (45), and the average current absolute value | ˆ Jz LWS|A(mS,mh 1), Eq. (49), reach their maximum values for the real-rapidity reduced subspace for which Mps =M1=Msp =L/2 −Sand thus Mn=0for n >1. Its average current absolute value, Eq. (48), can be written as | ˆ Jz LWS|A(mS,mS)=lmS| ˆ Jz(lmS)| Mb S Mp S .(50) The sum lmSin this expression runs over the set of n =1 band occupancy configurations that generate the Mb S Mp SLWSs with the same spin Swhose Mp S, Mh S, and Mb Snumbers are given in Eq. (44). That at fixed mS=2S/L the average current absolute value | ˆ Jz LWS|A(mS,mh 1)≈Jm S2Mps in Eq. (49) where Mps =1, ..., Msp reaches the largest value for the real-rapidity reduced subspace for which Mps =Msp plays a key role in our analysis. This implies that in each LWS fixed-Ssubspace the set of average current absolute values, Eqs. (48) and (49), for which mh 1>m Sobey the inequality | ˆ Jz LWS|A(mS,mh 1)<| ˆ Jz LWS|A(mS,mS)for mS<m h 1<1.(51) Here | ˆ Jz LWS|A(mS,mS)is the corresponding real-rapidity reduced subspace average current absolute value, Eq. (50). We call | ˆ Jz LWS|A(mS)the average current absolute value of a LWS fixed-Ssubspace. It reads | ˆ Jz LWS|A(mS)=lr| ˆ Jz(lr,S)| Nsinglet(S) =lr| ˆ Jz(lr,S)| {Mn}mS∞ n=1Mb n Mn.(52) As in Eq. (31), the sum lrin this expression runs over all n =1, ..., ∞band occupancy configurations that generate the Nsinglet(S) ={Mn}mS∞ n=1Mb n MnLWSs with the same spin S. As in that equation, the summation {Mn}mSis thus over all sets of n-band pseudoparticle numbers {Mn}that obey the sum rule ∞ n=1n Mn=Msp =L/2 −S, Eq. (18). This corresponds to the set of all energy eigenstates with the same number Msp =L/2 −Sof spin–singlet pairs and different numbers Mps =1, ..., Msp of pseudoparticles. That the inequalities, Eq. (51), are valid for all reduced subspaces of any LWS fixed-S subspace for which mh 1>m Sstraightforwardly implies the validity of the following related inequality, | ˆ Jz LWS|A(mS,mS)≥|ˆ Jz LWS|A(mS).(53) 82 J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 Since that validity refers to all S>0 values, it ensures the validity, within the TL, of the following important inequality used below in the analysis of Sec. 7, SlmS | ˆ Jz(lmS)|2 (2S)2 SMb S Mp S≥Slr| ˆ Jz(lr,S)|2 (2S)2 S{Mn}mS∞ n=1Mb n Mn.(54) Before presenting such an analysis, a more general inequality accounting for the effects of the string deformations is introduced in the ensuing section. 6. The effects of the string deformations on the spin currents at finite magnetic field At finite magnetic field only the deviations Dn,l jthat occur in the strings themselves, Eq. (16), may have effects in the TL on the spin currents and other quantities. The set of these complex rapidities with the same real part of form n,l j=n j+i(n +1 −2l) +Dn,l jremain being labeled by the quantum numbers n =1, ..., ∞and l=1, ..., nthat refer to the number of bound spin– singlet pairs and each of these pairs, respectively. Physically, this means that, as in the case of an ideal string, the distorted string associated with that set of complex rapidities also describes an independent configuration within which n =1, ..., ∞spin–singlet pairs are bound. The set of TBA complex rapidities with the same real part, Eq. (17), obey the symmetry relation n,l j=(n,n+1−l j)∗. The two complex rapidities n,l jand n,l jassociated with two spin–singlet pairs labeled by the quantum numbers land l=n +1 −l, respectively, being related as n,l j=(n,l j)∗for l=1, ..., nis actually a necessary condition for the binding of the l=1, ..., nspin–singlet pairs within the n-pair configuration. Importantly and due to self-conjugacy, the deviations Dn,l j=Rn,l j+iδn,l jin Eq. (16) for the set of complex rapidities with the same real part associated with a distorted string are also such that Dn,l j=(Dn,n+1−l j)∗. This reveals that the symmetry n,l j=(n,n+1−l j)∗prevails under string deformations. This ensures that as for the ideal strings, the imaginary parts of the nreal rapidities with the same real part associated with deformed strings also describe the binding within the corresponding n-pair configurations of l=1, ..., nspin–singlet pairs. The V-strings deformations [39] have in the TL and finite magnetic field no effects on the spin currents. At finite magnetic field the EKS-strings collapse of narrow pairs, described below within our representation in terms spin–singlet pair unbinding processes, is in the TL the only aberration from the ideal strings [39] that may have effects on the spin currents. This refers only to the currents of |Sz| >0 energy and momentum eigenstates described by groups of real and complex rapidities. Here we identify such effects and justify why in the TL they have no impact whatsoever in the high-temperature stiffness upper bounds introduced in the ensuing section. The general consensus is that the use ideal strings for energy and momentum eigenstates described by groups of real and complex rapidities leads in the TL to exact results as long as either the temperature or the magnetic field are nonzero [39]. Consistently, although the collapse of narrow pairs is indeed found to enhance the spin currents absolute values of a few states, it does no change in the TL the stiffness upper bounds used in our study. The string deviations from the TBA ideal strings do not change the value of the number of spin–singlet pairs. Hence their density is also exactly given by msp =(1 −mS)/2for the corresponding LWSs and non-LWSs. Narrow pairs refer to a string deformation originated by a deviation Dn,l jthat renders the separation between two rapidities n,l jand n,l+1 jin the imaginary direction less than i. Such a separation may become narrower and eventually merge and J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 83 split back onto the horizontal axis [39]. Such a process is what is called the collapse of a narrow pair. Within our representation in terms of the model physical spins 1/2, it then refers to an elementary process that leads to the unbinding of two spin–singlet pairs. On the one hand, for the set of n >2 complex rapidities with the same real part associated with nbound pairs, it leads to the partition of the corresponding n-pair configuration into a n-pair configuration where n=n −2. The latter is described by a smaller number n=n −2of complex rapidities with the same real part in a string of smaller length n=n −2. The process also generates two unbound spin–singlet pairs described by real rapidities. On the other hand, for n =2 complex rapidities with the same real part it leads in turn to the unbinding of the two spin–singlet pairs of the corresponding n =2 pair configuration. This gives rise solely to the two unbound spin–singlet pairs described by real rapidities. Hence the collapse of a narrow pair is a process that causes an increase in the value of the number of strings of all lengths, Mst =Mp+MB st , Eq. (15). It does not change though that of spin–singlet pairs, Msp =L/2 −S. Specifically, it always leads to a positive deviation δMp=2 in the value of the number of spin-band pseudoparticles and corresponding unbound spin–singlet pairs. Moreover, it gives rise to a negative deviation δMB sp =−2in the value of the number of bound spin–singlet pairs. There is as well either an additional negative deviation δMB st =−1or no deviation δMB st =0in the number MB st of independent configurations with bound spin–singlet pairs within them. This depends on whether the deformed n-pair configuration that suffers the collapse of a narrow pair has n =2or n >2 spin–singlet pairs bound within it, respectively. We denote by | ˆ Jz LWS|AD(mS)the average current absolute value of the LWS fixed-Ssubspace spanned by energy eigenstates for which some of the complex strings are deformed. It is given by | ˆ Jz LWS|AD(mS)=lrD| ˆ Jz(lrD,S)| Nsinglet(S) .(55) The sum lrDin this expression runs over all L −2Spaired physical spins 1/2 occupancy configurations that generate the Nsinglet(S) LWSs with the same spin Sand thus the same number Msp =L/2 −Sof spin–singlet pairs. As given in Eq. (53), within the TBA the average of the current absolute values is largest in the fixed-Ssubspaces spanned by energy and momentum eigenstates described only by groups of real rapidities. Such an average is larger than that in the fixed-Ssubspaces spanned by all energy and momentum eigenstates of spin S. The main point is that a larger fraction of unbound spin–singlet pairs relative to bound spin–singlet pairs at the fixed number Msp =L/2 −Sof such pairs tends to enhance the spin current absolute values. A generalization of the inequality, Eq. (53), which accounts for the effects of the collapse of narrow pairs and thus of spin–singlet pair unbinding processes, involves the average current absolute value, Eq. (55), and reads | ˆ Jz LWS|A(mS,mS)≥|ˆ Jz LWS|AD(mS)≥|ˆ Jz LWS|A(mS).(56) On the one hand, the validity of the inequality | ˆ Jz LWS|A(mS,mS)≥|ˆ Jz LWS|AD(mS)in this equation follows from the energy and momentum eigenstates described by real rapidities having no strings of length n >1 and thus being string-deformation free. This is because all their Msp =L/2 −Sspin–singlet pairs are unbound. The binding of spin–singlet pairs within n-pair configurations for which n >2in states with groups of real and complex rapidities lessens the 84 J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 current absolute values. The unbinding of spin–singlet pairs under string deformations only partially neutralizes this effect. Indeed, it does not refer to all spin–singlet pairs bound within n-pair configurations for which n >2. In contrast, for the energy and momentum eigenstates described by real rapidities all Msp =L/2 −Sspin–singlet pairs are unbound. On the other hand, the inequality | ˆ Jz LWS|AD(mS)≥|ˆ Jz LWS|A(mS)in Eq. (56) is valid because the collapse of narrow pairs caused by complex rapidity string deformations may unbind some spin–singlet pairs. This effect tends to enhance the average of the current absolute values in the fixed-Ssubspaces whose strings of some states are deformed. This effect is though very small in the TL. Indeed most string deformations involve small variations in the string fine structure that do not lead to the collapse of narrow pairs and in the TL have no effects on the spin currents absolute values. Since the inequalities in Eq. (56) are valid for all S>0 values, the following important inequality, which is an extension of that given in Eq. (54), holds, SlmS | ˆ Jz(lmS)|2 (2S)2 SMb S Mp S≥SlrD| ˆ Jz(lrD,S)|2 (2S)2 SNsinglet(S) .(57) 7. High-temperature stiffness upper bounds within the thermodynamic limit The high-temperature stiffness upper bounds introduced in this section rely on replacing averages of the spin current absolute values in the full LWS spin-Ssubspaces by those in the corresponding smaller LWS real-rapidity reduced subspaces. It follows from the inequalities, Eqs. (54) and (57), that our final results are independent from the use in the TL of ideal or deformed strings for the states described by groups of real and complex rapidities. For simplicity, we use the number notation in Eq. (44), within which Mp S(qj) =M1(qj), qj=(2π/L) Ij∈[−qb, qb], Ij=I1 j, and qb=qb 1=π(Mb S−1)/L. Each LWS real-rapidity reduced subspace is then spanned by Mb S Mp Senergy and momentum eigenstates with the same S value. A first spin stiffness upper bound, Du1(T ) ≥D(T ), is derived from the direct use in the high-temperature stiffness expression, Eq. (31), of the inequalities in Eqs. (54) and (57). This leads to Du1(T ) =(2Sz)2 2LT L/2 S=|Sz|lmS | ˆ Jz(lmS)|2 (2S)2 L/2 S=|Sz|Mb S Mp S .(58) The sums lSin this expression run over the real-rapidity LWSs whose number is Mb S Mp Sthat span each LWS real-rapidity reduced subspace. The spin currents ˆ Jz(lmS)are given by ˆ Jz(lmS)= Mb S  j=1 Mp S(qj)jS 1(qj), (59) J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 85 where Mb S j=1Mp S(qj) =Mp S. The elementary current jS 1(qj)in this expression is that in Eq. (A.20) of Appendix A.2. It reads jS 1(qj) =j1(qj)for qj∈[−qb, qb]and M1=Mps =Msp where j1(qj)is the elementary current, Eq. (36) for n =1. For the present real-rapidity LWSs one has that mh 1=mS. Hence the limits given in Eq. (38) apply. The elementary current jS 1(qj)changes thus from jS 1(qj) =−π 2Jsinqjfor qj∈[−π/2, π/2]as mS→0to jS 1(qj) =−2Jsinqjfor qj∈[−π, π]as mS→1. It can be written as jS 1(qj) =−jS 1sS 1(qj)where |sS 1(qj)| ≤1for qj∈−π 2(1−mS), π 2(1−mS). As justified in Appendix A.2, the elementary current coefficient jS 1>0in that expression reaches its largest value jS 1=2Jfor the whole mS∈[0, 1]range for mS→1. Moreover, in that Appendix it is found that the replacement in jS 1(qj) =−jS 1sS 1(qj)of jS 1and sS 1(qj)by 2Jand sinqj, respectively, ensures that | Mb S j=1Mp S(qj) 2Jsinqj| ≥| Mb S j=1Mp S(qj) jS 1(qj)|for all real-rapidity LWSs and the whole mS∈[0, 1]interval. This thus implies the validity of the following inequality, L/2  S=|Sz| lmS J2 ∗(lmS) (2S)2≥ L/2  S=|Sz| lmS | ˆ Jz(lmS)|2 (2S)2,(60) where J∗(lmS) =− Mb S j=1Mp S(qj) 2Jsinqj. Our second stiffness upper-bound, Du2(T ) ≥D(T ), is thus obtained by replacing in Eq. (58) the factor on the right-hand side of Eq. (60) by that on its left-hand side. This accounts for replacing the exact elementary spin current jS 1(qj)by a upper-bound elementary spin current given by j(q j)=−2Jsinqj.(61) Under this replacement, the sum lrSin Eq. (58) can be performed. Such a sum is carried out in Appendix B, with the result, Du2(T ) =− S J2(Sz)2 LT S2Msp +2S+sin(2πS/L) sin(2π/L) Msp+2(S−1) Msp−1 SMsp+2S Msp  =− S J2(Sz)2 LT S2L/2+S+sin(2πS/L) sin(2π/L) L/2+S−2 L/2−S−1 SL/2+S L/2−S,(62) for T→∞. Here the summations refer to − S=L/2−1 S=|Sz|and S=L/2 S=|Sz|, respectively, and for simplicity we have chosen Lto be even so that Szand Sare integers. (In the present TL this reaches again the same final results as for Lodd.) The following behaviors of the spin stiffness upper bound Du2(T ), Eq. (62), corresponding to m 1 and (1 −m) 1are derived in Appendix B, Du2(T ) =J2cu2 2Tm2≈J2 2Tm2,for m1, =J2 2T(1−m) , for (1−m) 1,(63) 86 J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 respectively, where cu2=9 4(√5−2)5 3+√3 2π≈1.032.(64) Finally, we emphasize that our T→∞upper bound, Eq. (62), has been inherently constructed to the exact T→∞stiffness reading, D(T ) =Du2(T ) =J2 2T(1−m) , (65) for (1 −m) 1 and D(T ) =J2c2 2Tm2,(66) for m 1. Here cis a mand Tindependent coefficient, c≈1 such that c2<c u2. The calculations of Appendix B that reached the expressions in Eqs. (65) and (66) correspond in these two limits to average current absolute values of the form | ˆ Jz LWS|A(mS,mS)= cJm S2Mps =cJm S√L−2Swhere c=1for (1 −mS) 1 and c≈1for mS1, consistently with Eq. (49) for mh 1=mSwhere mS=mfor LWSs. 8. Concluding remarks The upper bound on high-temperature spin stiffness derived in this paper, Eqs. (62)–(64), vanishes as m2in the m →0 limit and is independent of the system size L. This ensures that the spin stiffness vanishes within the grand-canonical ensemble as h →0for high temperature T→∞ in the TL. We believe that our result is exact in these limits. The possibility of the absence of ballistic spin transport for the whole finite-temperature range T>0within the grand-canonical ensemble in the limit of zero magnetic field remains though an interesting unsolved problem. Concerning the relation of our results to previous results on the spin stiffness of the spin-1/2 XXX chain, the upper bound of Ref. [26] is valid for the whole temperature range T>0 and vanishes as m2Lin the m →0 limit. This latter behavior reveals that within the canonical ensemble the model spin stiffness vanishes as m →0for finite temperature within the TL. However and as mentioned above, it leaves out, marginally, the grand canonical ensemble in which m2 =O(1/L). The large overestimate of the current absolute values used in deriving the stiffness upper bound of that reference, whose limiting values are given in Eq. (7), leads for high temperature to an extra factor of the order O((1 −m)L) relative to our upper bound, Eq. (62). This refers to an overestimate of the method used in Ref. [26] that has ignored the factor 1/2Mps =1/√(1−mS)L in the corresponding spin current average value, first expression of Eq. (49) for mh 1=mSwhere mS=mfor LWSs. We note that our result on vanishing spin stiffness as h →0in the TL crucially depends on the existence of a global SU(2)symmetry where the current under consideration is a part of the symmetry operator algebra. We thus expect that our result should be extendable to other integrable models with similar one or several global SU(2)symmetries, such as e.g. the fermionic 1D Hubbard model. In conclusion, in this paper we addressed the important fundamental and highly debated question on the possibility of ballistic spin transport within the grand-canonical ensemble for h →0 J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 93 mechanism that justifies the use of the function sS 1(qj) =− sinqj. First we discuss the suitable use of a odd function, sS 1(qj) =−sS 1(−qj), for that elementary current. We then justify the specific choice, sS 1(qj) =− sinqj. On the one hand, that we use a odd function for sS 1(qj)is all right for LWSs with symmetrical compact and symmetrical non-compact distributions such that Mp S(qj) =Mp S(−qj). On the other hand, analysis of the BA equation reveals that the exact function sS 1(qj)in Eq. (A.20) such that |sS 1(qj)| ≤1is not a odd function of qjfor general LWSs with asymmetrical compact and asymmetrical non-compact distributions such that Mp S(qj) = Mp S(−qj). Nonetheless, the use of a odd function sS 1(qj)for these states enhances in general their current absolute values, | Mb S j=1Mp S(qj) jS 1(qj)|. Our following analysis applies to general LWSs described only by groups of real rapidities. Those do not necessarily have compact Mp S(qj)occupancies. Hence rather than the elementary current jS 1(qj)given in Eq. (A.14), which is specific to such occupancies, here we use the more general elementary current jS 1(qj) =−2Jsin k1(qj) 2πσ1(k1(qj)) . It is that given in Eq. (36) for n =1 and LWSs described only by groups of real rapidities. For all such LWSs the BA equation is of the same form, Eq. (12) and Eq. (20) for n =1, for large finite Land the TBA, respectively. It can be written as qj=k1(qj)−2 L Mb S  j=1 Mp S(qj)arctantan(k1(qj)/2)−tan(k1(qj)/2) 2,(A.23) where j=1, ..., Mb S. If the momentum distribution is an even function, Mp S(qj) =Mp S(−qj), one finds that k1(0) =0at qj=0. The elementary current, jS 1(qj) =−2Jsin k1(qj) 2πσ1(k1(qj)) , is then a odd function. This follows from the distribution 2πσ1(k) turning out to be an even function in that case. The latter distribution can be written as 2πσb 1(k) ¯ Mp S(k) and equivalently as 2πσb(k) ¯ Mp S(k). Here 2πσb 1(k) is the distribution, Eq. (37) for n =1, and 2πσb(kj)is the solution of Eq. (34). For the present case of real rapidities they are the same distributions. Moreover, ¯ Mp S(kj) =Mp S(qj). In the general case of LWSs for which the momentum distribution Mp S(qj)is not an even function, Mp S(qj) =Mp S(−qj), the corresponding elementary current jS 1(qj)is not a odd function. Consistently, the n =1 band momentum qj=0 then corresponds to a finite momentum rapidity k1(0)given by k1(0)=2 L Mb S  j=1 Mp S(qj)arctantan(k1(qj)/2)+tan(k1(0)/2) 2,(A.24) such that k1(0) <π(1 −mS)/2. This implies that there is a positive or negative qjinterval: qj∈[0,q0]→k1 j∈[−k1(0), k1(0)]for q0>0 qj∈[q0,0]→k1 j∈[−k1(0), k1(0)]for q0<0,(A.25) 94 J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 where q0=2 L ι=±1 Mb S  j=1 Mp S(qj)arctantan(k1(qj)/2)+(ι) tan(k1(0)/2) 2,(A.26) in which the elementary current, jS 1(qj) =−2Jsin k1(qj) 2πσ1(k1(qj)) , has opposite signs for the two subintervals k1 j∈[−k1(0), 0]and k1 j∈[0, k1(0)], respectively. This refers to the corresponding momentum rapidity interval k1 j∈[−k1(0), k1(0)]. Indeed the distribution 2πσ1(k1(qj)) =2πσ1(k1 j)is for all LWSs such that 2πσ1(k1 j) ≥0. And this applies to its whole range k1 j∈[−π, π]and thus corresponding qjrange qj∈−π(1−mS)/2,π(1−mS)/2. In the qjinterval qj∈[0, q0]for q0>0 and qj∈[q0, 0]for q0<0the band momentum qj has the same sign. However, the elementary current jS 1(qj)has opposite signs in two momentum qjsubintervals of these intervals. For example, for q0>0 such subintervals read qj∈[0, q(0)] and qj∈[q(0), q0], respectively. Here, q(0)=2 L Mb S  j=1 Mp S(qj)arctantan(k1(qj)/2) 2,(A.27) is the qjvalue at which the momentum rapidity vanishes, k1(q(0)) =0. The function sS 1(qj)in Eq. (A.20) such that |sS 1(qj)| ≤1 has the same signs as k1 j. It follows that the current contributions from occupancies in such q0>0 subintervals, qj∈[0, q(0)]and qj∈[q(0), q0], tend to cancel. This would not be so if sS 1(qj)was a odd function. Moreover, the canceling momentum rapidity interval k1 j∈[−k1(0), k1(0)]corresponds to qjalternative positive qj∈[0, q0]and negative qj∈[q0, 0]intervals if the asymmetric distribution Mp S(qj) has integrated larger values for qj>0 and qj<0, respectively. Hence the use of a suitably chosen odd function sS 1(qj)enhances indeed the current absolute values | Mb S j=1Mp S(qj) jS 1(qj)| of most LWSs. Finally, we justify the choice of the specific odd function, sS 1(qj) =− sinqj. As follows from Eq. (38) for mh 1=mS, one finds for all LWSs described only by groups of real rapidities that in the limits mS1 and (1 −mS) 1 their elementary currents jS 1(qj)are exactly given by jS 1(qj)=−Jπ 2sin(qj), for mS1 =−2Jsin(qj), for (1−mS)1,(A.28) respectively. The simplest odd function sS 1(qj) =−sS 1(−qj)that in these two limits reaches the exact behavior of the elementary currents carried by such LWSs is indeed sS 1(qj) = − sinqj. Additionally, we have confirmed that this choice enhances the current absolute values | Mb S j=1Mp S(qj) jS 1(qj)|of most LWSs. Importantly, it enhances in all LWS fixed-Ssubspaces under consideration the quantity lrS| ˆ Jz(lrS, S)|2/(2S)2on the right-hand side of Eq. (58). Combining the above arguments and properties justifies why the replacement of the exact elementary functions jS 1(qj)by j(q j) =−2Jsinqj, Eq. (61), in the current absolute values | Mb S j=1Mp S(qj) jS 1(qj)|of all LWSs described only by groups of real rapidities leads to the inequality, Eq. (60). J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 95 Appendix B. Derivation of the second stiffness upper bound Here the sum lr1in Eq. (58) is performed by the use of the upper-bound elementary spin current j(q j) =−2Jsinqj, Eq. (61). To reach this goal we first consider the Mp S-dependent sums, for fixed Mb S. Those give a upper bound on corresponding sums over lSin Eq. (58), namely, 1 4J2 lS| ˆ Jz(lS,S)|2≤I(Mp S). (B.1) Here I(Mp s)= b1,b2...bMb S∈{0,1} δMp S,lbl Mb S  k=1 bksinqk 2 ,(B.2) and bj≡M(qj)are binary occupation numbers, which we sum over. The δ-constrain can be analytically treated by means of a counting field parameter λ. This is done by defining ˜ I(λ)= Mb S  MS=0 eλMp SI(Mp S). (B.3) We then find immediately that ˜ I(λ)= Mb S  k=1 Mb S  l=1 sinqksinql b1,b2...bMb S bkbl j eλbj = k sin2qkeλ(1+eλ)Mb S−1 + k=l sinqksinqle2λ(1+eλ)Mb S−2 = k sin2qk(eλ(1+eλ)Mb S−1−e2λ(1+eλ)Mb S−2) = k sin2qkeλ(1+eλ)Mb S−2 = k sin2qkMb S−1  Mp S=1Mb S−2 Mp S−1eMp Sλ.(B.4) Indeed, due to δ-constraint one has that eλMp S=Mb S k=1eλbk. We have been using the property that 0 =(bksinqk)2=ksin2qk+k=lsinqksinql. From it we find I(Mp s), I(0)=I(Mb S)=0,(B.5) 96 J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 I(Mp S)=⎛ ⎝ Mb S  k=1 sin2qk⎞ ⎠Mb S−2 Mp S−1for 1 ≤Mp S≤Mb S−1.(B.6) Furthermore, we can explicitly calculate the sum over sin2qk. This gives  k sin2qk= Mb S  k=1 sin2π L(2k−Mb S−1) =1 2Mb S−sin(2πMb S/L) sin(2π/L)  =L 4+S 2+1 2 sin(2πS/L) sin(2π/L) .(B.7) From the use of the estimates in Eq. (B.1) with Eqs. (B.6) and (B.7) in the expression for the stiffness, Eq. (31), we finally arrive at the simple bound given in Eq. (62), where a single sum over Sremains. Next we confirm the behaviors reported in Eq. (63), which are reached by the stiffness upper bound, Eq. (62), in the m →0 and m →1 limits as L →∞. Concerning the m →0 limit, within the TL one may replace L/2+S−2 L/2−S−1on the right-hand side of Eq. (62) by a simpler expression, L/2+S L/2−S. Hence the following identity can be used, L/2  S=0L/2+S L/2−S=fL+1, lim L→∞ L/2−1 S=1ϕ(S/L)L/2+S L/2−S L/2 S=0L/2+S L/2−S=ϕ(1/3). (B.8) Here fjis the j-th Fibonacci number, defined by f0=f1=1, fj+1=fj+fj−1, and ϕ(x) is an arbitrary smooth function on (0, 1), possibly with poles at 0or 1. In our case, ϕ(x) =1 x21 2+x+1 2πsin(2πx).(B.9) The replacement of L/2+S−2 L/2−S−1by L/2+S L/2−Son the right-hand side of Eq. (62) amounts though by multiplying it by an additional factor, lim L→∞ SL/2+S−2 L/2−S−1 SL/2+S L/2−S=lim L→∞ fL−2 fL+1=√5−2.(B.10) From the combination of such procedures, we arrive at the following final compact upper bound valid for m =−2Sz/L →0in the TL, D(T ) ≤9 2(√5−2)5 3+√3 2πJ2 TSz L2 .(B.11) This is the expression given in Eq. (63) for m 1. J.M.P. Carmelo, T. Prosen / Nuclear Physics B 914 (2017) 62–98 97 Note that the lower limit of the sum in Eq. (B.8) can in the TL be pulled up to the S=|Sz| for any |Sz| ≤L/3. This is so that the sum still starts before the maximum of the binomial symbol, which in the TL can be approximated with a Gaussian. This yields the same asymptotic inequality, Eq. (54). Finally, we evaluate the behavior of the stiffness upper bound, Eq. (62), in the regime m →1, i.e., −Sz=L/2 −δ, where δL. 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