Thermodynamic behavior of a one-dimensional Bose gas at low temperature

Thermodynamic behaviour of a one-dimensional Bose gas at low temperature Giulia De Rosi,1, ∗Grigori E. Astrakharchik,2, †and Sandro Stringari1, ‡ 1INO-CNR BEC Center and Dipartimento di Fisica, Universit`a di Trento, Via Sommarive 14, I-38123 Povo, Italy 2Departament de F´ısica, Universitat Polit`ecnica de Catalunya, 08034 Barcelona, Spain (Dated: September 20, 2017) We show that the chemical potential of a one-dimensional (1D) interacting Bose gas exhibits a non-monotonic temperature dependence which is peculiar of superfluids. The effect is a direct consequence of the phononic nature of the excitation spectrum at large wavelengths exhibited by 1D Bose gases. For low temperatures T, we demonstrate that the coefficient in T2expansion of the chemical potential is entirely defined by the zero-temperature density dependence of the sound velocity. We calculate that coefficient along the crossover between the Bogoliubov weaklyinteracting gas and the Tonks-Girardeau gas of impenetrable bosons. Analytic expansions are provided in the asymptotic regimes. The theoretical predictions along the crossover are confirmed by comparison with the exactly solvable Yang-Yang model in which the finite-temperature equation of state is obtained numerically by solving Bethe-ansatz equations. A 1D ring geometry is equivalent to imposing periodic boundary conditions and arising finite-size effects are studied in details. At T= 0 we calculated various thermodynamic functions, including the inelastic structure factor, as a function of the number of atoms, pointing out the occurrence of important deviations from the thermodynamic limit. PACS numbers: PACS numbers I. INTRODUCTION It is well known that the thermodynamic behaviour of a superfluid is dominated, at low temperature, by the thermal excitation of phonons [1]. This explains, in particular, the peculiar behaviour exhibited by the specific heat as well as by other fundamental thermodynamic functions. A non trivial (and less investigated in the literature) consequence of superfluidity shows up in the non-monotonic behaviour of the chemical potential [2]. At low temperature Tthe chemical potential increases with Tas a consequence of the thermal excitation of phonons. At high temperature, in the ideal gas classical regime, the chemical potential is instead a decreasing function of T. This non-monotonic behaviour has been recently measured in a strongly interacting atomic Fermi gas [3], where it was shown that the chemical potential exhibits a maximum in the vicinity of the superfluid critical temperature. It is consequently interesting to explore the lowtemperature thermodynamic behaviour of other systems, like one-dimensional (1D) interacting Bose gases, which are known to exhibit a phononic excitation spectrum, despite the fact that they cannot be considered superfluids according to standard definition. By investigating the drag flow caused by a moving external perturbation, Astrakharchik and Pitaevskii [4] have in fact shown that 1D Bose gases interacting with contact potential exhibit a traditional superfluid behaviour, characterized by the ∗[email protected] †[email protected] ‡[email protected] absence of friction force, only in the weakly interaction regime, where Bogoliubov theory applies and the gas can be locally considered Bose-Einstein condensed, despite the absence of true long range order. In this work, we investigate the low-temperature expansion of the chemical potential µof a 1D Bose gas with contact repulsive interaction for the whole crossover, ranging from the weakly to the strongly interaction limits. A major motivation is given by the possibility of comparing the low-T expansion of the chemical potential with the numerical results now available within the YangYang theory [5,6], along the whole interaction strength crossover. Previous comparisons were in fact available only in the case of the Tonks-Girardeau limit [7], corresponding to the ideal Fermi gas, where the low-T expansion corresponds to the Sommerfeld expansion. We find that for all intermediate interaction regimes, described at T= 0 by Lieb-Liniger (LL) theory, the increase of the chemical potential at low temperature follows the µ∝T2 law and is actually caused by the phononic nature of the long wavelength elementary excitations, as in usual superfluids [2]. The relevant coefficient fixing the T2law depends on the density derivative of the T= 0 sound velocity which is calculated using Lieb-Liniger theory. This feature strengthens the analogy with superfluids even in 1D dimension. Importantly, our results can be also generalized to every Luttinger liquid at low temperature whose macroscopic elementary excitations can be described in terms of phonons. Recently, a ring geometry has been experimentally realized for a microscopic system of N= 8 −20 atoms [8]. Motivated by the experimental progress, we study in details also the behavior of a gas containing a finite number of atoms in a ring, focusing on the deviations of its thermodynamic behavior from the one in the large N 2 limit. Our system is a uniform gas of bosons interacting with a repulsive contact interaction H=−~2 2m N X i=1 ∂2 ∂x2 i + 2c N X i>j δ(xi−xj) (1) where the interaction parameter cis related to the 1D coupling constant g1D =−2~2/(ma1D) through c= mg1D/~2, where a1D is the 1D scattering length. The system (1) has been realized experimentally for the whole interaction crossover by suitably tuning the interaction strength [9–11], described by the dimensionless parameter γ=c n=−2 na1D (2) from weak (γ→0) to strong (γ1) interactions [10,12– 15]. The Bogoliubov (BG) perturbative theory can be used in the limit of weak interactions. In the TonksGirardeau (TG) limit of strong repulsions the bosons are impenetrable and their wave function can be mapped onto that of an ideal Fermi gas [16]. The paper is organized as follows. In Sec. II we derive the low-temperature expansion of the chemical potential, starting from the free energy of an ideal phononic gas. This assumption is fully justified by the low-momenta behavior of the Lieb-Liniger excitation spectrum. The low-temperature expansion exhibits aT2-dependence on temperature, with the coefficient related to the density derivative of the LL sound velocity at zero temperature. The Bethe-ansatz results for the chemical potential are shown to agree very well with the lowtemperature expansion, for the whole BG-TG crossover. In Sec. III we investigate the BG weakly-interacting gas. By considering the quantum fluctuation contribution in the ground-state energy at T= 0, we explore the behavior of the chemical potential and of the sound velocity. While this correction is important at T= 0, it does not affect the low-temperature expansion of the chemical potential. Similarly to Sec. III, we calculate in Sec. IV the first corrections in the interaction parameter γto the TG strongly interacting gas. The starting point is the expansion, for large values of γ, of the ground-state energy of a hard-sphere gas. In Sec. Vwe derive the low-temperature expansions of both the adiabatic and the isothermal inverse compressibilities. The coefficients of the T2laws are studied as a function of the interaction parameter γand analytically calculated in the BG and TG limits. In Sec. VI we consider a ring configuration with a finite number of particles at zero temperature and calculate the finite-size corrections with respect to the thermodynamic limit for the energy, the chemical potential and the sound velocity. Results for the static inelastic structure factor for a finite number of particles are also reported. In Sec. VII, we draw our final conclusions. II. LOW-TEMPERATURE EXPANSION OF THE CHEMICAL POTENTIAL It is well known that at T= 0 the elementary excitations of an interacting 1D Bose gas have a phononic character at small momenta [17,18], characterized by the linear dispersion relation (p)p→0=vsp . (3) At T= 0 the sound velocity is related to the density dependence of the chemical potential according to the relation vs(γ) = rn m ∂µ(T= 0, γ) ∂n ,(4) where µis the chemical potential and n=N/L denotes the linear density. The density dependence of the chemical potential at zero temperature can be calculated within the Lieb–Liniger model. The ratio between the sound velocity and the Fermi velocity vF=π~n/m is known as the Luttinger parameter, KL=vF/vs, and it plays an important role in defining the long-range properties of one-dimensional systems. Figure 1shows the dependence of the sound velocity on the interaction parameter γfor the Lieb-Liniger model, described by Hamiltonian (1). There is a smooth crossover between the meanfield BG value defined as mv2 s=g1Dnfor weak interactions to the Tonks-Girardeau (ideal Fermi gas) value vs=vFin the limit of strong repulsion. 10−310−210−1100101102103 γ 0.0 0.2 0.4 0.6 0.8 1.0 1.5 vs(γ)/vF vs(γ)/vF vBG s(γ)/vF vTG s/vF vs(γ1)/vF vs(γ1)/vF FIG. 1. (Color online) Sound velocity vsin units of Fermi velocity vF(solid line) as a function of the interaction parameter γ, calculated by solving the Lieb-Liniger equations. The Bogoliubov (dotted line, vBG s(γ)/vF=√γ/π) and Tonks-Girardeau (dashed line, vTG s=vF) limits, including their first-order corrections (thin solid line vs(γ 1)/vF=vBG s(γ)/vFp1−√γ/(2π) and thin dot-dashed line vs(γ1)/vF=p1−8/γ, respectively) are present too, see Secs. III and IV. For larger momenta the 1D excitation spectrum is characterized by a continuous structure, bounded by two branches of elementary excitations [5,17,18], which have been the object of recent measurements [19,20]. For 3 small values of γ, the Lieb-I particle-like branch corresponds to the Bogoliubov excitation spectrum [17,18, 21]. The Lieb-II hole-like branch is instead associated in the weakly-interacting regime with the dark soliton dispersion predicted by Gross-Pitaevskii theory [18,21,22]. The two branches merge into the phononic spectrum for pmvs, Fig. 2. 0 0.25 0.5 0.75 1 p/pF 0 0.5 1 1.5 (p)/EF Bogoliubov GP soliton Phonons 0 0.25 0.5 0.75 1 p/pF 0 1 2 3 (p)/EF Lieb I Lieb II Phonons FIG. 2. (Color online) Lieb–Liniger excitation spectrum in the BG regime with γ= 4.52 (left) and in the deep TG regime with γ=∞(right). The units are the Fermi energy EFand the Fermi momentum pF. The shaded region represents the continuum of the excitations and is delimited by the upper (Lieb I) and the lower (Lieb II) branch of the spectrum. On the left, the Lieb I and II branches are not reported. On the left, the dashed line gives the Bogoliubov dispersion and the dotted line gives the mean field soliton spectrum. In the limit γ→0, the Lieb I branch tends to be equal to the Bogoliubov dispersion, while the Lieb II one coincides with the soliton spectrum. The solid line is the Lieb–Liniger phononic spectrum calculated with γ= 4.52. On the right, Lieb I and Lieb II branches are reported and they coincide with the particle and hole ideal Fermi gas excitations, respectively. The solid line is the phononic spectrum calculated with the Fermi velocity. At low temperature (kBTmv2 s) we expect that the thermodynamic behaviour of the system can be calculated in terms of a gas of non interacting phonons. The free energy A=E−TS of this gas is then given by A(T, L) = E0+kBTL 2π~Z+∞ −∞ log h1−e−β(p)idp (5) where (p) is dispersion (3) and we have added the energy E0calculated at T= 0 with the Lieb–Liniger theory. Notice that the thermal contribution to Ais affected by two-body interactions through the dependence of (p) on the interaction parameter γ. The integral of Eq. (5) yields the following low-temperature expansion for the free energy A(T, L) = E0−π 6 (kBT)2L ~vs ,(6) which differs from the usual T4behaviour exhibited by three-dimensional (3D) superfluids [18] because of the 1D structure of the integral (5). Starting from result (6) for the free energy, one can calculate the low-temperature expansion of the chemical potential: µ(T, γ) = ∂A ∂N T,L =EF"α(γ) + β(γ)T TF2#(7) where we have introduced the energy scale EF=kBTF= ~2π2n2/(2m) given by the Fermi energy of a 1D Fermi gas, because it exhibits the same density dependence of the quantum degeneracy temperature of the system. We have also defined the relevant dimensionless parameters of the expansion α(γ) = µ(T= 0, γ) EF (8) and β(γ) = πEF 6~v2 s ∂vs ∂n ,(9) which are functions of the interaction parameter γand can be calculated at zero temperature using Lieb–Liniger theory. It is worth noticing that the parameter β(γ), which is the most relevant because it fixes the leading coefficient of the low-Texpansion, depends on the density derivative of the sound velocity. The two numerical functions α(γ), Eq. (8), and β(γ), Eq. (9), have been calculated within LL theory and their values are reported in Figs. 3and 4with their BG and TG limits. In particular, the TG limits for α(γ) and β(γ) reproduce the low-temperature Sommerfeld expansion of the chemical potential for the 1D ideal Fermi gas, Eq. (21). 10−310−210−1100101102103 γ 10−3 10−2 10−1 100 101 α(γ) α(γ) αBG(γ) αTG α(γ1) α(γ1) FIG. 3. (Color online) α(γ) (solid line) with the leading dependence (dashed line, αTG = 1 and dotted line, αBG(γ) = 2γ/π2) and first order corrections [thin dot-dashed line, α(γ1) = 1 −16/(3γ) and thin solid line, α(γ1) = αBG(γ)(1 −√γ/π)] for Tonks-Girardeau and Bogoliubov limits, respectively. In Figs. 5-6we report the temperature dependence of the chemical potential of the system described by Hamiltonian (1) as obtained numerically from the Bethe-ansatz (BA) approach first developed by Yang-Yang [5–7,23] for several characteristic values of γ. The Yang-Yang 4 10−310−210−1100101102103 γ 10−1 100 101 102 β(γ) β(γ) βBG(γ) βTG FIG. 4. (Color online) β(γ) (solid line) with the TonksGirardeau (dashed line, βTG =π2/12) and Bogoliubov (dotted line, βBG(γ) = π3/(24√γ)) limits. description has been probed experimentally [24,25] and allows not only to investigate the thermodynamics, but also the Luttinger liquid physics and the quantum criticality of the system [26–28]. The numerical results for the thermodynamics have been derived recently in an analytic fashion by using the polylog functions at finite temperature by Guan and Batchelor [26,28]. The crossover from mean-field to Tonks-Girardeau regimes (see Fig. 1) introduces two distinct energy scales. Correspondingly, we rescale the chemical potential in units of the Fermi energy EFin Fig. 5and in units of the mean-field zero-temperature chemical potential µBG(T= 0) = g1Dnin Fig. 6. The first choice provides natural units in the TG regime in which strongly repulsive bosons behave similarly to an ideal Fermi gas (IFG) in the limit of γ→ ∞. In this regime, the chemical potential as a function of Tis calculated by inverting the Fermi–Dirac distribution (upper dashed line in Fig. 5): nIFG(p) = 1 e 1 kBTp2 2m−µ+ 1 ; (10) and, despite the absence of superfluidity, it still exhibits the quadratic low-temperature dependence µ∝T2, which follows from the low-temperature Sommerfeld expansion, Eq. (21). By reducing the interaction parameter γ, the system becomes softer and the limit of vanishing interactions, γ→0, corresponds to an ideal Bose gas (IBG) with the chemical potential µ(T) fixed by the relationship (lower dashed line in Fig. 5): nIBG(p) = 1 e 1 kBTp2 2m−µ−1 .(11) Notice that, because of the absence of Bose-Einstein condensation [29,30], the chemical potential of the 1D ideal Bose gas is always negative and approaches the value µ= 0 as T→0. Remarkably, for all finite interaction strengths the temperature dependence is not monotonic. Moreover, the initial increase is perfectly described by the quadratic low-temperature expansion (7), thereby proving that the model based on a gas of independent phonons well accounts for the thermodynamic behaviour of the 1D interacting Bose gas. This is a non trivial result due to the complex structure of the elementary excitations at larger wave vectors exhibiting a double branch converging into the phonon law (3) only at small momenta. We notice also that the chemical potential for high temperatures, which is a decreasing function of T, can be considered as a shift of the ideal Bose chemical potential, Eq. (11), for every value of γ. The behavior of the chemical potential in the weaklyinteracting regime (γ1) is best seen in Fig. 6. For low temperatures Tµthe gas behaves like a quasicondensate, exhibiting typical features of superfluids. For µTTF, the gas is a thermal degenerate gas, while for TTFthe gas behaves classically with µ < 0. A similar classification of the quantum degeneracy states in 1D trapped configurations was first proposed in [31]. 0 0.511.522.533.5 4 T/TF −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 µ/EF γ= 0.1 γ= 1 γ= 10 γ= 100 γ= 1000 IFG IBG BA Phonons FIG. 5. (Color online) Chemical potential as a function of temperature Tin Fermi units for several values of γand at a fixed density n|a1D|= 2/γ. The solid lines represent the Bethe–ansatz (BA) solutions for different values of γ. The dot-dashed lines are the low–temperature expansions of the chemical potential taking into account only the phononic contribution, Eq. (7). The phononic expansions for γ≥1000 are equal to the analytical Sommerfeld expansion of Eq. (21). Both the chemical potentials as a function of Tfor the ideal Fermi (upper dashed line) and ideal Bose (lower dashed line) gas are also reported, Eq. (10) and (11), respectively. Although there is no phase transition in 1D systems at finite T, in the canonical ensemble, there exists a critical point, corresponding to the value µ= 0 of the chemical potential, which separates the vacuum from the filled “Fermi sea” of repulsive bosons at T= 0. In particular, a universality class is present in the temperature regime T |µ|and near the critical point µ= 0 [26–28]. Figure 6is similar to Fig. 5, but with the chemical potential expressed in units of the BG chemical potential at zero temperature: µBG(T= 0) = g1Dnand the temperature in units of: TBG(γ) = mv2 F√γ πkB (12) 5 which has been introduced as an appropriate temperature scale for visualizing the behavior of the chemical potential at low temperature. With the new units, the phononic expansion (7) takes the form: µ(T, γ) = g1Dn"α(γ)π2 2γ+ 2β(γ)T TBG 2#.(13) 0 0.2 0.4 0.6 0.8 1 T/TBG(γ) 0.4 0.6 0.8 1 1.2 1.4 µ/(g1Dn) γ= 0.001 γ= 0.01 γ= 0.1 γ= 1 γ= 0.001 γ= 0.01 γ= 0.1 γ= 1 BA Phonons FIG. 6. (Color online) Chemical potential as a function of temperature in BG units for several values of γ. The solid lines represent the Bethe–ansatz (BA) solutions for different values of γ. The dashed lines are the low–temperature expansions of the chemical potential taking into account only the phononic contribution, Eq. (13). We notice also that the phononic expansion (13) does not hold for very small γ, like γ= 0.001, for which value the low–temperature expansion is not reported. Figures 5and 6point out in a clear way the nonmonotonic behavior of the chemical potential µas a function of Tfor a fixed value of the density. This is a general feature exhibited by superfluids [2] and it is shown here that it characterizes also interacting 1D Bose gas for all finite values of the interaction parameter γ. Both figures show also that the phononic expansion describes very well the low-T thermodynamics for all values of γ, although the region of the applicability of the phononic description depends on γ. As pointed out in Ref. [32], for small values of the interaction parameter γ, higher-order corrections beyond the linear phononic contribution in the excitation spectrum (3) might be important. III. BOGOLIUBOV REGIME γ→0 In the mean-field theory, the chemical potential is linear in density, µBG(T= 0) = g1Dnand the velocity of sound takes the value vBG s(γ) = ~n√γ/m =vF√γ/π, see Fig. 1. The first correction to the mean-field expression for the equation of state comes from the quantum fluctuations [18,33,34]. With respect to the 3D case, in 1D this calculation is simpler because it does not require the renormalization of the scattering length due to the absence of ultraviolet divergencies in the calculation of the ground-state energy. Therefore in 1D one can consider all ranges of momenta and one finds [17]: E0 N=1 2g1Dn+2 2N +∞ X p>0(p)−g1Dn−p2 2m(14) where (p) = sg1Dn mp2+p2 2m2 (15) is the Bogoliubov excitation spectrum. By considering the thermodynamic limit of Eq. (14) and by solving the integral in momentum space, one finally finds the first-order correction in the interaction parameter for the ground state energy [17] E0 N(γ1) = ~2n2 2mγ1−4 3π√γ.(16) The same result can be also found by performing a power series expansion of the Lieb-Liniger equations [35,36]. The correction is negative as it comes from second order perturbation theory and, contrary to the higher dimensions, in 1D there is no renormalization of the coupling constant thus no additional terms have to be added. Equation (16) allows one to calculate the higher-order corrections for the other thermodynamic quantities at T= 0. For the chemical potential, one finds the result µ(γ1) ≈~2n2γ m1−√γ π(17) which implies the result α(γ1) ≈αBG(γ)1−√γ π(18) for the expansion of the coefficient α(γ), where αBG(γ) = 2γ/π2is the mean-field value. The corresponding result has been plotted in Fig. 3and well reproduces the exact value of α(γ) up to values γ∼1. From Eq. (4) and Eq. (17), one can calculate also the correction to the sound velocity [17] vs(γ1) ≈vBG s(γ)r1−√γ 2π(19) which is also reported in Fig. 1, yielding the expression β(γ1) ≈βBG(γ),(20) for the coefficient β(γ), Eq. (9) with βBG(γ) = π3/(24√γ) the Bogoliubov value. Notice that, differently from the case of α(γ) [see Eq. (18)], the first correction βBG(γ) vanishes because of an exact cancellation between the corrections provided by the terms ∂vs/∂n and v2 sof Eq. (9). This explains why the Bogoliubov approximation describes correctly the value of β(γ) for a large interval of values of γ, up to γ∼1 (see Fig. 4). 6 IV. TONKS-GIRARDEAU REGIME γ→ ∞ In the TG limit of strong repulsion, γ→ ∞, the energetic properties are the same as in an ideal Fermi gas. The thermodynamic quantities do not depend on the coupling constant g1D, but only on the density n, encoded in the Fermi energy EF. This regime can be interpreted as that of a unitary Bose gas with the Bertsch parameter equal to 1 as the chemical potential is equal to the Fermi energy [µTG(T= 0) = EF]. Similarly, the sound velocity is equal to the Fermi velocity vTG s=vF=p2EF/m, see Fig. 1. The low-temperature expansion of the chemical potential in this limit is equal to the first terms of the Sommerfeld expansion (dot-dashed line for γ= 1000 in Fig. 5) of the 1D ideal Fermi gas, as already pointed out in [7]: µSomm(T) = EF"1 + π2 12 T TF2#(21) which contains the TG limits of α(γ) and β(γ) parameters, Figs. 3and 4. Leading corrections to the ground-state energy in the TG regime arise from the “excluded volume” and can be obtained from the equation of state of hard spheres (i. e. impenetrable) bosons with diameter a1D >0 [16]: E0 N=π2~2 6m n2 (1 −na1D)2.(22) In the limit of point-like bosons a1D = 0, Eq. (22) reproduces the ground state energy of the ideal Fermi gas, ETG =π2~2n2/(6m). Expanding the denominator in Eq. (22) generates a power series with integer coefficients, E/ETG = 1 + 2na1D + 3(na1D)2+ 4(na1D)3+···. It is interesting to notice that for a δ-interacting potential the momentum-dependent s-wave scattering length, a1D(k) = arctan(ka1D)/k =a1D −(1/3)k2a2 1D, does not affect first and second corrections in na1D but induces a negative correction in front of the third correction. Indeed, the universality of the first and the second corrections becomes evident by comparing low-density expansion of the equation of state for hard spheres, Eq. (22), and contact δ-potential obtained by solving Bethe equations recursively [37], E0 N=π2~2n2 6m1+2na1D +3(na1D)2+4−4π2 15 (na1D)3. (23) The non-universal correction depends on the shape of the potential and for the LL model it has a non-integer coefficient, which qualitatively can be understood by noting that the typical value of the scattering momentum in TG regime is proportional to kF=π~n/m, which is consistent with π2terms appearing in expansion (23). The universal terms are the same both in the super TonksGirardeau [38] (a1D >0) and the strongly repulsive [36] (a1D <0) regimes. From Eq. (23), by introducing the parameter (2) and by considering only the leading term, one finds E0 N(|γ|  1) ≈π2~2n2 6m1−4 γ.(24) From Eq. (24), one easily calculates the correction of the chemical potential at T= 0, Eq. (8): µ(|γ|  1) ≈EF1−16 3γ(25) which implies the result [36] α(|γ|  1) ≈1−16 3γ(26) for α(γ), including the first correction to the TG result αTG = 1. Prediction (26) is reported in Fig. 3for positive values of γ, its accuracy being good for values of γlarger than ∼10. From Eq. (4) and Eq. (25), one can calculate also the first correction, at large γ, to the sound velocity [36,39]: vs(|γ|  1) ≈vFr1−8 γ(27) which is reported in Fig. 1. For the coefficient β(γ), which provides the T2-correction in the expansion of the chemical potential, we find again an exact cancellation between the 1/γ correction (provided by the term ∂vs/∂n) and v2 sentering the expression (9) for β(γ), similarly to what happens in the small γexpansion discussed in the previous Section III in the case of the Bogoliubov gas. We then find that the Tonks-Girardeau expression βTG =π2/12 provides an accurate estimate of β(γ) for values of γlarger than ∼10 (see Fig. 4). More accurate analytical expressions for the above thermodynamical quantities, which allow to probe the whole range of interaction strength with excellent accuracy, are reported in [32,40–42]. V. LOW-TEMPERATURE EXPANSION OF THE INVERSE COMPRESSIBILITY Here we derive the dependence of the adiabatic and isothermal inverse compressibilities on the interaction parameter γin the limit of low temperature. A. Adiabatic inverse compressibility and sound velocity From the Gibbs–Duhem relation dP =ndµ+sdT, one finds ∂P ∂n ¯s =n∂µ ∂n¯s +n¯s∂T ∂n ¯s (28) 7 where sis the entropy density and ¯s=s/n is the entropy per particle. At low temperature the entropy per particle of a non– interacting gas of phonons takes the form [18] ¯s(T) = πk2 BT 3~vsn,(29) which depends on the T= 0 value (4) of the sound velocity. Use of relation (29) permits to express the dependence of the second contribution to the adiabatic inverse compressibility on the r.h.s. of Eq. (28) on the interaction parameter γ ∂T ∂n ¯s =3~vs¯s πk2 B1 + 6~nvsβ(γ) πEF,(30) in terms of the coefficient β(γ), Eq. (9) related to the density derivative of the sound velocity at constant entropy. The first contribution on the r.h.s. of Eq. (28) can be obtained by using Eqs. (7) and (29), ∂µ ∂n¯s =m nv2 s+(kBT)2 nEF12n~vs πEF β2(γ)−γ∂β(γ) ∂γ . (31) From the above equations one finally finds the low temperature expansion ∂P(T, γ) ∂n ¯s =∂P(γ) ∂n T=0 +EFδ(γ)T TF2 (32) of the adiabatic inverse compressibility, where ∂P(γ) ∂n T=0 =mv2 s(γ) (33) is its T= 0 value and we have defined the positive quantity δ(γ) = 24 π2β2(γ)vs(γ) vF−γ∂β(γ) ∂γ +π2 6 vF vs(γ)+ 2β(γ), (34) which is reported in Fig. 7together with its asymptotic limits in the Bogoliubov and Tonks–Girardeau regimes. B. Isothermal inverse compressibility By fixing the temperature Tin Eq. (28) and by considering the low-temperature expansion of the chemical potential (7), one can also calculate the low-temperature expression for the isothermal inverse compressibility ∂P(T, γ) ∂n T =∂P(γ) ∂n T=0 +EFη(γ)T TF2 (35) where we have defined the negative dimensionless coefficient η(γ) = −2β(γ)−γ∂β(γ) ∂γ .(36) 10−310−210−1100101102103 γ 100 101 102 103 δ(γ) δ(γ) δBG(γ) δTG FIG. 7. (Color online) Value of the dimensionless coefficient δ(γ) of the low–temperature expansion of the adiabatic inverse compressibility (solid line). The BG and TG analytical limits are also shown: δBG(γ) = 5π3/(16√γ) (dotted line) and δTG =π2/2 (dashed line). Notice that the thermal corrections to the isothermal and adiabatic inverse compressibilities have opposite sign, being the coefficient η(γ) always negative. The absolute value of η(γ) is reported in Fig. 8together with the asymptotic limits in the Bogoliubov and Tonks– Girardeau regimes. The negative value of η(γ) is the consequence of the peculiar temperature dependence of the free energy (6). 10−310−210−1100101102103 γ 100 101 102 |η(γ)| |η(γ)| |ηBG(γ)| |ηTG| FIG. 8. (Color online) Absolute value of the dimensionless coefficient η(γ) of the low–temperature expansion of the isothermal inverse compressibility (solid line). The BG and TG analytical limits are also shown: ηBG(γ) = −π3/(16√γ) (dotted line) and ηTG =−π2/6 (dashed line). VI. GAS ON A RING The physics in one dimension is unusual in many aspects. The mean–field regime is reached at large densities contrarily to what happens in three dimensions where the weakly–interacting limit corresponds to small densities, according to the limit na3→0. For a fixed number of particles Nthe mean–field limit in one dimension, 8 n|a1D|→∞, can be obtained either increasing the linear density n=N/L, by decreasing the system size L, or by increasing the s–wave scattering length a1D, i.e. decreasing the coupling constant g1D=−2~2/(ma1D). Asymptotically, at a certain point, the size of the system Lwill become comparable to the healing length ξ=s~2 2mg1Dn(37) and finite–size effects will become important. This should be contrasted to the three-dimensional case where the mean–field regime is instead achieved by increasing the system size Lwhich consequently becomes larger than the healing length. Finite–size effects depend on the system geometry. Interestingly, periodic boundary conditions, commonly used as a mathematical tool in the three–dimensional world, in one dimension can be explicitly realized in a ring and have consequently a direct physical interest. This is another peculiarity of the one–dimensional world. In the following we calculate the finite–size dependence of thermodynamic quantities for a gas confined in a ring whose properties are then equivalent to the ones of a linear 1D system satisfying periodic boundary conditions (PBC). If one considers a plane wave ∝eikz and one imposes PBC, one finds that the momentum is quantized according to pi=~ki=2π~ni L(38) where ni= 0,±are integers. Moreover, in 1D, all the integrals in momentum space, defined in the thermodynamic limit (N, L →+∞,n= finite), are replaced by a sum over the discretized momenta (38) as: Z+∞ −∞ dp →2π~ L +∞ X p=−∞ .(39) In the following, we calculate the finite-size corrections in both BG and TG regimes at zero temperature, as well as the static inelastic structure factor for a finite number of particles. A. Bogoliubov regime at T= 0 Let us consider the T= 0 ground-state energy per particle given by E0 N=1 2g1Dn+1 2N +∞ X p=−∞ (p)−g1Dn−p2 2m(40) corresponding to the Bogoliubov regime of small γ, where (p) is provided by the Bogoliubov spectrum (15). Equation (40) differs from Eq. (14) because it contains the p= 0 term in the sum. This term has been included in order to avoid self-interaction effects in the leading mean-field term of Eq. (14) which should be replaced by g1D(N−1)/(2L). By introducing the discretized values of p(38), the energy can be rewritten in the form E0 N=1 2g1Dn[1 + √γG(y)] (41) where we have introduced the dimensionless variable y=γN2,(42) depending on the interaction parameter γand the function G(y) = 2 y√y +∞ X ni=0 h2πnipy+ (πni)2−2(πni)2−yi+1 √y, (43) where the adding of the quantity 1/√yensures that the term ni= 0 in the sum is counted just once. By using the Euler-Maclaurin expansion (see Appendix A), one can calculate the expression for the series (43) for large values of y: G(y1) ≈ − 4 3π−π 3y.(44) In Fig. 9we report the comparison of the series (43) with its expansion (44). We notice that the two curves agree in an excellent way for y > 10. The thermodynamic limit −4/(3π) is also reported. 100101102103104 y=γN2 −1.2 −1 −0.8 −0.6 −0.4 −0.2 G(y) G(y) G(y1) −4/(3π) FIG. 9. (Color online) Comparison of the numerical series G(y) (43) (solid line) and its analytical expansion (44) (dashed line) holding for y1. The dot-dashed line represents the thermodynamic value. For large number of particles, the ground-state energy per particle (41) then takes the form: E0 N(γN21, γ 1) ≈1 2g1Dn1−4 3π√γ−π 3N2√γ (45) and, in the thermodynamic limit, reproduces Eq. (16). The condition y=γN21 is equivalent to requiring that the healing length (37) be smaller than the size Lof the system. 9 The ground-state energy contains three contributions: the leading term corresponds to the usual mean field energy, the second contribution arises from the quantum fluctuations and is a one-dimensional analog of the LeeHuang-Yang correction in 3D, while the last term accounts for finite-size effects and depends explicitly on the interaction parameter γ. Finite size corrections can be sizeable, as clearly shown by Fig. 10 where we report the energy per particle as a function of yfor the thermodynamic limit (16) (dotdashed line), the Bethe-ansatz (BA) calculation (circle), the Bogoliubov expression (41) (solid line) and the expansion (45) (dashed line). The figure reveals a general good agreement between the BA and the Bogoliubov predictions (41), except for γ= 1, where Eq. (41), being based on the Bogoliubov approach, is no longer adequate. 100101102103104 y=N2γ 0.5 0.6 0.7 0.8 0.9 1 (E0/N)/(g1Dn/2) γ= 0.01 γ= 0.1 γ= 1 γ= 0.01 γ= 0.1 γ= 1 y1 1−4√γ/(3π) BA FIG. 10. (Color online) Comparison of the ground state energy per particle, in BG units, as a function of y=γN2in the thermodynamic limit of Bogoliubov theory (16) (dot-dashed line), the Bethe-ansatz (BA) calculation (circle), the Bogoliubov expression (41) (solid line) and the y1 expansion (45) (dashed line), for several values of the interaction parameter γ. The chemical potential can be obtained by deriving Eq. (40) with respect to N, at fixed L. One finds µ=∂E0 ∂N L =g1Dn"1 + 1 2N +∞ X p=−∞ p2 2m 1 (p)−1# (46) which can be rewritten as µ=g1Dn[1 + √γF(y)], where yis provided by Eq. (42) and we have introduced the series F(y) = 1 √y +∞ X ni=0 πni py+ (niπ)2−1!+1 2√y(47) depending on the quantized momenta (38) and such that the zero-momentum term is accounted for once. The Euler-Maclaurin expression, applied to the sum (47), yields F(y1) ≈ −1 π−π 12y(48) holding in the y1 limit. In Fig. 11 we report the comparison of the series (47) with its expansion (48) holding for y1. The two curves agree very well for y > 10. 100101102103 y=γN2 −0.6 −0.5 −0.4 −0.3 F(y) F(y) F(y1) −1/π FIG. 11. (Color online) Comparison of the numerical series F(y) (47) (solid line) and its analytical expansion (48) (dashed line) holding for y1. The dot-dashed line represents the thermodynamic value. Using Eq. (48), one can finally write the following expansion for the chemical potential µ(γN21, γ 1) ≈g1Dn1−√γ π−π 12N2√γ. (49) In Fig. 12 we report the results for the chemical potential as a function of y(42) for the thermodynamic limit (17) (dot-dashed line), the Bethe-ansatz calculation (symbols) and the Bogoliubov expression (46) (solid line). The y1 expansion (49) practically coincides with the full series (46). The square symbol corresponds to the “forward” definition µ+=E0(N+ 1) −E0(N) of the chemical potential, the star symbol to the “backward” expression µ−=E0(N)−E0(N−1), while the circles to the “symmetric” value ¯µ= (µ++µ−)/2. While the three definitions of the chemical potential coincide in the thermodynamic limit N→ ∞, they are different in a finite system [44]. In particular, the symmetric definition ¯µwell agrees with the calculation (46), based on the differential definition µ= (∂E0/∂N)L, except for the γ= 1 case. From Eq. (46), one can also calculate the sound velocity (4), corresponding to the density derivative of the chemical potential for a fixed value of L. The resulting expression, vs(γ) = vBG s(γ)v u u t1−g1Dn 2N +∞ X p=−∞ p2 2m21 3(p)(50) with vBG s(γ) the sound velocity defined in the Bogoliubov regime, used in Fig. 1. The above expression can be rewritten as vs(γ) = vBG s(γ)p1−√γH(y) where we have defined the series H(y) = √y 2 +∞ X ni=0 πni [y+ (πni)2]3/2,(51)