Beyond-Luttinger-Liquid Thermodynamics of a One-Dimensional Bose Gas with Repulsive Contact Interactions

PHYSICAL REVIEW RESEARCH 1, 033083 (2019)

Beyond-Luttinger-liquid thermodynamics of a one-dimensional Bose gas with repulsive contact interactions

Giulia De Rosi ,1,2,* Pietro Massignan ,1,2,† Maciej Lewenstein ,1,3 and Grigori E. Astrakharchik 2,‡ 1ICFO – Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Avenida Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain 2Departament de Física, Universitat Politècnica de Catalunya, Campus Nord B4-B5, 08034 Barcelona, Spain 3ICREA – Institució Catalana de Recerca i Estudis Avançats, Passeig Lluís Companys 23, 08010 Barcelona, Spain

(Received 22 May 2019; published 7 November 2019)

We present a thorough study of the thermodynamics of a one-dimensional repulsive Bose gas, focusing in particular on corrections beyond the Luttinger-liquid description. We compute the chemical potential, pressure, and contact as a function of temperature and gas parameter with an exact thermal Bethe ansatz. In addition, we provide interpretations of the main features in the analytically tractable regimes, based on a variety of approaches (Bogoliubov, hard core, Sommerfeld, and virial). The beyond-Luttinger-liquid thermodynamic effects are found to be nonmonotonic as a function of gas parameter. Such behavior is explained in terms of nonlinear dispersion and “negative excluded volume” effects, for weak and strong repulsion, respectively, responsible for the opposite sign corrections in the thermal next-to-leading term of the thermodynamic quantities at low temperatures. Our predictions can be applied to other systems including super Tonks-Girardeau gases, dipolar and Rydberg atoms, helium, quantum liquid droplets in bosonic mixtures, and impurities in a quantum bath.

DOI: 10.1103/PhysRevResearch.1.033083

I. INTRODUCTION Thermodynamic quantities can be measured more easily, but these generally exhibit a monotonic behavior to lowest or- Gapless systems in one spatial dimension often feature a der in temperature. For example, the phononic excitations linear phononic spectrum at low momenta and this strongly are responsible for the linear increase with temperature of constrains the low-temperature thermodynamics. A unified the specific heat [5] and for the quadratic growth of the description of the various quantum degeneracy regimes is then chemical potential [21,22], for every interaction strength. As obtained within Luttinger-liquid (LL) theory, which relates the temperature is increased, however, higher momenta get the low-temperature properties of the system to the Luttinger explored and the deviation of the spectrum from the simple parameter, i.e., the ratio of the Fermi velocity and the zero- linear behavior becomes important [23,24], resulting in a temperature sound velocity, itself a function of the interaction continuous structure bounded by two branches of elementary strength [1–4]. In the weakly repulsive or Gross-Pitaevskii excitations [25]. In the GP regime, the upper particlelike (GP) regime, a gas of bosons with short-range interactions branch corresponds to the Bogoliubov spectrum [26], while admits a mean-field description [5]. In the opposite limit of the lower holelike one is instead associated with the dark very strong repulsion, the gas approaches the Tonks-Girardeau soliton dispersion predicted by Gross-Pitaevskii theory [27]. (TG) limit, where bosons become impenetrable and the sys- In the opposite TG regime, the upper and the lower branches tem wave function can be mapped onto that of an ideal Fermi coincide with the particle and hole excitations of the ideal gas (IFG), resulting in indistinguishable thermodynamics [6]. Fermi gas, respectively [5]. Such complex structure has not Seminal experiments have explored this continuous interac- permitted, so far, an easy physical interpretation of its effects tion crossover in the past few years [7–13]. on the corresponding thermodynamic behavior. This impor- The landscape of physical regimes in a one-dimensional tant gap is filled by the present work. As we will demonstrate, (1D) Bose gas is even richer at higher temperature [14–16]. the resulting thermal corrections are no longer monotonic and The correlation functions behave differently in the various permit one to classify the regimes of interaction. regimes [17–20], but those are hard to access experimentally. In this paper, we provide a detailed study of the beyond- Luttinger-liquid thermodynamics [24,28]ina1DBosegas with short-range (contact) repulsion. First, we solve numer- ically the thermal Bethe ansatz (TBA) equations within the *[email protected] †[email protected] Yang-Yang theory, which provide an exact answer to the prob- ‡[email protected] lem at all temperatures T and interaction strengths [29,30], and we compute key thermodynamic quantities, such as the μ C Published by the American Physical Society under the terms of the chemical potential , pressure P, and Tan’s contact . Then, Creative Commons Attribution 4.0 International license. Further we gain further insight into the problem by investigating distribution of this work must maintain attribution to the author(s) analytically different tractable regions, including low and and the published article’s title, journal citation, and DOI. high temperatures, and weak and strong interactions. We

2643-1564/2019/1(3)/033083(9) 033083-1 Published by the American Physical Society GIULIA DE ROSI et al. PHYSICAL REVIEW RESEARCH 1, 033083 (2019) demonstrate that the Bogoliubov (BG) theory correctly de- given by the solution of the thermal Bethe ansatz equations, scribes thermodynamic properties at low temperatures and are shown as symbols in Figs. 1–3. The results are reported weak interactions. For strong repulsion, we show that the as ratios of the observables to their values given by the LL leading interaction effects at both low and high temperatures theory. With this choice, at low T , they all converge to unity stem from a “negative excluded volume” correction derived for any value of the interaction strength γ , while, at higher T , from the hard-core (HC) model. Moreover, we demonstrate any deviation from Luttinger-liquid line quantifies beyond-LL that the contact is proportional to the chemical potential in the behavior which, instead, is strongly affected by γ . In these GP limit at low temperatures and to the pressure in the TG figures, we rescale the temperature by the chemical potential 2 2 2 regime for any T . Finally, we show that the leading beyond- at zero temperature mv , defining T¯ ≡ kBT/(mv ). Since mv LL correction vs temperature in the investigated quantities is also the typical energy associated with phonons, depending (i.e., μ, P, and C) is negative in the GP regime and is positive on γ through the sound velocity v, such temperature unit is in the TG limit. The same trend is also visible in the first the proper one for the LL description holding in the whole correction to the leading classical gas contribution at high T . interaction crossover. In the rest of the paper, we provide the understanding of dominant effects in the regimes which may II. MODEL be treated analytically. The Hamiltonian of a 1D gas of N bosons with contact III. CHEMICAL POTENTIAL repulsive interactions is given by γ 2 N ∂2 N Let us start by considering weak interactions ( 1). h¯ 2 H =− + g δ(x − x ), (1) At low temperatures kBT mv , the gas behaves like a 2m ∂x2 i j i=1 i i> j quasicondensate, exhibiting features of superfluids [36] with phononic excitations [22]. In this regime, the thermodynamics 2 where m is the atom mass, g =−2¯h /(ma) is the coupling can be understood via Bogoliubov theory in terms of a gas constant, and a < 0 is the 1D s-wave scattering length. The of noninteracting bosonic quasiparticles [5]. The thermal free interaction strength is determined by the dimensionless quan- energy is γ =− / tity 2 (na) which depends on the gas parameter na, +∞ with n = N/L the linear density and L the length of the dp −β(p) ABG = A − E0 = kBTL ln[1 − e ], (6) system. There is a crossover between the weak (γ 1) −∞ 2πh¯ and strong (γ 1) interaction limits. A peculiar feature of 2 one dimension is that the high-density n|a|1regimeis where (p) = p2v2 + [p2/(2m)] is the T = 0 BG spec- described by the Bogoliubov theory contrarily to the usual trum [25,31], which depends on γ through v.FromEq.(6), three-dimensional case. Instead, the low-density n|a|1 the temperature shift of the chemical potential is found to be limit corresponds to a unitary Bose gas where the system (1) ∂v +∞ dp ∂(p) 1 possesses the same thermodynamic properties of an IFG. μ = μ − μ = . BG 0 ∂ π ∂ β(p) − At zero temperature, the system reduces to the Lieb- n L −∞ 2 h¯ v e 1 (7) Liniger model, whose ground-state energy E0, chemical potential μ = (∂E /∂N) , , and sound velocity v = Within the LL theory, one retains only the phononic part √ 0 0 a L ≈ | | n/m(∂μ /∂n) can be found from Bethe ansatz as a func- of the BG dispersion, (p) v p , and obtains the universal 0 a μ = μ ¯ 2 μ = π 2 2 ∂ /∂ / tion of the interaction strength γ [5,25,31]. The speed of result LL ¯ T , with ¯ m v ( v n)L (6¯h)[22]. ≈ sound smoothly changes from the mean-field value v = Expanding the BG spectrum to higher momenta, (p) √ GP | | + 2/ 2 2 gn/m to the Fermi velocity v = h¯πn/m in the TG regime. v p [1 p (8m v )], allows one to compute the first correc- F ¯ 4 Within the canonical ensemble, the complete thermody- tion beyond LL, which is O(T ) (see Appendix B1): namics of the system is obtained starting from the Helmholtz 2 2 6 μBG = μLL[1 − π T¯ /4] + O(T¯ ). (8) free energy A = E − TS, with E the energy and S the entropy. 2 2 2 This allows for the calculation of the chemical potential At mv kBT mvF , where mvF provides the degeneracy temperature, the gas is in the thermal degenerate state. μ = (∂A/∂N) , , , (2) T a L 2 At even higher temperatures kBT mvF , the gas behaves pressure classically with negative chemical potential. In the GP regime, the dominant contribution to thermodynamics is determined P =−(∂A/∂L)T,a,N = nμ − A/L, (3) by single-particle excitations. A reliable description in this and Tan’s contact parameter [32,33] case is provided by Hartree-Fock theory, which yields the C = / 2 ∂ /∂ . μ = μ + μ (4m h¯ )( A a)T,L,N (4) chemical potential GP IBG 2gn [5], with IBG the chemical potential of the ideal Bose gas (IBG). Hence, we Simple scaling considerations [34,35] lead to a series of exact perform the virial expansion of the equation of state in terms β μ − thermodynamic relations holding for any value of temperature of a small effective fugacity,z ˜ = e ( GP 2gn) 1. At leading and interaction strength (see Appendix A), order in temperature, one obtains μ ≈ μGP, with C h¯2a A TS E P √ − = + − μ = − . 0 3 2 2 (5) μGP = kBT [ln(nλ) − nλ/ 2] + O(T ), (9) N 4m N N N n 2 The chemical potential, pressure, and contact across the where λ = 2πh¯ /(mkBT ) is the thermal wavelength. whole spectrum of temperature and interaction strength, as Equation (9) is an expansion for small gas parameter nλ 1

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For strong interactions (γ 1), the thermodynamics at any temperature may be addressed by making an analogy with the hard-core model [37]. Its free energy is obtained from that of an ideal Fermi gas, subtracting from the system size an “excluded volume” Na, where a is the diameter of the HC,

AHC(L) = AIFG(L → Lˆ ≡ L − Na). (10) The scattering length a is positive for hard-core potentials, and the available phase space is diminished by Na.For the repulsive δ potential in Eq. (1), instead, the scattering length is negative and the phase space is increased by N|a| effectively inducing “negative excluded volume.” Although the HC equation of state applies for a > 0, its continuation to a < 0atT = 0 differs from the Lieb-Liniger equation of state 4 FIG. 1. Thermal shift of the chemical potential, μ = μ − μ0, only by terms O(na) , with such deviation attributed to the vs temperature. The symbols denote numerical TBA results for different phase shift dependence on the scattering momentum several interaction strengths γ , and the lines correspond to various for δ-function and hard-core potentials [38]. We ﬁnd that the theories described in the text. The temperature is normalized to the negative excluded volume correction turns out to be dominant 2 typical energy associated with phonons, T¯ = kBT/(mv ), and we for γ 1 and permits one to describe the thermodynamics of μ measure shifts in terms of the LL result LL, so that deviations δ-interacting gas even at high T , as shown in Fig. 1. Similarly, from unity directly quantify beyond-LL effects. In these units, the we expect that the “positive excluded volume” correction will BG result, given by Eq. (7), is independent of γ . be important for the thermodynamics of short-range gases with a > 0 in a strongly correlated metastable state (super and it holds if λ is much larger than the interaction range. Tonks-Girardeau gas [39,40]). Equation (9) depends on the coupling constant g only through Following the Sommerfeld expansion of the IFG free en- the O(T 0 ) term (see Appendix B2). For smaller γ (i.e., larger ergy [41] (see Appendix C1) and taking into account the densities), higher values of T¯ are needed for the agreement of excluded volume correction for a HC gas through Eq. (10), Eq. (9) with TBA, as may be seen in Fig. 1. we arrive at

2 π 2τˆ 2 π 4τˆ 4 6 7π 6τˆ 6 10 μ = Eˆ 1 + anˆ + (1 + 2anˆ) + 1 + anˆ + 1 + anˆ + O(ˆτ 8) , (11) HC F 3 12 36 5 144 9

whereτ ˆ = kBT/EˆF , with an effective Fermi energy EˆF = whose effects become important at lower T . The TG regime h¯2π 2nˆ2/(2m) depending on the rescaled densityn ˆ = n/(1 − (γ =+∞) is recovered from the HC model when a = 0 and an) which takes into account the negative excluded vol- it possesses the same thermodynamic properties of an IFG. ume and is applicable for n|a|1. An alternative derivation of Eq. (11)uptoO(T 4) order was already presented in Ref. [42]. We further note that our result of the IFG IV. PRESSURE Sommerfeld expansion in Appendix C1 corrects a minor Let us now consider the thermal shift of the pressure, P = 4 = misprint in the O(T )termofRef.[21]. From the T 0 P − P0, with P0 = nμ0 − E0/L the pressure at T = 0. Within contribution of Eq. (11), we calculate the HC sound velocity BG theory, we start from Eqs. (6) and (7) and approximate 2 2 2 2 vHC = vF /(1 − an) . By comparing Eqs. (8) and (11), one (p) ≈ v|p|[1 + p /(8m v )] to obtain (see Appendix B1) 2 notices that the O(T )-phononic contribution is always pos- 2 2 itive, while quantum statistical effects are responsible for an π T¯ 1 + 5χγ = − + ¯ 6 . opposite sign in BG and HC theories in the beyond-LL O(T 4 ) PBG PLL 1 O(T ) (13) 20 1 + χγ term. At high T , we apply the virial expansion to the equation 2 The leading-order (Luttinger-liquid) result is PLL = P¯T¯ , of state of an IFG, and we get the corresponding expansion of 2 3 where P¯ = πm v (1 + χγ )/(6¯h) and χγ = [∂v/∂n] n/v de- the free energy (see Appendix C2). Using Eqs. (10) and (2), L pends on γ through v. The LL result in Eq. (13) corrects a we derive the virial expansion of the chemical potential of a misprint in Ref. [43]. The virial expansion is used to obtain hard-core gas, the pressure in the GP regime at high T , resulting in anˆ nˆλ √ μ = λ + + + √ + 0 . HC kBT ln(ˆn ) anˆ 1 O(T ) (12) P = nk T [1 − nλ/(2 2)] + O(T 0 ), (14) 2 2 GP B Equations (9) and (12) share the classical gas logarith- whose detailed derivation is reported in Appendix B2. mic term, while the second perturbative contribution O(nλ) Within the HC approach, the Sommerfeld expansion of the exhibits an opposite sign emerging from quantum statistics, free energy, given by Eq. (10), and Eqs. (11) and (3) provide

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FIG. 2. Thermal shift of the pressure. Symbols denote numerical FIG. 3. Thermal shift of the Tan’s contact. Symbols denote nu- TBA results, and lines correspond to various theories, as in Fig. 1. merical TBA results and lines correspond to various theories, as Deviations from unity of P/PLL indicate beyond-LL effects. in Fig. 1. Deviations from unity of C/CLL indicate beyond- Luttinger-liquid corrections. the following low-T behavior of the pressure: ˆ = − π 2 π 4 π 6 HC excluded volume L L Na, which transforms the a 2 2 4 35 6 8 P = nÊˆ 1 + τˆ + τˆ + τˆ + O(ˆτ ) , dependence, given by Eq. (4), in a Lˆ one: CHC ∝ PHC = HC 3 F 4 20 432 −(∂AHC/∂Lˆ )T,L,N , holding at any T . Equation (18) can be (15) derived directly from Eq. (5)byusingEHC = LPˆ HC/2. which is consistent with Ref. [42] and includes the O(T 6) term. The high-T pressure is instead derived from the virial VI. EXPERIMENTAL CONSIDERATIONS expansion of Eq. (10), and Eqs. (12) and (3): √ The pressure, chemical potential, free energy, energy, and P = nkˆ T [1 + nˆλ/(2 2)] + O(T 0 ). (16) entropy as a function of T have been measured by using in HC B situ absorption imaging in three-dimensional ultracold gases Equation (16) provides the first quantum correction to the [50–52]. A similar experimental technique has been applied Tonks equation P = nkˆ BT , which describes a classical HC to a 1D Bose gas to extract the chemical potential as a gas [42,44,45], and a higher-order interaction correction to the function of temperature T and interaction strength γ [16], virial result of the Yang-Yang theory [30], whose calculation resulting in an excellent agreement with TBA. Finally, Tan’s is reported in Appendix D. contact parameter can be extracted from radio-frequency spectroscopy [53–55], Bragg spectroscopy [56], and from V. TAN’S CONTACT the large-momentum tail of the momentum distribution n(k) [57,58]. In a system with zero-range interaction, the Tan’s contact defined in Eq. (4) provides a relation between the equation of state and short-distance (large-momentum) properties, such as VII. CONCLUSIONS the interaction energy, the pair correlation function, and the We provided a complete study of the chemical potential, relation between pressure and energy density [35,46–49], as pressure, and contact as a function of temperature and in- shown, for example, in Eq. (5). teraction strength for a 1D Bose gas with repulsive contact Let us compute here the thermal contribution to the contact, interactions. Exact results were obtained within thermal Bethe 2 C = C − C0, where C0 = 4m/h¯ (∂E0/∂a)L,N is the contact ansatz theory and the main features were described analyti- at T = 0. Within the BG theory, from Eq. (6) one obtains cally. Beyond-Luttinger-liquid effects were explained in terms C = (C¯/μ¯ )μ , (17) of a nonlinear Bogoliubov dispersion relation for weak inter- BG BG actions and negative excluded volume for strong repulsion. 3 2 2 3 with C¯ = πm v Nγ (∂v/∂γ )n/(3¯h ) entering in the LL re- The beyond-LL effects are responsible for an opposite sign in 2 sult, CLL = C¯T¯ . It can be shown that Eq. (17) is consistent the thermal next-to-leading term of the low-T thermodynamic with Eq. (5). At high T and in the GP regime, C does not behavior, being negative in the GP limit and positive in the TG depend on T since, within the Hartree-Fock approximation, regime. The same trend is also visible in the first correction to the free energy depends on a only through the T = 0term the leading classical gas contribution at high T . Finally, we (see Appendix B2). found that the Tan’s contact parameter is proportional to the With the HC approach, we find, from Eqs. (10) and (4), chemical potential and to the pressure for weak and strong interactions, respectively. C = 4mNP /h¯2, (18) HC HC Looking forward, our work may stimulate further theoret- where the temperature dependence is encoded in PHC, which ical and experimental investigations aimed at the characteri- is given in Eqs. (15) and (16) for low and high T, respectively. zation of quantum degeneracy regimes, the beyond-Luttinger- The relation between contact and pressure emerges from the liquid physics, and the microscopic nature of 1D Bose gases.

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Our predictions are relevant for the investigation of the prop- From Eq. (A1), one can deduce the scaling law erties of impurities immersed in helium [59], in a 1D Bose gas [60], and in other 1D quantum liquids [61,62] as a function of A( 2T, −1a, n) A(T, a, n) = 2 , (A2) T and the interaction strength of the bath. Also, the knowledge N N of thermodynamics is crucial for the description of harmonically trapped gases [14,33,63,64], especially for the investiga- where is an arbitrary, dimensionless parameter. Taking the tion of breathing modes [40,65–70] whose frequency values derivative of Eq. (A2) with respect to at = 1 yields are affected by the thermodynamic properties [71,72]. Other interesting extensions of our work include multicomponent ∂ ∂ ∂ A(T, a, n) 2T − a + n systems [73–75] and configurations with a well-defined num- ∂T ∂a ∂n N ber of atoms [76]. The nonlinear Bogoliubov dispersion ef- a,L,N T,L,N T,a,L A(T, a, n) fects are expected to be seen not only for contact interactions, = 2 . (A3) but also in other short-range interacting systems provided that N the density is high enough to be in the mean-field regime, but not yet so large that the finite-range effects are visible. On the From Eq. (A3) and by using Eqs. (2)–(4), and A = E − TS =−∂ /∂ other hand, the “excluded volume” correction is expected to where S ( A T )a,L,N , we find Eq. (5). be applicable essentially to any short-range interacting system (integrable or not) at low density. For example, the excluded volume effects should be as well visible in the a > 0regime APPENDIX B: WEAKLY-INTERACTING BOSE GAS in (i) metastable states of gas with short-range interactions, In this Appendix, we report the full calculation regarding i.e., for the super Tonks-Girardeau gas [39,40], (ii) gases the low- and high-temperature expansions of a weakly inter- with finite-range interactions such as dipolar atoms [77–79], acting Bose gas. Rydberg atoms [80], bosonic 4He (liquid) in a certain density range [81], and fermionic 3He (gas) at low densities [82]. Our results can be extended to 1D quantum liquid droplets in 1. Low-temperature expansion from nonlinear bosonic mixtures [83] in order to explore thermal effects. In Bogoliubov dispersion relation particular, 1D enhances quantum fluctuations [84,85], which The low-momentum expansion of the Bogoliubov spec- are responsible for droplet stability, and is achieved in current trum x = (p) = v|p|[1 + p2/(8m2v2 )] > 0 may be inverted experiments [86]. to find the only real and positive solution p. Hence, for the free energy, given by Eq. (6), we get the integral, ACKNOWLEDGMENTS +∞ 1 − x G.D.R.’s project has received funding from the European dx ln 1 − e kBT 3p2 (x) Union’s Horizon 2020 research and innovation program under 0 v 1 + 8(mv)2 the Marie Skłodowska-Curie grant agreements UltraLiquid π 2 (k T ) π 4 (k T )3 No. 797684 and PROBIST No. 754510, and from the “A. ≈− B + B , (B1) della Riccia” Foundation. G.D.R. and M.L. acknowledge 6 v 120 m2v5 the Spanish Ministry MINECO (National Plan 15 Grants: FISICATEAMO No. FIS2016-79508-P, SEVERO OCHOA where the analytic solution may be found expanding the inte- | | | | No. SEV-2015-0522, FPI), European Social Fund, Fundació grand for p m v , which is justified at low temperatures. Cellex, Generalitat de Catalunya (AGAUR Grant No. 2017 We find the low-T expansion of the free energy, within the SGR 1341 and CERCA/Program), ERC AdG OSYRIS and Bogoliubov theory: NOQIA, EU FETPRO QUIC, the National Science Centre, and Poland-Symfonia Grant No. 2016/20/W/ST4/00314. ABG = A − E0 P.M. is financially supported by the “Ramón y Cajal” 2 2 2 program. G.E.A. and P.M. acknowledge funding from the π (kBT ) L π (kBT ) =− 1 − + O(T 4 ) , (B2) Spanish MINECO (Grant No. FIS2017-84114-C2-1-P). The 6 h¯v 20 m2v4 Barcelona Supercomputing Center (The Spanish National Supercomputing Center - Centro Nacional de Supercom- where E0 is the ground-state energy calculated within the putación) is acknowledged for the provided computational Lieb-Liniger model at zero temperature [31]. facilities (Grant No. RES-FI-2019-1-0006). With a similar procedure for the chemical potential, given by Eq. (7), we get the integral, APPENDIX A: THERMODYNAMIC RELATIONS +∞ p2(x) 1 In this Appendix, we provide details about the derivation dx 3p2 (x) x 0 + kBT − of the thermodynamic relations. vx 1 8(mv)2 e 1 Let us consider the following general expression of the free π 2 (k T )2 π 4 (k T )4 energy per particle required by dimensional analysis [34,35]: ≈ B − B , (B3) 6 v3 24 m2v7 A(T, a, n) T ∝ n2 f na, . (A1) N n2 which provides the low-T expansion given by Eq. (8).

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2. High-temperature virial expansion 1. Low-temperature Sommerfeld expansion within the Hartree-Fock theory Let us briefly review the Sommerfeld expansion [41], The equation of state for a 1D weakly interacting Bose gas which enables one to calculate integrals of the form with density n and pressure P can be derived from +∞ nλ = g1/2(˜z) d H( ) f ( ) 0 λ = 2λ + , P gn kBTg3/2(˜z) (B4) μ +∞ 2i−1 2i d = dH() + a (k T ) H()|=μ, (C2) wherez ˜ = eβ(μ−2gn) is the effective fugacity within i B d2i−1 0 i=1 the Hartree-Fock theory [5], the Bose functions are = +∞ i/ ν gν (z) i=1 z i , and the thermal wavelength is where λ = 2πh¯2/(mk T ). B 1 By inverting the expression for n in Eq. (B4)interms f () = −μ (C3) ofz ˜ 1, and by expanding it for small values of the gas e kBT + 1 parameter nλ 1, we obtain √ is the Fermi-Dirac distribution. Let us, for example, consider (nλ)2 3 − 1 the 1D density of states of the IFG, z˜ = nλ − √ + √ (nλ)3 + O[(nλ)4]. (B5) 2 3 = √1 , By using the definition ofz ˜ in Eq. (B5) and a further expan- H( ) (C4) 2 EF sion for nλ 1, we finally find the virial expansion of the 2 2 2 chemical potential: where EF = kBTF = h¯ π n /(2m) is the Fermi energy. In √ λ − Eq. (C2), we have introduced the dimensionless number n 3 3 4 2 μGP = kBT ln (nλ) − √ + √ (nλ) 2 4 3 1 √ a = 2 − ζ (2i), (C5) i 2(i−1) 2 3 − 5 2 − √ (nλ)3 + O[(nλ)4] + 2gn. (B6) 6 2 where ζ (i) is the Riemann zeta function. By considering Eq. (B5) in the equation of state of P,given At very low T , the chemical potential of the IFG ap- μ → + δ by Eq. (B4), we derive the expansion of the pressure: proaches the Fermi energy, and hence we set EF (1 ) √ with 0 δ 1. Then, we consider the Sommerfeld expan- λ γ − 3 n 3 3 4 2 sion, given by Eq. (C2), up to the O(δ ) order, corresponding P = nk T − √ + + √ (nλ) +∞ GP B 1 π = = 2 2 2 6 3 to the integer i 3, and we require 0 d H( ) f ( ) 1, ensuring the correct normalization. We solve the resulting + O[(nλ)3] . (B7) equation for the real solution δ and we expand again in series, getting the chemical potential From Eqs. (B6), (B7), and (3), we calculate the high-T π 2 π 4 7π 6 behavior of the free energy, μ = E 1 + τ 2 + τ 4 + τ 6 + O(τ 8) , (C6) IFG F 12 36 144 nλ = λ − − √ AGP kBTN ln (n ) 1 τ = / 2 2 with kBT EF . √ The Sommerfeld expansion, given by Eq. (C2), allows one − λ 2 3 3 4 (n ) 3 to obtain the low-temperature behavior kBT μ of the Fermi + √ + O[(nλ) ] + gnN. (B8) μ 1 μ ν π 2 k T 2 4 3 3 functions fν (e kBT ) ≈ ( ) [1 + ν(ν − 1)( B ) ], ν(ν) kBT 6 μ We notice that at this level of approximation, the interactions where (ν) is the Euler Gamma function. By using the latter appear in Eqs. (B6) and (B8) only through their contribution expression in Eq. (C1), one recovers the result, given by at zero temperature. Eq. (C6), and obtains the low-temperature expansion of the pressure, APPENDIX C: IDEAL FERMI GAS 2 π 2 π 4 35π 6 P = nE 1 + τ 2 + τ 4 + τ 6 + O(τ 8) . In this Appendix, we provide the details of the Sommerfeld IFG 3 F 4 20 432 and virial expansions of an ideal Fermi gas. (C7) The equation of state of a 1D IFG with density n and pressure P can be derived from From Eqs. (C6), (C7), and (3), we calculate the low-T expansion of the free energy: nλ = f1/2(z) Pλ = k Tf/ (z), (C1) π 2 π 4 π 6 B 3 2 EF N 2 4 7 6 8 AIFG = 1 − τ − τ − τ + O(τ ) . μ/(kBT ) 3 4 60 432 where we have defined the fugacity z = e and the Fermi = +∞ − i−1 i/ ν functions fν (z) i=1 ( 1) z i . (C8)

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2. High-temperature virial expansion APPENDIX D: YANG-YANG VIRIAL EXPANSION By inverting the equation for the density n for z 1, given OF THE PRESSURE by Eq. (C1), and by expanding for nλ 1, we find In this Appendix, we report the calculation of the Yang- √ Yang virial expansion of the pressure in the weak and strong (nλ)2 3 − 1 z(nλ) = nλ + √ + √ (nλ)3 + O[(nλ)4], (C9) repulsion limits. 2 3 Let us consider the virial expansion of the pressure in the Yang-Yang model [30], from which, using the definition of z and a further expansion λ x for n 1, we find the virial expansion of the chemical P 1 2 2 2 1 = 1 + √ + ex dy e−y − √ (nλ), potential, nk T π √ B 2 2 0 2 λ − n 3 3 4 2 (D1) μIFG = kBT ln(nλ) + √ + √ (nλ) √ 2 4 3 where x = γ (nλ)/(2 2π ). Since, in Eq. (D1), only O(nλ) √ 2 3 − 5 terms are taken into account, we stop expansions at order + √ (nλ)3 + O[(nλ)4] . (C10) O(x). 6 2 In the weakly interacting regime (x 1), we get If we consider Eq. (C9) in the equation of P, given by PGP nλ γ Eq. (C1), we derive the high-temperature behavior of the = 1 − √ + (nλ)2 + O[(nλ)3], (D2) nk T 2π pressure, B 2 2 √ which for γ = 0 reproduces only the first correction of the λ − n 3 3 4 2 virial expansion of an ideal Bose gas. We notice here that PIFG = nkBT 1 + √ + √ (nλ) 2 2 6 3 the expression we derived in Eq. (B7) is more accurate than √ the one in Eq. (D2), as the former contains an extra O[(nλ)2] 2 3 − 5 + √ (nλ)3 + O[(nλ)4] , (C11) term independent of γ , which only emerges from the next- 8 2 to-leading order of the Yang-Yang virial expansion, given by and of the free energy, Eq. (D1). √ In the strong repulsive regime (x 1), we obtain nλ 3 3 − 4 (nλ)2 = λ − + √ + √ PHC nλ AIFG kBTN ln (n ) 1 = (1 + na) + √ + O[(nλ)2], (D3) 2 2 4 3 3 nk T √ B 2 2 2 3 − 5 (nλ)3 which provides the first terms of the virial expansion of the + √ + O[(nλ)4] . (C12) 6 2 4 hard-core model, given by Eq. (16).

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