Thermodynamics of a one-dimensional Bose gas with repulsive contact interactions

Thermodynamics of a one-dimensional Bose gas with repulsive contact interactions

Giulia De Rosi ICFO – Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain    Pietro Massignan ICFO – Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain Departament de Física, Universitat Politècnica de Catalunya, Campus Nord B4-B5, 08034 Barcelona, Spain    Maciej Lewenstein ICFO – Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain ICREA – Institució Catalana de Recerca i Estudis Avançats, Pg. Lluís Companys 23, 08010 Barcelona, Spain    Grigori E. Astrakharchik Departament de Física, Universitat Politècnica de Catalunya, Campus Nord B4-B5, 08034 Barcelona, Spain
July 1, 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, the pressure and the contact, as a function of temperature and gas parameter with 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 non-monotonic as a function of gas parameter. Such behavior is explained in terms of non-linear 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.

Gapless systems in one spatial dimension often feature a linear phononic spectrum at low momenta, and this strongly constrains the low-temperature thermodynamics. A unified description of the various quantum degeneracy regimes is then obtained within Luttinger Liquid (LL) theory, which relates the low-temperature properties of the system to the Luttinger parameter, i.e., the ratio of the Fermi velocity and the zero-temperature sound velocity, itself a function of the interaction strength Haldane (1981); Voit (1995); Giamarchi (2003); Cazalilla et al. (2011). In the weakly-repulsive or Gross-Pitaevskii (GP) regime, a gas of bosons with short-range interactions admits a mean-field description Pitaevskii and Stringari (2016). In the opposite limit of very strong repulsion, the gas approaches the Tonks-Girardeau (TG) limit, where bosons become impenetrable and the system wave function can be mapped onto that of an ideal Fermi gas (IFG), resulting in indistinguishable thermodynamics Girardeau (1960). Seminal experiments have explored this continuous interaction crossover in the past few years Paredes et al. (2004); Kinoshita et al. (2004); Tolra et al. (2004); Kinoshita et al. (2005); Haller et al. (2011); Jacqmin et al. (2011); Guarrera et al. (2012).

The landscape of physical regimes in a one-dimensional (1D) Bose gas is even richer at higher temperature Petrov et al. (2000); Vogler et al. (2013); Salces-Carcoba et al. (2018). The correlation functions behave differently in the various regimes Kheruntsyan et al. (2003); Astrakharchik and Giorgini (2003); Deuar et al. (2009); Panfil and Caux (2014), but those are hard to access experimentally. Thermodynamic quantities can be measured more easily, but these generally exhibit a monotonic behavior to lowest order in temperature. For example, the phononic excitations are responsible for the linear increase with temperature of the specific heat Pitaevskii and Stringari (2016), and for the quadratic growth of the chemical potential Lang et al. (2015); De Rosi et al. (2017), for every interaction strength. As the temperature is increased, however, higher momenta get explored and the deviation of the spectrum from the simple linear behavior becomes important Meinert et al. (2015); Fabbri et al. (2015). As we will demonstrate, the resulting thermal corrections are no longer monotonic and permit to classify the regimes of interaction.

In this Letter, we provide a detailed study of the “beyond Luttinger Liquid” thermodynamics Imambekov et al. (2012); Fabbri et al. (2015) in a 1D Bose gas with short-range (contact) repulsion. First, we solve numerically the thermal Bethe-Ansatz (TBA) equations within the Yang-Yang theory, which provide an exact answer to the problem at all temperatures and interaction strengths Yang and Yang (1969); Yang (1970), and we compute key thermodynamic quantities, such as the chemical potential , the pressure , and the Tan’s contact . Then, we gain further insight into the problem by investigating analytically different tractable regions, including low- and high-temperature, and weak- and strong-interactions. We demonstrate that the Bogoliubov (BG) theory correctly describes thermodynamic properties at low temperatures and weak interactions. For strong repulsion, we show that the leading interaction effects at both low and high temperatures stem from a “negative excluded volume” correction derived from the hard-core (HC) model. Moreover, we demonstrate that the contact is proportional to the chemical potential in the GP limit at low temperatures and to the pressure in the TG regime for any temperature. Finally, we show that the leading beyond-LL correction vs. temperature in the investigated quantities (i.e. , , and ) is negative in the GP regime and is positive in the TG limit. Remarkably, the same trend is also visible in the first correction to the leading classical gas contribution at high temperatures.


The Hamiltonian of a 1D gas of bosons with contact repulsive interactions is given by


where is the atom mass, is the coupling constant, and is the 1D -wave scattering length. The interaction strength is determined by the dimensionless quantity which depends on the gas parameter , with the linear density and the length of the system. There is a crossover between the weak () and strong () interaction limits. A peculiar feature of one dimension is that the high-density regime is described by the Bogoliubov theory contrarily to the usual three-dimensional case. Instead, the low-density limit corresponds to a unitary Bose gas where the system (1) possesses the same thermodynamic properties of an IFG.

At zero temperature the system reduces to the Lieb-Liniger model, whose ground-state energy , chemical potential and sound velocity can be found from Bethe-Ansatz as a function of the interaction strength  Lieb and Liniger (1963); Lieb (1963); Pitaevskii and Stringari (2016). The sound speed smoothly changes from the mean-field value to the Fermi velocity in the TG regime.

Within the canonical ensemble, the complete thermodynamics of the system is obtained starting from the Helmholtz free energy , with the energy and the entropy. This allows for the calculation of the chemical potential


the pressure


and the Tan’s contact parameter Braaten et al. (2011); Yao et al. (2018)


Simple considerations on scale invariance Fetter and Walecka (1971); Barth and Zwerger (2011) lead to a series of exact thermodynamic relations holding for any value of temperature and interaction strength 111 See Supplemental Material for details about the derivation of thermodynamic relations, the low- and high-temperature expansions for weakly interacting bosons and for the ideal Fermi gas, as well as the weakly and strongly repulsive limits of the Yang-Yang virial expansion of the pressure. :


The chemical potential, the pressure, and the contact across the whole spectrum of temperature and interaction strength, as given by the solution of the thermal Bethe-Ansatz equations, are shown as symbols in Figs. 1, 2, and 3. In the rest of the paper, we provide a deeper understanding of dominant effects in the various regimes which may be treated analytically.

Chemical Potential.—

Figure 1: Thermal shift of the chemical potential vs. temperature. The symbols denote numerical TBA results for several interaction strengths , and the lines correspond to various theories described in the text. The temperature is normalized to the typical energy associated with phonons, , and we measure shifts in terms of the LL result , so that deviations from unity directly quantify beyond-LL effects. In these units the BG result, Eq. (7), is represented by a universal curve, independent on .

Let us start by considering weak interactions (). At low temperatures , where is the chemical potential, the gas behaves like a quasicondensate, exhibiting features of superfluids Astrakharchik and Pitaevskii (2004) with phononic excitations De Rosi et al. (2017). In this regime, the thermodynamics can be understood via Bogoliubov theory in terms of a gas of non-interacting bosonic quasi-particles Pitaevskii and Stringari (2016). The thermal free energy is:


where is the BG spectrum Lieb and Liniger (1963); Lieb (1963), which depends on through . From Eq. (6), the temperature shift of the chemical potential is found to be


Within the LL theory, one retains only the phononic part of the BG dispersion, , and obtains the universal result , with and De Rosi et al. (2017). Expanding the BG spectrum to higher momenta, , allows to compute the first correction beyond LL, which is Note1 ():


At , where provides the degeneracy temperature, the gas is in the thermal degenerate state.

At even higher temperatures , the gas behaves classically with negative chemical potential. In the GP regime, the dominant contribution to thermodynamics is determined by single-particle excitations. A reliable description in this case is provided by Hartree-Fock theory, which yields the chemical potential Pitaevskii and Stringari (2016), with the chemical potential of the ideal Bose gas (IBG). Hence, we perform the virial expansion of the equation of state in terms of a small effective fugacity . At leading order in temperature one obtains , with Note1 ():


where is the thermal wavelength. Eq. (9) is an expansion for small gas parameter and it holds if is much larger than the interaction range. Equation (9) depends on the coupling constant only through the term. For smaller (i.e. larger densities), higher values of are needed for the agreement of Eq. (9) with TBA, as may be seen in Fig. 1.

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


The scattering length is positive for hard-core potentials, and the available phase space is diminished by . For the repulsive -potential in Eq. (1), instead, the scattering length is negative and the phase space is increased by effectively inducing “negative excluded volume”. Although the HC equation of state applies for , its continuation to at differs from the Lieb-Liniger equation of state only by terms , with such deviation attributed to the different phase shift dependence on the scattering momentum for -function and hard-core potentials Astrakharchik et al. (2010). We find that the “negative excluded volume” correction turns out to be dominant for and permits to describe the thermodynamics of -interacting gas even at high , as shown in Fig. 1. Similarly, we expect that the “positive excluded volume” correction will be important for the thermodynamics of short-range gases with in a strongly-correlated metastable state (super Tonks-Girardeau gas Astrakharchik et al. (2005); Haller et al. (2009)).

Following the Sommerfeld expansion of the IFG free energy Ashcroft and Mermin (1976); Note1 (), and taking into account the excluded volume correction for a HC gas through Eq. (10), we arrive to:


where , with an effective Fermi energy depending on the rescaled density which takes into account the “negative excluded volume” and is applicable for . An alternative derivation of Eq. (11) up to -order was already presented in Ref. Wadati et al. (2005). We further note that our result of the IFG Sommerfeld expansion Note1 () corrects a minor misprint in the -term of Ref. Lang et al. (2015). From the contribution of Eq. (11), we calculate the HC sound velocity . By comparing Eqs. (8) and (11), one notices that the -phononic contribution is always positive, while quantum statistical effects are responsible for an opposite sign in BG and HC theories in the beyond-LL -term.

At high , we apply the virial expansion to the equation of state of an IFG, and we get the corresponding expansion of the free energy Note1 (). Using Eqs. (10) and (2), we derive the virial expansion of the chemical potential of a hard-core gas:


Equations (9) and (12) share the classical gas logarithmic term, while the second perturbative contribution exhibits an opposite sign emerging from quantum statistics, whose effects become important at lower . The TG regime () is recovered from the HC model when , and it possesses the same thermodynamic properties of an IFG.

Figure 2: Thermal shift of the pressure. Symbols denote numerical TBA results, and lines correspond to various theories, as in Fig. 1. Deviations from unity of indicate beyond-Luttinger-liquid corrections.


Let us now consider the thermal shift of the pressure, , with the pressure at . Within BG theory, we start from Eqs. (6)-(7) and approximate to obtain Note1 ():


The leading-order (Luttinger liquid) result is , where and depends on through . The LL result in Eq. (13) corrects a minor misprint in Ref. Yu-Zhu et al. (2015). The virial expansion can be used to obtain the pressure in the GP regime at high temperatures, resulting in Note1 ():


Within the HC approach, the Sommerfeld expansion of the free energy, Eq. (10), and Eqs. (11) and (3), provide the following low- behavior of the pressure


which is consistent with Ref. Wadati et al. (2005) and includes the -term. The high- pressure is instead derived from the virial expansion of Eq. (10), and Eqs. (12) and (3):


Equation (16) provides the first quantum correction to the Tonks equation , which describes a classical HC gas Tonks (1936); Mattis (1993); Wadati et al. (2005), and a higher-order interaction correction to the virial result of the Yang-Yang theory Yang (1970); Note1 ().

Figure 3: Thermal shift of the Tan’s contact. Symbols denote numerical TBA results, and lines correspond to various theories, as in Fig. 1. Deviations from unity of indicate beyond-Luttinger-Liquid corrections.

Tan’s Contact.—

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 the interaction energy, the pair correlation function, and the relation between pressure and energy density Olshanii and Dunjko (2003); Tan (2008a, b, c); Barth and Zwerger (2011), as shown for example in Eq. (5).

Let us compute here the thermal contribution to the contact, , where is the contact at . Within the BG theory, from Eq. (6) one obtains:


with entering in the LL result . It can be shown that Eq. (17) is consistent with Eq. (5). At high and in the GP regime, does not depend on , since within the Hartree-Fock approximation the free-energy depends on only through the term Note1 ().

With the HC approach, we find from Eqs. (10) and (4):


where the temperature dependence is encoded in , which is given in Eqs. (15) and (16) for low- and high-T, respectively. The relation between contact and pressure emerges from the HC excluded volume , which transforms the -dependence, Eq. (4), in a -one: , holding at any . Moreover, Eq. (18) can be derived directly from Eq. (5) by using .

Experimental Observation.—

The pressure, the chemical potential, the free energy, the energy and the entropy as a function of have been measured by using in-situ absorption imaging in three-dimensional ultracold gases Ho and Zhou (2009); Nascimbène et al. (2010); Ku et al. (2012). A similar experimental technique has been applied to 1D Bose gas to extract the chemical potential as a function of temperature and interaction strength Salces-Carcoba et al. (2018), resulting in an excellent agreement with TBA. Finally, Tan’s contact parameter can be extracted from radio-frequency spectroscopy Wild et al. (2012); Sagi et al. (2012); Yan et al. (2019), Bragg spectroscopy Hoinka et al. (2013) and from the large-momentum tail of the momentum distribution Stewart et al. (2010); Chang et al. (2016).


We provided a detailed study of the chemical potential, the pressure and the contact as a function of temperature and interaction strength for a 1D Bose gas with repulsive contact interactions. Exact results were obtained within thermal Bethe-Ansatz theory and the main characteristic features were explained analytically. Beyond-Luttinger-Liquid effects were explicitly highlighted and explained in terms of non-linear Bogoliubov dispersion relation for weak interactions and “negative excluded volume” for strong repulsion. The beyond-LL effects are responsible for an opposite sign in the thermal next-to-leading term of the low- thermodynamic behavior, being negative in the GP limit and positive in the TG regime. The same trend is also visible in the first correction to the leading classical gas contribution at high temperatures. Finally, we found that the Tan’s contact parameter is proportional to the chemical potential and to the pressure for weak and strong interactions, respectively.

In outlook, we hope that our work can stimulate further theoretical and experimental investigations aiming at the characterization of quantum degeneracy regimes, the beyond-Luttinger-Liquid physics and the microscopic nature of 1D Bose gases. Our predictions are relevant for the investigation of the properties of impurities immersed in helium Bardeen et al. (1967), in a 1D Bose gas Reichert et al. (2019) and in other 1D quantum liquids Recati et al. (2005); Schecter and Kamenev (2014) as a function of and the interaction strength of the bath. Also, the knowledge of thermodynamics is crucial for the description of harmonically trapped gases Petrov et al. (2000); Vignolo and Minguzzi (2013); Xu and Rigol (2015); Yao et al. (2018), especially for the investigation of breathing modes Moritz et al. (2003); Haller et al. (2009); Hu et al. (2014); Fang et al. (2014); Chen et al. (2015); Gudyma et al. (2015); De Rosi and Stringari (2016) whose frequency values are affected by the thermodynamic properties Astrakharchik (2005); De Rosi and Stringari (2015). Other interesting extensions of our work include multicomponent systems Yang et al. (2015); Decamp et al. (2016a, b) and configurations with a well-defined number of atoms Labuhn et al. (2016). The “excluded volume” effects should be as well visible in regime in (i) metastable states of gas with short-range interactions, i.e. for the super Tonks-Girardeau gas Astrakharchik et al. (2005); Haller et al. (2009) (ii) gases with finite-range interactions such as dipolar atoms Arkhipov et al. (2005); Citro et al. (2007); Girardeau and Astrakharchik (2012), Rydberg atoms Osychenko et al. (2011), bosonic He (liquid) in a certain density range Bertaina et al. (2016) and fermionic He (gas) at low densities Astrakharchik and Boronat (2014). Our results can be extended to 1D quantum liquid droplets in bosonic mixtures Petrov (2015) in order to explore thermal effects. In particular, 1D enhances quantum fluctuations Petrov and Astrakharchik (2016); Zin et al. (2018), which are responsible for droplet stability, and it is achieved in current experiments Cheiney et al. (2018).

G. D. R. is supported by the EC through the PROBIST program (GA 754510) of the H2020 Marie Skłodowska-Curie COFUND Action and by the “A. della Riccia” Foundation. G. D. R. and M. L. acknowledge the Spanish Ministry MINECO (National Plan 15 Grant: FISICATEAMO No. FIS2016-79508-P, SEVERO OCHOA No. SEV-2015-0522, FPI), European Social Fund, Fundació Cellex, Generalitat de Catalunya (AGAUR Grant No. 2017 SGR 1341 and CERCA/Program), ERC AdG OSYRIS and NOQIA, EU FETPRO QUIC, the National Science Centre and Poland-Symfonia Grant No. 2016/20/W/ST4/00314. P. M. is supported by the “Ramón y Cajal” program. G. E. A.  and P. M. acknowledge funding from the Spanish MINECO (FIS2017-84114-C2-1-P). The Barcelona Supercomputing Center (The Spanish National Supercomputing Center - Centro Nacional de Supercomputación) is acknowledged for the provided computational facilities (RES-FI-2019-1-0006).


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Supplemental Material for
Thermodynamics of a one-dimensional Bose gas with repulsive contact interactions
In this Supplemental Material, we provide details about the derivation of the thermodynamic relations (Sec. S1), the low-and the high-temperature expansions of a weakly-interacting Bose gas (Sec. S2), the Sommerfeld and the virial expansions of an ideal Fermi gas (Sec. S3), and the Yang-Yang virial expansion of the pressure in the weak and strong repulsion limits (Sec. S4).

Appendix S1 Thermodynamic relations

Let us consider the following general expression of the free energy per particle required by dimensional analysis Fetter and Walecka (1971); Barth and Zwerger (2011):


From Eq. (S1), one can deduce the scaling law


where is an arbitrary, dimensionless parameter. Taking the derivative of Eq. (S2) with respect to at yields


From Eq. (S3) and by using Eqs. (2)-(4) and where , we find Eq. (5).

Appendix S2 Weakly-interacting Bose gas

s2.1 Low-temperature expansion from non-linear Bogoliubov dispersion relation

The low-momentum expansion of the Bogoliubov spectrum may be inverted to find the only real and positive solution . Hence, for the free energy, Eq. (6), we get the integral:


where the analytic solution may be found expanding the integrand for , which is justified at low temperatures. We find the low- expansion of the free energy, within the Bogoliubov theory:


where is the ground-state energy calculated within the Lieb-Liniger model at zero temperature Lieb and Liniger (1963).

With a similar procedure for the chemical potential, Eq. (7), we get the integral:


which provides the low- expansion Eq. (8).

s2.2 High-temperature virial expansion within the Hartree-Fock theory

The equation of state for a 1D weakly interacting Bose gas with density and pressure can be derived from


where is the effective fugacity within the Hartree-Fock theory Pitaevskii and Stringari (2016), the Bose functions are , and the thermal wavelength is .

By inverting the expression for in Eq. (S7) in terms of , and by expanding it for small values of the gas parameter , we obtain


By using the definition of in Eq. (S8) and a further expansion for , we finally find the virial expansion of the chemical potential:


By considering Eq. (S8) in the equation of state of , Eq. (S7), we derive the expansion of the pressure:


From Eqs. (S9)-(S10) and Eq. (3), we calculate the high- behavior of the free energy:


We notice that, at this level of approximation, the interactions appear in Eqs. (S9) and (S11) only through their contribution at zero-temperature.

Appendix S3 Ideal Fermi gas

The equation of state of a 1D IFG with density and pressure can be derived from


where we have defined the fugacity and the Fermi functions .

s3.1 Low-temperature Sommerfeld expansion

Let us briefly review the Sommerfeld expansion Ashcroft and Mermin (1976) which enables to calculate integrals of the form:




is the Fermi-Dirac distribution. Let us for example consider the 1D density of states of the IFG:


where is the Fermi energy. In Eq. (S13), we have introduced the dimensionless number


where is the Riemann zeta function.

At very low , the chemical potential of the IFG approaches the Fermi energy, hence we set with . Then, we consider the Sommerfeld expansion, Eq. (S13), up to the -order, corresponding to the integer , and we require , ensuring the correct normalization. We solve the resulting equation for the real solution and we expand again in series, getting the chemical potential


with .

The Sommerfeld expansion, Eq. (S13), allows one to obtain the low-temperature behavior of the Fermi functions , where is the Euler Gamma function. By using the latter expression in Eq. (S12), one recovers the result, Eq. (S17), and obtains the low-temperature expansion of the pressure:


From Eqs. (S17) and (S18) and Eq. (3), we calculate the low- expansion of the free energy:


s3.2 High-temperature virial expansion

By inverting the equation for the density for , Eq. (S12), and by expanding for , we find


from which, using the definition of and a further expansion for , we find the virial expansion of the chemical potential:


If we consider Eq. (S20) in the equation of , Eq. (S12), we derive the high-temperature behavior of the pressure


and of the free energy:


Appendix S4 Yang-Yang virial expansion of the pressure

Let us consider the virial expansion of the pressure in the Yang-Yang model Yang (1970):


where . Since in Eq. (S24), only -terms are taken into account, we stop expansions at order .

In the weakly interacting regime (), we get


which for reproduces only the first correction of the virial expansion of an ideal Bose gas. We notice here that the expression we derived in Eq. (S10) is more accurate than the one in Eq. (S25), as the former contains an extra -term independent on , which only emerges from the next-to-leading order of the Yang-Yang virial expansion, Eq. (S24).

In the strong repulsive regime (), we obtain


which provides the first terms of the virial expansion of the hard-core model, Eq. (16).

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