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Energy-pressure relation for low-dimensional gases

A particularly simple relation of proportionality between internal energy and pressure holds for scale invariant thermodynamic systems, including classical and quantum Bose and Fermi ideal gases. One can quantify the deviation from such a relation by introducing the internal energy shift as the difference between the internal energy of the system and the corresponding value for scale invariant gases. We discuss general thermodynamic properties associated to the scale invariance, provide criteria for which the internal energy shift density is a bounded function of temperature. We then study the internal energy shift and deviations from the energy-pressure proportionality in low dimensional models of gases interpolating between the ideal Bose and the ideal Fermi gases, focusing on the Lieb-Liniger model in 1d and on the anyonic gas in 2d. In 1d the internal energy shift is determined from the thermodynamic Bethe ansatz integral equations and an explicit relation for it is given at high temperature. Our results show that the internal energy shift is positive, it vanishes in the two limits of zero and infinite coupling (respectively the ideal Bose and the Tonks Girardeau gas) and it has a maximum at a finite, temperature depending, value of the coupling. Remarkably, at fixed coupling the energy shift density saturates to a finite value for infinite temperature. In 2d we consider systems of Abelian anyons and non Abelian Chern-Simons particles: as it can be seen also directly from a study of the virial coefficients, in the usually considered hard-core limit the internal energy shift vanishes and the energy is just proportional to the pressure, with the proportionality constant being simply the area of the system. Soft-core boundary conditions at coincident points for the two-body wavefunction introduce a length scale, and induce a non-vanishing internal energy shift.


I. INTRODUCTION
At thermodynamic equilibrium, pure homogeneous fluids (found in regions of the phase diagram hosting a single phase) are characterized by an equation of state f (V, P, T ) = 0 relating pressure, volume and temperature. In general, an equation of state can be solved with respect to any of the three quantities V , P , or T , thus providing different ways to characterize the equilibrium properties of the system: for example, the partial derivatives of the form V = V (T, P ) have the physical meaning of thermal expansion coefficient α V ≡ (1/V )(∂V /∂T ) P and isothermal compressibility β T ≡ −(1/V )(∂V /∂P ) T [1][2][3][4].
The equations of state for classical and quantum ideal gases are the starting point for understanding the thermodynamics of interacting gases [1][2][3][4]: in particular, the equation of state for the classical ideal gas is approximately valid for the low-density region of any real gas. In general, the internal energy of an interacting gas is a function of both temperature and pressure as a result of forces between the molecules. If such forces did not exist, no energy would be required to alter the average intermolecular distance, i.e. no energy would be required to implement volume and pressure changes in a gas at constant temperature.
It follows that in the absence of molecular interactions, the internal energy of a gas would depend on its temperature only. These considerations lead to the definition of an ideal gas as the one whose macroscopic behavior is characterized by the two equations: P V = N k B T and E = E(T ), where E is the internal energy.
The determination of the deviation of thermodynamic properties of non-ideal gases from the ideal behavior is in general a long-standing problem: a commonly used approach to quantify such a deviation is to define the shift of thermodynamic quantities as the difference with respect to the corresponding value of the same quantities in the ideal case. Historically, several techniques have been developed in order to encode deviations from the ideal gas law.
Equations of states which are cubic in the volume feature a simple formulation together with the ability to represent for instance both liquid and vapor behavior. The first cubic equation of states was the Van der Waals equation [5] (v denotes the volume per particle), accounting for attractive intermolecular (or Van der Waals) forces and a finite excluded volume through its positive constants a and b respectively.
In the high-temperature regime, the deviations from the ideal equation of state can be expressed in a more general way, called virial expansion, and obtained by expressing the pressure as a power series in the density ρ in the form where B n (T ) is the n-th virial coefficient [1][2][3][4]. Many other similar functional forms have been proposed in various contexts for the equation of state of interacting gases, with the virial equations among the first to receive a firm theoretical foundation. In fact, virial expansions can be derived from first principles of statistical mechanics and such a derivation has also the merit to enlighten the physical significance of the various coefficients: the second virial term above written arises on account of interactions between pairs of molecules and, more generally, the n-th term depends upon the interactions among k-bodies, k ranging from 2 to n. In this paper we focus on the study of the energy-pressure relation in low-dimensional systems: for ideal gases, the internal energy is simply proportional to the product P V of pressure and volume, with the proportionality constant depending on the dimensionality d of the system. As we discuss in Section II this simple relation between energy and pressure holds for any scale-invariant thermodynamic systems, i.e. for systems having a N -body Hamiltonian H N that scales as H N → λ −α H N under a dilatation of λ-linear scaling factor, and boundary conditions at coincident points on the wave-function ψ N (x 1 , · · · , x N ) which are true also for any rescaled wave-functionψ N (x 1 , · · · , x N ) ≡ ψ N (λx 1 , · · · , λx N ) . The first condition means that H N is an homogeneous function of the coordinates: ideal classical and quantum gases are particular cases of this class of systems, since their Hamiltonian is scale-invariant with α = 2.
To quantify deviations from the ideal energy-pressure relation, in the following we introduce the internal energy shift as the difference between the internal energy of the system and the corresponding value of the scale-invariant (including ideal) gases. Low-dimensional quantum systems provide a natural playground for the study of the internal energy shift, since in 1d and 2d systems it is possible to naturally interpolate from the thermodynamic properties of an ideal Bose gas to those of an ideal Fermi gas, and determine how deviations from the ideal gas behavior affect thermodynamic quantities. We will consider the Lieb-Liniger (LL) model in 1d and the anyonic gas in 2d. For these two systems the physical nature of the interpolation between the Bose and Fermi statistics seems to be formally different: • in the LL model (a 1d model of interacting bosons), the interpolation between ideal bosonic and fermionic behavior is driven by the increase of the repulsive interaction among the particles.
• in 2d anionic gases, one can instead explicitly interpolate between the two canonical bosonic and fermionic statistics by tuning the statistical parameter.
However, the anyonic statistics incorporates the effects of interaction in microscopic bosonic or fermionic systems (statistical transmutation) and, from this point of view, it is again the variation of the underlying microscopic interactions that induces the interpolation between Bose and Fermi ideal gas.
So, our first paradigmatic example of interpolating behavior between ideal Bose and Fermi gases will be the LL model of one-dimensional bosons interacting via a pairwise δ-potential: the equilibrium properties of this model can be exactly solved via Bethe ansatz both at zero [6] and finite temperature [7]. In the exact solution of this model, a crucial role is played by the coupling γ, which turns out to be proportional to the strength of the two-body δpotential: the limit of vanishing γ corresponds to an ideal 1d Bose gas; on the other side, the limit of infinite γ corresponds to the Tonks-Girardeau (TG) gas [8][9][10], having (local) expectation values and thermodynamic quantities of a 1d ideal Fermi gas [11][12][13][14]. Two features makes the LL model attractive for the purposes of studying the internal energy: first, its integrability [15,16], crucial for getting non-perturbative exact results all along the crossover from weak to strong coupling regimes; second, its experimental realization by means of ultracold atom set-ups [11][12][13][14]17], where bosons are confined within 1d atom waveguides which freeze almost all transverse degrees of freedom [18][19][20]. The coupling strength of the LL system can be tuned through the Feshbach resonance mechanism [21].
Our second paradigmatic example will be the 2d ideal anyonic gases in which we will study the energy-pressure relation in the interpolation between 2d Bose and Fermi gases induced by the pure statistical Aharonov-Bohm interactions. We will consider Abelian and non-Abelian Chern-Simons particle systems, and both models admit a soft-core generalization that can be understood as the result of an additional contact interaction besides the pure statistical one. As it is well known, quantum two-dimensional systems of indistinguishable particles have the peculiarity of admitting generalized braiding statistics, because of the nontrivial topological structure of braiding transformations defined over the space-time ambient manifold. Ordinary bosonic and fermionic quantum statistics in 2d admit the generalization represented by Abelian anyons, where an elementary braiding operation is encoded in terms of a multiplicative phase factor acting on the multi-anyonic scalar wavefunction [22][23][24][25][26]. A different generalization of the standard quantum statistics is represented by non-Abelian anyons, described by a multi-component many-body wavefunction and corresponding to higher-dimensional representations of the braid group: non-Abelian anyons generalize the parastatistics, exactly in the same manner in which Abelian anyons generalize Bose and Fermi statistics.
Thermodynamic properties of ideal Abelian anyonic gas (assuming hard-core boundary conditions for the wavefunction at coincident points) were studied in the low-density regime [27]: the exact expression therein obtained for the second virial coefficient is periodic and non-analytic as a function of the statistical parameter. Different approaches have been subsequently used in order to approximate the values of a few higher virial coefficients, including the semiclassical approximation [28] and Monte Carlo computations [29] (for more references see [25,30]). The thermodynamics of a system of free non-Abelian anyons appears as a harder task and, so far, only results about the second virial coefficient are available [31][32][33][34]. In Section VI we also study the shift of the internal energy of soft-core anyonic gases: a family of models for "colliding" anyons (featuring generalized soft-core boundary conditions) can be introduced as the set of well-defined self-adjoint extensions of the Schrödinger anyonic Hamiltonian. The mathematical arguments underlying the possibility of such a generalization were discussed in [35], and the second virial coefficient of soft-core Abelian anyons was studied in [36][37][38]. The corresponding self-adjoint extensions for the non-Abelian anyonic theory have been as well discussed [39][40][41][42]. The model of soft-core anyons is here considered as an explicit example of scaling symmetry breaking due to the presence of an intrinsic length scale.
Among all thermodynamic properties of ideal classical and quantum gases, the linear relation between internal energy and pressure is particularly simple, and in this paper we study how it is affected by the various interactions represented by the low-dimensional models above mentioned. For extensive computer simulations of energy-pressure relation in 3D classical systems of interacting particles, see [43][44][45][46][47][48] (and [49] for the definition of "Roskilde systems").
The paper is organized as follows: in Section II we show that a simple relation of pro-portionality between internal energy and pressure holds for scale-invariant thermodynamic systems, including classical and quantum (Bose and Fermi) ideal gases, and we discuss some simple consequences of scale-invariance in generic dimensionality, including some useful properties of isoentropic transformations. In Section III we set criteria under which the internal energy shift per particle of an imperfect gas at fixed density saturates towards a finite value as the temperature becomes very large. These criteria are expressed in terms of the second virial coefficient and by distinguishing the different dimensionalities. Section IV is devoted to define the models we are going to study in next Sections: the LL models and the different anyonic models. The internal energy shift of the LL model is studied in Section V by using thermodynamic Bethe ansatz integral equations: the comparison with the 1d hard-core bosons is also discussed. Section VI deals with the internal energy shift of anyonic gases, and we present results for both the hard-and the soft-core anyonic gases.
Our conclusions are drawn in Section VII, while more technical material is presented in the Appendices.

II. SCALE-INVARIANT SYSTEMS
A proportionality between internal energy E and pressure P holds for any scale-invariant thermodynamic system. Indeed, let us consider a (classical or quantum) system of N particles in a volume V with Hamiltonian H N (V ). It is intended that in this Section and the next, we denote by V the length L in 1d and the area A in 2d. We define a classical system to be scale-invariant when the Hamiltonian transforms as under a dilatation of a λ-linear scaling factor such that the coordinate x of the particles transforms as x → λx (the momentum p transforms correspondingly as p → λ −1 p): therefore the Hamiltonian is an homogeneous function of its spatial coordinates. For quantum systems, we define them to be scale-invariant if they fulfill condition (2) and respect at the same time scale-invariant boundary conditions for the N -body wave-function ψ N at contact points, i.e conditions on the wave-function ψ N (x 1 , · · · , x N ) which are true also for any rescaled wavefunctionψ N (x 1 , · · · , x N )) ≡ ψ N (λx 1 , · · · , λx N ), where λ = 0 is a real constant. A typical example of scale-invariant boundary conditions for the N -body wave-function at contact points is given by the hard-core condition.
In the canonical ensemble the pressure P and the internal energy E are defined as where as usual β = 1/k B T and Z is the partition function: For any d-dimensional scale-invariant system of volume V , the map leaves log Z invariant in the thermodynamic limit (since βH N → βH N ). With the notation whence relations (3) imply Notice that Eq. (7) is valid both for classical and quantum scale-invariant systems, and follows from the invariance of the partition function under map (5): therefore the scaleinvariance of boundary conditions at contact points is required in the quantum case. So, for instance, in Section VI, we will show that, for the 2d ideal anyonic gas, Eq.(7) only holds in the case of hard-core boundary conditions while it is violated in the soft-core case.
From Section IV onwards, we study some low-dimensional quantum systems, since we are primarily interested in interpolating between the two ordinary quantum statistics.
From the considerations above, it follows that any quadratic scale-invariant Hamiltonian fulfills the scaling property H N → λ −2 H N under a dilatation of a λ-linear scaling factor and therefore enjoys the property Few examples of systems for which property (8) is known are the following: i) d-dimensional ideal classical and quantum (Bose and Fermi) gas have quadratic dispersion relations, and they all obey the well known relation E = (d/2)P V , as it can be also deduced from the virial theorem [50]; ii) as reviewed in Section VI, the 1d LL Bose gas has total internal energy E = P L/2 (since d = 1 and α = 2) for any temperature in both its scale-invariant limit regimes: the non-interacting limit (γ → 0) and the fermion-like Tonks-Girardeau limit (γ → ∞), which correspond respectively to the zero and infinite coupling associated with the δ-like contact interactions; iii) for the 3d Fermi gas at the unitary limit the relation E = (3/2)P V holds as well [51] (see the discussion in [52]).
As discussed in Section VI, also 2d hard-core ideal anyonic gases obey Eq. (7) for general values of the statistics parameters.
We derive now some scaling properties for scale-invariant d-dimensional systems undergoing adiabatic reversible thermodynamic processes (as above, the argument is carried out in the quantum case for the sake of generality). Let us consider the scale-invariant thermodynamic system confined in a region subjected to a quasi-static scaling transformation of the volume and the temperature (V, T ) → (λ d V, λ −α T ), under which the ratios E i /k B T are left invariant (same proof of (67)), as long as the N -particle Hamiltonian H N gains a λ −α factor under a λ-factor scaling of its spatial coordinates. The total entropy S of the system remains invariant under such a process: indeed the energy scales proportionally to λ −α (because of the transformation (T, of temperature and energy levels), exactly as required for any isoentropic process fulfilling relation (7). This last statement results from E = d α P V and P = −∂ V E(N, S, V ), which give (α/d)E isoentr /V = −(dE isoentr /dV ) and therefore: i.e. E ∝ λ −α , along the series of equilibrium state of a given isoentropic process. We conclude that adiabatic reversible expansions and compressions (as well as arbitrary isoentropic processes followed by thermal relaxation) of scale-invariant systems are characterized by the following transformations for internal energy and temperature: It is worth to point out three immediate consequences of (11): • as ideal gases, scale-invariant systems undergoing an isoentropic process comply with the invariance of P Vγ, whereγ ≡ 1 + α/d.
• isoentropic transformations of scale-invariant systems let the dilution parameter x ≡ ρλ d T invariant, by taking into account a generalized definition of the thermal wavelength depending on the dispersion relation and the dimensionality [53].
• the internal energy associated with equilibrium states of an isoentropic process is proportional to the temperature: where y remains constant along the isoentropic curve. Notice that the factor y would depend solely on the dilution parameter x = ρ λ d T for a given system if Eq. (12) is considered over the entire phase diagram.
It is worth to point out that for scale-invariant Hamiltonian systems, the dependence of virial coefficients upon the temperature is very simple, i.e.
As it will be discussed in Section VI, the fact that the B k (T ) respect Eq. (13) implies the validity of the relation (7) at all orders of the virial expansion (within its radius of convergence).
We conclude this Section by showing that it is also possible to deduce a property of the internal pressure for scale-invariant systems. The internal pressure π T is defined in general as the volume derivative of the internal energy in isothermal processes [54]: the internal pressure is a measure of attractiveness for molecular interactions and is related to the (thermodynamic) pressure P by the expression Eq. (15) is usually referred to as the thermodynamic equation of state, because it expresses the internal pressure just in terms of fundamental thermodynamic parameters P, V, T . For general scale-invariant systems, plugging (7) and (13) into the definition (14) [or equivalently Eq. (13) into Eq. (15)] leads to the following result in the dilute regime: where P ideal is the pressure of the ideal Boltzmann gas having the same V, T .
Relation (16) (14) can be exactly regarded in the dilute limit as the interaction contribution acting in favor to(/against) the external pressure P and counterbalancing the thermal contribution P Boltzmann .

III. ENERGY-PRESSURE RELATION FOR IMPERFECT GASES AT HIGH TEMPERATURE
Hereafter we denote by ρ the number density, by E res the internal energy shift and by e res (ρ, T ) the internal energy shift density (per particle) of a generic classical or quantum imperfect gas, defined as The internal energy shift represents a measure of the deviation from the relation (7) derived in Section II for scale-invariant systems whose Hamiltonians are homogeneous functions of the coordinates.
In many textbooks, deviations from ideal gas behavior are quantified by introducing the so-called departure functions (or residual thermodynamic quantities) (see, for instance, [55]). Such departure functions are obtained by taking the difference between the considered quantity and the corresponding value for the ideal gas, when two among the P , V and T parameters are kept fixed, typically P and T [55]. The quantity defined in (17)  that the only scale-invariant system whose conventionally defined departure internal energy vanishes is the ideal gas). An example of a system which is scale-invariant but with nonvanishing (conventionally defined) departure internal energy is the hard-core anyonic gas, as discussed in Section VI: on the contrary, the one defined in (17) can be considered as the correct residual quantity measuring deviations from scale-invariance. To avoid possible misunderstandings, we decided to refer to E res [defined in (17)] as the internal energy shift rather than departure internal energy.
In the low-density regime, the thermodynamic quantities can be associated with the virial coefficients {B n (T )} of the equation of state P = P (ρ, T ). Following statistical mechanics textbooks [1][2][3][4], in the d-dimensional case the following virial expansions for the pressure P , the Helmholtz free energy A H , the Gibbs free energy G, the entropy S, the internal energy E and the enthalpy H are obtained: Helmholtz free energy : Gibbs free energy : Entropy : Internal energy : Below we state the necessary conditions (proven in Appendix A) under which the energy shift of a (classical or quantum) gas remains bounded in the limit of high temperatures, i.e.
For simplicity, hereafter, we limit ourselves to the case of quadratic dispersion relation α = 2, for which such conditions are (with c 1 and c 2 real coefficients): • For d = 1: In Section V the explicit expression of B 2 for the LL model as a function of the coupling constant γ [56] is reported: for finite γ, it is in general c 2 = 0, so that e res is bounded.
• For d = 2: In Section VI the 2D anyonic gas is studied, and shown to have vanishing internal energy shift in the high-temperature limit. This is in agreement with (21) because, referring as α to the statistical parameter, B h.c. [34,38], where c 1 , c 1 , c 2 are suitable functions, hence both are subleading w.r.t. β log β in the β → 0 limit.
• For d > 2 : and in this case lim

IV. THE MODELS
In this Section we recall the main properties of the LL and anyonic models studied in the next Sections: in Subsection IV A we introduce the 1d Lieb-Liniger model, in Subsection IV B we outline the main thermodynamic properties of an ideal gas of Abelian anyons (and its soft-core generalization), while in Subsection IV C we briefly introduce the system of Non-Abelian Chern Simons (NACS) particles, i.e. a model of non-Abelian anyons.

A. Lieb-Liniger model
The LL Bose gas is described by an Hamiltonian for N non-relativistic bosons of mass m in one dimension interacting via a pairwise δ-potential [6] having the form where λ is the strength of the δ-like repulsion (we consider here only positive or vanishing values of λ: λ ≥ 0). The effective coupling constant of the LL model is given by the dimensionless quantity where ρ = N/L is the density of the gas. We also use the notation so that γ = c/ρ. The limit γ 1 corresponds to the weak coupling limit: in this regime the Bogoliubov approximation gives a good estimate of the ground-state energy of the system [6]. For large γ one approaches instead the Tonks-Girardeau limit [9].
In the LL model temperatures are usually expressed in units of the quantum degeneracy temperature T D as The thermodynamic Bethe ansatz integral equations relate at temperature T the pseudoenergies ε(k) to density f (k) of the occupied levels [7,15,16]. One has the following set of coupled equations whereμ is the chemical potential. At T = 0 the energy level density gets a compact support, so that Eq. (27c) becomes where the boundary value K has to be determined from the condition If one measures energies in units of k B T D and wave-vectors in units of ρ, by defining the scaled wave-vector K ≡ k/ρ, the scaled pseudo-energies E(K) ≡ ε(k)/k B T D and the scaled potential µ ≡μ/k B T D , Eqs. (27) read One sees that scaled quantities depends only on γ and τ .
Once the TBA integral equations (30) are solved, thermodynamic quantities as the free energy can be computed. In Section V we report both the expressions of internal energy and pressure, and we study the internal energy shift.

B. Abelian Anyons
The dynamics of a systems of N identical Abelian anyons is expressed by [25] where with θ ij the relative angle between the particles i and j. The study of the thermodynamics for a system of identical Abelian anyons has been developed starting with [27], in which the exact quantum expression for the second virial coefficient has been derived: Eq. (32) holds provided that hard-core wavefunction boundary conditions are assumed, i.e. lim x i →x j ψ N (x 1 , · · · , x N ) = 0 for any 1 ≤ i < j ≤ N , being ψ N the N -body wavefunction in the bosonic gauge [25].
In (32) α = 2j + δ, where α represents the statistical parameter of anyons [25], j is an integer and |δ| ≤ 1. We remind that α = 1 and α = 0 correspond respectively to free 2d spin-less fermions and bosons, and that λ T is the thermal wavelength defined as The virial expansion is expressed in powers of the number density ρ; in the dilute regime, the second virial coefficient gives the leading contribution to the deviation of the energypressure relation from the non-interacting case, as a result of rewriting the grand canonical partition function as a cluster expansion [1,3]. About the higher virial coefficients of the ideal anyonic gas, only numerical approximations of the first few ones are available so far [25], and they are limited to the hard-core case.
The relative two-body Hamiltonian for a free system of anyons with statistical parameter α, written in the bosonic description, is of the form [25] H where A = (A 1 , A 2 ) and A i ≡ ij x j r 2 (i = 1, 2 and ij is the completely antisymmetric tensor).
By relaxing the regularity condition on the wavefunctions at contact points, it is possible to obtain the one-parameter family of soft-core boundary conditions (35), according to the method of self-adjoint extensions [57]. The s-wave solutions of the radial Schrödinger equation correspond to a one-parameter family of boundary conditions [36,38]: and correspondingly read as where σ = ±1 and κ is a momentum scale introduced by the boundary condition.
We refer to as the hard-core parameter of the gas. If σ = −1, in addition to the solution (36), there is a bound state with energy E B = −εk B T = −κ 2 /M and wavefunction The second virial coefficient for Abelian anyons in this general case has been computed through different approaches in [37,38,58], and is given by dt e −εt t |α|−1 1 + 2σ cos πα t |α| + t 2|α| , (39) where θ(x) is the Heaviside step function and B h.c.

C. Non-Abelian Anyons
The SU (2) non-Abelian Chern-Simons (NACS) spin-less particles are point-like sources mutually interacting via a topological non-Abelian Aharonov-Bohm effect [62]. These particles carry non-Abelian charges and non-Abelian magnetic fluxes, so that they acquire fractional spins and obey braid statistics as non-Abelian anyons.
Details on NACS statistical mechanics [39,[63][64][65][66] are given in Appendix B for general soft-core boundary conditions [40,41]. For non-Abelian anyons, the independence on the statistics of the virial coefficients in a strong magnetic field has been established in [67] while the theory of non-relativistic matter with non-Abelian Chern-Simons gauge interaction in (2+1) dimensions was studied in [68]. The N -body Hamiltonian for ideal non-Abelian Chern-Simons quantum particles can be written as [39] where M α is the mass of the α-th particles, ∇z α = ∂ ∂zα and In Eq. (40) α = 1, . . . , N labels the particles, (x α , y α ) = (z α +z α , −i(z α −z α ))/2 are their spatial coordinates, andQ a 's are the isovector operators in a representation of isospin l. From a field-theoretical viewpoint, the quantum number l labels the irreducible representations of the group of the rotations induced by the coupling of the NACS particle matter field with the non-Abelian gauge field: as a consequence, the values of l are of course quantized and vary over all the non-negative integer and half-integer numbers; l = 0 corresponds to a system of ideal bosons. As usual, a basis of isospin eigenstates can be labeled by l and the magnetic quantum number m = −l, −l + 1, · · · , l − 1, l.
The thermodynamics depends in general on the value of the isospin quantum number l, the Chern-Simons coupling κ, and the temperature T . In order to enforce the gauge covariance of the theory, the parameter κ in (40) has to fulfill the condition 4πκ = integer [69]. Therefore we adopt the notation: Similarly to the Abelian anyons case, the s−wave general solution of the radial Schrödinger equation (B12), derived from the projection of (40) over a generic two-particle isospin channel (j, j z ), belongs to a one-parameter family accounting for the range of possible boundary conditions, and reads where σ = ±1, and κ j,jz is a momentum scale introduced by the boundary condition.
We refer to the (2l + 1) 2 quantities as the hard-core parameters of the system [34], with the hard-core limit corresponding to σ = +1, ε j,jz → ∞ for all j, j z .
We conclude this Section by observing that, according to the regularization used in [27,70], the second virial coefficient is defined as where B (n.i.) 2 (l, T ) is the second virial coefficient for the system with particle isospin l and without statistical interaction (κ → ∞). In Appendix C, B (l, T ) is the (convergent) variation of the divergent partition function for the two-body relative Hamiltonian, between the interacting case in exam and the non-interacting limit (κ → ∞).

V. INTERNAL ENERGY SHIFT FOR THE LIEB-LINIGER BOSE GAS
Before studying the energy shift of the Lieb-Liniger model, we consider by comparison the 1d hard-core bosons model described by the Hamiltonian: where The thermodynamics of the 1d hard-core Bose gas has been determined and studied by Thermodynamics Bethe Ansatz [71,72]: the relation between pressure and internal energy is a Bernoulli equation [73] from which it follows that In this case the internal energy shift is negative, due to the fact that the pressure increases for the effect of the excluded volume. Furthermore E res vanishes for a → 0, as it should.
With regard to the low-dimensional models considered in Sections IV, V, VI, the reader will notice that in 2d the hard-core condition results in a vanishing internal energy shift, while it does not do likewise in 1d (46)(47)(48)(49); however, non-hard-core boundary conditions either in 1d (23) and 2d (36,43) result in a positive energy shift. Furthermore, unlike the non-hard-core case, the dependence (49) of the internal energy shift on the temperature is given only by T -dependence of the pressure.
Let us now address the internal energy shift of the LL model: using the results of Section IV A, the pressure and the energy are given by At T = 0 the energy per particle is given by where E(γ) is given by while the function g(t) is solution of the linear integral equation with ≡ c/K determined from the condition = γ 1 −1 g(t)dt. It is well known that E → 0 for γ → 0 and E → π 2 /3 for γ → ∞; furthermore E ≈ γ for γ 1 and E ≈ (π 2 /3) (1 − 4γ) for γ 1 [16].
At T = 0 the pressure P = −(∂E/∂L) N is then It is immediately seen that the shift is positive and that it vanishes for γ = 0 (1d ideal Bose gas) and for γ → ∞ (TG gas, having the equation of state of the 1d ideal Fermi gas). Furthermore , for γ 1 (53) and a maximum of the shift appears at a finite value of γ (at γ ≈ 4.7). The plot of e res (T = 0) in units of k B T D is the black lowest curve in Fig.1.
At finite temperature one gets A plot of Eq. (54) as a function of the coupling γ for different scaled temperatures is again in Fig.1. Even though the shift depends (rather weakly) on the temperature, the same structure occurring at zero temperature is seen: a maximum appears at a finite value of γ, i.e. between the two ideal bosonic and fermionic limits.
The high-temperature limit can be explicitly studied: indeed the second virial coefficient [16,56] written in scaled units is where λ T = 2π 2 /mk B T is the thermal De Broglie wavelength (33). Using the virial expansion (18) (valid for τ 4π) one gets at the first non-trivial order (i.e., B 2 ): Using the relation ρλ T = 2 π/τ , from (55)-(56) one gets then where we have introduced the error function Erf(x) = 2 √ π x 0 dy e −y 2 [74]. Using the asymptotic expansion One explicitly sees that e res → 0 in the two ideal limits γ → 0 and γ → ∞ and that there is maximum between them (roughly γ max ∼ √ τ ). From (58) we also see that if one fixes a finite coupling γ and increases the temperature (i.e., τ ), then the internal energy shift density approaches the value k B T D γ. Remarkably, the internal energy shift is finite also for infinite temperature: this is shown in Fig.2 where e res /k B T D is plotted as a function of the scaled temperature τ for different values γ, showing that the asymptotic value γ is reached for large temperatures.

VI. ENERGY-PRESSURE RELATION FOR ANYONIC MODELS
In this Section we study the Abelian and non-Abelian anyonic gases introduced in Section II and we discuss their internal energy shift: we show that in the hard-core case the energypressure obey (7), therefore in this case the gases have vanishing internal energy shift. The soft-core condition introduces instead a scale and this gives raise to a positive internal energy shift.
In the first part of this Section we treat together the Abelian and non-Abelian gases: the Hamiltonians for Abelian/non-Abelian anyons are defined respectively in (31) and (40): they are homogeneous with respect to the particles coordinates and they scale as The hard-core condition at coincident points (in both Abelian and the non-Abelian models) is a particular case of a scale-invariant boundary condition, because all finite-λ-scalings of the N -body eigenfunctionsψ are hard-core eigenfunctions too, and vanishing whenever any coordinate sits on the boundary of the rescaled volume. Denoting by A the area of the system, in the hard-core case the coordinate scaling results in a dilatation of the energy spectrum: then the map (A, β) → (λ 2 A, λ 2 β) let (log Z) invariant. As a consequence of (3) and (6) we obtain the exact identity: in agreement with the more general relation (7). Equivalently, hard-core anyonic gases fulfill of course: The validity of (62) for the particular cases represented by 2D Bose and 2D (spin-less) Fermi ideal gases is remarked in [25]. Thermodynamic relations (62) where r 2 i , and H N , H 1 transform according to under the canonical scaling transformation (r i , p i ) → (λ r i , 1 λ p i ); as a consequence, the following relation holds for the regularized Hamiltonian, expressed in terms of scaling for the regularizing frequency and the spatial coordinates: Notice that the harmonic regularization breaks the scale-invariance, which is retrieved in the ω → 0 limit. Now we apply the hard-core condition. For any eigenfunction ψ n (r i ) of H N,ω (r i , p i ) fulfilling the hard-core boundary condition, we correspondingly getψ n (r i ) ≡ ψ n ( √ γ r i ) (also fulfilling the hard-core boundary condition) as eigenfunction of H N,γ ω (r i , p i ), so that, denoting the hard-core condition by the superscript "h.c.", the frequency acts barely as a dilatation for the energy spectrum: whence the N -body partition function Assuming valid the existence of the virial expansion, one has that the thermodynamic imply that the coefficients B N +1 (T )λ −2N so that B N +1 (T )λ −2N T has to be independent of temperature, thus hard-core ideal anyonic gases fulfill From this relation follows that for these systems the last three identities of (18) take the form Entropy : Internal energy : Enthalpy : We point out that the corresponding entropy and heat capacity at constant volume are unaffected by the statistical interaction at the lowest order of virial expansion (being independent of B 2 ). Using formula (C2) for B 2 , one can obtain the leading deviation of the various thermodynamic quantities from their ideal gas value.
An important consequence of (72) is that from (18) and (73) one gets again Eq.(62) at all orders of the virial expansion for hard-core Abelian and non-Abelian anyonic gases (within the convergence radii of these expansion) [75], in agreement with the general relation 7 The scaling properties for isoentropic processes derived in Sec. II apply in particular to Abelian and non-Abelian anyonic gases with hard-core conditions. Isoentropic processes between initial and final states at equilibrium of hard-core anyonic gases are characterized by the following relation between internal energy and temperature: As a consequence, for hard-core anyonic gases subjected to an isotropic transformation one gets P ∝ A −2 , in agreements with (62) and (74), and the dilution parameter x ≡ ρλ 2 T remains invariant along isoentropic curves. Furthermore, according to (12), the internal energy associated with equilibrium states of an isoentropic process takes the form E = y × N k B T , where y remains constant along the isoentropic curve, while it depends solely on the dilution parameter x = ρ λ 2 T over the entire phase diagram. Since we are in two dimensions, y is just the compressibility factor.
Remarkably, Eq. (12) traces the study of free anyonic thermodynamics back to the determination of how the compressibility factor y(x) depends on the dilution parameter x, and this is a genuine consequence of the scaling symmetry, valid therefore also beyond the radius of convergence of the virial expansion.
For the family of systems represented by Abelian anyons gases, where α will denote henceforth the statistical parameter as in Subsection IV B, the factor y can be parametrized as y = y(x, α). The cases y(x, 0) (2D Bose gas) and y(x, 1) (2D Fermi gas) can be traced back to the analysis in Chapter 4 of Ref. [25]; as the dilution parameter x is swept from 0 to ∞ the gas moves from ideality to an increasingly dense regime, and y(x, 0) monotonically decreases from 1 to 0, while y(x, 1) monotonically increases from 1 to ∞; low/high density limit behaviors immediately follow from [25]: where B n denote here the Bernoulli numbers [74]. As expected, the general behavior at intermediate α is non-trivial, while the basic qualitative statements about y(x, α) are that y(0, α) = 1 for any α (limit of ideal gas) and since the dominance of the second virial coefficient in a very dilute regime. This approximate behavior interpolates the curves y(x, 0) and y(x, 1), and the sign of its slope at x = 0 switches at α = 1 − 1/2, i.e., within the dilute regime approximation the statistical energy is negative for 0 ≤ α < 1 − 1/2, positive for 1 − 1/2 < α ≤ 1.
A remarkable perturbative result is argued in Eq. (22) of [76] about the ground state energy for Abelian anyons, which, by assuming the continuity of E(N, A, T ) at T = 0, reads here y(x, α) ∼ α 2 x, x 1 and α 1 .
Let us pause here for a comment about the classical limit of the hard-core anyonic system.
In the picture of anyons as charge-flux composites written for instance in the bosonic bases, one is free to consider arbitrarily large magnetic fluxes Φ = α h/q, q = being the charge of the particles. The kinetic terms alone would yield the Bose statistics for the quantum case, and the Boltzmann statistics for its classical limit. The quadratic terms in α should be regarded as self-energies of the vortices (in both cases). Finally, the momentum-flux terms correspond to the magnetic inter-particle interaction, and they are responsible for the non-trivial anyonic thermodynamics, which is periodic in the flux variable α = qΦ/h. Correspondingly  [25] as a uniform upward spectral shift whose α-dependence is periodic with finite period ∆α = 2, therefore this shift becomes irrelevant (to the density of states) for any values of the flux in the classical limit k B T ω 0 .

A. Soft-core anyons
The scale-invariance in force for hard-core anyons does not apply in presence of soft-core boundary conditions, in which case we will compare internal energy and pressure within the dilute regime (up to the first order in the dilution parameter ρλ 2 T ). Let us define the relative internal energy shift density e rel as the dimensionless quantity For Abelian anyons, Eqs. (39)-(78) give where The resulting shift is whose leading term in ρλ 2 T is illustrated in Fig. 3. The plot of the shift e rel (α, T, ε, σ = 1) exhibits a smooth behavior in the bosonic points and a cusp in the fermionic ones, as soon as the hard-core condition is relaxed. We observe that the restriction of e rel (α, T, ε) over the interval α ∈ [0, 1] is not a monotonic function of α for any ε. The proportionality E = P A remains valid at the bosonic points also in the soft-core case [i.e. e rel (α = 2n, T, ε) = 0], and the monotonicity of the shift e rel as a function of α occurs for any ε ∈ [ε − , ε + ], ε − ≈ 0.13, ε + ≈ 3.0, and in particular the relative shift is maximal at the fermionic points for any ε ∈ [ε − , ε + ], while outside of this interval the shift due to the soft-core boundary conditions reaches its maximum at an internal point α max (ε), featuring the properties α max (ε) → 1 − for ε → (ε ± ) ± , and α max (ε) → 0 + for ε → 0 or ε → ∞ .
This feature can be contrasted with the decay of e rel (α = ±1, ε, T ), as log ε → ±∞, for Abelian anyons in the fermionic limit: in this case, the energy shift is a power law in ε (and precisely linear) for ε 1, while instead it decays exponentially in ε for ε 1, as seen in Fig.6. The shift e rel (α = ±1, ε, T ) reaches its maximum 2/e ≈ 0.74 (in units of (ρλ 2 T )) at ε = 1. conventionally-defined departure internal energy is the hard-core anyonic gas, as discussed in Section VI: summarizing, the quantity (17) can be regarded as a good measure of the deviation from scale-invariance for non-ideal gases.
In particular, we have provided criteria for which the internal energy shift density of an imperfect (classical or quantum) gas is a bounded function of temperature. We have also shown that for general scale-invariant systems the dependence of virial coefficients upon the temperature is very simple, and is expressed by Eq. (13).
We have considered deviations from the energy-pressure proportionality in lowdimensional models of gases which interpolate between the ideal Bose and the ideal Fermi gases, focusing the attention on the Lieb-Liniger model in 1d and on the anyonic gas in 2d.
In 1d the internal energy shift is determined from the thermodynamic Bethe ansatz integral equations and an explicit relation for it is provided at high temperature: the internal energy shift is positive, it vanishes in the two limits of zero and infinite coupling (respectively the ideal Bose and the Tonks-Girardeau gas) and it has a maximum at a finite, temperaturedepending, value of the coupling. Remarkably, at fixed coupling the internal energy shift density saturates to a finite value for infinite temperature.
In 2d we have considered systems of Abelian anyons and non-Abelian Chern-Simons particles and we have showed that the relation between the internal energy and the pressure of anyonic gas is exactly the same found for 2D Bose and Fermi ideal gases as long as the hard-core case is considered. Soft-core boundary conditions introduce a length scale and determine a non-vanishing internal energy shift: we have provided details about this shift in the dilute limit. Asymptotic expressions with respect to the hard-core parameter ε are derived for both Abelian and non-Abelian soft-core anyonic gases. We use Eqs. (18) in order to write the internal energy shift in the dilute limit for quadratic dispersion relation of the particles where f (x) denotes the second virial coefficient as a function of its unique variable x ≡ 1/(k B T ). We are interested in the boundedness of e res as the high-temperature limit x → 0 is approached, i.e.
and Ψ H (z 1 , . . . , z N ) stands for the wavefunction of the N -body system of the NACS particles in the holomorphic gauge. Ψ A (z 1 , . . . , z N ) obeys the braid statistics [65] due to the transformation function U (z 1 , . . . , z N ), while Ψ H (z 1 , . . . , z N ) satisfies ordinary statistics: Ψ A (z 1 , . . . , z N ) is commonly referred to as the NACS particle wavefunction in the anyon gauge.
The statistical mechanics of the NACS particles can be studied in the low-density regime in terms of the cluster expansion of the grand partition function Ξ The virial expansion (with the pressure expressed in powers of the density ρ = N A ) is given as where B n (T ) is the n-th virial coefficient, which can be expressed in terms of the first cluster coefficients b 1 , · · · b n . The second virial coefficient B 2 (T ) turns out to be [3] where A is the area and Z N = Tr e −βH N the N -particle partition function. We assume that the NACS particles have equal masses and belong to the same isospin multiplet {|l, m >} with m = −l, . . . , l. The quantity Z 1 = Tr e −βH 1 is then given by where B B 2 (ω j , T, ε j,jz ) is the soft-core expression entering Eq. (39): B B 2 (ω j , T, ε j,jz ) = B h.c. 2 (δ j , T )−2λ 2 T e ε j,jz θ(−σ) + δ j σ π (sin πδ j ) ∞ 0 dte −ε j,jz t t |δ j |−1 1 + 2σ(cos πδ j ) t |δ j | + t 2|δ j | , with δ j ≡ (ω j + 1) mod 2 − 1, and B F 2 (ω j , T, ε j,jz ) is the previous expression evaluated for ω j → ω j + 1: with Γ j ≡ ω j mod 2 − 1.

(C10)
The depletion of B 2 in Eq. (C10), with respect to its hard-core value − 1 24 λ 2 T , arises from the anyonic collisions allowed by the soft-core conditions.