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Journal of Physics Conference Series

Entropy and specific heat of a critical quantum system with long-range interaction

Ekaterina S. Pisanova

Published Mar 1, 2019• 0 citations
Publisher

Abstract

The entropy and the specific heat of a quantum spherical model with a long-range interaction (decreasing at large distances r as r –d–σ where d is the space dimensionality and 0 < σ ≤ 2) are studied in the quantum critical region at the upper quantum critical dimension d = 3σ/2. The problem of obtaining the temperature-dependent corrections to the ground state free energy involves the solution of a transcendental equation, the exact solution of which is expressed in terms of the Lambert W-function. The free energy, the entropy and the specific heat in the quantum critical region are derived in terms of the Lambert W-function. For systems in real physical dimensions (chains, thin layers, i.e. films and three-dimensional systems) the exact results for the entropy and the specific heat obtained in terms of the Lambert W-function and the leading asymptotic ones are compared on the basis of the calculated relative errors. It can be concluded that for a class of quantum models at the upper critical dimension the Lambert W-function is a very effective tool for an exact computation of low-temperature critical properties, especially in the finite-temperature quantum critical region.

DOI
10.1088/1742-6596/1186/1/012015

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The entropy and the specific heat of a quantum spherical model with a long-range interaction (decreasing at large distances r as r –d–σ where d is the space dimensionality and 0 < σ ≤ 2) are studied in the quantum critical region at the upper quantum critical dimension d = 3σ/2. The problem of obtaining the temperature-dependent corrections to the ground state free energy involves the solution of a transcendental equation, the exact solution of which is expressed in terms of the Lambert W-function. The free energy, the entropy and the specific heat in the quantum critical region are derived in terms of the Lambert W-function. For systems in real physical dimensions (chains, thin layers, i.e. films and three-dimensional systems) the exact results for the entropy and the specific heat obtained in terms of the Lambert W-function and the leading asymptotic ones are compared on the basis of the calculated relative errors. It can be concluded that for a class of quantum models at the upper critical dimension the Lambert W-function is a very effective tool for an exact computation of low-temperature critical properties, especially in the finite-temperature quantum critical region.
The pure quantum limit of an exactly solvable lattice model describing structural phase transitions in an anharmonic crystal with long-range interaction is considered. At the upper quantum critical dimension the free energy density at T = 0 in the neighbourhood of the quantum critical point is exactly calculated in terms of the Lambert W-function. For the three real physical dimensions, the exact results, obtained here, and the asymptotic ones are compared. It is pointed out that the Lambert W-function turns out to be a very effective tool for an exact computation of non-universal characteristics in the upper critical dimensions, especially in a broader neighbourhood of the critical region.
Abstract The entropy, the internal energy and the specific heat of a d-dimensional quantum spherical model with a long-range interaction, the decreasing of which is controlled by a parameter σ (0 < σ ≤ 2), are studied close to its quantum critical point in the context of the finite-size scaling (FSS) theory for d = σ. The obtained specific heat critical exponent σ does not depend on σ and satisfies the quantum hyperscaling relation. Based on the derived FSS forms and asymptotes of the universal scaling functions of the considered quantities, the leading temperature dependencies of the entropy, the internal energy and the specific heat are obtained in both the renormalized classical and the quantum disordered regions of the phase diagram. It has been shown that when the temperature T → +0, the entropy and the specific heat in the renormalized classical region tend to zero as powers of T, while in the quantum disordered region they tend to zero exponentially.
A d-dimensional quantum spherical model with a long-range interaction (decreasing at long distances r as r−d−σ, 0 <σ ≤ 2) is considered close to its quantum critical point. The low-temperature behaviors of the entropy, the internal energy and the specific heat capacity are studied for σ/2 < d < 3σ/2 in the context of the finite-size scaling (FSS) theory. For a σ-dimensional system and a d-dimensional system with short-range interaction (σ = 2) the results for the corresponding critical amplitudes as functions of the dimensionality of the system are presented.
An exactly solvable lattice model describing structural phase transitions in an anharmonic crystal with long-range interaction (decreasing at large distances r as r−d−σ, where d is the space dimensionality and 0 < σ ≤ 2) is considered near to its zero-temperature critical point. The low-temperature behavior of the bulk specific heat capacity at the lower classical critical dimension (d = σ) is studied in different regions of the (T, λ)-phase diagram, where T is the temperature and λ is a parameter which switches on quantum fluctuations. From the results obtained one can see that when T → 0+ the specific heat capacity tends to zero in a different way in the regions: (a) renormalized classical region - as T raised to the second power and (b) quantum disordered region - exponentially.
We study the interacting homogeneous Bose gas in two spatial dimensions in the thermodynamic limit at fixed density. We shall be concerned with some mathematical aspects of this complicated problem in many-body quantum mechanics. More specifically, we consider the dilute limit where the scattering length of the interaction potential, which is a measure for the effective range of the potential, is small compared to the average distance between the particles. We are interested in a setting with positive (i.e., non-zero) temperature. After giving a survey of the relevant literature in the field, we provide some facts and examples to set expectations for the two-dimensional system. The crucial difference to the three-dimensional system is that there is no Bose–Einstein condensate at positive temperature due to the Hohenberg–Mermin–Wagner theorem. However, it turns out that an asymptotic formula for the free energy holds similarly to the three-dimensional case. We motivate this formula by considering a toy model with δ interaction potential. By restricting this model Hamiltonian to certain trial states with a quasi-condensate we obtain an upper bound for the free energy that still has the quasi-condensate fraction as a free parameter. When minimizing over the quasi-condensate fraction, we obtain the Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity, which plays an important role in our rigorous contribution. The mathematically rigorous result that we prove concerns the specific free energy in the dilute limit. We give upper and lower bounds on the free energy in terms of the free energy of the non-interacting system and a correction term coming from the interaction. Both bounds match and thus we obtain the leading term of an asymptotic approximation in the dilute limit, provided the thermal wavelength of the particles is of the same order (or larger) than the average distance between the particles. The remarkable feature of this result is its generality: the correction term depends on the interaction potential only through its scattering length and it holds for all nonnegative interaction potentials with finite scattering length that are measurable. In particular, this allows to model an interaction of hard disks.
Abstract We investigate the critical behavior in the quantum regime of the Hubbard model in d-dimensional hypercubic lattices with infinite-range Coulomb interaction. The Haldane–Wu mutual exclusion statistics approach, which comprises three species of exclusions, allows us to derive the singular part of the free energy, entropy, and specific heat in the quantum regime g > T, where g is the scaling interaction parameter and T is the temperature. We present results from the analytic series expansions of observables near the quantum critical point: T = g = μ = 0 , where µ is the chemical potential. The signature of the mutual exclusion statistics manifests through the appearance of polylogarithms related to Lerch functions in the coefficients of the free energy series in powers of T / g . The calculated critical exponents from the analytic series are consistent with our scaling analysis in the quantum regime. In particular, the specific heat is shown to display a variety of critical behaviors, C / T ∼ g − α g , g → 0 + , depending on the lattice dimension: from a singular behavior in d = 1 ( α g = 1 / 2 ) to a constant value in d = 2 ( α g = 0 ), and regular behavior for d > 2 ( α g = 1 − d / 2 < 0 ).