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arXiv:cond-mat/9806190 · Journal of Physics A Mathematical and General

On the finite-temperature generalization of theC-theorem and the interplay between classical and quantum fluctuations

Daniel M. Danchev, N. S. Tonchev

Published Sep 28, 1999• 4 citations
arXivHTMLPDF

Abstract

The behaviour of the finite-temperature C -function, defined by Neto and Fradkin (1993 Nucl. Phys. B 400 525), is analysed within a d -dimensional exactly solvable lattice model, recently considered by Vojta (1996 Phys. Rev. B 53 710), which is of the same universality class as the quantum nonlinear O( n ) sigma model in the limit n . The scaling functions of C for the cases d = 1 (absence of long-range order), d = 2 (existence of a quantum critical point), d = 4 (existence of a line of finite-temperature critical points that ends up with a quantum critical point) are derived and analysed. The locations of regions where C is monotonically increasing (which depend significantly on d ) are exactly determined. The results are interpreted within the finite-size scaling theory that has to be modified for d = 4.

DOI
10.1088/0305-4470/32/41/302

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The behaviour of the finite-temperature C -function, defined by Neto and Fradkin (1993 Nucl. Phys. B 400 525), is analysed within a d -dimensional exactly solvable lattice model, recently considered by Vojta (1996 Phys. Rev. B 53 710), which is of the same universality class as the quantum nonlinear O( n ) sigma model in the limit n . The scaling functions of C for the cases d = 1 (absence of long-range order), d = 2 (existence of a quantum critical point), d = 4 (existence of a line of finite-temperature critical points that ends up with a quantum critical point) are derived and analysed. The locations of regions where C is monotonically increasing (which depend significantly on d ) are exactly determined. The results are interpreted within the finite-size scaling theory that has to be modified for d = 4.
We revisit the two-dimensional quantum Ising model by computing renormalization group flows close to its quantum critical point. The low but finite temperature regime in the vicinity of the quantum critical point is squashed between two distinct non-Gaussian fixed points: the classical fixed point dominated by thermal fluctuations and the quantum critical fixed point dominated by zero-point quantum fluctuations. Truncating an exact flow equation for the effective action, we derive a set of renormalization group equations and analyze how the interplay of quantum and thermal fluctuations, both non-Gaussian in nature, influences the shape of the phase boundary and the region in the phase diagram where critical fluctuations occur. The solution of the flow equations makes this interplay transparent: we detect finite temperature crossovers by computing critical exponents and we confirm that the power law describing the finite temperature phase boundary as a function of control parameter is given by the correlation length exponent at zero temperature as predicted in an ϵϵϵ-expansion with ϵ=1ϵ=1ϵ=1 by Sachdev [Phys. Rev. B 55, 142 (1997)].
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.
The scaling of the transition temperature into an ordered phase close to a quantum critical point as well as the order parameter fluctuations inside the quantum critical region provide valuable information about universal properties of the underlying quantum critical point. Here, we employ quantum Monte Carlo simulations to examine these relations in detail for two-dimensional quantum systems that exhibit a finite-temperature Ising-transition line in the vicinity of a quantum critical point that belongs to the universality class of either (i) the three-dimensional Ising model for the case of the quantum Ising model in a transverse magnetic field on the square lattice or (ii) the chiral Ising transition for the case of a half-filled system of spinless fermions on the honeycomb lattice with nearest-neighbor repulsion. While the first case allows large-scale simulations to assess the scaling predictions to a high precision in terms of the known values for the critical exponents at the quantum critical point, for the later case, we extract values of the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\eta}$, related to the order parameter fluctuations, which we discuss in relation to other recent estimates from ground-state quantum Monte Carlo calculations as well as analytical approaches.
The quantum critical behavior of the (2 + 1)-dimensional Gross-Neveu model in the vicinity of its zero-temperature critical point is considered.The model is known to be renormalisable in the large-N limit, which offers the possibility to obtain expressions for various thermodynamic functions in closed form.We have used the concept of finite-size scaling to extract information about the leading temperature behavior of the free energy and the mass term, defined by the fermionic condensate and determined the crossover lines in the coupling (g)-temperature (T ) plane.These are given by T ∼ |g -gc|, where gc denotes the critical coupling at zero temperature.According to our analysis no spontaneous symmetry breaking survives at finite temperature.We have found that the leading temperature behavior of the fermionic condensate is proportional to the temperature with the critical amplitude √ 5 3 π.The scaling function of the singular part of the free energy is found to exhibit a maximum at ln 2 2π corresponding to one of the crossover lines.The critical amplitude of the singular part of the free energy is given by the universal number 1 3 1 2π ζ(3) -Cl2 π 3 = -0.274543 . .., where ζ(z) and Cl2(z) are the Riemann zeta and Clausen's functions, respectively.Interpreted in terms of the thermodynamic Casimir effect, this result implies an attractive Casimir "force".This study is expected to be useful in shedding light on a broader class of four fermionic models.
We prove a C-theorem within the framework of two dimensional quantum field theories at finite temperature. There exists a function C(g) of coupling constants which is non-increasing along renormalization group trajectories and non-decreasing along temperature trajectory and stationary only at the fixed points. The connection between the C-theorem at zero temperature and the C-theorem at finite temperature is discussed. We also consider the thermodynamical aspects of the C-theorem. If we define the C-function in an arbitrary number of dimensions in anology to the two dimensional case, we can show that its behavior is not universal. The phase transitions destroy the monotonic properties of the C-function. The proof of the C-theorem is also presented within the framework of the Kallen-Lehmann spectral representation at finite temperature.
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.