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Low Temperature Physics

Scaling functions in the universality class of two-dimensional models with a one-component order parameter

S. B. Rutkevich

Published Jan 1, 1994• 0 citations
Publisher

Abstract

Thermodynamic studies are carried out for the fermion model, which is equivalent in the critical region to the two-dimensional Ising model in a uniform magnetic field. Natural assumptions concerning the structure of the ground state lead to the equation of state in the form of a similitude relation. The obtained expression for the scaling function gives correct values of critical indices β = 1/8, δ = 15, and γ = γ′=7/4.

DOI
10.1063/10.0033580

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Thermodynamic studies are carried out for the fermion model, which is equivalent in the critical region to the two-dimensional Ising model in a uniform magnetic field. Natural assumptions concerning the structure of the ground state lead to the equation of state in the form of a similitude relation. The obtained expression for the scaling function gives correct values of critical indices β = 1/8, δ = 15, and γ = γ′=7/4.
The fermionic model is considered equivalent in the critical region to the two-dimensional Ising model in homogeneous magnetic field. The ground state of the fermionic Hamiltonian is obtained from the variational principle of the quantum mechanics, where probe states are taken in the BCS-like form. In this approximation the explicit representation of the scaling function is derived which is characterized by the correct values of critical indexes and the susceptibility critical amplitudes ratio C + /C - =12 pi .
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 effect of an external field on the phase transition is studied in the continuous version of the model of free fermions without dislocations. The critical indices δ = ∞, β = + 0, γ = 3/2 and the phase diagram are obtained as well as an exact expression for the free energy in the strong field limit. The results are compared with scaling properties in the six-vertex model. While the similarity of scaling in zero field and a coincidence of the values of β and δ indicate that both models have the same class of universality, considerable differences are observed in other characteristics like the index γ, phase diagram, and scaling function. This “discrepancy” is explained by the fact that the external field violates the symmetry in fermion and six-vertex models in different ways. The possibility of experimental observation of the obtained scaling dependences in two-dimensional crystals undergoing a transition to the incommensurate phase is discussed.
In this thesis, we present a recent study of the one-dimensional attractively interacting two-component Fermi gas with an external magnetic field. We analytically study the quantum phase diagram, equation of state, quantum criticality, and universal scalings of the model, using the thermodynamic Bethe ansatz equations. We present a high precision equation of state from which universal scaling functions are derived; we show that the thermodynamic properties display universal scaling behaviour at quantum criticality. Expression of high-order-accuracy entropy and specific heat are also analytically derived. Moreover, we present an independent confirmation of the equation of state through an efficient numerical approach of treating the thermodynamic Bethe ansatz equations for which the contributions from the spin strings with length larger than a value of ncn_cnc​ are calculated analytically.
"Critical point phenomena in magnetic systems are studied with the aid of the scaling laws. These laws relate the critical indices which describe the behavior of thermodynamic and correlation functions near T. Kadanoff's derivation of the scaling 1aw is reviewed. An experimental survey of ferromagnetic and antiferromagnetic data near T is presented. Susceptibility, correlation length, magnectization, specific heat and a magnetic equation of state are discussed. It is found that the ""critical region"" can extend as far as [equation] although frequently it is restricted to E < 10-2 The critical indices S, y, v, ~ and ~, are close to the three dimensional ISing model values. Values for 0, y' and ~ are not known as conclusively. Specific heat behavior is discussed in terms -~ of a three-parameter fit, namely C = aE + b. Varying b allows one to estimate a possible range of a's. The experimental information is consistent with the scaling law relations 2 - alpha = 2 - alpha' = dV = d gamma / (2-pi) = gamma + 2B = B(d+1) The scaling law theory is extended to predict the temperature, field and spatial dependence of various spin correlation functions. These predictions are checked by analytic calculations for the two dimensional Ising model in zero field. Results are presented for the energy density-energy density and energy density-spin correlation functions in the limit E « 1, R » 1, but the product ER arbitrary. Results are also presented for a group of two, three and four spin correlations at cT. In each case the scaling law predictions are verified."
Order parameter fluctuations for the two-dimensional Ising model in the region of the critical temperature are presented. A locus of temperatures T(*) (L) and a locus of magnetic fields B(*) (L) are identified, for which the probability density function is similar to that for the two-dimensional XY model in the spin wave approximation. The characteristics of the fluctuations along these points are largely independent of universality class. We show that the largest range of fluctuations relative to the variance of the distribution occurs along these loci of points, rather than at the critical temperature itself and we discuss this observation in terms of intermittency. Our motivation is the identification of a generic form for fluctuations in correlated systems in accordance with recent experimental and numerical observations. We conclude that a universality-class-dependent form for the fluctuations is a particularity of critical phenomena related to the change in symmetry at a phase transition.