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Physical review. B, Solid state

Scaling behavior at zero-temperature critical points

George Allen Baker Jr., Jill C. Bonner

Published Nov 1, 1975• 29 citations
Publisher

Abstract

A scaling form for the logarithm of the partition function suitable for a zero-temperature critical point is obtained and found to hold for the spherical model in less than two dimensions and the classical $n$-component Heisenberg linear chain. Nevertheless, several cases are found where the critical-exponent relations involving the specific heat fail. These anomalous cases do not imply a breakdown of the scaling implicit in the basic formulation of renormalization-group theory.

DOI
10.1103/physrevb.12.3741

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A scaling form for the logarithm of the partition function suitable for a zero-temperature critical point is obtained and found to hold for the spherical model in less than two dimensions and the classical nnn-component Heisenberg linear chain. Nevertheless, several cases are found where the critical-exponent relations involving the specific heat fail. These anomalous cases do not imply a breakdown of the scaling implicit in the basic formulation of renormalization-group theory.
We explicitly compute the critical exponents associated with logarithmic corrections (the so-called hatted exponents) starting from the renormalization group equations and the mean field behavior for a wide class of models at the upper critical behavior (for short and long range φ n -theories) and below it.This allows us to check the scaling relations among these critical exponents obtained by analysing the complex singularities (Lee-Yang and Fisher zeroes) of these models.Moreover, we have obtained an explicit method to compute the ϙ exponent [defined by ξ ∼ L(log L) ϙ] and, finally, we have found a new derivation of the scaling law associated with it.
A ddd--dimensional quantum model in the spherical approximation confined to a general geometry of the form Ld−d′×∞d′×LτzL^{d-d^{\prime}} \times\infty^{d^{\prime}}\times L_τ^{z}Ld−d′×∞d′×Lτz​ (LLL--linear space size and LτL_τLτ​--temporal size) and subjected to periodic boundary conditions is considered. Because of its close relation with the quantum rotors model it can be regarded as an effective model for studying the low-temperature behavior of the quantum Heisenberg antiferromagnets. Due to the remarkable opportunity it offers for rigorous study of finite-size effects at arbitrary dimensionality this model may play the same role in quantum critical phenomena as the popular Berlin-Kac spherical model in classical critical phenomena. Close to the zero-temperature quantum critical point, the ideas of finite-size scaling are utilized to the fullest extent for studying the critical behavior of the model. For different dimensions $1
In chapter one, we explain briefly the continuum limit, scaling, and high temperature expansion of critical phenomena, Monte Carlo algorithms and fitting. In chapter two, different continuum limits of the Ising model in dimensions (D) 2, 3 and 4 are investigated numerically. The data indicate that triviality occurs for D = 4 and fails for D < 4 in each limit. In chapter three, a relation between the critical exponents of the leading and confluent scaling terms is derived using the finite size scaling argument. We also determine the new scaling variable of the 4D Ising model based on a new Monte Carlo simulation data. In chapter four, a Monte Carlo study of two dimensional diluted Ising systems is reported. It is shown that regular dilution does not affect critical exponents, while a random one does, with critical exponents varying continuously with impurity concentration. The importance of fluctuations in producing such effects is emphasized. In chapter five, a different point of view regarding the critical exponent of the specific heat of the 3D Ising model is presented. Based on the analysis of high temperature expansion, finite size scaling and Monte Carlo data in the symmetric phase of the 3D Ising model, it is shown that logarithmic scaling behavior of specific heat is more consistent than power scaling behavior.
The basic concepts and formulation of the renormalization group are explained beginning at an elementary level. Discussion is in the framework of classical statistical mechanics with emphasis on applications to the theory of critical phenomena. The details are worked out in the large n limit for 2 < d < 4, where n is the number of components of the fluctuating field of interest and d is the dimension of the thermodynamical system. In the large n limit, the infinite sum of ``tree graphs'' offers an exact and analytically tractable description of the renormalization group. It illustrates many concepts including the fixed point, the critical surface in the space of coupling parameters, and critical exponents. Most important, it illustrates the origin and the limitation of the scaling hypothesis. The critical behavior of various correlation functions and the free energy is examined. Attention is paid to terms often ignored in qualitative scaling arguments. We have attempted to make this paper self-contained and of pedagogical value to a wide audience.
If the zero-field transition in high-temperature superconductors such as ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ is a critical point in the universality class of the three-dimensional XY\mathrm{XY}XY model, then the general theory of critical phenomena predicts the existence of a critical region in which thermodynamic functions have a characteristic scaling form. We report an attempt to calculate the universal scaling function associated with the specific heat, for which experimental data have become available in recent years. Scaling behavior is extracted from a renormalization-group analysis, and the 1/N1/N1/N expansion is adopted as a means of approximation. The estimated scaling function is qualitatively similar to that observed experimentally, and also to the lowest-Landau-level scaling function used by some authors to provide an alternative interpretation of the same data. Unfortunately, the 1/N1/N1/N expansion is not sufficiently reliable at small values of N for a quantitative fit to be feasible.