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arXiv:0810.3601 · Physical Review B

Second-order critical lines of spin- S Ising models in a splitting field using Grassmann techniques

Jean-Yves Fortin, Maxime Clusel

Published Nov 17, 2008• 2 citations
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Abstract

We propose a method to study the second-order critical lines of classical spin-$S$ Ising models on two-dimensional lattices in a crystal or splitting field, using an exact expression for the bare mass of the underlying field theory. Introducing a set of anticommuting variables to represent the partition function, we derive an exact and compact expression for the bare mass of the model including all local multifermion interactions. By extension of the Ising and Blume-Capel models, we extract the free-energy singularities in the low-momentum limit corresponding to a vanishing bare mass. The loci of these singularities define the critical lines depending on the spin $S$, in good agreement with previous numerical estimations. This scheme appears to be general enough to be applied in a variety of classical Hamiltonians.

DOI
10.1103/physrevb.78.172402

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We propose a method to study the second-order critical lines of classical spin-SSS Ising models on two-dimensional lattices in a crystal or splitting field, using an exact expression for the bare mass of the underlying field theory. Introducing a set of anticommuting variables to represent the partition function, we derive an exact and compact expression for the bare mass of the model including all local multifermion interactions. By extension of the Ising and Blume-Capel models, we extract the free-energy singularities in the low-momentum limit corresponding to a vanishing bare mass. The loci of these singularities define the critical lines depending on the spin SSS, in good agreement with previous numerical estimations. This scheme appears to be general enough to be applied in a variety of classical Hamiltonians.
We review the applications of the integral over anticommuting Grassmann variables (nonquantum fermionic fields) to the analytic solutions and the field-theoretical formulations for the 2D Ising models. The 2D Ising model partition function QQQ is presentable as the fermionic Gaussian integral. The use of the spin-polynomial interpretation of the 2D Ising problem is stressed, in particular. Starting with the spin-polynomial interpretation of the local Boltzmann weights, the Gaussian integral for QQQ appears in the universal form for a variety of lattices, including the standard rectangular, triangular, and hexagonal lattices, and with the minimal number of fermionic variables (two per site). The analytic solutions for the correspondent 2D Ising models then follow by passing to the momentum space on a lattice. The symmetries and the question on the location of critical point have an interesting interpretation within this spin-polynomial formulation of the problem. From the exact lattice theory we then pass to the continuum-limit field-theoretical interpretation of the 2D Ising models. The continuum theory captures all relevant features of the original models near TcT_cTc​. The continuum limit corresponds to the low-momentum sector of the exact theory responsible for the critical-point singularities and the large-distance behaviour of correlations. The resulting field theory is the massive two-component Majorana theory, with mass vanishing at TcT_cTc​. By doubling of fermions in the Majorana representation, we obtain as well the 2D Dirac field theory of charged fermions for 2D Ising models. The differences between particular 2D Ising lattices are merely adsorbed, in the field-theoretical formulation, in the definition of the effective mass.
We review problems involving the use of Grassmann techniques in the field of classical spin systems in two dimensions.These techniques are useful to perform exact correspondences between classical spin Hamiltonians and field-theory fermionic actions.This contributes to a better understanding of critical behavior of these models in term of non-quadratic effective actions which can be seen as an extension of the free fermion Ising model.Within this method, identification of bare masses allows for an accurate estimation of critical points or lines and which is supported by Monte-Carlo results and diagrammatic techniques.
We use Grassmann algebra to study the phase transition in the two-dimensional ferromagnetic Blume-Capel model from a fermionic point of view. This model presents a phase diagram with a second order critical line which becomes first order through a tricritical point, and was used to model the phase transition in specific magnetic materials and liquid mixtures of He3^33-He4^44. In particular, we are able to map the spin-1 system of the BC model onto an effective fermionic action from which we obtain the exact mass of the theory, the condition of vanishing mass defines the critical line. This effective action is actually an extension of the free fermion Ising action with an additional quartic interaction term. The effect of this term is merely to render the excitation spectrum of the fermions unstable at the tricritical point. The results are compared with recent numerical Monte-Carlo simulations.
We study classical Ising spin-1/2 models on a two-dimensional (2D) square lattice with ferromagnetic or antiferromagnetic nearest-neighbor interactions, under the effect of a pure imaginary magnetic field. The complex Boltzmann weights of spin configurations cannot be interpreted as a probability distribution, which prevents application of standard statistical algorithms. In this work, the mapping of the Ising spin models under consideration onto symmetric vertex models leads to real (positive or negative) Boltzmann weights. This enables us to apply accurate numerical methods based on the renormalization of the density matrix, namely, the corner transfer matrix renormalization group and the higher-order tensor renormalization group. For the 2D antiferromagnet, varying the imaginary magnetic field, we calculate with high accuracy the curve of critical points related to the symmetry breaking of magnetizations on the interwoven sublattices. The critical exponent β and the anomaly number c are shown to be constant along the critical line, equal to their values β=1/8 and c=1/2 for the 2D Ising model in a zero magnetic field. The 2D ferromagnets behave in analogy with their 1D counterparts defined on a chain of sites, namely, there exists a transient temperature which splits the temperature range into its high-temperature and low-temperature parts. The free energy and the magnetization are well defined in the high-temperature region. In the low-temperature region, the free energy exhibits singularities at the Yang-Lee zeros of the partition function and the magnetization is also ill-defined: It varies chaotically with the size of the system. The transient temperature is determined as a function of the imaginary magnetic field by using the fact that from the high-temperature side both the first derivative of the free energy with respect to the temperature and the magnetization diverge at this temperature.
The Blume-Caple model on the triangular lattice is considered in this paper. Using the (perturbative) real-space renormalization group (RSRG) technique we analytically obtain the running of the parameters of the model, i.e., the exchange integral J and the crystal field Δ. The observations show that the first-order calculations are insufficient to capture the tricritical phenomena, and the RG analysis based on second-order expansion of the partition function show qualitative agreement with the previous works. This reveals that the second-order perturbation works appropriately, to obtain the qualitative features of the phase diagram of the model, e.g., a tricritical point which is unstable towards the Ising critical point from one side (which forms the line of second order, separating the ferromagnetic and paramagnetic phases) and the critical point at and from the other side (which forms the line of first-order transition separating ferromagnetic and zero state phases). This analysis shows also that there is another unstable critical point at , with the RG eigenvalue in the Δ-direction (towards S = 0 zero state and paramagnetic phases).