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Physical review. B, Condensed matter

Quantum studies of the two-dimensional axial next-nearest-neighbor Ising model: A BCS-type theory

Santiago D’Elía

Published Jan 1, 1991• 3 citations
Publisher

Abstract

The Hamiltonian limit of the two-dimensional axial next-nearest-neighbor Ising model is mapped into a cutoff interacting field fermion theory. The single-particle spectrum is defined by a set of integral equations with a singular kernel at the ferromagnetic critical line (FCL). The interaction operator V${\mathrm{^}}_{\mathrm{in}}$ is calculated using single-particle states with definite momentum and parity. There are various interaction processes associated with V${\mathrm{^}}_{\mathrm{in}}$. One of these is, at the FCL, an attractive pairing force V${\mathrm{^}}_{\mathit{p}}$. Allowance for V${\mathrm{^}}_{\mathit{p}}$ leads to the formation of a condensate of Cooper pairs, each pair having opposite momenta and parities. This is shown by using Green's functions.

DOI
10.1103/physrevb.43.1202

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The Hamiltonian limit of the two-dimensional axial next-nearest-neighbor Ising model is mapped into a cutoff interacting field fermion theory. The single-particle spectrum is defined by a set of integral equations with a singular kernel at the ferromagnetic critical line (FCL). The interaction operator V{\mathrm{^}}_{\mathrm{in}} is calculated using single-particle states with definite momentum and parity. There are various interaction processes associated with V{\mathrm{^}}_{\mathrm{in}}. One of these is, at the FCL, an attractive pairing force V{\mathrm{^}}_{\mathit{p}}. Allowance for V{\mathrm{^}}_{\mathit{p}} leads to the formation of a condensate of Cooper pairs, each pair having opposite momenta and parities. This is shown by using Green's functions.
We study the large-S limit of a one-dimensional quantum spin model in a transverse magnetic field. For S=1/2, this is dual to the quantum Hamiltonian limit of the two-dimensional axial next-nearest-neighbor Ising model. The phase diagram has three regions; paramagnetic, ferromagnetic, and a modulated phase of period 2. The Peschel-Emery curve is discovered to be a disorder line. The long-distance correlation functions behave differently on the two sides of the line, and the correlation length goes to zero upon approaching it from either side.
We investigate the dynamics of a two-dimensional axial next-nearest-neighbor Ising model following a quench to zero temperature. The Hamiltonian is given by H= -J_(0) summation operator(L)_(i,j=1)S_(i,j)S_(i+1,j)-J_(1)summation operator_(i,j=1)(S_{i,j}S_{i,j+1}-kappaS_{i,j}S_{i,j+2}) . For kappa<1 , the system does not reach the equilibrium ground state but slowly evolves to a metastable state. For kappa>1 , the system shows a behavior similar to that of the two-dimensional ferromagnetic Ising model in the sense that it freezes to a striped state with a finite probability. The persistence probability shows algebraic decay here with an exponent theta=0.235+/-0.001 while the dynamical exponent of growth z=2.08+/-0.01 . For kappa=1 , the system belongs to a completely different dynamical class; it always evolves to the true ground state with the persistence and dynamical exponent having unique values. Much of the dynamical phenomena can be understood by studying the dynamics and distribution of the number of domain walls. We also compare the dynamical behavior to that of a Ising model in which both the nearest and next-nearest-neighbor interactions are ferromagnetic.
The magnetic critical exponent of the two-dimensional axial next-nearest-neighbor Ising model is estimated by means of the finite-size scaling of the order parameter. For a finite system without ordering field, this magnitude is computed from the two leading eigenvectors of the transfer matrix. The ferromagnetic-paramagnetic transition temperatures estimated with this method exhibit a small lattice-size dependence. Moreover, they are in good agreement with those obtained using phenomenological renormalization, and also with analytical approximations. The correlation-length critical exponent is calculated by means of phenomenological renormalization. The results found with these numerical techniques indicate that the ferromagnetic-paramagnetic transition of the model seems to be continuous, in contradiction with a previous result obtained using the persistence-length criterion, which indicated that the transition is always first order. We found that the phase transition is first order only when the ratio of the next-nearest-neighbor and nearest-neighbor interactions equals one-half.
We investigate the effect of including doman-wall interactions in the two-dimensional axial next-nearest-neighbor Ising model. At low temperatures this problem is reduced to a one-dimensional system of interacting fermions which can be treated exactly. It is found that the critical boundaries of the low-temperature phases are in good agreement with those obtained using a free-fermion approximation. In contrast with the monotonic behavior derived from the free-fermion approach, the wall density or wave number displays reentrant phenomena when the ratio of the next-nearest-neighbor and nearest-neighbor interactions is greater than 1/2.
The two-dimensional axial-next-nearest-neighbor Ising model is studied numerically using finite-lattice results. The row-to-row transfer matrix is mapped into a simple one-dimensional spin-12\frac{1}{2}21​ quantum Hamiltonian. The leading eigenvalues and the corresponding eigenvectors are calculated numerically for chains of finite lengths. Using a finite-lattice renormalization-group transformation the high- (low-) temperature region is analyzed with the use of the Hamiltonian for the original (dual) model. Although there is no clear evidence for a massless phase, the phase boundaries obtained in this way are a strong indication of the presence of a sinusoidal phase between the paramagnetic and the 〈2〉〈2〉〈2〉 antiphase, in fair agreement with previous analytical results.