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Physical review. B, Condensed matter

Hamiltonian limit of the two-dimensional axial next-nearest-neighbor Ising model: A many-body approach

Santiago D’Elía

Published Aug 1, 1992• 1 citation
Publisher

Abstract

The Hamiltonian limit of the two-dimensional axial next-nearest-neighbor Ising model is studied with use of an interacting-fermion technique. Details of the ferromagnetic side of the corresponding phase diagram are analyzed, especially the characterization of spatially modulated structures and the type of the phase transition.
DOI
10.1103/physrevb.46.3128

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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.
The three-dimensional axial next-nearest-neighbor Ising model is studied by a modified tensor product variational approach. A global phase diagram is constructed with numerous commensurate and incommensurate magnetic phases. The devil's stairs behavior for the model is confirmed. The wavelength of the spin modulated phases increases to infinity at the boundary with the ferromagnetic phase. Widths of the commensurate phases are considerably narrower than those calculated by mean-field approximations.
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.
The critical behavior at the phase transition of the ferromagnetic two-dimensional anisotropic Ising model with next-nearest neighbor (NNN) couplings in the presence of the field is determined using mainly Monte Carlo (MC) method. This method is used to investigate the phase diagram of the model and to verify the existence of a divergence at null temperature which often appears in two-dimensional systems. We analyze also the influence of the report of the NNN interactions [Formula: see text] and the magnetic field [Formula: see text] on the critical temperature of the system, and we show that the critical temperature depends on the magnetic field for positive values of the interaction. Finally, we have investigated other thermodynamical qualities such as the magnetic susceptibility [Formula: see text]. It has been shown that their thermal behavior depends qualitatively and quantitatively on the strength of NNN interactions and the magnetic field.