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Journal of Physics C Solid State Physics

The spin-S quantum Ising model at T=O

G. Kamieniarz, Leon Kowalewski, Włodzimierz Piechocki

Published Jul 10, 1983• 5 citations
Publisher

Abstract

The Ising model with a transverse field for a general spin S is investigated within the framework of the Green function method in the paramagnetic region at T=O. The analysis of self-consistent equations gives a description of soft-mode phase transitions as well as extrapolated values of the critical fields and critical energy gap exponents.

DOI
10.1088/0022-3719/16/19/016

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Physical Review B•Oct 18, 2006•11 citations
The Ising model with a transverse field for a general spin S is investigated within the framework of the Green function method in the paramagnetic region at T=O. The analysis of self-consistent equations gives a description of soft-mode phase transitions as well as extrapolated values of the critical fields and critical energy gap exponents.
An Ising model in a transverse field with general spin S is considered within RPA Green-function theory in the paramagnetic region. Expressions for the temperature renormalised collective excitation spectrum with a gap decreasing as the exchange interaction increases, the transverse correlation function, and the critical exchange-field ratio for magnetic ordering, are derived. Zero-temperature expansions are given in the limit of weak interaction. Critical values at T=0K are calculated for several values of S and an asymptotical expression for S>>1 is given.
Susceptibility of the transverse field Ising model on the square lattice is calculated numerically in the paramagnetic phase in a wide range of temperatures and transverse fields. An expression with one constant πππ, that determines both the critical exponent γγγ and the critical transverse field, compellingly represents the data asymptotically near the quantum critical point, except for a narrow classical region close to the phase transition line, and shows two crossovers as temperature varies.
The phase transition in the Ising model has been studied as a function of an applied transverse field, by Green's function and series-expansion methods. The critical indices are probably independent of the applied field except at T=0T=0T=0 where they appear to be related to those of the Ising model in one higher dimension.
The ground state of an XY\mathrm{XY}XY model changes at a critical transverse field, above which there is a gap from the fully polarized ground state to the first excitation. We investigate the phase transition from this ``Mott insulator'' phase as the transverse field approaches its critical value. We calculate exactly some Green's functions and critical exponents for ddd-dimensional ferromagnetic lattices and for the Sherrington-Kirkpatrick spin-glass model. Finite-dimensional spin glasses are also discussed.
In this work we investigate the quantum phase transitions at zero temperature of the one-dimensional transverse Ising model with an extra term containing four-spin interactions. The competition between the energy couplings of the model leads to an interesting zero-temperature phase diagram. We use a modified Lanczos method to determine the ground state and the first excited state energies of the system, with sizes of up to 20 spins. We apply finite size scaling to the energy gap to obtain the boundary region where ferromagnetic to paramagnetic transition takes place. We also find the critical exponent associated with the correlation length. We find a degenerate ⟨3,1⟩⟨3,1⟩⟨3,1⟩ phase region. The first-order transition boundary between this phase and the paramagnetic phase is determined by analyzing the behavior of the transverse spin susceptibility as the system moves from one region to the other.