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Journal of Applied Physics

Phase transitions in (1+1)-dimensional transverse Ising models for spin S≥1/2

K. A. Penson, M. Kolb

Published Apr 15, 1985• 0 citations
Publisher

Abstract

The phase transitions in the ground state of (1+1)-dimensional Ising models in a transverse field are studied for spin S=1/2, 1, 3/2, and 2 using the finite-size scaling method. The critical fields, thermal and correlation exponents are calculated as a function of S. The critical fields are monotonic functions of S but the exponents are S-independent and take their values for S=1/2. A novel calculational method based on universality of ratios of finite-size correlation length amplitudes is proposed and tested. It results in a much improved accuracy of extrapolation procedures. Such an universality appears to be a consequence of recently postulated conformal invariance of correlation functions of a class of two-dimensional classical models.

DOI
10.1063/1.335135

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The phase transitions in the ground state of (1+1)-dimensional Ising models in a transverse field are studied for spin S=1/2, 1, 3/2, and 2 using the finite-size scaling method. The critical fields, thermal and correlation exponents are calculated as a function of S. The critical fields are monotonic functions of S but the exponents are S-independent and take their values for S=1/2. A novel calculational method based on universality of ratios of finite-size correlation length amplitudes is proposed and tested. It results in a much improved accuracy of extrapolation procedures. Such an universality appears to be a consequence of recently postulated conformal invariance of correlation functions of a class of two-dimensional classical models.
The phase transitions in the ground state of (1+1)-dimensional Ising models in a transverse field are investigated for spin S=12,1,32,and2S=\frac{1}{2}, 1, \frac{3}{2}, \mathrm{and} 2S=21​,1,23​,and2. The critical fields and "thermal" and correlation exponents are calculated as a function of SSS by the finite-size scaling method. The universality of the transition with respect to SSS is fully confirmed. Furthermore, we elaborate on a recent conjecture by Luck and others relating the correlation exponent $\ensuremath{\eta}$ to the amplitude of a correlation length for a two-dimensional (2D) classical isotropic system. The extension of this relation to quantum chains as extremely anisotropic 2D systems has been considered. We find that while no simple prescription can be given to calculate various correlation exponents from the amplitudes, the ratios of the amplitudes of different quantities still have universal character.
We analyze the fidelity per lattice site for two different ground states of the one-dimensional quantum Ising model in a transverse field near the critical point. It is found that, in the thermodynamic limit, the fidelity per lattice site is singular, and the derivative of its logarithmic function with respect to the transverse field strength is logarithmically divergent at the critical point. The scaling behavior is confirmed numerically by performing a finite-size scaling analysis for systems of different sizes, consistent with the conformal invariance at the critical point. This allows us to extract the correlation length critical exponent, which turns out to be universal in the sense that the correlation length critical exponent does not depend on either the anisotropic parameter or the transverse field strength.
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.
We study the quantum transition at T=0 in the spin-1/2 Ising spin glass in a transverse field in two dimensions. The world line path integral representation of this model corresponds to an effective classical system in 2+1 dimensions, which we study by Monte Carlo simulations. Values of the critical exponents are estimated by a finite-size scaling analysis. We find that the dynamical exponent, z, and the correlation length exponent, \ensuremath{\nu}, are given by z=1.5\ifmmode\pm\else\textpm\fi{}0.05 and \ensuremath{\nu}=1.0\ifmmode\pm\else\textpm\fi{}0.1. Both the linear and nonlinear susceptibility are found to diverge at the critical point.
We have adjusted the density-matrix renormalization method to handle two-dimensional systems of limited width. The key ingredient for this extension is the incorporation of symmetries in the method. The advantage of our approach is that we can force certain symmetry properties to the resulting ground-state wave function. Combining the results obtained for system sizes up to $30\ifmmode\times\else\texttimes\fi{}6$ and finite-size scaling, we derive the phase-transition point and the critical exponent for the gap in the Ising model in a transverse field on a two-dimensional square lattice.