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Studies in Applied Mathematics

Magnetic Susceptibility of the Random One‐ Dimensional Transverse Ising Model

Eytan Barouch, Barry M. McCoy

Published Mar 1, 1972• 6 citations
Publisher

Abstract

We study the 1‐ dimensional Ising model with random coupling constants in the presence of a transverse magnetic field h. In the nonrandom (translationally invariant) case it is known that there is a value of h for which the magnetic susceptibility diverges logarithmically. We demonstrate that in the random case this logarithmic divergence is smoothed out into an infinitely differentiable essential singularity.

DOI
10.1002/sapm197251157

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We study the 1‐ dimensional Ising model with random coupling constants in the presence of a transverse magnetic field h. In the nonrandom (translationally invariant) case it is known that there is a value of h for which the magnetic susceptibility diverges logarithmically. We demonstrate that in the random case this logarithmic divergence is smoothed out into an infinitely differentiable essential singularity.
We study the quantum phase transition in the two-dimensional random Ising model in a transverse field by Monte Carlo simulations. We find results similar to those known analytically in one dimension. At the critical point the dynamical exponent is infinite and the typical correlation function decays with a stretched exponential dependence on distance. Away from the critical point there are Griffiths-McCoy singularities, characterized by a single continuously varying exponent, ${z}^{\ensuremath{'}}$, which diverges at the critical point, as in one dimension. Consequently, the zero temperature susceptibility diverges for a range of parameters about the transition.
We consider the effect of a random longitudinal field on the Ising model in a transverse magnetic field. For spatial dimension d>2d>2d>2, there is at low strength of randomness and transverse field a phase with true long-range order that is destroyed at higher values of the randomness or transverse field. The properties of the quantum phase transition at zero temperature are controlled by a fixed point with no quantum fluctuations. This fixed point also controls the classical finite-temperature phase transition in this model. Many critical properties of the quantum transition are therefore identical to those of the classical transition. In particular, we argue that the dynamical scaling is activated, i.e., the logarithm of the diverging time scale rises as a power of the diverging length scale.
Abstract The magnetic analog of the Grüneisen parameter, that is, the magnetocaloric effect, is a valuable tool for studying field‐tuned quantum phase transitions. We determine the magnetic Grüneisen parameter of the one‐dimensional random transverse‐field Ising model, focusing on its low‐temperature behavior at the exotic infinite‐randomness quantum critical point and in the associated quantum Griffiths phases. We present extensive numerical simulations showing that the magnetic Grüneisen parameter diverges logarithmically with decreasing temperature in the quantum Griffiths phase. It changes sign right at criticality. These results confirm a recent strong‐disorder renormalization group theory. We also compare our findings to the behavior of the clean transverse‐field Ising chain.
We analyse the one dimensional X-Y model with random coupling constants in a z-axis magnetic field. For the non-random system at zero temperature, there is a thermodynamic singularity at finite field, corresponding to a change in the type of spin ordering. Some thermodynamic quantities diverge and the correlation length becomes infinite. For the random system however, we can show that all the thermodynamic quantities are smoothly varying and the correlation length is bounded above at the critical point.
The zero-field transverse susceptibility of a one-dimensional random bond Ising model with alternating spin 1/2 and S is exactly obtained. The integral form of the susceptibility is written. The thermal curve of the susceptibility becomes irregular when weak interactions appear with a small probability. The leading-order terms of an expansion, in terms of the moment of the distribution of a random variable, are obtained. The first-order term vanishes and the second-order term is written explicitly. The expansion up to infinite order is exactly obtained in the case of uniform spin S = 1/2. The classical limit S → ∞ is also considered. It is found that the low-temperature limit T → 0 of the susceptibility is independent of S .
We study critical behaviors of the reduced fidelity susceptibility for two neighboring sites in the one-dimensional transverse field Ising model. It is found that the divergent behaviors of the susceptibility take the form of square of logarithm, in contrast with the global ground-state fidelity susceptibility which is power divergence. In order to perform a scaling analysis, we take the square root of the susceptibility and determine the scaling exponent analytically and the result is further confirmed by numerical calculations.