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Journal of Physics C Solid State Physics

Critical behaviour of the Ising model with a transverse field

J. Oitmaa, Michael Plischke

Published Jun 14, 1976• 19 citations
Publisher

Abstract

Ten terms have been obtained in the high-temperature series for the fluctuation in long-range order of the Ising model in a transverse magnetic field on the three cubic lattices. This model is relevant to real materials such as singlet-singlet magnetic systems, hydrogen-bonded ferroelectrics and cooperative Jahn-Teller systems. The critical temperature of the system was obtained over essentially the entire range of the transverse field for which a phase transition exists. No crossover to the expected d=4 critical behaviour was observed at T=0.

DOI
10.1088/0022-3719/9/11/017

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Ten terms have been obtained in the high-temperature series for the fluctuation in long-range order of the Ising model in a transverse magnetic field on the three cubic lattices. This model is relevant to real materials such as singlet-singlet magnetic systems, hydrogen-bonded ferroelectrics and cooperative Jahn-Teller systems. The critical temperature of the system was obtained over essentially the entire range of the transverse field for which a phase transition exists. No crossover to the expected d=4 critical behaviour was observed at T=0.
The Ising model (exchange constant J) with a transverse field Gamma can be regarded as a simple model of hydrogen bonded ferroelectrics and of certain magnetic systems. The transition temperature T C decreases as Gamma increases and falls to zero at a critical field Gamma C . This transition is studied in the quadratic and simple cubic lattices with nearest neighbour interactions by a power series expansion of the susceptibility at high T up to fifth order in (J/kT) to all orders in ( Gamma /kT). The coefficients are calculated by diagram counting and time ordering perturbation methods. The critical Gamma against T curve is derived and the critical exponent gamma of the susceptibility. This appears to vary smoothly with Gamma but it is argued that this is an artifact due to the use of a short series and that the critical exponents are probably independent of Gamma as long as T C remains finite but change discontinuity at T C =0.
Both weak-coupling and strong-coupling series expansions are calculated for the Ising model in a transverse field at zero temperature in three dimensions. Series are obtained for the ground-state energy, the magnetization, the susceptibility, and the energy gaps, on the simple cubic, the body-centred cubic and the face-centred cubic lattices. The analysis of the critical behaviour is consistent with the behaviour predicted by renormalization group theory for the four-dimensional simple Ising model. There is a remarkable degree of universality between all three lattices.
The critical behaviour of the susceptibility of the Ising model with a transverse field is investigated by means of the perturbation expansions. Exact expressions for the coefficients up to seventh order in J / k T are given as functions of Δ/ k T for the quadratic, triangular, simple cubic, b.c.c. and f.c.c. lattices. The universality of the critical index is concluded from the analysis of the coefficients and the critical temperature is obtained as function of the transverse field for the five kinds of lattices.
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.