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Physical review. B, Condensed matter

Spectral and magnetic transitions in the quantum Ising model

Indubala I. Satija

Published Apr 1, 1990• 16 citations
Publisher

Abstract

The quantum Ising model in a transverse field is studied where the exchange interaction is a modulating function of sites with many harmonics. The magnetically disordered phase is found to exhibit a ``mixed'' spectrum containing both critical and localized states, while the magnetically ordered phase has a pure spectrum with all states localized. Therefore, the spectral transition, although broadened, occurs simulataneously with the magnetic transition, and the magnetic disorder is accompanied by spectral disorder.

DOI
10.1103/physrevb.41.7235

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The quantum Ising model in a transverse field is studied where the exchange interaction is a modulating function of sites with many harmonics. The magnetically disordered phase is found to exhibit a ``mixed'' spectrum containing both critical and localized states, while the magnetically ordered phase has a pure spectrum with all states localized. Therefore, the spectral transition, although broadened, occurs simulataneously with the magnetic transition, and the magnetic disorder is accompanied by spectral disorder.
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The quantum antiferromagnetic spin-1/2 Ising model on a triangular lattice and analogous fully frustrated Ising model on a square lattice with quantum fluctuation induced by the application of the transverse magnetic field are studied at finite temperatures by constructing an exact mapping onto a purely classical model with a more complex interaction. It is shown that in weak fields the temperatures of the phase transitions separating the critical phase from the ordered and disordered phases in both models are proportional to the magnitude of the field.
The quantum Ising model is perhaps the simplest possible model of a quantum magnetic material. Despite its simplicity, its versatility and wide range of applications, from quantum computation, to combinatorial optimization, to biophysics, make it one of the most important models of modern physics. In this thesis, we develop a general framework for studying quantum Ising systems with an arbitrary single ion Hamiltonian, with emphasis on the effects of quantum fluctuations, and the quantum phase transition between paramagnetic and ferromagnetic states that occurs when a magnetic field is applied transverse to the easy axis of the system. The magnetic insulating crystal LiHoF₄ is a physical realization of the quantum Ising model, with the additional features that the dominant coupling between spins is the long range dipolar interaction, and each electronic spin is strongly coupled to a nuclear degree of freedom. These nuclear degrees of freedom constitute a spin bath environment acting on the system. In this thesis, we present an effective low temperature Hamiltonian for LiHoF₄ that incorporates both these features, and we analyze the effects of the nuclear spin bath on the system. We find the lowest energy crystal field excitation in the system is gapped at the quantum critical point by the presence of the nuclear spins, with spectral weight being transferred down to a lower energy electronuclear mode that fully softens to zero at the quantum critical point. Furthermore, we present a toy model, the spin half spin half model, that illustrates the effects of an anisotropic hyperfine interaction on a quantum Ising system. We find the critical transverse field is increased when the longitudinal hyperfine coupling is dominant, as well as an enhancement of both the longitudinal electronic susceptibility and an applied longitudinal field. In addition, we present a field theoretic formalism for incorporating the effects of fluctuations beyond the random phase approximation in general quantum Ising systems. We find that any regular on site interaction, such as a nuclear spin bath, does not fundamentally alter the critical properties of a quantum Ising system. This formalism is used to calculate corrections to the magnetization of LiHoF₄.
We study the 1D quasiperiodic quantum Ising model in a transverse field, which exhibits long-range order above a critical coupling and describes the phase diagram of the 2D classical Ising-model quasiperiodic in one direction. We numerically determine the correlation length, magnetization, and energy spectrum of this system. The site dependency of the magnetization is described by a new order parameter, the width of the magnetization. The scalings \ensuremath{\delta} and f(\ensuremath{\alpha}) associated with the Cantor spectrum are found to exhibit special features at the onset of long-range order.
We study the one-dimensional quantum Ising model in a transverse magnetic field where the exchange couplings are ordered according to the Thue-Morse (TM) sequence. At zero temperature, this model is equivalent to a two-dimensional classical Ising model in a magnetic field with TM aperiodicity along one direction. We compute the order parameter (magnetization) of the chain and the scaling behavior of the energy spectrum when the system undergoes a phase transition. Analogous to the quasiperiodic (QP) quantum Ising chain, the onset of long-range order is signaled by a nonanaliticity in the exponent \ensuremath{\delta} which describes the scaling of the total bandwidth with the size of the chain. The critical spin-coupling can be computed analytically and it is found to be lower than the QP case. Furthermore, the energy bands are found to be narrower than the corresponding QP chain. The former and latter results are consistent with the fact that the present structure has a degree of ordering intermediate between QP and random.