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International Journal of Modern Physics B

SOME EXACT CALCULATIONS OF AN ALTERNATING TRANSVERSE ISING MODEL

HIDENORI SUZUKI, Masuo Suzuki

Published Oct 20, 2002• 5 citations
Publisher

Abstract

The alternating transverse Ising model with A and B sublattices is solved exactly in one and two dimensions, when a transverse field applied only to the A sublattice. The critical point of the honeycomb lattice is given as a function of the alternating transverse field. Moreover, the zero-temperature phase transition in the one-dimensional model with another alternating transverse field is discussed rigorously.

DOI
10.1142/s0217979202014826

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The alternating transverse Ising model with A and B sublattices is solved exactly in one and two dimensions, when a transverse field applied only to the A sublattice. The critical point of the honeycomb lattice is given as a function of the alternating transverse field. Moreover, the zero-temperature phase transition in the one-dimensional model with another alternating transverse field is discussed rigorously.
This chapter is devoted to a discussion of quantum phase transitions in regularly alternating spin-1/2 Ising chain in a transverse field. After recalling some generally-known topics of the classical (temperature-driven) phase transition theory and some basic concepts of the quantum phase transition theory I pass to the statistical mechanics calculations for a one-dimensional spin-1/2 Ising model in a transverse field, which is the simplest possible system exhibiting the continuous quantum phase transition. The essential tool for these calculations is the Jordan-Wigner fermionization. The latter technique being completed by the continued fraction approach permits to obtain analytically the thermodynamic quantities for a `slightly complicated' model in which the intersite exchange interactions and on-site fields vary regularly along a chain. Rigorous analytical results for the ground-state and thermodynamic quantities, as well as exact numerical data for the spin correlations computed for long chains (up to a few thousand sites) demonstrate how the regularly alternating bonds/fields effect the quantum phase transition. I discuss in detail the case of period 2, swiftly sketch the case of period 3 and finally summarize emphasizing the effects of periodically modulated Hamiltonian parameters on quantum phase transitions in the transverse Ising chain and in some related models.
The spin 3/2 transverse Ising model with anisotropic crystal field on honeycomb lattice (z=3) has been studied by using the effective field theory with correlations. The general expressions for evaluating the second order phase transition temperatures are obtained. The results show that both the transverse field and crystal field have important effects on the phase diagram of the Ising system. The transition temperatures do not always fall to zero when Ω/|J|0 457. They approach a constant value 0 525 for large negative D(D→-∞) without transverse field (Ω=0). On the other hand, it is shown that all phase transformations are second order.
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.
A newly developed zero-temperature real-space renormalization-group method is applied to the quantum Ising model with a transverse field on different two-dimensional lattices: hexagonal, triangular, and quadratic. The order parameter, spin-spin correlation functions, the critical fields, and the critical exponents describing the low-field-large-field transition are calculated. The influence of size and shape of the spin blocks used in the iterative process is discussed. The results for triangular and hexagonal lattices are new; for the quadratic lattice the calculations show an encouraging agreement with previous series-expansion results. The expected equivalence to the critical behavior of the three-dimensional Ising model is discussed: A reasonable agreement is obtained for the hexagonal lattice.
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.
A zero-temperature real-space renormalization-group method is presented and applied to the quantum Ising model with a transverse field in one dimension. The transition between the low-field and high-field regimes is studied. Magnetization components, spin correlation functions, and critical exponents are derived and checked against the exact results. It is shown that increasing the size of the blocks in the iterative procedure yields more accurate results, especially for the critical "magnetic" exponents near the transition.