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Renormalization and scaling of a two-dimensional Ising system in a transverse field at Tc>0

Stella, A.L., Toigo, F.

Published Jan 1, 1977• 0 citations
OpenAlex

Abstract

A quantum generalization of the Niemeijer and van Leeuwen (N-vL) renormalization group transformation is constructed which allows to study the critical properties (at Tc>0) of a two-dimensional Ising system with a transverse field on a triangular lattice. Explicit calculations are performed in a second order cumulant expansion. Only one fixed point is found corresponding to the same Ising-like hamiltonian given by N-vl. The linearized transformation has a zero eigenvalue associated with Γ, the transverse field strength. The critical properties of the system are briefly discussed; in particular we show that the singular behaviour of the transverse susceptibility at Γ = 0, which turns out to be of the same kind as that of the energy density, is explained by the existence of such a zero eigenvalue. Our arguments suggest a natural extension of this result to three dimensions.

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A quantum generalization of the Niemeijer and van Leeuwen (N-vL) renormalization group transformation is constructed which allows to study the critical properties (at Tc>0) of a two-dimensional Ising system with a transverse field on a triangular lattice. Explicit calculations are performed in a second order cumulant expansion. Only one fixed point is found corresponding to the same Ising-like hamiltonian given by N-vl. The linearized transformation has a zero eigenvalue associated with Γ, the transverse field strength. The critical properties of the system are briefly discussed; in particular we show that the singular behaviour of the transverse susceptibility at Γ = 0, which turns out to be of the same kind as that of the energy density, is explained by the existence of such a zero eigenvalue. Our arguments suggest a natural extension of this result to three dimensions.
The Niemeijer-Van Leeuwen cumulant expansion is used in the renormalization-group analysis of the critical-point behavior of the Ising model. Second- and third-order results are obtained for the simple cubic and square lattices, respectively, for a variety of cell sizes, using a parameter-dependent renormalization transformation introduced by Kadanoff and Houghton. Although the results in two dimensions are in general good, there is no evidence of any convergence, with the best values for the critical exponents and critical temperature being given by the second-order theory. In three dimensions the values found for the critical temperature and magnetic scaling index for a cell of 27 spins are reasonable, but the corresponding thermal scaling index is quite poor.
The combined Suzuki-Trotter decomposition and Niemeijer--van Leuween real-space renormalization-group techniques are used to study the critical properties of a two-dimensional Ising system with a transverse field. The inverse critical temperature as a function of the external field and the temperature dependence of the transverse component of the magnetization are found. It is also shown that any real-space renormalization-group procedure based on the simple generalization of the Niemeijer--van Leeuwen majority rule for one of the components of the total-cell spin does not preserve the symmetry of the quantum spin space.
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.
The equivalence of the critical behavior of the ddd-dimensional Ising model at a finite temperature to that of a ddd- 1 dimensional system at zero temperature as a function of a transverse magnetic field is used to calculate the critical exponents for the three-dimensional model by applying the Niemeijer-Van Leeuwen renormalization-group method to the two-dimensional problem.
The field theoretical renormalisation group method, which correctly describes the critical as well as the multicritical (at zero temperature) behaviours of the quantum mechanical d-dimensional Ising model in a transverse magnetic field, is proposed. Due to the introduction of the additional renormalisation the singularities associated with zero temperature have been removed. The crossover scaling form of the inverse correlation length and longitudinal susceptibility are considered for the following spatial dimensions d:2<d<3, d=3 and 3<d<4. For 2<d<3, d=3 and 3<d<4 the calculations are performed to the first order in 3-d and 4-d, respectively.