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Journal of Physics A Mathematical and General

CAM approach to the ground-state phase transition in the two-dimensional transverse Ising model

Yoshihiko Nonomura, Masuo Suzuki

Published Nov 7, 1992• 5 citations
Publisher

Abstract

The double-cluster approximation (DCA) and the multi-effective-field approximation (MEFA) are formulated for the two-dimensional S=1/2 transverse Ising model on a square lattice in the ground state. These two cluster-effective-field approximations expand the sizes of clusters effectively. Combining some suitable series of these approximations with the coherent-anomaly method (CAM), the authors have analysed the critical phenomenon of this model. They have obtained g * c =1.528+or-0.016 and gamma =1.25+or-0.08 using the DCA, and gamma =1.26+or-0.10 using the MEFA. These estimates are consistent with the other results, g * c approximately=1.52 and gamma approximately=1.24. Their scheme is expected to be useful in the study of frustrated systems, because various shapes of clusters can be used in their calculations.

DOI
10.1088/0305-4470/25/21/008

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The double-cluster approximation (DCA) and the multi-effective-field approximation (MEFA) are formulated for the two-dimensional S=1/2 transverse Ising model on a square lattice in the ground state. These two cluster-effective-field approximations expand the sizes of clusters effectively. Combining some suitable series of these approximations with the coherent-anomaly method (CAM), the authors have analysed the critical phenomenon of this model. They have obtained g * c =1.528+or-0.016 and gamma =1.25+or-0.08 using the DCA, and gamma =1.26+or-0.10 using the MEFA. These estimates are consistent with the other results, g * c approximately=1.52 and gamma approximately=1.24. Their scheme is expected to be useful in the study of frustrated systems, because various shapes of clusters can be used in their calculations.
A cluster-effective-field theory, called the double-cluster approximation, is formulated for quantum spin systems at T=0. Combining this approximation with the coherent-anomaly method (CAM), critical phenomena in such systems can be analysed. To test their theory, the authors have estimated the critical points and critical exponents of the one-dimensional S=1/2 transverse Ising model and the one-dimensional S=1/2 XZ model. The estimates are in good agreement with the exact solutions. These results show that the CAM approach is useful in treating zero-temperature phase transitions in quantum spin systems.
How to impose boundary conditions is crucial to finite-cluster calculations of quantum spin systems, but only open-boundary clusters have been used in cluster-effective-field approximations up to now. In the present paper, periodic-boundary clusters are also considered for the formulation of cluster-effective-field approximations in frustrated quantum spin systems. Namely, the open-boundary-condition double-cluster approximation (OBC-DCA) and the periodic-boundary-condition double-cluster approximation (PBC-DCA) are applied to the one-dimensional S= 1/2 frustrated XXZ model at the ground state. These two approximations are compared using the coherent-anomaly method (CAM). Within the limitation of cluster sizes in exact-diagonalization calculations, the PBC-DCA can reproduce the true phase boundary of the Neel-dimer transition. On the other hand, the OBC-DCA severely underestimates the existence of magnetic orders as quantum fluctuation is increased. These findings suggest that previous cluster-effective-field studies based on open-boundary clusters should be reconsidered.
We have formulated the Bethe-like approximation for the one-dimensional S=12S=\frac{1}{2}S=21​ transverse Ising model in the ground state, and estimated the critical phenomenon of its correlation function using the coherent-anomaly method (CAM). We have calculated up to the 21-spin approximations, and obtained g c * =0.5000±0.0006, ν=1.01±0.01 and ην=0.26±0.01.
Two systematic series of multi-body effective field approximations are constructed and the CAM analyses are performed for the Ising mode on the two-dimensional square lattice. In the first series a two-body effective field is introduced in addition to the usual one-body one and a set of strips are used. The critical temperature and the critical index of the susceptibility are estimated as T* c approximately=2.2697 J/k B and gamma approximately=1.749. In the second series only a three-line strip is used and applied with increasing numbers of multi-body effective fields. The expected CAM scaling behaviour has also been observed.
A Kosterlitz-Thouless (KT)-type transition in the one-dimensional S =1/2 XXZ model at the ground state is analyzed using the double-cluster approximation (DCA) and the coherent-anomaly method (CAM). It is numerically shown that the periodic-boundary condition (PBC) is suitable for the construction of a systematic series of cluster approximations in quantum antiferromagnets. The transition point and the exponent of the essential singularity are estimated as g c * ≃0.9969 and σ≃0.527, which are in good agreement with the exact values, g c * =1 and σ=0.5, respectively. Such accuracy of estimation is due to the consideration of a power-law factor in front of the exponential singularity of \barχ, which had been neglected until now.
The coherent-anomaly method proposed by one of the authors (M.S.) combined with a transfer-matrix version of the double-cluster approximation is applied to antiferromagnetic Ising models. We determined both the critical temperature and the critical exponent γ as functions of an external magnetic field. For a square lattice, our results are in a good agreement with the ones obtained recently by the phenomenological renormalization method. For triangle and simple cubic lattices, our results are less definite but still in a qualitative agreement with values predicted by other methods.