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Journal of the Physical Society of Japan

Coherent Anomaly in Kosterlitz-Thouless-Type Transition in theS=1/2 XXZ Chain at the Ground State

Yoshihiko Nonomura, Masuo Suzuki

Published Nov 15, 1993• 5 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.62.3774

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Journal of the Physical Society of Japan•Dec 15, 1988•12 citations
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.
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.
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.
A series of effective-field approximations are formulated for the Kosterlitz-Thouless transition in the sine-Gordon model by means of the cumulant expansion and the variational method. Effective-field essential singularities xi approximately exp( xi n (K c (n) -K)-1) are obtained for the correlation length as first derived by Saito in a single approximation. However, a systematic variance of the effective-field critical coefficient xi n approximately 2ln(J/2y 0 )/(n+1) pi is found when the order of approximation n increases. The true critical exponent nu of the Kosterlitz-Thouless transition is thus revealed to be less than the effective-field exponent nu 0 , nu < nu 0 =1, from Suzuki's coherent-anomaly method. The phase transition in the two-dimensional XY model is studied from its relation to the sine-Gordon model. The critical exponent eta c , of the spin-spin correlation function at the critical point is found to be eta c =1/(4+1/ pi 2 ).
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 new scheme of the coherent-anomaly method (CAM) is proposed to study critical phenomena in the models for which a mean-field description gives spurious first-order phase transition. A canonical series of mean-field-type approximations are constructed so that the spurious discontinuity should vanish asymptotically as the approximate critical temperature approachs the true value. The true value of the critical exponents β and γ are related to the coherent-anomaly exponents defined among the classical approximations. The formulation is demonstrated in the two-dimensional q -state Potts models for q =3 and 4. The result shows that the present method enables us to estimate the critical exponents with high accuracy by using the date of the cluster-mean-field approximations.