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Journal of Physics A Mathematical and General

Cluster-effective-field approximations in frustrated quantum spin systems: CAM analysis of Neel-dimer transition in the S= 1/2 frustrated XXZ chain at the ground state

Yoshihiko Nonomura, Masuo Suzuki

Published Feb 21, 1994• 9 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/27/4/009

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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 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 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.
For the quantum spin clusters of tens of sites, this paper calculates the total spin of ground state by exact diagonalization. In the non frustrated case, the ground state total spin takes not the lowest possible total spin but a higher value. It means the existence of Neel order in the ground state even for the clusters of nano scale. If introducing frustrations into clusters, it can induce a quantum phase transition from Neel order to spin disorder. This transition can be described and marked through the change of the ground state total spin. Calculations show that it should be the first order transition since the existence of level crossing at the critical J 2/J 1. From the numerical results of clusters, it can be inferred that for the infinite two dimensional J 1 J 2 model, the critical J 2/J 1 should be equal to 0 376 2 (±0.000 2). This result is more accurate than the previous data by analytical approximations and scaling analysis based on finite systems.
We provide microscopic diagrammatic derivations of the molecular coherent potential approximation (MCA) and dynamical cluster approximation (DCA) and show that both are $\ensuremath{\Phi}$ derivable. The MCA (DCA) maps the lattice onto a self-consistently embedded cluster with open (periodic) boundary conditions, and therefore violates (preserves) the translational symmetry of the original lattice. As a consequence of the boundary conditions, the MCA (DCA) converges slowly (quickly) with corrections O(1/Lc)\mathcal{O}{(1/L}_{c})O(1/Lc​) [O(1/Lc2)],[\mathcal{O}{(1/L}_{c}^{2})],[O(1/Lc2​)], where Lc{L}_{c}Lc​ is the linear size of the cluster. These analytical results are demonstrated numerically for the one-dimensional symmetric Falicov-Kimball model.