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Nuclear Physics B

Application of the projector Monte Carlo method to the transfer matrix of the classical XY model

Daniel R. Stump

Published Jan 1, 1986• 4 citations
Publisher
DOI
10.1016/0550-3213(86)90409-8

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Classical 1D XY model with next-nearest-neighbor couplings: Transfer matrix solution

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We discuss the critical behavior of several models, all related to the 19-vertex model. These models combine Ising and Gaussian degrees of freedom and have central charge c=3/2 at the critical point. We present arguments why the frustrated XY model, which also has Gaussian and Ising degrees of freedom, might fall into a different universality class. We use the Monte Carlo transfer-matrix method of Bl\"ote and Nightingale to obtain accurate numerical results for the central charge and exponents of the frustrated XY model. In particular, we test the prediction for the central charge c to be equal to 3/2 also in this model. We find c=1.66\ifmmode\pm\else\textpm\fi{}0.04 for the central charge of the frustrated XY model at the critical point. It is not clear at this stage to which universality class this critical behavior belongs.
An improved transfer matrix method is proposed for calculating the partition function of quantum spin systems, which is expressed by the M-th approximant of the Suzuki-Trotter transformation with Trotter number M. The point is to incorporate the symmetry property of the Trotter subsystem in calculation. The size of our transfer matrix is reduced very much. Thermodynamics is described by the maximum eigenvalue of the transfer matrix. Our method is applied to the XY model with spin 1/2. We consider the cases of pair and tri spin decompositions with Trotter number up to 4 (M≤4). We can do our calculation analytically in those cases. In the pair spin decomposition we find results which have been obtained by the Monte Carlo simulation and by a transfer matrix method by computer. We find nice results in the tri spin decomposition. Our result in the case M=4 is as good as the result in M=8 by the previous Monte Carlo simulation on comparing them with the exact result of Katsura.
The principle and the efficiency of the Monte Carlo transfer-matrix algorithm are discussed. Enhancements of this algorithm are illustrated by applications to several phase transitions in lattice spin models. We demonstrate how the statistical noise can be reduced considerably by a similarity transformation of the transfer matrix using a variational estimate of its leading eigenvector, in analogy with a common practice in various quantum Monte Carlo techniques. Here we take the two-dimensional coupled XY-Ising model as an example. Furthermore, we calculate interface free energies of finite three-dimensional O(n) models, for the three cases n=1, 2, and 3. Application of finite-size scaling to the numerical results yields estimates of the critical points of these three models. The statistical precision of the estimates is satisfactory for the modest amount of computer time spent. \textcopyright{} 1996 The American Physical Society.