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arXiv:1209.2442 · Europhysics Letters (EPL)

Equivalence between the two-dimensional Ising model and the quantum XY chain with randomness and with open boundary

Kazuhiko Minami

Published Oct 23, 2014• 4 citations
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Abstract

The equivalence between the two-dimensional Ising model and the one-dimensional quantum XY model is generalized to the cases with alternating/random interactions and with periodic/free boundary conditions. It is proved that the eigenstate for the maximum eigenvalue of the transfer matrix of the two-dimensional Ising model corresponds to the ground state of the Hamiltonian of the one-dimensional XY model, which certifies direct relations between the physical quantities of the two models.

DOI
10.1209/0295-5075/108/30001

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The equivalence between the two-dimensional Ising model and the one-dimensional quantum XY model is generalized to the cases with alternating/random interactions and with periodic/free boundary conditions. It is proved that the eigenstate for the maximum eigenvalue of the transfer matrix of the two-dimensional Ising model corresponds to the ground state of the Hamiltonian of the one-dimensional XY model, which certifies direct relations between the physical quantities of the two models.
By relating a transfer matrix associated with a classical system to the Hamiltonian of its corresponding quantum mechanical system, it is proved that the two-dimensional Ising model in the absence of a magnetic field is equivalent to the ground state of the linear XY-model in the presence of a magnetic field, under appropriate relations among coupling parameters appearing in the two Hamiltonians. Simple relations are established for spin correlations in both systems. Consequently, the Ising system and the ground state of the XY-model are shown to exhibit similar singularities with respect to temperature and a magnetic field, respectively. It is also proved that the two-dimensional dimer problem is equivalent to the ground state of a generalized XY-model, which is solved exactly in general.
We have performed an analytical study of quantum-classical equivalence for quantum XY -spin chains with arbitrary interactions to explore the classical counterpart of the factorizing magnetic fields that drive the system into a separable ground state. We demonstrate that the factorizing line in the parameter space of a quantum model is equivalent to the so-called natural boundary that emerges in mapping the quantum XY -model onto the two-dimensional classical Ising model. As a result, we show that the quantum systems with the non-factorizable ground state could not be mapped onto the classical Ising model. Based on the presented correspondence we suggest a promising method for obtaining the factorizing field of quantum systems through the commutation of the quantum Hamiltonian and the transfer matrix of the classical model.
The one-dimensional S=1 quantum spin system has been mapped onto the equivalent two-dimensional Ising model on the basis of the equivalence theorem of Suzuki. The classical representation of the partition function has been expressed in terms of the maximum eigenvalue of the transfer matrix and evaluated by means of the numerically exact transfer-matrix method. The physical meaning of two kinds of correlation functions in RSD and CBD has been made clear in terms of the concept of the maximum eigenstate. Calculations have been made for both ferromagnetic and antiferromagnetic Heisenberg chains with several single-site anisotropy values, both positive and negative. Good agreements are found with some existing numerical results.
Abstract The connection matrix of the Ising model on a ddd-dimensional hypercube is investigated. In addition to the interactions between the nearest neighbors, the interactions between the next-nearest neighbors are taken into account. For such a matrix, the exact relations for the eigenvalues and eigenvectors, which are reasonably simply expressed through the corresponding characteristics of the one-dimensional Ising model, are obtained. Both periodic and free boundary conditions are considered.
The eigenvalue problem for the transfer matrix of the two‐dimensional Ising lattice, with free boundaries perpendicular to the transfer direction, is solved. The mathematical mechanism which produces the asymptotic degeneracy of the eigenvalue spectrum below the bulk transition temperature is different from the case with cyclic boundary conditions. The results are used to calculate the two‐spin correlation function between two spins in the same edge of the lattice, and between two spins in opposite edges.