Abstract
We exploit mappings between quantum and classical systems in order to obtain a class of two-dimensional classical systems characterised by long-range interactions and with critical properties equivalent to those of the class of one-dimensional quantum systems treated by the authors in a previous publication. In particular, we use three approaches: the Trotter-Suzuki mapping, the method of coherent states, and a calculation based on commuting the quantum Hamiltonian with the transfer matrix of a classical system. This enables us to establish universality of certain critical phenomena by extension from the results in the companion paper for the classical systems identified.
Highlights
Mappings between statistical mechanical models have provided new pathways to compute thermodynamic properties of systems which were previously intractable [1,2,3]
In this paper we exploit these quantum to classical (QC) mappings for the opposite reason: to take advantage of known ground state critical behaviour in a general class of quantum spin chains with long-range interactions to determine the finite temperature critical properties of an equivalent class of classical spin systems
In a companion paper to appear in [6] we computed the critical exponents s, ], and z, corresponding to the energy gap, correlation length, and dynamic exponent, respectively, for a class of quantum spin chains, establishing universality for this class of systems
Summary
Mappings between statistical mechanical models have provided new pathways to compute thermodynamic properties of systems which were previously intractable [1,2,3]. Suzuki [3] introduced a powerful method based on Trotter’s formula Another technique exploits the fact that if the quantum Hamiltonian commutes with the transfer matrix of a classical system, they are equivalent. Igloi and Lajko [16] showed that the quantum Ising model with site-dependent coupling parameters in a transverse magnetic field is equivalent to an Ising model on a square lattice with a diagonally layered structure Some of these systems are not translation invariant, but they all have nearest neighbour spin-spin interactions; to our knowledge there is no system with long-range interactions for which classical-quantum equivalence has previously been proved. (iii) The simultaneous diagonalisation of the quantum Hamiltonian and the transfer matrix for the classical system (Section 4)
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