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Journal of Physics A Mathematical and General

Exact results for quantum chains with multisite interactions

Loïc Turban

Published Aug 21, 1985• 11 citations
Publisher

Abstract

Generalised spin-1/2 and Potts quantum chains with m- (n-) site interactions for the z(x) component of the site variables are shown to be self-dual. Through duality, with m arbitrary, n=2 versions of the models are transformed into m independent Ising or Potts chains in a transverse field, allowing one to get the exact values of the critical exponents for any m.

DOI
10.1088/0305-4470/18/12/029

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Journal of Physics A Mathematical and General•May 1, 1982•0 citations
Generalised spin-1/2 and Potts quantum chains with m- (n-) site interactions for the z(x) component of the site variables are shown to be self-dual. Through duality, with m arbitrary, n=2 versions of the models are transformed into m independent Ising or Potts chains in a transverse field, allowing one to get the exact values of the critical exponents for any m.
Abstract A one-dimensional (1D) q -state Potts model with N sites, m -site interaction K in a field H is studied for arbitrary values of m . Exact results for the partition function and the two-point correlation function are obtained at H = 0. The system in a field is shown to be self-dual. Using a change of Potts variables, it is mapped onto a standard 2D Potts model, with first-neighbour interactions K and H , on a cylinder with helical boundary conditions (BC). The 2D system has a length N / m and a transverse size m . Thus the Potts chain with multi-site interactions is expected to develop a 2D critical singularity along the self-duality line, ( e q K − 1 ) ( e q H − 1 ) = q , when N / m → ∞ and m → ∞ .
Considers the pure Z(N) spin systems (including the standard Ising and Potts models) as well as generalised gauge systems (plaquettes or more complex simplex) in d-dimensional hypercubic lattices. These models are self-dual, and the authors show how this duality can be thought of as a series-parallel transformation. The simplicity of the equations enables conjectures to be made on the approximate critical frontier of the diluted version of the above systems, including some particular asymptotic behaviours which are exact. As an illustration the d=2 diluted Z(4) spin system is discussed in some detail: for those regions where exact results are available the agreement is satisfactory.
Introduces a family of D-dimensional q-state quantum Potts models with multi-spin interactions, which are self-dual for any D. These models are introduced using their (d=D+1)-dimensional classical formulations and the transfer matrix technique. The self-duality is proved on the quantum Hamiltonian. In the limit q to infinity the ground state energy is obtained exactly and the phase transition is first-order. A 1/q expansion allows one to get an approximate expression for the line in the (q,D)-plane separating regions where the transition changes from first- to second-order.