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Journal of Physics C Solid State Physics

Self-dual Ising chain in a transverse field with multispin interactions

Loïc Turban

Published Feb 10, 1982• 48 citations
Publisher

Abstract

A class of one-dimensional Ising models in a transverse field with multispin interactions is introduced and shown to be self-dual, thus allowing the determination of the critical point when unique.
DOI
10.1088/0022-3719/15/4/006

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Abstract We study the spin-1/2 Ising chain with multispin interactions K involving the product of m successive spins, for general values of m . Using a change of spin variables the zero-field partition function of a finite chain is obtained for free and periodic boundary conditions and we calculate the two-spin correlation function. When placed in an external field H the system is shown to be self-dual. Using another change of spin variables the one-dimensional Ising model with multispin interactions in a field is mapped onto a zero-field rectangular Ising model with first-neighbour interactions K and H . The 2D system, with size m × N / m , has the topology of a cylinder with helical BC. In the thermodynamic limit N / m → ∞ , m → ∞ , a 2D critical singularity develops on the self-duality line, sinh 2 K sinh 2 H = 1 .
The phase transition in the Ising model has been studied as a function of an applied transverse field, by Green's function and series-expansion methods. The critical indices are probably independent of the applied field except at T=0T=0T=0 where they appear to be related to those of the Ising model in one higher dimension.
We consider self-dual transverse-field Ising spin chains with mmm-spin interaction, where the phase transition is of second and first order, for m< 3m<~3m< 3 and m>3m>3m>3, respectively. We present a statistical analysis of the spectra of the Hamiltonians on relatively large L< 18L<~18L< 18 finite lattices. Outside the critical point we found level repulsion close to the Wigner distribution and the same rigidity as for the Gaussian orthogonal ensemble. At the transition point the level statistics in the self-dual sector is shown to be the superposition of two independent Wigner distributions. This is explained by the existence of an extra symmetry, which is connected to level crossing in the thermodynamic limit. Our study has given no evidence for the possible integrability of the models for m>2m>2m>2, even at the transition point.