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Physical review. B, Condensed matter

Critical dynamics, Lifshitz tricriticality, and supersymmetry: The Ising model on the hcp lattice

Eytan Domany, James E. Gubernatis

Published Sep 1, 1985• 20 citations
Publisher

Abstract

Critical dynamics in d dimensions can be mapped onto a supersymmetric equilibrium problem in d+1 with a Lifshitz tricritical point. The existence of such a point is demonstrated for the Ising model on the hcp lattice by Monte Carlo simulation.
DOI
10.1103/physrevb.32.3354

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The d-dimensional time-dependent Ginzburg-Landau (TDGL) model is mapped onto a special (d+1)-dimensional model which exhibits a Lifshitz tricritical point (LTP). Many of the LTP critical properties follow from those of the TDGL model, and are shown to belong to a novel universality class of LTP's which results from a (previously ignored) relevant, nonlocal, quartic spin operator. These properties are analyzed with the use of scaling, an \ensuremath{\epsilon} expansion, and the n\ensuremath{\rightarrow}\ensuremath{\infty} limit.
A renormalization-group analysis of critical behavior near a Lifshitz tricritical point (LTP) is presented, with emphasis on the role played by new, momentum-dependent quartic terms. These result in new stable fixed points which determine the critical behavior. For some values of n (the number of order-parameter components) and m (the dimensionality of the ``soft'' subspace, characterized by quartic momentum-dependent inverse correlation functions), the renormalization-group recursion relations have two stable and accessible fixed points. However, one of these can never be reached in practice, due to a thermodynamic instability which results in a first-order phase transition. For m=d-1, one of the fixed points describes the critical dynamics of the usual n-vector spin model in (d-1) dimensions. This dynamic fixed point also characterizes LTP behavior for large n and n=1. In all other cases, the LTP has new exponents, which are not related to the dynamic model. Our results may be relevant to Lifshitz tricritical behavior in RbCaF3{\mathrm{RbCaF}}_{3}RbCaF3​ and in some liquid-crystal systems.
We introduce and analyze a quantum spin or Majorana chain with a tricritical Ising point separating a critical phase from a gapped phase with order-disorder coexistence. We show that supersymmetry is not only an emergent property of the scaling limit but also manifests itself on the lattice. Namely, we find explicit lattice expressions for the supersymmetry generators and currents. Writing the Hamiltonian in terms of these generators allows us to find the ground states exactly at a frustration-free coupling. These confirm the coexistence between two (topologically) ordered ground states and a disordered one in the gapped phase. Deforming the model by including explicit chiral symmetry breaking, we find the phases persist up to an unusual chiral phase transition where the supersymmetry becomes exact even on the lattice.