Ordering of geometrically frustrated classical and quantum Ising magnets

TL;DR

This paper provides a comprehensive analysis of classical and quantum frustrated Ising models on a triangular lattice, deriving effective Hamiltonians and phase diagrams through a novel string mapping approach, and comparing predictions with simulations.

Contribution

It introduces an exact mapping of frustrated Ising models to elastic string lattices, enabling derivation of phase diagrams and universality classes from microscopic parameters.

Findings

Derived the phase diagram including critical points.

Established the universality of height profile stiffness.

Predicted ordered phase structure from entropy considerations.

Abstract

A systematic study of both classical and quantum geometric frustrated Ising models with a competing ordering mechanism is reported in this paper. The ordering comes in the classical case from a coupling of 2D layers and in the quantum model from the quantum dynamics induced by a transverse field. By mapping the Ising models on a triangular lattice to elastic lattices of non-crossing strings, we derive an exact relation between the spin variables and the displacement field of the strings. Using this map both for the classical (2+1)D stacked model and the quantum frustrated 2D system, we obtain a microscopic derivation of an effective Hamiltonian which was proposed before on phenomenological grounds within a Landau-Ginzburg-Wilson approach. In contrast to the latter approach, our derivation provides the coupling constants and hence the entire transverse field--versus--temperature phase diagram can be deduced, including the universality classes of both the quantum and the finite--temperature transitions. The structure of the ordered phase is obtained from a detailed entropy argument. We compare our predictions to recent simulations of the quantum system and find good agreement. We also analyze the connections to a dimer model on the hexagonal lattice and its height profile representation, providing a simple derivation of the continuum free energy and a physical explanation for the universality of the stiffness of the height profile for anisotropic couplings.