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arXiv:1206.5740 · Physical Review Letters

Emergent Critical Phase and Ricci Flow in a 2D Frustrated Heisenberg Model

Peter P. Orth, Premala Chandra, Piers Coleman, Jörg Schmalian

Published Dec 4, 2012• 19 citations
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Abstract

We introduce a two-dimensional frustrated Heisenberg antiferromagnet on interpenetrating honeycomb and triangular lattices. Classically the two sublattices decouple, and "order from disorder" drives them into a coplanar state. Applying Friedan's geometric approach to nonlinear sigma models, we obtain the scaling of the spin stiffnesses governed by the Ricci flow of a four-dimensional metric tensor. At low temperatures, the relative phase between the spins on the two sublattices is described by a six-state clock model with an emergent critical phase.

DOI
10.1103/physrevlett.109.237205

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We introduce a two-dimensional frustrated Heisenberg antiferromagnet on interpenetrating honeycomb and triangular lattices. Classically the two sublattices decouple, and "order from disorder" drives them into a coplanar state. Applying Friedan's geometric approach to nonlinear sigma models, we obtain the scaling of the spin stiffnesses governed by the Ricci flow of a four-dimensional metric tensor. At low temperatures, the relative phase between the spins on the two sublattices is described by a six-state clock model with an emergent critical phase.
We consider the spin-1/2 antiferromagnetic Heisenberg model on a bilayer honeycomb lattice including interlayer frustration in the presence of an external magnetic field. In the vicinity of the saturation field, we map the low-energy states of this quantum system onto the spatial configurations of hard hexagons on a honeycomb lattice. As a result, we can construct effective classical models (lattice-gas as well as Ising models) on the honeycomb lattice to calculate the properties of the frustrated quantum Heisenberg spin system in the low-temperature regime. We perform classical Monte Carlo simulations for a hard-hexagon model and adopt known results for an Ising model to discuss the finite-temperature order-disorder phase transition that is driven by a magnetic field at low temperatures. We also discuss an effective-model description around the ideal frustration case and find indications for a spin-flop-like transition in the considered isotropic spin model.
We investigate two-dimensional spin models and begin with an introduction to critical phenomena with emphasis on classical systems with boundaries and conformal invariance. Then we introduce the antiferromagnetic quantum Heisenberg model with frustrating interactions. Conformal invariance of a statistical mechanical system at criticality has powerful implications in two dimensions. With the use of conformal field theory, we investigate finite-size and boundary effects for a class of N-critical models that describe the simplest type of multicritical behaviour. In Papers I and II, correlation functions for the order parameter and other operators of these multicritical Ising models are derived for restricted domains (semi-infinite plane, infinite strip, disc and rectangle) with free or fixed boundary conditions. These yield the corresponding universal surface exponents and allow for a detailed analysis of correlations lengths, structure factors and response functions. The interplay between duality and boundary conditions is also made explicit. In Paper III, we analyse structure factors at criticality and show how apparent power- law scaling at intermediate momenta - induced by finite size and boundaries - depends on the boundary conditions applied. The predictions are confirmed by using correlation functions calculated in Papers I and II. Two-dimensional Heisenberg antiferromagnets have attracted much attention because of their relation to high-Tc superconductors. The second part of the thesis (Papers IV and V) treats spin-1/2 antiferromagnets defined on square- and honeycomb lattices with frustrating interactions. By using a Schwinger-boson mean-field theory, we calculate low-temperature quantum corrections to thermodynamic parameters, such as spin-wave velocity and transverse susceptibility. Their dependence on frustration is explicated and used to estimate the stability of the Néel state via a mapping to the nonlinear .sigma. model that describes the low-energy limit of the antiferromagnet.
In an extensive computational experiment, we test Polyakov's conjecture that under certain circumstances an isotropic Heisenberg model can develop algebraic spin correlations. We demonstrate the emergence of a multispin U(1) order parameter in a Heisenberg antiferromagnet on interpenetrating honeycomb and triangular lattices. The correlations of this relative phase angle are observed to decay algebraically at intermediate temperatures in an extended critical phase. Using finite-size scaling we show that both phase transitions are of the Berezinskii-Kosterlitz-Thouless type, and at lower temperatures we find long-range Z(6) order.
We study the spin-1/2 Ising-XXZ model on a decorated honeycomb lattice composed of five spins per unit cell, one Ising spin, and four Heisenberg spins. This model involving the Heisenberg exchange interaction is one of the few models that can be exactly solvable through the generalized star-triangle transformation. The significance of this model is its close relationship to the fully decorated quantum Heisenberg honeycomb lattice since 4/5 of the particles are Heisenberg spins. We investigate the phase diagram at zero temperature and identify a relevant quantum spin frustrated phase resulting from the contribution of quantum Heisenberg exchange interaction. We obtain an exact residual entropy for the quantum spin frustrated phase, which coincides with the residual entropy of the antiferromagnetic spin-1/2 Ising model on a triangular lattice. We also thoroughly explore its thermodynamic properties, focusing mainly on the frustrated region such as entropy, specific heat, spontaneous magnetization, and critical temperature under several conditions.
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 two-dimensional (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 noncrossing strings, we derive an exact relation between the spin variables and the displacement field of the strings. Using this map for both the classical (2+1)D(2+1)\mathrm{D}(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.