Geometrical frustration | EPFL Graph Search

In condensed matter physics, the term geometrical frustration (or in short: frustration) refers to a phenomenon where atoms tend to stick to non-trivial positions or where, on a regular crystal lattice, conflicting inter-atomic forces (each one favoring rather simple, but different structures) lead to quite complex structures. As a consequence of the frustration in the geometry or in the forces, a plenitude of distinct ground states may result at zero temperature, and usual thermal ordering may be suppressed at higher temperatures. Much studied examples are amorphous materials, glasses, or dilute magnets. The term frustration, in the context of magnetic systems, has been introduced by Gerard Toulouse in 1977. Frustrated magnetic systems had been studied even before. Early work includes a study of the Ising model on a triangular lattice with nearest-neighbor spins coupled antiferromagnetically, by G. H. Wannier, published in 1950. Related features occur in magnets with competing intera

Quadrupolar Ordering in Two-Dimensional Spin-One Systems

Tamás András Tóth

The principal aim of this thesis is to gain a better understanding of the competition between magnetic and quadrupolar degrees of freedom on two-dimensional lattices. Recent experimental investigations of the material NiGa2S4 revealed several anomalous properties that might be accounted for within the framework of quadrupolar ordering. Exhibiting both a ferroquadrupolar and an antiferroquadrupolar phase, the S = 1 bilinear-biquadratic Heisenberg model on the triangular lattice is a possible candidate for describing the low-temperature behaviour of the system. In this work, we put forward a more realistic model that includes single-ion anisotropy. We perform a thorough investigation of the variational phase diagram of this model and we show that it exhibits a variety of unconventional phases. We derive the excitation spectrum of the quadrupolar phases in the phase diagram and we point out that ferroquadrupolar order is particularly sensitive to the nature of anisotropy. Finally, we study quantum effects in the perturbative limit of large anisotropy and we argue that the non-trivial degeneracy of the mean-field solution is lifted by an emergent supersolid phase. We also discuss our results in the context of NiGa2S4. In the second part of the thesis, we aim at gaining an insight into the interplay between geometrical frustration and quadrupolar degrees of freedom by mapping out the phase diagram of the spin-one bilinear-biquadratic model on the square lattice. Our variational approach reveals a remarkable 1/2-magnetization plateau of mixed quadrupolar and magnetic character above the classically degenerate "semi-ordered" phase, and this finding is corroborated by exact diagonalization of finite clusters. "Order-by-disorder" phenomenon gives rise to a state featuring three-sublattice antiferroquadrupolar order below the plateau, which is truly surprising given the bipartite nature of the square lattice. We place particular emphasis on investigating the properties of the SU(3) Heisenberg model, which is shown to feature a subtle competition between quantum and thermal fluctuations. Our results suggest a suppression of two-sublattice Néel order in a finite window below the SU(3) point. Experimental implications for the Mott-insulating states of three-flavour fermionic atoms in optical lattices are discussed.

EPFL2011

Tensor network investigation of frustrated Ising models

Jeanne Colbois

In spin systems, geometrical frustration describes the impossibility of minimizing simultaneously all the interactions in a Hamiltonian, often giving rise to macroscopic ground-state degeneracies and emergent low-temperature physics. In this thesis, combining tensor network (TN) methods to Monte Carlo (MC) methods and ground-state energy lower bound approaches, we study two-dimensional frustrated classical Ising models. In particular, we focus on the determination of the residual entropy in the presence of farther-neighbor interactions in kagome lattice Ising antiferromagnets (KIAFM).In general, using MC to determine the residual entropy is a significant challenge requiring ad-hoc updates, a precise evaluation of the energy at all temperatures to allow for thermodynamic integration, and a good control of the finite-size scaling behavior. As an alternative, we turn to TNs; however, we argue that, in the presence of frustration and macroscopic ground-state degeneracy, standard algorithms fail to converge at low temperatures on the usual TN formulation of partition functions. Inspired by methods for constructing ground-state energy lower bounds, we propose a systematic way to find the ground-state local rule using linear programming. Characterizing the rules as tiles that can be tessellated to form ground states of the model gives rise to a natural contractible TN formulation of the partition function. This method provides a direct access to the ground-state properties of frustrated models and, in particular, allows an extremely precise determination of their residual entropy.We then study two models inspired by artificial spin systems on the kagome lattice with out-of-plane (OOP) anisotropy. The first model is motivated by experiments on an array of chirally coupled nanomagnets. We argue that the farther-neighbor to nearest-neighbor couplings ratios in this system are much smaller than in the dipolar case, J2/J1 being of the order of 2%. A comparison of the experimental correlations with the results of extensive TN and MC simulations shows that (1) the experimental second- and third-neighbor correlations are inverted as compared to those of a pure nearest-neighbor model at equilibrium (even with a magnetic field), and (2) second-neighbor couplings as small as 1% of the nearest-neighbor couplings will affect the spin-spin correlations even at fairly high temperatures.Motivated by dipolar coupled artificial spin systems, we turn to the progressive lifting of the ground-state degeneracy of the KIAFM. We provide a detailed study of the ground-state phases of this model with up to third neighbor interactions, for arbitrary J2, J3 such that J1 >> J2, J3, obtaining exact results for the ground-state energies. When all couplings are antiferromagnetic, we exhibit three macroscopically degenerate ground-state phases and establish their residual entropy using our TN approach. Furthermore, in the phase corresponding to the dipolar KIAFM truncated to third neighbors, we use the ground-state tiles to establish the existence of a mapping to the ground-state manifold of the triangular Ising antiferromagnet.

EPFL2022