The study of correlation effects in topological phases of matter can benefit from a multidisciplinary approach that combines techniques drawn from condensed matter, high-energy physics and quantum information science. In this work, we exploit these connections to study the strongly-interacting limit of certain lattice Hubbard models of topological insulators, which map onto four-Fermi quantum field theories with a Wilson-type discretisation and have been recently shown to be at reach of cold-atom quantum simulators based on synthetic spin-orbit coupling. We combine large-S and tensor-network techniques to explore the possible spontaneous symmetry-breaking phases that appear when the interactions of the topological insulators are sufficiently large. In particular, we show that varying the Wilson parameter r of the lattice discretisations leads to a novel Heisenberg–Ising compass model with critical lines that flow with the value of r.
The study of correlation effects in topological phases of matter can benefit from a multidisciplinary approach that combines techniques drawn from condensed matter, high-energy physics and quantum information science. In this work, we exploit these connections to study the strongly-interacting limit of certain lattice Hubbard models of topological insulators, which map onto four-Fermi quantum field theories with a Wilson-type discretisation and have been recently shown to be at reach of cold-atom quantum simulators based on synthetic spin-orbit coupling. We combine large-S and tensor-network techniques to explore the possible spontaneous symmetry-breaking phases that appear when the interactions of the topological insulators are sufficiently large. In particular, we show that varying the Wilson parameter r of the lattice discretisations leads to a novel Heisenberg–Ising compass model with critical lines that flow with the value of r.
Four-Fermi quantum field theories in (2+1) dimensions lie among the simplest models in high-energy physics, the understanding of which requires a non-perturbative lattice formulation addressing their strongly-coupled fixed points. These lattice models are also relevant in condensed matter, as they offer a neat playground to explore strong correlations in the quantum anomalous Hall (QAH) effect. We give a detailed description of our multidisciplinary approach to understand the fate of the QAH phases as the four-Fermi interactions are increased, which combines strong-coupling and effective-potential techniques, unveiling a rich phase diagram with large-N Chern insulators and Lorentz-breaking fermion condensates. Moreover, this toolbox can be enlarged with recent advances in quantum information science, as we show that tensor-network algorithms based on projected entangled pairs can be used to improve our understanding of the strong-coupling limit. We also present a detailed scheme that uses ultra-cold atoms in optical lattices with synthetic spin–orbit coupling to build quantum simulators of these four-Fermi models. This yields a promising alternative to characterize the strongly-coupled fixed points and, moreover, could also explore real-time dynamics and finite-fermion densities.
We show that synthetic spin-orbit coupling for ultracold atoms in optical Raman potentials can be exploited to build versatile quantum simulators of correlated Chern insulators connected to strongly coupled four-Fermi field theories similar to the Gross-Neveu model in (2 + 1) dimensions.Exploiting this multidisciplinary perspective, we identify a large-N quantum anomalous Hall (QAH) effect in absence of any external magnetic field, and use it to delimit regions in parameter space where these correlated topological phases appear, the boundaries of which are controlled by strongly coupled fixed points of these four-Fermi relativistic field theories.We further show how, for strong interactions, the QAH effect gives way to magnetic phases described by a two-dimensional quantum compass model in a transverse field.We present a detailed description of the phase diagram using the large-N effective potential, and variational techniques such as projected entangled pairs.
In recent years, topology has become increasingly prevalent in condensed matter physics. It has allowed us to understand, and even predict, a variety of striking and remarkable physical phenomena. The study of strongly interacting systems has similarly lavished us with a diverse range of exotic phases and unconventional transitions, many of which are still poorly understood. In this thesis we will explore the interplay between topology and interactions in an effort to uncover new and novel phases. First we study how interactions impact the quantum phase transition between a topologically non-trivial phase and a trivial phase. The combination of interactions and the low-energy degrees of freedom associated with the transition leads to the emergence of a dome of lattice-symmetry breaking nematic order. Such behaviour is reminiscent of a number of strongly correlated electronic systems. We move on to study the strongly interacting limit of one of the earliest and best-known non-interacting topological phases, Haldane's model of a Chern insulator. Recently realized with ultracold atoms in a shaken optical lattice, the model has a non-trivial topological invariant associated with its band structure. In the strongly interacting limit the spin degrees of freedom are all that survive and we find a rich phase diagram of magnetically ordered phases, using a combination of both classical and quantum techniques. Supplementing the model with an additional term we can `quantum-melt' one of these ordered states to produce a disordered, liquid state that we positively identify as a chiral spin liquid, a highly entangled state of matter with fractionalised excitations. We generalise this mechanism to other two dimensional lattices, uncovering a possible unifying framework with which to understand the emergence of chiral spin liquids in lattice spin models. Finally, motivated by groundbreaking experiments in the ultracold atoms community, we investigate a model of two-component bosons with an artificial spin-orbit coupling. The interplay between the lattice, interactions and spin-orbit coupling produces a variety of unusual superfluid phases. Using a novel Monte Carlo technique we reveal the finite temperature phase diagram that appears close to the Mott transition.
Understanding the robustness of topological phases of matter in the presence of interactions poses a difficult challenge in modern condensed matter, showing interesting connections to high-energy physics. In this work, we leverage these connections to present a complete analysis of the continuum long-wavelength description of a generic class of correlated topological insulators: Wilson-Hubbard topological matter. We show that a Wilsonian renormalization group (RG) approach, combined with the so-called topological Hamiltonian, provide a quantitative route to understand interaction-induced topological phase transitions that occur in Wilson-Hubbard matter. We benchmark two-loop RG predictions for a quasi-1D Wilson-Hubbard model by means of exhaustive numerical simulations based on matrix product states (MPS). The agreement of the RG predictions with MPS simulations motivates the extension of the RG calculations to higher-dimensional topological insulators.
For a long time, quantum states of matter have been successfully characterized by the Ginzburg-Landau formalism that was able to classify all different types of phase transitions. This view changed with the discovery of the quantum Hall effect and topological insulators. The latter are materials that host metallic edge states in an insulating bulk, some of which are protected by the existing symmetries. Complementary to the search of topological phases in condensed matter, great efforts have been made in quantum simulations based on cold atomic gases. Sophisticated laser schemes provide optical lattices with different geometries and allow to tune interactions and the realization of artificial gauge fields. At the same time, new concepts coming from quantum information, based on entanglement, are pushing the frontier of our understanding of quantum phases as a whole. The concept of entanglement has revolutionized the description of quantum many-body states by describing wave functions with tensor networks (TN) that are exploited for numerical simulations based on the variational principle. This thesis falls within the framework of the studies in condensed matter physics: it focuses indeed on the so-called synthetic realization of quantum states of matter, more specifically, of topological ones, which may have on the long-run outfalls towards robust quantum computers. We propose a theoretical investigation of cold atoms in optical lattice pierced by effective (magnetic) gauge fields and subjected to experimentally relevant interactions, by adding a modern numerical approach based on TN algorithms. More specifically, this work will focus on (i) interacting topological phases in quasi-1D systems and, in particular, the Creutz-Hubbard model, (ii) the connection between condensed matter and high energy physics studying the Gross-Neveu model and the discretization of Wilson-Hubbard model, (iii) implementing tensor network-based algorithms. Durante mucho tiempo, los estados cuánticos de la materia se han caracterizado con éxito por el formalismo de Ginzburg-Landau que permitió de clasificar todos los diferentes tipos de transiciones de fase. Esta visión cambió con el descubrimiento del efecto Hall cuántico y los aislantes topológicos. Estos últimos son materiales que albergan estados de borde metálicos en una masa aislante, algunos de los cuales están protegidos por las simetrías existentes. Conjuntamente a la búsqueda de fases topológicas en materia condensada, se han hecho grandes esfuerzos en simulaciones cuánticas basadas en gases atómicos fríos. Los sofisticados esquemas láser proporcionan redes ópticas con diferentes geometrías y permiten ajustar las interacciones y la realización de campos de gauge artificial. Al mismo tiempo, los nuevos conceptos que provienen de la información cuántica, basados en el entanglement, están empujando la frontera de nuestra comprensión de las fases cuánticas en su conjunto. El concepto de entanglement ha revolucionado la descripción de los estados cuánticos de muchos cuerpos al describir las funciones de onda con redes tensoras (TN) que se explotan para simulaciones numéricas basadas en el principio de variación. Esta tesis se enmarca en los estudios de física de la materia condensada: en particular, se centra en la llamada realización sintética de los estados cuánticos de la materia, más específicamente, de los topológicos, que pueden tener en las salidas a largo plazo hacia computadoras cuánticas robustas. Se propone una investigación teórica de los átomos fríos en la red óptica con campos de gauge efectivos y sometidos a interacciones relevantes experimentalmente, agregando un enfoque numérico moderno basado en algoritmos TN. Más específicamente, este trabajo se centrará en (i) fases topológicas en los sistemas cuasi-1D y, en particular, el modelo Creutz-Hubbard, (ii) la conexión entre la materia condensada y la física de alta energía estudiando el modelo Gross-Neveu y el discretización del modelo Wilson-Hubbard, (iii) implementación de algoritmos basados en redes tensoras.
We show that a Wilson-type discretization of the Gross–Neveu model, a fermionic N-flavor quantum field theory displaying asymptotic freedom and chiral symmetry breaking, can serve as a playground to explore correlated symmetry-protected phases of matter using techniques borrowed from high-energy physics. A large-N study, both in the Hamiltonian and Euclidean formalisms, yields a phase diagram with trivial, topological, and symmetry-broken phases separated by critical lines that meet at a tri-critical point. We benchmark these predictions using tools from condensed matter and quantum information science, which show that the large-N method captures the essence of the phase diagram even at N=1. Moreover, we describe a cold-atom scheme for the quantum simulation of this lattice model, which would allow to explore the single-flavor phase diagram.