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arXiv:2110.01621 · Physical review. B./Physical review. B

Topological phases in interacting spin-1 systems

A. Alnor, Thomas Bækkegaard, Nikolaj Thomas Zinner

Published Nov 7, 2022• 1 citation
arXivHTMLPDF

Abstract

Different topological phases of quantum systems have become areas of increased focus in recent decades. In particular, the question of how to realize and manipulate systems with a nontrivial first Chern number is pursued both experimentally and theoretically. Here we go beyond typical spin-1/2 systems and consider both single and coupled spin-1 systems as a means of realizing higher first Chern numbers and studying the emergence of different topological phases. We show that rich topological phase diagrams can be realized by coupling two spin-1 systems using both numerical and analytical methods. Furthermore, we consider a concrete realization of spin-1 systems using a superconducting circuit. This realization includes nonstandard spin-spin interaction terms that may endanger the topological properties. We argue, however, that the realistic circuit Hamiltonian including all terms should be expected to show the rich phase structure as well. This puts our theoretical predictions within reach of state-of-the-art experimental setups.

DOI
10.1103/physrevb.106.174505

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Different topological phases of quantum systems have become areas of increased focus in recent decades. In particular, the question of how to realize and manipulate systems with a nontrivial first Chern number is pursued both experimentally and theoretically. Here we go beyond typical spin-1/2 systems and consider both single and coupled spin-1 systems as a means of realizing higher first Chern numbers and studying the emergence of different topological phases. We show that rich topological phase diagrams can be realized by coupling two spin-1 systems using both numerical and analytical methods. Furthermore, we consider a concrete realization of spin-1 systems using a superconducting circuit. This realization includes nonstandard spin-spin interaction terms that may endanger the topological properties. We argue, however, that the realistic circuit Hamiltonian including all terms should be expected to show the rich phase structure as well. This puts our theoretical predictions within reach of state-of-the-art experimental setups.
Superconducting circuits, owing to their high tunability, provide a controllable platform for studying topological systems that are difficult to manipulate in natural materials. The topological properties of spin particles have been extensively studied and remain of significant interest. Previous studies have primarily focused on spin-1/2 models. In this article, we explore the topological properties of the spin-J system by measuring the Chern number, which characterizes nontrivial phases and topological transitions. We experimentally demonstrate topological transitions in high-spin systems. Furthermore, the feasibility of the adiabatic evolution scheme was examined in our study. Experimental results proved that our scheme can be applied to explore topological phenomenon in higher-dimensional systems.
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.
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The recent discovery of topological insulators has led to a tremendous interest in the exploration of topological phases of matter which do not fit into Landau's symmetry breaking paradigm. Numerous exotic topological materials are theoretically predicted. Some of them have been experimentally reported, but many remain not. In this thesis, we explore topological phases of matter from three aspects: their classification, realization and application. We first review some basic classification theories, which provide us a "big picture" and lay the foundation for the rest of the thesis. We then move on to propose a systematic method based on quaternion algebra to construct toy tight-binding Hamiltonians for all the exotic phases in a recently developed periodic table for topological insulators and superconductors. We also introduce two peculiar families of topological phases that are beyond the table---the Hopf and four-dimensional topological insulators without time reversal symmetry. Prototypical Hamiltonians are constructed and their topological properties, such as robust edge states, are numerically studied. Motivated by rapid experimental progress in engineering spin-orbit coupling and artificial gauge field for cold atoms, we continue the thesis by proposing a feasible experimental scheme to realize a three-dimensional chiral topological insulator with cold fermionic atoms in an optical lattice. To unambiguously probe topological phases, we also bring forth systematic and generic methods to measure the characteristic topological invariants, for both free and strongly interacting systems. Moreover, we demonstrate that a kaleidoscope of knot and link structures is encoded in the spin texture of Hopf insulators and show how to observe different knots and links in cold atoms via time-of-flight images. The last part of the thesis is about the application of topological materials. After a demonstration of how to create, braid and detect Majorana fermions with cold atoms, we put forward a proposal to construct a self-test quantum random number generator by using Majorana fermions. Majorana random number generators are able to generate certifiable true random numbers with unconditional security. They offer a new perspective to the utilization of topological materials and may have vital applications in cryptography and related areas.
An interesting route to the realization of topological Chern bands in ultracold atomic gases is through the use of optical flux lattices. These models differ from the tight-binding real-space lattice models of Chern insulators that are conventionally studied in solid-state contexts. Instead, they involve the coherent coupling of internal atomic (spin) states, and can be viewed as tight-binding models in reciprocal space. By changing the form of the coupling and the number NNN of internal spin states, they give rise to Chern bands with controllable Chern number and with nearly flat energy dispersion. We investigate in detail how interactions between bosons occupying these bands can lead to the emergence of fractional quantum Hall states, such as the Laughlin and Moore-Read states. In order to test the experimental realization of these phases, we study their stability with respect to band dispersion and band mixing. We also probe interesting topological phases that emerge in these systems when the Chern number is greater than 1.