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arXiv:2206.15199

A Tale of Two Quantum Compass Models

Soumya Sur, M. S. Laad, Arya Subramonian, Syed R. Hassan

Published Jun 30, 2022• 0 citations
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Abstract

We investigate two variants of quantum compass models (QCMs). The first, an orbital-only honeycomb QCM, is shown to exhibit a quantum phase transition (QPT) from a $XX$- to $ZZ$-ordered phase in the $3d$-Ising universality class, in accord with earlier studies. In a fractionalized parton construction, this describes a ``superfluid-Mott insulator'' transition between a higher-order topological superfluid and the toric code, the latter described as a $p$-wave resonating valence bond state of the partons. The second variant, the spinless fermion QCM on a square lattice, is of interest in the context of cold-atom lattices with higher-angular momentum states on each atom. We explore finite-temperature orbital order-disorder transitions in the itinerant and localized limits using complementary methods. In the itinerant limit, we uncover an intricate temperature ($T$)-dependent dimensional crossover from a high-$T$ quasi-$1d$ insulator-like state, via an incoherent bad-metal-like state at intermediate $T$, to a $2d$ symmetry-broken insulator at low $T$, well below the ``orbital'' ordering scale. Finally, we discuss how engineering specific, tunable, and realistic perturbations in both these variants can act as a playground for simulating a variety of exotic QPTs between topologically ordered and trivial phases. In the cold-atom context, we propose a novel way to engineer a possible realisation of the exotic exciton Bose liquid phase at a QPT between a Bose superfluid and a charge density wave insulator. We argue that advances in the design of Josephson junction arrays and manipulating cold-atom lattices offer the hope of simulating such novel phases of matter in the foreseeable future.

DOI
10.48550/arxiv.2206.15199

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We investigate two variants of quantum compass models (QCMs). The first, an orbital-only honeycomb QCM, is shown to exhibit a quantum phase transition (QPT) from a XXXXXX- to ZZZZZZ-ordered phase in the 3d3d3d-Ising universality class, in accord with earlier studies. In a fractionalized parton construction, this describes a superfluid-Mott insulator'' transition between a higher-order topological superfluid and the toric code, the latter described as a $p$-wave resonating valence bond state of the partons. The second variant, the spinless fermion QCM on a square lattice, is of interest in the context of cold-atom lattices with higher-angular momentum states on each atom. We explore finite-temperature orbital order-disorder transitions in the itinerant and localized limits using complementary methods. In the itinerant limit, we uncover an intricate temperature ($T$)-dependent dimensional crossover from a high-$T$ quasi-$1d$ insulator-like state, via an incoherent bad-metal-like state at intermediate $T$, to a $2d$ symmetry-broken insulator at low $T$, well below the orbital'' ordering scale. Finally, we discuss how engineering specific, tunable, and realistic perturbations in both these variants can act as a playground for simulating a variety of exotic QPTs between topologically ordered and trivial phases. In the cold-atom context, we propose a novel way to engineer a possible realisation of the exotic exciton Bose liquid phase at a QPT between a Bose superfluid and a charge density wave insulator. We argue that advances in the design of Josephson junction arrays and manipulating cold-atom lattices offer the hope of simulating such novel phases of matter in the foreseeable future.
Compass models are theories of matter in which the couplings between the internal spin (or other relevant field) components are inherently spatially (typically, direction) dependent. Compass-type interactions appear in diverse physical systems including Mott insulators with orbital degrees of freedom (where interactions sensitively depend on the spatial orientation of the orbitals involved), the low energy effective theories of frustrated quantum magnets, systems with strong spin-orbit couplings (such as the iridates), vacancy centers, and cold atomic gases. Kitaev's models, in particular the compass variant on the honeycomb lattice, realize basic notions of topological quantum computing. The fundamental inter-dependence between internal (spin, orbital, or other) and external (i.e., spatial) degrees of freedom which underlies compass models generally leads to very rich behaviors including the frustration of (semi-)classical ordered states on non-frustrated lattices and to enhanced quantum effects prompting, in certain cases, the appearance of zero temperature quantum spin liquids. As a consequence of these frustrations, new types of symmetries and their associated degeneracies may appear. These intermediate symmetries lie midway between the extremes of global symmetries and local gauge symmetries and lead to effective dimensional reductions. We review compass models in a unified manner, paying close attention to exact consequences of these symmetries, and to thermal and quantum fluctuations that stabilize orders via order out of disorder effects. We review non-trivial statistics and the appearance of topological quantum orders in compass systems in which, by virtue of their intermediate symmetry standard orders do not arise. This is complemented by a survey of numerical results. Where appropriate theoretical and experimental results are compared.
Quantum spin models with spatially dependent interactions, known as compass models, play an important role in study of frustrated quantum magnetism and Mott insulators with orbital degrees of freedom. One example is Kitaev model on honeycomb lattice with spin liquid ground states and anyonic excitations. It is highly desirable to go beyond this exactly solvable limit by generalizing Kitaev model to connect with specific materials or other known models. A conventional way to achieve this is through hybridization with other models. For example, Kitaev-Heisenberg model was proposed to describe iridates. Here we pursue an alternative direction and propose a new dubbed the tripod model, which contains a continuum of compass-type models. It smoothly interpolates Ising Kitaev and quantum 120∘120^\circ120∘ model by tuning a single parameter θ′\theta'θ′, angle between three legs of a tripod in spin space. Not only does tripod model establish a natural connection between three paradigmatic models, but it also enables study of quantum phase transitions between them. We solve this model numerically by tensor networks in thermodynamic limit and find a spin-disordered (spin liquid) phase between an antiferromagnetic phase and a dimer phase. Our results unambiguously establish long-range order of geometrically frustrated 120∘120^\circ120∘ model and provide an intuitive picture for phase transition between spin-liquid phase and dimer phase. The tripod model proposed here could be realized in cold atoms experiments by introducing non-Abelian artificial gauge fields for fermionic Hubbard model.
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.
In this thesis we apply recently developed, as well as sophisticated quantum Monte Carlo methods to numerically investigate models of strongly correlated electron systems on honeycomb structures. The latter are of particular interest owing to their unique properties when simulating electrons on them, like the relativistic dispersion, strong quantum fluctuations and their resistance against instabilities. This work covers several projects including the advancement of the weak-coupling continuous time quantum Monte Carlo and its application to zero temperature and phonons, quantum phase transitions of valence bond solids in spin-1/2 Heisenberg systems using projector quantum Monte Carlo in the valence bond basis, and the magnetic field induced transition to a canted antiferromagnet of the Hubbard model on the honeycomb lattice. The emphasis lies on two projects investigating the phase diagram of the SU(2) and the SU(N)-symmetric Hubbard model on the hexagonal lattice. At sufficiently low temperatures, condensed-matter systems tend to develop order. An exception are quantum spin-liquids, where fluctuations prevent a transition to an ordered state down to the lowest temperatures. Previously elusive in experimentally relevant microscopic two-dimensional models, we show by means of large-scale quantum Monte Carlo simulations of the SU(2) Hubbard model on the honeycomb lattice, that a quantum spin-liquid emerges between the state described by massless Dirac fermions and an antiferromagnetically ordered Mott insulator. This unexpected quantum-disordered state is found to be a short-range resonating valence bond liquid, akin to the one proposed for high temperature superconductors. Inspired by the rich phase diagrams of SU(N) models we study the SU(N)-symmetric Hubbard Heisenberg quantum antiferromagnet on the honeycomb lattice to investigate the reliability of 1/N corrections to large-N results by means of numerically exact QMC simulations. We study the melting of phases as correlations increase with decreasing N and determine whether the quantum spin liquid found in the SU(2) Hubbard model at intermediate coupling is a specific feature, or also exists in the unconstrained t-J model and higher symmetries.
We study quantum phases and transitions in ultra-cold quantum gases, beyond the scope of conventional condensed matter systems. The study focuses on three major thrusts. The first thrust is the study of cold ensembles of bosonic or fermionic atoms in the higher orbital bands of optical lattices. Orbital and topological quantum physics is systematically explored for various lattice geometries, to be outlined below. The second is the low dimensional spin-imbalanced fermions in the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. We analyze its thermal instability towards Berezinsky-Kosterlitz-Thouless (BKT) phase transitions driven by different topological defects at finite temperature. The third is the theoretical search for a mechanism to achieve a Bose-Einstein crystal phase for bosons in the ground state. A new class of two-body interactions characterized by strong momentum-dependence is found to give rise to such a quantum state. It is different than but related to the long-debated concept of supersolidity. The first thrust forms the main part of the dissertation. Meta-stable phases of high orbital bosons and fermions have been explored for different lattice geometries such as square and bipartite lattices. Exotic Mott and superfluid phases are studied in both one and two dimensions. Quantum and thermal phase transitions, associated with time reversal symmetry breaking and their experimental signatures are discussed. In particular a quantum ``disordered" superfluid phase, beyond the mean field description, is found in one dimension. The challenging problem of probing time reversal symmetry breaking is solved. Besides, a one-dimensional fermionic optical ladder with coupled s- and p-orbitals is shown to mimic spin-orbit interactions and exhibit topological insulator and topological superconducting phases. Charge fractionalization is shown to be realizable in such an sp-orbital ladder.
In this thesis we will study recent examples of exotic, topological, and many body localized quantum phase transitions. In Chapter 2 we study the quantum phase transition between the Z_2 spin liquid and valence bond solid (VBS) orders on a triangular lattice. We find a possible nematic Z_2 spin liquid intermediate phase and predict a continuous 3d XY* transition to the neighboring columnar and resonating-plaquette VBS phases. In Chapter 3 we demonstrate that an extended Kane-Mele Hubbard model on a bilayer honeycomb lattice has two novel quantum phase transitions. The first is a quantum phase transition between the weakly interacting gapless Dirac fermion phase and a strongly interacting fully gapped and symmetric trivial phase, which cannot be described by the standard Gross-Neveu model. The second is a quantum critical point between a quantum spin Hall insulator with spin S^z conservation and the previously mentioned strongly interacting fully gapped phase. We argue that the first quantum phase transition is related to the Z_16 classification of the topological superconductor ^3He-B phase with interactions, while the second quantum phase transition is a topological phase transition described by a bosonic O(4) nonlinear sigma model field theory with a Theta-term. In Chapter 4 we propose that if the highest and lowest energy eigenstates of a Hamiltonian belong to different SPT phases, then this Hamiltonian can't be fully many body localized. In Chapter 5 we study the disordered XYZ spin chain and its marginally many body localized critical lines, which we find to be characterized by an effective central charge c'=ln2 and continuously varying critical exponents.