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arXiv:2209.11243 · Physical review. B./Physical review. B

Symmetry fractionalization in the gauge mean-field theory of quantum spin ice

Félix Desrochers, Li Ern Chern, Yong Baek Kim

Published Feb 6, 2023• 19 citations
arXivHTMLPDF

Abstract

Symmetry fractionalization is a ubiquitous feature of topologically ordered states that can be used to classify different symmetry-enriched topological phases and reveal some of their unique experimental signatures. Despite its vast popularity, there is currently no available framework to study symmetry fractionalization of quantum spin ice (QSI)---a $\text{U}(1)$ quantum spin liquid (QSL) on the pyrochlore lattice supporting emergent photons---within the most widely used theoretical framework to describe it, gauge mean-field theory (GMFT). In this work, we provide an extension of GMFT that allows for the classification of space-time symmetry fractionalization. The construction classifies all GMFT Ans\"atze that yield physical wave functions invariant under given symmetries and a specific low-energy gauge structure. As an application of the framework, we first show that the only two Ans\"atze with emergent U(1) gauge fields that respect all space group symmetries are the well-known 0- and $\ensuremath{\pi}$-flux states. We then showcase how the framework may describe QSLs beyond the currently known ones by classifying chiral U(1) QSI. We find a new chiral QSL described by $\ensuremath{\pi}/2$ fluxes of the emergent gauge field threading the hexagonal plaquettes of the pyrochlore lattice. We finally discuss how the different ways translation symmetries fractionalize for all these states lead to unique experimentally relevant signatures and compute their respective inelastic neutron scattering cross-section to illustrate the argument.

DOI
10.1103/physrevb.107.064404

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Symmetry fractionalization is a ubiquitous feature of topologically ordered states that can be used to classify different symmetry-enriched topological phases and reveal some of their unique experimental signatures. Despite its vast popularity, there is currently no available framework to study symmetry fractionalization of quantum spin ice (QSI)---a U(1)\text{U}(1)U(1) quantum spin liquid (QSL) on the pyrochlore lattice supporting emergent photons---within the most widely used theoretical framework to describe it, gauge mean-field theory (GMFT). In this work, we provide an extension of GMFT that allows for the classification of space-time symmetry fractionalization. The construction classifies all GMFT Ans"atze that yield physical wave functions invariant under given symmetries and a specific low-energy gauge structure. As an application of the framework, we first show that the only two Ans"atze with emergent U(1) gauge fields that respect all space group symmetries are the well-known 0- and \ensuremathπ\ensuremath{\pi}\ensuremathπ-flux states. We then showcase how the framework may describe QSLs beyond the currently known ones by classifying chiral U(1) QSI. We find a new chiral QSL described by \ensuremathπ/2\ensuremath{\pi}/2\ensuremathπ/2 fluxes of the emergent gauge field threading the hexagonal plaquettes of the pyrochlore lattice. We finally discuss how the different ways translation symmetries fractionalize for all these states lead to unique experimentally relevant signatures and compute their respective inelastic neutron scattering cross-section to illustrate the argument.
The presence of multiple competing interactions in spin systems can lead to novel phases known as quantum spin liquids (QSLs). QSLs are disordered spin systems with long-range entanglement that realize the deconfined phase of emergent lattice gauge theories and host fractional excitations. A notable example of great experimental interest is quantum spin ice (QSI), a three-dimensional QSL that realizes an analog of quantum electrodynamics with emergent photons and interacting electric and magnetic charges. The current leading candidate materials for realizing QSI are dipolar-octupolar (DO) cerium-based pyrochlore compounds. These are believed to stabilize a U(1)U(1)U(1) QSL known as π\piπ-flux QSI, where translation symmetry acting on spinon excitations fractionalizes. In this thesis, we explore the physics of QSI, relying heavily on parton constructions, to make experimentally relevant predictions for DO materials. First, we explore the physics of symmetry fractionalization in QSI, whereby lattice symmetries are realized projectively when acting on fractional quasiparticles. Symmetry fractionalization is examined within the framework of gauge mean-field theory (GMFT), a parton construction designed explicitly for application to QSI. This study shows that the pyrochlore lattice space group strongly constrains possible symmetry fractionalization classes. It systematically explains how all space group transformations, beyond translation, fractionalize in π\piπ-flux QSI. Next, we apply this framework to DO materials and show how the equal-time structure factor predicted for π\piπ-flux QSI is consistent with recent energy-integrated inelastic neutron scattering experiments on certain candidate materials. We also predict that the dynamical spin structure factor of π\piπ-flux QSI is characterized by a broad continuum with three distinctive peaks of decreasing intensity that are approximately equally spaced. We argue that these peaks should be accessible with currently available experimental apparatus and offer a smoking-gun signature of π\piπ-flux QSI. We confirm those predictions by extending the exclusive boson representation and speculate on the thermodynamic behavior of cerium candidates. We close by commenting on recent experimental breakthroughs and possible future directions.
Quantum spin liquids (QSLs) have generated considerable excitement as phases of matter with emergent gauge structures and fractionalized excitations. In this context, phase transitions out of QSLs have been widely discussed as Higgs transitions from deconfined to confined phases of a lattice gauge theory. However, the possibility of a wider range of novel phases, occurring between these two limits, has yet to be systematically explored. In this Research Letter, we develop a formalism which allows for interactions between fractionalized quasiparticles coming from the constraint on the physical Hilbert space and can be used to search for exotic, hidden phases. Taking pyrochlore spin ice as a starting point, we show how a U(1) QSL can give birth to abundant daughter phases, without the need for fine-tuning of parameters. These include a (charged) Z2{\mathbb{Z}}_{2}Z2​ QSL and a supersolid. We discuss implications for experiment, and numerical results which support our analysis. These results are of broad relevance to QSLs subject to a parton description and offer a useful perspective in the search for exotic hidden phases in quantum magnets.
We classify distinct types of quantum number fractionalization occurring in two-dimensional topologically ordered phases, focusing in particular on phases with Z2{\mathbb{Z}}_{2}Z2​ topological order, that is, on gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquids. We find that the fractionalization class of each anyon is an equivalence class of projective representations of the symmetry group, corresponding to elements of the cohomology group H2(G,Z2){H}^{2}(G,{\mathbb{Z}}_{2})H2(G,Z2​). This result leads us to a symmetry classification of gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquids, such that two phases in different symmetry classes cannot be connected without breaking symmetry or crossing a phase transition. Symmetry classes are defined by specifying a fractionalization class for each type of anyon. The fusion rules of anyons play a crucial role in determining the symmetry classes. For translation and internal symmetries, braiding statistics plays no role, but can affect the classification when point group symmetries are present. For square lattice space group, time-reversal, and SO(3)\mathrm{SO}(3)SO(3) spin rotation symmetries, we find $2\phantom{\rule{0.16em}{0ex}}098\phantom{\rule{0.16em}{0ex}}176\ensuremath{\approx}{2}^{21}$ distinct symmetry classes. Our symmetry classification is not complete, as we exclude, by assumption, permutation of the different types of anyons by symmetry operations. We give an explicit construction of symmetry classes for square lattice space group symmetry in the toric code model. Via simple examples, we illustrate how information about fractionalization classes can, in principle, be obtained from the spectrum and quantum numbers of excited states. Moreover, the symmetry class can be partially determined from the quantum numbers of the four degenerate ground states on the torus. We also extend our results to arbitrary Abelian topological orders (limited, though, to translations and internal symmetries), and compare our classification with the related projective symmetry group classification of parton mean-field theories. Our results provide a framework for understanding and probing the sharp distinctions among symmetric Z2{\mathbb{Z}}_{2}Z2​ spin liquids and are a first step toward a full classification of symmetric topologically ordered phases.
Abstract We present a systematic classification and variational study of chiral quantum spin liquids on the pyrochlore lattice based on fermionic parton constructions. Focusing on chiral U ( 1 ) and Z 2 spin-liquid Ansätze , we characterize their symmetry properties, flux structures, and low-energy spinon spectra within a projective symmetry group framework, and incorporate gauge fluctuations through Gutzwiller-projected wave functions studied by variational Monte Carlo. From the equal-time spin structure factor, we develop correlation-based diagnostics that distinguish gauge-dominated Coulomb phases from states with substantial matter-field and short-range contributions. Distinct chiral flux sectors, though close in energy, exhibit markedly different degrees of emergent U ( 1 ) gauge-field dominance, reflected in the geometry and contrast of pinch-point singularities. Although these states are not competitive ground states of the nearest-neighbor Heisenberg model, they define a physically meaningful family of proximate chiral phases relevant to extended pyrochlore Hamiltonians.
We present a systematic framework to classify symmetry-enriched topological quantum spin liquids in two spatial dimensions. This framework can deal with all topological quantum spin liquids, which may be either Abelian or non-Abelian and chiral or nonchiral. It can systematically treat a general symmetry, which may include both lattice symmetry and internal symmetry, may contain antiunitary symmetry, and may permute anyons. The framework applies to all types of lattices and can systematically distinguish different lattice systems with the same symmetry group using their quantum anomalies, which are sometimes known as Lieb-Schultz-Mattis anomalies. We apply this framework to classify U ( 1 ) 2 N chiral states and non-Abelian Ising ( ν ) states enriched by a p 6 × SO ( 3 ) or p 4 × SO ( 3 ) symmetry and Z N topological orders and U ( 1 ) 2 N × U ( 1 ) − 2 N topological orders enriched by a p 6 m × SO ( 3 ) × Z 2 T , p 4 m × SO ( 3 ) × Z 2 T , p 6 m × Z 2 T , or p 4 m × Z 2 T symmetry, where p 6 , p 4 , p 6 m , and p 4 m are lattice symmetries while SO(3) and Z 2 T are spin rotation and time-reversal symmetries, respectively. In particular, we identify symmetry-enriched topological quantum spin liquids that are not easily captured by the usual parton-mean-field approach, including examples with the familiar Z 2 topological order. Published by the American Physical Society 2024
Simplified models for quantum spin-ice materials, such as the XXZ model on the pyrochlore lattice, may host interesting new spin-liquid ground states. In this work, the authors provide a complete classification of nearest-neighbor chiral Z2{\mathbb{Z}}_{2}Z2​ spin-liquid states on the pyrochlore lattice using a projective symmetry group analysis. Moreover, the newly classified states are applied to the XXZ model within a self-consistent mean-field scheme. Observable signatures in neutron scattering experiments are discussed.