arXiv subjects

Michael Hermele

Publications and source records attributed to Michael Hermele.

At least 37 records · Page 2Linked to original sources

arXiv2020

Odd Fracton Theories, Proximate Orders, and Parton Constructions

The Lieb-Schultz-Mattis (LSM) theorem implies that gapped phases of matter must satisfy non-trivial conditions on their low-energy properties when a combination of lattice translation and $U(1)$ symmetry are imposed. We describe a framework to characterize the action of symmetry on fractons and other sub-dimensional fractional excitations, and use this together with the LSM theorem to establish that X-cube fracton order can occur only at integer or half-odd-integer filling. Using explicit parton constructions, we demonstrate that "odd" versions of X-cube fracton order can occur in systems at half-odd-integer filling, generalizing the notion of odd $Z_2$ gauge theory to the fracton setting. At half-odd-integer filling, exiting the X-cube phase by condensing fractional quasiparticles leads to symmetry-breaking, thereby allowing us to identify a class of conventional ordered phases proximate to phases with fracton order. We leverage a dual description of one of these ordered phases to show that its topological defects naturally have restricted mobility. Condensing pairs of these defects then leads to a fracton phase, whose excitations inherit these mobility restrictions.

cond-mat.str-el↗

arXiv2020

Fractons from vector gauge theory

Motivated by the prediction of fractonic topological defects in a quantum crystal, we utilize a reformulated elasticity duality to derive a description of a fracton phase in terms of coupled vector U(1) gauge theories. The fracton order and restricted mobility emerge as a result of an unusual Gauss law where electric field lines of one gauge field act as sources of charge for others. At low energies this vector gauge theory reduces to the previously studied fractonic symmetric tensor gauge theory. We construct the corresponding lattice model and a number of generalizations, which realize fracton phases via a condensation of string-like excitations built out of charged particles, analogous to the p-string condensation mechanism of the gapped X-cube fracton phase.

cond-mat.str-el↗

arXiv2019

Fracton fusion and statistics

We introduce and develop a theory of fusion and statistical processes of gapped excitations in Abelian fracton phases. The key idea is to incorporate lattice translation symmetry via its action on superselection sectors, which results in a fusion theory endowed with information about the non-trivial mobility of fractons and sub-dimensional excitations. This results in a description of statistical processes in terms of local moves determined by the fusion theory. Our results can be understood as providing a characterization of translation-invariant fracton phases. We obtain simple descriptions of the fusion theory in the X-cube and checkerboard fracton models, as well as for gapped electric and magnetic excitations of some gapless U(1) tensor gauge theories. An alternate route to the X-cube model fusion theory is provided by starting with a system of decoupled two-dimensional toric code layers, and giving a description of the p-string condensation mechanism within our approach. We discuss examples of statistical processes of fractons and sub-dimensional excitations in the X-cube and checkerboard models. As an application of the ideas developed, we prove that the X-cube and semionic X-cube models realize distinct translation-invariant fracton phases, even when the translation symmetry is broken corresponding to an arbitrary but finite enlargement of the crystalline unit cell.

cond-mat.str-el↗

arXiv2019

Local constraints can globally shatter Hilbert space: a new route to quantum information protection

We show how local constraints can globally "shatter" Hilbert space into subsectors, leading to an unexpected dynamics with features reminiscent of both many body localization and quantum scars. A crisp example of this phenomenon is provided by a "fractonic circuit" - a model of quantum circuit dynamics in one dimension constrained to conserve both charge and dipole moment. We show how the Hilbert space of the fractonic circuit dynamically fractures into disconnected emergent subsectors within a particular charge and dipole symmetry sector. A large number of the emergent subsectors, exponentially many in the size of the system, have dimension one and exhibit strictly localized quantum dynamics---even in the absence of spatial disorder and in the presence of temporal noise. Exponentially large localized subspaces can be proven to exist for any one dimensional fractonic circuit with finite spatial range, and provide a potentially new route for the robust storage of quantum information. Other emergent subsectors display non-trivial dynamics and may be constructed by embedding finite sized non-trivial blocks into the localized subspace. The shattering of a particular symmetry sector into a distribution of dynamical subsectors with varying sizes leads to the coexistence of high and low entanglement states, i.e. this provides a general mechanism for the production of quantum many body scars. We discuss the detailed pattern of fracturing and its implications. We also discuss other mechanisms for similarly shattering Hilbert space.

cond-mat.stat-mech↗

arXiv2019

Cage-Net Fracton Models

We introduce a class of gapped three-dimensional models, dubbed "cage-net fracton models," which host immobile fracton excitations in addition to non-Abelian particles with restricted mobility. Starting from layers of two-dimensional string-net models, whose spectrum includes non-Abelian anyons, we condense extended one-dimensional "flux-strings" built out of point-like excitations. Flux-string condensation generalizes the concept of anyon condensation familiar from conventional topological order and allows us to establish properties of the fracton ordered (equivalently, flux-string condensed) phase, such as its ground state wave function and spectrum of excitations. Through the examples of doubled Ising and SU(2)$_k$ cage-net models, we demonstrate the existence of strictly immobile Abelian fractons and of non-Abelian particles restricted to move only along one dimension. In the doubled Ising cage-net model, we show that these restricted-mobility non-Abelian excitations are a fundamentally three-dimensional phenomenon, as they cannot be understood as bound states amongst two-dimensional non-Abelian anyons and Abelian particles. We further show that the ground state wave function of such phases can be understood as a fluctuating network of extended objects -- cages -- and strings, which we dub a cage-net condensate. Besides having implications for topological quantum computation in three dimensions, our work may also point the way towards more general insights into quantum phases of matter with fracton order.

cond-mat.str-el↗

arXiv2018

Topological states from topological crystals

We present a scheme to explicitly construct and classify general topological states jointly protected by an onsite symmetry group and a spatial symmetry group. We show that all these symmetry protected topological states can be adiabatically deformed (allowing for stacking of trivial states) into a special class of states we call topological crystals. A topological crystal in, for example, three dimensions is a real-space assembly of finite-sized pieces of topological states in one and two dimensions protected by the local symmetry group alone, arranged in a configuration invariant under the spatial group and glued together such there is no open edge or end. As a demonstration of principle, we explicitly enumerate all inequivalent topological crystals for non-interacting time-reversal symmetric electronic insulators with significant spin-orbit coupling and any one of the 230 space groups in three dimensions. Because every topological crystalline insulator can be deformed into a topological crystal, the enumeration of the latter gives topological crystalline insulators a full classification and for each class an explicit real-space construction. We also extend these results to give a unified classification including both strong topological insulators and topological crystalline insulators.

cond-mat.mes-hall↗

arXiv2018

Fracton topological order from Higgs and partial confinement mechanisms of rank-two gauge theory

Fractons are gapped point-like excitations in $d=3$ topological ordered phases whose motion is constrained. They have been discovered in several gapped models but a unifying physical mechanism for generating them is still missing. It has been noticed that in symmetric-tensor ${\rm U}(1)$ gauge theories, charges are fractons and cannot move freely due to, for example, the conservation of not only the charge but also the dipole moment. To connect these theories with fully gapped fracton models, we study Higgs and partial confinement mechanisms in rank-2 symmetric-tensor gauge theories, where charges or magnetic excitations, respectively, are condensed. Specifically, we describe two different routes from the rank-2 ${\rm U}(1)$ scalar charge theory to the X-cube fracton topological order, finding that a combination of Higgs and partial confinement mechanisms is necessary to obtain the fully gapped fracton model. On the other hand, the rank-2 $\mathbb{Z}_2$ scalar charge theory, which is obtained from the former theory upon condensing charge-2 matter, is equivalent to four copies of the $d=3$ toric code and does not support fracton excitations. We also explain how the checkerboard fracton model can be viewed as a rank-2 $\mathbb{Z}_2$ gauge theory with two different Gauss' law constraints on different lattice sites.

cond-mat.str-el↗

arXiv2018

Fractons

We review what is known about fracton phases of quantum matter. Fracton phases are characterized by excitations that exhibit restricted mobility, being either immobile under local Hamiltonian dynamics, or mobile only in certain directions. They constitute a new class of quantum state of matter, which does not wholly fit into any of the existing paradigms, but which connects to areas including glassy quantum dynamics, topological order, spin liquids, elasticity theory, quantum information theory, and gravity. We begin by discussing gapped fracton phases, which may be described using exactly solvable lattice spin models. We introduce the basic phenomena, and discuss the geometric and topological response of fracton phases. We also discuss connections to generalized gauge theories, and explain how gapped fracton phases may be obtained from more familiar theories. We then introduce the framework of tensor gauge theory, which provides a powerful complementary perspective on fracton phases. We discuss how tensor gauge theory encodes the fracton phenomenon, and how it allows us to access gapless fracton phases. We discuss the basic properties of gapless fracton phases, and their connections to elasticity theory and gravity. We also discuss what is known about the dynamics and thermodynamics of fractons at non-zero density, before concluding with a brief survey of some open problems.

cond-mat.str-el↗

arXiv2018

Topological Entanglement Entropy of Fracton Stabilizer Codes

Entanglement entropy provides a powerful characterization of two-dimensional gapped topological phases of quantum matter, intimately tied to their description by topological quantum field theories (TQFTs). Fracton topological orders are three-dimensional gapped topologically ordered states of matter, but the existence of a TQFT description for these phases remains an open question. We show that three-dimensional fracton phases are nevertheless characterized, at least partially, by universal structure in the entanglement entropy of their ground state wave functions. We explicitly compute the entanglement entropy for two archetypal fracton models --- the `X-cube model' and `Haah's code' --- and demonstrate the existence of a topological contribution that scales linearly in subsystem size. We show via Schrieffer-Wolff transformations that the topological entanglement of fracton models is robust against arbitrary local perturbations of the Hamiltonian. Finally, we argue that these results may be extended to characterize localization-protected fracton topological order in excited states of disordered fracton models.

cond-mat.str-el↗

arXiv2017

Building crystalline topological phases from lower-dimensional states

We study the classification of symmetry protected topological (SPT) phases with crystalline symmetry (cSPT phases). Focusing on bosonic cSPT phases in two and three dimensions, we introduce a simple family of cSPT states, where the system is comprised of decoupled lower-dimensional building blocks that are themselves SPT states. We introduce a procedure to classify these block states, which surprisingly reproduces a classification of cSPT phases recently obtained by Thorngren and Else using very different methods, for all wallpaper and space groups. The explicit constructions underlying our results clarify the physical properties of the phases classified by Thorngren and Else, and expose additional structure in the classification. Moreover, the states we classify can be completely characterized by point group SPT (pgSPT) invariants and related weak pgSPT invariants that we introduce. In many cases, the weak invariants can be visualized in terms of translation-symmetric stacking of lower-dimensional pgSPT states. We apply our classification to propose a Lieb-Shultz-Mattis type constraint for two-dimensional spin systems with only crystalline symmetry, and establish this constraint by a dimensional reduction argument. Finally, the surprising matching with the Thorngren-Else classification leads us to conjecture that all SPT phases protected only by crystalline symmetry can be built from lower-dimensional blocks of invertible topological states. We argue that this conjecture holds if we make a certain physically reasonable but unproven assumption.

cond-mat.str-el↗

arXiv2017

Surface field theories of point group symmetry protected topological phases

We identify field theories that describe the surfaces of three-dimensional bosonic point group symmetry protected topological (pgSPT) phases. The anomalous nature of the surface field theories is revealed via a dimensional reduction argument. Specifically, we study three different surface field theories. The first field theory is quantum electrodynamics in three space-time dimensions (QED3) with four flavors of fermions. We show this theory can describe the surfaces of a majority of bosonic pgSPT phases protected by a single mirror reflection, or by $C_{nv}$ point group symmetry for $n=2,3,4,6$. The second field theory is a variant of QED3 with charge-1 and charge-3 Dirac fermions. This field theory can describe the surface of a reflection symmetric pgSPT phase built by placing an $E_{8}$ state on the mirror plane. The third field theory is an ${\rm O}(4)$ non-linear sigma model with a topological theta-term at $θ=π$, or, equivalently, a non-compact ${\rm CP}^1$ model. Using a coupled wire construction, we show this is a surface theory for bosonic pgSPT phases with ${\rm U}(1) \times \mathbb{Z}_{2}^{P}$ symmetry. For the latter two field theories, we discuss the connection to gapped surfaces with topological order. Moreover, we conjecture that the latter two field theories can describe surfaces of more general bosonic pgSPT phases with $C_{nv}$ point group symmetry.

cond-mat.str-el↗

arXiv2017

Fracton topological order via coupled layers

In this work, we develop a coupled layer construction of fracton topological orders in $d=3$ spatial dimensions. These topological phases have sub-extensive topological ground-state degeneracy and possess excitations whose movement is restricted in interesting ways. Our coupled layer approach is used to construct several different fracton topological phases, both from stacked layers of simple $d=2$ topological phases and from stacks of $d=3$ fracton topological phases. This perspective allows us to shed light on the physics of the X-cube model recently introduced by Vijay, Haah, and Fu, which we demonstrate can be obtained as the strong-coupling limit of a coupled three-dimensional stack of toric codes. We also construct two new models of fracton topological order: a semionic generalization of the X-cube model, and a model obtained by coupling together four interpenetrating X-cube models, which we dub the "Four Color Cube model." The couplings considered lead to fracton topological orders via mechanisms we dub "p-string condensation" and "p-membrane condensation," in which strings or membranes built from particle excitations are driven to condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the phases we construct to be easily studied in terms of more familiar degrees of freedom. Our work raises the possibility of studying fracton topological phases from within the framework of topological quantum field theory, which may be useful for obtaining a more complete understanding of such phases.

cond-mat.str-el↗

arXiv2017

Topological phases protected by point group symmetry

We consider symmetry protected topological (SPT) phases with crystalline point group symmetry, dubbed point group SPT (pgSPT) phases. We show that such phases can be understood in terms of lower-dimensional topological phases with on-site symmetry, and can be constructed as stacks and arrays of these lower-dimensional states. This provides the basis for a general framework to classify and characterize bosonic and fermionic pgSPT phases, that can be applied for arbitrary crystalline point group symmetry and in arbitrary spatial dimension. We develop and illustrate this framework by means of a few examples, focusing on three-dimensional states. We classify bosonic pgSPT phases and fermionic topological crystalline superconductors with $Z_2^P$ (reflection) symmetry, electronic topological crystalline insulators (TCIs) with ${\rm U}(1) \times {Z}_2^P$ symmetry, and bosonic pgSPT phases with $C_{2v}$ symmetry, which is generated by two perpendicular mirror reflections. We also study surface properties, with a focus on gapped, topologically ordered surface states. For electronic TCIs we find a $Z_8 \times Z_2$ classification, where the $Z_8$ corresponds to known states obtained from non-interacting electrons, and the $Z_2$ corresponds to a "strongly correlated" TCI that requires strong interactions in the bulk. Our approach may also point the way toward a general theory of symmetry enriched topological (SET) phases with crystalline point group symmetry.

cond-mat.str-el↗

arXiv2016

Theory of quantum kagome ice and vison zero modes

We derive an effective $Z_2$ gauge theory to describe the quantum kagome ice (QKI) state that has been observed by Carrasquilla $\textit{et. al.}$ in Monte Carlo studies of the $S = 1/2$ kagome XYZ model in a Zeeman field. The numerical results on QKI are consistent with, but do not confirm or rule out, the hypothesis that it is a $Z_2$ spin liquid. Our effective theory allows us to explore this hypothesis and make a striking prediction for future numerical studies, namely that symmetry-protected vison zero modes arise at lattice disclination defects, leading to a Curie defect term in the spin susceptibility, and a characteristic $(N_{dis} - 1) \ln 2$ contribution to the entropy, where $N_{dis}$ is the number of disclinations. Only the $Z_2$ Ising symmetry is required to protect the vison zero modes. This is remarkable because a unitary $Z_2$ symmetry cannot be responsible for symmetry-protected degeneracies of local degrees of freedom. We also discuss other signatures of symmetry fractionalization in the $Z_2$ spin liquid, and phase transitions out of the $Z_2$ spin liquid to nearby ordered phases.

cond-mat.str-el↗

arXiv2016

Flux-fusion anomaly test and bosonic topological crystalline insulators

We introduce a method, dubbed the flux-fusion anomaly test, to detect certain anomalous symmetry fractionalization patterns in two-dimensional symmetry enriched topological (SET) phases. We focus on bosonic systems with Z2 topological order, and symmetry group of the form G = U(1) $\rtimes$ G', where G' is an arbitrary group that may include spatial symmetries and/or time reversal. The anomalous fractionalization patterns we identify cannot occur in strictly d=2 systems, but can occur at surfaces of d=3 symmetry protected topological (SPT) phases. This observation leads to examples of d=3 bosonic topological crystalline insulators (TCIs) that, to our knowledge, have not previously been identified. In some cases, these d=3 bosonic TCIs can have an anomalous superfluid at the surface, which is characterized by non-trivial projective transformations of the superfluid vortices under symmetry. The basic idea of our anomaly test is to introduce fluxes of the U(1) symmetry, and to show that some fractionalization patterns cannot be extended to a consistent action of G' symmetry on the fluxes. For some anomalies, this can be described in terms of dimensional reduction to d=1 SPT phases. We apply our method to several different symmetry groups with non-trivial anomalies, including G = U(1) X Z2T and G = U(1) X Z2P, where Z2T and Z2P are time-reversal and d=2 reflection symmetry, respectively.

cond-mat.str-el↗

arXiv2016

Fractionalizing glide reflections in two-dimensional Z2 topologically ordered phases

We study the fractionalization of space group symmetries in two-dimensional topologically ordered phases. Specifically, we focus on Z2-fractionalized phases in two dimensions whose deconfined topological excitations transform trivially under translational symmetries, but projectively under glide reflections, whose quantum numbers are hence fractionalized. We accomplish this by generalizing the dichotomy between even and odd gauge theories to incorporate additional symmetries inherent to non-symmorphic crystals. We show that the resulting fractionalization of point group quantum numbers can be detected in numerical studies of ground state wave functions. We illustrate these ideas using a microscopic model of a system of bosons at integer unit cell filling on a lattice with space group p4g, that can be mapped to a half-magnetization plateau for an S =1/2 spin system on the Shastry-Sutherland lattice.

cond-mat.str-el↗

arXiv2016

Symmetry fractionalization and anomaly detection in three-dimensional topological phases

In a phase with fractional excitations, topological properties are enriched in the presence of global symmetry. In particular, fractional excitations can transform under symmetry in a fractionalized manner, resulting in different Symmetry Enriched Topological (SET) phases. While a good deal is now understood in $2D$ regarding what symmetry fractionalization patterns are possible, the situation in $3D$ is much more open. A new feature in $3D$ is the existence of loop excitations, so to study $3D$ SET phases, first we need to understand how to properly describe the fractionalized action of symmetry on loops. Using a dimensional reduction procedure, we show that these loop excitations exist as the boundary between two $2D$ SET phases, and the symmetry action is characterized by the corresponding difference in SET orders. Moreover, similar to the $2D$ case, we find that some seemingly possible symmetry fractionalization patterns are actually anomalous and cannot be realized strictly in $3D$. We detect such anomalies using the flux fusion method we introduced previously in $2D$. To illustrate these ideas, we use the $3D$ $Z_2$ gauge theory with $Z_2$ global symmetry as an example, and enumerate and describe the corresponding SET phases. In particular, we find four non-anomalous SET phases and one anomalous SET phase, which we show can be realized as the surface of a $4D$ system with symmetry protected topological order.

cond-mat.str-el↗

arXiv2015

High-energy electronic excitations in Sr$_2$IrO$_4$ observed by Raman scattering

Spin-orbit interaction in Sr$_2$IrO$_4$ leads to the realization of the $J_{\mathrm{eff}}$ = 1/2 state and also induces an insulating behavior. Using large-shift Raman spectroscopy, we found two high-energy excitations of the d-shell multipletat at 690 meV and 680 meV with $A_{1g}$ and $B_{1g}$ symmetry respectively. As temperature decreases, the $A_{1g}$ and $B_{1g}$ peaks narrow, and the $A_{1g}$ peak shifts to higher energy while the energy of the $B_{1g}$ peak remains the same. When 25$\%$ of Ir is substituted with Rh the $A_{1g}$ peak softens by 10$\%$ but the $B_{1g}$ peak does not. We show that both pseudospin-flip and non-pseudosin-flip dd electronic transitions are Raman active, but only the latter are observed.

cond-mat.str-el↗