Abstract: Arrays of Rydberg atoms constitute a highly tunable, strongly interacting venue for the pursuit of exotic states of matter. We develop a strategy for accessing a family of fractionalized phases known as quantum spin liquids in two-dimensional Rydberg arrays. We specifically use effective field theory methods to study arrays assembled from Rydberg chains tuned to an Ising phase transition that famously hosts emergent fermions propagating within each chain. This highly entangled starting point allows us to naturally access spin liquids familiar from Kitaev's honeycomb model — albeit from an entirely different framework. In particular, we argue that finite-range repulsive Rydberg interactions, which frustrate nearby symmetry-breaking orders, can enable coherent propagation of emergent fermions between the chains in which they were born. Delocalization of emergent fermions across the full two-dimensional Rydberg array yields a gapless ℤ₂ spin liquid with a single massless Dirac cone. Here, the Rydberg occupation numbers exhibit universal power-law correlations that provide a straightforward experimental diagnostic of this phase. We further show that explicitly breaking symmetries perturbs the gapless spin liquid into gapped, topologically ordered descendants: Breaking lattice symmetries generates toric-code topological order, whereas introducing Floquet-mediated chirality generates non-Abelian Ising topological order. In the toric-code phase, we analytically construct microscopic incarnations of non-Abelian defects, which can be created and transported by dynamically controlling the atom positions in the array. Our work suggests that appropriately tuned Rydberg arrays provide a cold-atoms counterpart of solid-state "Kitaev materials" and, more generally, it spotlights a different angle for pursuing experimental platforms for Abelian and non-Abelian fractionalization.

Publication: Physical Review B Vol.: 106 No.: 11 ISSN: 2469-9950

ID: CaltechAUTHORS:20221031-575177800.11

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Abstract: Fracton order features point excitations that either cannot move at all or are only allowed to move in a lower-dimensional submanifold of the whole system. In this paper, we generalize the (2+1)-dimensional [(2+1)D] U(1) Chern-Simons (CS) theory, a powerful tool in the study of (2+1)D topological orders, to include infinite gauge field components and find that they can describe interesting types of (3+1)-dimensional fracton order beyond what is known from exactly solvable models and tensor gauge theories. On the one hand, they can describe foliated fractonic systems for which increasing the system size requires insertion of nontrivial (2+1)D topological states. The CS formulation provides an easier approach to study the phase relation among foliated models. More interestingly, we find simple examples that lie beyond the foliation framework, characterized by 2D excitations of infinite order and irrational braiding statistics. This finding extends our realm of understanding of possible fracton phenomena.

Publication: Physical Review B Vol.: 105 No.: 19 ISSN: 2469-9950

ID: CaltechAUTHORS:20210106-102305508

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Abstract: Rydberg chains provide an appealing platform for probing conformal field theories (CFTs) that capture universal behavior in a myriad of physical settings. Focusing on a Rydberg chain at the Ising transition separating charge density wave and disordered phases, we establish a detailed link between microscopics and low-energy physics emerging at criticality. We first construct lattice incarnations of primary fields in the underlying Ising CFT including chiral fermions, a nontrivial task given that the Rydberg chain Hamiltonian does not admit an exact fermionization. With this dictionary in hand, we compute correlations of microscopic Rydberg operators, paying special attention to finite, open chains of immediate experimental relevance. We further develop a method to quantify how second-neighbor Rydberg interactions tune the sign and strength of four-fermion couplings in the Ising CFT. Finally, we determine how the Ising fields evolve when four-fermion couplings drive an instability to Ising tricriticality. Our results pave the way to a thorough experimental characterization of Ising criticality in Rydberg arrays, and can inform the design of novel higher-dimensional phases based on coupled critical chains.

Publication: Physical Review B Vol.: 104 No.: 23 ISSN: 2469-9950

ID: CaltechAUTHORS:20211207-393208000

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Abstract: Theoretical high-energy and condensed-matter physics share various ideas and tools. New connections between the two have been established through quantum information, providing exciting prospects for theoretical advances and even potential experimental studies. Six scientists discuss different directions of research in this inter-disciplinary field.

Publication: Nature Reviews Physics Vol.: 3 No.: 6 ISSN: 2522-5820

ID: CaltechAUTHORS:20210518-074149060

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Abstract: The X-cube model, a prototypical gapped fracton model, was shown in Ref. [1] to have a foliation structure. That is, inside the 3+1 D model, there are hidden layers of 2+1 D gapped topological states. A screw dislocation in a 3+1 D lattice can often reveal nontrivial features associated with a layered structure. In this paper, we study the X-cube model on lattices with screw dislocations. In particular, we find that a screw dislocation results in a finite change in the logarithm of the ground state degeneracy of the model. Part of the change can be traced back to the effect of screw dislocations in a simple stack of 2+1 D topological states, hence corroborating the foliation structure in the model. The other part of the change comes from the induced motion of fractons or sub-dimensional excitations along the dislocation, a feature absent in the stack of 2+1D layers.

Publication: SciPost Physics Vol.: 10 No.: 4 ISSN: 2542-4653

ID: CaltechAUTHORS:20211222-591602700

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Abstract: In the study of three-dimensional gapped models, two-dimensional gapped states should be considered as a free resource. This is the basic idea underlying the notion of “foliated fracton order” proposed in Shirley et al. [Phys. Rev. X 8, 031051 (2018)]. We have found that many of the known type-I fracton models, although they appear very different, have the same foliated fracton order, known as “X-cube” order. In this paper, we identify three-dimensional fracton models with different kinds of foliated fracton order. Whereas the X-cube order corresponds to the gauge theory of a simple paramagnet with subsystem planar symmetry, the different orders correspond to twisted versions of the gauge theory for which the system prior to gauging has nontrivial order protected by the planar subsystem symmetry. We present constructions of the twisted models and demonstrate that they possess nontrivial order by studying their fractional excitation contents.

Publication: Physical Review B Vol.: 102 No.: 11 ISSN: 2469-9950

ID: CaltechAUTHORS:20190807-104405031

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Abstract: Fractons are a new type of quasiparticle which are immobile in isolation, but can often move by forming bound states. Fractons are found in a variety of physical settings, such as spin liquids and elasticity theory, and exhibit unusual phenomenology, such as gravitational physics and localization. The past several years have seen a surge of interest in these exotic particles, which have come to the forefront of modern condensed matter theory. In this review, we provide a broad treatment of fractons, ranging from pedagogical introductory material to discussions of recent advances in the field. We begin by demonstrating how the fracton phenomenon naturally arises as a consequence of higher moment conservation laws, often accompanied by the emergence of tensor gauge theories. We then provide a survey of fracton phases in spin models, along with the various tools used to characterize them, such as the foliation framework. We discuss in detail the manifestation of fracton physics in elasticity theory, as well as the connections of fractons with localization and gravitation. Finally, we provide an overview of some recently proposed platforms for fracton physics, such as Majorana islands and hole-doped antiferromagnets. We conclude with some open questions and an outlook on the field.

Publication: International Journal of Modern Physics A Vol.: 35 No.: 6 ISSN: 0217-751X

ID: CaltechAUTHORS:20200409-115738925

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Abstract: Fractional excitations in fracton models exhibit novel features not present in conventional topological phases: their mobility is constrained, there are an infinitude of types, and they bear an exotic sense of ‘braiding’. Hence, they require a new framework for proper characterization. Based on our definition of foliated fracton phases in which equivalence between models includes the possibility of adding layers of gapped 2D states, we propose to characterize fractional excitations in these phases up to the addition of quasiparticles with 2D mobility. That is, two quasiparticles differing by a set of quasiparticles that move along 2D planes are considered to be equivalent; likewise, ‘braiding’ statistics are measured in a way that is insensitive to the attachment of 2D quasiparticles. The fractional excitation types and statistics defined in this way provide a universal characterization of the underlying foliated fracton order which can subsequently be used to establish phase relations. We demonstrate as an example the equivalence between the X-cube model and the semionic X-cube model both in terms of fractional excitations and through an exact mapping.

Publication: Annals of Physics Vol.: 410ISSN: 0003-4916

ID: CaltechAUTHORS:20181023-102132294

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Abstract: We establish the presence of foliated fracton order in the Majorana checkerboard model. In particular, we describe an entanglement renormalization group transformation which utilizes toric code layers as resources of entanglement and furthermore discuss entanglement signatures and fractional excitations of the model. In fact, we give an exact local unitary equivalence between the Majorana checkerboard model and the semionic X-cube model augmented with decoupled fermionic modes. This mapping demonstrates that the model lies within the X-cube foliated fracton phase.

Publication: Physical Review B Vol.: 100 No.: 8 ISSN: 2469-9950

ID: CaltechAUTHORS:20190624-080757714

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Abstract: Based on several previous examples, we summarize explicitly the general procedure to gauge models with subsystem symmetries, which are symmetries with generators that have support within a sub-manifold of the system. The gauging process can be applied to any local quantum model on a lattice that is invariant under the subsystem symmetry. We focus primarily on simple 3D paramagnetic states with planar symmetries. For these systems, the gauged theory may exhibit foliated fracton order and we find that the species of symmetry charges in the paramagnet directly determine the resulting foliated fracton order. Moreover, we find that gauging linear subsystem symmetries in 2D or 3D models results in a self-duality similar to gauging global symmetries in 1D.

Publication: SciPost Physics Vol.: 6 No.: 4 ISSN: 2542-4653

ID: CaltechAUTHORS:20181023-140032523

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Abstract: In this work, we show that the checkerboard model exhibits the phenomenon of foliated fracton order. We introduce a renormalization-group transformation for the model that utilizes toric code bilayers as an entanglement resource and show how to extend the model to general three-dimensional manifolds. Furthermore, we use universal properties distilled from the structure of fractional excitations and ground-state entanglement to characterize the foliated fracton phase and find that it is the same as two copies of the X-cube model. Indeed, we demonstrate that the checkerboard model can be transformed into two copies of the X-cube model via an adiabatic deformation.

Publication: Physical Review B Vol.: 99 No.: 11 ISSN: 2469-9950

ID: CaltechAUTHORS:20181023-135513684

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Abstract: Chaotic dynamics in closed local quantum systems scrambles quantum information, which is manifested quantitatively in the decay of the out-of-time-ordered correlators (OTOC) of local operators. How is information scrambling affected when the system is coupled to the environment and suffers from dissipation? In this paper, we address this question by defining a dissipative version of OTOC and numerically study its behavior in a prototypical chaotic quantum chain in the presence of dissipation. We find that dissipation leads to not only the overall decay of the scrambled information due to leaking but also structural changes so that the ‘information light cone’ can only reach a finite distance even when the effect of overall decay is removed. Based on this observation we conjecture a modified version of the Lieb-Robinson bound in dissipative systems.

Publication: Physical Review B Vol.: 99 No.: 1 ISSN: 2469-9950

ID: CaltechAUTHORS:20190109-090932499

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Abstract: Fracton models exhibit a variety of exotic properties and lie beyond the conventional framework of gapped topological order. In a previous work, we generalized the notion of gapped phase to one of foliated fracton phase by allowing the addition of layers of gapped two-dimensional resources in the adiabatic evolution between gapped three-dimensional models. Moreover, we showed that the X-cube model is a fixed point of one such phase. In this paper, according to this definition, we look for universal properties of such phases which remain invariant throughout the entire phase. We propose multi-partite entanglement quantities, generalizing the proposal of topological entanglement entropy designed for conventional topological phases. We present arguments for the universality of these quantities and show that they attain non-zero constant value in non-trivial foliated fracton phases.

Publication: SciPost Physics Vol.: 6 No.: 1 ISSN: 2542-4653

ID: CaltechAUTHORS:20180924-141823095

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Abstract: Matrix product representation provides a useful formalism to study not only entangled states but also entangled operators in one dimension. In this paper, we focus on unitary transformations and show that matrix product operators that are unitary provide a necessary and sufficient representation of one-dimensional (1D) unitaries that preserve locality. That is, we show that matrix product operators that are unitary are guaranteed to preserve locality by mapping local operators to local operators, while at the same time all locality-preserving unitaries can be represented in a matrix product way. Moreover, we show that matrix product representation gives a straightforward way to extract the index defined by Gross, Nesme, Vogts, and Werner in [D. Gross et al., Commun. Math. Phys. 310, 419 (2012)] for classifying 1D locality-preserving unitaries. The key to our discussion is a set of “fixed-point” conditions which characterize the form of the matrix product unitary operators after blocking sites. Finally, we show that if the unitary condition is only required for certain system sizes, then matrix product formalism allows more possibilities. In particular, we give an example of a simple matrix product operator which is unitary only for odd system sizes, does not preserve locality, and carries a “fractional” index.

Publication: Physical Review B Vol.: 98 No.: 24 ISSN: 2469-9950

ID: CaltechAUTHORS:20171108-141516563

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Abstract: The tensor network representation of many-body quantum states, given by local tensors, provides a promising numerical tool for the study of strongly correlated topological phases in two dimensions. However, the representation may be vulnerable to instabilities caused by small variations in the local tensors. For example, the topological order in the tensor network representations of the toric code ground state has been shown in Chen, Zeng, Gu, Chuang, and Wen, Phys. Rev. B 82, 165119 (2010)to be unstable if the variations break certain Z_2 symmetry of the tensor. In this work, we ask whether other types of topological orders suffer from similar kinds of instability and if so, what is the underlying physical mechanism and whether we can protect the order by enforcing certain symmetries on the tensor. We answer these questions by showing that the tensor network representations of all string-net models are indeed unstable, but the matrix product operator (MPO) symmetries of the tensors identified in Şahinoğlu, Williamson, Bultinck, Mariën, Haegeman, Schuch, and Verstraete, arXiv:1409.2150 can help to protect the order. In particular, we show that a subset of variations that break the MPO symmetries lead to instability by inducing the condensation of bosonic quasiparticles, which destroys the topological order in the wave function. Therefore such variations must be forbidden for the encoded topological order to be reliably extracted from the local tensors. On the other hand, if a tensor network based variational algorithm is used to simulate the phase transition due to boson condensation, such variation directions may prove important to access the continuous transition correctly.

Publication: Physical Review B Vol.: 98 No.: 12 ISSN: 2469-9950

ID: CaltechAUTHORS:20161107-095433292

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Abstract: In this paper, we present an exactly solvable model for two-dimensional topological superconductors with helical Majorana edge modes protected by time-reversal symmetry. Our construction is based on the idea of decorated domain walls and makes use of the Kasteleyn orientation on a two-dimensional lattice, which was used for the construction of the symmetry protected fermion phase with Z_2 symmetry by Tarantino et al. and Ware et al. By decorating the time-reversal domain walls with spinful Majorana chains, we are able to construct a commuting projector Hamiltonian with zero correlation length ground state wave function that realizes a strongly interacting version of the two-dimensional topological superconductor. From our construction, it can be seen that the T_2 = −1 transformation rule for the fermions is crucial for the existence of such a nontrivial phase; with T_2 = 1, our construction does not work.

Publication: Physical Review B Vol.: 98 No.: 9 ISSN: 2469-9950

ID: CaltechAUTHORS:20171023-101545053

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Abstract: Fractons are gapped pointlike 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 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 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 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 Z_2 gauge theory with two different Gauss' law constraints on different lattice sites.

Publication: Physical Review B Vol.: 98 No.: 3 ISSN: 2469-9950

ID: CaltechAUTHORS:20180710-075128971

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Abstract: Fracton models, a collection of exotic gapped lattice Hamiltonians recently discovered in three spatial dimensions, contain some “topological” features: They support fractional bulk excitations (dubbed fractons) and a ground-state degeneracy that is robust to local perturbations. However, because previous fracton models have been defined and analyzed only on a cubic lattice with periodic boundary conditions, it is unclear to what extent a notion of topology is applicable. In this paper, we demonstrate that the X-cube model, a prototypical type-I fracton model, can be defined on general three-dimensional manifolds. Our construction revolves around the notion of a singular compact total foliation of the spatial manifold, which constructs a lattice from intersecting stacks of parallel surfaces called leaves. We find that the ground-state degeneracy depends on the topology of the leaves and the pattern of leaf intersections. We further show that such a dependence can be understood from a renormalization group transformation for the X-cube model, wherein the system size can be changed by adding or removing 2D layers of topological states. Our results lead to an improved definition of fracton phase and bring to the fore the topological nature of fracton orders.

Publication: Physical Review X Vol.: 8 No.: 3 ISSN: 2160-3308

ID: CaltechAUTHORS:20180829-095556028

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Abstract: Symmetry fractionalization describes the fascinating phenomena that excitations in a 2D topological system can transform under symmetry in a fractional way. For example in fractional quantum Hall systems, excitations can carry fractional charges while the electrons making up the system have charge one. An important question is to understand what symmetry fractionalization (SF) patterns are possible given different types of topological order and different global symmetries. A lot of progress has been made recently in classifying the SF patterns, providing deep insight into the strongly correlated experimental signatures of systems like spin liquids and topological insulators. We review recent developments on this topic. First, it was shown that the SF patterns need to satisfy some simple consistency conditions. More interestingly, it was realized that some seemingly consistent SF patterns are actually ‘anomalous’, i.e. they cannot be realized in strictly 2D systems. We review various methods that have been developed to detect such anomalies. Applying such an understanding to 2D spin liquid allows one to enumerate all potentially realizable SF patterns and propose numerical and experimental probing methods to distinguish them. On the other hand, the anomalous SF patterns were shown to exist on the surface of 3D systems and reflect the nontrivial order in the 3D bulk. We review examples of this kind where the bulk states are topological insulators, topological superconductors, or have other symmetry protected topological orders.

Publication: Reviews in Physics Vol.: 2ISSN: 2405-4283

ID: CaltechAUTHORS:20161107-083133279

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Abstract: In many-body localized systems, propagation of information forms a light cone that grows logarithmically with time. However, local changes in energy or other conserved quantities typically spread only within a finite distance. Is it possible to detect the logarithmic light cone generated by a local perturbation from the response of a local operator at a later time? We numerically calculate various correlators in the random-field Heisenberg chain. While the equilibrium retarded correlator A(t = 0)B(t > 0) is not sensitive to the unbounded information propagation, the out-of-time-ordered correlator A(t = 0)B(t > 0)A(t = 0)B(t > 0) can detect the logarithmic light cone. We relate out-of-time-ordered correlators to the Lieb-Robinson bound in many-body localized systems, and show how to detect the logarithmic light cone with retarded correlators in specially designed states. Furthermore, we study the temperature dependence of the logarithmic light cone using out-of-time-ordered correlators.

Publication: Annalen der Physik Vol.: 529 No.: 7 ISSN: 0003-3804

ID: CaltechAUTHORS:20161114-153129646

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Abstract: In this work, we develop a coupled layer construction of fracton topological orders in d=3 spatial dimensions. These topological phases have subextensive 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.

Publication: Physical Review B Vol.: 95 No.: 24 ISSN: 2469-9950

ID: CaltechAUTHORS:20170622-102731134

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Abstract: Three-dimensional gauge theories with a discrete gauge group can emerge from spin models as a gapped topological phase with fractional point excitations (gauge charge) and loop excitations (gauge flux). It is known that 3D gauge theories can be “twisted,” in the sense that the gauge flux loops can have nontrivial braiding statistics among themselves and such twisted gauge theories are realized in models discovered by Dijkgraaf and Witten. A different framework to systematically construct three-dimensional topological phases was proposed by Walker and Wang and a series of examples have been studied. Can the Walker-Wang construction be used to realize the topological order in twisted gauge theories? This is not immediately clear because the Walker-Wang construction is based on a loop condensation picture while the Dijkgraaf-Witten theory is based on a membrane condensation picture. In this paper, we show that the answer to this question is Yes, by presenting an explicit construction of the Walker-Wang models which realize both the twisted and untwisted gauge theories with gauge group Z_2×Z_2. We identify the topological order of the models by performing modular transformations on the ground-state wave functions and show that the modular matrices exactly match those for the Z_2×Z_2 gauge theories. By relating the Walker-Wang construction to the Dijkgraaf-Witten construction, our result opens up a way to study twisted gauge theories with fermonic charges, and correspondingly strongly interacting fermionic symmetry protected topological phases and their surface states, through exactly solvable models.

Publication: Physical Review B Vol.: 95 No.: 11 ISSN: 2469-9950

ID: CaltechAUTHORS:20170206-091125776

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Abstract: 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 3DZ_2 gauge theory with Z_2 global symmetry as an example, and enumerate and describe the corresponding SET phases. In particular, we find four nonanomalous 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.

Publication: Physical Review B Vol.: 94 No.: 19 ISSN: 1098-0121

ID: CaltechAUTHORS:20160510-094301838

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Abstract: 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 ℤ_2 topological order and a symmetry group of the form G=U(1)⋊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 nontrivial 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 nontrivial anomalies, including G=U(1)×ℤ^T_2 and G=U(1)×ℤ^P_2, where ℤ^T_2 and ℤ^P_2 are time-reversal and d=2 reflection symmetry, respectively.

Publication: Physical Review X Vol.: 6 No.: 4 ISSN: 2160-3308

ID: CaltechAUTHORS:20160623-123437341

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Abstract: The boundary of a fractionalized topological phase can be gapped by condensing a proper set of bosonic quasiparticles. Interestingly, in the presence of a global symmetry, such a boundary can have different symmetry transformation properties. Here we present an explicit example of this kind, in the double semion state with time reversal symmetry. We find two distinct cases where the semionic excitations on the boundary can transform either as time reversal singlets or as time reversal (Kramers) doublets, depending on the coherent phase factor of the Bose condensate. The existence of these two possibilities are demonstrated using both field-theory argument and exactly solvable lattice models. Furthermore, we study the domain walls between these two types of gapped boundaries and find that the application of time reversal symmetry tunnels a semion between them.

Publication: Physical Review B Vol.: 93 No.: 23 ISSN: 2469-9950

ID: CaltechAUTHORS:20160623-125613186

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Abstract: It is well known that unitary symmetries can be “gauged,” i.e., defined to act in a local way, which leads to a corresponding gauge field. Gauging, for example, the charge-conservation symmetry leads to electromagnetic gauge fields. It is an open question whether an analogous process is possible for time reversal which is an antiunitary symmetry. Here, we discuss a route to gauging time-reversal symmetry that applies to gapped quantum ground states that admit a tensor network representation. The tensor network representation of quantum states provides a notion of locality for the wave function coefficient and hence a notion of locality for the action of complex conjugation in antiunitary symmetries. Based on that, we show how time reversal can be applied locally and also describe time-reversal symmetry twists that act as gauge fluxes through nontrivial loops in the system. As with unitary symmetries, gauging time reversal provides useful access to the physical properties of the system. We show how topological invariants of certain time-reversal symmetric topological phases in D=1, 2 are readily extracted using these ideas.

Publication: Physical Review X Vol.: 5 No.: 4 ISSN: 2160-3308

ID: CaltechAUTHORS:20160104-140519845

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Abstract: In addition to possessing fractional statistics, anyon excitations of a 2D topologically ordered state can realize symmetry in distinct ways, leading to a variety of symmetry-enriched topological (SET) phases. While the symmetry fractionalization must be consistent with the fusion and braiding rules of the anyons, not all ostensibly consistent symmetry fractionalizations can be realized in 2D systems. Instead, certain “anomalous” SETs can only occur on the surface of a 3D symmetry-protected topological (SPT) phase. In this paper, we describe a procedure for determining whether a SET of a discrete, on-site, unitary symmetry group G is anomalous or not. The basic idea is to gauge the symmetry and expose the anomaly as an obstruction to a consistent topological theory combining both the original anyons and the gauge fluxes. Utilizing a result of Etingof, Nikshych, and Ostrik, we point out that a class of obstructions is captured by the fourth cohomology group H^4 (G,U(1)), which also precisely labels the set of 3D SPT phases, with symmetry group G. An explicit procedure for calculating the cohomology data from a SET is given, with the corresponding physical intuition explained. We thus establish a general bulk-boundary correspondence between the anomalous SET and the 3D bulk SPT whose surface termination realizes it. We illustrate this idea using the chiral spin liquid [U(1)_2] topological order with a reduced symmetry Z_2 ×Z_2 ⊂SO(3) , which can act on the semion quasiparticle in an anomalous way. We construct exactly solved 3D SPT models realizing the anomalous surface terminations and demonstrate that they are nontrivial by computing three-loop braiding statistics. Possible extensions to antiunitary symmetries are also discussed.

Publication: Physical Review X Vol.: 5 No.: 4 ISSN: 2160-3308

ID: CaltechAUTHORS:20151113-155112359

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Abstract: Topological quantum states cannot be created from product states with local quantum circuits of constant depth and are in this sense more entangled than topologically trivial states, but how entangled are they? Here we quantify the entanglement in one-dimensional topological states by showing that local quantum circuits of linear depth are necessary to generate them from product states. We establish this linear lower bound for both bosonic and fermionic one-dimensional topological phases and use symmetric circuits for phases with symmetry. We also show that the linear lower bound can be saturated by explicitly constructing circuits generating these topological states. The same results hold for local quantum circuits connecting topological states in different phases.

Publication: Physical Review B Vol.: 91 No.: 19 ISSN: 1098-0121

ID: CaltechAUTHORS:20150618-130311270

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Abstract: We construct an exactly soluble Hamiltonian on the D=3 cubic lattice, whose ground state is a topological phase of bosons protected by time-reversal symmetry, i.e., a symmetry-protected topological (SPT) phase. In this model, excitations with anyonic statistics are shown to exist at the surface but not in the bulk. The statistics of these surface anyons is explicitly computed and shown to be identical to the three-fermion Z2 model, a variant of Z2 topological order which cannot be realized in a purely D=2 system with time-reversal symmetry. Thus the model realizes a novel surface termination for three-dimensional (3D) SPT phases, that of a fully symmetric gapped surface with topological order. The 3D phase found here was previously proposed from a field theoretic analysis but is outside the group cohomology classification that appears to capture all SPT phases in lower dimensions. Such phases may potentially be realized in spin-orbit-coupled magnetic insulators, which evade magnetic ordering. Our construction utilizes the Walker-Wang prescription to create a 3D confined phase with surface anyons, which can be extended to other topological phases.

Publication: Physical Review B Vol.: 90 No.: 24 ISSN: 1098-0121

ID: CaltechAUTHORS:20150115-095144828

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Abstract: Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phases which is protected by SO(3) spin rotation symmetry. The topological insulator is another example of SPT phases which are protected by U(1) and time-reversal symmetries. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: Distinct d-dimensional bosonic SPT phases with on-site symmetry G (which may contain antiunitary time-reversal symmetry) can be labeled by the elements in H^(1+d)[G,UT(1)], the Borel (1+d)-group-cohomology classes of G over the G module UT(1). Our theory, which leads to explicit ground-state wave functions and commuting projector Hamiltonians, is based on a new type of topological term that generalizes the topological θ term in continuous nonlinear σ models to lattice nonlinear σ models. The boundary excitations of the nontrivial SPT phases are described by lattice nonlinear σ models with a nonlocal Lagrangian term that generalizes the Wess-Zumino-Witten term for continuous nonlinear σ models. As a result, the symmetry G must be realized as a non-on-site symmetry for the low-energy boundary excitations, and those boundary states must be gapless or degenerate. As an application of our result, we can use H^(1+d)[U(1)⋊ Z^(T)_(2),U_T(1)] to obtain interacting bosonic topological insulators (protected by time reversal Z2T and boson number conservation), which contain one nontrivial phase in one-dimensional (1D) or 2D and three in 3D. We also obtain interacting bosonic topological superconductors (protected by time-reversal symmetry only), in term of H^(1+d)[Z^(T)_(2),U_T(1)], which contain one nontrivial phase in odd spatial dimensions and none for even dimensions. Our result is much more general than the above two examples, since it is for any symmetry group. For example, we can use H1+d[U(1)×Z2T,UT(1)] to construct the SPT phases of integer spin systems with time-reversal and U(1) spin rotation symmetry, which contain three nontrivial SPT phases in 1D, none in 2D, and seven in 3D. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: (G_H,G_Ψ,H^(1+d)[G_Ψ,U_T(1)]), where G_H is the symmetry group of the Hamiltonian and G_Ψ the symmetry group of the ground states.

Publication: Physical Review B Vol.: 87 No.: 15 ISSN: 1098-0121

ID: CaltechAUTHORS:20130509-104730026

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