arXiv subjects

Lukasz Fidkowski

Publications and source records attributed to Lukasz Fidkowski.

At least 37 records · Page 2Linked to original sources

arXiv2018

Surface Topological Order and a new 't Hooft Anomaly of Interaction Enabled 3+1D Fermion SPTs

Symmetry protected topological (SPT) phases are well understood in the context of free fermions and in the context of interacting but essentially bosonic models. Recently it has been realized that intrinsically fermionic SPTs exist which only appear in interacting models. Here we show that the 3+1 dimensional realizations of these phases have surface states characterized by a new 't Hooft anomaly, captured by a $H^3(G, Z_2)$ class. This is encoded in the anomalous action of symmetry on the surface states with topological order, which must necessarily permute the anyons. We discuss in detail an example with symmetry group $G = Z_2 \times Z_4$. Using a network model of the surface we derive a candidate surface topological order given by a $Z_4$ gauge theory. We relate our findings to anomalies valued in $H^3$ with various coefficients introduced previously in both bosonic and fermionic settings, and describe a general framework that unifies these various anomalies.

cond-mat.str-el↗

arXiv2017

Radical chiral Floquet phases in a periodically driven Kitaev model and beyond

Time periodic driving serves not only as a convenient way to engineer effective Hamiltonians, but also as a means to produce intrinsically dynamical phases that do not exist in the static limit. A recent example of the latter are 2D chiral Floquet (CF) phases exhibiting anomalous edge dynamics that pump discrete packets of quantum information along one direction. In non-fractionalized systems with only bosonic excitations, this pumping is quantified by a dynamical topological index that is a rational number -- highlighting its difference from the integer valued invariant underlying equilibrium chiral phases (e.g. quantum Hall systems). Here, we explore CF phases in systems with emergent anyon excitations that have fractional statistics (Abelian topological order). Despite the absence of mobile non-Abelian particles in these systems, external driving can supply the energy to pump otherwise immobile non-Abelian defects (sometimes called twist defects or genons) around the boundary, thereby transporting an irrational fractional number of quantum bits along the edge during each drive period. This enables new CF phases with chiral indices that are square roots of rational numbers, inspiring the label: "radical CF phases". We demonstrate an unexpected bulk-boundary correspondence, in which the radical CF edge is tied to bulk dynamics that exchange electric and magnetic anyon excitations during each period. We construct solvable, stroboscopically driven versions of Kitaev's honeycomb spin model that realize these radical CF phases, and discuss their stability against heating in strongly disordered many-body localized settings or in the limit of rapid driving as an exponentially long-lived pre-thermal phenomena.

cond-mat.str-el↗

arXiv2017

Eigenstate phases with finite on-site non-Abelian symmetry

We study the eigenstate phases of disordered spin chains with on-site finite non-Abelian symmetry. We develop a general formalism based on standard group theory to construct local spin Hamiltonians invariant under any on-site symmetry. We then specialize to the case of the simplest non-Abelian group, $S_3$, and numerically study a particular two parameter spin-1 Hamiltonian. We observe a thermal phase and a many-body localized phase with a spontaneous symmetry breaking (SSB) from $S_3$ to $\mathbb{Z}_3$ in our model Hamiltonian. We diagnose these phases using full entanglement distributions and level statistics. We also use a spin-glass diagnostic specialized to detect spontaneous breaking of the $S_3$ symmetry down to $\mathbb{Z}_3$. Our observed phases are consistent with the possibilities outlined by Potter and Vasseur [Phys. Rev. B 94, 224206 (2016)], namely thermal/ ergodic and spin-glass many-body localized (MBL) phases. We also speculate about the nature of an intermediate region between the thermal and MBL+SSB regions where full $S_3$ symmetry exists.

cond-mat.dis-nn↗

arXiv2017

An infinite family of 3d Floquet topological paramagnets

We uncover an infinite family of time-reversal symmetric 3d interacting topological insulators of bosons or spins, in time-periodically driven systems, which we term Floquet topological paramagnets (FTPMs). These FTPM phases exhibit intrinsically dynamical properties that could not occur in thermal equilibrium, and are governed by an infinite set of $Z_2$-valued topological invariants, one for each prime number. The topological invariants are physically characterized by surface magnetic domain walls that act as unidirectional quantum channels, transferring quantized packets of information during each driving period. We construct exactly solvable models realizing each of these phases, and discuss the anomalous dynamics of their topologically protected surface states. Unlike previous encountered examples of Floquet SPT phases, these 3d FTPMs are not captured by group cohomology methods, and cannot be obtained from equilibrium classifications simply by treating the discrete time-translation as an ordinary symmetry. The simplest such FTPM phase can feature anomalous $Z_2$ (toric code) surface topological order, in which the gauge electric and magnetic excitations are exchanged in each Floquet period, which cannot occur in a pure 2d system without breaking time reversal symmetry.

cond-mat.str-el↗

arXiv2017

Interacting invariants for Floquet phases of fermions in two dimensions

We construct a many-body quantized invariant that sharply distinguishes among two dimensional non-equilibrium driven phases of interacting fermions. This is an interacting generalization of a band-structure Floquet quasi-energy winding number, and describes chiral pumping of quantum information along the edge. In particular, our invariant sharply distinguishes between a trivial and anomalous Floquet Anderson insulator in the interacting, many-body localized setting. It also applies more generally to models where only fermion parity is conserved, where it differentiates between trivial models and ones that pump Kitaev Majorana chains to the boundary, such as ones recently introduced in the context of emergent fermions arising from eigenstate $\Z_2$ topological order. We evaluate our invariant for the edge of such a system with eigenstate $\Z_2$ topological order, and show that it is necessarily nonzero when the Floquet unitary exchanges electric and magnetic excitations, proving a connection between bulk anyonic symmetry and edge chirality.

cond-mat.str-el↗

arXiv2016

Symmetry enriched string-nets: Exactly solvable models for SET phases

We construct exactly solvable models for a wide class of symmetry enriched topological (SET) phases. Our construction applies to 2D bosonic SET phases with finite unitary onsite symmetry group $G$ and we conjecture that our models realize every phase in this class that can be described by a commuting projector Hamiltonian. Our models are designed so that they have a special property: if we couple them to a dynamical lattice gauge field with gauge group $G$, the resulting gauge theories are equivalent to modified string-net models. This property is what allows us to analyze our models in generality. As an example, we present a model for a phase with the same anyon excitations as the toric code and with a $\mathbb{Z}_2$ symmetry which exchanges the $e$ and $m$ type anyons. We further illustrate our construction with a number of additional examples.

cond-mat.str-el↗

arXiv2016

Twisted Interferometry: the topological perspective

Three manifold topology is used to analyze the effect of anyonic interferometers in which the probe anyons' path along an arm crosses itself, leading to a "twisted" or braided space-time trajectory for the probe anyons. In the case of Ising non-Abelian anyons, twisted interferometry is shown to be able to generate a topologically protected $π/8$-phase gate, which cannot be generated from quasiparticle braiding.

quant-ph↗

arXiv2016

Chiral Floquet Phases of Many-body Localized Bosons

We construct and classify chiral topological phases in driven (Floquet) systems of strongly interacting bosons, with finite-dimensional site Hilbert spaces, in two spatial dimensions. The construction proceeds by introducing exactly soluble models with chiral edges, which in the presence of many-body localization (MBL) in the bulk are argued to lead to stable chiral phases. These chiral phases do not require any symmetry, and in fact owe their existence to the absence of energy conservation in driven systems. Surprisingly, we show that they are classified by a quantized many-body index, which is well defined for any MBL Floquet system. The value of this index, which is always the logarithm of a positive rational number, can be interpreted as the entropy per Floquet cycle pumped along the edge, formalizing the notion of quantum-information flow. We explicitly compute this index for specific models, and show that the nontrivial topology leads to edge thermalization, which provides an interesting link between bulk topology and chaos at the edge. We also discuss chiral Floquet phases in interacting fermionic systems and their relation to chiral bosonic phases.

cond-mat.dis-nn↗

arXiv2016

Realizing anomalous anyonic symmetries at the surfaces of 3d gauge theories

The hallmark of a 2 dimensional topologically ordered phase is the existence of deconfined `anyon' excitations that have exotic braiding and exchange statistics, different from those of ordinary bosons or fermions. As opposed to conventional Landau-Ginzburg-Wilson phases, which are classified on the basis of the spontaneous breaking of an underlying symmetry, topologically ordered phases, such as those occurring in the fractional quantum Hall effect, are absolutely stable, not requiring any such symmetry. Recently, though, it has been realized that symmetries, which may still be present in such systems, can give rise to a host of new, distinct, many-body phases, all of which share the same underlying topological order. A useful approach to classifying SETs is to determine all possible such symmetry actions on the anyons that are algebraically consistent with the anyons' statistics. Remarkably, however, there exist symmetry actions that, despite being algebraically consistent, cannot be realized in any physical system, and hence do not lead to valid 2d SETs. One class of such `anomalous' SETs, characterized by certain disallowed symmetry fractionalization patters, finds a physical interpretation as an allowed surface state of certain 3d short-range entangled phases, but another, characterized by some seemingly valid but anomalous permutation actions of the symmetry on the anyons, has so far eluded a physical interpretation. In this work, we find a physical realization for these anomalously permuting SETs as surface theories of certain 3d long-range entangled phases, completing our understanding of general anomalous SETs in 2 dimensions.

cond-mat.str-el↗

arXiv2016

Discrete spin structures and commuting projector models for 2d fermionic symmetry protected topological phases

We construct exactly solved commuting projector Hamiltonian lattice models for all known 2+1d fermionic symmetry protected topological phases (SPTs) with on-site unitary symmetry group $G_f = G \times \mathbb{Z}_2^f$, where $G$ is finite and $\mathbb{Z}_2^f$ is the fermion parity symmetry. In particular, our models transcend the class of group supercohomology models, which realize some, but not all, fermionic SPTs in 2+1d. A natural ingredient in our construction is a discrete form of the spin structure of the 2d spatial surface $M$ on which our model is defined, namely a `Kasteleyn' orientation of a certain graph associated with the lattice. As a special case, our construction yields commuting projector models for all $8$ members of the $\mathbb{Z}_8$ classification of 2d fermionic SPTs with $G = \mathbb{Z}_2$.

cond-mat.str-el↗

arXiv2016

Symmetry fractionalization and twist defects

Topological order in two dimensions can be described in terms of deconfined quasiparticle excitations - anyons - and their braiding statistics. However, it has recently been realized that this data does not completely describe the situation in the presence of an unbroken global symmetry. In this case, there can be multiple distinct quantum phases with the same anyons and statistics, but with different patterns of symmetry fractionalization - termed symmetry enriched topological (SET) order. When the global symmetry group $G$, which we take to be discrete, does not change topological superselection sectors - i.e. does not change one type of anyon into a different type of anyon - one can imagine a local version of the action of $G$ around each anyon. This leads to projective representations and a group cohomology description of symmetry fractionalization, with $H^2(G,{\cal A})$ being the relevant group. In this paper, we treat the general case of a symmetry group $G$ possibly permuting anyon types. We show that despite the lack of a local action of $G$, one can still make sense of a so-called twisted group cohomology description of symmetry fractionalization, and show how this data is encoded in the associativity of fusion rules of the extrinsic `twist' defects of the symmetry. Furthermore, building on work of Hermele, we construct a wide class of exactly solved models which exhibit this twisted symmetry fractionalization, and connect them to our formal framework.

cond-mat.str-el↗

arXiv2015

Exactly Soluble Model of a 3D Symmetry Protected Topological Phase of Bosons with Surface Topological Order

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 anyonic excitations 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 3-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 SPT phases, which only becomes available in D=3: that of a fully symmetric gapped surface with topological order. The 3D phase found here is also outside the group cohomology classification that appears to capture all SPT phases in lower dimensions. It is identified with a phase previously predicted from a field theoretic analysis, whose surface is roughly `half' of a Kitaev E8 phase. Our construction utilizes the Walker-Wang prescription to create a 3D confined phase with surface anyons,which can be extended to other symmetry classes and topological orders.

cond-mat.str-el↗

arXiv2015

Anomalous Symmetry Fractionalization and Surface Topological Order

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 an SET of a discrete, onsite, 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 are captured by the fourth cohomology group $H^4( G, \,U(1))$, which also precisely labels the set of 3D SPT phases, with symmetry group $G$. 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 $\mathbb{Z}_2 \times \mathbb{Z}_2 \subset SO(3)$, 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 non-trivial by computing three loop braiding statistics. Possible extensions to anti-unitary symmetries are also discussed.

cond-mat.str-el↗

arXiv2014

Interaction effects on 3D topological superconductors: surface topological order from vortex condensation, the 16 fold way and fermionic Kramers doublets

Three dimensional topological superconductors with time reversal symmetry (class DIII) are indexed by an integer $ν$, the number of surface Majorana cones, according to the free fermion classification. The superfluid B phase of He$^3$ realizes the $ν=1$ phase. Recently, it has been argued that this classification is reduced in the presence of interactions to Z$_{16}$. This was argued from the symmetry respecting surface topological orders of these states, which provide a non-perturbative definition of the bulk topological phase. Here, we verify this conclusion by focusing on the even index case, $ν=2m$, where a vortex condensation approach can be used to explicitly derive the surface topological orders. We show a direct relation to the well known result on one dimensional topological superconductors (class BDI), where interactions reduce the free fermion classification from Z down to Z$_8$. Finally, we discuss in detail the fermionic analog of Kramers time reversal symmetry, which allows semions of some surface topological orders to transform as $T^2=\pm i$.

cond-mat.str-el↗

arXiv2014

Symmetry-induced anyon "breeding" in fractional quantum Hall states

An exotic feature of the fractional quantum Hall effect is the emergence of anyons, which are quasiparticle excitations with fractional statistics. In the presence of a symmetry, such as $U(1)$ charge conservation, it is well known that anyons can carry fractional symmetry quantum numbers. In this work we reveal a different class of symmetry realizations: i.e. anyons can "breed" in multiples under symmetry operation. We focus on the global Ising ($Z_2$) symmetry and show examples of these unconventional symmetry realizations in Laughlin-type fractional quantum Hall states. One remarkable consequence of such an Ising symmetry is the emergence of anyons on the Ising symmetry domain walls. We also provide a mathematical framework which generalizes this phenomenon to any Abelian topological orders.

cond-mat.str-el↗

arXiv2013

Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model

Three dimensional topological superconductors (TScs) protected by time reversal (T) symmetry are characterized by gapless Majorana cones on their surface. Free fermion phases with this symmetry (class DIII) are indexed by an integer n, of which n=1 is realized by the B-phase of superfluid Helium-3. Previously it was believed that the surface must be gapless unless time reversal symmetry is broken. Here we argue that a fully symmetric and gapped surface is possible in the presence of strong interactions, if a special type of topological order appears on the surface. The topological order realizes T symmetry in an anomalous way, one that is impossible to achieve in purely two dimensions. For odd n TScs, the surface topological order must be non-Abelian. We propose the simplest non-Abelian topological order that contains electron like excitations, SO(3)_6, with four quasiparticles, as a candidate surface state. Remarkably, this theory has a hidden T invariance which however is broken in any 2D realization. By explicitly constructing an exactly soluble Walker-Wang model we show that it can be realized at the surface of a short ranged entangled 3D fermionic phase protected by T symmetry, with bulk electrons trasforming as Kramers pairs, i.e. T^2=-1 under time reversal. We also propose an Abelian theory, the semion-fermion topological order, to realize an even n TSc surface, for which an explicit model is derived using a coupled layer construction. We argue that this is related to the n=2 TSc, and use this to build candidate surface topological orders for n=4 and n=8 TScs. The latter is equivalent to the three fermion state which is the surface topological order of a Z2 bosonic topological phase protected by T invariance. One particular consequence of this is that an n=16 TSc admits a trivially gapped T-symmetric surface.

cond-mat.str-el↗

arXiv2013

Symmetry Enforced Non-Abelian Topological Order at the Surface of a Topological Insulator

The surfaces of three dimensional topological insulators (3D TIs) are generally described as Dirac metals, with a single Dirac cone. It was previously believed that a gapped surface implied breaking of either time reversal $\mathcal T$ or U(1) charge conservation symmetry. Here we discuss a novel possibility in the presence of interactions, a surface phase that preserves all symmetries but is nevertheless gapped and insulating. Then the surface must develop topological order of a kind that cannot be realized in a 2D system with the same symmetries. We discuss candidate surface states - non-Abelian Quantum Hall states which, when realized in 2D, have $σ_{xy}=1/2$ and hence break $\mathcal T$ symmetry. However, by constructing an exactly soluble 3D lattice model, we show they can be realized as $\mathcal T$ symmetric surface states. The corresponding 3D phases are confined, and have $θ=π$ magnetoelectric response. Two candidate states have the same 12 particle topological order, the (Read-Moore) Pfaffian state with the neutral sector reversed, which we term T-Pfaffian topological order, but differ in their $\mathcal T$ transformation. Although we are unable to connect either of these states directly to the superconducting TI surface, we argue that one of them describes the 3D TI surface, while the other differs from it by a bosonic topological phase. We also discuss the 24 particle Pfaffian-antisemion topological order (which can be connected to the superconducting TI surface) and demonstrate that it can be realized as a $\mathcal T$ symmetric surface state.

cond-mat.str-el↗

arXiv2013

Twisted Interferometry

We propose and analyze the effect of anyonic interferometers that are designed such that the probe anyons traveling in a given path through the interferometer twist or braid around each other. These "twisted" interferometers are found to provide operational utility that may not be available from anyon braiding operations and standard (untwisted) anyonic interferometry measurements. In particular, it enables Ising anyons to generate "magic states," which can be used to implement pi/8-phase gates. We consider the possible implementations of such twisted interferometers in quantum Hall systems, 2D p_x + i p_y superconductors, and 2D Majorana heterostructures, and discuss obstacles and challenges associated with implementation.

quant-ph↗