Translation symmetry-enriched toric code insulator Pok Man Tam,1 J¨ornW. F. Venderbos,2, 3 and Charles L. Kane1 1Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA 2Department of Physics, Drexel University, Philadelphia, PA 19104, USA 3Department of Materials Science & Engineering, Drexel University, Philadelphia, PA 19104, USA We introduce a two-dimensional electronic insulator that possesses a toric code topological order enriched by translation symmetry. This state can be realized from disordering a weak topological superconductor by double-vortex condensation. It is termed the toric code insulator, whose anyonic excitations consist of a charge-e chargon, a neutral fermion and two types of visons. There are two types of visons because they have constrained motion as a consequence of the fractional Josephson effect of one-dimensional topological superconductor. Importantly, these two types of visons are related by a discrete translation symmetry and have a mutual semionic braiding statistics, leading to a symmetry-enrichment akin to the type in Wen's plaquette model and Kitaev's honeycomb model. We construct this state using a three-fluid coupled-wire model, and analyze the anyon spectrum and braiding statistics in detail to unveil the nature of symmetry-enrichment due to translation. We also discuss potential material realizations and present a band-theoretic understanding of the state, fitting it into a general framework for studying fractionalizaton in strongly-interacting weak topological phases. I. INTRODUCTION as topological orders [10, 11]. Well-known examples of topological order include fractional quantum Hall states Over the past decades, symmetry and topology have and quantum spin liquid states [12, 13]. In the case of the emerged as two central and interwoven organizing prin- former, recent experimental evidence for the fractional ciples in the study of condensed matter physics. In the statistics of Laughlin quasiparticles has been reported case of weakly interacting systems, symmetry-protected [14, 15]. Perhaps the simplest example of topological or- topological (SPT) phases|a class which includes topo- der, however, is of the Z2 type, which was first studied in logical insulators (TIs) and superconductors (TSCs)| the context of frustrated quantum antiferromagnets [16{ have been predicted theoretically and in a number of 19] and later reconsidered in the form of Kitaev's toric cases discovered experimentally [1,2]. Internal symme- code toy model, as well as Wen's plaquette model [20, 21]. tries, such as time-reversal and particle-hole symmetries, Given the compelling appeal of the toric code model, in give rise to so-called strong SPT phases with protected this paper we refer to the Z2 topological order as toric gapless boundary modes [3,4]. In addition to strong code topological order. It features four types of anyons: 1, e, m and f = e m, where e and m are self-bosons SPT phases, there exist weak SPT phases, which can be × viewed and constructed as stacks of strong SPT phases which obey a mutual π-braiding (semionic) statistics as from a lower dimension. Importantly, the distinction of well as the Z2 fusion rule. Remarkably, as pointed out weak SPTs from a trivial phase requires an additional by Hansson et al. [22], toric code topological order not discrete translation symmetry along the stacking direc- only arises in spin systems, but also in conventional su- tion, which prevents hybridization of pairs of stacked lay- perconductors, where f is interpreted as the Bogoliubov ers. Prototypical examples of weak SPTs include the fermion and m as the superconducting vortex. It is then three-dimensional (3D) weak TI [5{7], which harbors natural to wonder to what extent and how the structure an even number of surface Dirac modes, and the two- of the topological order is modified in unconventional dimensional (2D) weak TSC [8,9], which harbors a pair superconductors. More specifically, it is natural to ask of counter-propagating Majorana edge modes. While the whether a 2D weak TSC can provide a platform for a basic properties of weak topological phases are well un- toric code topological order enriched by translation sym- derstood in the weakly-interacting regime, less is known metry. Here we answer this question in the affirmative arXiv:2107.04030v1 [cond-mat.str-el] 8 Jul 2021 about the effect of strong interactions, which may lead by constructing an insulator from a strongly interacting to exotic correlated phases and phenomena either on the weak TSC and showing that the resulting anyon spec- boundary or in the bulk. The purpose of this paper is trum exhibits symmetry-enrichment. to study the effect of strong interactions on weak topo- In general, a symmetry-enriched topological (SET) or- logical phases, and in particular to address the interplay der can exhibit many interesting properties in addition to between translation symmetry and topology. To this end, the fusion and braiding properties of anyons [23{28]. For we focus on the paradigmatic example of a strongly in- instance, it can feature fractionalized symmetry quantum teracting weak TSC in two dimensions. numbers, such as the electric charge of Laughlin quasi- Strong interactions can give rise to correlated quan- particles. In this paper, we focus on another aspect: the tum phases with emergent fractionalized quasiparticles permutation of anyon types by a symmetry transforma- known as anyons. Such quantum phases are referred to tion. This has been famously demonstrated in spin sys- 2 <latexit sha1_base64="OVwIzM+6tBhkSzt+Mvb4k2/ze70=">AAACLXicbVDLSgNBEJz1bXytevQyGAIKIewGUY+iHjwqmBjIhtA7mWSHzD6Y6Q2EJT/kxV8RwYMiXv0NJ5uAMVowUF3VTU+Xn0ih0XHerIXFpeWV1bX1wsbm1vaOvbtX13GqGK+xWMaq4YPmUkS8hgIlbySKQ+hL/uD3r8b+w4ArLeLoHocJb4XQi0RXMEAjte3rkpd5A1BJINqeCuIy9TDgCHnhjcozrha9EH78vPRGhSM4bttFp+LkoH+JOyVFMsVt237xOjFLQx4hk6B103USbGWgUDDJRwUv1TwB1ocebxoaQch1K8uvHdGSUTq0GyvzIqS5OjuRQaj1MPRNZwgY6HlvLP7nNVPsnrcyESUp8ohNFnVTSTGm4+hoRyjOUA4NAaaE+StlAShgaAIumBDc+ZP/knq14p5WqncnxYvLaRxr5IAckiPikjNyQW7ILakRRh7JM3kj79aT9Wp9WJ+T1gVrOrNPfsH6+gY0Qaiv</latexit> <latexit sha1_base64="8SddBbpKHVMZZT6kTE7cAwoOOxE=">AAACLXicbVDLSgNBEJz1bXytevQyGAIKIewGUY+iHjwqmBjIhtA7mWSHzD6Y6Q2EJT/kxV8RwYMiXv0NJ5uAMVowUF3VTU+Xn0ih0XHerIXFpeWV1bX1wsbm1vaOvbtX13GqGK+xWMaq4YPmUkS8hgIlbySKQ+hL/uD3r8b+w4ArLeLoHocJb4XQi0RXMEAjte3rkpd5A1BJINqeCuIy9TDgCHnhjcozrha9EH78vPRGhSP/uG0XnYqTg/4l7pQUyRS3bfvF68QsDXmETILWTddJsJWBQsEkHxW8VPMEWB96vGloBCHXrSy/dkRLRunQbqzMi5Dm6uxEBqHWw9A3nSFgoOe9sfif10yxe97KRJSkyCM2WdRNJcWYjqOjHaE4Qzk0BJgS5q+UBaCAoQm4YEJw50/+S+rVintaqd6dFC8up3GskQNySI6IS87IBbkht6RGGHkkz+SNvFtP1qv1YX1OWhes6cw++QXr6xs1xqiw</latexit> <latexit sha1_base64="QVHdQG3fPeKcA/0jIqnziISe3+8=">AAAB+HicbVDLSsNAFJ3UV62PRl26GSxC3ZSkiLosunFZwT6gCWEynTRDJzNhZlKooV/ixoUibv0Ud/6N0zYLbT1w4XDOvdx7T5gyqrTjfFuljc2t7Z3ybmVv/+Cwah8dd5XIJCYdLJiQ/RApwignHU01I/1UEpSEjPTC8d3c702IVFTwRz1NiZ+gEacRxUgbKbCr3gTJNKaBJ2NR1xeBXXMazgJwnbgFqYEC7cD+8oYCZwnhGjOk1MB1Uu3nSGqKGZlVvEyRFOExGpGBoRwlRPn54vAZPDfKEEZCmuIaLtTfEzlKlJomoelMkI7VqjcX//MGmY5u/JzyNNOE4+WiKGNQCzhPAQ6pJFizqSEIS2puhThGEmFtsqqYENzVl9dJt9lwrxrNh8ta67aIowxOwRmoAxdcgxa4B23QARhk4Bm8gjfryXqx3q2PZWvJKmZOwB9Ynz9325L3</latexit> tems such as Wen's plaquette model and Kitaev's honey- <latexit sha1_base64="apv1tuNdPNtjZ5cmtEn/XEtLBzY=">AAAB9XicbVBNSwMxEJ31s9avqkcvwSJ4KrtF1ItQ9OKxgv2A7lqyabYbmk2WJFsppf/DiwdFvPpfvPlvTNs9aOuDgcd7M8zMC1POtHHdb2dldW19Y7OwVdze2d3bLx0cNrXMFKENIrlU7RBrypmgDcMMp+1UUZyEnLbCwe3Ubw2p0kyKBzNKaZDgvmARI9hY6dEfYpXGrOurWF673VLZrbgzoGXi5aQMOerd0pffkyRLqDCEY607npuaYIyVYYTTSdHPNE0xGeA+7VgqcEJ1MJ5dPUGnVumhSCpbwqCZ+ntijBOtR0loOxNsYr3oTcX/vE5moqtgzESaGSrIfFGUcWQkmkaAekxRYvjIEkwUs7ciEmOFibFBFW0I3uLLy6RZrXgXler9ebl2k8dRgGM4gTPw4BJqcAd1aAABBc/wCm/Ok/PivDsf89YVJ585gj9wPn8AUEiSZA==</latexit> (a) (b) ' (t) ' =0 comb model [21, 29], in which e-particles transform into ⇢ ⇢ m-particles (and vice versa) under a discrete translation. <latexit sha1_base64="39BrfJx30nCFSs5IcuKjFKgMf0c=">AAACLXicbVDLSgNBEJz1bXytevQyGAIKIewGUY+iHjwqmBjIhtA7mWSHzD6Y6Q2EJT/kxV8RwYMiXv0NJ5uAMVowUF3VTU+Xn0ih0XHerIXFpeWV1bX1wsbm1vaOvbtX13GqGK+xWMaq4YPmUkS8hgIlbySKQ+hL/uD3r8b+w4ArLeLoHocJb4XQi0RXMEAjte3rkpd5A1BJINqeCuIy9TDgCHnhjcozrha9EH78vPRGhSN23LaLTsXJQf8Sd0qKZIrbtv3idWKWhjxCJkHrpusk2MpAoWCSjwpeqnkCrA893jQ0gpDrVpZfO6Ilo3RoN1bmRUhzdXYig1DrYeibzhAw0PPeWPzPa6bYPW9lIkpS5BGbLOqmkmJMx9HRjlCcoRwaAkwJ81fKAlDA0ARcMCG48yf/JfVqxT2tVO9OiheX0zjWyAE5JEfEJWfkgtyQW1IjjDySZ/JG3q0n69X6sD4nrQvWdGaf/IL19Q03S6ix</latexit> This phenomenon is sometimes known as \weak symme- (c) <latexit sha1_base64="zOIYcVY1RSY7Cdof1QY2EtCF0V4=">AAACK3icbVDLSgNBEJyNrxhfUY9eBkPAg4TdIOpRIoLHCCYRsiH0TibZwdkHM71CWPI/XvwVD3rwgVf/w8kmYEwsGKiu6qany4ul0GjbH1ZuaXlldS2/XtjY3NreKe7uNXWUKMYbLJKRuvNAcylC3kCBkt/FikPgSd7y7i/HfuuBKy2i8BaHMe8EMAhFXzBAI3WLtbKbug+gYl90XeVHx9RFnyNkhTs6nnG1GATw62elOypcdYslu2JnoIvEmZISmaLeLb64vYglAQ+RSdC67dgxdlJQKJjko4KbaB4Du4cBbxsaQsB1J81uHdGyUXq0HynzQqSZOjuRQqD1MPBMZwDo63lvLP7ntRPsn3dSEcYJ8pBNFvUTSTGi4+BoTyjOUA4NAaaE+StlPihgaOItmBCc+ZMXSbNacU4r1ZuT0kVtGkeeHJBDckQcckYuyDWpkwZh5JE8kzfybj1Zr9an9TVpzVnTmX3yB9b3DxH7qC4=</latexit> try breaking", where the pattern of quasiparticles breaks E <latexit sha1_base64="twskZYsDHHfDZ0O6x83k9L5KqSs=">AAACK3icbVDJSgNBEO1xjeMW9eilMQiCEmZE1GOIF48RTCJkQqjpdDKNPQvdNUIY5n+8+Cse9OCCV//DziRgXB40vHqviup6fiKFRsd5s+bmFxaXlksr9ura+sZmeWu7peNUMd5ksYzVjQ+aSxHxJgqU/CZRHEJf8rZ/ezH223dcaRFH1zhKeDeEYSQGggEaqVeu73uZdwcqCUTPU0F8RD0MOEJRePnRjKvFMIRvvyi93D7slStO1SlA/xJ3Sipkikav/OT1Y5aGPEImQeuO6yTYzUChYJLntpdqngC7hSHvGBpByHU3K27N6b5R+nQQK/MipIU6O5FBqPUo9E1nCBjo395Y/M/rpDg472YiSlLkEZssGqSSYkzHwdG+UJyhHBkCTAnzV8oCUMDQxGubENzfJ/8lreOqe1o9vjqp1OrTOEpkl+yRA+KSM1Ijl6RBmoSRe/JIXsir9WA9W+/Wx6R1zprO7JAfsD6/AOqEqBQ=</latexit>