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arXiv:2012.12280 · Physical Review Research

Theory of weak symmetry breaking of translations in Z2 topologically ordered states and its relation to topological superconductivity from an exact lattice Z2 charge-flux attachment

Peng Rao, Inti Sodemann

Published May 14, 2021• 18 citations
arXivHTMLPDF

Abstract

We study Z 2 topologically ordered states enriched by translational symmetry by employing a recently developed two-dimensional (2D) bosonization approach that implements an exact Z 2 charge-flux attachment in the lattice.Such states can display "weak symmetry breaking" of translations, in which both the Hamiltonian and ground state remain fully translational invariant but the symmetry is "broken" by its anyon quasiparticles, in the sense that its action maps them into a different superselection sector.We demonstrate that this phenomenon occurs when the fermionic spinons form a weak topological superconductor in the form of a 2D stack of one-dimensional Kitaev wires, leading to the amusing property that there is no local operator that can transport the π -flux quasiparticle across a single Kitaev wire of fermonic spinons without paying an energy gap in spite of the vacuum remaining fully translational invariant.We explain why this phenomenon occurs hand in hand with other previously identified peculiar features such as ground-state degeneracy dependence on the size of the torus and the appearance of dangling boundary Majorana modes in certain Z 2 topologically ordered states.Moreover, by extending the Z 2 charge-flux attachment to open lattices and cylinders, we construct a plethora of exactly solvable models providing an exact description of their dispersive Majorana gapless boundary modes.We also review the Z × (Z 2 ) 3 classification of 2D BdG Hamiltonians (class D) enriched by translational symmetry and provide arguments on its robust stability against interactions and self-averaging disorder that preserve translational symmetry.

DOI
10.1103/physrevresearch.3.023120

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We study Z 2 topologically ordered states enriched by translational symmetry by employing a recently developed two-dimensional (2D) bosonization approach that implements an exact Z 2 charge-flux attachment in the lattice.Such states can display "weak symmetry breaking" of translations, in which both the Hamiltonian and ground state remain fully translational invariant but the symmetry is "broken" by its anyon quasiparticles, in the sense that its action maps them into a different superselection sector.We demonstrate that this phenomenon occurs when the fermionic spinons form a weak topological superconductor in the form of a 2D stack of one-dimensional Kitaev wires, leading to the amusing property that there is no local operator that can transport the π -flux quasiparticle across a single Kitaev wire of fermonic spinons without paying an energy gap in spite of the vacuum remaining fully translational invariant.We explain why this phenomenon occurs hand in hand with other previously identified peculiar features such as ground-state degeneracy dependence on the size of the torus and the appearance of dangling boundary Majorana modes in certain Z 2 topologically ordered states.Moreover, by extending the Z 2 charge-flux attachment to open lattices and cylinders, we construct a plethora of exactly solvable models providing an exact description of their dispersive Majorana gapless boundary modes.We also review the Z × (Z 2 ) 3 classification of 2D BdG Hamiltonians (class D) enriched by translational symmetry and provide arguments on its robust stability against interactions and self-averaging disorder that preserve translational symmetry.
One-dimensional systems with topological order are intimately related to the appearance of zero-energy modes localized on their boundaries. The most common example is the Kitaev chain, which displays Majorana zero-energy modes and it is characterized by a twofold ground-state degeneracy related to the global Z2{\mathbb{Z}}_{2}Z2​ symmetry associated with fermionic parity. By extending the symmetry to the ZN{\mathbb{Z}}_{N}ZN​ group, it is possible to engineer systems hosting topological parafermionic modes. In this work, we address one-dimensional systems with a generic discrete symmetry group GGG. We define a ladder model of gauge fluxes that generalizes the Ising and Potts models and displays a symmetry broken phase. Through a non-Abelian Jordan-Wigner transformation, we map this flux ladder into a model of dyonic operators, defined by the group elements and irreducible representations of GGG. We show that the so-obtained dyonic model has topological order, with zero-energy modes localized at its boundary. These dyonic zero-energy modes are in general weak topological modes, but strong dyonic zero modes appear when suitable position-dependent couplings are considered.
A time-reversal invariant Kitaev-type model is introduced in which spins (Dirac matrices) on the square lattice interact via anisotropic nearest-neighbor and next-nearest-neighbor exchange interactions. The model is exactly solved by mapping it onto a tight-binding model of free Majorana fermions coupled with static Z2{\mathbb{Z}}_{2}Z2​ gauge fields. The Majorana fermion model can be viewed as a model of time-reversal-invariant superconductor and is classified as a member of symmetry class DIII in the Altland-Zirnbauer classification. The ground-state phase diagram has two topologically distinct gapped phases which are distinguished by a Z2{\mathbb{Z}}_{2}Z2​ topological invariant. The topologically nontrivial phase supports both a Kramers' pair of gapless Majorana edge modes at the boundary and a Kramers' pair of zero-energy Majorana states bound to a 0-flux vortex in the $\ensuremath{\pi}$-flux background. Power-law decaying correlation functions of spins along the edge are obtained by taking the gapless Majorana edge modes into account. The model is also defined on the one-dimension ladder, in which case again the ground-state phase diagram has Z2{\mathbb{Z}}_{2}Z2​ trivial and nontrivial phases.
Understanding the interplay between symmetries and phases lies at the heart of characterizing states of matter. The conventional Landau's paradigm associates local ordered states with symmetry breaking, but this paradigm fails to include gapped topological phases, which can only be distinguished by the global topological invariant and the present of gapless edge modes at open boundaries. In this thesis, variety of states of matter is enriched from two aspects: a new paradigm of phases that incorporates both symmetry broken phase and topological phase, and exotic topological phases that are realized in different degree of freedoms of the same system. The two aspects are explored by studying corresponding one dimensional fermionic models. Besides gapped phases, exotic quasiparticle interferences of gapless topological systems are also investigated. \n \nThe first chapter serves as a theoretical basis for the next two chapters by giving an introduction to topological phases in one dimension. In particular, a bosonization method is introduced to describe topological degeneracy and edge zero modes, which is favoured by the ability to deal with interactions as well as to uncover the phenomena of spin-charge separation. \n \nIn the second chapter a new paradigm of phases is proposed and investigated that incorporates both symmetry broken phases and symmetry-protected topological phases. Through exact-solvable models bond spin density wave(BSDW) states are proposed as candidates for this new paradigm, which are ZZ2\\ZZ_2ZZ2​ symmetry breaking due to spontaneous dimerization and yet each is topological nontrivial with edge zero modes of fractional fermions. More general lattice models are constructed, and it is shown that BSDW exists in a considerable regime of parameters for a family of lattice models, from both mean-field analysis and bosonization. \n \nIn the third chapter, we give an effective field description for spin-charge separated edge modes consisting of both Majorana modes and fractional fermions, and demonstrate how to harvest both simultaneously in a spinless fermionic ladder with particle-hole symmetry and single-chain fermion parity. Further, the idea of spin-charge separated edge modes is applied to reinterpret the prominent Haldane phase of spin-1 chain in its equivalent spin-frac12\\frac{1}{2}frac12 fermionic model. \n \nIn contrast to the above two theoretically oriented topics that aim to add variety to topological phases, the third topic establishes a bridge between abstract concepts for topological systems to remarkable features that can be detected in experiments. The last chapter reveals how topological charges for topological Fermi points, through exhibiting exotic quasiparticle interference in the presence of impurities, could be visualized in STS measurements. \nThe generalized joint density of states(GJDOS) is applied to address effects from the geometry of band dispersion and topological charges separately. Both analytical results and numerical simulation suggest distinct and robust quasiparticle interference features as indicators for topological charges that can be accessed by FT-STS experiments.
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-eee 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.
We develop an exact map of all states and operators from two-dimensional lattices of spins-1/21/21/2 into lattices of fermions and bosons with mutual semionic statistical interaction that goes beyond previous dualities of Z2{\mathbb{Z}}_{2}Z2​ lattice gauge theories because it does not rely on imposing local conservation laws and captures the motion of ``charges'' and ``fluxes'' on equal footing. This map allows to explicitly compute the Berry phases for the transport of fluxes in a large class of symmetry-enriched topologically ordered states with emergent Z2{\mathbb{Z}}_{2}Z2​ gauge fields that includes chiral, nonchiral, Abelian or non-Abelian, that can be perturbatively connected to models where the visons are static and the emergent fermionic spinons have a noninteracting dispersion. The numerical complexity of computing such vison phases reduces, therefore, to computing overlaps of ground states of free-fermion Hamiltonians. Among other results, we establish numerically the conditions under which the Majorana-carrying flux excitation in Ising-topologically ordered states enriched by translations acquires the 0 or $\ensuremath{\pi}$ phase when moving around a single plaquette.
The simplest continuum model of a one-dimensional noninteracting superconducting fermionic symmetry-protected topological (SPT) phase is studied in great detail using analytical methods. In a first step, we present a full exact diagonalization of the fermionic Bogoliubov--de Gennes Hamiltonian for a system of finite length and with open boundaries. In particular, we derive exact analytical expressions for the Majorana zero modes emerging in the topologically nontrivial phase, revealing their spatial localization, their transformation properties under symmetry operations, and the exact finite-size energy splitting of the associated quasidegenerate ground states. We then proceed to analyze the model via exact operator bosonization in both open and closed geometries. In the closed wire geometry, we demonstrate fermion parity switching from twisting boundary conditions in the topologically nontrivial phase. For the open wire, on the other hand, we first take a semiclassical approach employing the Mathieu equation to study the two quasidegenerate ground states as well as their energy splitting at finite system sizes. We then finally derive the exact forms of the Majorana zero modes in the bosonic language using vertex-algebra techniques. These modes are verified to be in exact agreement with the results obtained from the fermionic description. The complementary viewpoints provided by the fermionic and bosonic formulations of the superconducting SPT phase are reconciled, allowing us to provide a complete and exact account of how Majorana zero modes manifest in a bosonized description of an SPT phase.