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arXiv:2406.02482 · Physical review. B./Physical review. B

Three-dimensional fracton topological orders with boundary Toeplitz braiding

Bo-Xi Li, Yao Zhou, Peng Ye

Published Nov 4, 2024• 5 citations
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Abstract

In this paper, we theoretically study a class of three-dimensional (3D) nonliquid states that show exotic boundary phenomena in the thermodynamical limit. More concretely, we focus on a class of 3D fracton topological orders formed via stacking 2D twisted ${\mathbb{Z}}_{N}$ topologically ordered layers along $z$ direction. Nearby layers are coupled while maintaining translation symmetry along $z$ direction. The effective field theory is given by the infinite-component Chern-Simons (iCS) field theory, with an integer-valued symmetric block-tridiagonal Toeplitz $K$ matrix whose size is thermodynamically large. With open boundary conditions (OBC) along $z$, certain choice of $K$ matrices exhibits exotic boundary ``Toeplitz braiding'', where the mutual braiding phase angle between two anyons at opposite boundaries oscillates and remains nonzero in the thermodynamic limit. In contrast, in trivial case, the mutual braiding phase angle decays exponentially to zero in the thermodynamical limit. As a necessary condition, this phenomenon requires the existence of boundary zero modes in the $K$-matrix spectrum under OBC. We categorize nontrivial $K$ matrices into two distinct types. Each type-I possesses two boundary zero modes, whereas each type-II possesses only one boundary zero mode. Interestingly, the integer-valued Hamiltonian matrix of the familiar 1D ``Su-Schrieffer-Heeger model'' can be used as a nontrivial $K$ matrix. Importantly, since large-gauge-invariance ensures integer quantized $K$-matrix entries, global symmetries are not needed to protect these zero modes. We also present numerical simulation as well as finite-size scaling, further confirming the above analytical results. Symmetry fractionalization in iCS field theory is also briefly discussed. Motivated by the present field-theoretical work, it will be interesting to construct 3D lattice models for demonstrating Toeplitz braiding, which is left to future investigation.

DOI
10.1103/physrevb.110.205108

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In this paper, we theoretically study a class of three-dimensional (3D) nonliquid states that show exotic boundary phenomena in the thermodynamical limit. More concretely, we focus on a class of 3D fracton topological orders formed via stacking 2D twisted ZN{\mathbb{Z}}_{N}ZN​ topologically ordered layers along zzz direction. Nearby layers are coupled while maintaining translation symmetry along zzz direction. The effective field theory is given by the infinite-component Chern-Simons (iCS) field theory, with an integer-valued symmetric block-tridiagonal Toeplitz KKK matrix whose size is thermodynamically large. With open boundary conditions (OBC) along zzz, certain choice of KKK matrices exhibits exotic boundary Toeplitz braiding'', where the mutual braiding phase angle between two anyons at opposite boundaries oscillates and remains nonzero in the thermodynamic limit. In contrast, in trivial case, the mutual braiding phase angle decays exponentially to zero in the thermodynamical limit. As a necessary condition, this phenomenon requires the existence of boundary zero modes in the $K$-matrix spectrum under OBC. We categorize nontrivial $K$ matrices into two distinct types. Each type-I possesses two boundary zero modes, whereas each type-II possesses only one boundary zero mode. Interestingly, the integer-valued Hamiltonian matrix of the familiar 1D Su-Schrieffer-Heeger model'' can be used as a nontrivial KKK matrix. Importantly, since large-gauge-invariance ensures integer quantized KKK-matrix entries, global symmetries are not needed to protect these zero modes. We also present numerical simulation as well as finite-size scaling, further confirming the above analytical results. Symmetry fractionalization in iCS field theory is also briefly discussed. Motivated by the present field-theoretical work, it will be interesting to construct 3D lattice models for demonstrating Toeplitz braiding, which is left to future investigation.
Infinite-component BFBFBF field theory: Connection of fracton order, Toeplitz braiding, and non-Hermitian amplification
Building on the infinite-component Chern--Simons theory of three-dimensional fracton phases by Ma et al. [Phys. Rev. B 105, 195124 (2022)] and the Toeplitz braiding of anyons by Li et al.~[Phys. Rev B 110, 205108 (2024)], we show that stacking (3+1)(3+1)(3+1)D BFBFBF topological field theories, which serve as low-energy effective descriptions of a class of three-dimensional topological orders, along a fourth spatial direction gives rise to an exotic class of four-dimensional fracton phases. Their low-energy physics is governed by a new field-theoretic framework, namely \textit{infinite-component BFBFBF} (iBFBFBF) \textit{theories}, characterized by asymmetric integer Toeplitz KKK matrices. Under open boundary conditions along the stacking direction, iBFBFBF theories with properly chosen KKK matrices exhibit a striking phenomenon termed \textit{Toeplitz particle--loop braiding}, where a particle and a loop placed on opposite three-dimensional boundaries acquire a strongly oscillating yet robustly nonvanishing braiding phase even at infinite separation. This nonlocal braiding admits a geometric interpretation: adiabatically transporting the particle induces a winding boundary trajectory on the opposite boundary that encircles the loop. We show that this robustness originates from boundary zero singular modes (ZSMs) of Toeplitz KKK matrices revealed by singular value decomposition, rather than from boundary zero eigenmodes responsible for previously known Toeplitz braiding of anyons, and that the same ZSM mechanism also underlies directional amplification in the rapidly developing field of non-Hermitian physics. We analytically and numerically study representative iBFBFBF theories with Hatano--Nelson--type and non-Hermitian Su--Schrieffer--Heeger--type KKK matrices, establishing a universal correspondence between ZSMs and Toeplitz particle--loop braiding.
Fracton order is an intriguing new type of order which shares many common features with topological order, such as topology-dependent ground-state degeneracies, and excitations with mutual statistics.However, it also has several distinctive geometrical aspects, such as excitations with restricted mobility, which naturally lead to effective descriptions in terms of higher-rank gauge fields.In this paper, we investigate possible effective field theories for three-dimensional fracton order, by presenting a general philosophy whereby topological-like actions for such higher-rank gauge fields can be constructed.Our approach draws inspiration from Chern-Simons and BF theories in 2+1 dimensions, and imposes constraints binding higher-rank gauge charge to higher-rank gauge flux.We show that the resulting fractonic Chern-Simons and BF theories reproduce many of the interesting features of their familiar two-dimensional cousins.We analyze one example of the resulting fractonic Chern-Simons theory in detail, and show that upon quantization it realizes a gapped fracton order with quasiparticle excitations that are mobile only along a subset of one-dimensional lines, and display a form of fractional self-statistics.The ground-state degeneracy of this theory is both topology and geometry dependent, scaling exponentially with the linear system size when the model is placed on a three-dimensional torus.By studying the resulting quantum theory on the lattice, we show that it describes a Z s generalization of the Chamon code.
Symmetry fractionalization on topological excitations is one of the most remarkable quantum phenomena in topological orders with symmetry, i.e., symmetry-enriched topological phases. While much progress has been theoretically and experimentally made in two dimensions (2D), the understanding on symmetry fractionalization in 3D is far from complete. A long-standing challenge is to understand symmetry fractionalization on looplike topological excitations, which are spatially extended objects. In this paper, we construct a powerful topological-field-theoretical framework approach for 3D topological orders, which leads to a systematic characterization and classification of symmetry fractionalization. For systems with Abelian gauge groups (Gg{G}_{g}Gg​) and Abelian symmetry groups (Gs{G}_{s}Gs​), we successfully establish equivalence classes that lead to a finite number of patterns of symmetry fractionalization, although there are notoriously infinite number of Lagrangian-descriptions of the system. Based on this, we compute topologically distinct types of fractional symmetry charges carried by particles. Then, for each type, we compute topologically distinct statistical phases of braiding processes among loop excitations and external symmetry fluxes. As a result, we are able to unambiguously list all physical observables for each pattern of symmetry fractionalization. We present detailed calculations on many concrete examples. As an example, we find that the symmetry fractionalization in an untwisted ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological order with Z2{\mathbb{Z}}_{2}Z2​ symmetry is classified by ${({\mathbb{Z}}_{2})}^{6}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}$. If the topological order is twisted, the classification reduces to (Z2)6{({\mathbb{Z}}_{2})}^{6}(Z2​)6 in which particle excitations always carry integer charge. Inspired by the field-theoretical analysis, we find that our classification of symmetry fractionalization with twist $\ensuremath{\omega}$ can be formally organized into an algebraic formalism: ${⨁}_{{\ensuremath{\nu}}_{i}}{\mathcal{H}}^{4}({G}_{g}{\ensuremath{\leftthreetimes}}_{{\ensuremath{\nu}}_{i}}{G}_{s},U(1))/{\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_{i}),$ where anomaly-free symmetry fractionalization on particles ${\ensuremath{\nu}}_{i}\ensuremath{\in}{\mathcal{H}}_{non}^{2}({G}_{s},{G}_{g})$. Despite the lack of detailed dependence of ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_{i})$ on $\ensuremath{\omega}$ and ${\ensuremath{\nu}}_{i}$, the present field-theoretical approach allows us to efficiently calculate and understand ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_{i})$. Several future directions are presented at the end of this paper.
Universal topological data of topologically ordered phases can be captured by topological quantum field theory in continuous space time by taking the limit of low energies and long wavelengths. While previous continuum field-theoretical studies of topological orders in three-dimensional (3D) real space focused on either self-statistics, braiding statistics, shrinking rules, fusion rules, or quantum dimensions, it has yet to systematically put all topological data together in a unified continuum field-theoretical framework. Here, we construct the topological BFBFBF field theory with twisted terms (e.g., AAdAAAdAAAdA and AABAABAAB) as well as a KKK-matrix BBBBBB term to simultaneously explore all such topological data and reach anomaly-free topological orders. Following the spirit of the famous KKK-matrix Chern-Simons theory of two-dimensional topological orders, we present general formulas and systematically show how the KKK-matrix BBBBBB term confines topological excitations, and how self-statistics of particles is transmuted between bosonic one and fermionic one. To reach anomaly-free topological orders, we explore, within the present continuum field-theoretical framework, how the principle of gauge invariance fundamentally influences possible realizations of topological data. More concretely, we present the topological actions of (i) particle-loop braidings with emergent fermions, (ii) multiloop braidings with emergent fermions, and (iii) Borromean rings braidings with emergent fermions, and calculate their universal topological data. Together with previous efforts, our paper paves the way toward a more systematic and complete continuum field-theoretical analysis of exotic topological properties of 3D topological orders. Several interesting future directions are also discussed.
We introduce hybrid fracton orders: three-dimensional gapped quantum phases that exhibit the phenomenology of both conventional three-dimensional topological orders and fracton orders. Hybrid fracton orders host both (i) mobile topological quasiparticles and loop excitations, as well as (ii) pointlike topological excitations with restricted mobility, with nontrivial fusion rules and mutual braiding statistics between the two sets of excitations. Furthermore, hybrid fracton phases can realize either conventional three-dimensional topological orders or fracton orders after undergoing a phase transition driven by the condensation of certain gapped excitations. Therefore they serve as parent orders for both long-range-entangled quantum liquid and nonliquid phases. We study the detailed properties of hybrid fracton phases through exactly solvable models in which the resulting orders hybridize a three-dimensional Z2{\mathbb{Z}}_{2}Z2​ topological order with (i) the X-cube fracton order or (ii) Haah's code. The hybrid orders presented here can also be understood as the deconfined phase of a gauge theory whose gauge group is given by an Abelian global symmetry GGG and subsystem symmetries of a normal subgroup NNN along lower-dimensional subregions.
While the topological order in two dimensions has been studied extensively since the discovery of the integer and fractional quantum Hall systems, topological states in three spatial dimensions are much less understood.In this paper, we propose a general formalism for constructing a large class of threedimensional topological states by stacking layers of 2D topological states and introducing coupling between them.Using this construction, different types of topological states can be obtained, including those with only surface topological order and no bulk topological quasiparticles, and those with topological order both in the bulk and at the surface.For both classes of states, we study its generic properties and present several explicit examples.As an interesting consequence of this construction, we obtain example systems with nontrivial braiding statistics between string excitations.In addition to studying the string-string braiding in the example system, we propose a topological field-theory description for the layer-constructed systems, which captures not only the string-particle braiding statistics but also the string-string braiding statistics when the coupling is twisted.Last, we provide a proof of a general identity for Abelian string statistics and discuss an example system with non-Abelian strings.