Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2511.09301 · Physical review. B./Physical review. B

Infinite-component $BF$ field theory: Nexus of fracton order, Toeplitz braiding, and non-Hermitian amplification

Bo-Xi Li, Peng Ye

cond-mat.str-elcond-mat.mes-hallhep-thquant-phSubmitted Nov 12, 2025• Announced Nov 13, 2025• 1 citation• Revised Jan 13, 2026 (v2)
v1 · Nov 12, 2025v2 · Jan 13, 2026v3 · Mar 28, 2026
arXivHTMLPDF

Abstract

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 $(3+1)$D $BF$ topological field theories 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 -- infinite-component $BF$ (i$BF$) theories -- characterized by asymmetric integer Toeplitz $K$ matrices. Under open boundary conditions, i$BF$ theories exhibit a striking phenomenon: \textit{Toeplitz particle-loop braiding}, where a particle and a loop placed on opposite three-dimensional boundaries acquire a finite braiding phase even at infinite separation. This nonlocal braiding admits a geometric interpretation: 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 $K$ matrices revealed by singular value decomposition (SVD), rather than from the eigenvalue zero modes responsible for previously known Toeplitz braiding of anyons. We analytically and numerically study representative i$BF$ theories with Hatano-Nelson-type and non-Hermitian Su-Schrieffer-Heeger--type $K$ matrices, establishing a universal correspondence between ZSMs and Toeplitz particle-loop braiding. Our results identify boundary zero singular modes as the mechanism behind Toeplitz particle-loop braiding and establish infinite-component $BF$ theory as a predictive framework for higher-dimensional fracton topological orders.

Journal reference
Phys. Rev. B 113, 125141 (2026)
DOI
10.1103/3dh7-ktmd
Comments
This paper is part of a series of work with arXiv:2406.02482

Related papers

Three-dimensional fracton topological orders with boundary Toeplitz braiding

Bo-Xi Li, Yao Zhou, Peng Ye

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.

arXiv:2406.02482•Physical review. B./Physical review. B•Nov 4, 2024•5 citations

Fractonic topological phases from coupled wires

Joseph Sullivan, Arpit Dua, Meng Cheng

In three dimensions, gapped phases can support "fractonic" quasiparticle excitations, which are either completely immobile or can only move within a low-dimensional submanifold, a peculiar topological phenomenon going beyond the conventional framework of topological quantum field theory.In this work we explore fractonic topological phases using three-dimensional coupled wire constructions, which have proven to be a successful tool to realize and characterize topological phases in two dimensions.We find that both gapped and gapless phases with fractonic excitations can emerge from the models.In the gapped case, we argue that fractonic excitations are mobile along the wire direction, but their mobility in the transverse plane is generally reduced.We show that the excitations in general have infinite-order fusion structure, distinct from previously known gapped fracton models.Like the two-dimensional coupled-wire constructions, many models exhibit gapless (or even chiral) surface states, which can be described by infinite-component Luttinger liquids.However, the universality class of the surface theory strongly depends on the surface orientation, thus revealing a different type of bulk-boundary correspondence unique to fracton phases.

arXiv:2010.15148•Physical Review Research•May 17, 2021•16 citations

Continuum field theory of three-dimensional topological orders with emergent fermions and braiding statistics

Zhifeng Zhang, Qing-Rui Wang, Peng Ye

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 $BF$ field theory with twisted terms (e.g., $AAdA$ and $AAB$) as well as a $K$-matrix $BB$ term to simultaneously explore all such topological data and reach anomaly-free topological orders. Following the spirit of the famous $K$-matrix Chern-Simons theory of two-dimensional topological orders, we present general formulas and systematically show how the $K$-matrix $BB$ 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.

arXiv:2307.09983•Physical Review Research•Nov 3, 2023•9 citations

Fractonic order in infinite-component Chern-Simons gauge theories

Xiuqi Ma, Wilbur E. Shirley, Meng Cheng +3 more

The Chern-Simons theory is a powerful tool in the study of 2+1D topological phases, especially quantum Hall systems. Here, the authors extend the framework by allowing an infinite number of gauge fields and use it to study the 3+1D fracton order. Fracton order has been studied mostly using lattice models and higher-rank gauge theories. The introduction of the Chern-Simons formulation allows the user to take advantage of theoretical tools developed over the past decades for quantum Hall systems. In particular, the authors are able to study foliated fracton order more systematically and identify a new type of fracton order, not possible in previous formulations.

arXiv:2010.08917•Physical review. B./Physical review. B•May 18, 2022•25 citations

Non-Hermitian Crystalline Braid Topology from Hermitian Projection: A Zero-Mode Resonance Mechanism

Stefan Đorđević, Vladimir Juričić

Non-Hermitian topological phases are typically engineered through gain and loss, nonreciprocity, or interaction with an environment. Here we show that they can instead emerge purely by projecting a fully Hermitian, topologically trivial parent lattice onto an embedded subsystem. The mechanism is general: when a zero mode of the eliminated degrees of freedom couples to the retained subsystem, the embedding self-energy develops a pole, the zero-frequency description becomes singular, and topology is carried by the finite-frequency projected Green's function. We realize the mechanism exactly in a trivial nearest-neighbor square lattice with an embedded one-dimensional zig-zag brane. In the periodic transverse geometry, the parity of the eliminated complement selects the outcome: even sectors reduce to a regular Schur complement and yield conventional SSH-type descendants, whereas odd sectors host a sublattice-imbalance zero mode and follow the resonant route. There, the complex bands braid through isolated finite-frequency exceptional points (EPs), while a parity symmetry inherited from the embedding, together with $\mathrm{TRS}^{\dagger}$, induces conjugated pseudo-Hermiticity and quantizes the complex Berry phase. The stable bulk invariant of the nondegenerate phases is this quantized complex Berry phase; adjacent sectors are separated by parity-paired exceptional points whose half-integer vorticities encode the local exchange of complex-energy strands.The absence of the non-Hermitian skin effect ensures that the invariant is defined directly on the ordinary Brillouin zone. A topolectrical implementation of the projected response predicts momentum-resolved transmission minima at the exceptional-point transition frequencies together with a characteristic low-frequency resonant admittance, providing an experimentally testable signature of the mechanism.

cond-mat.mes-hallCross-listedarXiv:2606.06626•Jun 8, 2026

Fracton topological order via coupled layers

Han Ma, Ethan Lake, Xie Chen +1 more

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.

arXiv:1701.00747•Physical review. B./Physical review. B•Jun 21, 2017•189 citations
Infinite-component BFBFBF field theory: Nexus 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 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 -- infinite-component BFBFBF (iBFBFBF) theories -- characterized by asymmetric integer Toeplitz KKK matrices. Under open boundary conditions, iBFBFBF theories exhibit a striking phenomenon: \textit{Toeplitz particle-loop braiding}, where a particle and a loop placed on opposite three-dimensional boundaries acquire a finite braiding phase even at infinite separation. This nonlocal braiding admits a geometric interpretation: 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 (SVD), rather than from the eigenvalue zero modes responsible for previously known Toeplitz braiding of anyons. 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. Our results identify boundary zero singular modes as the mechanism behind Toeplitz particle-loop braiding and establish infinite-component BFBFBF theory as a predictive framework for higher-dimensional fracton topological orders.
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 KKK-matrix spectrum under OBC. We categorize nontrivial KKK 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.
In three dimensions, gapped phases can support "fractonic" quasiparticle excitations, which are either completely immobile or can only move within a low-dimensional submanifold, a peculiar topological phenomenon going beyond the conventional framework of topological quantum field theory.In this work we explore fractonic topological phases using three-dimensional coupled wire constructions, which have proven to be a successful tool to realize and characterize topological phases in two dimensions.We find that both gapped and gapless phases with fractonic excitations can emerge from the models.In the gapped case, we argue that fractonic excitations are mobile along the wire direction, but their mobility in the transverse plane is generally reduced.We show that the excitations in general have infinite-order fusion structure, distinct from previously known gapped fracton models.Like the two-dimensional coupled-wire constructions, many models exhibit gapless (or even chiral) surface states, which can be described by infinite-component Luttinger liquids.However, the universality class of the surface theory strongly depends on the surface orientation, thus revealing a different type of bulk-boundary correspondence unique to fracton phases.
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.
The Chern-Simons theory is a powerful tool in the study of 2+1D topological phases, especially quantum Hall systems. Here, the authors extend the framework by allowing an infinite number of gauge fields and use it to study the 3+1D fracton order. Fracton order has been studied mostly using lattice models and higher-rank gauge theories. The introduction of the Chern-Simons formulation allows the user to take advantage of theoretical tools developed over the past decades for quantum Hall systems. In particular, the authors are able to study foliated fracton order more systematically and identify a new type of fracton order, not possible in previous formulations.
Non-Hermitian topological phases are typically engineered through gain and loss, nonreciprocity, or interaction with an environment. Here we show that they can instead emerge purely by projecting a fully Hermitian, topologically trivial parent lattice onto an embedded subsystem. The mechanism is general: when a zero mode of the eliminated degrees of freedom couples to the retained subsystem, the embedding self-energy develops a pole, the zero-frequency description becomes singular, and topology is carried by the finite-frequency projected Green's function. We realize the mechanism exactly in a trivial nearest-neighbor square lattice with an embedded one-dimensional zig-zag brane. In the periodic transverse geometry, the parity of the eliminated complement selects the outcome: even sectors reduce to a regular Schur complement and yield conventional SSH-type descendants, whereas odd sectors host a sublattice-imbalance zero mode and follow the resonant route. There, the complex bands braid through isolated finite-frequency exceptional points (EPs), while a parity symmetry inherited from the embedding, together with TRS†\mathrm{TRS}^{\dagger}TRS†, induces conjugated pseudo-Hermiticity and quantizes the complex Berry phase. The stable bulk invariant of the nondegenerate phases is this quantized complex Berry phase; adjacent sectors are separated by parity-paired exceptional points whose half-integer vorticities encode the local exchange of complex-energy strands.The absence of the non-Hermitian skin effect ensures that the invariant is defined directly on the ordinary Brillouin zone. A topolectrical implementation of the projected response predicts momentum-resolved transmission minima at the exceptional-point transition frequencies together with a characteristic low-frequency resonant admittance, providing an experimentally testable signature of the mechanism.
In this work, we develop a coupled layer construction of fracton topological orders in d=3d=3d=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=2d=2d=2 topological phases and from stacks of d=3d=3d=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.