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:1310.5708 · Physical Review B

Bulk-edge correspondence in (2 + 1)-dimensional Abelian topological phases

Jennifer Cano, Meng Cheng, Michael Mulligan, Chetan Nayak, Eugeniu Plamadeala, Jon Yard

Published Mar 14, 2014• 123 citations
arXivHTMLPDF

Abstract

The same bulk two-dimensional topological phase can have multiple distinct, fully chiral edge phases. We show that this can occur in the integer quantum Hall states at $\ensuremath{\nu}=8$ and 12, with experimentally testable consequences. We show that this can occur in Abelian fractional quantum Hall states as well, with the simplest examples being at $\ensuremath{\nu}=\frac{8}{7},\frac{12}{11},\frac{8}{15},\frac{16}{5}$. We give a general criterion for the existence of multiple distinct chiral edge phases for the same bulk phase and discuss experimental consequences. Edge phases correspond to lattices while bulk phases correspond to genera of lattices. Since there are typically multiple lattices in a genus, the bulk-edge correspondence is typically one-to-many; there are usually many stable fully chiral edge phases corresponding to the same bulk. We explain these correspondences using the theory of integral quadratic forms. We show that fermionic systems can have edge phases with only bosonic low-energy excitations and discuss a fermionic generalization of the relation between bulk topological spins and the central charge. The latter follows from our demonstration that every fermionic topological phase can be represented as a bosonic topological phase, together with some number of filled Landau levels. Our analysis shows that every Abelian topological phase can be decomposed into a tensor product of theories associated with prime numbers $p$ in which every quasiparticle has a topological spin that is a ${p}^{n}$th root of unity for some $n$. It also leads to a simple demonstration that all Abelian topological phases can be represented by $\mathrm{U}{(1)}^{N}$ Chern-Simons theory parameterized by a $K$ matrix.

DOI
10.1103/physrevb.89.115116

Related papers

The Magical Geometry of 1D Quantum Liquids

Eugeniu Plamadeala

We investigate the edge properties of Abelian topological phases in two spatial dimensions. We discover that many of them support multiple fully chiral edge phases, with surprising and measurable experimental consequences. Using the machinery of conformal field theory and integral quadratic forms we establish that distinct chiral edge phases correspond to genera of positive-definite integral lattices. This completes the notion of bulk-boundary correspondence for topological phases. We establish that by tuning inter-channel interactions the system can be made to transition between the different edge phases without closing the bulk gap.Separately we construct a family of one-dimensional models, called Perfect Metals, with no relevant mass-generating operators. These theories describe stable quantum critical phases of interacting fermions, bosons or spins in a quantum nanowire. These models rigorously answer a long-standing question about the existence of stable metallic phases in one and two spatial dimensions in the presence of generic disorder. Separately, they are the first example of a stable phase of an infinite parallel array of coupled Luttinger liquids. We perform a detailed study of the transport properties of Perfect Metals and show that in addition to violating the Wiedemann-Franz law, they naturally exhibit low power-law dependence of electric and thermal conductivities on temperature all the way to zero temperature. We dub this phenomenological set of properties a hyperconductor because in some sense, hyperconductors are better conductors that superconductors, which may have thermal conductivities that are exponentially small in temperature.

•Jan 1, 2016•0 citations

Non-Abelian twist to integer quantum Hall states

Pedro L. S. Lopes, Victor L. Quito, Bo Han +1 more

At a fixed magnetic filling fraction, fractional quantum Hall states may display a plethora of interaction-induced competing phases. Effective Chern-Simons theories have suggested the existence of multiple interaction-induced short-range entangled phases also at integer quantum Hall plateaus. Among these, a bosonic phase has been proposed with edge modes carrying representations based on the ${E}_{8}$ exceptional Lie algebra. Through a theoretical coupled-wire construction, we provide an explicit microscopic model for this ${E}_{8}$ Abelian quantum Hall state, at filling $\ensuremath{\nu}=16$, and discuss how it is intimately related to topological paramagnets in $3+1$ dimensions. Still using coupled wires, we partition the ${E}_{8}$ state into a pair of non-Abelian, long-range entangled states. These two states occur at filling $\ensuremath{\nu}=8$, demonstrating that even topological order may also exist at integer Hall plateaus. These phases are bosonic, carry chiral edge theories with either ${G}_{2}$ or ${F}_{4}$ internal symmetries, and host Fibonacci anyonic excitations in the bulk. This suggests that the $\ensuremath{\nu}=8$ quantum Hall plateau may provide an unexpected platform to realize decoherence-free quantum computation by anyon braiding. We also find that these topological ordered phases are related by a notion of particle-hole conjugation based on the ${E}_{8}$ state that exchanges the ${G}_{2}$ and ${F}_{4}$ Fibonacci states. We argue that these phases can be tracked down by their electric and thermal Hall transport satisfying a distinctive Wiedemann-Franz law $\left({\ensuremath{\kappa}}_{xy}/{\ensuremath{\sigma}}_{xy}\right)/\left[\left({\ensuremath{\pi}}^{2}{k}_{B}^{2}T\right)/3{e}^{2}\right]<1$, even at integer magnetic filling factors.

arXiv:1901.09043•Physical review. B./Physical review. B•Aug 6, 2019•17 citations

Theory and classification of interacting integer topological phases in two dimensions: A Chern-Simons approach

Yuan-Ming Lu, Ashvin Vishwanath

We study topological phases of interacting systems in two spatial dimensions in the absence of topological order (i.e., with a unique ground state on closed manifolds and no fractional excitations). These are the closest interacting analogs of integer quantum Hall states, topological insulators, and superconductors. We adapt the well-known Chern-Simons $K$-matrix description of quantum Hall states to classify such ``integer'' topological phases. Our main result is a general formalism that incorporates symmetries into the $K$-matrix description. Remarkably, this simple analysis yields the same list of topological phases as a recent group cohomology classification, and in addition provides field theories and explicit edge theories for all these phases. The bosonic topological phases, which only appear in the presence of interactions and which remain well defined in the presence of disorder, include (i) bosonic insulators with a Hall conductance quantized to even integers, (ii) a bosonic analog of quantum spin Hall insulators, and (iii) a bosonic analog of a chiral topological superconductor, whose $K$ matrix is the Cartan matrix of Lie group ${E}_{8}$. We also discuss interacting fermion systems where symmetries are realized in a projective fashion, where we find the present formalism can handle a wider range of symmetries than a recent group super-cohomology classification. Lastly, we construct microscopic models of these phases from coupled one-dimensional systems.

arXiv:1205.3156•Physical Review B•Sep 14, 2012•507 citations

Short-range entangled bosonic states with chiral edge modes and T duality of heterotic strings

Eugeniu Plamadeala, Michael Mulligan, Chetan Nayak

We consider states of bosons in two dimensions that do not support anyons in the bulk, but nevertheless have stable chiral edge modes that are protected even without any symmetry. Such states must have edge modes with central charge $c=8k$ for integer $k$. While there is a single such state with $c=8$, there are, naively, two such states with $c=16$, corresponding to the two distinct even unimodular lattices in 16 dimensions. However, we show that these two phases are the same in the bulk, which is a consequence of the uniqueness of signature $(8k+n,n)$ even unimodular lattices. The bulk phases are stably equivalent, in a sense that we make precise. However, there are two different phases of the edge corresponding to these two lattices, thereby realizing a novel form of the bulk-edge correspondence. Two distinct fully chiral edge phases are associated with the same bulk phase, which is consistent with the uniqueness of the bulk since the transition between them, which is generically first order, can occur purely at the edge. Our construction is closely related to $T$ duality of toroidally compactified heterotic strings. We discuss generalizations of these results.

arXiv:1304.0772•Physical Review B•Jul 26, 2013•26 citations

Bulk-edge correspondence in fractional Chern insulators

Zhao Liu, Dmitry L. Kovrizhin, Emil J. Bergholtz

It has been recently realized that strong interactions in topological Bloch bands give rise to the appearance of novel states of matter. Here we study connections between these systems---fractional Chern insulators and the fractional quantum Hall states---via generalization of a gauge-fixed Wannier-Qi construction in the cylinder geometry. Our setup offers a number of important advantages compared to the earlier exact diagonalization studies on a torus. Most notably, it gives access to edge states and to a single-cut orbital entanglement spectrum, hence to the physics of bulk-edge correspondence. It is also readily implemented in the state-of-the-art density matrix renormalization group method that allows for numerical simulations of significantly larger systems. We demonstrate our general approach on examples of flat-band models on ruby and kagome lattices at bosonic filling fractions $\ensuremath{\nu}=1/2$ and $\ensuremath{\nu}=1$, which show the signatures of (non)-Abelian phases, and establish the correspondence between the physics of edge states and the entanglement in the bulk. Notably, we find that the non-Abelian $\ensuremath{\nu}=1$ phase can be stabilized by purely on-site interactions in the presence of a confining potential.

arXiv:1304.1323•Physical Review B•Aug 26, 2013•36 citations

Chiral edge states in (2+1)-dimensional topological phases

Carlos A. Hernaski, Pedro R. S. Gomes

Chiral edge states of [Formula: see text]-dimensional Abelian and non-Abelian topological phases can be represented by chiral conformal field theories with integer and noninteger values of central charge, respectively. In this work we describe certain edge states in terms of constrained fermionic fields that realize chiral coset CFT structures. This construction arises naturally in the so-called quantum wires approach for topological phases and allows for representing fractionalized edge states directly in terms of fermionic degrees of freedom. At the same time, the constrained fermions description introduces some subtleties concerning gauge anomalies since it involves the coupling of chiral fermions to gauge fields. We describe in this paper how to handle these issues.

arXiv:1707.08048•International Journal of Modern Physics A•Oct 9, 2019•0 citations
The same bulk two-dimensional topological phase can have multiple distinct, fully chiral edge phases. We show that this can occur in the integer quantum Hall states at \ensuremathν=8\ensuremath{\nu}=8\ensuremathν=8 and 12, with experimentally testable consequences. We show that this can occur in Abelian fractional quantum Hall states as well, with the simplest examples being at \ensuremathν=87,1211,815,165\ensuremath{\nu}=\frac{8}{7},\frac{12}{11},\frac{8}{15},\frac{16}{5}\ensuremathν=78​,1112​,158​,516​. We give a general criterion for the existence of multiple distinct chiral edge phases for the same bulk phase and discuss experimental consequences. Edge phases correspond to lattices while bulk phases correspond to genera of lattices. Since there are typically multiple lattices in a genus, the bulk-edge correspondence is typically one-to-many; there are usually many stable fully chiral edge phases corresponding to the same bulk. We explain these correspondences using the theory of integral quadratic forms. We show that fermionic systems can have edge phases with only bosonic low-energy excitations and discuss a fermionic generalization of the relation between bulk topological spins and the central charge. The latter follows from our demonstration that every fermionic topological phase can be represented as a bosonic topological phase, together with some number of filled Landau levels. Our analysis shows that every Abelian topological phase can be decomposed into a tensor product of theories associated with prime numbers ppp in which every quasiparticle has a topological spin that is a pn{p}^{n}pnth root of unity for some nnn. It also leads to a simple demonstration that all Abelian topological phases can be represented by U(1)N\mathrm{U}{(1)}^{N}U(1)N Chern-Simons theory parameterized by a KKK matrix.
We investigate the edge properties of Abelian topological phases in two spatial dimensions. We discover that many of them support multiple fully chiral edge phases, with surprising and measurable experimental consequences. Using the machinery of conformal field theory and integral quadratic forms we establish that distinct chiral edge phases correspond to genera of positive-definite integral lattices. This completes the notion of bulk-boundary correspondence for topological phases. We establish that by tuning inter-channel interactions the system can be made to transition between the different edge phases without closing the bulk gap.Separately we construct a family of one-dimensional models, called Perfect Metals, with no relevant mass-generating operators. These theories describe stable quantum critical phases of interacting fermions, bosons or spins in a quantum nanowire. These models rigorously answer a long-standing question about the existence of stable metallic phases in one and two spatial dimensions in the presence of generic disorder. Separately, they are the first example of a stable phase of an infinite parallel array of coupled Luttinger liquids. We perform a detailed study of the transport properties of Perfect Metals and show that in addition to violating the Wiedemann-Franz law, they naturally exhibit low power-law dependence of electric and thermal conductivities on temperature all the way to zero temperature. We dub this phenomenological set of properties a hyperconductor because in some sense, hyperconductors are better conductors that superconductors, which may have thermal conductivities that are exponentially small in temperature.
At a fixed magnetic filling fraction, fractional quantum Hall states may display a plethora of interaction-induced competing phases. Effective Chern-Simons theories have suggested the existence of multiple interaction-induced short-range entangled phases also at integer quantum Hall plateaus. Among these, a bosonic phase has been proposed with edge modes carrying representations based on the E8{E}_{8}E8​ exceptional Lie algebra. Through a theoretical coupled-wire construction, we provide an explicit microscopic model for this E8{E}_{8}E8​ Abelian quantum Hall state, at filling $\ensuremath{\nu}=16$, and discuss how it is intimately related to topological paramagnets in 3+13+13+1 dimensions. Still using coupled wires, we partition the E8{E}_{8}E8​ state into a pair of non-Abelian, long-range entangled states. These two states occur at filling $\ensuremath{\nu}=8$, demonstrating that even topological order may also exist at integer Hall plateaus. These phases are bosonic, carry chiral edge theories with either G2{G}_{2}G2​ or F4{F}_{4}F4​ internal symmetries, and host Fibonacci anyonic excitations in the bulk. This suggests that the $\ensuremath{\nu}=8$ quantum Hall plateau may provide an unexpected platform to realize decoherence-free quantum computation by anyon braiding. We also find that these topological ordered phases are related by a notion of particle-hole conjugation based on the E8{E}_{8}E8​ state that exchanges the G2{G}_{2}G2​ and F4{F}_{4}F4​ Fibonacci states. We argue that these phases can be tracked down by their electric and thermal Hall transport satisfying a distinctive Wiedemann-Franz law $\left({\ensuremath{\kappa}}_{xy}/{\ensuremath{\sigma}}_{xy}\right)/\left[\left({\ensuremath{\pi}}^{2}{k}_{B}^{2}T\right)/3{e}^{2}\right]<1$, even at integer magnetic filling factors.
We study topological phases of interacting systems in two spatial dimensions in the absence of topological order (i.e., with a unique ground state on closed manifolds and no fractional excitations). These are the closest interacting analogs of integer quantum Hall states, topological insulators, and superconductors. We adapt the well-known Chern-Simons KKK-matrix description of quantum Hall states to classify such ``integer'' topological phases. Our main result is a general formalism that incorporates symmetries into the KKK-matrix description. Remarkably, this simple analysis yields the same list of topological phases as a recent group cohomology classification, and in addition provides field theories and explicit edge theories for all these phases. The bosonic topological phases, which only appear in the presence of interactions and which remain well defined in the presence of disorder, include (i) bosonic insulators with a Hall conductance quantized to even integers, (ii) a bosonic analog of quantum spin Hall insulators, and (iii) a bosonic analog of a chiral topological superconductor, whose KKK matrix is the Cartan matrix of Lie group E8{E}_{8}E8​. We also discuss interacting fermion systems where symmetries are realized in a projective fashion, where we find the present formalism can handle a wider range of symmetries than a recent group super-cohomology classification. Lastly, we construct microscopic models of these phases from coupled one-dimensional systems.
We consider states of bosons in two dimensions that do not support anyons in the bulk, but nevertheless have stable chiral edge modes that are protected even without any symmetry. Such states must have edge modes with central charge c=8kc=8kc=8k for integer kkk. While there is a single such state with c=8c=8c=8, there are, naively, two such states with c=16c=16c=16, corresponding to the two distinct even unimodular lattices in 16 dimensions. However, we show that these two phases are the same in the bulk, which is a consequence of the uniqueness of signature (8k+n,n)(8k+n,n)(8k+n,n) even unimodular lattices. The bulk phases are stably equivalent, in a sense that we make precise. However, there are two different phases of the edge corresponding to these two lattices, thereby realizing a novel form of the bulk-edge correspondence. Two distinct fully chiral edge phases are associated with the same bulk phase, which is consistent with the uniqueness of the bulk since the transition between them, which is generically first order, can occur purely at the edge. Our construction is closely related to TTT duality of toroidally compactified heterotic strings. We discuss generalizations of these results.
It has been recently realized that strong interactions in topological Bloch bands give rise to the appearance of novel states of matter. Here we study connections between these systems---fractional Chern insulators and the fractional quantum Hall states---via generalization of a gauge-fixed Wannier-Qi construction in the cylinder geometry. Our setup offers a number of important advantages compared to the earlier exact diagonalization studies on a torus. Most notably, it gives access to edge states and to a single-cut orbital entanglement spectrum, hence to the physics of bulk-edge correspondence. It is also readily implemented in the state-of-the-art density matrix renormalization group method that allows for numerical simulations of significantly larger systems. We demonstrate our general approach on examples of flat-band models on ruby and kagome lattices at bosonic filling fractions $\ensuremath{\nu}=1/2$ and $\ensuremath{\nu}=1$, which show the signatures of (non)-Abelian phases, and establish the correspondence between the physics of edge states and the entanglement in the bulk. Notably, we find that the non-Abelian $\ensuremath{\nu}=1$ phase can be stabilized by purely on-site interactions in the presence of a confining potential.
Chiral edge states of [Formula: see text]-dimensional Abelian and non-Abelian topological phases can be represented by chiral conformal field theories with integer and noninteger values of central charge, respectively. In this work we describe certain edge states in terms of constrained fermionic fields that realize chiral coset CFT structures. This construction arises naturally in the so-called quantum wires approach for topological phases and allows for representing fractionalized edge states directly in terms of fermionic degrees of freedom. At the same time, the constrained fermions description introduces some subtleties concerning gauge anomalies since it involves the coupling of chiral fermions to gauge fields. We describe in this paper how to handle these issues.