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arXiv:1304.0772 · Physical Review B

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

Eugeniu Plamadeala, Michael Mulligan, Chetan Nayak

Published Jul 26, 2013• 26 citations
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Abstract

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.

DOI
10.1103/physrevb.88.045131

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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.
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 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.
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.
Symmetry protected topological (SPT) states are short-range entangled states with symmetry. The boundary of a SPT phases has either gapless excitations or degenerate ground states, around a gapped bulk. Recently, we proposed a systematic construction of SPT phases in interacting bosonic systems, however it is not very clear what is the form of the low-energy excitations on the gapless edge. In this paper, we answer this question for two-dimensional (2D) bosonic SPT phases with ZN{\mathbb{Z}}_{N}ZN​ and U(1)\text{U}(1)U(1) symmetry. We find that while the low-energy modes of the gapless edges are nonchiral, symmetry acts on them in a ``chiral'' way, i.e., acts on the right movers and the left movers differently. This special realization of symmetry protects the gaplessness of the otherwise unstable edge states by prohibiting a direct scattering between the left and right movers. Moreover, understanding of the low-energy effective theory leads to experimental predictions about the SPT phases. In particular, we find that all the 2D U(1)\text{U}(1)U(1) SPT phases have even integer quantized Hall conductance.
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.
Protected chiral edge modes are a well-known signature of topologically ordered phases, such as the fractional quantum Hall states (FQHS). Using the framework of projected entangled pair states (PEPS) on the square lattice, the authors construct a family of chiral spin-12\frac{1}{2}21​ quantum spin liquids with Z2{\mathbb{Z}}_{2}Z2​ gauge symmetry and analyze in full detail the properties of the edge modes. Surprisingly, the results show that the latter can be well described by a chiral conformal field theory of free bosons (SU(2)1{}_{1}1​), as is the case for the $\ensuremath{\nu}$=12\frac{1}{2}21​ (bosonic) gapped Laughlin state, despite the fact that the numerical data here suggest a critical bulk. The authors thus propose that our family of PEPS physically describes a boundary between a chiral topological phase and a trivial phase and might be closely connected to an (unknown) analogous FQHS.
We study effectively one-dimensional systems that emerge at the edge of a two-dimensional topologically ordered state, or at the boundary between two topologically ordered states. We argue that anyons of the bulk are associated with emergent symmetries of the edge, which play a crucial role in the structure of its phase diagram. Using this symmetry principle, transitions between distinct gapped phases at the boundaries of Abelian states can be understood in terms of symmetry-breaking transitions or transitions between symmetry-protected topological phases. Yet more exotic phenomena occur when the bulk hosts non-Abelian anyons. To demonstrate these principles, we explore the phase diagrams of the edges of a single and a double layer of the toric code, as well as those of domain walls in a single- and double-layer Kitaev spin liquid (KSL). In the case of the KSL, we find that the presence of a non-Abelian anyon in the bulk enforces Kramers-Wannier self-duality as a symmetry of the effective boundary theory. These examples illustrate a number of surprising phenomena, such as spontaneous duality breaking, two-sector phase transitions, and unfreezing of marginal operators at a transition between different gapless phases.