Double Point Contact in the k=3 Read-Rezayi State

We compute the dependence of the tunneling current in a double point contact in the k=3 Read-Rezayi state (which is conjectured to describe an incompressible quantum hall fluid at filling fraction nu=12/5) on voltage, separation between the two contacts, and temperature. Using the tunneling hamiltonian of cond-mat/0607431, we show that the effect of quasiholes in the bulk region between the two contacts is simply an overall constant multiplying the interference term. This is the same effect as found for the differential conductivity in cond-mat/0601242; the difference is that we do an actual edge theory calculation and compute the full current-voltage curve at weak tunneling.Comment: 7 pages, 3 figure

A new equivalence between fused RSOS and loop models

We consider the topological theories of cond-mat/0404617 and cond-mat/0610583 and study ground state amplitudes of string net configurations which consist of large chunks $G$ of (trivalent) regular lattice. We evaluate these amplitudes in two different ways: first we use the Turaev-Viro prescription to write the amplitude as a sum over labelings of the faces of $G$, and second we use the local rules that constrain the amplitude (the $F$-matrix) to resolve subgraphs in creative ways. In the case of the Doubled Fibonacci theory this second way allows us to produce loop models. In particular, we show that the hard hexagon model is equivalent to an anisotropic loop model. Many other interesting equivalences can presumably be obtained.Comment: 4 pages, 7 figure

Solving the eigenvalue problem arising from the adjoint sector of the c=1 matrix model

We solve the non-local eigenvalue problem that arose from consideration of the adjoint sector of the c=1 matrix model in hep-th/0503112. We obtain the exact wavefunction and a scattering phase that matches the string theory calculation.Comment: 10 pages, 1 figur

Local Commuting Projector Hamiltonians and the Quantum Hall Effect

We prove that neither Integer nor Fractional Quantum Hall Effects with nonzero Hall conductivity are possible in gapped systems described by Local Commuting Projector Hamiltonians

D-Brane Instability as a Large N Phase Transition

In AdS/CFT analyticity suggests that certain singular behaviors expected at large 't Hooft coupling should continue smoothly to weak 't Hooft coupling where the gauge theory is tractable. This may provide a window into stringy singularity resolution and is a promising technique for studying the signature of the black hole singularity discussed in hep-th/0306170. We comment briefly on its status. Our main goal, though, is to study a simple example of this technique. Gross and Ooguri (hep-th/9805129) have pointed out that the D-brane minimal surface spanning a pair of 't Hooft loops undergoes a phase transition as the distance between the loops is varied. We find the analog of this behavior in the weakly coupled Super Yang Mills theory by computing 't Hooft loop expectation values there.Comment: 18 pages, 8 figure

Discrete spin structures and commuting projector models for 2d fermionic symmetry protected topological phases

We construct exactly solved commuting projector Hamiltonian lattice models for all known 2+1d fermionic symmetry protected topological phases (SPTs) with on-site unitary symmetry group $G_f = G \times \mathbb{Z}_2^f$, where $G$ is finite and $\mathbb{Z}_2^f$ is the fermion parity symmetry. In particular, our models transcend the class of group supercohomology models, which realize some, but not all, fermionic SPTs in 2+1d. A natural ingredient in our construction is a discrete form of the spin structure of the 2d spatial surface $M$ on which our model is defined, namely a `Kasteleyn' orientation of a certain graph associated with the lattice. As a special case, our construction yields commuting projector models for all $8$ members of the $\mathbb{Z}_8$ classification of 2d fermionic SPTs with $G = \mathbb{Z}_2$.Comment: 13 pages, 12 figures. V2: Corrected typos and added citation

Realizing anomalous anyonic symmetries at the surfaces of 3d gauge theories

The hallmark of a 2 dimensional topologically ordered phase is the existence of deconfined `anyon' excitations that have exotic braiding and exchange statistics, different from those of ordinary bosons or fermions. As opposed to conventional Landau-Ginzburg-Wilson phases, which are classified on the basis of the spontaneous breaking of an underlying symmetry, topologically ordered phases, such as those occurring in the fractional quantum Hall effect, are absolutely stable, not requiring any such symmetry. Recently, though, it has been realized that symmetries, which may still be present in such systems, can give rise to a host of new, distinct, many-body phases, all of which share the same underlying topological order. A useful approach to classifying SETs is to determine all possible such symmetry actions on the anyons that are algebraically consistent with the anyons' statistics. Remarkably, however, there exist symmetry actions that, despite being algebraically consistent, cannot be realized in any physical system, and hence do not lead to valid 2d SETs. One class of such `anomalous' SETs, characterized by certain disallowed symmetry fractionalization patters, finds a physical interpretation as an allowed surface state of certain 3d short-range entangled phases, but another, characterized by some seemingly valid but anomalous permutation actions of the symmetry on the anyons, has so far eluded a physical interpretation. In this work, we find a physical realization for these anomalously permuting SETs as surface theories of certain 3d long-range entangled phases, completing our understanding of general anomalous SETs in 2 dimensions.Comment: 16 pages, new introductio

Superpotentials for Quiver Gauge Theories

We compute superpotentials for quiver gauge theories arising from marginal D-Brane decay on collapsed del Pezzo cycles S in a Calabi-Yau X. This is done using the machinery of A-infinity products in the derived category of coherent sheaves of X, which in turn is related to the derived category of S and quiver path algebras. We confirm that the superpotential is what one might have guessed from analyzing the moduli space, i.e., it is linear in the fields corresponding to the Ext2's of the quiver and that each such Ext2 multiplies a polynomial in Ext1's equal to precisely the relation represented by the Ext2.Comment: 25 pages, LaTeX2e, xypi

Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model

Three dimensional topological superconductors (TScs) protected by time reversal (T) symmetry are characterized by gapless Majorana cones on their surface. Free fermion phases with this symmetry (class DIII) are indexed by an integer n, of which n=1 is realized by the B-phase of superfluid Helium-3. Previously it was believed that the surface must be gapless unless time reversal symmetry is broken. Here we argue that a fully symmetric and gapped surface is possible in the presence of strong interactions, if a special type of topological order appears on the surface. The topological order realizes T symmetry in an anomalous way, one that is impossible to achieve in purely two dimensions. For odd n TScs, the surface topological order must be non-Abelian. We propose the simplest non-Abelian topological order that contains electron like excitations, SO(3)_6, with four quasiparticles, as a candidate surface state. Remarkably, this theory has a hidden T invariance which however is broken in any 2D realization. By explicitly constructing an exactly soluble Walker-Wang model we show that it can be realized at the surface of a short ranged entangled 3D fermionic phase protected by T symmetry, with bulk electrons trasforming as Kramers pairs, i.e. T^2=-1 under time reversal. We also propose an Abelian theory, the semion-fermion topological order, to realize an even n TSc surface, for which an explicit model is derived using a coupled layer construction. We argue that this is related to the n=2 TSc, and use this to build candidate surface topological orders for n=4 and n=8 TScs. The latter is equivalent to the three fermion state which is the surface topological order of a Z2 bosonic topological phase protected by T invariance. One particular consequence of this is that an n=16 TSc admits a trivially gapped T-symmetric surface.Comment: v3, fixed typos and added appendi

Symmetry Enforced Non-Abelian Topological Order at the Surface of a Topological Insulator

The surfaces of three dimensional topological insulators (3D TIs) are generally described as Dirac metals, with a single Dirac cone. It was previously believed that a gapped surface implied breaking of either time reversal $\mathcal T$ or U(1) charge conservation symmetry. Here we discuss a novel possibility in the presence of interactions, a surface phase that preserves all symmetries but is nevertheless gapped and insulating. Then the surface must develop topological order of a kind that cannot be realized in a 2D system with the same symmetries. We discuss candidate surface states - non-Abelian Quantum Hall states which, when realized in 2D, have $\sigma_{xy}=1/2$ and hence break $\mathcal T$ symmetry. However, by constructing an exactly soluble 3D lattice model, we show they can be realized as $\mathcal T$ symmetric surface states. The corresponding 3D phases are confined, and have $\theta=\pi$ magnetoelectric response. Two candidate states have the same 12 particle topological order, the (Read-Moore) Pfaffian state with the neutral sector reversed, which we term T-Pfaffian topological order, but differ in their $\mathcal T$ transformation. Although we are unable to connect either of these states directly to the superconducting TI surface, we argue that one of them describes the 3D TI surface, while the other differs from it by a bosonic topological phase. We also discuss the 24 particle Pfaffian-antisemion topological order (which can be connected to the superconducting TI surface) and demonstrate that it can be realized as a $\mathcal T$ symmetric surface state.Comment: 17 pages, 6 figures. v2: clarified the connection of T-Pfaffian state to free fermion TI, references update