arXiv subjects

Paul Fendley

Publications and source records attributed to Paul Fendley.

At least 55 records · Page 3Linked to original sources

arXiv2012

Parafermionic edge zero modes in Z_n-invariant spin chains

A sign of topological order in a gapped one-dimensional quantum chain is the existence of edge zero modes. These occur in the Z_2-invariant Ising/Majorana chain, where they can be understood using free-fermion techniques. Here I discuss their presence in spin chains with Z_n symmetry, and prove that for appropriate coupling they are exact, even in this strongly interacting system. These modes are naturally expressed in terms of parafermions, generalizations of fermions to the Z_n case. I show that parafermionic edge zero modes do not occur in the usual ferromagnetic and antiferromagnetic cases, but rather only when the interactions are chiral, so that spatial-parity and time-reversal symmetries are broken.

cond-mat.str-el↗

arXiv2012

Entanglement scaling in two-dimensional gapless systems

We numerically determine subleading scaling terms in the ground-state entanglement entropy of several two-dimensional (2D) gapless systems, including a Heisenberg model with Néel order, a free Dirac fermion in the π-flux phase, and the nearest-neighbor resonating-valence-bond wavefunction. For these models, we show that the entanglement entropy between cylindrical regions of length x and L - x, extending around a torus of length L, depends upon the dimensionless ratio x/L. This can be well-approximated on finite-size lattices by a function ln(sin(πx/L)), akin to the familiar chord-length dependence in one dimension. We provide evidence, however, that the precise form of this bulk-dependent contribution is a more general function in the 2D thermodynamic limit.

cond-mat.str-el↗

arXiv2011

Boson pairing and unusual criticality in a generalized XY model

We discuss the unusual critical behavior of a generalized XY model containing both 2π-periodic and π-periodic couplings between sites. The presence of vortices and half-vortices allows for single-particle condensate and pair-condensate phases. Using a field theoretic formulation and worm algorithm Monte Carlo simulations, we show that in two dimensions it is possible for the system to pass directly from the disordered (high temperature) phase to the single particle (quasi)-condensate via an Ising transition, a situation reminiscent of the `deconfined criticality' scenario.

cond-mat.stat-mech↗

arXiv2011

The eight-vertex model and lattice supersymmetry

We show that the XYZ spin chain along the special line of couplings J_xJ_y+J_xJ_z+J_yJ_z=0 possesses a hidden N=(2,2) supersymmetry. This lattice supersymmetry is non-local and changes the number of sites. It extends to the full transfer matrix of the corresponding eight-vertex model. In particular, it is shown how to derive the supercharges from Baxter's Bethe ansatz. This analysis leads to new conjectures concerning the ground state for chains of odd length. We also discuss a correspondence between the spectrum of this XYZ chain and that of a manifestly supersymmetric staggered fermion chain.

math-ph↗

arXiv2011

An integrable modification of the critical Chalker-Coddington network model

We consider the Chalker-Coddington network model for the Integer Quantum Hall Effect, and examine the possibility of solving it exactly. In the supersymmetric path integral framework, we introduce a truncation procedure, leading to a series of well-defined two-dimensional loop models, with two loop flavours. In the phase diagram of the first-order truncated model, we identify four integrable branches related to the dilute Birman-Wenzl-Murakami braid-monoid algebra, and parameterised by the loop fugacity $n$. In the continuum limit, two of these branches (1,2) are described by a pair of decoupled copies of a Coulomb-Gas theory, whereas the other two branches (3,4) couple the two loop flavours, and relate to an $SU(2)_r \times SU(2)_r / SU(2)_{2r}$ Wess-Zumino-Witten (WZW) coset model for the particular values $n= -2\cos[π/(r+2)]$ where $r$ is a positive integer. The truncated Chalker-Coddington model is the $n=0$ point of branch 4. By numerical diagonalisation, we find that its universality class is neither an analytic continuation of the WZW coset, nor the universality class of the original Chalker-Coddington model. It constitutes rather an integrable, critical approximation to the latter.

cond-mat.dis-nn↗

arXiv2010

Ground-state properties of a supersymmetric fermion chain

We analyze the ground state of a strongly interacting fermion chain with a supersymmetry. We conjecture a number of exact results, such as a hidden duality between weak and strong couplings. By exploiting a scale free property of the perturbative expansions, we find exact expressions for the order parameters, yielding the critical exponents. We show that the ground state of this fermion chain and another model in the same universality class, the XYZ chain along a line of couplings, are both written in terms of the same polynomials. We demonstrate this explicitly for up to N = 24 sites, and provide consistency checks for large N. These polynomials satisfy a recursion relation related to the Painlevé VI differential equation, and using a scale-free property of these polynomials, we derive a simple and exact formula for their limit as N goes to infinity.

cond-mat.stat-mech↗

arXiv2010

Exact and simple results for the XYZ and strongly interacting fermion chains

We conjecture exact and simple formulas for physical quantities in two quantum chains. A classic result of this type is Onsager, Kaufman and Yang's formula for the spontaneous magnetization in the Ising model, subsequently generalized to the chiral Potts models. We conjecture that analogous results occur in the XYZ chain when the couplings obey J_xJ_y + J_yJ_z + J_x J_z=0, and in a related fermion chain with strong interactions and supersymmetry. We find exact formulas for the magnetization and gap in the former, and the staggered density in the latter, by exploiting the fact that certain quantities are independent of finite-size effects.

cond-mat.stat-mech↗

arXiv2010

Spin Hamiltonians with resonating-valence-bond ground states

Quantum dimer models exhibit quantum critical points and liquid states when the ground state is the resonating-valence bond (RVB) state. We construct SU(2)-invariant spin-1/2 Hamiltonians with the same RVB ground state. The main technical obstacle overcome is the fact that different "dimer" configurations in the spin model are not orthogonal to each other. We show that the physics depends on how dimers are related to the spins, and find a Hamiltonian that may be quantum critical.

cond-mat.str-el↗

arXiv2009

Link invariants, the chromatic polynomial and the Potts model

We study the connections between link invariants, the chromatic polynomial, geometric representations of models of statistical mechanics, and their common underlying algebraic structure. We establish a relation between several algebras and their associated combinatorial and topological quantities. In particular, we define the chromatic algebra, whose Markov trace is the chromatic polynomial χ_Q of an associated graph, and we give applications of this new algebraic approach to the combinatorial properties of the chromatic polynomial. In statistical mechanics, this algebra occurs in the low temperature expansion of the Q-state Potts model. We establish a relationship between the chromatic algebra and the SO(3) Birman-Murakami-Wenzl algebra, which is an algebra-level analogue of the correspondence between the SO(3) Kauffman polynomial and the chromatic polynomial.

math.CO↗

arXiv2009

Boundary Conformal Field Theory and Tunneling of Edge Quasiparticles in non-Abelian Topological States

We explain how (perturbed) boundary conformal field theory allows us to understand the tunneling of edge quasiparticles in non-Abelian topological states. The coupling between a bulk non-Abelian quasiparticle and the edge is due to resonant tunneling to a zero mode on the quasiparticle, which causes the zero mode to hybridize with the edge. This can be reformulated as the flow from one conformally-invariant boundary condition to another in an associated critical statistical mechanical model. Tunneling from one edge to another at a point contact can split the system in two, either partially or completely. This can be reformulated in the critical statistical mechanical model as the flow from one type of defect line to another. We illustrate these two phenomena in detail in the context of the nu=5/2 quantum Hall state and the critical Ising model. We briefly discuss the case of Fibonacci anyons and conclude by explaining the general formulation and its physical interpretation.

cond-mat.mes-hall↗

arXiv2008

Charge frustration and quantum criticality for strongly correlated fermions

We study a model of strongly correlated electrons on the square lattice which exhibits charge frustration and quantum critical behavior. The potential is tuned to make the interactions supersymmetric. We establish a rigorous mathematical result which relates quantum ground states to certain tiling configurations on the square lattice. For periodic boundary conditions this relation implies that the number of ground states grows exponentially with the linear dimensions of the system. We present substantial analytic and numerical evidence that for open boundary conditions the system has gapless edge modes.

cond-mat.str-el↗

arXiv2008

Non-equilibrium transport through a point contact in the $ν=5/2$ non-Abelian quantum Hall state

We analyze charge-$e/4$ quasiparticle tunneling between the edges of a point contact in a non-Abelian model of the $ν=5/2$ quantum Hall state. We map this problem to resonant tunneling between attractive Luttinger liquids and use the time-dependent density-matrix renormalization group (DMRG) method to compute the current through the point contact in the presence of a {\it finite voltage difference} between the two edges. We confirm that, as the voltage is decreases, the system is broken into two pieces coupled by electron hopping. In the limits of small and large voltage, we recover the results expected from perturbation theory about the infrared and ultraviolet fixed points. We test our methods by finding the analogous non-equilibrium current through a point contact in a $ν=1/3$ quantum Hall state, confirming the Bethe ansatz solution of the problem.

cond-mat.mes-hall↗

arXiv2008

Tutte chromatic identities from the Temperley-Lieb algebra

This paper introduces a conceptual framework, in the context of quantum topology and the algebras underlying it, for analyzing relations obeyed by the chromatic polynomial χ(Q) of planar graphs. Using it we give new proofs and substantially extend a number of classical results concerning the combinatorics of the chromatic polynomial. In particular, we show that Tutte's golden identity is a consequence of level-rank duality for SO(N) topological quantum field theories and Birman-Murakami-Wenzl algebras. This identity is a remarkable feature of the chromatic polynomial relating χ({ϕ+2}) for any triangulation of the sphere to (χ({ϕ+1}))^2 for the same graph, where ϕdenotes the golden ratio. The new viewpoint presented here explains that Tutte's identity is special to these values of the parameter Q. A natural context for analyzing such properties of the chromatic polynomial is provided by the {\em chromatic algebra}, whose Markov trace is the chromatic polynomial of an associated graph. We use it to show that another identity of Tutte's for the chromatic polynomial at Q=ϕ+1 arises from a Jones-Wenzl projector in the Temperley-Lieb algebra. We generalize this identity to each value Q= 2+2\cos(2πj/(n+1)) for j< n positive integers. When j=1, these Q are the Beraha numbers, where the existence of such identities was conjectured by Tutte. We present a recursive formula for this sequence of chromatic polynomial relations.

math.CO↗

arXiv2008

Topological order from quantum loops and nets

I define models of quantum loops and nets which have ground states with topological order. These make possible excited states comprised of deconfined anyons with non-abelian braiding. With the appropriate inner product, these quantum loop models are equivalent to net models whose topological weight involves the chromatic polynomial. A useful consequence is that the models have a quantum self-duality, making it possible to find a simple Hamiltonian preserving the topological order. For the square lattice, this Hamiltonian has only four-spin interactions.

cond-mat.stat-mech↗

arXiv2008

Critical points in coupled Potts models and critical phases in coupled loop models

We show how to couple two critical Q-state Potts models to yield a new self-dual critical point. We also present strong evidence of a dense critical phase near this critical point when the Potts models are defined in their completely packed loop representations. In the continuum limit, the new critical point is described by an SU(2) coset conformal field theory, while in this limit of the the critical phase, the two loop models decouple. Using a combination of exact results and numerics, we also obtain the phase diagram in the presence of vacancies. We generalize these results to coupling two Potts models at different Q.

cond-mat.stat-mech↗

arXiv2008

Order parameter statistics in the critical quantum Ising chain

In quantum spin systems obeying hyperscaling, the probability distribution of the total magnetization takes on a universal scaling form at criticality. We obtain this scaling function exactly for the ground state and first excited state of the critical quantum Ising spin chain. This is achieved through a remarkable relation to the partition function of the anisotropic Kondo problem, which can be computed by exploiting the integrability of the system.

cond-mat.stat-mech↗

arXiv2007

Cooper pairs and exclusion statistics from coupled free-fermion chains

We show how to couple two free-fermion chains so that the excitations consist of Cooper pairs with zero energy, and free particles obeying (mutual) exclusion statistics. This behavior is reminiscent of anyonic superconductivity, and of a ferromagnetic version of the Haldane-Shastry spin chain, although here the interactions are local. We solve this model using the nested Bethe ansatz, and find all the eigenstates; the Cooper pairs correspond to exact-string or ``0/0'' solutions of the Bethe equations. We show how the model possesses an infinite-dimensional symmetry algebra, which is a supersymmetric version of the Yangian symmetry algebra for the Haldane-Shastry model.

cond-mat.stat-mech↗

arXiv2007

Quantum loop models and the non-abelian toric code

I define quantum loop models whose degrees of freedom are Ising spins on the square lattice as in the toric code, but where the excitations should have non-abelian statistics. The inner product is topological, allowing a direct implementation of the anyonic fusion matrix on the lattice. It also makes deconfined anyons possible for a variety of values of the weight per loop $d$ in the ground state. For d=\sqrt{2}, a gapped non-abelian topological phase can occur with only four-spin interactions.

cond-mat.stat-mech↗