arXiv subjects

Steven H. Simon

Publications and source records attributed to Steven H. Simon.

At least 109 records · Page 6Linked to original sources

arXiv2013

Tunnelling current through fractional quantum Hall interferometers

We calculate the tunnelling current through a Fabry-Pérot interferometer in the fractional quantum Hall regime. Within linear response theory (weak tunnelling but arbitrary source-drain voltage) we find a general expression for the current due to tunnelling of quasiparticles in terms of Carlson's $R$ function. Our result is valid for fractional quantum Hall states with an edge theory consisting of a charged channel and any number of neutral channels, with possibly different edge velocities and different chiralities. We analyse the case with a single neutral channel in detail, which applies for instance to the edge of the Moore-Read state. In addition we consider an asymmetric interferometer with different edge lengths between the point contacts on opposite edges, and we study the behaviour of the current as a function of varying edge length. Recent experiments attempted to measure the Aharanov-Bohm effect by changing the area inside the interferometer using a plunger gate. Theoretical analyses of these experiments have so far not taken into account the accompanying change in the edge lengths. We show that the tunnelling current exhibits multiple osculations as a function of this edge length, with frequencies proportional to the injected edge current and inversely proportional to the edge velocities. In particular the edge velocities can be measured by looking at the Fourier spectrum of the edge current. We provide a numerical scheme to calculate and plot the $R$ function, and include sample plots for a variety of edge states with parameter values which are experimentally relevant.

cond-mat.mes-hall↗

arXiv2013

Spin-singlet Gaffnian wave function for fractional quantum Hall systems

We characterize in detail a wave function conceivable in fractional quantum Hall systems where a spin or equivalent degree of freedom is present. This wave function combines the properties of two previously proposed quantum Hall wave functions, namely the non-Abelian spin-singlet state and the nonunitary Gaffnian wave function. This is a spin-singlet generalization of the spin-polarized Gaffnian, which we call the "spin-singlet Gaffnian" (SSG). In this paper we present evidence demonstrating that the SSG corresponds to the ground state of a certain local Hamiltonian, which we explicitly construct, and, further, we provide a relatively simple analytic expression for the unique ground-state wave functions, which we define as the zero energy eigenstates of that local Hamiltonian. In addition, we have determined a certain nonunitary, rational conformal field theory which provides an underlying description of the SSG and we thus conclude that the SSG is ungapped in the thermodynamic limit. In order to verify our construction, we implement two recently proposed techniques for the analysis of fractional quantum Hall trial states: The "spin dressed squeezing algorithm", and the "generalized Pauli principle".

cond-mat.str-el↗

arXiv2012

Exactly Solvable Lattice Models with Crossing Symmetry

We show how to compute the exact partition function for lattice statistical-mechanical models whose Boltzmann weights obey a special "crossing" symmetry. The crossing symmetry equates partition functions on different trivalent graphs, allowing a transformation to a graph where the partition function is easily computed. The simplest example is counting the number of nets without ends on the honeycomb lattice, including a weight per branching. Other examples include an Ising model on the Kagome' lattice with three-spin interactions, dimers on any graph of corner-sharing triangles, and non-crossing loops on the honeycomb lattice, where multiple loops on each edge are allowed. We give several methods for obtaining models with this crossing symmetry, one utilizing discrete groups and another anyon fusion rules. We also present results indicating that for models which deviate slightly from having crossing symmetry, a real-space decimation (renormalization-group-like) procedure restores the crossing symmetry.

cond-mat.stat-mech↗

arXiv2012

Three-dimensional topological lattice models with surface anyons

We study a class of three dimensional exactly solvable models of topological matter first put forward by Walker and Wang [arXiv:1104.2632v2]. While these are not models of interacting fermions, they may well capture the topological behavior of some strongly correlated systems. In this work we give a full pedagogical treatment of a special simple case of these models, which we call the 3D semion model: We calculate its ground state degeneracies for a variety of boundary conditions, and classify its low-lying excitations. While point defects in the bulk are confined in pairs connected by energetic strings, the surface excitations are more interesting: the model has deconfined point defects pinned to the boundary of the lattice, and these exhibit semionic braiding statistics. The surface physics is reminiscent of a $ν=1/2$ bosonic fractional quantum Hall effect in its topological limit, and these considerations help motivate an effective field theoretic description for the lattice models as variants of $bF$ theories. Our special example of the 3D semion model captures much of the behavior of more general `confined Walker-Wang models'. We contrast the 3D semion model with the closely related 3D version of the toric code (a lattice gauge theory) which has deconfined point excitations in the bulk and we discuss how more general models may have some confined and some deconfined excitations. Having seen that there exist lattice models whose surfaces have the same topological order as a bosonic fractional quantum Hall effect on a confining bulk, we construct a lattice model whose surface has similar topological order to a fermionic quantum hall effect. We find that in these models a fermion is always deconfined in the three dimensional bulk.

cond-mat.str-el↗

arXiv2012

Fractional Quantum Hall Effect of Lattice Bosons Near Commensurate Flux

We study interacting bosons on a lattice in a magnetic field. When the number of flux quanta per plaquette is close to a rational fraction, the low energy physics is mapped to a multi-species continuum model: bosons in the lowest Landau level where each boson is given an internal degree of freedom, or pseudospin. We find that the interaction potential between the bosons involves terms that do not conserve pseudospin, corresponding to umklapp processes, which in some cases can also be seen as BCS-type pairing terms. We argue that in experimentally realistic regimes for bosonic atoms in optical lattices with synthetic magnetic fields, these terms are crucial for determining the nature of allowed ground states. In particular, we show numerically that certain paired wavefunctions related to the Moore-Read Pfaffian state are stabilized by these terms, whereas certain other wavefunctions can be destabilized when umklapp processes become strong.

cond-mat.mes-hall↗

arXiv2012

Spinful Composite Fermions in a Negative Effective Field

In this paper we study fractional quantum Hall composite fermion wavefunctions at filling fractions ν= 2/3, 3/5, and 4/7. At each of these filling fractions, there are several possible wavefunctions with different spin polarizations, depending on how many spin-up or spin-down composite fermion Landau levels are occupied. We calculate the energy of the possible composite fermion wavefunctions and we predict transitions between ground states of different spin polarizations as the ratio of Zeeman energy to Coulomb energy is varied. Previously, several experiments have observed such transitions between states of differing spin polarization and we make direct comparison of our predictions to these experiments. For more detailed comparison between theory and experiment, we also include finite-thickness effects in our calculations. We find reasonable qualitative agreement between the experiments and composite fermion theory. Finally, we consider composite fermion states at filling factors ν= 2+2/3, 2+3/5, and 2+4/7. The latter two cases we predict to be spin polarized even at zero Zeeman energy.

cond-mat.mes-hall↗

arXiv2012

Importance of interband transitions for the fractional quantum Hall effect in bilayer graphene

Several recent works have proposed that electron-electron interactions in bilayer graphene can be tuned with the help of external parameters, making it possible to stabilize different fractional quantum Hall states. In these prior works, phase diagrams were calculated based on a single Landau level approximation. We go beyond this approximation and investigate the influence of polarization effects and virtual interband transitions on the stability of fractional quantum Hall states in bilayer graphene. We find that for realistic values of the dielectric constant, the phase diagram is strongly modified by these effects. We illustrate this by evaluating the region of stability of the Pfaffian state.

cond-mat.str-el↗

arXiv2012

Evaluation of ranks of real space and particle entanglement spectra for large systems

We devise a way to calculate the dimensions of symmetry sectors appearing in the Particle Entanglement Spectrum (PES) and Real Space Entanglement Spectrum (RSES) of multi-particle systems from their real space wave functions. We first note that these ranks in the entanglement spectra equal the dimensions of spaces of wave functions with a number of particles fixed. This also yields equality of the multiplicities in the PES and the RSES. Our technique allows numerical calculations for much larger systems than were previously feasible. For somewhat smaller systems, we can find approximate entanglement energies as well as multiplicities. We illustrate the method with results on the RSES and PES multiplicities for integer quantum Hall states, Laughlin and Jain composite fermion states and for the Moore-Read state at filling $ν=5/2$, for system sizes up to 70 particles.

cond-mat.str-el↗

arXiv2012

Telegraph noise and the Fabry-Perot quantum Hall interferometer

We consider signatures of abelian and nonabelian quasiparticle statistics in quantum Hall Fabry-Perot interferometers. When quasiparticles enter and exit the interference cell, for instance due to glassy motion in the dopant layer, the anyonic phase can be observed in phase jumps. In the case of the nonabelian nu=5/2 state, if the interferometer is small, we argue that free Majorana states in the interference cell are either strongly coupled to one another or are strongly coupled to the edge. We analyze the expected phase jumps and in particular suggest that changes in the parity of the Majorana ground state should gives rise to characteristic jumps of pi in the interference phase.

cond-mat.mes-hall↗

arXiv2011

Wilson Line Picture of Levin-Wen Partition Functions

Levin and Wen [Phys. Rev. B 71, 045110 (2005)] have recently given a lattice Hamiltonian description of doubled Chern-Simons theories. We relate the partition function of these theories to an expectation of Wilson loops that form a link in 2+1 dimensional spacetime known in the mathematical literature as Chain-Mail. This geometric construction gives physical interpretation of the Levin-Wen Hilbert space and Hamiltonian, its topological invariance, exactness under coarse-graining, and how two opposite chirality sectors of the doubled theory arise.

cond-mat.str-el↗

arXiv2011

Condensation of achiral simple currents in topological lattice models: a Hamiltonian study of topological symmetry breaking

We describe a family of phase transitions connecting phases of differing non-trivial topological order by explicitly constructing Hamiltonians of the Levin-Wen[PRB 71, 045110] type which can be tuned between two solvable points, each of which realizes a different topologically ordered phase. We show that the low-energy degrees of freedom near the phase transition can be mapped onto those of a Potts model, and we discuss the stability of the resulting phase diagram to small perturbations about the model. We further explain how the excitations in the condensed phase are formed from those in the original topological theory, some of which are split into multiple components by condensation, and we discuss the implications of our results for understanding the nature of general achiral topological phases in 2+1 dimensions in terms of doubled Chern-Simons theories.

cond-mat.str-el↗

arXiv2010

Phase transitions in topological lattice models via topological symmetry breaking

We study transitions between phases of matter with topological order. By studying these transitions in exactly solvable lattice models we show how universality classes may be identified and critical properties described. As a familiar example to elucidate our results concretely, we describe in detail a transition between a fully gapped achiral 2D $p$-wave superconductor ($p+ip$ for pseudospin up/$p-ip$ for pseudospin down) to an $s$-wave superconductor which we show to be in the 2D transverse field Ising universality class.

cond-mat.str-el↗

arXiv2010

A Random Matrix--Theoretic Approach to Handling Singular Covariance Estimates

In many practical situations we would like to estimate the covariance matrix of a set of variables from an insufficient amount of data. More specifically, if we have a set of $N$ independent, identically distributed measurements of an $M$ dimensional random vector the maximum likelihood estimate is the sample covariance matrix. Here we consider the case where $N<M$ such that this estimate is singular and therefore fundamentally bad. We present a radically new approach to deal with this situation. Let $X$ be the $M\times N$ data matrix, where the columns are the $N$ independent realizations of the random vector with covariance matrix $Σ$. Without loss of generality, we can assume that the random variables have zero mean. We would like to estimate $Σ$ from $X$. Let $K$ be the classical sample covariance matrix. Fix a parameter $1\leq L\leq N$ and consider an ensemble of $L\times M$ random unitary matrices, $\{Φ\}$, having Haar probability measure. Pre and post multiply $K$ by $Φ$, and by the conjugate transpose of $Φ$ respectively, to produce a non--singular $L\times L$ reduced dimension covariance estimate. A new estimate for $Σ$, denoted by $\mathrm{cov}_L(K)$, is obtained by a) projecting the reduced covariance estimate out (to $M\times M$) through pre and post multiplication by the conjugate transpose of $Φ$, and by $Φ$ respectively, and b) taking the expectation over the unitary ensemble. Another new estimate (this time for $Σ^{-1}$), $\mathrm{invcov}_L(K)$, is obtained by a) inverting the reduced covariance estimate, b) projecting the inverse out (to $M\times M$) through pre and post multiplication by the conjugate transpose of $Φ$, and by $Φ$ respectively, and c) taking the expectation over the unitary ensemble. We have a closed analytical expression for $\mathrm{invcov}_L(K)$ and $\mathrm{cov}_L(K)$ in terms of its eigenvalue decomposition.

math.PR↗

arXiv2010

Trial wavefunctions for the Goldstone mode in ν=1/2+1/2 quantum Hall bilayers

Based on the known physics of the excitonic superfluid or 111 state of the quantum Hall ν=1/2+1/2 bilayer, we create a simple trial wavefunction ansatz for constructing a low energy branch of (Goldstone) excitations by taking the overall ground state and boosting one layer with respect to the other. This ansatz works extremely well for any interlayer spacing. For small d this is simply the physics of the Goldstone mode, whereas for large d this is a reflection of composite fermion physics. We find hints that certain aspects of composite fermion physics persist to low d whereas certain aspects of Goldstone mode physics persist to high d. Using these results we show nonmonotonic behavior of the Goldstone mode velocity as a function of d.

cond-mat.mes-hall↗

arXiv2010

Space-Time Geometry of Topological phases

The 2+1 dimensional lattice models of Levin and Wen [PRB 71, 045110 (2005)] provide the most general known microscopic construction of topological phases of matter. Based heavily on the mathematical structure of category theory, many of the special properties of these models are not obvious. In the current paper, we present a geometrical space-time picture of the partition function of the Levin-Wen models which can be described as doubles (two copies with opposite chiralities) of underlying Anyon theories. Our space-time picture describes the partition function as a knot invariant of a complicated link, where both the lattice variables of the microscopic Levin-Wen model and the terms of the hamiltonian are represented as labeled strings of this link. This complicated link, previously studied in the mathematical literature, and known as Chain-Mail, can be related directly to known topological invariants of 3-manifolds such as the so called Turaev-Viro invariant and the Witten-Reshitikhin-Turaev invariant. We further consider quasi-particle excitations of the Levin-Wen models and we see how they can be understood by adding additional strings to the Chain-Mail link representing quasi-particle world-lines. Our construction gives particularly important new insight into how a doubled theory arises from these microscopic models.

cond-mat.str-el↗

arXiv2010

Breaking of Particle-Hole Symmetry by Landau Level Mixing in the nu=5/2 Quantized Hall State

We perform numerical studies to determine if the fractional quantum Hall state observed at filling nu=5/2 is the Moore-Read wavefunction or its particle hole conjugate, the so-called AntiPfaffian. Using a truncated Hilbert space approach we find that for realistic interactions, including Landau-level mixing, the ground state remains fully polarized and the AntiPfaffian is strongly favored.

cond-mat.mes-hall↗

arXiv2009

S3 Quantum Hall Wavefunctions

We construct a family of quantum Hall Hamiltonians whose ground states, at least for small system sizes, give correlators of the S3 conformal field theories. The ground states are considered as trial wavefunctions for quantum Hall effect of bosons at filling fraction nu=3/4 interacting either via delta function interaction or delta function plus dipole interaction. While the S3 theories can be either unitary or nonunitary, we find high overlaps with exact diagonalizations only for the nonunitary case, suggesting that these wavefunctions may correspond to critical points, possibly analogous to the previously studied Gaffnian wavefunction. These wavefunctions give an explicit example which cannot be fully characterized by their thin-torus limit or by their pattern of zeros.

cond-mat.str-el↗

arXiv2009

Skyrmions in the Moore-Read state at nu=5/2

We study charged excitations of the non-abelian Moore-Read liquid at filling factor nu=5/2, allowing for spin depolarization. Using a combination of numerical studies, and taking account of non-zero well widths, we find that at sufficiently low Zeeman energy it is energetically favourable for charge e/4 quasiholes to bind into "skyrmions" of charge e/2. We show that skyrmion formation is further promoted by disorder, and argue that this can lead to a depolarized nu=5/2 ground state in realistic experimental situations. We comment on the consequences for the activated transport.

cond-mat.str-el↗