Bulletin of the American Physical Society
APS March Meeting 2013
Volume 58, Number 1
Monday–Friday, March 18–22, 2013; Baltimore, Maryland
Session N23: Fractional Quantum Hall Theory I |
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Sponsoring Units: FIAP Chair: B. Andrei Bernevig, Princeton University Room: 325 |
Wednesday, March 20, 2013 11:15AM - 11:27AM |
N23.00001: When is a ``wavefunction'' not a wavefunction?: a quantum-geometric reinterpretation of the Laughlin state F.D.M. Haldane The Laughlin state is the fundamental model for the description of fractional quantum Hall (FQH) fluids and was presented as a ``lowest Landau-level (LLL) Schr\"odinger wavefunction'', i.e., of the form $f(z_1,\ldots ,z_N)\exp -\sum_i z_i^*z_i/2$, where $z_i$ = $(x_i + iy_i)/\surd 2\ell_B$, and $|z_i-z_0|^2$ = constant is the shape of a Landau orbit. Its characterization as a LLL wavefunction was generally accepted without question, and leads to ``explanations'' of its success in terms of judicious placement of its zeroes. However, the Laughlin state also occurs in the n=2 LL, and now has been found in Chern-insulator lattice systems. Numerical studies confirm that (without direct reference to which LL is partially-occupied) its success can be explained solely in terms of the short-range repulsion between the non-commuting guiding centers of Landau orbits. These (as a ``quantum geometry'') do not by themselves have a Schr\"odinger (as opposed to Heisenberg) description. A reexamination shows that the variable ``z'' describes the shape of an emergent geometry of the FQH fluid derived from the Coulomb interaction, not the Landau-orbit shape, and that the holomorphic function is a coherent state representation of a Heisenberg state, not a Schr\"odinger wavefunction. [Preview Abstract] |
Wednesday, March 20, 2013 11:27AM - 11:39AM |
N23.00002: The correct theory of the quantum Hall effect fractions Keshav Shrivastava The effective charge,e*$=$(1/2)ge, obtained by introducing the symmetric g values, g$=$(2J$+$1)/(2L$+$1) with J given by L and S with both signs for S, and the Bohr magneton, used in the cyclotron frequency leads to factors of the type (1/2)g(n$+$1/2) in the eigen values which give the correct description of the modified Landau levels. The resistivity after introducing the flux quantization is modified by the effective charge which gives the plateaux. The helicity of every electron is defined by the sign of p.s where p is the linear momentum and s is the spin. Hence the $+$s particles move in the direction opposite to those of -s. The principal fractions, two-particle states and resonances explain most of the data. The remaining data is explained by the formation of electron clusters with spin different from 1/2. In this way all of the 101 or more fractions of the experimental data are correctly derived from the theory. The theory does not depend seriously on the dimensionality so it explains the graphite as well as the graphene.\\[4pt] [1] K. N. Shrivastava, AIP Conf. Proc. 1482,335-339(2012); AIP Conf. Proc. 1150, 59-67(2009); International J. Mod. Phys. B 25, 1301-1357(2011).\\[0pt][2] Maher M. A. Ali and K. N. Shrivastava, AIP Conf.Proc. 1482, 43-46(2012). [Preview Abstract] |
Wednesday, March 20, 2013 11:39AM - 11:51AM |
N23.00003: Nematic order and a new field theory of the quantum Hall effect Joseph Maciejko, Benjamin Hsu, Yeje Park, Steve Kivelson, Shivaji Sondhi Motivated by recent experimental and theoretical studies of anisotropic versions of the fractional quantum Hall (FQH) effect, we construct an effective field theory for a continuous quantum phase transition between an isotropic FQH state and a nematic FQH state. The theory parallels earlier work on FQH ferromagnets. The $SO(3)$ order parameter $\mathbf{n}$ of the ferromagnet is replaced by the Landau-de Gennes nematic tensor order parameter $Q_{ab}$ which can be mapped to a $SO(2,1)$ Lorentz vector. We construct an analog of the $CP^1$ representation of a ferromagnet in terms of complex $SU(1,1)$ spinors. We identify these vector and spinor order parameters with the unimodular metric and zweibein fields appearing in Haldane's recent geometrical description of the FQH effect, where the metric field $g_{ab}$ is given by the matrix exponential of the nematic order parameter $Q_{ab}$. Our theory predicts that if the gap of the Girvin-MacDonald-Platzman collective mode can be made to collapse at zero momentum in a FQH system, an instability to a FQH nematic state should occur. [Preview Abstract] |
Wednesday, March 20, 2013 11:51AM - 12:03PM |
N23.00004: Elementary formula for the Hall conductivity of interacting systems Titus Neupert, Luiz Santos, Claudio Chamon, Christopher Mudry We proof a formula for the Hall conductivity of interacting electrons under the assumption that the ground state manifold has finite degeneracy and discrete translation symmetry is neither explicitly nor spontaneously broken. Via an algebraic regularization, our derivation makes use of the noncommutative relations obeyed by the components of the position and density operators in topological band structures. We discuss the implications of our result in the context of fractional Chern insulators. [Preview Abstract] |
Wednesday, March 20, 2013 12:03PM - 12:15PM |
N23.00005: Fractional Quantum Hall Effect from Phenomenological Bosonization Vladimir Zyuzin In this work we propose a model of the fractional quantum Hall effect within conventional one-dimensional bosonization. It is shown that in this formalism the resulting bosonized fermion operator corresponding to momenta of Landau gauge wave function is effectively two-dimensional. At special filling factors the bulk gets gapped, and the theory is described by a sine-Gordon model. The edges are shown to be gapless, chiral, and carrying a fractional charge. The hierarchy of obtained fractional charges is consistent with existing experiments and theories. It is also possible to draw a connection to composite fermion description and to the Laughlin many-body wave function. [Preview Abstract] |
Wednesday, March 20, 2013 12:15PM - 12:27PM |
N23.00006: Axial anomaly of Lifshitz fermions with arbitrary anisotropic scaling $z$ in $2n$ spacetime dimensions Xueda Wen We calculate the axial anomaly of a Lifshitz fermion with arbitrary anisotropy scaling exponent $z$ which is coupled to gauge fields in $2n$ spacetime dimensions. We find that the result is identical to the relativistic case. The conclusion is verified with both path integral methods and spectral methods in $2n$ spacetime dimensions. Our work is a generalization of I. Bakas' work (arXiv:1110.1332) which focuses on (3+1) dimensions. In addition, we discuss the application of our conclusion to transport processes in quantum Hall systems as well as Weyl semi-metals. [Preview Abstract] |
Wednesday, March 20, 2013 12:27PM - 12:39PM |
N23.00007: Metallic phase of the quantum Hall effect in four-dimensional space Jonathan Edge, Jakub Tworzydlo, Carlo Beenakker We study the phase diagram of the quantum Hall effect in four-dimensional (4D) space. Unlike in 2D, in 4D there exists a metallic as well as an insulating phase, depending on the disorder strength. The critical exponent $\nu\approx 1.2$ of the diverging localization length at the quantum Hall insulator-to-metal transition differs from the semiclassical value $\nu=1$ of 4D Anderson transitions in the presence of time-reversal symmetry. Our numerical analysis is based on a mapping of the 4D Hamiltonian onto a 1D dynamical system, providing a route towards the experimental realization of the 4D quantum Hall effect. [Preview Abstract] |
Wednesday, March 20, 2013 12:39PM - 12:51PM |
N23.00008: Correlations in incompressible quantum liquid states: constructions of electronic trial wavefunctions John Quinn Numerical studies indicate that incompressible quantum Hall states occur when the relation between the single particle angular momentum $l$ and the number $N$ of electrons in the partially filled Landau level is $2l = \nu^{-1}N-c_\nu$. Here, $\nu$ is the filling factor and $c_\nu$ is a ``finite size shift.'' The values of $c_\nu$ found numerically depend on correlations, and for $\nu=p/q\leq 1/2$ are given by $c_\nu = q+1-p$. This finite size shift points the way to constructing electronic trial wavefunctions. A trial wavefunction can always be written $\Psi = FC$, where $F = \prod_{i < j}z_{ij}$ and $C(z_{ij})$ is a symmetric correlation function caused by interactions. For the Moore-Read state, $C_{MR}(z_{ij})$ is a product of $F$ and the antisymmetric Pfaffian. $C_{MR}$ is not the only possible correlation function for this state. Another choice is the quadratic function $C_Q = S \left\{\prod_{i < j\in g_A} \prod_{k < l\in g_B}(z_{ij}z_{kl})^2\right\}$, where $S$ is a symmetrizing operator, and $g_A$ and $g_B$ each contain $N/2$ particles resulting from a partition of N into two sets. For the Jain states (e.g. $\nu=2/5$), different partitioning of $N$ particles into sets of unequal size gives appropriate correlation functions. [Preview Abstract] |
Wednesday, March 20, 2013 12:51PM - 1:03PM |
N23.00009: Exactly solvable $U(1)\times U(1)$ boson models for integer and fractional quantum Hall insulators in two dimensions Olexei Motrunich, Scott Geraedts We present a solvable boson model with $U(1)\times U(1)$ symmetry in (2+1) dimensions that realizes insulating phases with a quantized Hall conductivity $\sigma_{xy}$. The model is short-ranged, with no topological terms, and can be realized by a local Hamiltonian. For one set of parameters, the model has a non-fractionalized phase with $\sigma_{xy}=2n$ in appropriate units, with $n$ an integer. In this case, the physical origin is dynamical binding between $n$ bosons of one species and a vortex of the other species and condensation of such composites. Other choices for the parameters of the model yield a phase with $\sigma_{xy}=2\frac{c}{d}$, where $c$ and $d$ are mutually prime integers. In this phase, $c$ bosons dynamically bind to $d$ vortices and such objects condense. The are two species of excitations that are bosonic by themselves but carry fractional charge $1/d$ and have mutual statistics $2\pi\frac{b}{d}$, where $b$ is an integer such that $ad-bc=1$, and $a$ is also an integer. The model can be studied using sign-free Monte Carlo. We have performed simulations which include a boundary between a quantum Hall insulator and a trivial insulator, and found gapless edge states on the boundary. [Preview Abstract] |
Wednesday, March 20, 2013 1:03PM - 1:15PM |
N23.00010: Matrix Product States and Fractional Quantum Hall B. Andrei Bernevig, Benoit Estienne, Nicolas Regnault, Zlatko Papic We present an exact matrix product state expansion (MPS) for a large series of Jack polynomial wavefunctions which serve as Fractional Quantum Hall ground-states of pseudopotential Hamiltonians. Using the basis of descendants in Virasoro and W algebras we build MPS descriptions of the (k,2) Jacks which include the Moore-Read state and the Gaffnian state, as well as MPS representation of the Z$_3$ Read-Rezayi state. We then give a general method for computing MPS representations for other non-abelian states and their quasiholes. [Preview Abstract] |
Wednesday, March 20, 2013 1:15PM - 1:27PM |
N23.00011: Benchmarking MPS for fractional quantum Hall states Nicolas Regnault, Benoit Estienne, Zlatko Papic, B. Andrei Bernevig We discuss the numerical apsects of the Matrix Product State (MPS) representation for a large series of Fractional Quantum Hall states. We benchmark the MPS for several model states such as the Read-Rezayi series using both overlap, energies, densities and pair correlation functions. We discuss how accurate this description is depending on the geometry (sphere, disk or cylinder). As an application, we use the MPS to compute the size of the quasiholes for the Read-Rezayi series. [Preview Abstract] |
Wednesday, March 20, 2013 1:27PM - 1:39PM |
N23.00012: Exact Matrix Product States for Quantum Hall Wave Functions Roger Mong, Michael Zaletel We show that the model wave functions used to describe the fractional quantum Hall effect have exact representations as matrix product states (MPS). These MPS can be implemented numerically in the orbital basis of both finite and infinite cylinders, which provides an efficient way of calculating arbitrary observables. We extend this approach to the charged excitations and numerically compute their Berry phases. Finally, we present an algorithm for numerically computing the real-space entanglement spectrum starting from an arbitrary orbital basis MPS, which allows us to study the scaling properties of the real-space entanglement spectra on infinite cylinders. The real-space entanglement spectrum obeys a scaling form dictated by the edge conformal field theory, allowing us to accurately extract the two entanglement velocities of the Moore-Read state. [Preview Abstract] |
Wednesday, March 20, 2013 1:39PM - 1:51PM |
N23.00013: Long-wavelength corrections to Hall conductivity in fractional quantum Hall fluids Bo Yang, F.D.M. Haldane Recent work by Hoyos and Son, then Bradlyn et al., has investigated the relation between the long-wavelength ($O(q^2)$) corrections to the Hall conductivity $\sigma_H({\mathbf q})$ and the Hall viscosity of quantum Hall states. These works assume the presence of Galilean and rotational invariance. However, these are not generic symmetries of electrons in condensed matter. We identify translation and (2D) inversion symmetry as the only generic symmetries of an ``ideal" quantum Hall liquid, as these are needed to guarantee the absence of any dissipationless ground state current density; then $\sigma_H({\mathbf q})$ = $\sigma_H(-{\mathbf q})$ characterizes the dissipation less current that flows in response to a spatially-non-uniform electric field. We consider the general problem for fractional quantum Hall (FQH) states without Galilean or rotational invariance, when the guiding-center contribution to the Hall viscosity becomes a non-trivial tensor property related to an emergent geometry of the FQH state, (Bo Yang et,al (PRB 85,165318). [Preview Abstract] |
Wednesday, March 20, 2013 1:51PM - 2:03PM |
N23.00014: Fractional topological superconductors with fractionalized Majorana fermions Abolhassan Vaezi In Ref[1], I introduced a two dimensional fractional topological superconductor (FTSC) as a strongly correlated topological state which can be achieved by inducing superconductivity into an Abelian fractional quantum Hall (FQH) state, through the proximity effect. When the proximity coupling is weak, the FTSC has the same topological order as its parent state, and thus Abelian. However, upon increasing the proximity coupling, the bulk gap of such an Abelian FTSC closes and reopens resulting in a new topological order: a non-Abelian FTSC. I show that the conformal field theory (CFT) that describes the edge state of non-Abelian FTSC is $U(1)/Z_2$ orbifold theory and use this to write down the ground-state wave-function. Further, I predict FTSC based on Laughlin state at $\nu=1/m$ filling to host vortices with fractionalized Majorana zero modes. These zero modes are non-Abelian quasi-particles which is evident in their quantum dimension of $d_m=\sqrt{2m}$. Using the multi-quasi-particle wave-function based on the edge CFT, I derive the braid matrix for the zero modes. Finally, the potential applications of the non-Abelian FTSCs in the topological quantum computation will be illustrated. [1] A. Vaezi, ArXiv:1204.6245 (2012) [Preview Abstract] |
Wednesday, March 20, 2013 2:03PM - 2:15PM |
N23.00015: Momentum polarization: an entanglement measure of topological spin Xiaoliang Qi Topologically ordered states are states of matter which are distinct from trivial states by topological properties such as ground state degeneracy and quasi-particles carrying fractional quantum numbers and fractional statistics. The topological spin is an important property of a topological quasi-particle, which is the Berry phase obtained in the adiabatic self-rotation of the quasi-particle by $2\pi$. In this paper we propose a new approach to compute the topological spin in candidate systems of two-dimensional topologically ordered states. We identify the topological spin with a new quantity, the momentum polarization defined on the cylinder geometry. We show that the momentum polarization is determined by the quantum entanglement between the two halves of the cylinder, and can be computed from the reduced density matrix. As an example we present numerical results for the honeycomb lattice Kitaev model, which correctly reproduces the expected spin $e^{i\frac{2\pi}{16}}$ of the Ising non-Abelian anyon ($\sigma$ particle). Our result provides a new efficient approach to characterize and identify topological states of matter from finite size numerics. [Preview Abstract] |
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