Bulletin of the American Physical Society
APS March Meeting 2014
Volume 59, Number 1
Monday–Friday, March 3–7, 2014; Denver, Colorado
Session D45: Quantum Hall Effect: Theory |
Hide Abstracts |
Sponsoring Units: FIAP Chair: Kun Yang, Florida State University Room: Mile High Ballroom 4D |
Monday, March 3, 2014 2:30PM - 2:42PM |
D45.00001: Anyonic Symmetries and Non-Abelian Topological Defects of Bosonic Abelian Fractional Quantum (Spin) Hall States in the $ADE$ Classification Mayukh Khan, Jeffrey Teo, Taylor Hughes We consider bosonic abelian Fractional Quantum Hall (FQH) and Fractional Quantum Spin Hall (FQSH) states with edge theories drawn from the $ADE$ Kac Moody algebras at level $1.$ This set of systems have `anyonic' symmetries that leave braiding and fusion invariant Remarkably, the group of anyonic symmetries for this class of models is isomorphic to the symmetries of the Dynkin diagrams of the particular ADE Lie Algebra under consideration. The triality symmetry of the Dynkin diagram of $so(8)$ leads to the largest anyonic symmetry group $S_3$ (the permutation group on 3 elements). Each element of the anyonic symmetry group corresponds to a distinct way of gapping out the edge (i.e., each element corresponds to a Lagrangian subgroup). Junctions between two distinct gapped edges host non abelian twist defects with quantum dimensions ($>1$). In the case of $so(8)$ we have more exotic twist defects with non-abelian fusion. [Preview Abstract] |
Monday, March 3, 2014 2:42PM - 2:54PM |
D45.00002: Experimental Proposal to Detect Topological Ground State Degeneracy Maissam Barkeshli, Yuval Oreg, Xiao-Liang Qi One of the most profound features of topologically ordered states of matter, such as the fractional quantum Hall (FQH) states, is that they possess topology-dependent ground state degeneracies that are robust to all local perturbations. Here we present the first proposal to directly detect these topological degeneracies in an experimentally accessible setup. The detection scheme uses nonlinear electrical conductance measurements in a double layer FQH system, with appropriately patterned top and bottom gates. We propose two experimental platforms; in the first, the detection of topo- logically degenerate states coincides with the detection of ZN parafermion zero modes. We map the relevant physics to a single-channel ZN quantum impurity model, providing a novel generalization of the Kondo model. Our proposal can also be adapted to detect the ZN parafermion zero modes recently discovered in FQH line junctions proximitized with superconductivity. [Preview Abstract] |
Monday, March 3, 2014 2:54PM - 3:06PM |
D45.00003: Chiral Luttinger liquids and a generalized Luttinger's theorem in fractional quantum Hall edges via finite-entanglement scaling Daniel Varjas, Michael Zaletel, Joel Moore We use bosonic field theories and the infinite system density matrix renormalization group (iDMRG) method to study infinite strips of fractional quantum Hall (FQH) states starting from microscopic Hamiltonians. Finite-entanglement scaling allows us to accurately measure chiral central charge, edge mode exponents and momenta without finite-size errors. We analyze states in the first and second level of the standard hierarchy and compare our results to predictions of the chiral Luttinger liquid ($\chi$LL) theory. The results confirm the universality of scaling exponents in chiral edges and demonstrate that renormalization is subject to universal relations in the non-chiral case. We prove a generalized Luttinger's theorem involving all singularities in the momentum-resolved density, which naturally arises when mapping Landau levels on a cylinder to a fermion chain and deepens our understanding of non-Fermi liquids in 1D. [Preview Abstract] |
Monday, March 3, 2014 3:06PM - 3:18PM |
D45.00004: Identifying Non-Abelian Topological Order through Minimal Entangled States Wei Zhu, Shoushu Gong, Duncan Haldane, D.N. Sheng The topological order is encoded in the pattern of long-range quantum entanglements, which cannot be mea- sured by any local observable. Here we perform an exact diagonalization study to establish the non-Abelian topological order for topological band models through entanglement entropy measurement. We focus on the quasiparticle statistics of the non-Abelian Moore-Read and Read-Rezayi states on the lattice models with bosonic particles. We identify multiple independent minimal entangled states (MESs) in the groundstate mani- fold on a torus. The extracted modular S matrix from MESs faithfully demonstrates the Majorana quasiparticle or Fibonacci quasiparticle statistics, including the quasiparticle quantum dimensions and the fusion rules for such systems. These findings unambiguously demonstrate the topological nature of the quantum states for these flatband models without using the knowledge of model wavefunctions. We also establish that MESs manifest the eigenstates of nonlocal Wilson loop operators for the non-Abelian topological states and encode the full information of the topological order. [Preview Abstract] |
Monday, March 3, 2014 3:18PM - 3:30PM |
D45.00005: Momentum polarization of non-Abelian topologically ordered states Yi Zhang, Xiao-liang Qi We study momentum polarization of non-Abelian topologically ordered states for the Gutzwiller projected Chern insulator wave function with Chern number C=2. The resulting quasiparticle topological spin and edge central charge confirm the field theory description of an SU(2) gauge field coupled to $\nu=2$ fermions and rule out other candidate theories. We also discuss characteristic differences and the quantum phase transition between this non-Abelian topological phase and an Abelian topological phase described by the projected wave function of two C=1 Chern insulators. [Preview Abstract] |
Monday, March 3, 2014 3:30PM - 3:42PM |
D45.00006: $Z_2$ fractional topological insulators in two dimensions Cecile Repellin, Andrei Bernevig, Nicolas Regnault The simplest example of a two dimensional fractional topological insulator (FTI) consists of two decoupled copies of a Laughlin state with opposite chiralities. Using a simple microscopic model at half filling, we study the stability of this type of FTI phase upon addition of two coupling terms of different nature: a Rashba term, and an interspin interaction term. Using exact diagonalization and entanglement spectrums, we numerically show that the FTI phase survives significant amplitudes of both the band structure and the interaction coupling terms, at different system sizes. We compare our system to a similar two component fractional Chern insulator. Our study shows that the time reversal invariant system survives the introduction of interaction coupling on a much larger scale than the time reversal symmetry breaking one, stressing the importance of time reversal symmetry in the FTI phase stability. [Preview Abstract] |
Monday, March 3, 2014 3:42PM - 3:54PM |
D45.00007: Discrete Symmetry Breaking in Fractional Chern Insulators Akshay Kumar, Rahul Roy, S.L. Sondhi We study the interplay between quantum hall ordering and spontaneous translational symmetry breaking in a multiple Chern number (C $>$ 1) band at partial filling. We begin with non-interacting fermions in a family of square lattice models with flat C=2 bands and a wide band gap, and add nearest neighbor density-density repulsive interactions. By means of Hartree-Fock theory supplemented by numerical exact diagonalization for a small system at 1/2 filling, we find that the system generically develops charge density wave order with two degenerate ground states. We note that this physics is especially transparent in the limit in which the C=2 band describes two decoupled C=1 bands. We discuss the nature of domain walls in this phase and note the close analogy to the quantum Hall Ising ferromagnet in the multivalley problem. Finally we discuss generalizations to other fillings and higher Chern numbers. [Preview Abstract] |
Monday, March 3, 2014 3:54PM - 4:06PM |
D45.00008: Models for the phase transition between a Fermi liquid and fractional Chern insulator Joel Moore, Michael Zaletel, Siddharth Parameswaran A partially filled band with nonzero Chern density can support fractional quantum Hall states (``fractional Chern insulators'') as a consequence of repulsive interactions between electrons. In the absence of this repulsion, the ground state is generically a simple band metal with an anomalous Hall effect. There are several possible scenarios for a second-order transition between metallic and quantum Hall states, which can be approached as a composite-fermion band crossing, a coupling between Luttinger liquids, or via a parton construction. We discuss the extent to which these scenarios lead to different predictions and test those predictions by density-matrix renormalization group calculations. [Preview Abstract] |
Monday, March 3, 2014 4:06PM - 4:18PM |
D45.00009: Fractional Chern insulators on finite cylinders and their bulk-edge correspondence Zhao Liu, Dmitry Kovrizhin, Emil Bergholtz, Ravindra Bhatt It has been recently realized that strong interactions in topological Bloch bands give rise to the appearance of novel states of matter. Here we study these systems -- fractional Chern insulators -- via generalization of a gauge-fixed Wannier-Qi construction in the cylinder geometry. Our setup offers a number of important advantages compared to the earlier exact diagonalization studies on a torus. Most notably, it gives access to edge states and to a single-cut orbital entanglement spectrum, hence to the physics of bulk-edge correspondence. It is also readily implemented using density matrix renormalization group method which allows for numerical simulations of significantly larger systems. Previously [Z. Liu et al., Phys. Rev. B {\bf 88}, 081106(R) (2013)], this approach was applied to bosons on the ruby lattice model at filling $\nu=1/2$ and $\nu=1$, which show the signatures of (non)-Abelian phases, and we establish the correspondence between the physics of edge states and entanglement in the bulk. Here, we generalize this to other fillings such as $\nu=2/3$. [Preview Abstract] |
Monday, March 3, 2014 4:18PM - 4:30PM |
D45.00010: The impossibility of exactly flat non-trivial Chern bands in strictly local periodic tight binding models Li Chen, Tahereh Mazaheri, Alexander Seidel, Xiang Tang We investigate the possibility of exactly flat non-trivial Chern bands in tight binding models with local (strictly short-ranged) hopping parameters. We demonstrate that while any two of three criteria can be simultaneously realized (exactly flat band, non-zero Chern number, local hopping), it is not possible to simultaneously satisfy all three. We discuss both the case of a single flat band, for which we give a rather elementary proof, as well as the case of multiple degenerate flat bands. In the latter case, our result follows by making use of K-theory. [Preview Abstract] |
Monday, March 3, 2014 4:30PM - 4:42PM |
D45.00011: Criticality in Translation-Invariant Parafermion Chains Wei Li, Shuo Yang, Hong-hao Tu, Meng Cheng Parafermionic zero modes have been recently proposed to emerge at certain topological defects in Abelian fractional quantum Hall systems. In this work, we investigate the phase diagram of a translationally invariant $Z_3$ parafermion chain, with nearest- and next-nearest-neighbor hopping terms. The model can be mapped to a $Z_3$ Potts model with nearest-neighbor couplings via a generalized Jordan-Wigner transformation. The phase diagram is obtained numerically via accurate density matrix renormalization group method, and six gapless phases with central charges being 4/5, 1 or 2 are found. By checking the energy derivatives, we observe continuous phase transitions between $c=1$ and $c=2$ phases, while the phase transition between $c=4/5$ and $c=1$ is conjectured to be of Kosterlitz-Thouless type. [Preview Abstract] |
Monday, March 3, 2014 4:42PM - 4:54PM |
D45.00012: Twist Defects in Topological Systems with Anyonic Symmetries Jeffrey Teo, Abhishek Roy, Xiao Chen Twist defects are point-like objects that support robust non-local storage of quantum information and non-abelian unitary operations. Unlike quantum deconfined anyonic excitations, they rely on symmetry rather than a non-abelian topological order. Zero energy Majorana bound states can arise at lattice defects, such as disclinations and dislocations, in a topological crystalline superconductor. More general parafermion bound state can appear as twist defects in a topological phase with an anyonic symmetry, such as a bilayer fractional quantum Hall state and the Kitaev toric code. They are however fundamentally different from quantum anyonic excitations in a true topological phase. This is demonstrated by their unconventional exchange and braiding behavior, which is characterized by a modified spin statistics theorem and modular invariance. Gauging anyonic symmetries by treating twist defects as quantum excitations provides a connection between some non-abelian topological states and abelian ones. [Preview Abstract] |
Monday, March 3, 2014 4:54PM - 5:06PM |
D45.00013: Convergence of Topological Entanglement Entropy for Finite Size Systems Clare Abreu, Raul Herrera, Edward Rezayi Quantum information theoretic concepts have been widely used to study topological phases of condensed matter, the prime examples of which are fractional quantum Hall states. Interest in these phases is driven in part by their potential use in fault-tolerant topological quantum computation. In particular, quantum entanglement has proven to be a useful tool to probe topological order. We present numerical studies for some model fractional quantum Hall states in spherical and toroidal geometries. We implement bipartitioning of the system with both orbital and real space cuts for small size systems. Additionally, we compare the topological entanglement entropies obtained from low-order Renyi entropies to the expected value to determine whether our results converge for small sizes. We extend these studies to generic Hamiltonians and discuss the prospect of obtaining the topological entanglement entropy from finite size calculations in these systems. [Preview Abstract] |
Monday, March 3, 2014 5:06PM - 5:18PM |
D45.00014: Matrix Product States for Non-Abelian Quasiholes Yang-Le Wu, B. Estienne, N. Regnault, B. Andrei Bernevig Exotic phases in fractional quantum Hall effect provide a potential platform for the realization of non-Abelian anyons. A large class of physically relevant trial wave functions for these strongly-correlated phases can be constructed from the many-point correlators in various chiral conformal field theories. It was recently realized that this construction can be naturally reformulated in terms of matrix product states and efficiently carried out on a computer, even for interacting conformal fields. In this talk, I will explain how to construct the matrix product state representation of quasihole wave functions, and employ this new numerical tool to examine the braiding statistics and the screening property of several non-Abelian quantum Hall states, including the Moore-Read and the Read-Rezayi states, as well as the Gaffnian wave function. [Preview Abstract] |
Monday, March 3, 2014 5:18PM - 5:30PM |
D45.00015: Critical integer quantum Hall topology in the integrable Maryland model Sriram Ganeshan, Kostyantyn Kechedzhi One-dimensional tight binding models such as Aubry-Andre-Harper (AAH) model (with onsite cosine potential) and the integrable Maryland model (with onsite tangent potential) have been the subjects of extensive theoretical research in localization studies. AAH can be directly mapped onto the two-dimensional Hofstadter model that manifests the integer quantum Hall topology on a lattice. However, no such connection has been made for the Maryland model (MM). In this talk, we present a generalized model that contains AAH and MM as the limiting cases with the MM lying precisely at a topological quantum phase transition (TQPT) point. A remarkable feature of this critical point is that the 1D MM retains well-defined energy gaps whereas the equivalent 2D model becomes gapless, signifying the 2D nature of the TQPT. The criticality allows us to associate topological invariants with the Maryland model in a restricted mathematical sense at the special filling factors that are adiabatically connected to the spectral gaps in the 1D Aubry-Andre-Harper model. Our theory presented in this work establishes deep and unexpected mathematical connections between 2D topological models and a family of 1D incommensurate localization models. [Preview Abstract] |
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