Bulletin of the American Physical Society
2009 APS March Meeting
Volume 54, Number 1
Monday–Friday, March 16–20, 2009; Pittsburgh, Pennsylvania
Session X23: Quantum Hall Effect: Quantum Computation |
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Sponsoring Units: DCMP Chair: Steven Simon, Oxford University Room: 325 |
Thursday, March 19, 2009 2:30PM - 2:42PM |
X23.00001: Pfaffian statistics through adiabatic transport in the 1D coherent state representation Alexander Seidel Recent work has shown that certain quantum Hall states and their excitations can be efficiently labeled by simple one-dimensional patterns. One way to achieve this is to observe that the low energy sector of these states is adiabatically connected to simple charge-density-wave patterns that appear when the system is deformed, e.g., into a thin torus. In this talk it will be shown that the patterns emerging in this limit already determine the non-abelian statistics of the $\nu=1$ Moore-Read state. Aside from the knowledge of these patterns, the method essentially relies only on the principle of adiabatic continuity, and topological as well as locality arguments. [References: A. Seidel, Phys. Rev. Lett. 101, 196802 (2008), A. Seidel, D.-H. Lee, Phys. Rev. B 76, 155101 (2007).] [Preview Abstract] |
Thursday, March 19, 2009 2:42PM - 2:54PM |
X23.00002: Formation of novel topological liquids in non-Abelian quantum Hall states with a finite density of interacting anyons Simon Trebst, Charlotte Gils, Eddy Ardonne, Andreas W. W. Ludwig, Matthias Troyer, Zhenghan Wang We discuss the nucleation of novel topological liquids within a `parent' non-Abelian quantum Hall state that arises from a macroscopic occupation of the parent liquid with interacting anyons. The edge states between the `nucleated' and the `parent' liquids are neutral, and correspond precisely to the gapless modes of chains of interacting anyons. Exploring the collective states of anyonic spin chains is thus, at the same time, a tool to systematically study topological phases which can occur inside non-Abelian quantum Hall liquids due to population of such liquids with a finite density of interacting non-Abelian anyons. We discuss several examples including the interface between the Moore-Read and Read-Rezayi quantum Hall states. [Preview Abstract] |
Thursday, March 19, 2009 2:54PM - 3:06PM |
X23.00003: Thermopower Measurements in the $\nu = 5/2$ FQHE J.P. Eisenstein, W.E. Chickering, L.N. Pfeiffer, K.W. West The fractional quantized Hall state at $\nu = 5/2$ is currently under intense scrutiny owing to its possible application to topological quantum computation. This connection relies on the possibility that the ground state of the 2D electron gas at $\nu = 5/2$ is the non-abelian Moore-Read paired composite fermion state. A key attribute of this state is its large degeneracy when quasiparticles are present. While most schemes for detecting this degeneracy rely on interferometric braiding operations, it may also have thermodynamic implications. In particular, the entropy of the 2D system at $\nu = 5/2$ should be significantly enhanced relative to conventional abelian quantum Hall states. In this contribution we report measurements of the low temperature thermopower of the 2DES at $\nu = 5/2$. Although thermopower is a transport parameter, it is closely related to the entropy of the system under study and there have been recent suggestions [1] that it may bear signatures of non-abelian quasiparticle statistics. Our measurements will highlight the similarities and differences between the thermopower at $\nu = 5/2$ and other quantum Hall states. This work is supported by Microsoft Project Q and DOE grant DE-FG03-99ER45766. [1] Kun Yang, arXiv:0807.3341 [Preview Abstract] |
Thursday, March 19, 2009 3:06PM - 3:18PM |
X23.00004: Spontaneous Particle-Hole Symmetry Breaking in the 5/2 Fractional Quantum Hall Effect Kwon Park, Michael Peterson, Sankar Das Sarma The essence of the $\nu=5/2$ fractional quantum Hall effect is believed to be well captured by the Moore-Read Pfaffian (or anti-Pfaffian) description. However, an important mystery regarding the formation of the Pfaffian state is the role of the three-body interaction Hamiltonian $H_3$ that produces the Pfaffian as an exact ground state and the concomitant particle-hole symmetry breaking. We show that a two-body interaction Hamiltonian $H_2$ constructed via particle-hole symmetrization of $H_3$ produces a ground state nearly exactly approximating the Pfaffian and anti-Pfaffian states, respectively, in the spherical geometry. More importantly, the ground state energy of $H_2$ is shown to exhibit a ``Mexican-hat'' structure as a function of particle number in the vicinity of half filling for a given flux indicating spontaneous particle-hole symmetry breaking. [Preview Abstract] |
Thursday, March 19, 2009 3:18PM - 3:30PM |
X23.00005: Jack Polynomials, W-algebras and application to Fractional Quantum Hall Effect Steven H. Simon, B. Andrei Bernevig, Victor Gurarie We examine Jack symmetric functions and certain W-algebras as schemes for generating fractioanl quantum Hall wavefunctions. We add substantially to the evidence that the Jack functions correspond to certain W-algebras, by calculating the central charge and scaling dimensions of some of the fields in both cases and showing that they match. Except for the Read-Rezayi series all of these Jack symmetric functions turn out to be nonunitary theories. We discuss the (perhaps optimistic) possibility that these approaches may have relevance to various physical quantum Hall systems. Open questions in the field, as well as why this is of importance to those concerned with real experiments, will also be discussed. [Preview Abstract] |
Thursday, March 19, 2009 3:30PM - 3:42PM |
X23.00006: Resources needed for peforming Shor's algorithm with Ising Anyons Mara Baraban, Steven Simon, Nick Bonesteel While Ising anyons, the presumed quasiparticles of the $\nu=5/2$ fractional quantum Hall state, do obey non-Abelian statistics, their braid group is not sufficiently rich to support universal quantum computation (UQC). Recently, Bravyi [1] proposed a method for combining topological and non-topological operations that exploits the topological protection of the Ising anyons to allow for UQC even with very low accuracy non-topological operations. Starting from Bravyi's proposal, we calculate the resources required to perform Shor's algorithm. We find that when parallelization is included, the required number of qubits grows as the number of gates ($\sim N^3$, where $N$ is the length of the number to be factored using Shor's algorithm) and that the total time required is nearly independent of $N$. Numerical work has further allowed us to determine how far apart the anyons must be in a realistic sample in order to perform topological operations. We thus estimate how large a coherent sample would be required for Ising anyons to successfully execute modular exponentiation. We compare our results to the requirements for performing Shor's algorithm via fully topological quantum computation with Fibonacci anyons, the presumed quasiparticle excitation of the $\nu=12/5$ fractional quantum Hall state. [1] S. Bravyi, Phys. Rev. A \textbf{73}, 042313 (2006) [Preview Abstract] |
Thursday, March 19, 2009 3:42PM - 3:54PM |
X23.00007: Multichannel Kondo Models in non-Abelian Quantum Hall Droplets Gregory A. Fiete, Waheb Bishara, Chetan Nayak We study the coupling between a quantum dot and the edge of a non-Abelian fractional quantum Hall state which is spatially separated from it by an integer quantum Hall state. Near a resonance, the physics at energy scales below the level spacing of the edge states of the dot is governed by a $k$-channel Kondo model when the quantum Hall state is a Read-Rezayi state at filling fraction $\nu=2+k/(k+2)$ or its particle-hole conjugate at $\nu=2+2/(k+2)$. The $k$-channel Kondo model is channel isotropic even without fine tuning in the former state; in the latter, it is generically channel anisotropic. In the special case of $k=2$, our results provide a new venue, realized in a mesoscopic context, to distinguish between the Pfaffian and anti-Pfaffian states at filling fraction $\nu=5/2$. [Preview Abstract] |
Thursday, March 19, 2009 3:54PM - 4:06PM |
X23.00008: Topological Phase Transition of the 5/2 Fractional Quantum Hall Effect Hao Wang, D.N. Sheng, F.D.M. Haldane We study the effect of three-body interaction on the fractional quantum Hall effect at filling factor 5/2 using exact diagonization method with torus geometry in a finite-size system. The calculation of the wave function overlap suggests that a repulsive three-body interaction will induce a phase transition to a Pfaffian state. Its particle-hole conjugate (anti-Pfaffian) state is found only in a very narrow region of the attractive three-body interaction while a stronger attraction will push the system into a compressible state. The results from the low energy spectrum and the particle-hole parity evolution further indicate a continuous phase transition between the Pfaffian and anti-Pfaffian states, with the pure Coulomb system sitting at the critical point of the transition. [Preview Abstract] |
Thursday, March 19, 2009 4:06PM - 4:18PM |
X23.00009: Topological order in two-dimensional fermion systems with $p_x+ip_y$ pairing Noah Bray-Ali, Letian Ding, Stephan Haas We numerically evaluate the ``entanglement spectrum'' (singular value decomposition of the density matrix) of paired states of fermions in two dimensions that break parity and time-reversal symmetries, focusing on the spinless $p_x+ip_y$ case in which the gap function has orbital angular momentum $\ell=1$. In the weak-pairing phase, the low-lying entanglement spectrum has a gapless structure, which we compare to that of the Moore-Read state, a nonabelian quantum hall fluid. In the strong-pairing phase, we find a different structure, which we compare to the $\ell=0$, $s$-wave case. At the weak-strong transition, we compute the entanglement entropy from the spectrum, and find a logarithmic correction to the generic, ``area'' law behavior. [Preview Abstract] |
Thursday, March 19, 2009 4:18PM - 4:30PM |
X23.00010: The fractional quantum Hall effect: The cases of 5/2 and 12/5 Keshav Shrivastava We find that there is a state of zero energy because of a zero value in (1/2)g. When negative sign is used, L=0, S=1/2, g=(2J+1)/(2L+1)=[2(L-S)+1]/[2L+1]=0 so that [n+(1/2)][(1/2)g]=0. For positive sign, L+S, L=0, g=2 so that [n+(1/2)][(1/2)g]=5/2 for n=2. Hence 0 and 5/2 become particle-hole conjugates. In this definition, the sign of the spin for the particle is different from that for the hole as required by the helicity, p.s. For negative sign, L=2, (1/2)g=2/5 and (n-n')[(1/2)g]=12/5 with n-n'=6. For the positive sign, (1/2)g=3/5 for L=2 and for n-n"=4, we get 12/5. Thus 12/5 can arise for up spin as well as for down spin for different Landau levels[1]. On the basis of a product of [n+(1/2)][(1/2)g] we are able to understand all of the fractions given by Pan et al[2]. [1] K. N. Shrivastava, Phys. Lett. A 113,435(1986); A326,469(2004); Mod. Phys. Lett. 13,1087(1999); 14,1009(2000); AIP Conf. Proc. 909, 43-49(2007); 909.50-56(2007);1017, 422-428(2008);1017,326-330(2008); 1017, 47-56(2008), Proc. SPIE(USA)7155,71552F1-8[7155{\_}86](2008). [2] Wei Pan et al, Phys. Rev. B 77, 075307(2008). [Preview Abstract] |
Thursday, March 19, 2009 4:30PM - 4:42PM |
X23.00011: Probing the neutral edge modes in transport across a point contact via thermal effects in the Read-Rezayi non-abelian quantum Hall states Eytan Grosfeld, Sourin Das Non-abelian quantum Hall states are characterized by the existence of neutral gapless edge modes, whose structure is intricately related to the existence of bulk quasi-particle excitations obeying non-abelian statistics. Detecting the presence of these neutral modes is thus an important step towards establishing the non-abelian nature of these quantum Hall states. While it is hard to couple to the neutral modes using an electric field, they will directly couple to a temperature gradient and respond by contributing to the thermal current. By obtaining an expression for the thermal current through a quantum point contact, we demonstrate that a measurement of the thermal current will reveal the presence of the neutral modes. In addition, since thermal measurements are difficult to implement, we propose a setup which uses no external heaters and relies solely on noise measurements to detect thermal effects. The idea is to have two point contacts in series separated by a distance set by the thermal equilibration length of the charge mode. We show that by using the first point contact as a heating device, the excess charge noise measured at the second point contact carries a non-trivial signature of the presence of the neutral mode hence leading to its indirect detection. [Preview Abstract] |
Thursday, March 19, 2009 4:42PM - 4:54PM |
X23.00012: Charge-statistics separation and probing non-Abelian states for quantum Hall plateau at $\nu$=5/2 Feifei Li, Dima Feldman Several states were proposed for quantum Hall plateau at $\nu$=5/2. We suggest a transport experiment that can distinguish six of the candidate states. The proposal involves measurements of current and shot noise in a geometry with three quantum Hall edges connected by two quantum point contacts. Unlike interference experiments, this approach can distinguish Pfaffian and anti-Pfaffian states as well as different states with identical Pfaffian or anti-Pfaffian statistics. Moreover, the transport is not sensitive to the fluctuations of the number of quasiparticles trapped in the system.\\[0pt] [1] D. E. Feldman and Feifei Li, Phys. Rev. B {\bf78}, 161304(R) (2008). [Preview Abstract] |
Thursday, March 19, 2009 4:54PM - 5:06PM |
X23.00013: Quantum Hall hierarchy revisited Susanne Viefers, Juha Suorsa, Hans Hansson, Maria Hermanns Using techniques from conformal field theory, we construct explicit candidate wave functions for the entire Abelian quantum Hall hierarchy, i.e. quasihole- and quasielectron condensates, as well as mixtures of these. The formalism presented here, generalizes and unifies our previous techniques, which were only able to address quasielectron condensates. In the special cases of the positive and negative Jain sequences $\nu = n/(2np \pm 1)$, our method exactly reproduces Jain's composite fermion wave functions. In general our results are consistent with Wen's topological classification of FQH states. [Preview Abstract] |
Thursday, March 19, 2009 5:06PM - 5:18PM |
X23.00014: Non-Abelian quasielectrons. Hans Hansson, Maria Hermanns, Susanne Viefers Using methods from conformal field theory, we construct trial wave functions for quasielectron excitations in both Abelian and non-Abelian quantum Hall states. We briefly explain the underlying theory, and present analytical and numerical results for the case of two and four quasielectrons in the non-Abelian Moore-Read pfaffian state. Our methods generalize to other non-Abelian states, and can possibly also give a description of condensates of non-Abelian quasielectrons.. [Preview Abstract] |
Thursday, March 19, 2009 5:18PM - 5:30PM |
X23.00015: Topological Entanglement Entropy of Realistic Quantum Hall States Barry Friedman, Greg Levine The entanglement entropy for the incompressible states of a realistic quantum Hall system was studied by direct diagonalization for square clusters with periodic boundary conditions. The subdominant term to the area law, the topological entanglement entropy, was extracted for filling factors 1/3 and 5/2. The result for filling factor 1/3 is consistent with the topological entanglement entropy for the Laughlin wave function while the 5/2 filling factor exhibits a topological entanglement entropy consistent with the Moore-Read wave function. Preliminary results for the topological entanglement entropy for other incompressible states in the second Landau level will be discussed. [Preview Abstract] |
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