Session R23: Fractional Quantum Hall Theory II

Interaction induced Landau level mixing in the fractional quantum Hall regime

We study Landau Level mixing in parabolic bands perturbatively to second order in the ratio of interaction to cyclotron energy, for the lowest ($N=0$) and first excited ($N=1$) Landau levels. The mixing is accounted for by constructing an effective Hamiltonian which includes two body and three body interactions. Our study builds upon two previous treatments~[1,2], using as a stepping stone the observation that the effective Hamiltonian is fully determined by the 2 and 3 body problems. For the $N=0$ problem we provide a compact and transparent derivation of the effective Hamiltonian using first quantization which captures a class of virtual processes omitted in earlier calculations of Landau-level mixing corrected Haldane pseudo-potentials. We will comment on potential application of our results for numerical studies.\\[4pt] [1] G. Murthy and R. Shankar, {\em PRB} {\bf 65}, 245309 (2002)\\[0pt] [2] W. Bishara and C. Nayak, {\em PRB} {\bf 80}, 121302 (2009).   

Can Ohmic contact in quantum Hall systems be considered a voltage probe?

Ohmic contacts are crucial elements of mesoscopic systems, which have no clear theoretical description yet. We propose a model of the Ohmic contact with a finite capacitance $C$ attached to a quantum Hall edge channel. It is shown that in contrast to na\"ive expectations the fluctuations of currents originating at such contact have non-equilibrium statistics. Consequently, the Ohmic contact can be considered a ``voltage probe'' only for certain values of the system parameters. In particular, the distribution function of outgoing electrons is close to the equilibrium one if the contact's temperature is much larger than $e^2/2\pi C$.   

Hamiltonian Formulation of the Hydrodynamics with Quantum Anomalies

The hydrodynamic limit of a charged massless chiral spinor under the presence of gauge field in 3 dimensions is consider in [1]. For this system, global gauge symmetry is anomalous. In order to satisfy the second law of thermodynamics, charge current and entropy flow have to be corrected. We present a Hamiltonian formulation of the relativistic hydrodynamics which accounts for these new terms; extending the analysis done in [2]. In this formulation, the limit when particles become massless can be performed in a straightforward way and it has the advantage of being the natural framework to quantization. We show that the Poisson's structure of the hydrodynamics of ideal relativistic fluid allows for a one-parameter deformation. The value of the parameter is fixed by quantum anomalies present in the underlying theory. This formulation allows for generalizations to hydrodynamics of systems with additional conserved quantities, and is found to be a higher dimensional analogous to quantum hall effect. [1] D.T. Son and P. Surowka, Phys.Rev.Lett. 103, 191601 (2009). [2] D.D. Holm and B.A. Kupershmidt, Phys Lett. 101A, 23 (1984)   

Quantum Hall viscosity of Hierarchy States

We describe a strategy for calculating the odd, non-dissipative viscosity for hierarchical QH states. Using previously developed techniques for expressing the wave functions on the plane in terms of conformal blocks, we can in simple cases construct the corresponding torus wave functions and show that they have good modular properties. Under certain assumptions, the QH viscosity can be directly extracted from these wave functions, and in the simplest case of the $\nu=2/5$ Jain states, we have verified the result numerically. Our results are consistent with the general formula, given by Read, relating the QH viscosity to the average orbital spin of the electrons.   

Extracting net current from an upstream neutral mode in the fractional quantum Hall regime

Upstream neutral modes, counter propagating to charge modes and carrying energy without net charge, had been predicted to exist in some of the fractional quantum Hall states and were recently observed via noise measurements. Understanding such modes will assist in identifying the wavefunction of these states, as well as shedding light on the role of Coulomb interactions within edge modes. In this work, performed mainly in the 2/3~state, we placed a quantum dot a few micrometers upstream of an ohmic contact, which served as a ``neutral modes source.'' We showed the neutral modes heat the input of the dot, causing a net thermo-electric current to flow through it. Heating of the electrons led to a decay of the neutral mode, manifested in the vanishing of the thermo-electric current at T \textgreater\ 100mK. This setup provides a straightforward method to investigate upstream neutral modes without turning to the more cumbersome noise measurements.   

Phase diagram of the composite fermion Wigner crystals

The energies of the Wigner crystal (WC) phase and the fractional quantum Hall (FQH) liquid have been compared in the past at some special filling factors. We deduce in this work the phase diagram of the WC phase as a function of the general filling factor by considering: (i) the WC of electrons; (ii) WCs of composite fermions (CFs) carrying $2p$ vortices; and (iii) FQH states supporting WC of CF quasiparticles or CF quasiholes. In particular, we find that the re-entrant insulating phase between 1/5 and 2/9 is a WC of composite fermions carrying two vortices. To distinguish the CF Wigner crystal from the electron WC, we compute a number of properties, including shear modulus, magnetophonon and magnetoplasmon dispersions, and melting temperatures. The width dependence of the phase diagram is also studied. A technical innovation that makes these comparisons feasible is to model the WC as the thermodynamic limit of the Thomson crystal on the surface of a sphere, which minimizes the Coulomb energy of classical charged particles.   

Hierarchy of fractional Chern insulators and competing compressible states

We study the phase diagram of interacting electrons in a dispersionless Chern band as a function of their filling. We find hierarchy multiplets of incompressible states at fillings $\nu$ = 1/3, 2/5, 3/7, 4/9, 5/9, 4/7, 3/5 as well as $\nu$ = 1/5, 2/7. These are accounted for by an analogy to Haldane pseudopotentials extracted from an analysis of the two-particle problem. Important distinctions to standard fractional quantum Hall physics are striking: absent particle-hole symmetry in a single band, an interaction-induced single-hole dispersion appears, which perturbs and eventually destabilizes incompressible states as $\nu$ increases. For this reason the nature of the state at $\nu$ = 2/3 is hard to pin down, while $\nu$ = 5/7, 4/5 do not seem to be incompressible in our system.   

Fractional Chern Insulators beyond Laughlin states

We report the first numerical observation of composite fermion (CF) states in fractional Chern insulators (FCI) using exact diagonalization. The ruby lattice Chern insulator model for both fermions and bosons exhibits a clear signature of CF states at filling factors $2/5$ and $3/7$ ($2/3$ and $3/4$ for bosons). The topological properties of these states are studied through several approaches. Quasihole and quasielectron excitations in FCI display similar features as their fractional quantum hall (FQH) counterparts. The entanglement spectrum of FCI groundstates shows an identical fingerprint to its FQH partner. We show that the correspondence between FCI and FQH obeys the emergent symmetry already established, proving the validity of this approach beyond the clustered states. We investigate other Chern insulator models and find similar signatures of CF states. However, some of these systems exhibit strong finite size effects.   

Crystal-symmetry preserving Wannier states for fractional Chern insulators

Recently, many numerical evidences of fractional quantum anomalous states (FQAH states), i.e. the fractional quantum Hall states (FQH states) on lattice, when a band with non-zero Chern number (We refer to it as a Chern band) is partially filled. Some trial wavefunction of FQAH states can be obtained by mapping the FQH wavefunctions defined in the continuum onto the lattice through the scheme proposed in Ref. [1] in which the single particle Landau orbits in the Landau levels are mapped to the one dimensional Wannier wavefunctions (which is a plane wave on the other direction) of the Chern bands with Chern number C=1. However, this mapping will generically break the lattice rotational symmetry. In this talk, we shall present a modified scheme to accommodate the mapping with the lattice rotational symmetry. The wavefunctions constructed through this modified scheme should serve as better trial wavefunctions to compare with the numerics. The focus of the talk shall be mainly on the C4 rotational symmetry of square lattices. Related issues on C6 symmetry of honeycomb lattice and higher Chern number bands will be discussed. [1] X.-L. Qi, Phys. Rev. Lett. 107, 126803 (2011)   

Establishing non-Abelian topological order in Gutzwiller projected Chern insulators via Entanglement Entropy and Modular S-matrix

We use entanglement entropy signatures to establish non-Abelian topological order in a new class of ground states, the projected Chern-insulator wave functions. The simplest instance is obtained by Gutzwiller projecting a filled band with Chern number C=2 which may also be viewed as the square of the band insulator Slater determinant. We demonstrate that this wave function is captured by the $SU(2)_2$ Chern Simons theory coupled to fermions. In addition to the expected torus degeneracy and topological entanglement entropy, we also show that the modular S-matrix, extracted from entanglement entropy calculations, provides direct access to the peculiar non-Abelian braiding statistics of Majorana fermions in this state. We also provide microscopic evidence for the generalization (expected from the field theory), that the N$^{\rm th}$ power of a Chern number $C$ Slater determinant realizes the topological order of the $SU(N)_C$ Chern Simons theory coupled to fermions, by studying the $SU(2)_3$ and the $SU(3)_2$ wave functions. An advantage of projected Chern insulator wave functions over lowest Landau level wave functions for the same phases is the relative ease with which physical properties, such as entanglement entropy, can be numerically calculated using Monte Carlo techniques.   

Galilean invariance and linear response theory for Fractional Quantum Hall Effect

We study a general effective field theory of Galilean invariant two-dimensional charged fluid in external electro-magnetic and gravitational fields. We find that combination of the generalized Galilean [1] and gauge invariance implies nontrivial Ward identities between gravitational and electro-magnetic linear responses in the system. This identity appears to hold in all orders of gradient expansion and it generalizes the relation between Hall viscosity and Hall conductivity recently found by Hoyos and Son. We also check the relation in the case of free electrons with integer filling of Landau levels where corresponding linear responses can be calculated directly. \\[4pt] [1] Carlos Hoyos, Dam Thanh Son ``Hall Viscosity and Electromagnetic Response''   

Critical behavior of the transport coefficients at the plateau-insulator transition in IQHE

Using the non-commutative Kubo formula for disordered lattice systems, we mapped the conductivity tensor $\sigma(E_F,T)$ as function of Fermi level $E_F$ and temperature T for the disordered Hofstadter model. Convergence and accuracy tests indicate that the simulations can be used to investigate the critical behavior near the plateau-insulator transition. Our analysis provides the first quantitative theoretical confirmation of the well established experimental facts about the critical behavior: 1) The semicircle law for the components of the conductivity tensor; 2) The existence of the quantized Hall insulator state characterized by zero direct and Hall conductivities, but with Hall resistivity quantized at $h/e^2$; 3) Single scaling behavior with exponents that are consistent with previous studies.   

Energy Scales of the Reentrant Integer Quantum Hall States in High Landau Levels

The reentrant integer quantum Hall states (RIQHS) have been identified with the electronic bubble phases. These bubble phases are exotic electronic solids similar to the Wigner crystal, but have more than one electron per lattice site. Recently we reported the presence of a peak in the temperature dependent magnetoresistence of the RIQHSs and we have associated this peak with the onset of the RIQHSs. We found that, contrary to the predictions of the bubble theory, the onset temperatures of the RIQHSs in the third Landau level are much higher than those in the second Landau level. We have extended such measurements of the onset temperatures to several high Landau levels. In this talk we will discuss the orbital dependence of the onset temperatures of RIQHSs and we will compare these quantitative results to the predictions of the bubble theory. This work was supported by the DOE BES contract no. DE-SC0006671.   

Boundary degeneracy of topological order states

It is known that topological ordered states have degenerate ground states on compact space manifold. Its ground state degeneracy on higher genus Riemann surface is encoded by the fusion rules of the fractionalized quasipartcles and the genus number. Here we study topologically ordered states on space manifold with boundary. We find that Bulk-Edge correspondence is not a complete story - edge theory information may not be fully-determined by the bulk theory. Ground state degeneracy of boundary states depends on boundary gapping conditions. Take Abelian topological order as an example, K matrix Chern-Simons theory, the boundary ground state degeneracy counts the number of group elements in a discrete finite quotient group from anyon transport and fusion algebra. We compare this result to Toric code model, Levin-Wen string-net model and flux insertion argument. By glueing the edges of a non-compact manifold to make it compact, we go back to demonstrate bulk ground state degeneracy from edge theory viewpoint, in terms of the Betti number and homology group, such as 2+1 D Chern-Simons or higher dimensional B-F theory.   

Monte Carlo Study of a $U(1)\times U(1)$ Loop Model with Modular Invariance

We study a $U(1)\times U(1)$ system in (2+1)-dimensions with long-range interactions and mutual statistics. The model has the same form after the application of operations from the modular group, a property which we call modular invariance. Using the modular invariance of the model, we propose a possible phase diagram. We obtain a sign-free reformulation of the model and study it in Monte Carlo. This study confirms our proposed phase diagram. We use the modular invariance to analytically determine the current-current correlation functions and conductivities in all the phases in the diagram, as well as at special ``fixed'' points which are unchanged by an operation from the modular group. We numerically determine the order of the phase transitions, and find segments of second-order transitions. For the statistical interaction parameter $\theta=\pi$, these second-order transitions are evidence of a critical loop phase obtained when both loops are trying to condense simultaneously. We also measure the critical exponents of the second-order transitions.