We propose a new approach to probing ergodicity and its breakdown in quantum many-body systems based on their response to a local perturbation. We study the distribution of matrix elements of a local operator between the systems eigenstates, finding
a qualitatively different behaviour in the many-body localized (MBL) and ergodic phases. To characterize how strongly a local perturbation modifies the eigenstates, we introduce the parameter ${cal G}(L)=langle ln (V_{nm}/delta) rangle$, which represents a disorder-averaged ratio of a typical matrix element of a local operator $V$ to the energy level spacing, $delta$; this parameter is reminiscent of the Thouless conductance in the single-particle localization. We show that the parameter ${cal G}(L)$ decreases with system size $L$ in the MBL phase, and grows in the ergodic phase. We surmise that the delocalization transition occurs when ${cal G}(L)$ is independent of system size, ${cal G}(L)={cal G}_csim 1$. We illustrate our approach by studying the many-body localization transition and resolving the many-body mobility edge in a disordered 1D XXZ spin-1/2 chain using exact diagonalization and time-evolving block decimation methods. Our criterion for the MBL transition gives insights into microscopic details of transition. Its direct physical consequences, in particular logarithmically slow transport at the transition, and extensive entanglement entropy of the eigenstates, are consistent with recent renormalization group predictions.
We study the nature of the u=5/2 quantum Hall state in wide quantum wells under the mixing of electronic subbands and Landau levels. We introduce a general method to analyze the Moore-Read Pfaffian state and its particle-hole conjugate, the anti-Pfa
ffian, under periodic boundary conditions in a quartered Brillouin zone scheme containing both even and odd numbers of electrons. We examine the rotational quantum numbers on the torus, and show spontaneous breaking of the particle-hole symmetry can be observed in finite-size systems. In the presence of electronic-subband and Landau-level mixing the particle-hole symmetry is broken in such a way that the anti-Pfaffian is unambiguously favored, and becomes more robust in the vicinity of a transition to the compressible phase, in agreement with recent experiments.
It was recently pointed out that topological liquid phases arising in the fractional quantum Hall effect (FQHE) are not required to be rotationally invariant, as most variational wavefunctions proposed to date have been. Instead, they possess a geome
tric degree of freedom corresponding to a shear deformation that acts like an intrinsic metric. We apply this idea to a system with an anisotropic band mass, as is intrinsically the case in many-valley semiconductors such as AlAs and Si, or in isotropic systems like GaAs in the presence of a tilted magnetic field, which breaks the rotational invariance. We perform exact diagonalization calculations with periodic boundary conditions (torus geometry) for various filling fractions in the lowest, first and second Landau levels. In the lowest Landau level, we demonstrate that FQHE states generally survive the breakdown of rotational invariance by moderate values of the band mass anisotropy. At 1/3 filling, we generate a variational family of Laughlin wavefunctions parametrized by the metric degree of freedom. We show that the intrinsic metric of the Laughlin state adjusts as the band mass anisotropy or the dielectric tensor are varied, while the phase remains robust. In the n=1 Landau level, mass anisotropy drives transitions between incompressible liquids and compressible states with charge density wave ordering. In n>=2 Landau levels, mass anisotropy selects and enhances stripe ordering with compatible wave vectors at partial 1/3 and 1/2 fillings.
We report a systematic study of the fractional quantum Hall effect (FQHE) using the density-matrix renormalization group (DMRG) method on two different geometries: the sphere and the cylinder. We provide convergence benchmarks based on model Hamilton
ians known to possess exact zero-energy ground states, as well as an analysis of the number of sweeps and basis elements that need to be kept in order to achieve the desired accuracy.The ground state energies of the Coulomb Hamiltonian at $ u=1/3$ and $ u=5/2$ filling are extracted and compared with the results obtained by previous DMRG implementations in the literature. A remarkably rapid convergence in the cylinder geometry is noted and suggests that this boundary condition is particularly suited for the application of the DMRG method to the FQHE.
We construct model wavefunctions for the collective modes of fractional quantum Hall systems. The wavefunctions are expressed in terms of symmetric polynomials characterized by a root partition and a squeezed basis, and show excellent agreement with
exact diagonalization results for finite systems. In the long wavelength limit, the model wavefunctions reduce to those predicted by the single-mode approximation, and remain accurate at energies above the continuum of roton pairs.
We develop a nonperturbative approach to the quantum Hall bilayer (QHB) at u=1 using trial wave functions. We predict phases of the QHB for arbitrary distance d and, our approach, in a dual picture, naturally introduces a new kind of quasiparticles
- neutral fermions. Neutral fermion is a composite of two merons of the same vorticity and opposite charge. For small d (i.e. in the superfluid phase), neutral fermions appear as dipoles. At larger d dipoles dissociate into the phase of the two decoupled Fermi-liquid-like states. This scenario is relevant for the experimental situation where impurities lock charged merons. In a translation invariant (clean) system, continuous creation and annihilation of meron-antimeron pairs evolves the QHB toward a paired phase. The quantum fluctuations fix the form of the pairing function to g(z)=1/z^*. A part of the description of the paired phase is the 2D superconductor i.e. BF Chern-Simons theory. The paired phase is not very distinct from the superfluid phase.
