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arXiv:1106.5514 · Physical Review B

Quantum Hall effect in a one-dimensional dynamical system

Jan P. Dahlhaus, Jonathan M. Edge, J. Tworzydło, C. W. J. Beenakker

Published Sep 22, 2011• 87 citations
arXivHTMLPDF

Abstract

We construct a periodically time-dependent Hamiltonian with a phase transition in the quantum Hall universality class. One spatial dimension can be eliminated by introducing a second incommensurate driving frequency, so that we can study the quantum Hall effect in a one-dimensional (1D) system. This reduction to 1D is very efficient computationally and would make it possible to perform experiments on the 2D quantum Hall effect using cold atoms in a 1D optical lattice.

DOI
10.1103/physrevb.84.115133

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We construct a periodically time-dependent Hamiltonian with a phase transition in the quantum Hall universality class. One spatial dimension can be eliminated by introducing a second incommensurate driving frequency, so that we can study the quantum Hall effect in a one-dimensional (1D) system. This reduction to 1D is very efficient computationally and would make it possible to perform experiments on the 2D quantum Hall effect using cold atoms in a 1D optical lattice.
Quasiperiodically driven fermionic systems can support topological phases not realized in equilibrium. The fermions are localized in the bulk, but support quantized energy currents at the edge. These phases were discovered through an abstract classification, and few microscopic models exist. We develop a coupled layer construction for tight-binding models of these phases in $d\ensuremath{\in}{1,2}$ spatial dimensions, with any number of incommensurate drive frequencies DDD. The models exhibit quantized responses associated with synthetic two- and four-dimensional quantum Hall effects in the steady state. A numerical study of the phase diagram for (d+D)=(1+2)(d+D)=(1+2)(d+D)=(1+2) shows (i) robust topological and trivial phases separated by a sharp phase transition, (ii) charge diffusion and a half-integer energy current between the drives at the transition, and (iii) a long-lived topological energy current which remains present when weak interactions are introduced.
In this thesis, we discuss quantum phase transitions in low-dimensional optical lattices, namely one- and two-dimensional lattices. The dimensional confinement is realized in experiments by suppressing the hopping in the extra dimensions through a deep potential barrier that prevents the atoms to tunnel across dimensions. The dimensionality considerably affects the physical behavior of the system. For instance, in one dimension, the celebrated Fermi-liquid theory does not describe the many-body behavior of a fermionic system. There are no single particle excitations, but instead collective behavior, and spin and charge degrees of freedom are decoupled . In the first part of this thesis, we consider one-dimensional fermionic gases under the effect of time-dependent driving forces. First, we describe a proposal to realize non-standard Hubbard models that display correlated-hopping terms by modulating interactions through a time-dependent magnetic field. The extra terms describe hopping processes that depend on the particle-density difference in neighboring sites. The effective model is obtained in arbitrary spatial dimensions, but only in one-dimension it is known to be exactly solvable for specific values of the parameters. Moreover, an SU(2) symmetry in the charge sector allows to construct non-trivial superconducting eigenstates (eta-states), discovered by Yang as excited states of the Hubbard model. Then, we study a one-dimensional bipartite optical lattice with fermionic atoms at half-filling. The lattice potential barriers have heights alternating in magnitude. As a consequence, the hopping amplitudes also alternate in magnitude and the elementary unit cell has two sites. This system is known to display Peierls dimerization at half-filling: the fermions form dimers in neighboring sites that inhibit transport and open a gap in the spectrum. After applying shaking on this system we discover that one can dynamically induce a phase transition to an unconventional metallic state with four Fermi-points. The second part of this thesis describes bosonic systems in two-dimensional optical lattices. First, we consider a square optical lattice with alternating deep-shallow wells arranged in a checkerboard pattern. This lattice is realized in the experimental laboratory of Prof. Hemmerich in Hamburg. In the experiment, the visibility is extracted from the time-of-flight imaging to quantify the phase coherence of the bosonic gas. We develop a mean-field theory and determine the phase diagram of this model. We interpret the experimental data by assuming that at large detuning, only the deeper wells are populated and form an imbalanced Mott insulator with empty shallow wells and integer density on the deeper wells. We support this hypothesis by large-scale Gutzwiller calculations. Then, we investigate a system of bosons in the Lieb lattice, which is the lattice geometry of the Cu-O plane in cuprates . We consider configurations of the optical potential where s and p orbitals sitting on neighboring wells become resonant and hybridize. The interest in this problem comes from the possibility to realize time-reversal broken phases, usually known as Varma phases for electronic systems, which have been proposed to explain the pseudo-gap regime of cuprate superconductors. We show the conditions under which the symmetry breaking in a four-band model can lead to states with loop currents. Moreover, the lowest branch of the excitation spectrum has a non-vanishing Chern number and therefore supports topological excitations.
The quantum kicked rotator is a periodically driven dynamical system with a metal--insulator transition. We extend the model so that it includes phase transitions between a metal and a topological insulator, in the universality class of the quantum spin Hall effect. We calculate the Z2{\mathbb{Z}}_{2}Z2​ topological invariant using a scattering formulation that remains valid in the presence of disorder. The scaling laws at the phase transition can be studied efficiently by replacing one of the two spatial dimensions with a second incommensurate driving frequency. We find that the critical exponent does not depend on the topological invariant, in agreement with earlier independent results from the network model of the quantum spin Hall effect.
We show a universality in the quantum Hall effect in that the flux-averaged Hall conductivity in any two- or three-dimensional system with or without disorder is given by a topological invariant quantized in units of e2h\frac{{e}^{2}}{h}he2​ at T=0T=0T=0 for every energy level in a finite system. Relevance to the observable Hall conductivity over the whole energy spectrum is presented together with numerical results for lattice systems.
This dissertation is about the physics of dilute gaseous Bose-Einstein condensates (BECs) confined to lower dimensions by optical lattices. The central theme of the effects of reduced dimensionality is explored within various one-dimensional (1D) and two-dimensional (2D) systems. We create a 2D BEC by adiabatically increasing the confinement of a trapping potential in one direction to the point where motion in that direction is frozen out. Doing this in two directions, we create a 1D BEC. Two experiments examine the ground state properties of a 1D and 2D system. In the 1D system (Chap. 9), a reduction in three-body recombination signals an increase in correlation resulting in a partial "fermionization" of the Bose gas. In the 2D system (Chap. 8), we measure temperature-dependent condensate phase fluctuations in the vicinity of the Berezinskii-Kosterlitz-Thouless transition. Other experiments investigate dynamic properties of reduced dimension systems. Strongly inhibited transport of a 1D gas in a lattice is observed in one experiment (Chap. 9). Another 2D experiment measures suppressed collisional decay rates due to the reduced dimensionality (Chap. 9). A final experiment (Chap. 7) examines quantum/classical correspondence in the effectively 1D dynamics of a 3D BEC. The dynamics is effectively 1D in the sense that the experiment is over before motion in the radial directions (which are not frozen out) can occur. This dissertation also describes the design and implementation of a novel 1D "accordion lattice" (Chaps. 5-6) which greatly facilitated the Berezinskii-Kosterlitz-Thouless experiment, the quantum/classical correspondence experiment, and a "superlattice" experiment conducted to assist in the calibration of the accordion lattice.
We propose a method by which the quantization of the Hall conductance can be directly measured in the transport of a one-dimensional atomic gas. Our approach builds on two main ingredients: (1) a constriction optical potential, which generates a mesoscopic channel connected to two reservoirs, and (2) a time-periodic modulation of the channel specifically designed to generate motion along an additional synthetic dimension. This fictitious dimension is spanned by the harmonic oscillator modes associated with the tightly confined channel, and hence, the corresponding “lattice sites” are intimately related to the energy of the system. We analyze the quantum-transport properties of this hybrid two-dimensional system, highlighting the appealing features offered by the synthetic dimension. In particular, we demonstrate how the energetic nature of the synthetic dimension combined with the quasienergy spectrum of the periodically driven channel allows for the direct and unambiguous observation of the quantized Hall effect in a two-reservoir geometry. Our work illustrates how topological properties of matter can be accessed in a minimal one-dimensional setup, with direct and practical experimental consequences.