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arXiv:1710.08931 · New Journal of Physics

Quasi-one-dimensional Hall physics in the Harper–Hofstadter–Mott model

Filip Kozarski, Dario Hügel, Lode Pollet

Published Feb 19, 2018• 2 citations
arXivHTMLPDF

Abstract

We study the ground-state phase diagram of the strongly interacting Harper–Hofstadter–Mott model at quarter flux on a quasi-one-dimensional lattice consisting of a single magnetic flux quantum in y -direction. In addition to superfluid phases with various density patterns, the ground-state phase diagram features quasi-one-dimensional analogs of fractional quantum Hall phases at fillings ν = 1/2 and 3/2, where the latter is only found thanks to the hopping anisotropy and the quasi-one-dimensional geometry. At integer fillings—where in the full two-dimensional system the ground-state is expected to be gapless—we observe gapped non-degenerate ground-states: at ν = 1 it shows an odd 'fermionic' Hall conductance, while the Hall response at ν = 2 consists of the transverse transport of a single particle–hole pair, resulting in a net zero Hall conductance. The results are obtained by exact diagonalization and in the reciprocal mean-field approximation.

DOI
10.1088/1367-2630/aab081

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We study the ground-state phase diagram of the strongly interacting Harper–Hofstadter–Mott model at quarter flux on a quasi-one-dimensional lattice consisting of a single magnetic flux quantum in y -direction. In addition to superfluid phases with various density patterns, the ground-state phase diagram features quasi-one-dimensional analogs of fractional quantum Hall phases at fillings ν = 1/2 and 3/2, where the latter is only found thanks to the hopping anisotropy and the quasi-one-dimensional geometry. At integer fillings—where in the full two-dimensional system the ground-state is expected to be gapless—we observe gapped non-degenerate ground-states: at ν = 1 it shows an odd 'fermionic' Hall conductance, while the Hall response at ν = 2 consists of the transverse transport of a single particle–hole pair, resulting in a net zero Hall conductance. The results are obtained by exact diagonalization and in the reciprocal mean-field approximation.
We derive the reciprocal cluster mean-field method to study the strongly interacting bosonic Harper-Hofstadter-Mott model. The system exhibits a rich phase diagram featuring band insulating, striped superfluid, and supersolid phases. Furthermore, for finite hopping anisotropy, we observe gapless uncondensed liquid phases at integer fillings, which are analyzed by exact diagonalization. The liquid phases at fillings $\ensuremath{\nu}=1,3$ exhibit the same band fillings as the fermionic integer quantum Hall effect, while the phase at $\ensuremath{\nu}=2$ is CT\mathcal{C}\mathcal{T}CT-symmetric with zero charge response. We discuss how these phases become gapped on a quasi-one-dimensional cylinder, leading to a quantized Hall response, which we characterize by introducing a suitable measure for nontrivial many-body topological properties. Incompressible metastable states at fractional filling are also observed, indicating competing fractional quantum Hall phases. The combination of reciprocal cluster mean-field and exact diagonalization yields a promising method to analyze the properties of bosonic lattice systems with nontrivial unit cells in the thermodynamic limit.
Since the discovery of the quantum Hall effect, the lattice geometry's influence on charged particles in magnetic fields has been the subject of extensive research. Prototypical models such as the non-interacting Harper-Hofstadter model (HHm) exhibit fractionalization of the Bloch bands with non-trivial topology, manifesting in quantum (spin) Hall phases. Ultracold atomic gases with artificial magnetic fields enabled the experimental study of the non-interacting model, while the effect of strong interactions on the band properties remains an open problem. In this work we study the strongly-interacting bosonic Harper-Hofstadter-Mott model (HHMm) using a reciprocal cluster mean-field (RCMF) method. We obtain a rich groundstate phase diagram featuring striped superfluid, supersolid, and symmetry protected topological (SPT) phases. At filling ν=2\nu=2ν=2 (in contrast to fillings ν=1,3\nu=1, 3ν=1,3) the SPT phase has no fermionic counterpart. Incompressible metastable states at fractional filling are also observed. The SPT phases in the HHMm are promising candidates for realizing strongly-correlated topological phases using cold atoms.
In this research, we study the bosonic fractional quantum Hall (FQH) states in a system of ultracold bosons in a two-dimensional optical lattice in the presence of a synthetic magnetic field, described by the bosonic Harper–Hofstadter Hamiltonian. We use the cluster Gutzwiller mean-field and exact diagonalization techniques in our work. We obtain incompressible states as ground states at various filling factors similar to those of the FQH states. We focus in particular on the ν = 1/2 FQH state, and it is characterized by the two-point correlation function and the many-body Chern number. We further investigate the effect of dipolar interaction on the ν = 1/2 FQH state. We find that the dipolar interaction stabilizes the FQH state against the competing superfluid state.
We prove that the colored Hofstadter butterfly has a many-body interpretation for a broad class of weakly interacting lattice fermion systems. Starting from a spectral gap of a Hofstadter-like one-particle Hamiltonian at arbitrary magnetic flux bbb, we construct an open region in the three-dimensional parameter space (b,μ,λ)(b,μ,λ)(b,μ,λ) of magnetic field, chemical potential, and interaction strength on which the infinite-volume interacting system has locally unique gapped ground states. The construction combines quasi-adiabatic continuation in the interaction strength with denominator-independent magnetic perturbation estimates, and therefore covers both commensurate and incommensurate fluxes, where no finite magnetic unit cell exists. On connected uniformly gapped regions meeting the non-interacting plane λ=0λ=0λ=0, we prove a many-body gap-labeling theorem: the Hall conductivity appearing in the macroscopic Ohm's law is constant and quantized, satisfying 2πσH∈Z2πσ^{\mathrm{H}}\in\mathbb{Z}2πσH∈Z. Thus the integer colors of the non-interacting Hofstadter butterfly persist as Hall-conductivity labels of interacting quantum Hall phases.
Quantum Hall systems have a one-body energy spectrum consisting of dispersion-less Landau levels. Electron-electron interactions thus dominate in partially filled Landau levels, which exhibit a myriad of strongly correlated phases such as quantum hall ferromagnets and fractional quantum Hall phases. We study two examples of these phenomena. In the first project, we explore the ground state of a system with an interface between two semi-infinite regions with fillings ν= 4 and ν= 3 respectively. The width of the interface can be controlled by varying the background potential, which provides an additional tuning parameter. For a certain range of interaction strengths, the ν= 4 bulk is unpolarized whereas the ν= 3 bulk is fully polarized. In the parameter space spanned by the interaction strength and width of the interface, we find two phases at the interface. Phase A has spin as a good quantum number, and the long-wavelength spin edge excitations are gapped. In phase B, spin rotation symmetry is spontaneously broken at the mean-field level. Using symmetry arguments we find the effective theory near the interface of phase B. This effective theory is known to have gapless long-wavelength spin excitations. In the second project, we study the ground state of a tunnel-decoupled double-layer graphene system when both layers are undoped. We find a simple Hamiltonian in the continuum limit from symmetries of the system. Using the Hartree-Fock approximation we find a state with inter-layer coherence with broken layer U(1) symmetry. This phase becomes magnetized in presence of a non-zero Zeeman field. A first-order phase transition can be driven from the ferromagnetic phase to the magnetized inter-layer coherent phase by increasing the Zeeman field. We predict the number of gapless modes in the bulk.
We investigate certain generalized flux phases arising in a mean-field approach to the t-J model. First we establish that the energy of noninteracting electrons moving in a uniform magnetic field has an absolute minimum as a function of the flux at exactly one flux quantum per particle. Using this result we show that if the hard-core nature of the hole bosons is taken into account, then the generalized flux states in the mean-field approximation constitute a solution where both the spinons and the holons experience an average flux of one flux quantum per particle. We elaborate the relation to the integer and fractional quantum Hall effect. Based on this analogy one expects that only states with certain fractional fillings (ν=1/odd integer) are stable against the fluctuations of the (gauge) field. We speculate that at arbitrary fillings the system minimizes its energy by phase separating into these stable states. Finally we work out the ground state energy for nonuniform (staggered) flux phases and establish that the often claimed equivalence of frustration and doping does not hold for these states.