Realization of fractional Chern insulators in the thin-torus limit with ultracold bosons

Grusdt, Fabian; Höning, Michael

doi:10.1103/physreva.90.053623

Cited by 62 publications

(73 citation statements)

References 77 publications

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“…For example, the Hofstadter Hamiltonian on a two-leg ladder can give rise to a topological phase provided an additional ad-hoc off-diagonal term is added [19], and similar topological phases are also predicted in the same ladder geometry when the arXiv:1708.02929v1 [cond-mat.quant-gas] 9 Aug 2017 magnetic flux per plaquette is spatial oscillating [20,21]. While these models go in the direction that one can have topological phases protected by the inversion symmetry by introducing a spatial modulation of the physical parameters [22][23][24][25], the phases discussed here represent, to the best of our knowledge, the first example of noninteracting topological states stabilized by the Hofstadter Hamiltonian in a ladder geometry with synthetic length L y > 2.…”

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confidence: 83%

Topological phases in frustrated synthetic ladders with an odd number of legs

et al. 2018

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The realization of the Hofstadter model in a strongly anisotropic ladder geometry has now become possible in one-dimensional optical lattices with a synthetic dimension. In this work, we show how the Hofstadter Hamiltonian in such ladder configurations hosts a topological phase of matter which is radically different from its two-dimensional counterpart. This topological phase stems directly from the hybrid nature of the ladder geometry, and is protected by a properly defined inversion symmetry. We start our analysis considering the paradigmatic case of a three-leg ladder which supports a topological phase exhibiting the typical features of topological states in one dimension: robust fermionic edge modes, a degenerate entanglement spectrum and a non-zero Zak phase; then, we generalize our findings -addressable in the state-of-the-art cold atom experiments -to ladders with an higher number of legs. The Hofstadter problem [1] describing a particle hopping on a two-dimensional lattice pierced by a magnetic field, is a paradigm of quantum mechanics. Formulated more then forty years ago, it embeds a multitude of seminal notions in modern condensed matter physics [2]: topological bands, edge excitations, fractal properties of the spectrum, just to mention some of them. Despite its apparent simplicity and the enormous body of investigation both theoretical and experimental [3][4][5][6][7], the Hofstadter problem still hides some surprises, as we are going to discuss in the following.The motivation of our work stems from the recent realization [8-10] of the Hofstadter Hamiltonian in optical lattices with a synthetic dimension. The possibility of engineering an additional (synthetic) few sites long dimension [11,12] with non-trivial boundary conditions by using some internal degrees of freedom of the atoms has encouraged the study of the Hofstadter Hamiltonian in a strongly anisotropic geometry [13].(a) Is this a new territory for the Hofstadter problem or are we bound to detect a smooth crossover from a two-to a one-dimensional behavior? A naïve expectation would induce to think that synthetic lattices can simulate a two-dimensional geometry when the transverse dimension is much longer than the correlation length of the system along the transverse dimension itself, and they turn out to be effectively one-dimensional when the number of sites in the transverse direction is small, such as in synthetic ladders. In this work, we show that in the presence of a selected applied magnetic field and for an odd number of legs (L y ), this simple picture fails and the Hofstadter Hamiltonian can host unexpected topological phases for suitable values of the flux piercing the ladder. These phases, addressable in the state-of-the-art cold-atom experiments [8,9], are protected by a hidden inversion symmetry and exhibit the typical features of topological states in one dimension, i.e., exponentially localized degenerate edge states, a non-zero Zak phase [14][15][16], and a degenerate entanglement spectrum [17,18]. Note, however, that they...

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confidence: 83%

Topological phases in frustrated synthetic ladders with an odd number of legs

et al. 2018

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“…There are three magnetic unit cells in the x direction and two magnetic unit cells in the y direction of the torus. In this low-dimensional realization, the ground state multiplet is composed of two charge density waves [42]. As θ y traverses [0,2π ), the energies of two quasidegenerate ground states are interchanged and return to their original values after θ y traverses another period from [2π,4π ) (see Fig.…”

Section: A Ground State Degeneracymentioning

confidence: 99%

“…It is known that the Laughlin state, which in two dimensions on a torus corresponds to a twofold degenerate ground state that cannot be resolved by local observables, becomes a pair of charge density waves in the thin torus geometry of Fig. 1(b) [41,42]. Using bosonization methods we characterize the bulk and the edge excitations of the precursor to the Laughlin state on a ladder.…”

Section: Introductionmentioning

confidence: 99%

Precursor of the Laughlin state of hard-core bosons on a two-leg ladder

et al. 2017

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We study hard core bosons on a two-leg ladder lattice under the orbital effect of a uniform magnetic field. At densities which are incommensurate with flux, the ground state is a Meissner state, or a vortex state, depending on the strength of the flux. When the density is commensurate with the flux, analytical arguments predict the possibility to stabilize a ground state of central charge c = 1, which is a precursor of the two-dimensional Laughlin state at ν = 1/2. This differs from the coupled wire construction of the Laughlin state in that there exists a nonzero backscattering term in the edge Hamiltonian. By using a combination of bosonization and density matrix renormalization group (DMRG) calculations, we construct a phase diagram versus density and flux from local observables and central charge. We delimit the region where the finite-size ground state displays signatures compatible with this precursor to the Laughlin state. We show how bipartite charge fluctuations allow access to the Luttinger parameter for the edge Luttinger liquid corresponding to the precursor Laughlin state. The properties studied with local observables are confirmed by the long distance behavior of correlation functions. Our findings are consistent with an exact-diagonalization calculation of the many body ground state transverse conductivity in a thin torus geometry for parameters corresponding to the precursor Laughlin state. The model considered is simple enough such that the precursor to the Laughlin state could be realized in current ultracold atom, Josephson junction array, and quantum circuit experiments.

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“…They reported an observation of the chiral currents flowing around the ladder due to the effective magnetic field 1,7 . Motivated by this experimental ability, the coupled wire realization of the bosonic Laughling ν = 1/2 fractional quantum Hall effect (FQHE) introduced by Kane et al 8 , was recently suggested in two-leg ladders for strong on-site interactions 9,10 .…”

Section: Introductionmentioning

confidence: 99%

Chiral currents in one-dimensional fractional quantum Hall states

2015

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We study bosonic and fermionic quantum two-leg ladders with orbital magnetic flux. In such systems, the ratio, ν, of particle density to magnetic flux shapes the phase-space, as in quantum Hall effects. In fermionic (bosonic) ladders, when ν equals one over an odd (even) integer, Laughlin fractional quantum Hall (FQH) states are stabilized for sufficiently long ranged repulsive interactions. As a signature of these fractional states, we find a unique dependence of the chiral currents on particle density and on magnetic flux. This dependence is characterized by the fractional filling factor ν, and forms a stringent test for the realization of FQH states in ladders, using either numerical simulations or future ultracold-atom experiments. The two-leg model is equivalent to a single spinful chain with spin-orbit interactions and a Zeeman magnetic field, and results can thus be directly borrowed from one model to the other.

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Realization of fractional Chern insulators in the thin-torus limit with ultracold bosons

Cited by 62 publications

References 77 publications

Topological phases in frustrated synthetic ladders with an odd number of legs

Topological phases in frustrated synthetic ladders with an odd number of legs

Precursor of the Laughlin state of hard-core bosons on a two-leg ladder

Chiral currents in one-dimensional fractional quantum Hall states

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