Fractional corner charges in a two-dimensional superlattice Bose-Hubbard model

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Fractional corner charges in a two-dimensional superlattice Bose-Hubbard model

Julian Bibo, Izabella Lovas, Yizhi You, Fabian Grusdt, and Frank Pollmann

Phys. Rev. B 102, 041126(R) – Published 30 July 2020

Abstract

We study higher order topology in the presence of strong interactions in a two-dimensional, experimentally accessible superlattice Bose-Hubbard model with alternating hoppings and strong on-site repulsion. We evaluate the phase diagram of the model around half-filling using the density renormalization group ansatz and find two gapped phases separated by a gapless superfluid region. We demonstrate that the gapped states realize two distinct higher order symmetry protected topological phases, which are protected by a combination of charge conservation and $C_{4}$ lattice symmetry. The phases are distinguished in terms of a many-body topological invariant and a quantized, experimentally accessible fractional corner charge that is robust against arbitrary, symmetry preserving edge manipulations. We support our claims by numerically studying the full counting statistics of the corner charge, finding a sharp distribution peaked around the quantized values. Our results allow for a direct comparison with experiments and represent a confirmation of theoretically predicted higher order topology in a strongly interacting system. Experimentally, the fractional corner charge can be observed in ultracold atomic settings using state of the art quantum gas microscopy.

Received 16 December 2019
Accepted 15 July 2020

DOI:https://doi.org/10.1103/PhysRevB.102.041126

Physics Subject Headings (PhySH)

Bose gases Symmetry protected topological states

Superfluidity Ultracold gases

Bose-Hubbard model Density matrix renormalization group

Condensed Matter, Materials & Applied PhysicsGeneral Physics

Authors & Affiliations

Julian Bibo ^1,2, Izabella Lovas^1,2, Yizhi You³, Fabian Grusdt^2,4,5, and Frank Pollmann^1,2

¹Department of Physics, T42, Technical University of Munich, D-85748 Garching, Germany
²Munich Center for Quantum Science and Technology (MQCST), Schellingstr. 4, D-80799 Munich, Germany
³Princeton Center for Theoretical Science, Princeton University, New Jersey 08544, USA
⁴Department of Physics and Institute for Advanced Study, Technical University of Munich, D-85748 Garching, Germany
⁵Department of Physics and Arnold Sommerfeld Center for Theoretical Physics (ASC), Ludwig-Maximilians-Universität München, Theresienstr. 37, D-80333 Munich, Germany

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Issue

Vol. 102, Iss. 4 — 15 July 2020

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Images

Figure 1
2D SL-BHM showing an HOSPT phase with quantized fractional corner charges. (a) The SL-BHM has a $2 \times 2$ unit cell, with two different hopping amplitudes inside and between unit cells, $t$ and $1 - t$ , respectively. The ground state is topological trivial (nontrivial) for $t = 1 (t = 0)$ . (b) Average occupation number of lattice sites in the topological phase with particle number $N = L^{2} / 2 - 2$ , displaying four holes localized around the corners, giving rise to fractional charges $Q_{corner} = 1 / 2$ . (c) The full distribution function of corner charges for the trivial (topological) phase, peaked around a quantized fractional part $0 (1 / 2)$ . We used the parameters $N = L^{2} / 2 (L^{2} / 2 - 2)$ , $t = 0.9 (0.1)$ , $U = 32$ , and $ξ_{env} = 3.2 (3.08)$ .
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Figure 2
Exactly solvable limits of both phases are shown for the trivial (TR) and topological phase (TO) in (a) and (b), respectively. In the topological phase, the corner of the lattice is completely decoupled from the rest of the system. Panel (c) shows the bipartite entanglement entropy $S$ against the interpolation parameter $t$ obtained by DMRG on an infinite cylinder with circumference $L_{y} = 6$ . The large entanglement is characteristic for the superfluid (SF) phase. The Hilbert space was truncated to maximal four bosons per site and we used $χ = 500$ states for the simulations.
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Figure 3
(a) Corner charge $Q_{corner}$ as a function of hopping $t$ at half-filling $N = N_{0}$ , quantized to $Q_{corner} = 0 (1 / 2)$ in the gapped trivial (topological) phase. We used DMRG on an $8 \times 8$ square lattice, taking $U = 32, ξ_{env} = 2.8$ . (b) Berry phase $γ$ as a function of the hopping parameter, quantized around $γ = 0 (π)$ in the gapped trivial (topological) phase. Result is obtained for a $4 \times 4$ square lattice with periodic boundary conditions at half-filling.
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Figure 4
Full counting statistics of edge and corner charges. (a) and (b) FCS of fractional edge charge $Q_{edge}$ of the 1D SL-BHM (defined in Ref. [8]) in the TR and TO phases, for two different system sizes $L = 24$ and $L = 200$ . The distribution is peaked around the quantized value $0 (1 / 2)$ in the TR (TO) phase and gets sharper with increasing $L$ , confirming that $Q_{edge}$ is a good quantum number in the thermodynamic limit. Parameters for the TR (TO) phase: $N = N_{0} (N_{0} - 1)$ , $t_{1} = 0.2 (1)$ , $t_{2} = 1 (0.2)$ , $U = 10$ . The envelopes are $ξ_{env} = 12 (12)$ for $L = 24$ and $ξ_{env} = 53 (45)$ for $L = 200$ , respectively. (c) and (d) FCS of fractional corner charge $Q_{corner}$ , measured in the TR (TO) phase of the 2D SL-BHM for system size $10 \times 10$ , peaked around $0 (1 / 2)$ . The width of the distribution should approach to zero in the thermodynamic limit, similar to the 1D case. Results were obtained for $N = N_{0} (N_{0} - 2)$ , $t = 0.9 (0.1)$ , $U = 32$ , and $ξ_{env} = 3.2 (3.08)$ .
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