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arXiv:1911.04149 · Physical review. B./Physical review. B

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

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

Published Jul 30, 2020• 18 citations
arXivHTMLPDF

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.

DOI
10.1103/physrevb.102.041126

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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 C4{C}_{4}C4​ 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.
We study the entanglement spectrum (ES) of two-dimensional Cn{C}_{n}Cn​-symmetric second-order topological insulators (TIs). We show that some characteristic higher-order topological observables, e.g., the filling anomaly and its associated fractional corner charge, can be determined from the ES of atomic and fragile TIs. By constructing the relationship between the configuration of Wannier orbitals and the number of protected in-gap states in the ES for different symmetric cuts in real space, we express the fractional corner charge in terms of the number of protected in-gap states of the ES. We show that our formula is robust in the presence of electron-electron interactions as long as the interactions preserve Cn{C}_{n}Cn​ rotation symmetry and charge-conservation symmetry. Moreover, we discuss the possible signatures of higher-order topology in the many-body ES. Our methods allow the identification of some classes of higher-order topology without requiring the usage of nested Wilson loops or nested entanglement spectra.
Strongly correlated analogs of topological insulators have been explored in systems with purely on-site symmetries, such as time-reversal or charge conservation. Here, we use recently developed tensor network tools to study a quantum state of interacting bosons which is featureless in the bulk, but distinguished from an atomic insulator in that it exhibits entanglement which is protected by its spatial symmetries. These properties are encoded in a model many-body wave function that describes a fully symmetric insulator of bosons on the honeycomb lattice at half filling per site. While the resulting integer unit cell filling allows the state to bypass ``no-go'' theorems that trigger fractionalization at fractional filling, it nevertheless has nontrivial entanglement, protected by symmetry. We demonstrate this by computing the boundary entanglement spectra, finding a gapless entanglement edge described by a conformal field theory as well as degeneracies protected by the nontrivial action of combined charge conservation and spatial symmetries on the edge. Here, the tight-binding representation of the space group symmetries plays a particular role in allowing certain entanglement cuts that are not allowed on other lattices of the same symmetry, suggesting that the lattice representation can serve as an additional symmetry ingredient in protecting an interacting topological phase. Our results extend to a related insulating state of electrons, with short-ranged entanglement and no band insulator analog.
Many-body higher-order topological invariant for CnC_nCn​-symmetric insulators
Higher-order topological insulators in two spatial dimensions display fractional corner charges. While fractional charges in one dimension are known to be captured by a many-body bulk invariant, computed by the Resta formula, a many-body bulk invariant for higher-order topology and the corresponding fractional corner charges remains elusive despite several attempts. Inspired by recent work by Tada and Oshikawa, we propose a well-defined many-body bulk invariant for CnC_nCn​ symmetric higher-order topological insulators, which is valid for both non-interacting and interacting systems. Instead of relating them to the bulk quadrupole moment as was previously done, we show that in the presence of CnC_nCn​ rotational symmetry, this bulk invariant can be directly identified with quantized fractional corner charges. In particular, we prove that the corner charge is quantized as e/ne/ne/n with CnC_nCn​ symmetry, leading to a Zn\mathbb{Z}_nZn​ classification for higher-order topological insulators in two dimensions.
The combination of topology and quantum criticality can give rise to an exotic mix of counterintuitive effects. Here, we show that unexpected topological properties take place in a paradigmatic strongly correlated Hamiltonian: the 1D extended Bose-Hubbard model. In particular, we reveal the presence of two distinct topological quantum critical points with localized edge states and gapless bulk excitations. Our results show that the topological critical points separate two phases, one topologically protected and the other topologically trivial, both characterized by a long-range ordered string correlation function. The long-range order persists also at the topological critical points and explains the presence of localized edge states protected by a finite charge gap. Finally, we introduce a superresolution quantum gas microscopy scheme for dipolar dysprosium atoms, which provides a reliable route towards the experimental study of topological quantum critical points.
The low energy behavior of a huge variety of one-dimensional interacting spinful fermionic systems exhibits spin-charge separation, described in the continuum limit by two sine-Gordon models decoupled in the charge and spin channels. Interaction is known to induce, besides the gapless Luttinger liquid phase, eight possible gapped phases, among which are the Mott, Haldane, charge-/spin-density, and bond-ordered wave insulators, and the Luther Emery liquid. Here we prove that some of these physically distinct phases have nontrivial topological properties, notably the presence of degenerate protected edge modes with fractionalized charge/spin. Moreover, we show that the eight gapped phases are in one-to-one correspondence with the symmetry-protected topological (SPT) phases classified by group cohomology theory in the presence of particle-hole symmetry P. The latter result is also exploited to characterize SPT phases by measurable nonlocal order parameters which follow the system evolution to the quantum phase transition. The implications on the appearance of exotic orders in the class of microscopic Hubbard Hamiltonians, possibly without P symmetry at higher energies, are discussed.
Higher-order topological insulators in two spatial dimensions display fractional corner charges. While fractional charges in one dimension are known to be captured by a many-body bulk invariant, computed by the Resta formula, a many-body bulk invariant for higher-order topology and the corresponding fractional corner charges remains elusive despite several attempts. Inspired by recent work by Tada and Oshikawa [arXiv:2302.00800], we propose a well-defined, many-body bulk invariant for Cn{C}_{n}Cn​-symmetric higher-order topological insulators that is valid for both noninteracting and interacting systems. Instead of relating them to the bulk quadrupole moment as was previously done, we show that in the presence of Cn{C}_{n}Cn​ rotational symmetry, this bulk invariant can be directly identified with quantized fractional corner charges. In particular, we prove that the corner charge is quantized as e/ne/ne/n with Cn{C}_{n}Cn​ symmetry, leading to a Zn{\mathbb{Z}}_{n}Zn​ classification for higher-order topological insulators in two dimensions.