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arXiv:1808.10248 · Physical Review Letters

Many-Body Chern Number without Integration

Koji Kudo, Haruki Watanabe, Toshikaze Kariyado, Yasuhiro Hatsugai

Published Apr 9, 2019• 62 citations
arXivHTMLPDF

Abstract

The celebrated work of Niu, Thouless, and Wu demonstrated the quantization of Hall conductance in the presence of many-body interactions by revealing the many-body counterpart of the Chern number. The generalized Chern number is formulated in terms of the twisted angles of the boundary condition, instead of the single particle momentum, and involves an integration over all possible twisted angles. However, this formulation is physically unnatural, since topological invariants directly related to observables should be defined for each Hamiltonian under a fixed boundary condition. In this work, we show via numerical calculations that the integration is indeed unnecessary-the integrand itself is effectively quantized and the error decays exponentially with the system size. This implies that the numerical cost in computing the many-body Chern number could, in principle, be significantly reduced as it suffices to compute the Berry connection for a single value of the twisted boundary condition if the system size is sufficiently large.

DOI
10.1103/physrevlett.122.146601

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The celebrated work of Niu, Thouless, and Wu demonstrated the quantization of Hall conductance in the presence of many-body interactions by revealing the many-body counterpart of the Chern number. The generalized Chern number is formulated in terms of the twisted angles of the boundary condition, instead of the single particle momentum, and involves an integration over all possible twisted angles. However, this formulation is physically unnatural, since topological invariants directly related to observables should be defined for each Hamiltonian under a fixed boundary condition. In this work, we show via numerical calculations that the integration is indeed unnecessary-the integrand itself is effectively quantized and the error decays exponentially with the system size. This implies that the numerical cost in computing the many-body Chern number could, in principle, be significantly reduced as it suffices to compute the Berry connection for a single value of the twisted boundary condition if the system size is sufficiently large.
The quantized Hall conductivity of integer and fractional quantum Hall (IQH and FQH) states is directly related to a topological invariant, the many-body Chern number. The conventional calculation of this invariant in interacting systems requires a family of many-body wave functions parameterized by twist angles to calculate the Berry curvature. In this paper, we demonstrate how to extract the Chern number given a single many-body wave function, without knowledge of the Hamiltonian. For FQH states, our method requires one additional integer invariant as input: the number of $2\ensuremath{\pi}$ flux quanta, sss, that must be inserted to obtain a topologically trivial excitation. As we discuss, sss can be obtained in principle from the degenerate set of ground state wave functions on the torus, without knowledge of the Hamiltonian. We perform extensive numerical simulations involving IQH and FQH states to validate these methods.
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Understanding correlation effects in topological phases of matter is at the forefront of current research in condensed matter physics. Here we try to clarify some subtleties in studying topological behaviors of interacting Weyl semimetals. It is well known that there exist two topological invariants defined to identify their topological character. One is the many-body Chern number, which can be directly linked to the Hall conductivity and thus to the two-particle correlations. The other is the topological index constructed from the single-particle Green's functions. Because the information on Green's functions is achieved more easily than the many-body wave functions, usually only the topological index is employed in the literature. However, the approach based on the single-particle Green's function can break down in the strongly correlated phase. For illustration, an exactly solvable two-orbital model with momentum-local two-body interactions is discussed, in which both topological invariants can be calculated analytically. We find that the topological index calculated from the Green's function formalism can be nonzero even for a nontopological strongly correlated phase with a vanishing many-body Chern number. In addition, we stress that the physical surface states implied by nonzero many-body Chern numbers should be the edge modes of particle-hole collective excitations, rather than those of a quasiparticle nature derived from the Green's function formalism. Our observations thus demonstrate the limitation of the validity of the Green's function formalism in the investigations of interacting topological materials.
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We investigate how topological Chern numbers can be defined when single-particle states hybridize with continua. We do so exemplarily in a bosonic Haldane model at zero temperature with an additional on-site decay of one boson into two and the conjugate fusion of two bosons into one. Restricting the Hilbert space to two bosons at maximum, the exact self-energy is accessible. We use the bilinear Hamiltonian H 0 corrected by the self-energy Σ to compute Chern numbers by two different approaches. The results are gauged against a full many-body calculation in the Hilbert space where possible. We establish numerically and analytically that the effective Hamiltonian H eff = H 0 ( k ⃗ ) + Σ ( ω , k ⃗ ) reproduces the correct many-body topology if the considered band does not overlap with the continuum. In case of overlaps, one can extend the definition of the Chern number to the non-Hermitian H eff and there is evidence that the Chern number changes at exceptional points. But the bulk-boundary correspondence appears to be no longer valid and edge modes delocalize. Published by the American Physical Society 2024