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arXiv:1906.11553 · Physical Review Letters

Finite-Temperature Topological Invariant for Interacting Systems

Razmik G. Unanyan, Maximilian Kiefer-Emmanouilidis, Michael Fleischhauer

Published Nov 16, 2020• 29 citations
arXivHTMLPDF

Abstract

We generalize the ensemble geometric phase, recently introduced to classify the topology of density matrices, to finite-temperature states of interacting systems in one spatial dimension (1D). This includes cases where the gapped ground state has a fractional filling and is degenerate. At zero temperature the corresponding topological invariant agrees with the well-known invariant of Niu, Thouless, and Wu. We show that its value at finite temperatures is identical to that of the ground state below some critical temperature T_{c} larger than the many-body gap. We illustrate our result with numerical simulations of the 1D extended superlattice Bose-Hubbard model at quarter filling. Here, a cyclic change of parameters in the ground state leads to a topological charge pump with fractional winding ν=1/2. The particle transport is no longer quantized when the temperature becomes comparable to the many-body gap, yet the winding of the generalized ensemble geometric phase is.

DOI
10.1103/physrevlett.125.215701

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We generalize the ensemble geometric phase, recently introduced to classify the topology of density matrices, to finite-temperature states of interacting systems in one spatial dimension (1D). This includes cases where the gapped ground state has a fractional filling and is degenerate. At zero temperature the corresponding topological invariant agrees with the well-known invariant of Niu, Thouless, and Wu. We show that its value at finite temperatures is identical to that of the ground state below some critical temperature T_{c} larger than the many-body gap. We illustrate our result with numerical simulations of the 1D extended superlattice Bose-Hubbard model at quarter filling. Here, a cyclic change of parameters in the ground state leads to a topological charge pump with fractional winding ν=1/2. The particle transport is no longer quantized when the temperature becomes comparable to the many-body gap, yet the winding of the generalized ensemble geometric phase is.
We show that a topological pump in a one-dimensional insulator can induce a strictly quantized transport in an auxiliary chain of noninteracting fermions weakly coupled to the first. The transported charge is determined by an integer topological invariant of the fictitious Hamiltonian of the insulator, given by the covariance matrix of single-particle correlations. If the original system consists of noninteracting fermions, this number is identical to the Thouless, Kohmoto, Nightingale, and den Nijs (TKNN) invariant of the original system and thus the coupling induces a transfer of topology to the auxiliary chain. When extended to particles with interactions, for which the TKNN number does not exist, the transported charge in the auxiliary chain defines a topological invariant for the interacting system. In certain cases this invariant agrees with the many-body generalization of the TKNN number introduced by Niu, Thouless, and Wu. We illustrate the topology transfer to the auxiliary system for the Rice-Mele model of noninteracting fermions at half filling and the extended superlattice Bose-Hubbard model at quarter filling. In the latter case the induced charge pump is fractional.
The ground states of noninteracting fermions in one-dimension with chiral symmetry form a class of topological band insulators, described by a topological invariant that can be related to the Zak phase. Recently, a generalization of this quantity to open systems---known as the ensemble geometric phase (EGP)---has emerged as a robust way to describe topology at nonzero temperature. By using this quantity, we explore the nature of topology allowed for dissipation beyond a Lindblad description, to allow for coupling to external baths at finite temperatures. We introduce two main aspects to the theory of open-system topology. First, we discover topological phase transitions as a function of the temperature TTT, manifesting as changes in differences of the EGP accumulated over a closed loop in parameter space. We characterize the nature of these transitions and reveal that the corresponding nonequilibrium steady state can exhibit a nontrivial structure---contrary to previous studies where it was found to be in a fully mixed state. Second, we demonstrate that the EGP itself becomes quantized when key symmetries are present, allowing it to be viewed as a topological marker which can undergo equilibrium topological transitions at nonzero temperatures.
In a recent paper [Bardyn et al., Phys. Rev. X 8, 011035 (2018)], it was shown that the generalization of many-body polarization to mixed states can be used to construct a topological invariant that is also applicable to finite-temperature and nonequilibrium Gaussian states of lattice fermions. The many-body polarization defines an ensemble geometric phase that is identical to the Zak phase of a fictitious Hamiltonian, whose symmetries determine the topological classification. Here we show that in the case of Gaussian states of bosons, the corresponding topological invariant is always trivial. This also applies to finite-temperature states of bosons in lattices with a topologically nontrivial band structure. As a consequence, there is no quantized topological charge pumping for translationally invariant bulk states of noninteracting bosons.
We consider a topologically nontrivial flat-band structure in one spatial dimension in the presence of nearest- and next-nearest-neighbor Hubbard interaction. The noninteracting band structure is characterized by a symmetry-protected topologically quantized Berry phase. At certain fractional fillings, a gapped phase with a filling-dependent ground-state degeneracy and fractionally charged quasiparticles emerges. At filling 13\frac{1}{3}31​, the ground states carry a fractional Berry phase in the momentum basis. These features at first glance suggest a certain analogy to the fractional quantum Hall scenario in two dimensions. We solve the interacting model analytically in the physically relevant limit of a large band gap in the underlying band structure, the analog of a lowest Landau level projection. Our solution affords a simple physical understanding of the properties of the gapped interacting phase. We pinpoint crucial differences to the fractional quantum Hall case by studying the Berry phase and the entanglement entropy associated with the degenerate ground states. In particular, we conclude that the ``fractional topological phase in one-dimensional flat bands'' is not a one-dimensional analog of the two-dimensional fractional quantum Hall states, but rather a charge density wave with a nontrivial Berry phase. Finally, the symmetry-protected nature of the Berry phase of the interacting phase is demonstrated by explicitly constructing a gapped interpolation to a state with a trivial Berry phase.
In 2+1-dimensions (2+1D), a gapped quantum phase with no symmetry (i.e. a topological order) can have a thermal Hall conductance κxy=cπ2kB23hTκ_{xy}=c \frac{π^2 k_B^2}{3h}Tκxy​=c3hπ2kB2​​T, where the dimensionless ccc is called chiral central charge. If there is a U1U_1U1​ symmetry, a gapped quantum phase can also have a Hall conductance σxy=νe2hσ_{xy}=ν\frac{e^2}{h}σxy​=νhe2​, where the dimensionless ννν is called filling fraction. In this paper, we derive some quantization conditions of ccc and ννν, via a cobordism approach to define Chern--Simons topological invariants which are associated with ccc and ννν. In particular, we obtain quantization conditions that depend on the ground state degeneracies on Riemannian surfaces, and quantization conditions that depend on the type of spacetime manifolds where the topological partition function is non-zero.
We compare three complementary diagnostics for mixed Gaussian states at nonzero temperature, focusing on the Su-Schrieffer-Heeger (SSH) chain and its inversion-symmetric extension. While the ensemble geometric phase, a mixed-state generalization of the Zak phase, remains well defined at nonzero temperature, the modulus of the corresponding expectation value vanishes in the thermodynamic limit, limiting its practical use. To develop diagnostics suitable for large systems, we introduce local twist operators acting on neighboring sites, whose expectation values provide local indicators of the underlying topological phase. The topological phase is identified from the relative magnitude of these expectation values, which only requires measuring two local expectation values at nonzero temperature, together with one additional nonlocal expectation value when next-nearest-neighbor hopping is included. In addition, we generalize the local chiral marker to mixed Gaussian states, fully determined by its single-particle correlation matrix, with a nonzero purity gap in their effective single-particle Hamiltonian. The presence of a purity gap ensures that the correlation matrix can be flattened to an effective projector. Evaluating the chiral marker with respect to the band-flattened correlation matrix yields a real-space topological invariant that coincides with the winding number in the zero-temperature limit. The ensemble geometric phase, the local twist operators, and the local chiral marker provide complementary methods to characterize topology in the SSH chain beyond pure states.