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arXiv:2212.00030 · Nature Communications

Interacting topological quantum chemistry in 2D with many-body real space invariants

Jonah Herzog-Arbeitman, Bogdan Andrei Bernevig, Zhi-Da Song

Published Feb 8, 2024• 25 citations
arXivHTMLPDF

Abstract

The topological phases of non-interacting fermions have been classified by their symmetries, culminating in a modern electronic band theory where wavefunction topology can be obtained from momentum space. Recently, Real Space Invariants (RSIs) have provided a spatially local description of the global momentum space indices. The present work generalizes this real space classification to interacting 2D states. We construct many-body local RSIs as the quantum numbers of a set of symmetry operators on open boundaries, but which are independent of the choice of boundary. Using the U(1) particle number, they yield many-body fragile topological indices, which we use to identify which single-particle fragile states are many-body topological or trivial at weak coupling. To this end, we construct an exactly solvable Hamiltonian with single-particle fragile topology that is adiabatically connected to a trivial state through strong coupling. We then define global many-body RSIs on periodic boundary conditions. They reduce to Chern numbers in the band theory limit, but also identify strongly correlated stable topological phases with no single-particle counterpart. Finally, we show that the many-body local RSIs appear as quantized coefficients of Wen-Zee terms in the topological quantum field theory describing the phase.

DOI
10.1038/s41467-024-45395-9

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The topological phases of non-interacting fermions have been classified by their symmetries, culminating in a modern electronic band theory where wavefunction topology can be obtained from momentum space. Recently, Real Space Invariants (RSIs) have provided a spatially local description of the global momentum space indices. The present work generalizes this real space classification to interacting 2D states. We construct many-body local RSIs as the quantum numbers of a set of symmetry operators on open boundaries, but which are independent of the choice of boundary. Using the U(1) particle number, they yield many-body fragile topological indices, which we use to identify which single-particle fragile states are many-body topological or trivial at weak coupling. To this end, we construct an exactly solvable Hamiltonian with single-particle fragile topology that is adiabatically connected to a trivial state through strong coupling. We then define global many-body RSIs on periodic boundary conditions. They reduce to Chern numbers in the band theory limit, but also identify strongly correlated stable topological phases with no single-particle counterpart. Finally, we show that the many-body local RSIs appear as quantized coefficients of Wen-Zee terms in the topological quantum field theory describing the phase.
The recently introduced topological quantum chemistry (TQC) framework has provided a description of universal topological properties of all possible band insulators in all space groups based on crystalline unitary symmetries and time reversal. While this formalism filled the gap between the mathematical classification and the practical diagnosis of topological materials, an obvious limitation is that it only applies to weakly interacting systems, which can be described within band theory. It is an open question to which extent this formalism can be generalized to correlated systems that can exhibit symmetry-protected topological phases which are not adiabatically connected to any band insulator. In this work, we address the many facets of this question by considering the specific example of an extended version of a Hubbard diamond chain. This model features a Mott insulator, a trivial insulating phase, and an obstructed atomic limit phase. Here we first discuss the nature of the Mott insulator and determine the phase diagram and topology of the interacting model with infinite density matrix renormalization group calculations, variational Monte Carlo simulations, and with many-body topological invariants. We then proceed by considering a generalization of the TQC formalism to Green's functions combined with the concept of a topological Hamiltonian to identify the topological nature of the phases. Here we use cluster perturbation theory to calculate the Green's functions. The results are benchmarked with the above-determined phase diagram, and we discuss the applicability and limitations of the approach and its possible extensions in the diagnosis of topological phases in materials, in contrast to the use of many-body topological invariants.
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.
We investigate topological invariants in strongly interacting many-body systems within holographic mean-field theory (H-MFT) framework. Analytic expressions for retarded Green's functions are obtained for all possible fermionic bilinear interactions in the limit of probe background limit AdS4\mathrm{AdS}_4AdS4​, from which we construct topological Hamiltonians. Integrating Berry curvature over the momentum domain for the gapped spectra yields well-defined and quantized Chern numbers, enabling a systematic classification of them across interaction types. These topological invariants remain robust under deformation parameters like interaction and temperature, indicating that H-MFT encodes effective single-particle-state topology near a quantum critical point in strongly correlated systems. We point out why topological number is defined in the holographic theories while it is not in the perturbative field theory.
We study two-particle states in a Su-Schrieffer-Heeger (SSH) chain with periodic boundary conditions and nearest-neighbor (NN) interactions. The system is mapped into a problem of a single particle in a two-dimensional (2D) SSH lattice with potential walls along specific edges. The 2D SSH model has a trivial Chern number but a non-trivial Zak's phase, the one-dimensional (1D) topological invariant, along specific directions of the lattice, which allow for the presence of topological edge states. Using center-of-mass and relative coordinates, we calculate the energy spectrum of these two-body states for strong interactions and find that, aside from the expected appearance of doublon bands, two extra in-gap bands are present. These are identified as bands of topological states localized at the edges of the internal coordinate, the relative distance between the two particles. As such, the topological states reported here are intrinsically many-body in what concerns their real space manifestation, having no counterpart in single-particle states derived from effective models. Finally, we compare the effect of Hubbard interactions with that of NN interactions to show how the presence of the topological bound states is specific to the latter case.
Topological flat bands (FBs) offer an ideal platform for realizing exotic topological phases and exploring quantum geometric effects, yet their realization with both exact flatness and stable topology in local lattice models has been hindered by fundamental no-go theorems. The obstruction is also manifested as the absence of exact Gaussian TNS representations for topological insulators and superconductors. Here, we overcome this barrier by demonstrating the existence of critical topological FBs (CTFBs) in finite-range hopping models. They saturate the no-go theorems via a unique structure of Bloch wavefunctions: While continuous over the whole Brillouin zone, the projector P(k)P(k)P(k) onto FBs is non-analytic at isolated band touching points. Filling such CTFBs yields short-range entangled topological states with power-law correlations due to the non-analyticity. We then establish a general symmetry-based principle to systematically construct CTFBs that carry desired topological invariants in given space groups. It utilizes the bipartite structure and requires no further fine-tuning, and the topology is robust against arbitrary symmetry-preserving gap-opening perturbations. Remarkably, independent tuning of the quantum geometry is allowed. Examples exhibiting Chern numbers 1, 2, 3, 6, in 2D, and strong Z2\mathbb{Z}_2Z2​ indices in 2D and 3D are demonstrated for concreteness. An automated algorithm further identifies more than 50,000 symmetry-indicated CTFBs, including crystalline and higher-order topological ones. In the end, we show that the bipartite structure naturally endows the filled CTFB states with exact TNS representations with finite bond dimensions. By bridging the topological band theory to TNS methods, CTFB provides a novel tractable starting point for exploring strongly correlated topological matter, with potential relevance to realistic material realizations.
Abstract The study of topological band insulators has revealed fascinating phases characterized by band topology indices and anomalous boundary modes protected by global symmetries. In strongly correlated systems, where the traditional notion of electronic bands becomes obsolete, it has been established that topological insulator phases persist as stable phases, separate from the trivial insulators. However, due to the inability to express the ground states of such systems as Slater determinants, the formulation of generic variational wave functions for numerical simulations is highly desirable. In this paper, we tackle this challenge for two-dimensional topological insulators by developing a comprehensive framework for fermionic tensor network states. Starting from simple assumptions, we obtain possible sets of tensor equations for any given symmetry group, capturing consistent relations governing symmetry transformation rules on tensor legs. We then examine the connection between these tensor equations and non-chiral topological insulators by constructing edge theories and extracting quantum anomaly data from each set of tensor equations. By exhaustively exploring all possible sets of equations, we achieve a systematic classification of non-chiral topological insulator phases. Imposing the solutions of a given set of equations onto local tensors, we obtain generic variational wavefunctions for the corresponding topological insulator phases. Our methodology provides an important step toward simulating topological insulators in strongly correlated systems. We discuss the limitations and potential generalizations of our results, paving the way for further advancements in this field.