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arXiv:2202.12207 · Physical review. B./Physical review. B

Real-space many-body marker for correlated Z2 topological insulators

Ivan Gilardoni, Federico Becca, Antimo Marrazzo, Alberto Parola

Published Oct 17, 2022• 11 citations
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Abstract

Taking the clue from the modern theory of polarization [Rev. Mod. Phys. 66, 899 (1994)], we identify an operator to distinguish between ${\mathbb{Z}}_{2}$-even (trivial) and ${\mathbb{Z}}_{2}$-odd (topological) insulators in two spatial dimensions. Its definition extends the position operator [Phys. Rev. Lett. 82, 370 (1999)], which was introduced in one-dimensional systems. We first show a few examples of noninteracting models where single-particle wave functions are defined and allow for a direct comparison with standard techniques on large system sizes. Then, we illustrate its applicability for an interacting model on a small cluster where exact diagonalizations are available. Its formulation in the Fock space allows a direct computation of expectation values over the ground-state wave function (or any approximation of it), thus, allowing us to investigate generic interacting systems, such as strongly correlated topological insulators.

DOI
10.1103/physrevb.106.l161106

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Taking the clue from the modern theory of polarization [Rev. Mod. Phys. 66, 899 (1994)], we identify an operator to distinguish between Z2{\mathbb{Z}}_{2}Z2​-even (trivial) and Z2{\mathbb{Z}}_{2}Z2​-odd (topological) insulators in two spatial dimensions. Its definition extends the position operator [Phys. Rev. Lett. 82, 370 (1999)], which was introduced in one-dimensional systems. We first show a few examples of noninteracting models where single-particle wave functions are defined and allow for a direct comparison with standard techniques on large system sizes. Then, we illustrate its applicability for an interacting model on a small cluster where exact diagonalizations are available. Its formulation in the Fock space allows a direct computation of expectation values over the ground-state wave function (or any approximation of it), thus, allowing us to investigate generic interacting systems, such as strongly correlated topological insulators.
We extend the previously defined many-body marker for two-dimensional Z2 topological insulators [I. Gilardoni et al., Phys. Rev. B 106, L161106 (2022)] to distinguish trivial, weak, and strong topological insulators in three dimensions, in the presence of inversion symmetry. The marker is written in terms of ground-state expectation values of position operators and can be employed to detect topological phases of interacting systems beyond mean-field approximations, e.g., within quantum Monte Carlo techniques. Here we show that the correct results of the noninteracting limit are reproduced by the many-body marker.
We extend the previously defined many-body marker for two-dimensional Z2\mathbb{Z}_2Z2​ topological insulators [I. Gilardoni {\it et al.}, Phys. Rev. B {\bf 106}, L161106 (2022)] to distinguish trivial, weak-, and strong-topological insulators in three dimensions, in presence of the inversion symmetry. The marker is written in term of ground-state expectation values of position operators and can be employed to detect topological phases of interacting systems beyond mean-field approximations, e.g., within quantum Monte Carlo techniques. Here, we show that the correct results of the non-interacting limit are reproduced by the many-body marker.
We show that the two-dimensional Z2{\mathbb{Z}}_{2}Z2​ invariant for time-reversal invariant insulators can be formulated in terms of the boundary-condition dependence of the ground state wave function for both noninteracting and strongly correlated insulators. By introducing a family of quasi-single-particle states associated to the many-body ground state of an insulator, we show that the Z2{\mathbb{Z}}_{2}Z2​ invariant can be expressed as the integral of a certain Berry connection over half the space of boundary conditions, providing an alternative expression to the formulations that appear in S.-S. Lee and S. Ryu [Phys. Rev. Lett. 100, 186807 (2008)]. We show the equivalence of the different many-body formulations of the invariant and show how they reduce to known band-theoretic results for Slater determinant ground states. Finally, we apply our results to analytically calculate the invariant for the Kane-Mele model with nonlocal (orbital) Hatsugai-Kohmoto (HK) interactions. This rigorously establishes the topological nontriviality of the Kane-Mele model with HK interactions and represents one of the few exact calculations of the Z2{\mathbb{Z}}_{2}Z2​ invariant for a strongly interacting system.
Topological states of matter were first introduced for noninteracting fermions on an infinite uniform lattice. Since then, substantial effort has been made to generalize these concepts to more complex settings. Recently, local markers have been developed that can describe the topological state of systems without translational symmetry and a well-defined gap. However, no local marker for interacting matter has been proposed yet that is capable of directly addressing an interacting system. Here, we suggest such a many-body local marker based on the single-particle Green's function. Using this marker we identify topological transitions in finite lattices of a Chern insulator with Anderson disorder and Hubbard interactions. Importantly, our proposal can be generalized to nonequilibrium systems.
Understanding how topology survives in strongly correlated systems remains a central challenge, as most topological diagnostics rely on non-interacting band structures. Here we present a framework to characterize interacting topological phases within an effective single-particle description derived from the single-particle Green's function. Using the Su-Schrieffer-Heeger model with Hatsugai-Kohmoto interactions as an analytically tractable example, we construct the one-body reduced density matrix from the Green's function and use it to define an effective winding number together with quantum volume, a measurement of state geometry. These quantities allow us to distinguish three insulating phases including correlated Mott states directly from single-particle observables. Our results show that interacting topology can be interpreted in terms of the spectral weight distribution of single-particle excitations, providing an intuitive and computationally accessible route to diagnose topological phases in correlated systems. This approach is compatible with modern many-body simulation techniques and opens a pathway toward the identification of interacting topological materials.
We present a rigorous but elementary index theory for a class of one-dimensional systems of interacting (and possibly disordered) fermions with U(1)⋊Z2 symmetry defined on the infinite chain. The class includes the Su–Schrieffer–Heeger (SSH) model [Su et al., “Solitons in polyacetylene,” Phys. Rev. Lett. 42, 1698 (1979); Su et al., “Soliton excitations in polyacetylene,” Phys. Rev. B 22, 2099 (1983); and Asbóth et al., A Short Course on Topological Insulators: Band-Structure Topology and Edge States in One and Two Dimensions, Lecture Notes in Physics (Springer, 2016)] as a special case. For any locally unique gapped (fixed-charge) ground state of a model in the class, we define a Z2 index in terms of the sign of the expectation value of the local twist operator. We prove that the index is topological in the sense that it is invariant under continuous modification of models in the class with a locally unique (fixed-charge) gapped ground state. This establishes that any path of models in the class that connects the two extreme cases of the SSH model must go through a phase transition. Our rigorous Z2 classification is believed to be optimal for the class of models considered here. We also show an interesting duality of the index and prove that any topologically nontrivial model in the class has a gapless edge excitation above the ground state when defined on the half-infinite chain. The results extend to other classes of models, including the extended Hubbard model. Our strategy to focus on the expectation value of local unitary operators makes the theory intuitive and conceptually simple. This paper also contains a careful discussion about the notion of unique gapped ground states of a particle system on the infinite chain. (There are two lecture videos in which the main results of this paper are discussed [H. Tasaki, “Rigorous index theory for one-dimensional interacting topological insulators: A brief introduction,” online lecture (21:41), November, 2021, seehttps://www.gakushuin.ac.jp/~881791/OL/#Index1DTI2021S and https://youtu.be/ypGVb3eYrpg and H. Tasaki, “Rigorous index theory for one-dimensional interacting topological insulators: With a pedagogical introduction to the topological phase transition in the SSH model,” online lecture (49:07), November, 2021, see https://www.gakushuin.ac.jp/~881791/OL/#Index1DTI2021L and https://youtu.be/yxZYOevV2Y].