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arXiv:1805.00839 · Physical review. B./Physical review. B

Topological invariants for the Haldane phase of interacting Su-Schrieffer-Heeger chains: Functional renormalization-group approach

Björn Sbierski, Christoph Karrasch

Published Oct 1, 2018• 25 citations
arXivHTMLPDF

Abstract

We present a functional renormalization-group approach to interacting topological Green's function invariants with a focus on the nature of transitions. The method is applied to chiral symmetric fermion chains in the Mott limit that can be driven into a Haldane phase. We explicitly show that the transition to this phase is accompanied by a zero of the fermion Green's function. Our results for the phase boundary are quantitatively benchmarked against density matrix renormalization-group data.

DOI
10.1103/physrevb.98.165101

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We present a functional renormalization-group approach to interacting topological Green's function invariants with a focus on the nature of transitions. The method is applied to chiral symmetric fermion chains in the Mott limit that can be driven into a Haldane phase. We explicitly show that the transition to this phase is accompanied by a zero of the fermion Green's function. Our results for the phase boundary are quantitatively benchmarked against density matrix renormalization-group data.
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A one-dimensional closed interacting Kitaev chain and the dimerized version are studied. The topological invariants in terms of Green’s function are calculated by the density matrix renormalization group method and the exact diagonalization method. For the interacting Kitaev chain, we point out that the calculation of the topological invariant in the charge density wave phase must consider the dimerized configuration of the ground states. The variation of the topological invariant is attributed to the poles of eigenvalues of the zero-frequency Green functions. For the interacting dimerized Kitaev chain, we show that the topological invariant defined by Green’s functions can distinguish more topological nonequivalent phases than the fermion parity .
The topological properties of the Su-Schrieffer-Heeger (SSH) model in the presence of nearest-neighbor interaction are investigated by means of a topological marker, generalized from a noninteracting one by utilizing the single-particle Green's function of the many-body ground state. We find that despite the marker not being perfectly quantized in the presence of interactions, it always remains finite in the topologically nontrivial phase while converging to zero in the trivial phase when approaching the thermodynamic limit, and hence correctly judges the topological phases in the presence of interactions. The marker also correctly captures the interaction-driven, second-order phase transitions between a topological phase and a Landau-ordered phase, which is a charge-density wave order in our model with a local-order parameter, as confirmed by the calculation of entanglement entropy and the many-body Zak phase. Our paper thus points to the possibility of generalizing topological markers to interacting systems through Green's function, which may be feasible for topological insulators in any dimension and symmetry class.
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.
We study topological properties of phase transition points of topological quantum phase transitions by assigning a topological invariant defined on a closed circle or surface surrounding the phase transition point in the parameter space of momentum and transition driving parameter. By applying our scheme to the Su-Schrieffer-Heeger model and Haldane model, we demonstrate that the topological phase transition can be well characterized by the defined topological invariant of the transition point, which reflects the change of topological invariants of topologically different phases across the phase transition point.
For DDD-dimensional weakly interacting topological insulators in certain symmetry classes, the topological invariant can be calculated from a DDD- or (D+1)(D+1)(D+1)-dimensional integration over a certain curvature function that is expressed in terms of single-particle Green's functions. Based on the divergence of curvature function at the topological phase transition, we demonstrate how a renormalization group approach circumvents these integrations and reduces the necessary calculation to that for the Green's function alone, rendering a numerically efficient tool to identify topological phase transitions in a large parameter space. The method further unveils a number of statistical aspects related to the quantum criticality in weakly interacting topological insulators, including correlation function, critical exponents, and scaling laws, that can be used to characterize the topological phase transitions driven by either interacting or noninteracting parameters. We use 1D class BDI and 2D class A Dirac models with electron-electron and electron-phonon interactions to demonstrate these principles and find that interactions may change the critical exponents of the topological insulators.