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arXiv:2105.09345 · Physical review. B./Physical review. B

Analytical approach for the Mott transition in the Kane-Mele-Hubbard model

Joel Hutchinson, Philipp W. Klein, Karyn Le Hur

Published Aug 13, 2021• 13 citations
arXivHTMLPDF

Abstract

The description of interactions in strongly correlated topological phases of matter remains a challenge. Here, we develop a stochastic functional approach for interacting topological insulators including both charge and spin channels. We find that the Mott transition of the Kane-Mele-Hubbard model may be described by the variational principle with one equation. We present different views of this equation from the electron Green's function, the free-energy, and the Hellmann-Feynman theorem. In particular, we show the stability of the transition line towards fluctuations, in good agreement with numerical results. The band gap remains finite at the transition and the Mott phase is characterized by antiferromagnetism in the $x\text{\ensuremath{-}}y$ plane. The interacting topological phase is described through a ${\mathbb{Z}}_{2}$ number related to helical edge modes. Our results then show that improving stochastic approaches can give further insight on the understanding of interacting phases of matter.

DOI
10.1103/physrevb.104.075120

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The description of interactions in strongly correlated topological phases of matter remains a challenge. Here, we develop a stochastic functional approach for interacting topological insulators including both charge and spin channels. We find that the Mott transition of the Kane-Mele-Hubbard model may be described by the variational principle with one equation. We present different views of this equation from the electron Green's function, the free-energy, and the Hellmann-Feynman theorem. In particular, we show the stability of the transition line towards fluctuations, in good agreement with numerical results. The band gap remains finite at the transition and the Mott phase is characterized by antiferromagnetism in the x\ensuremath-yx\text{\ensuremath{-}}yx\ensuremath-y plane. The interacting topological phase is described through a Z2{\mathbb{Z}}_{2}Z2​ number related to helical edge modes. Our results then show that improving stochastic approaches can give further insight on the understanding of interacting phases of matter.
The dimerized Kane-Mele model with/without strong interaction is studied using analytical methods. The boundary of the topological phase transition of the model without strong interaction is obtained. Our results show that the occurrence of the transition only depends on dimerized parameter [Formula: see text]. From the one-particle spectrum, we obtain the completed phase diagram including the quantum spin Hall state and the topologically trivial insulator. Then, using different mean field methods, we investigate the Mott transition and the magnetic transition of the strongly correlated dimerized Kane-Mele model. In the region between the two transitions, the topological Mott insulator with characteristics of Mott insulators and topological phases may be the most interesting phase. In this work, the effects of hopping anisotropy and Hubbard interaction U on the boundaries of the two transitions are observed in detail. The completed phase diagram of the dimerized Kane-Mele-Hubbard model is also obtained in this work. Quantum fluctuations have extremely important influences on a quantum system. However, investigations are under the framework of mean field treatment in this work and the effects of fluctuations in this model will be discussed in the future.
Topological insulators present a bulk gap, but allow for dissipationless spin transport along the edges. These exotic states are characterized by the Z2 topological invariant and are protected by time-reversal symmetry. The Kane–Mele model is one model to realize this topological class in two dimensions, also called the quantum spin Hall state. In this brief review article, we provide a pedagogical introduction to the influence of correlation effects in the quantum spin Hall states, with special focus on the half-filled Kane–Mele–Hubbard model, solved by means of unbiased determinant quantum Monte Carlo (QMC) simulations. We explain the idea of identifying the topological insulator via π-flux insertion, the Z2 invariant and the associated behavior of the zero-frequency Green's function, as well as the spin Chern number in parameter-driven topological phase transitions. The examples considered are two descendants of the Kane–Mele–Hubbard model, the generalized and dimerized Kane–Mele–Hubbard model. From the Z2 index, spin Chern numbers and the Green's function behavior, one can observe that correlation effects induce shifts of the topological phase boundaries. Although the implementation of these topological quantities has been successfully employed in QMC simulations to describe the topological phase transition, we also point out their limitations as well as suggest possible future directions in using numerical methods to characterize topological properties of strongly correlated condensed matter systems.
We investigate the magnetic and conduction properties of Kane-Mele Hubbard model in quasi one-dimensional honeycomb ribbon systems at half-filling by varying the strength of both spin-orbit interaction and on-site Coulomb correlation term. We use the numerical many-body configuration interaction (CI) method to investigate the dispersions of charge and spin gaps along with the momentum resolved spin-density profile over the full Brillouin zone. While the spin sector retains its topological nature at all values of spin-orbit coupling and Hubbard term, we report a new signature of the topological phase transition in the charge sector. This phase transition from a topological band insulating phase to a antiferromagnetically ordered Mott insulating phase is characterized by a shift of the many-body charge gap minima from Brillouin zone boundary to Dirac point. Our results provide a better understanding of the shifting of the gap-closing point in the momentum space which was reported in an earlier mean-field study of the same model and suggests an alternative numerical route to detect topological phase transition in strongly-correlated systems in terms of their momentum space behaviors.
We study a generalized Kane-Mele-Hubbard model with third-neighbor hopping, an interacting two-dimensional model with a topological phase transition as a function of third-neighbor hopping, by means of the determinant projector quantum Monte Carlo method. This technique is essentially numerically exact on models without a fermion sign problem, such as the one we consider. We determine the interaction dependence of the Z2{Z}_{2}Z2​ topological insulator/trivial insulator phase boundary by calculating the Z2{Z}_{2}Z2​ invariants directly from the single-particle Green's function. The interactions push the phase boundary to larger values of third-neighbor hopping, thus, stabilizing the topological phase. The observation of boundary shifting entirely stems from quantum fluctuations. We also identify qualitative features of the single-particle Green's function which are computationally useful in numerical searches for topological phase transitions without the need to compute the full topological invariant.
Topological insulators are materials that behave as insulators in the bulk, but present conducting edge states.Despite the growing interest in such materials due to their possible applications on spintronic devices and quantum computing, several open questions still remain regarding the roles of both electronic interactions and dimensionality in these systems.The goal of this work is to tackle some of these questions.To do so, we study a quasi-1D system presenting a topologically non-trivial phase and electronic interactions: the so-called Kane-Mele-Hubbard ladders.Here, we apply the density matrix renormalization group (DMRG) algorithm in the tensor network formalism in order to calculate some physical properties of this system.The DMRG calculations allow us to reach the system's ground state and enable the efficient computation of observables and of the entanglement entropy.The Kane-Mele-Hubbard model is an interacting version of the Kane-Mele model, which displays topologically non-trivial phases.The interest on this system is due to the combination of effects caused by the spin-orbit coupling and the electronic interactions.For the non-interacting case we have a topological insulator, whereas in the highly interacting picture we have a Mott insulator with localized electrons.We explore the phase transition between these regimes, looking at the conditions needed for the appearance of spin-polarized edge states.We also propose the entanglement entropy between two halves of the system as an indicator of the topological phase transition, as it presents a particular pattern when the phase transition takes place.
We study the two-dimensional Kane-Mele-Hubbard model at half filling by means of quantum Monte Carlo simulations. We present a refined phase boundary for the quantum spin liquid. The topological insulator at finite Hubbard interaction strength is adiabatically connected to the groundstate of the Kane-Mele model. In the presence of spin-orbit coupling, magnetic order at large Hubbard UUU is restricted to the transverse direction. The transition from the topological band insulator to the antiferromagnetic Mott insulator is in the universality class of the three-dimensional XY model. The numerical data suggest that the spin liquid to topological insulator and spin liquid to Mott insulator transitions are both continuous.