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arXiv:1202.3203 · Physical Review B

Correlation effects in two-dimensional topological insulators

Yasuhiro Tada, Robert Peters, Masaki Oshikawa, Akihisa Koga, N. Kawakami, Satoshi Fujimoto

Published Apr 23, 2012• 73 citations
arXivHTMLPDF

Abstract

We investigate correlation effects in two-dimensional topological insulators (TI). In the first part, we discuss finite size effects for interacting systems of different sizes in a ribbon geometry. For large systems, there are two pairs of well separated massless modes on both edges. For these systems, we analyze the finite size effects using a standard bosonization approach. For small systems, where the edge states are massive Dirac fermions, we use the inhomogeneous dynamical mean-field theory (DMFT) combined with iterative perturbation theory as an impurity solver to study interaction effects. We show that the finite size gap in the edge states is renormalized for weak interactions, which is consistent with a Fermi-liquid picture for small size TIs. In the second part, we investigate phase transitions in finite size TIs at zero temperature focusing on the effects of possible interedge umklapp scattering for the edge states within the inhomogeneous DMFT using the numerical renormalization group. We show that correlation effects are effectively stronger near the edge sites because the coordination number is smaller than in the bulk. Therefore the localization of the edge states around the edge sites, which is a fundamental property in TIs, is weakened for strong coupling strengths. However, we find no signs for ``edge Mott insulating states'' and the system stays in the topological insulating state, which is adiabatically connected to the noninteracting state for all interaction strengths smaller than the critical value. Increasing the interaction further, a nearly homogeneous Mott insulating state is stabilized.

DOI
10.1103/physrevb.85.165138

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•Mar 25, 2019•0 citations
We investigate correlation effects in two-dimensional topological insulators (TI). In the first part, we discuss finite size effects for interacting systems of different sizes in a ribbon geometry. For large systems, there are two pairs of well separated massless modes on both edges. For these systems, we analyze the finite size effects using a standard bosonization approach. For small systems, where the edge states are massive Dirac fermions, we use the inhomogeneous dynamical mean-field theory (DMFT) combined with iterative perturbation theory as an impurity solver to study interaction effects. We show that the finite size gap in the edge states is renormalized for weak interactions, which is consistent with a Fermi-liquid picture for small size TIs. In the second part, we investigate phase transitions in finite size TIs at zero temperature focusing on the effects of possible interedge umklapp scattering for the edge states within the inhomogeneous DMFT using the numerical renormalization group. We show that correlation effects are effectively stronger near the edge sites because the coordination number is smaller than in the bulk. Therefore the localization of the edge states around the edge sites, which is a fundamental property in TIs, is weakened for strong coupling strengths. However, we find no signs for ``edge Mott insulating states'' and the system stays in the topological insulating state, which is adiabatically connected to the noninteracting state for all interaction strengths smaller than the critical value. Increasing the interaction further, a nearly homogeneous Mott insulating state is stabilized.
We address how the nature of linearly dispersing edge states of two-dimensional (2D) topological insulators evolves with increasing electron-electron correlation engendered by a Hubbard-like on-site repulsion UUU in finite ribbons of two models of topological band insulators. Using an inhomogeneous cluster slave-rotor mean-field method developed here, we show that electronic correlations drive the topologically nontrivial phase into a Mott insulating phase via two different routes. In a synchronous transition, the entire ribbon attains a Mott insulating state at one critical UUU that depends weakly on the width of the ribbon. In the second, asynchronous route, Mott localization first occurs on the edge layers at a smaller critical value of electronic interaction, which then propagates into the bulk as UUU is further increased until all layers of the ribbon become Mott localized. We show that the kind of Mott transition that takes place is determined by certain properties of the linearly dispersing edge states which characterize the topological resilience to Mott localization.
One of the remarkable interaction effects on topological insulators is the reduction of topological classification in free-fermion systems. We address this issue in a bilayer honeycomb lattice model by taking into account temperature effects on the reduction. Our analysis, based on the real-space dynamical mean-field theory, elucidates the following results. (i) Even when the reduction occurs, the winding number defined by the Green's function can take a nontrivial value at zero temperature. (ii) The winding number taking the nontrivial value becomes consistent with the absence of gapless edge modes due to Mott behaviors emerging only at the edges. (iii) Temperature effects can restore the gapless edge modes, provided that the energy scale of interactions is smaller than the bulk gap. In addition, we observe the topological edge Mott behavior only in some finite-temperature region.
The density of states, the dynamic (optical) conductivity, and the phase diagram of the paramagnetic two-dimensional Anderson-Hubbard model with strong correlations and disorder are analyzed within the generalized dynamical mean field theory (DMFT + Σ approximation). Strong correlations are accounted by the DMFT, while disorder is taken into account via the appropriate generalization of the self-consistent theory of localization. We consider the two-dimensional system with the rectangular “bare” density of states (DOS). The DMFT effective single-impurity problem is solved by numerical renormalization group (NRG). The “correlated metal,” Mott insulator, and correlated Anderson insulator phases are identified from the evolution of the density of states, optical conductivity, and localization length, demonstrating both Mott-Hubbard and Anderson metal-insulator transitions in two-dimensional systems of finite size, allowing us to construct the complete zero-temperature phase diagram of the paramagnetic Anderson-Hubbard model. The localization length in our approximation is practically independent of the strength of Hubbard correlations. But the divergence of the localization length in a finite-size two-dimensional system at small disorder signifies the existence of an effective Anderson transition.
In this article, we study the influence of long-range Coulomb interaction on three-dimensional second-order topological insulator (TI) by renormalization group theory. We find that both the analysis method and conclusions in the recent Letter Phys. Rev. Lett. {\bf 127}, 176601 (2021) are unreliable. There are two problems in this Letter. Firstly, the characteristic described by the RG flows m→∞m\rightarrow\inftym→∞ and D→0D\rightarrow0D→0 can not be used as the criterion for transition from second-order TI to TI, since this characteristic could be essentially not induced by Coulomb interaction but only results from the trivial power counting contribution of fermion action. Indeed, this characteristic is satisfied even for free second-order TI. Second, the flow of BBB is not paid attention, which is very important and should be seriously studied. In this article, we analyze carefully the corrections for the flows of the model parameters induced by Coulomb interaction. We find that the sign of mmm changes but the sign of BBB holds if the initial Coulomb strength is large enough, while the sign of mmm holds but the sign of BBB changes if the initial Coulomb strength takes small values. These results indicate that second-order TI is unstable to trivial band insulator not only under strong Coulomb interaction but also under weak Coulomb interaction. We also study the effects of disorder scattering in second-order TI by renormalization group theory. According to the criterion in Phys. Rev. Lett. {\bf 127}, 176601 (2021), weak disorder drives second-order TI to TI. However, we find that second-order TI is robust against weak disorder, since weak disorder does not give qualitative modification for second-order TI. This result is consistent with recent studies based on other methods. Interplay of Coulomb interaction and disorder in second-order TI is also investigated.
Topological insulators (TIs) are an exciting discovery because of their robustness against disorder and interactions. Recently, second-order TIs have been attracting increasing attention, because they host topologically protected 1D hinge states in 3D or 0D corner states in 2D. A significantly critical issue is whether the second-order TIs also survive interactions, but it is still unexplored. We study the effects of weak Coulomb interactions on a 3D second-order TI, with the help of renormalization-group calculations. We find that the 3D second-order TIs are always unstable, suffering from two types of topological phase transitions. One is from second-order TI to TI, the other is to normal insulator. The first type is accompanied by emergent time-reversal and inversion symmetries and has a dynamical critical exponent κ=1. The second type does not have the emergent symmetries but has nonuniversal dynamical critical exponents κ<1. Our results may inspire more inspections on the stability of higher-order topological states of matter and related novel quantum criticalities.
The Lieb lattice possesses three bands and with intrinsic spin orbit coupling λ\lambdaλ, supports topologically non-trivial band insulating phases. At half filling the lower band is fully filled, while the upper band is empty. The chemical potential lies in the flat band (FB) located at the middle of the spectrum, thereby stabilizing a flat band insulator. At this filling, we introduce on-site Hubbard interaction UUU on all sites. Within a slave rotor mean field theory we show that, in spite of the singular effect of interaction on the FB, the three bands remain stable up to a fairly large critical correlation strength (UcritU_{crit}Ucrit​), creating a correlated flat band insulator. Beyond UcritU_{crit}Ucrit​, there is a sudden transition to a Mott insulating state, where the FB is destroyed due to complete transfer of spectral weight from the FB to the upper and lower bands. We show that all the correlation driven insulating phases host edge modes with linearly dispersing bands along with a FB passing through the Dirac point, exhibiting that the topological nature of the bulk band structure remains intact in presence of strong correlation. Furthermore, in the limiting case of UUU introduced only on one sublattice where λ=0\lambda=0λ=0, we show that the Lieb lattice can support mixed edge modes containing contributions from both spinons and electrons, in contrast to purely spinon edge modes arising in the topological Mott insulator.