Topological insulators have become one of the most active research areas in condensed matter physics. This article reviews progress on the topic of electronic correlations effects in the two-dimensional case, with a focus on systems with intrinsic sp
in-orbit coupling and numerical results. Topics addressed include an introduction to the noninteracting case, an overview of theoretical models, correlated topological band insulators, interaction-driven phase transitions, topological Mott insulators and fractional topological states, correlation effects on helical edge states, and topological invariants of interacting systems.
The adiabatic insertion of a pi flux into a quantum spin Hall insulator gives rise to localized spin and charge fluxon states. We demonstrate that pi fluxes can be used in exact quantum Monte Carlo simulations to identify a correlated Z_2 topological
insulator using the example of the Kane-Mele-Hubbard model. In the presence of repulsive interactions, a pi flux gives rise to a Kramers doublet of spinon states with a Curie law signature in the magnetic susceptibility. Electronic correlations also provide a bosonic mode of magnetic excitons with tunable energy that act as exchange particles and mediate a dynamical interaction of adjustable range and strength between spinons. pi fluxes can therefore be used to build models of interacting spins. This idea is applied to a three-spin ring and to one-dimensional spin chains. Due to the freedom to create almost arbitrary spin lattices, correlated topological insulators with pi fluxes represent a novel kind of quantum simulator potentially useful for numerical simulations and experiments.
We investigate electronic correlation effects on edge states of quantum spin-Hall insulators within the Kane-Mele-Hubbard model by means of quantum Monte Carlo simulations. Given the U(1) spin symmetry and time-reversal invariance, the low-energy the
ory is the helical Tomanaga-Luttinger model, with forward scattering only. For weak to intermediate interactions, this model correctly describes equal-time spin and charge correlations, including their doping dependence. As apparent from the Drude weight, bulk states become relevant in the presence of electron-electron interactions, rendering the forward-scattering model incomplete. Strong correlations give rise to slowly decaying transverse spin fluctuations, and inelastic spin-flip scattering strongly modifies the single-particle spectrum, leading to graphene-like edge state signatures. The helical Tomanaga-Luttinger model is completely valid only asymptotically in the weak-coupling limit.
We consider models of heavy fermions in the strong coupling or local moment limit and include phonon degrees of freedom on the conduction electrons. Due to the large mass or low coherence temperature of the heavy fermion state, it is shown that such
a regime is dominated by vertex corrections which leads to the complete failure of the Migdal theorem. Even at weak electron-phonon couplings, binding of the conduction electrons competes with the Kondo effect and substantially reduces the coherence temperature, ultimately leading to the Kondo breakdown. Those results are obtained using a combination of the slave boson method and Migdal-Eliashberg approximation as well as the dynamical mean-field theory approximation.
We use a recently developed extension of the weak coupling diagrammatic determinantal quantum Monte Carlo method to investigate the spin, charge and single particle spectral functions of the one-dimensional quarter-filled Holstein model with phonon f
requency $omega_0 = 0.1 t$. As a function of the dimensionless electron-phonon coupling we observe a transition from a Luttinger to a Luther-Emery liquid with dominant $2k_f$ charge fluctuations. Emphasis is placed on the temperature dependence of the single particle spectral function. At high temperatures and in both phases it is well accounted for within a self-consistent Born approximation. In the low temperature Luttinger liquid phase we observe features which compare favorably with a bosonization approach retaining only forward scattering. In the Luther-Emery phase, the spectral function at low temperatures shows a quasiparticle gap which matches half the spin gap whereas at temperatures above which this quasiparticle gap closes, characteristic features of the Luttinger liquid model are apparent. Our results are based on lattice simulations on chains up to L=20 for two-particle properties and on CDMFT calculations with clusters up to 12 sites for the single-particle spectral function.
Gaussian Quantum Monte Carlo (GQMC) is a stochastic phase space method for fermions with positive weights. In the example of the Hubbard model close to half filling it fails to reproduce all the symmetries of the ground state leading to systematic er
rors at low temperatures. In a previous work [Phys. Rev. B {bf 72}, 224518 (2005)] we proposed to restore the broken symmetries by projecting the density matrix obtained from the simulation onto the ground state symmetry sector. For ground state properties, the accuracy of this method depends on a {it large overlap} between the GQMC and exact density matrices. Thus, the method is not rigorously exact. We present the limits of the approach by a systematic study of the method for 2 and 3 leg Hubbard ladders for different fillings and on-site repulsion strengths. We show several indications that the systematic errors stem from non-vanishing boundary terms in the partial integration step in the derivation of the Fokker-Planck equation. Checking for spiking trajectories and slow decaying probability distributions provides an important test of the reliability of the results. Possible solutions to avoid boundary terms are discussed. Furthermore we compare results obtained from two different sampling methods: Reconfiguration of walkers and the Metropolis algorithm.
We investigate the charge- and spin dynamical structure factors for the 2D one-band Hubbard model in the strong coupling regime within an extension of the Dynamical Cluster Approximation (DCA) to two-particle response functions. The full irreducible
two-particle vertex with three momenta and frequencies is approximated by an effective vertex dependent on the momentum and frequency of the spin/charge excitation. In the spirit of the DCA, the effective vertex is calculated with quantum Monte Carlo methods on a finite cluster. On the basis of a comparison with high temperature auxiliary field quantum Monte Carlo data we show that near and beyond optimal doping, our results provide a consistent overall picture of the interplay between charge, spin and single-particle excitations.
