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arXiv:1609.04708 · Physical review. B./Physical review. B

Nonequilibrium dynamical cluster approximation study of the Falicov-Kimball model

Andreas J. Herrmann, Naoto Tsuji, Martin Eckstein, Philipp Werner

Published Dec 6, 2016• 13 citations
arXivHTMLPDF

Abstract

We use a nonequilibrium implementation of the dynamical cluster approximation (DCA) to study the effect of short-range correlations on the dynamics of the two-dimensional Falicov-Kimball model after an interaction quench. As in the case of single-site dynamical mean-field theory, thermalization is absent in DCA simulations, and for quenches across the metal-insulator boundary, nearest-neighbor charge correlations in the nonthermal steady state are found to be larger than in the thermal state with identical energy. We investigate to what extent it is possible to define an effective temperature of the trapped state after a quench. Based on the ratio between the lesser and retarded Green's function, we conclude that a roughly thermal distribution is reached within the energy intervals corresponding to the momentum-patch dependent subbands of the spectral function. The effectively different chemical potentials of these distributions, however, lead to a very hot, or even negative, effective temperature in the energy intervals between these subbands.

DOI
10.1103/physrevb.94.245114

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We use a nonequilibrium implementation of the dynamical cluster approximation (DCA) to study the effect of short-range correlations on the dynamics of the two-dimensional Falicov-Kimball model after an interaction quench. As in the case of single-site dynamical mean-field theory, thermalization is absent in DCA simulations, and for quenches across the metal-insulator boundary, nearest-neighbor charge correlations in the nonthermal steady state are found to be larger than in the thermal state with identical energy. We investigate to what extent it is possible to define an effective temperature of the trapped state after a quench. Based on the ratio between the lesser and retarded Green's function, we conclude that a roughly thermal distribution is reached within the energy intervals corresponding to the momentum-patch dependent subbands of the spectral function. The effectively different chemical potentials of these distributions, however, lead to a very hot, or even negative, effective temperature in the energy intervals between these subbands.
We recently introduced the dynamical cluster approximation (DCA), a technique that includes short-ranged dynamical correlations in addition to the local dynamics of the dynamical mean-field approximation while preserving causality. The technique is based on an iterative self-consistency scheme on a finite-size periodic cluster. The dynamical mean-field approximation (exact result) is obtained by taking the cluster to a single site (the thermodynamic limit). Here, we provide details of our method, explicitly show that it is causal, systematic, $\ensuremath{\Phi}$ derivable, and that it becomes conserving as the cluster size increases. We demonstrate the DCA by applying it to a quantum Monte Carlo and exact enumeration study of the two-dimensional Falicov-Kimball model. The resulting spectral functions preserve causality, and the spectra and the charge-density-wave transition temperature converge quickly and systematically to the thermodynamic limit as the cluster size increases.
We present the exact solution of the Falicov-Kimball model after a sudden change of its interaction parameter using nonequilibrium dynamical mean-field theory. For different interaction quenches between the homogeneous metallic and insulating phases the system relaxes to a nonthermal steady state on time scales on the order of variant Planck's over 2pi/bandwidth, showing collapse and revival with an approximate period of h/interaction if the interaction is large. We discuss the reasons for this behavior and provide a statistical description of the final steady state by means of generalized Gibbs ensembles.
We study dynamical properties of the one- and two-dimensional Falicov-Kimball model using lattice Monte Carlo simulations. In particular, we calculate the spreading of charge correlations in the equilibrium model and after an interaction quench. The results show a reduction of the light-cone velocity with interaction strength at low temperature, while the phase velocity increases. At higher temperature, the initial spreading is determined by the Fermi velocity of the noninteracting system and the maximum range of the correlations decreases with increasing interaction strength. Charge order correlations in the disorder potential enhance the range of the correlations. We also use the numerically exact lattice Monte Carlo results to benchmark the accuracy of equilibrium and nonequilibrium dynamical cluster approximation calculations. It is shown that the bias introduced by the mapping to a periodized cluster is substantial, and that from a numerical point of view, it is more efficient to simulate the lattice model directly.
The dynamical cluster approximation (DCA) is a quantum cluster extension to the single-site dynamical mean-field theory that incorporates spatially nonlocal dynamic correlations systematically and nonperturbatively. The DCA+{\mathrm{DCA}}^{+}DCA+ algorithm addresses the cluster shape dependence of the DCA and improves the convergence with cluster size by introducing a lattice self-energy with continuous momentum dependence. However, we show that the DCA+{\mathrm{DCA}}^{+}DCA+ algorithm is plagued by a fundamental problem when its self-consistency equations are formulated using the bare Green's function of the cluster. This problem is most severe in the strongly correlated regime at low doping, where the DCA+{\mathrm{DCA}}^{+}DCA+ self-energy becomes overly metallic and local, and persists to cluster sizes where the standard DCA has long converged. In view of the failure of the DCA+{\mathrm{DCA}}^{+}DCA+ algorithm, we propose to complement DCA simulations with a post-interpolation procedure for single-particle and two-particle correlation functions to preserve continuous momentum dependence and the associated benefits in the DCA. We demonstrate the effectiveness of this practical approach with results for the half-filled and hole-doped two-dimensional Hubbard model.
We study the effects of correlated hopping in the two-dimensional (2D) Falicov-Kimball model by means of an extension of the dynamical mean-field approximation (DMFA). The extension is based on the projection technique and the DMFA, in which nonlocal correlations are taken into account through static quantities of a relevant subspace. The effect of the correlated hopping is to introduce nonlocal self-energy components which remain even at D → ∞ . We show that the sum rules of the spectral function and its moments are preserved. The spectral function obtained reveals significant nonlocal contributions which are absent in the DMFA.
The Hubbard model (HM) and the Falicov-Kimball model (FKM), which are two standard models of strongly correlated electrons, provide an interesting benchmark for the physics of locally correlated systems. We study the intermediate model, mass-imbalanced Hubbard model at half-filling on a 2D square lattice which connects continuously the HM to the FKM.We employ dynamical mean field theory (DMFT) and dynamical cluster approximation (DCA) using continuous-time auxiliary field method (CT-AUX) as self-consistent impurity solver to study the single particle spectral function and self-energy allowing us to map-out the phase diagram of the system. Solutions to the mass-imbalanced model cannot enforce paramagnetic self-consistency and therefore at low temperature result in paramagnetic (PM) to antiferromagnetic (AFM) transition. In addition, in the PM state and at finite temperature, we observed a spin dependent crossover from Fermi liquid (FL) regime to non-Fermi liquid (nFL) regime in which the FL regime is destroyed, monotonically, induced not only by the interaction but also by the mass imbalance. We present the results for small cluster DCA and explore its dependency on cluster size as well.