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arXiv:1405.6728 · Physical Review B

Efficient DMFT impurity solver using real-time dynamics with matrix product states

Martin Ganahl, Markus Aichhorn, Hans Gerd Evertz, Patrik Thunström, Karsten Held, Frank Verstraete

Published Oct 20, 2015• 88 citations

Abstract

We propose to calculate spectral functions of quantum impurity models using the time evolving block decimation (TEBD) for matrix product states. The resolution of the spectral function is improved by a so-called linear prediction approach. We apply the method as an impurity solver within the dynamical mean-field theory (DMFT) for the single- and two-band Hubbard model on the Bethe lattice. For the single-band model, we observe sharp features at the inner edges of the Hubbard bands. A finite-size scaling shows that they remain present in the thermodynamic limit. We analyze the real time-dependence of the double occupation after adding a single electron and observe oscillations at the same energy as the sharp feature in the Hubbard band, indicating a long-lived coherent superposition of states that correspond to the Kondo peak and the side peaks. For a two-band Hubbard model, we observe an even richer structure in the Hubbard bands, which cannot be related to a multiplet structure of the impurity, in addition to sharp excitations at the band edges of a type similar to the single-band case.

DOI: 10.1103/physrevb.92.155132

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We improve a recently developed expansion technique for calculating real-frequency spectral functions of any one-dimensional model with short-range interactions, by postprocessing computed Chebyshev moments with linear prediction. This can be achieved at virtually no cost, and in sharp contrast to existing methods based on the dampening of the moments, improves the spectral resolution rather than lowering it. We validate the method for the exactly solvable resonating level model and the single impurity Anderson model. It is capable of resolving sharp Kondo resonances, as well as peaks within the Hubbard bands when employed as an impurity solver for dynamical mean-field theory. Our method works at zero temperature and allows for arbitrary discretization of the bath spectrum. It achieves similar precision as the dynamical density matrix renormalization group, at lower cost. We also propose an alternative expansion, of $\mathbb{1}\ensuremath{-}exp(\ensuremath{-}\ensuremath{\tau}H)$ instead of the usual

H

, which opens the possibility of using established methods for the time evolution of matrix product states to calculate the spectral functions directly.

We implement the recently developed influence functional matrix product states approach as impurity solver in equilibrium and nonequilibrium dynamical mean field theory (DMFT) calculations of the single-band Hubbard model. The method yields numerically exact descriptions of metallic states without sharp spectral features, at a moderate numerical cost. Systems with narrow quasiparticle or spin-polaron peaks, as well as low-temperature Mott insulators provide more challenges, since these simulations require long time contours or high bond dimensions. A promising field of application is the DMFT simulation of nonequilibrium steady states, which we demonstrate with results for photo-doped Mott systems with long-lived doublon and holon populations.

The Dynamical Mean Field Theory (DMFT) is a powerful tool for calculating highly correlated systems (both bosonic and fermionic) in a state of thermodynamic equilibrium. However, in the case of non-equilibrium states, the method has significant limitations that do not allow obtaining correct results. The stumbling block here is the impurity solver: a method for calculating the dynamics of an open system. In this work we present the prototype of a universal impurity solver for the DMFT method which can solve both equilibrium and non-equilibrium problems. We analyse spectral functions of Bose-Hubbard model on the Bethe lattice in different regimes and show that our solver gives correct and physical results.

The dynamical mean-field theory (DMFT) has become a standard technique for the study of strongly correlated models and materials overcoming some of the limitations of density functional approaches based on local approximations. An important step in this method involves the calculation of response functions of a multiorbital impurity problem which is related to the original model. Recently there has been considerable progress in the development of techniques based on the density matrix renormalization group (DMRG) and related matrix product states (MPS) implying a substantial improvement to previous methods. In this paper we review some of the standard algorithms and compare them to the newly developed techniques, showing examples for the particular case of the half-filled two-band Hubbard model.

The dynamical mean-field theory (DMFT) has become a standard technique for the study of strongly correlated models and materials overcoming some of the limitations of density functional approaches based on local approximations. An important step in this method involves the calculation of response functions of a multiorbital impurity problem which is related to the original model. Recently there has been considerable progress in the development of techniques based on the density matrix renormalization group (DMRG) and related matrix product states (MPS) implying a substantial improvement to previous methods. In this paper we review some of the standard algorithms and compare them to the newly developed techniques, showing examples for the particular case of the half-filled two-band Hubbard model.

We introduce a neural network impurity solver for real-frequency DMFT that employs a multihead cross-attention mechanism to map hybridization functions to spectral functions, conditioned on impurity parameters. Trained on high-quality MPS data from complex contour time evolution and incorporating derivative constraints with respect to the complex-time angle, our model achieves smooth generalization to the real-frequency axis. Benchmarking on the single-band Hubbard model for the Bethe lattice demonstrates quantitative accuracy across metallic, strongly correlated, and insulating regimes.