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Papers in Physics

An efficient impurity-solver for the dynamical mean field theory algorithm

Y. Núñez-Fernández, K. Hallberg

Published Aug 1, 2017• 0 citations
Publisher

Abstract

One of the most reliable and widely used methods to calculate electronic structure of strongly correlated models is the Dynamical Mean Field Theory (DMFT) developed over two decades ago. It is a non-perturbative algorithm which, in its simplest version, takes into account strong local interactions by mapping the original lattice model on to a single impurity model. This model has to be solved using some many-body technique. Several methods have been used, the most reliable and promising of which is the Density Matrix Renormalization technique. In this paper, we present an optimized implementation of this method based on using the star geometry and correction-vector algorithms to solve the related impurity Hamiltonian and obtain dynamical properties on the real frequency axis. We show results for the half-filled and doped one-band Hubbard models on a square lattice. Received: 31 March 2017, Accepted: 6 June 2017; Edited by: D. Domínguez; Reviewed by: A. Feiguin, Northeastern University, Boston, United States; DOI: http://dx.doi.org/10.4279/PIP.090005 Cite as: Y Núñez Fernández, K Hallberg, Papers in Physics 9, 090005 (2017) This paper, by Y Núñez Fernández, K Hallberg, is licensed under the Creative Commons Attribution License 3.0.

DOI
10.4279/pip.090005

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One of the most reliable and widely used methods to calculate electronic structure of strongly correlated models is the Dynamical Mean Field Theory (DMFT) developed over two decades ago. It is a non-perturbative algorithm which, in its simplest version, takes into account strong local interactions by mapping the original lattice model on to a single impurity model. This model has to be solved using some many-body technique. Several methods have been used, the most reliable and promising of which is the Density Matrix Renormalization technique. In this paper, we present an optimized implementation of this method based on using the star geometry and correction-vector algorithms to solve the related impurity Hamiltonian and obtain dynamical properties on the real frequency axis. We show results for the half-filled and doped one-band Hubbard models on a square lattice. Received: 31 March 2017, Accepted: 6 June 2017; Edited by: D. Domínguez; Reviewed by: A. Feiguin, Northeastern University, Boston, United States; DOI: http://dx.doi.org/10.4279/PIP.090005 Cite as: Y Núñez Fernández, K Hallberg, Papers in Physics 9, 090005 (2017) This paper, by Y Núñez Fernández, K Hallberg, is licensed under the Creative Commons Attribution License 3.0.
We implement an efficient numerical method to calculate response functions of complex impurities based on the Density Matrix Renormalization Group (DMRG) and use it as the impurity-solver of the Dynamical Mean Field Theory (DMFT). This method uses the correction vector to obtain precise Green's functions on the real frequency axis at zero temperature. By using a self-consistent bath configuration with very low entanglement, we take full advantage of the DMRG to calculate dynamical response functions paving the way to treat large effective impurities such as those corresponding to multi-orbital interacting models and multi-site or multi-momenta clusters. This method leads to reliable calculations of non-local self energies at arbitrary dopings and interactions and at any energy scale.
We implement an efficient numerical method to calculate response functions of complex impurities based on the Density Matrix Renormalization Group (DMRG) and use it as the impurity-solver of the Dynamical Mean Field Theory (DMFT). This method uses the correction vector to obtain precise Green's functions on the real frequency axis at zero temperature. By using a self-consistent bath configuration with very low entanglement, we take full advantage of the DMRG to calculate dynamical response functions paving the way to treat large effective impurities such as those corresponding to multi-orbital interacting models and multi-site or multi-momenta clusters. This method leads to reliable calculations of non-local self energies at arbitrary dopings and interactions and at any energy scale.
Numerical studies on strongly correlated fermionic systems are very complicated and still provide essential problems. The main reason is the exponential growth of the un- derlying Hilbert state space with the system size and the fermionic sign problem for Monte Carlo studies. Among the most widely employed numerical techniques for study- ing two-dimensional quantum many-body systems are cluster extensions of the dynamical mean-field theory (DMFT), e.g. dynamical cluster approximation (DCA). They map an infinitely large multi-dimensional lattice problem to a one-dimensional impurity problem. In 2015 it was shown that the density matrix renormalisation group (DMRG) used as an impurity solver for DMFT (DMFT+DMRG) on the imaginary-frequency axis allows to solve multi-site and multi-band problems extremely fast compared to other solvers. Within this thesis, we further develop this DMRG+DMFT approach to apply the method on real material settings. The step from artificial, completely degenerate multi-band mod- els with simple dispersion relations on a Bethe lattice, studied in 2015, to systems with realistic band structures and lifted degeneracies involves more challenges than originally suspected. In this thesis, we will first recapitulate relevant methods for our approach like matrix prod- uct states, the density matrix renormalisation group and several time evolution methods. In this context we will present several improvements ranging from optimised time evo- lutions to entanglement based optimisations of tensor networks. Second, we will present a very detailed description of the dynamical mean field theory. We will focus on both methodological aspects and implementation details. This chapter is intended to allow other researcher to implement their own DMFT code using DMRG as an impurity solver. Third, we will discuss three different models to show the extent of problems DMRG+ DMFT is able to solve. We will focus on multi-site DCA calculations in the case of the two-dimensional Hubbard model and show that DMRG allows to tackle systems with intermediate interaction strengths at low temperatures, which are unsolvable with other solvers. In the second case, the real material Sr2VO4, we will show the first two-site DCA results for a realistic three-band model. In contrast to assumptions, partly reintroducing the momentum dependence of the self-energy does not improve agreement between exper- imental observations and theoretical results. Finally, we will move on to another realistic three-band model, which describes Sr2RuO4, to show how to deal with the influence of spin-orbit coupling on DMFT. We will present the first low-temperature results for this material and will confirm previous results of simplified model calculations.
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 develop a precise and reliable numerical method for the calculation of electronic properties of the two orbital Hubbard model in a square lattice at half filling, based on the Dynamical Mean Field Theory (DMFT). We use the Density Matrix Renormalization Group (DMRG) as the impurity solver for the DMFT's selfconsistent equations to obtain accurate values for the Green's functions on the real axis. This way, reliable densities of states are obtained that do not need to resort to analytical continuation methods as those using quantum Monte Carlo techniques. Large system sizes can be achieved with increasing accuracy.
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.