Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2311.02623

Towards out of equilibrium impurity solver for the Dynamical Mean Field Theory

Karpov, A., G. Sultanov, Evgeny Polyakov, Alexey N. Rubtsov

Published Nov 5, 2023• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2311.02623

Related papers

Dynamical mean-field theory for the Bose-Hubbard model

Wen-Jun Hu, Ning-Hua Tong

The dynamical mean-field theory (DMFT), which is successful in the study of strongly correlated fermions, was recently extended to boson systems [K. Byczuk and D. Vollhardt, Phys. Rev. B 77, 235106 (2008)]. In this paper, we employ the bosonic DMFT to study the Bose-Hubbard model which describes on-site interacting bosons in a lattice. Using exact diagonalization as the impurity solver, we get the DMFT solutions for the Green's function, the occupation density, as well as the condensate fraction on a Bethe lattice. Various phases are identified: the Mott insulator, the Bose-Einstein condensed (BEC) phase, and the normal phase. At finite temperatures, we obtain the crossover between the Mott-type regime and the normal phase, as well as the BEC-to-normal phase transition. Phase diagrams on the $\ensuremath{\mu}/U\text{\ensuremath{-}}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}/U$ plane and on the $T/U\text{\ensuremath{-}}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}/U$ plane are produced ($\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}$ is the scaled hopping amplitude). We compare our results with the previous ones, and discuss the implication of these results to experiments.

arXiv:0907.2928•Physical Review B•Dec 11, 2009•49 citations

Strong-coupling solver for the quantum impurity model

Xi Dai, Kristjan Haule, Gabriel Kotliar

We propose a fast impurity solver for the general quantum impurity model based on the perturbation theory around the atomic limit, which can be used in combination with the local density approximation (LDA) and the dynamical mean-field theory (DMFT). We benchmark the solver in the two-band Hubbard model within DMFT against quantum Monte Carlo (QMC) and numerical renormalization-group (NRG) results. We find that the solver works very well in the paramagnetic Mott insulator phase. We also apply this impurity solver to the DMFT study of the antiferromagnetic phase transition in the unfrustrated Bethe lattice. The Neel temperature obtained by the fast impurity solver agrees very well with the QMC results in the large Hubbard U limit. The method is a promising tool to be used in combination with the $\mathrm{LDA}+\mathrm{DMFT}$ to study Mott insulators starting from first principles.

arXiv:cond-mat/0412505•Physical Review B•Jul 8, 2005•26 citations

Non-Equilibrium Dynamical Mean Field Theory with Quantum Monte Carlo

Qiaoyuan Dong

This work is motivated by the fact that the investigation of non-equilibrium phenomena in strongly correlated electron systems has developed into one of the most active and exciting branches of condensed matter physics as it provides rich new insights that could not be obtained from the study of equilibrium situations. However, a theoretical description of those phenomena is missing. Therefore, in this thesis, we develop a numerical method that can be used to study two minimal models -- the Hubbard model and the Anderson impurity model with general parameter range and time dependence. We begin by introducing the theoretical framework and the general features of the Hubbard model. We then describe the dynamical mean field theory (DMFT), which was first invented by Georges in 1992. It provides a feasible way to approach strongly correlated electron systems and reduces the complexity of the calculations via a mapping of lattice models onto quantum impurity models subject to a self-consistency condition. We employ the non-equilibrium extension of DMFT and map the Hubbard model to the single impurity Anderson model (SIAM). Since the fundamental component of the DMFT method is a solver of the single impurity Anderson model, we continue with a description of the formalism to study the real-time dynamics of the impurity model staring at its thermal equilibrium state. We utilize the non-equilibrium strong-coupling perturbation theory and derive semi-analytical approximation methods such as the non-crossing approximation (NCA) and the one-crossing approximation (OCA). We then use the Quantum Monte-Carlo method (QMC) as a numerically exact method and present proper measurements of local observables, current and Green's functions. We perform simulations of the current after a quantum quench from equilibrium by rapidly applying a bias voltage in a wide range of initial temperatures. The current exhibits short equilibrium times and saturates upon the decrease of temperature at all times, indicating Kondo behavior both in the transient regime and in the steady state. However, this bare QMC solver suffers from a dynamical sign problem for long time propagations. To overcome the limitations of this bare treatment, we introduce the ``Inchworm algorithm'', based on iteratively reusing the information obtained in previous steps to extend the propagation to longer times and stabilize the calculations. We show that this algorithm greatly reduces the required order for each simulation and re-scales the exponential challenge to quadratic in time. We introduce a method to compute Green's functions, spectral functions, and currents for inchworm Monte Carlo and show how systematic error assessments in real time can be obtained. We illustrate the capabilities of the algorithm with a study of the behavior of quantum impurities after an instantaneous voltage quench from a thermal equilibrium state. We conclude with the applications of the unbiased inchworm impurity solver to DMFT calculations. We employ the methods for a study of the one-band paramagnetic Hubbard model on the Bethe lattice in equilibrium, where the DMFT approximation becomes exact. We begin with a brief introduction of the Mott metal insulator phase diagram. We present the results of both real time Green's functions and spectral functions from our nonequilibrium calculations. We observe the metal-insulator crossover as the on-site interaction is increased and the formation of a quasi-particle peak as the temperature is lowered. We also illustrate the convergence of our algorithms in different aspects.

•Jan 1, 2017•0 citations

Dynamical mean-field theory: from quantum impurity physics to lattice problems

R. Bulla

Since the first investigation of the Hubbard model in the limit of infinite dimensions by Metzner and Vollhardt, dynamical mean-field theory (DMFT) has become a very powerful tool for the investigation of lattice models of correlated electrons. In DMFT the lattice model is mapped on an effective quantum impurity model in a bath which has to be determined self-consistently. This approach led to significant progress in our understanding of typical correlation problems such as the Mott transition; furthermore, the combination of DMFT with ab-initio methods now allows for a realistic treatment of correlated materials. The focus of these lecture notes is on the relation between quantum impurity physics and the physics of lattice models within DMFT. Issues such as the observability of impurity quantum phase transitions in the corresponding lattice models are discussed in detail.

arXiv:cond-mat/0412314•The Philosophical Magazine A Journal of Theoretical Experimental and Applied Physics•Mar 10, 2006•14 citations

Hierarchical equations of motion for an impurity solver in dynamical mean-field theory

Dong Hou, Rulin Wang, Xiao Hong Zheng +3 more

A nonperturbative quantum impurity solver is proposed based on a formally exact hierarchical equations of motion (HEOM) formalism for open quantum systems. It leads to quantitatively accurate evaluation of physical properties of strongly correlated electronic systems, in the framework of dynamical mean-field theory (DMFT). The HEOM method is also numerically convenient to achieve the same level of accuracy as that using the state-of-the-art numerical renormalization group impurity solver at finite temperatures. The practicality of the HEOM+DMFT method is demonstrated by its applications to the Hubbard models with Bethe and hypercubic lattice structures. We investigate the metal-insulator transition phenomena, and address the effects of temperature on the properties of strongly correlated lattice systems.

arXiv:1309.1060•Physical Review B•Jul 29, 2014•49 citations

RECENT APPLICATIONS OF THE DMRG METHOD

K. Hallberg

Since its inception, the DMRG method has been a very powerful tool for the calculation of physical properties of low-dimensional strongly correlated systems. It has been adapted to obtain dynamical properties and to consider finite temperature, time-dependent problems, bosonic degrees of freedom, the treatment of classical problems and non-equilibrium systems, among others. We will briefly review the method and then concentrate on its latest developments, describing some recent successful applications. In particular we will show how the dynamical DMRG can be used together with the Dynamical Mean Field Theory (DMFT) to solve the associated impurity problem in the infinite-dimensional Hubbard model. This method is used to obtain spectral properties of strongly correlated systems. With this algorithm, more complex problems having a larger number of degrees of freedom can be considered and finite-size effects can be minimized.

International Journal of Modern Physics B•Jul 30, 2006•0 citations
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), which is successful in the study of strongly correlated fermions, was recently extended to boson systems [K. Byczuk and D. Vollhardt, Phys. Rev. B 77, 235106 (2008)]. In this paper, we employ the bosonic DMFT to study the Bose-Hubbard model which describes on-site interacting bosons in a lattice. Using exact diagonalization as the impurity solver, we get the DMFT solutions for the Green's function, the occupation density, as well as the condensate fraction on a Bethe lattice. Various phases are identified: the Mott insulator, the Bose-Einstein condensed (BEC) phase, and the normal phase. At finite temperatures, we obtain the crossover between the Mott-type regime and the normal phase, as well as the BEC-to-normal phase transition. Phase diagrams on the $\ensuremath{\mu}/U\text{\ensuremath{-}}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}/U$ plane and on the $T/U\text{\ensuremath{-}}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}/U$ plane are produced ($\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}$ is the scaled hopping amplitude). We compare our results with the previous ones, and discuss the implication of these results to experiments.
We propose a fast impurity solver for the general quantum impurity model based on the perturbation theory around the atomic limit, which can be used in combination with the local density approximation (LDA) and the dynamical mean-field theory (DMFT). We benchmark the solver in the two-band Hubbard model within DMFT against quantum Monte Carlo (QMC) and numerical renormalization-group (NRG) results. We find that the solver works very well in the paramagnetic Mott insulator phase. We also apply this impurity solver to the DMFT study of the antiferromagnetic phase transition in the unfrustrated Bethe lattice. The Neel temperature obtained by the fast impurity solver agrees very well with the QMC results in the large Hubbard U limit. The method is a promising tool to be used in combination with the LDA+DMFT\mathrm{LDA}+\mathrm{DMFT}LDA+DMFT to study Mott insulators starting from first principles.
This work is motivated by the fact that the investigation of non-equilibrium phenomena in strongly correlated electron systems has developed into one of the most active and exciting branches of condensed matter physics as it provides rich new insights that could not be obtained from the study of equilibrium situations. However, a theoretical description of those phenomena is missing. Therefore, in this thesis, we develop a numerical method that can be used to study two minimal models -- the Hubbard model and the Anderson impurity model with general parameter range and time dependence. We begin by introducing the theoretical framework and the general features of the Hubbard model. We then describe the dynamical mean field theory (DMFT), which was first invented by Georges in 1992. It provides a feasible way to approach strongly correlated electron systems and reduces the complexity of the calculations via a mapping of lattice models onto quantum impurity models subject to a self-consistency condition. We employ the non-equilibrium extension of DMFT and map the Hubbard model to the single impurity Anderson model (SIAM). Since the fundamental component of the DMFT method is a solver of the single impurity Anderson model, we continue with a description of the formalism to study the real-time dynamics of the impurity model staring at its thermal equilibrium state. We utilize the non-equilibrium strong-coupling perturbation theory and derive semi-analytical approximation methods such as the non-crossing approximation (NCA) and the one-crossing approximation (OCA). We then use the Quantum Monte-Carlo method (QMC) as a numerically exact method and present proper measurements of local observables, current and Green's functions. We perform simulations of the current after a quantum quench from equilibrium by rapidly applying a bias voltage in a wide range of initial temperatures. The current exhibits short equilibrium times and saturates upon the decrease of temperature at all times, indicating Kondo behavior both in the transient regime and in the steady state. However, this bare QMC solver suffers from a dynamical sign problem for long time propagations. To overcome the limitations of this bare treatment, we introduce the ``Inchworm algorithm'', based on iteratively reusing the information obtained in previous steps to extend the propagation to longer times and stabilize the calculations. We show that this algorithm greatly reduces the required order for each simulation and re-scales the exponential challenge to quadratic in time. We introduce a method to compute Green's functions, spectral functions, and currents for inchworm Monte Carlo and show how systematic error assessments in real time can be obtained. We illustrate the capabilities of the algorithm with a study of the behavior of quantum impurities after an instantaneous voltage quench from a thermal equilibrium state. We conclude with the applications of the unbiased inchworm impurity solver to DMFT calculations. We employ the methods for a study of the one-band paramagnetic Hubbard model on the Bethe lattice in equilibrium, where the DMFT approximation becomes exact. We begin with a brief introduction of the Mott metal insulator phase diagram. We present the results of both real time Green's functions and spectral functions from our nonequilibrium calculations. We observe the metal-insulator crossover as the on-site interaction is increased and the formation of a quasi-particle peak as the temperature is lowered. We also illustrate the convergence of our algorithms in different aspects.
Since the first investigation of the Hubbard model in the limit of infinite dimensions by Metzner and Vollhardt, dynamical mean-field theory (DMFT) has become a very powerful tool for the investigation of lattice models of correlated electrons. In DMFT the lattice model is mapped on an effective quantum impurity model in a bath which has to be determined self-consistently. This approach led to significant progress in our understanding of typical correlation problems such as the Mott transition; furthermore, the combination of DMFT with ab-initio methods now allows for a realistic treatment of correlated materials. The focus of these lecture notes is on the relation between quantum impurity physics and the physics of lattice models within DMFT. Issues such as the observability of impurity quantum phase transitions in the corresponding lattice models are discussed in detail.
A nonperturbative quantum impurity solver is proposed based on a formally exact hierarchical equations of motion (HEOM) formalism for open quantum systems. It leads to quantitatively accurate evaluation of physical properties of strongly correlated electronic systems, in the framework of dynamical mean-field theory (DMFT). The HEOM method is also numerically convenient to achieve the same level of accuracy as that using the state-of-the-art numerical renormalization group impurity solver at finite temperatures. The practicality of the HEOM+DMFT method is demonstrated by its applications to the Hubbard models with Bethe and hypercubic lattice structures. We investigate the metal-insulator transition phenomena, and address the effects of temperature on the properties of strongly correlated lattice systems.
Since its inception, the DMRG method has been a very powerful tool for the calculation of physical properties of low-dimensional strongly correlated systems. It has been adapted to obtain dynamical properties and to consider finite temperature, time-dependent problems, bosonic degrees of freedom, the treatment of classical problems and non-equilibrium systems, among others. We will briefly review the method and then concentrate on its latest developments, describing some recent successful applications. In particular we will show how the dynamical DMRG can be used together with the Dynamical Mean Field Theory (DMFT) to solve the associated impurity problem in the infinite-dimensional Hubbard model. This method is used to obtain spectral properties of strongly correlated systems. With this algorithm, more complex problems having a larger number of degrees of freedom can be considered and finite-size effects can be minimized.