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:1205.1817 · Physical Review B

Imaginary-time quantum many-body theory out of equilibrium. II. Analytic continuation of dynamic observables and transport properties

Andreas Dirks, Jong E. Han, Mark Jarrell, Thomas Pruschke

Published Jun 28, 2013• 3 citations
arXivHTMLPDF

Abstract

Within the imaginary-time theory for nonequilibrium in quantum dot systems the calculation of dynamical quantities like Green's functions is possible via a suitable quantum Monte Carlo algorithm. The challenging task is to analytically continue the imaginary-time data for both complex voltage and complex frequency onto the real variables. To this end a function-theoretical description of dynamical observables is introduced and discussed within the framework of the mathematical theory of several complex variables. We construct a feasible maximum-entropy algorithm for the analytical continuation by imposing a continuity assumption on the analytic structure and provide results for spectral functions in stationary nonequilibrium and current-voltage characteristics for different values of the dot charging energy.

DOI
10.1103/physrevb.87.235140

Related papers

Continuous-time quantum Monte Carlo and maximum entropy approach to an imaginary-time formulation of strongly correlated steady-state transport

Andreas Dirks, Philipp Werner, Mark Jarrell +1 more

Recently, Han and Heary [Phys. Rev. Lett. 99, 236808 (2007)] proposed an approach to steady-state quantum transport through mesoscopic structures, which maps the nonequilibrium problem onto a family of auxiliary quantum impurity systems subject to imaginary voltages. We employ continuous-time quantum Monte-Carlo solvers to calculate accurate imaginary time data for the auxiliary models. The spectral function is obtained from a maximum entropy analytical continuation in both Matsubara frequency and complexified voltage. To enable the analytical continuation we construct a kernel which is compatible with the analytical structure of the theory. While it remains a formidable task to extract reliable spectral functions from this unbiased procedure, particularly for large voltages, our results indicate that the method in principle yields results in agreement with those obtained by other methods.

arXiv:1002.4081•Physical Review E•Aug 5, 2010•30 citations

Imaginary-time quantum many-body theory out of equilibrium: Formal equivalence to Keldysh real-time theory and calculation of static properties

Jong E. Han, Andreas Dirks, Thomas Pruschke

We discuss the formal relationship between the real-time Keldysh and imaginary-time theory for nonequilibrium in quantum dot systems. The latter can be reformulated using the recently proposed Matsubara-voltage approach. We establish general conditions for correct analytic continuation procedure on physical observables, and apply the technique to the calculation of static quantities in steady-state nonequilibrium for a quantum dot subject to a finite bias voltage and external magnetic field. Limitations of the Matsubara voltage approach are also pointed out.

arXiv:1205.1816•Physical Review B•Oct 17, 2012•16 citations

CT-QMC and Maximum Entropy Approach to a Scattering-States Formulation of Strongly Correlated Steady-State Transport

Andreas Dirks, Philipp Werner, Mark Jarrell +1 more

In a recent work [17] Han and Heary introduced a formalism approaching steady-state quantum transport through mesoscopic structures which maps the non-equilibrium problem onto a family of auxiliary equilibrium quantum impurity systems by introducing imaginary voltages. We apply continuous-time quantum Monte-Carlo solvers to obtain precise and unbiased imaginary-time data for these auxiliary models. Physical observables are obtained by an analytical continuation in both Matsubara frequency and complexified voltage using a Maximum Entropy Method (MEM). The MEM is introduced by means of a kernel function compatible with the analytical structure of the theory. While it remains a yet challenging task to obtain reliable spectral functions, this unbiased procedure seems to indicate that the formalism yields results which are compatible with those of other methods.

Journal of Physics Conference Series•Jan 1, 2011•1 citation

Imaginary-Time Approach to the Kondo Effect out of Equilibrium

Andreas Dirks

An artificial quantum dot which is attached to two leads exhibits a non-equilibrium Kondo effect due to the presence of a finite source-drain voltage. The construction of a theory which provides numerically computable expectation values of physical observables for this non-perturbative quantum effect within an appropriate non-equilibrium quantum statistical ensemble is an unsolved problem. The present thesis investigates an imaginary-time approach by Han and Heary [1] by means of its formal structure and computational feasibility. The theory maps the non-equilibrium system to an infinite set of auxiliary effective-equilibrium systems which can be examined with conventional techniques. An analytic continuation is necessary to reobtain non-equilibrium expectation values from effective-equilibrium ones. It gives rise to an inverse problem. We show that continuous-time quantum Monte-Carlo (CT-QMC) methods provide a numerically controlled solution of the set of auxiliary systems, in particular for parameter values which are deep inside the non-equilibrium Kondo regime. For static local observables on the quantum dot we provide reasonable numerical estimates with clear signatures of non-equilibrium Kondo physics, using a standard MaxEnt approach to the inverse problem of analytic continuation. In order to compute local dynamical observables and transport characteristics, a theoretical framework based on the function theory of several complex variables is developed. It defines the inverse problem for the MaxEnt procedure. Resulting predictions for spectral functions and I-V characteristics in the weak- and intermediate-coupling regime are in good agreement with other methods. [1] J. E. Han, R. J. Heary, Phys. Rev. Lett. 99, 236808 (2007)

•Jan 1, 2012•0 citations

Quantum time correlation functions from complex time Monte Carlo simulations: A maximum entropy approach

Goran Krilov, Eunji Sim, B. J. Berne

We present a way of combining real-time path integral Monte Carlo simulations with a maximum entropy numerical analytic continuation scheme in a new approach for calculating time correlation functions for finite temperature many body quantum systems. The real-time dynamics is expressed in the form of the symmetrized time correlation function, which is suitable for Monte Carlo methods, and several simulation techniques are presented for evaluating this function accurately up to moderate values of time. The symmetrized time correlation function is then analytically continued in combination with imaginary time data to obtain the real-time correlation function. We test this approach on several exactly solvable problems, including two one-dimensional systems, as well two cases of vibrational relaxation of a system coupled to a dissipative environment. The computed time correlation functions are in good agreement with exact results over several multiples of the thermal time βℏ, and exhibit a significant improvement over analytic continuation of imaginary time correlation functions. Moreover, we show how the method can be systematically improved.

The Journal of Chemical Physics•Jan 15, 2001•77 citations

Imaginary-time formulation of steady-state nonequilibrium in quantum dot models

Jong E. Han

We examine the recently proposed imaginary-time formulation for strongly correlated steady-state nonequilibrium for its range of validity and discuss significant improvements in the analytic continuation of the Matsubara voltage as well as the fermionic Matsubara frequency. The discretization error in the conventional Hirsch-Fye algorithm has been compensated in the Fourier transformation with reliable small frequency behavior of self-energy. Here we give detailed discussions for generalized spectral representation ansatz by including high-order vertex corrections and its numerical analytic continuation procedures. The differential conductance calculations agree accurately with available data from other nonequilibrium transport theories. It is verified that, at finite source-drain voltage, the Kondo resonance is destroyed at bias comparable to the Kondo temperature. Calculated scaling coefficients of differential conductance for zero-bias anomaly fall within the range of experimental estimates.

arXiv:1001.4989•Physical Review B•Jun 4, 2010•23 citations
Within the imaginary-time theory for nonequilibrium in quantum dot systems the calculation of dynamical quantities like Green's functions is possible via a suitable quantum Monte Carlo algorithm. The challenging task is to analytically continue the imaginary-time data for both complex voltage and complex frequency onto the real variables. To this end a function-theoretical description of dynamical observables is introduced and discussed within the framework of the mathematical theory of several complex variables. We construct a feasible maximum-entropy algorithm for the analytical continuation by imposing a continuity assumption on the analytic structure and provide results for spectral functions in stationary nonequilibrium and current-voltage characteristics for different values of the dot charging energy.
Recently, Han and Heary [Phys. Rev. Lett. 99, 236808 (2007)] proposed an approach to steady-state quantum transport through mesoscopic structures, which maps the nonequilibrium problem onto a family of auxiliary quantum impurity systems subject to imaginary voltages. We employ continuous-time quantum Monte-Carlo solvers to calculate accurate imaginary time data for the auxiliary models. The spectral function is obtained from a maximum entropy analytical continuation in both Matsubara frequency and complexified voltage. To enable the analytical continuation we construct a kernel which is compatible with the analytical structure of the theory. While it remains a formidable task to extract reliable spectral functions from this unbiased procedure, particularly for large voltages, our results indicate that the method in principle yields results in agreement with those obtained by other methods.
We discuss the formal relationship between the real-time Keldysh and imaginary-time theory for nonequilibrium in quantum dot systems. The latter can be reformulated using the recently proposed Matsubara-voltage approach. We establish general conditions for correct analytic continuation procedure on physical observables, and apply the technique to the calculation of static quantities in steady-state nonequilibrium for a quantum dot subject to a finite bias voltage and external magnetic field. Limitations of the Matsubara voltage approach are also pointed out.
In a recent work [17] Han and Heary introduced a formalism approaching steady-state quantum transport through mesoscopic structures which maps the non-equilibrium problem onto a family of auxiliary equilibrium quantum impurity systems by introducing imaginary voltages. We apply continuous-time quantum Monte-Carlo solvers to obtain precise and unbiased imaginary-time data for these auxiliary models. Physical observables are obtained by an analytical continuation in both Matsubara frequency and complexified voltage using a Maximum Entropy Method (MEM). The MEM is introduced by means of a kernel function compatible with the analytical structure of the theory. While it remains a yet challenging task to obtain reliable spectral functions, this unbiased procedure seems to indicate that the formalism yields results which are compatible with those of other methods.
An artificial quantum dot which is attached to two leads exhibits a non-equilibrium Kondo effect due to the presence of a finite source-drain voltage. The construction of a theory which provides numerically computable expectation values of physical observables for this non-perturbative quantum effect within an appropriate non-equilibrium quantum statistical ensemble is an unsolved problem. The present thesis investigates an imaginary-time approach by Han and Heary [1] by means of its formal structure and computational feasibility. The theory maps the non-equilibrium system to an infinite set of auxiliary effective-equilibrium systems which can be examined with conventional techniques. An analytic continuation is necessary to reobtain non-equilibrium expectation values from effective-equilibrium ones. It gives rise to an inverse problem. We show that continuous-time quantum Monte-Carlo (CT-QMC) methods provide a numerically controlled solution of the set of auxiliary systems, in particular for parameter values which are deep inside the non-equilibrium Kondo regime. For static local observables on the quantum dot we provide reasonable numerical estimates with clear signatures of non-equilibrium Kondo physics, using a standard MaxEnt approach to the inverse problem of analytic continuation. In order to compute local dynamical observables and transport characteristics, a theoretical framework based on the function theory of several complex variables is developed. It defines the inverse problem for the MaxEnt procedure. Resulting predictions for spectral functions and I-V characteristics in the weak- and intermediate-coupling regime are in good agreement with other methods. [1] J. E. Han, R. J. Heary, Phys. Rev. Lett. 99, 236808 (2007)
We present a way of combining real-time path integral Monte Carlo simulations with a maximum entropy numerical analytic continuation scheme in a new approach for calculating time correlation functions for finite temperature many body quantum systems. The real-time dynamics is expressed in the form of the symmetrized time correlation function, which is suitable for Monte Carlo methods, and several simulation techniques are presented for evaluating this function accurately up to moderate values of time. The symmetrized time correlation function is then analytically continued in combination with imaginary time data to obtain the real-time correlation function. We test this approach on several exactly solvable problems, including two one-dimensional systems, as well two cases of vibrational relaxation of a system coupled to a dissipative environment. The computed time correlation functions are in good agreement with exact results over several multiples of the thermal time βℏ, and exhibit a significant improvement over analytic continuation of imaginary time correlation functions. Moreover, we show how the method can be systematically improved.
We examine the recently proposed imaginary-time formulation for strongly correlated steady-state nonequilibrium for its range of validity and discuss significant improvements in the analytic continuation of the Matsubara voltage as well as the fermionic Matsubara frequency. The discretization error in the conventional Hirsch-Fye algorithm has been compensated in the Fourier transformation with reliable small frequency behavior of self-energy. Here we give detailed discussions for generalized spectral representation ansatz by including high-order vertex corrections and its numerical analytic continuation procedures. The differential conductance calculations agree accurately with available data from other nonequilibrium transport theories. It is verified that, at finite source-drain voltage, the Kondo resonance is destroyed at bias comparable to the Kondo temperature. Calculated scaling coefficients of differential conductance for zero-bias anomaly fall within the range of experimental estimates.