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:2502.00109 · Physical Review Letters

Semigroup Influence Matrices for Nonequilibrium Quantum Impurity Models

Michael Sonner, Valentin Link, Dmitry A. Abanin

Published Sep 22, 2025• 9 citations
arXivHTMLPDF

Abstract

We introduce a framework for describing the real-time dynamics of quantum impurity models out of equilibrium which is based on the influence matrix approach. By replacing the dynamical map of a large fermionic quantum environment with an effective semigroup influence matrix (SGIM) which acts on a reduced auxiliary space, we overcome the limitations of previous proposals, achieving high accuracy at long evolution times. This SGIM corresponds to a uniform matrix-product state representation of the influence matrix and can be obtained by an efficient algorithm presented in this Letter. We benchmark this approach by computing the spectral function of the single impurity Anderson model with high resolution. Further, the spectrum of the effective dynamical map allows us to obtain relaxation rates of the impurity toward equilibrium following a quantum quench. Finally, for a quantum impurity model with on-site two-fermion loss, we compute the spectral function and confirm the emergence of Kondo physics at large loss rates.

DOI
10.1103/5gfn-l7w7

Related papers

Nonequilibrium quantum impurity problems via matrix-product states in the temporal domain

Julian Thoenniss, Alessio Lerose, Dmitry A. Abanin

Describing a quantum impurity coupled to one or more noninteracting fermionic reservoirs is a paradigmatic problem in quantum many-body physics. While historically the focus has been on the equilibrium properties of the impurity-reservoir system, recent experiments with mesoscopic and cold-atomic systems enabled studies of highly nonequilibrium impurity models, which require novel theoretical techniques. We propose an approach to analyze impurity dynamics based on the matrix-product state (MPS) representation of the Feynman-Vernon influence functional (IF). The efficiency of such a MPS representation rests on the moderate value of the temporal entanglement (TE) entropy of the IF, viewed as a fictitious ``wave function'' in the time domain. We obtain explicit expressions of this wave function for a family of one-dimensional reservoirs, and analyze the scaling of TE with the evolution time for different reservoir's initial states. While for initial states with short-range correlations we find temporal area-law scaling, Fermi-sea-type initial states yield logarithmic scaling with time, closely related to the real-space entanglement scaling in critical $1d$ systems. Furthermore, we describe an efficient algorithm for converting the explicit form of general reservoirs' IF to MPS form. Once the IF is encoded by a MPS, arbitrary temporal correlation functions of the interacting impurity can be efficiently computed, irrespective of the form of impurity interactions and bath geometry. The approach introduced here can be applied to a number of experimental setups, including highly nonequilibrium transport via quantum dots and real-time formation of impurity-reservoir correlations, as well as in nonequilibrium dynamical mean-field theory.

arXiv:2205.04995•Physical review. B./Physical review. B•May 1, 2023•49 citations

Multibath Influence Matrices: Universal Scaling from Real-Time Dynamics

Matan Lotem, Michael Sonner, Valentin Link +2 more

Diverging timescales are the hallmark of critical dynamics and the bottleneck to their classical and quantum simulation. To tame this, we compress a spacetime tensor network: temporally into semigroup influence matrices and spatially via matrix-product states. Benchmarking on the two-impurity Anderson model with its four fermionic species, we compute spectral functions to map the evolution from a Kondo resonance, across a non-Fermi-liquid quantum critical point, and into a gapped singlet phase. Resolving transient through asymptotic dynamics in sudden quenches and Kibble-Zurek ramps, we obtain the universal two-channel-Kondo exponent. Together, these results establish multibath influence matrices as a practical tool for real-time dynamics in strongly correlated multi-orbital systems.

cond-mat.str-elCross-listedarXiv:2607.16411•Jul 21, 2026•0 citations

Efficient method for quantum impurity problems out of equilibrium

Julian Thoenniss, Michael Sonner, Alessio Lerose +1 more

We introduce an efficient method to simulate the dynamics of an interacting quantum impurity coupled to noninteracting fermionic reservoirs. Viewing the impurity as an open quantum system, we describe the reservoirs by their Feynman-Vernon influence functionals (IFs). The IFs are represented as matrix-product states in the temporal domain, which enables an efficient computation of the dynamics for arbitrary interactions. We apply our method to study quantum quenches and transport in an Anderson impurity model, including highly nonequilibrium setups, and find a favorable performance compared to state-of-the-art methods. The computational resources required for an accurate computation of the dynamics scale polynomially with evolution time, indicating that a broad class of out-of-equilibrium quantum impurity problems are efficiently solvable. This approach will provide additional insights into the dynamical properties of mesoscopic devices and correlated materials.

arXiv:2211.10272•Physical review. B./Physical review. B•May 23, 2023•55 citations

Real-time impurity solver using Grassmann time-evolving matrix product operators

Ruofan Chen, Xiansong Xu, Chu Guo

An emergent and promising tensor-network-based impurity solver is to represent the Feynman-Vernon influence functional as a matrix product state, where the bath is integrated out analytically. Here we present an approach to calculate the equilibrium impurity spectral function based on the recently proposed Grassmann time-evolving matrix product operators method. The central idea is to perform a quench from a separable impurity-bath initial state as in the nonequilibrium scenario. The retarded Green's function $G(t+{t}_{0},{t}^{\ensuremath{'}}+{t}_{0})$ is then calculated after an equilibration time ${t}_{0}$ such that the impurity and bath are approximately in thermal equilibrium. There are two major advantages of this method. First, since we focus on real-time dynamics, we do not need to perform the numerically ill-posed analytic continuation as in imaginary- time evolution-based methods. Second, the required bond dimension of the matrix product state in real-time calculations is observed to be much smaller than that in imaginary-time calculations, leading to a significant improvement in numerical efficiency. The accuracy of this method is demonstrated using the single-orbital Anderson impurity model and benchmarked against the continuous-time quantum Monte Carlo method.

arXiv:2401.04880•Physical review. B./Physical review. B•Apr 2, 2024•10 citations

Variational principle for quantum impurity systems in and out of equilibrium: Application to Kondo problems

Yuto Ashida, Tao Shi, Mari Carmen Bañuls +2 more

We provide a detailed formulation of the recently proposed variational approach [Y. Ashida, T. Shi, M.-C. Ba\~nuls, J. I. Cirac, and E. Demler, Phys. Rev. Lett. 121, 026805 (2018)] to study ground-state properties and out-of-equilibrium dynamics for generic quantum spin-impurity systems. Motivated by the original ideas of Tomonaga, Lee, Low, and Pines, we construct a canonical transformation that completely decouples the impurity from the bath degrees of freedom. By combining this transformation with a Gaussian ansatz for the fermionic bath, we obtain a family of variational many-body states that can efficiently encode the strong entanglement between the impurity and fermions of the bath. We give a detailed derivation of equations of motions in the imaginary- and real-time evolutions on the variational manifold. We benchmark our approach by applying it to investigate ground-state and dynamical properties of the anisotropic Kondo model and compare results with those obtained using the matrix-product state (MPS) ansatz. We show that our approach can achieve an accuracy comparable to MPS-based methods with several orders of magnitude fewer variational parameters than the corresponding MPS ansatz. Comparisons to the Yosida ansatz and the exact solution from the Bethe ansatz are also discussed. We use our approach to investigate the two-lead Kondo model and analyze its long-time spatiotemporal behavior and the conductance behavior at finite bias and magnetic fields. The obtained results are consistent with the previous findings in the Anderson model and the exact solutions at the Toulouse point.

arXiv:1802.03861•Physical review. B./Physical review. B•Jul 11, 2018•37 citations

Mapping the non-equilibrium interacting Anderson Impurity Model to an effective Gaussian theory

Emmanuel Bogacz, Graham Kells, Andrew K. Mitchell

Quantum impurity models with strong electron correlations, such as the paradigmatic Anderson Impurity Model (AIM), are central to our understanding of a range of physical phenomena including local moment formation, Coulomb blockade and Kondo screening. They describe magnetic atoms and molecules on surfaces, quantum dot circuits, and correlated materials through dynamical mean field theory. The physics of such systems in strongly non-equilibrium conditions is particularly complex and challenging to capture, whereas Gaussian models of free fermions can be easily solved. Here we show that the time-evolving dynamics of the AIM after a quench can be described by a completely non-interacting version of the model, at the expense of coupling to additional static auxiliary degrees of freedom. Starting from the full solution of the quenched AIM using ED and DMRG, we study the properties of this mapping using numerical optimization, and uncover intriguing structure in the auxiliary system. The method allows us to understand interacting non-equilibrium dynamics through the simpler lens of an effective non-interacting system of larger dimension.

cond-mat.str-elCross-listedarXiv:2606.19206•Jun 18, 2026•0 citations
We introduce a framework for describing the real-time dynamics of quantum impurity models out of equilibrium which is based on the influence matrix approach. By replacing the dynamical map of a large fermionic quantum environment with an effective semigroup influence matrix (SGIM) which acts on a reduced auxiliary space, we overcome the limitations of previous proposals, achieving high accuracy at long evolution times. This SGIM corresponds to a uniform matrix-product state representation of the influence matrix and can be obtained by an efficient algorithm presented in this Letter. We benchmark this approach by computing the spectral function of the single impurity Anderson model with high resolution. Further, the spectrum of the effective dynamical map allows us to obtain relaxation rates of the impurity toward equilibrium following a quantum quench. Finally, for a quantum impurity model with on-site two-fermion loss, we compute the spectral function and confirm the emergence of Kondo physics at large loss rates.
Describing a quantum impurity coupled to one or more noninteracting fermionic reservoirs is a paradigmatic problem in quantum many-body physics. While historically the focus has been on the equilibrium properties of the impurity-reservoir system, recent experiments with mesoscopic and cold-atomic systems enabled studies of highly nonequilibrium impurity models, which require novel theoretical techniques. We propose an approach to analyze impurity dynamics based on the matrix-product state (MPS) representation of the Feynman-Vernon influence functional (IF). The efficiency of such a MPS representation rests on the moderate value of the temporal entanglement (TE) entropy of the IF, viewed as a fictitious ``wave function'' in the time domain. We obtain explicit expressions of this wave function for a family of one-dimensional reservoirs, and analyze the scaling of TE with the evolution time for different reservoir's initial states. While for initial states with short-range correlations we find temporal area-law scaling, Fermi-sea-type initial states yield logarithmic scaling with time, closely related to the real-space entanglement scaling in critical 1d1d1d systems. Furthermore, we describe an efficient algorithm for converting the explicit form of general reservoirs' IF to MPS form. Once the IF is encoded by a MPS, arbitrary temporal correlation functions of the interacting impurity can be efficiently computed, irrespective of the form of impurity interactions and bath geometry. The approach introduced here can be applied to a number of experimental setups, including highly nonequilibrium transport via quantum dots and real-time formation of impurity-reservoir correlations, as well as in nonequilibrium dynamical mean-field theory.
Diverging timescales are the hallmark of critical dynamics and the bottleneck to their classical and quantum simulation. To tame this, we compress a spacetime tensor network: temporally into semigroup influence matrices and spatially via matrix-product states. Benchmarking on the two-impurity Anderson model with its four fermionic species, we compute spectral functions to map the evolution from a Kondo resonance, across a non-Fermi-liquid quantum critical point, and into a gapped singlet phase. Resolving transient through asymptotic dynamics in sudden quenches and Kibble-Zurek ramps, we obtain the universal two-channel-Kondo exponent. Together, these results establish multibath influence matrices as a practical tool for real-time dynamics in strongly correlated multi-orbital systems.
We introduce an efficient method to simulate the dynamics of an interacting quantum impurity coupled to noninteracting fermionic reservoirs. Viewing the impurity as an open quantum system, we describe the reservoirs by their Feynman-Vernon influence functionals (IFs). The IFs are represented as matrix-product states in the temporal domain, which enables an efficient computation of the dynamics for arbitrary interactions. We apply our method to study quantum quenches and transport in an Anderson impurity model, including highly nonequilibrium setups, and find a favorable performance compared to state-of-the-art methods. The computational resources required for an accurate computation of the dynamics scale polynomially with evolution time, indicating that a broad class of out-of-equilibrium quantum impurity problems are efficiently solvable. This approach will provide additional insights into the dynamical properties of mesoscopic devices and correlated materials.
An emergent and promising tensor-network-based impurity solver is to represent the Feynman-Vernon influence functional as a matrix product state, where the bath is integrated out analytically. Here we present an approach to calculate the equilibrium impurity spectral function based on the recently proposed Grassmann time-evolving matrix product operators method. The central idea is to perform a quench from a separable impurity-bath initial state as in the nonequilibrium scenario. The retarded Green's function $G(t+{t}_{0},{t}^{\ensuremath{'}}+{t}_{0})$ is then calculated after an equilibration time t0{t}_{0}t0​ such that the impurity and bath are approximately in thermal equilibrium. There are two major advantages of this method. First, since we focus on real-time dynamics, we do not need to perform the numerically ill-posed analytic continuation as in imaginary- time evolution-based methods. Second, the required bond dimension of the matrix product state in real-time calculations is observed to be much smaller than that in imaginary-time calculations, leading to a significant improvement in numerical efficiency. The accuracy of this method is demonstrated using the single-orbital Anderson impurity model and benchmarked against the continuous-time quantum Monte Carlo method.
We provide a detailed formulation of the recently proposed variational approach [Y. Ashida, T. Shi, M.-C. Ba\~nuls, J. I. Cirac, and E. Demler, Phys. Rev. Lett. 121, 026805 (2018)] to study ground-state properties and out-of-equilibrium dynamics for generic quantum spin-impurity systems. Motivated by the original ideas of Tomonaga, Lee, Low, and Pines, we construct a canonical transformation that completely decouples the impurity from the bath degrees of freedom. By combining this transformation with a Gaussian ansatz for the fermionic bath, we obtain a family of variational many-body states that can efficiently encode the strong entanglement between the impurity and fermions of the bath. We give a detailed derivation of equations of motions in the imaginary- and real-time evolutions on the variational manifold. We benchmark our approach by applying it to investigate ground-state and dynamical properties of the anisotropic Kondo model and compare results with those obtained using the matrix-product state (MPS) ansatz. We show that our approach can achieve an accuracy comparable to MPS-based methods with several orders of magnitude fewer variational parameters than the corresponding MPS ansatz. Comparisons to the Yosida ansatz and the exact solution from the Bethe ansatz are also discussed. We use our approach to investigate the two-lead Kondo model and analyze its long-time spatiotemporal behavior and the conductance behavior at finite bias and magnetic fields. The obtained results are consistent with the previous findings in the Anderson model and the exact solutions at the Toulouse point.
Quantum impurity models with strong electron correlations, such as the paradigmatic Anderson Impurity Model (AIM), are central to our understanding of a range of physical phenomena including local moment formation, Coulomb blockade and Kondo screening. They describe magnetic atoms and molecules on surfaces, quantum dot circuits, and correlated materials through dynamical mean field theory. The physics of such systems in strongly non-equilibrium conditions is particularly complex and challenging to capture, whereas Gaussian models of free fermions can be easily solved. Here we show that the time-evolving dynamics of the AIM after a quench can be described by a completely non-interacting version of the model, at the expense of coupling to additional static auxiliary degrees of freedom. Starting from the full solution of the quenched AIM using ED and DMRG, we study the properties of this mapping using numerical optimization, and uncover intriguing structure in the auxiliary system. The method allows us to understand interacting non-equilibrium dynamics through the simpler lens of an effective non-interacting system of larger dimension.