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arXiv:2607.16411

Multibath Influence Matrices: Universal Scaling from Real-Time Dynamics

Matan Lotem, Michael Sonner, Valentin Link, Alessio Lerose, Dmitry A. Abanin

cond-mat.str-elquant-phSubmitted Jul 17, 2026• Announced Jul 21, 2026• 0 citations
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Abstract

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.

Comments
4.5 pages + End Matter, 3 figures

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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 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.
Understanding the time evolution of strongly correlated open quantum systems (OQSs) in response to perturbations (quenches) is of fundamental importance to the precise control of quantum devices. It is, however, rather challenging in multi-impurity quantum systems because such evolution often involves multiple intricate dynamical processes. In this work, we apply the numerically exact hierarchical equations of motion approach to explore the influence of two different types of perturbations, i.e., sudden swapping of the energy levels of impurity systems and activating the inter-impurity spin-exchange interaction, on the dissipation dynamics of the Kondo-correlated two-impurity Anderson model over a wide range of energetic parameters. By evaluating the time-dependent impurity spectral function and other system properties, we analyze the time evolution of the Kondo state in detail and conclude a phenomenologically scaling relation for Kondo dynamics driven by these perturbations. The evolutionary scaling relationship is not only related to the Kondo characteristic energy TK but also significantly affected by the simultaneous non-Kondo dynamic characteristic energy. We expect these results will inspire subsequent theoretical studies on the dynamics of strongly correlated OQSs.
Using the non-perturbative numerical renormalisation group (NRG) we analyse purely fermionic as well as more complex Bose-Fermi quantum impurity systems on their equilibrium properties and beyond: We present an extensive exploration of real-time dynamics for various quenches using the time-dependent NRG. For the purely fermionic system, we use the Single-impurity Anderson model (SIAM) in which the impurity is a localised orbital hybridising with a surrounding conduction band. The model enables charge scattering with a pseudo-gap scattering rate and features a quantum critical point separating the local moment (LM) phase with an unscreened spin moment from a symmetric strong coupling (SSC) phase with a fractionally screened spin moment. The real-time dynamics of the local double occupancy nicely thermalises for quenches regarding the SSC phase. Whereas for quenches within or into the LM phase the double occupancy systematically deviates from thermal equilibrium for long times. A comparison with results of a recently published time-dependent Gutwiller approach reveals clear discrepancies since this approach restricts the formation of the local moment only on the impurity site. We extend the system by an additional bosonic bath leading to the Bose-Fermi Anderson model (BFAM) and study the influence of the additional bosonic degrees of freedom on the real-time dynamics. In equilibrium the bosonic bath features an additional quantum critical point leading to a localised phase once the coupling to a sub-ohmic bosonic bath exceeds a critical value. In general the coupled bosonic bath leads to a renormalisation of the Coulomb interaction. We show that the dynamics is influenced by the specific choice of the bath exponents. For a F-type combination the non-equilibrium dynamics can be completely reproduced by an effective purely fermionic SIAM while for a B-type combination the nature of the bosonic bath and the coupling strength has a profound impact on the electron dynamics which can only be partially understood by an effective SIAM. Furthermore, we investigate the spin dynamics of an initially polarised impurity spin which is coupled to a fermionic and a bosonic bath leading to the Bose-Fermi Kondo model (BFKM). Through the coupling to the fermionic bath the polarisation vanishes with time. We show that in analogy to the spin-boson model (SBM) an initially coupling to the bosonic bath affects the real-time dynamics fundamentally leading to a remaining polarisation for long times.
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.
In this work, we present and analyze two tensor network-based influence functional approaches for simulating the real-time dynamics of quantum impurity models such as the Anderson model. Via comparison with recent numerically exact simulations, we show that such methods accurately capture the long-time nonequilibrium quench dynamics. The two parameters that must be controlled in these tensor network influence functional approaches are a time discretization (Trotter) error and a bond dimension (tensor network truncation) error. We show that the actual numerical uncertainties are controlled by an intricate interplay of these two approximations, which we demonstrate in different regimes. Our work opens the door to using these tensor network influence functional methods as general impurity solvers.
Developing numerical exact solvers for open quantum systems is a challenging task due to the non-perturbative and non-Markovian nature when coupling to structured environments. The Feynman-Vernon influence functional approach is a powerful analytical tool to study the dynamics of open quantum systems. Numerical treatments of the influence functional including the quasi-adiabatic propagator technique and the tensor-network-based time-evolving matrix product operator method have proven to be efficient in studying open quantum systems with bosonic environments. However, the numerical implementation of the fermionic path integral suffers from the Grassmann algebra involved. In this work, we present a detailed introduction to the Grassmann time-evolving matrix product operator method for fermionic open quantum systems. In particular, we introduce the concepts of Grassmann tensor, signed matrix product operator, and Grassmann matrix product state to handle the Grassmann path integral. Using the single-orbital Anderson impurity model as an example, we review the numerical benchmarks for structured fermionic environments for real-time nonequilibrium dynamics, real-time and imaginary-time equilibration dynamics, and its application as an impurity solver. These benchmarks show that our method is a robust and promising numerical approach to study strong coupling physics and non-Markovian dynamics. It can also serve as an alternative impurity solver to study strongly correlated quantum matter with dynamical mean-field theory.