Semi-group influence matrices for non-equilibrium quantum impurity models

TL;DR

This work introduces the semi-group influence matrix (SGIM), a uniform matrix-product-state representation that replaces the complex fermionic environment in non-equilibrium quantum impurity models with an efficient, time-translation-invariant semi-group acting on a reduced auxiliary space. The SGIM enables long-time, numerically exact real-time dynamics by contracting a uniform MPS via iTEBD, with the memory effects controlled by the auxiliary dimension $\chi_{\mathrm{aux}}$ and a memory cutoff $N_c$ effectively absorbed into a truncation tolerance. Benchmarking on the single-impurity Anderson model yields high-resolution spectral functions that capture the Kondo resonance and Friedel sum rule, while non-equilibrium quenches reveal Kondo-scale relaxation governed by the spectrum of the SGIM. The framework also naturally handles dissipative impurity dynamics, revealing the emergence of a Kondo peak under strong two-fermion loss, demonstrating the method's broad applicability to driven and open quantum impurity problems with potential impact on impurity solvers in non-equilibrium DMFT and multi-orbital systems.

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 semi-group 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 paper. 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 towards 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.

Figures (7)

• Figure 1: (a) Real time evolution of a QIM with two baths $\sigma=\uparrow,\downarrow$ expressed as tensor contraction corresponding to Eq. \ref{['eq:dmevo']}. (b) Tensor network representation of an infinite Gaussian IM corresponding to Eq. \ref{['eq:F_inf_def']}, decomposed into nearest neighbor gates. (c) Semi-group influence matrix as obtained from contracting (b) using iTEBD.
• Figure 2: Upper: Spectral function $A(\omega)$ of the SIAM \ref{['eq:SIAM']} with semicircular bath $J_{\uparrow/\downarrow}(\omega)={D}/(2\pi)\sqrt{1-\omega^2/D^2}$, $\mu=0,\,\beta=100/D$ and $U=2D$. Lower: Left: spectral function at low frequencies. Friedel sum rule requires $\pi DA(0)=2$ at $\beta=\infty$. Right: Imaginary part of the self energy $\Sigma(\omega)$ at low frequencies (reference data from Ref. cao2024Dynamical). We used a Trotter time-step $\delta t=0.05/D$.
• Figure 3: SIAM dynamics for a flatband bath $J_{\uparrow/\downarrow}(\omega)={2\Gamma}/[2\pi(1+\mathrm{e}^{\nu(\omega-\omega_c)})(1+\mathrm{e}^{-\nu(\omega+\omega_c)})]$ with $\nu=\omega_c=10\Gamma$, $\beta=100/\Gamma$, $\mu=0$. Left: Quench from a polarized impurity with $U=8\Gamma$. We achieve convergence with $\delta t=0.025/\Gamma,\,\chi_\mathrm{aux}=305$ (dashed: $\chi_\mathrm{aux}=154$). Insert: Asymptotic exponential decay of $\sigma_z=\rho_{\uparrow}-\rho_{\downarrow}$ compared to the prediction from the spectral gap. Right: Spin relaxation rate $\Gamma_D$ (computed via the spectral gap and real time evolution) for different interaction strengths $U$, and fit to the Schrieffer Wolff formula hewson1997Thecao2024Dynamical.
• Figure 4: Spectral function for the dissipative SIAM with the same bath used in Fig. \ref{['fig:hubbard_kondo_quench']} and onsite energy $\varepsilon_d=-3\Gamma$. As the loss rate $\gamma$ is increased, a sharp Kondo peak appears at $\mu=0$. For these computations we used a Trotter time-step of $\delta t=0.002/\Gamma$ and an auxiliary dimension of $χ_\mathrm{aux}=239$.
• Figure S5: Evolution of the iMPS bond dimension during computations of influence matrices for the quench dynamics from Fig. \ref{['fig:hubbard_kondo_quench']} (main text), with step size $\Gamma\delta t=0.025$ and different relative SVD cutoff tolerances. The final bond dimension at network layer zero correspond to the auxiliary space dimension $\chi_\mathrm{aux}$. The effective memory cutoff at which the bond dimension starts growing beyond one depends on the specified relative SVD tolerance. The table on the right shows the memory cutoffs as well as exemplary computation times for the contraction on consumer hardware (Apple M4 with linear algebra via OpenBLAS).