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:2312.11705 · Physical review. B./Physical review. B

Complex time evolution in tensor networks and time-dependent Green's functions

M. Grundner, Philipp Westhoff, Fabian B. Kugler, Olivier Parcollet, Ulrich Schollwöck

Published Apr 8, 2024• 17 citations
arXivHTMLPDF

Abstract

Real-time calculations in tensor networks are strongly limited in time by entanglement growth, restricting the achievable frequency resolution of Green's functions, spectral functions, self-energies, and other related quantities. By extending the time evolution to contours in the complex plane, entanglement growth is curtailed, enabling numerically efficient high-precision calculations of time-dependent correlators and Green's functions with detailed frequency resolution. Various approaches to time evolution in the complex plane and the required postprocessing for extracting the pure real-time and frequency information are compared. We benchmark our results on the examples of the single-impurity Anderson model using matrix product states and of the three-band Hubbard-Kanamori and Dworin-Narath models using a tree tensor network. Our findings indicate that the proposed methods are also applicable to challenging realistic calculations of materials.

DOI
10.1103/physrevb.109.155124

Related papers

Dynamics of one-dimensional spin models via complex-time evolution of tensor networks

Jeonghyeok Cha, Hyun‐Yong Lee, Heung‐Sik Kim

Studying the real-time dynamics of strongly correlated systems poses significant challenges, which have recently become more manageable thanks to advances in density matrix renormalization group (DMRG) and tensor network methods. A notable development in this area is the introduction of a complex-time evolution scheme for tensor network states, originally suggested for solving Anderson impurity model and designed to suppress the growth of entanglement under time evolution. In this study, we employ the complex-time evolution scheme to investigate the dynamics of one-dimensional spin systems, specifically the transverse-field Ising model (TFIM) and the XXZ model. Our analysis revisits the dynamic critical exponent z of the TFIM and explores the dynamical structure factor in both gapped and gapless states of the XXZ model. Importantly, the complex-time evolution reproduces the results of real-time evolution while mitigating the rapid growth of quantum entanglement typically associated with the latter. These results demonstrate that the combination of complex-time evolution and extrapolation provides a robust and efficient framework for studying the dynamics of complex quantum systems, enabling more comprehensive insights into their behavior.

arXiv:2503.19269•Scientific Reports•Oct 27, 2025•2 citations

Spectral Decomposition and High-Accuracy Green’s Functions: Overcoming the Nyquist-Shannon Limit via Complex-time Krylov Expansion

Sebastian Paeckel

The accurate computation of low-energy spectra of strongly correlated quantum many-body systems, typically accessed via Green's functions, is a long-standing problem posing enormous challenges to numerical methods. When the spectral decomposition is obtained from Fourier transforming a time series, the Nyquist-Shannon theorem limits the frequency resolution Δω according to the numerically accessible time domain size T via Δω=2π/T. In tensor network methods, increasing the domain size is exponentially hard due to the ubiquitous spread of correlations, limiting the frequency resolution and thereby restricting this ansatz class mostly to one-dimensional systems with small quasiparticle velocities. Here, we show how this limitation can be overcome by augmenting the time series with complex-time Krylov states. With the example of the critical S-1/2 Heisenberg model and light bipolarons in the two-dimensional Su-Schrieffer-Heeger model, we demonstrate the enormous improvements in accuracy, which can be achieved using this method.

arXiv:2411.09680•Physical Review Letters•Mar 16, 2026•2 citations

Spectroscopy and complex-time correlations using minimally entangled typical thermal states

Zhenjiu Wang, Paul McClarty, Dobromila Dankova +2 more

Tensor network states have enjoyed great success in capturing aspects of strong correlation physics. However, obtaining dynamical correlators at nonzero temperatures is generically hard even using these methods. Here, we introduce a practical approach to computing such correlators using minimally entangled typical thermal states (METTS). While our primary method directly computes dynamical correlators of physical operators in real time, we propose extensions where correlations are evaluated in the complex-time plane. The imaginary time component bounds the rate of entanglement growth and strongly alleviates the computational difficulty allowing the study of larger system sizes. To extract the physical correlators, one must take the limit of purely real-time evolution. We present two routes for obtaining this information: (i) via an correlation function in complex time combined with a stochastic analytic continuation method to obtain the real-time limit and (ii) a correlation function that asymptotically captures the desired correlation function quantitatively. We show that these numerical techniques capture the finite-temperature dynamics of the Shastry-Sutherland model—a model of interacting spins one-half in two spatial dimensions.

arXiv:2405.18484•Physical review. B./Physical review. B•Sep 22, 2025•2 citations

Dispersion relations in two- and three-dimensional quantum systems

Valeriia Bilokon, Elvira Bilokon, I. V. Lukin +2 more

Extracting momentum-resolved excitation spectra in strongly correlated quantum systems remains a major challenge, especially beyond one spatial dimension. We present an efficient tensor-network approach to compute dispersion relations via imaginary-time evolution within the infinite projected entangled-pair states framework. Benchmarking on the transverse-field Ising model, the method successfully captures dispersion relations in both paramagnetic and ferromagnetic phases for two- and three-dimensional lattices, achieving strong agreement with series-expansion methods, where these are applicable. Crucially, this work presents the first demonstration of dispersion relation calculations for three-dimensional quantum lattice models—a long-standing computational challenge that opens previously inaccessible research frontiers. The method demonstrates remarkable efficiency, requiring only modest computational resources while maintaining high accuracy across wide parameter ranges. Its broad applicability makes it a powerful tool for quantum simulation, photonic material design, and quantum information platforms requiring precise momentum-resolved spectra.

arXiv:2509.15483•Physical Review Research•Feb 17, 2026•0 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

Multiorbital dynamical mean-field theory with a complex-time solver

Yang Yu, Lei Zhang, Emanuel Gull +2 more

We present the combination of a complex-time tensor-network impurity solver with an analytic continuation scheme based on exponential fitting as an efficient framework for single and multiorbital dynamical mean-field calculations. By performing time evolution along a complex-time contour, the approach balances computational cost with the difficulty of spectral recovery, offering greater flexibility than methods confined to the real or imaginary axis. By complementing the complex-time evolution with an exponential fitting scheme, we faithfully extract real-time information at negligible cost. The resulting method obtains high-resolution spectra at a significantly lower computational cost than real-time evolution, offering a promising tool for studies of strongly correlated materials.

arXiv:2512.23237•Physical Review Research•Apr 2, 2026•0 citations
Real-time calculations in tensor networks are strongly limited in time by entanglement growth, restricting the achievable frequency resolution of Green's functions, spectral functions, self-energies, and other related quantities. By extending the time evolution to contours in the complex plane, entanglement growth is curtailed, enabling numerically efficient high-precision calculations of time-dependent correlators and Green's functions with detailed frequency resolution. Various approaches to time evolution in the complex plane and the required postprocessing for extracting the pure real-time and frequency information are compared. We benchmark our results on the examples of the single-impurity Anderson model using matrix product states and of the three-band Hubbard-Kanamori and Dworin-Narath models using a tree tensor network. Our findings indicate that the proposed methods are also applicable to challenging realistic calculations of materials.
Studying the real-time dynamics of strongly correlated systems poses significant challenges, which have recently become more manageable thanks to advances in density matrix renormalization group (DMRG) and tensor network methods. A notable development in this area is the introduction of a complex-time evolution scheme for tensor network states, originally suggested for solving Anderson impurity model and designed to suppress the growth of entanglement under time evolution. In this study, we employ the complex-time evolution scheme to investigate the dynamics of one-dimensional spin systems, specifically the transverse-field Ising model (TFIM) and the XXZ model. Our analysis revisits the dynamic critical exponent z of the TFIM and explores the dynamical structure factor in both gapped and gapless states of the XXZ model. Importantly, the complex-time evolution reproduces the results of real-time evolution while mitigating the rapid growth of quantum entanglement typically associated with the latter. These results demonstrate that the combination of complex-time evolution and extrapolation provides a robust and efficient framework for studying the dynamics of complex quantum systems, enabling more comprehensive insights into their behavior.
The accurate computation of low-energy spectra of strongly correlated quantum many-body systems, typically accessed via Green's functions, is a long-standing problem posing enormous challenges to numerical methods. When the spectral decomposition is obtained from Fourier transforming a time series, the Nyquist-Shannon theorem limits the frequency resolution Δω according to the numerically accessible time domain size T via Δω=2π/T. In tensor network methods, increasing the domain size is exponentially hard due to the ubiquitous spread of correlations, limiting the frequency resolution and thereby restricting this ansatz class mostly to one-dimensional systems with small quasiparticle velocities. Here, we show how this limitation can be overcome by augmenting the time series with complex-time Krylov states. With the example of the critical S-1/2 Heisenberg model and light bipolarons in the two-dimensional Su-Schrieffer-Heeger model, we demonstrate the enormous improvements in accuracy, which can be achieved using this method.
Tensor network states have enjoyed great success in capturing aspects of strong correlation physics. However, obtaining dynamical correlators at nonzero temperatures is generically hard even using these methods. Here, we introduce a practical approach to computing such correlators using minimally entangled typical thermal states (METTS). While our primary method directly computes dynamical correlators of physical operators in real time, we propose extensions where correlations are evaluated in the complex-time plane. The imaginary time component bounds the rate of entanglement growth and strongly alleviates the computational difficulty allowing the study of larger system sizes. To extract the physical correlators, one must take the limit of purely real-time evolution. We present two routes for obtaining this information: (i) via an correlation function in complex time combined with a stochastic analytic continuation method to obtain the real-time limit and (ii) a correlation function that asymptotically captures the desired correlation function quantitatively. We show that these numerical techniques capture the finite-temperature dynamics of the Shastry-Sutherland model—a model of interacting spins one-half in two spatial dimensions.
Extracting momentum-resolved excitation spectra in strongly correlated quantum systems remains a major challenge, especially beyond one spatial dimension. We present an efficient tensor-network approach to compute dispersion relations via imaginary-time evolution within the infinite projected entangled-pair states framework. Benchmarking on the transverse-field Ising model, the method successfully captures dispersion relations in both paramagnetic and ferromagnetic phases for two- and three-dimensional lattices, achieving strong agreement with series-expansion methods, where these are applicable. Crucially, this work presents the first demonstration of dispersion relation calculations for three-dimensional quantum lattice models—a long-standing computational challenge that opens previously inaccessible research frontiers. The method demonstrates remarkable efficiency, requiring only modest computational resources while maintaining high accuracy across wide parameter ranges. Its broad applicability makes it a powerful tool for quantum simulation, photonic material design, and quantum information platforms requiring precise momentum-resolved spectra.
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 present the combination of a complex-time tensor-network impurity solver with an analytic continuation scheme based on exponential fitting as an efficient framework for single and multiorbital dynamical mean-field calculations. By performing time evolution along a complex-time contour, the approach balances computational cost with the difficulty of spectral recovery, offering greater flexibility than methods confined to the real or imaginary axis. By complementing the complex-time evolution with an exponential fitting scheme, we faithfully extract real-time information at negligible cost. The resulting method obtains high-resolution spectra at a significantly lower computational cost than real-time evolution, offering a promising tool for studies of strongly correlated materials.