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arXiv:2311.10909 · Physical review. B./Physical review. B

Dynamical correlation functions from complex time evolution

Xiaodong Cao, Yi Lu, Edwin Miles Stoudenmire, Olivier Parcollet

Published Jun 5, 2024• 10 citations
arXivHTMLPDF

Abstract

As quantum systems evolve in time, their entanglement usually grows quickly, making tensor network methods inefficient. ``Imaginary time'' evolution is efficient, but extrapolating back to the real axis is ill-conditioned. Here, the authors show that complex time, a mixture of the two, remains efficient while allowing a controlled reconstruction technique. Therefore, one can reach significantly longer times, which they demonstrate by obtaining high-accuracy results for impurity models at low energies.

DOI
10.1103/physrevb.109.235110

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•Jan 1, 2019•1 citation
As quantum systems evolve in time, their entanglement usually grows quickly, making tensor network methods inefficient. ``Imaginary time'' evolution is efficient, but extrapolating back to the real axis is ill-conditioned. Here, the authors show that complex time, a mixture of the two, remains efficient while allowing a controlled reconstruction technique. Therefore, one can reach significantly longer times, which they demonstrate by obtaining high-accuracy results for impurity models at low energies.
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.
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.
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.
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One of the major ways in which quantum mechanics differs from classical mechanics is the existence of special quantum correlations - entanglement. Typical quantum states are highly entangled, making them complex and inefficient to represent. Physically interesting states are unusual, they are only weakly entangled. By restricting ourselves to weak entanglement, efficient representations of quantum states can be found. A tensor network is constructed by taking objects called tensors that encode spatially local information and gluing them together to create a large network that describes a complex quantum state. The manner in which the tensors are connected defines the entanglement structure of the quantum state. Tensors networks are therefore a natural framework for describing physical behaviour of complex quantum systems. In this thesis we utilize tensor networks to solve a number of interesting problems. Firstly, we study a Feynman path integral written over tensor network states. As a sum over classical trajectories, a Feynman path integral can struggle to capture entanglement. Combining the path integral with tensor networks overcomes this.We consider the effect of quadratic fluctuations on the tensor network path integral and calculate corrections to observables numerically and analytically. We also study the time evolution of complex quantum systems. By projecting quantum dynamics onto a classical phase space defined using tensor networks, we relate thermal behaviour of quantum systems to classical chaos. In doing so we demonstrate a relationship between entanglement growth and chaos. By studying the dynamics of coupled quantum chains we also gain insight into how quantum correlations spread over time. As noted, tensor networks are remarkably efficient. In the final section of this thesis we use tensor networks to create compressed machine learning algorithms. Their efficiency means that tensor networks can use 505050 times fewer parameters with no significant decrease in performance.