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

Dynamics of Pure States and Linear Response for Binary Operators

Jonas Richter, Robin Steinigeweg

Published Nov 2, 2017• 1 citation
OpenAlex

Abstract

We study static and dynamical expectation values for a class of pure states. This class mimics non-equilibrium density matrices via the concept of quantum typicality and allows for a controlled preparation of expectation values close to or far away from their equilibrium values at arbitrary temperatures. While these pure states do not require specific types of observables, we investigate so-called binary operators such as local fermionic occupation numbers or local energy densities in the isotropic Heisenberg spin-1/2 chain. For these two examples, we use numerical forward-propagation methods in real and imaginary time to demonstrate the accuracy of the pure-state approach within and outside the linear-response regime. At high temperatures, our numerical results show that the dynamics in all regimes is generated by a single correlation function. These numerical results are in excellent agreement with our analytical predictions for binary operators.

Related papers

Relation between far-from-equilibrium dynamics and equilibrium correlation functions for binary operators

Jonas Richter, Robin Steinigeweg

Linear response theory (LRT) is one of the main approaches to the dynamics of quantum many-body systems. However, this approach has limitations and requires, e.g., that the initial state is (i) mixed and (ii) close to equilibrium. In this paper, we discuss these limitations and study the nonequilibrium dynamics for a certain class of properly prepared initial states. Specifically, we consider thermal states of the quantum system in the presence of an additional static force which, however, become nonequilibrium states when this static force is eventually removed. While for weak forces the relaxation dynamics is well captured by LRT, much less is known in the case of strong forces, i.e., initial states far away from equilibrium. Summarizing our main results, we unveil that, for high temperatures, the nonequilibrium dynamics of so-called binary operators is always generated by an equilibrium correlation function. In particular, this statement holds true for states in the far-from-equilibrium limit, i.e., outside the linear response regime. In addition, we confirm our analytical results by numerically studying the dynamics of local fermionic occupation numbers and local energy densities in the spin-1/2 Heisenberg chain. Remarkably, these simulations also provide evidence that our results qualitatively apply in a more general setting, e.g., in the anisotropic XXZ model where the local energy is a non-binary operator, as well as for a wider range of temperature. Furthermore, exploiting the concept of quantum typicality, all of our findings are not restricted to mixed states, but are valid for pure initial states as well.

arXiv:1711.00672•Physical review. E•Jan 11, 2019•13 citations

Selected applications of typicality to real-time dynamics of quantum many-body systems

Tjark Heitmann, Jonas Richter, Dennis Schubert +1 more

Abstract Loosely speaking, the concept of quantum typicality refers to the fact that a single pure state can imitate the full statistical ensemble. This fact has given rise to a rather simple but remarkably useful numerical approach to simulate the dynamics of quantum many-body systems, called dynamical quantum typicality (DQT). In this paper, we give a brief overview of selected applications of DQT, where particular emphasis is given to questions on transport and thermalization in low-dimensional lattice systems like chains or ladders of interacting spins or fermions. For these systems, we discuss that DQT provides an efficient means to obtain time-dependent equilibrium correlation functions for comparatively large Hilbert-space dimensions and long time scales, allowing the quantitative extraction of transport coefficients within the framework of, e. g., linear response theory (LRT). Furthermore, it is discussed that DQT can also be used to study the far-from-equilibrium dynamics resulting from sudden quench scenarios, where the initial state is a thermal Gibbs state of the pre-quench Hamiltonian. Eventually, we summarize a few combinations of DQT with other approaches such as numerical linked cluster expansions or projection operator techniques. In this way, we demonstrate the versatility of DQT.

arXiv:2001.05289•Zeitschrift für Naturforschung A•Apr 27, 2020•46 citations

Regression Relation for Pure Quantum States and Its Implications for Efficient Computing

Tarek A. Elsayed, Boris V. Fine

We obtain a modified version of the Onsager regression relation for the expectation values of quantum-mechanical operators in pure quantum states of isolated many-body quantum systems. We use the insights gained from this relation to show that high-temperature time correlation functions in many-body quantum systems can be controllably computed without complete diagonalization of the Hamiltonians, using instead the direct integration of the Schrödinger equation for randomly sampled pure states. This method is also applicable to quantum quenches and other situations describable by time-dependent many-body Hamiltonians. The method implies exponential reduction of the computer memory requirement in comparison with the complete diagonalization. We illustrate the method by numerically computing infinite-temperature correlation functions for translationally invariant Heisenberg chains of up to 29 spins 1/2. Thereby, we also test the spin diffusion hypothesis and find it in a satisfactory agreement with the numerical results. Both the derivation of the modified regression relation and the justification of the computational method are based on the notion of quantum typicality.

arXiv:1208.4652•Physical Review Letters•Feb 15, 2013•96 citations

Real time evolution at finite temperatures with operator space matrix product states

Iztok Pižorn, Viktor Eisler, Sabine Andergassen +1 more

We propose a method to simulate the real time evolution of one-dimensional quantum many-body systems at finite temperature by expressing both the density matrices and the observables as matrix product states. This allows the calculation of expectation values and correlation functions as scalar products in operator space. The simulations of density matrices in inverse temperature and the local operators in the Heisenberg picture are independent and result in a grid of expectation values for all intermediate temperatures and times. Simulations can be performed using real arithmetics with only polynomial growth of computational resources in inverse temperature and time for integrable systems. The method is illustrated for the XXZ model and the single impurity Anderson model.

arXiv:1305.0504•New Journal of Physics•Jul 4, 2014•26 citations

Real time evolution at finite temperatures with operator space matrix product states

Iztok Pižorn, Viktor Eisler, Sabine Andergassen +1 more

We propose a method to simulate the real time evolution of one-dimensional quantum many-body systems at finite temperature by expressing both the density matrices and the observables as matrix product states. This allows the calculation of expectation values and correlation functions as scalar products in operator space. The simulations of density matrices in inverse temperature and the local operators in the Heisenberg picture are independent and result in a grid of expectation values for all intermediate temperatures and times. Simulations can be performed using real arithmetics with only polynomial growth of computational resources in inverse temperature and time for integrable systems. The method is illustrated for the XXZ model and the single impurity Anderson model.

•Jul 1, 2014•28 citations

Typical Pure States and Nonequilibrium Processes in Quantum Many-Body Systems

Takaaki Monnai, Ayumu Sugita

We show that it is possible to calculate equilibrium expectation values of many-body correlated quantities such as the characteristic functions and probability distributions with the use of only a single typical pure state. It also means that we can apply the pure state approach to Heisenberg operators and their spectral fluctuation, and hence to nonequilibrium processes starting from equilibrium. In particular, we can accurately analyze the full statistics of entropy production in nonequilibrium mesoscopic systems. In this way, we can access the full information on higher-order fluctuations in the large deviation regime far from equilibrium.

Journal of the Physical Society of Japan•Aug 7, 2014•30 citations
We study static and dynamical expectation values for a class of pure states. This class mimics non-equilibrium density matrices via the concept of quantum typicality and allows for a controlled preparation of expectation values close to or far away from their equilibrium values at arbitrary temperatures. While these pure states do not require specific types of observables, we investigate so-called binary operators such as local fermionic occupation numbers or local energy densities in the isotropic Heisenberg spin-1/2 chain. For these two examples, we use numerical forward-propagation methods in real and imaginary time to demonstrate the accuracy of the pure-state approach within and outside the linear-response regime. At high temperatures, our numerical results show that the dynamics in all regimes is generated by a single correlation function. These numerical results are in excellent agreement with our analytical predictions for binary operators.
Linear response theory (LRT) is one of the main approaches to the dynamics of quantum many-body systems. However, this approach has limitations and requires, e.g., that the initial state is (i) mixed and (ii) close to equilibrium. In this paper, we discuss these limitations and study the nonequilibrium dynamics for a certain class of properly prepared initial states. Specifically, we consider thermal states of the quantum system in the presence of an additional static force which, however, become nonequilibrium states when this static force is eventually removed. While for weak forces the relaxation dynamics is well captured by LRT, much less is known in the case of strong forces, i.e., initial states far away from equilibrium. Summarizing our main results, we unveil that, for high temperatures, the nonequilibrium dynamics of so-called binary operators is always generated by an equilibrium correlation function. In particular, this statement holds true for states in the far-from-equilibrium limit, i.e., outside the linear response regime. In addition, we confirm our analytical results by numerically studying the dynamics of local fermionic occupation numbers and local energy densities in the spin-1/2 Heisenberg chain. Remarkably, these simulations also provide evidence that our results qualitatively apply in a more general setting, e.g., in the anisotropic XXZ model where the local energy is a non-binary operator, as well as for a wider range of temperature. Furthermore, exploiting the concept of quantum typicality, all of our findings are not restricted to mixed states, but are valid for pure initial states as well.
Abstract Loosely speaking, the concept of quantum typicality refers to the fact that a single pure state can imitate the full statistical ensemble. This fact has given rise to a rather simple but remarkably useful numerical approach to simulate the dynamics of quantum many-body systems, called dynamical quantum typicality (DQT). In this paper, we give a brief overview of selected applications of DQT, where particular emphasis is given to questions on transport and thermalization in low-dimensional lattice systems like chains or ladders of interacting spins or fermions. For these systems, we discuss that DQT provides an efficient means to obtain time-dependent equilibrium correlation functions for comparatively large Hilbert-space dimensions and long time scales, allowing the quantitative extraction of transport coefficients within the framework of, e. g., linear response theory (LRT). Furthermore, it is discussed that DQT can also be used to study the far-from-equilibrium dynamics resulting from sudden quench scenarios, where the initial state is a thermal Gibbs state of the pre-quench Hamiltonian. Eventually, we summarize a few combinations of DQT with other approaches such as numerical linked cluster expansions or projection operator techniques. In this way, we demonstrate the versatility of DQT.
We obtain a modified version of the Onsager regression relation for the expectation values of quantum-mechanical operators in pure quantum states of isolated many-body quantum systems. We use the insights gained from this relation to show that high-temperature time correlation functions in many-body quantum systems can be controllably computed without complete diagonalization of the Hamiltonians, using instead the direct integration of the Schrödinger equation for randomly sampled pure states. This method is also applicable to quantum quenches and other situations describable by time-dependent many-body Hamiltonians. The method implies exponential reduction of the computer memory requirement in comparison with the complete diagonalization. We illustrate the method by numerically computing infinite-temperature correlation functions for translationally invariant Heisenberg chains of up to 29 spins 1/2. Thereby, we also test the spin diffusion hypothesis and find it in a satisfactory agreement with the numerical results. Both the derivation of the modified regression relation and the justification of the computational method are based on the notion of quantum typicality.
We propose a method to simulate the real time evolution of one-dimensional quantum many-body systems at finite temperature by expressing both the density matrices and the observables as matrix product states. This allows the calculation of expectation values and correlation functions as scalar products in operator space. The simulations of density matrices in inverse temperature and the local operators in the Heisenberg picture are independent and result in a grid of expectation values for all intermediate temperatures and times. Simulations can be performed using real arithmetics with only polynomial growth of computational resources in inverse temperature and time for integrable systems. The method is illustrated for the XXZ model and the single impurity Anderson model.
We propose a method to simulate the real time evolution of one-dimensional quantum many-body systems at finite temperature by expressing both the density matrices and the observables as matrix product states. This allows the calculation of expectation values and correlation functions as scalar products in operator space. The simulations of density matrices in inverse temperature and the local operators in the Heisenberg picture are independent and result in a grid of expectation values for all intermediate temperatures and times. Simulations can be performed using real arithmetics with only polynomial growth of computational resources in inverse temperature and time for integrable systems. The method is illustrated for the XXZ model and the single impurity Anderson model.
We show that it is possible to calculate equilibrium expectation values of many-body correlated quantities such as the characteristic functions and probability distributions with the use of only a single typical pure state. It also means that we can apply the pure state approach to Heisenberg operators and their spectral fluctuation, and hence to nonequilibrium processes starting from equilibrium. In particular, we can accurately analyze the full statistics of entropy production in nonequilibrium mesoscopic systems. In this way, we can access the full information on higher-order fluctuations in the large deviation regime far from equilibrium.