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:1107.4136 · Physical Review B

Real-time energy dynamics in spin- 12 Heisenberg chains

Stephan Langer, Markus Heyl, Ian P. McCulloch, Fabian Heidrich‐Meisner

Published Nov 14, 2011• 60 citations
arXivHTMLPDF

Abstract

We study the real-time dynamics of the local energy density in the spin-$1/2$ $\mathrm{XXZ}$ chain starting from initial states with an inhomogeneous profile of bond energies. Numerical simulations of the dynamics of the initial states are carried out using the adaptive time-dependent density matrix renormalization group method. We analyze the time dependence of the spatial variance associated with the local energy density to classify the dynamics as either ballistic or diffusive. Our results are consistent with ballistic behavior both in the massless and the massive phase. We also study the same problem within Luttinger liquid theory and obtain that energy wave packets propagate with the sound velocity. We recover this behavior in our numerical simulations in the limit of very weakly perturbed initial states.

DOI
10.1103/physrevb.84.205115

Related papers

Real-time study of diffusive and ballistic transport in spin- 1 2 chains using the adaptive time-dependent density matrix renormalization group method

S. Langer, Fabian Heidrich‐Meisner, Jochen Gemmer B +2 more

Using the adaptive time-dependent density matrix renormalization group method, we numerically study the spin dynamics and transport in one-dimensional spin-1/2 systems at zero temperature. Instead of computing transport coefficients from linear-response theory, we study the real-time evolution of the magnetization starting from spatially inhomogeneous initial states. In particular, we are able to analyze systems far away from equilibrium with this setup. By computing the time dependence of the variance of the magnetization, we can distinguish diffusive from ballistic regimes, depending on model parameters. For the example of the anisotropic spin-1/2 chain and at half filling, we find the expected ballistic behavior in the easy-plane phase, while in the massive regime the dynamics of the magnetization is diffusive. Our approach allows us to tune the deviation of the initial state from the ground state and the qualitative behavior of the dynamics turns out to be valid even for highly perturbed initial states in the case of easy-plane exchange anisotropies. We further cover two examples of nonintegrable models, the frustrated chain and the two-leg spin ladder, and we encounter diffusive transport in all massive phases. In the former system, our results indicate ballistic behavior in the critical phase. We propose that the study of the time dependence of the spatial variance of particle densities could be instrumental in the characterization of the expansion of ultracold atoms in optical lattices as well.

arXiv:0812.4252•Physical Review B•Jun 9, 2009•118 citations

Real-time and real-space spin and energy dynamics in one-dimensional spin- 1 2 systems induced by local quantum quenches at finite temperatures

Christoph Karrasch, Joel E. Moore, Fabian Heidrich‐Meisner

We study the spin and energy dynamics in one-dimensional spin-$\frac{1}{2}$ systems induced by local quantum quenches at finite temperatures using a time-dependent density matrix renormalization group method. System sizes are chosen large enough to ensure that the time-dependent data for the accessible time scales represent the behavior in the thermodynamic limit. As a main result, we observe a ballistic spreading of perturbations of the energy density in the integrable spin-$\frac{1}{2}$ XXZ chain for all temperatures and exchange anisotropies, related to the divergent thermal conductivity in this model and the exact conservation of the energy current. In contrast, the spin dynamics is ballistic in the massless phase, but shows a diffusive behavior at high temperatures in the easy-axis phase in the case of a vanishing background spin density. We extract a quantitative estimate for the spin-diffusion constant from the time dependence of the spatial variance of the spin density, which agrees well with values obtained from current-current correlation functions using an Einstein relation. Interestingly, the diffusion constant approaches a constant value deep in the easy-axis regime. As an example for nonintegrable models, we consider two-leg ladders, for which we observe indications of diffusive energy and spin dynamics. The relevance of our results for recent experiments with quantum magnets and bosons in optical lattices is discussed.

arXiv:1312.2938•Physical Review B•Feb 28, 2014•138 citations

Dynamics of the spin- 12 Heisenberg chain initialized in a domain-wall state

Grégoire Misguich, Kirone Mallick, P. L. Krapivsky

We study the dynamics of an isotropic spin-$\frac{1}{2}$ Heisenberg chain starting in a domain-wall initial condition where the spins are initially up on the left half-line and down on the right half-line. We focus on the long-time behavior of the magnetization profile. We perform extensive time-dependent density-matrix renormalization-group simulations (up to $t=350$) and find that the data are compatible with a diffusive behavior. Subleading corrections decay slowly blurring the emergence of the diffusive behavior. We also compare our results with two alternative scenarios: superdiffusive behavior and enhanced diffusion with a logarithmic correction. We finally discuss the evolution of the entanglement entropy.

arXiv:1708.01843•Physical review. B./Physical review. B•Nov 27, 2017•86 citations

Transport and real-time dynamics in one-dimensional quantum magnets and ultra-cold atomic gases

Langer, Stephan Markus

The goal of this thesis is to study the transport properties and real-time dynamics of quantum magnets and ultra-cold atomic gases in one spatial dimension using numerical methods. The focus will be on the discussion of diffusive versus ballistic dynamics along with a detailed analysis of characteristic velocities in ballistic regimes. For the simulation of time-dependent density profiles we use the adaptive time-dependent density matrix renormalization group (DMRG). This numerical method allows for the simulation of time-dependent wave functions close to as well as far from equilibrium in a controlled manner. The studies of one-dimensional quantum magnets are partially motivated by the experimental evidence for a highly anisotropic and for insulators comparably high thermal conductivity of certain cuprates. We use linear response theory to study transport coefficients at arbitrary temperatures by diagonalizing small systems exactly and then calculating the current-current correlation functions. As first application we discuss the spin transport in the spin-$1/2$ Heisenberg chain with anisotropic exchange interactions (XXZ-chain). The second application of exact diagonalization, here in combination with time-dependent DMRG, is a discussion of the transverse components of the current-current correlation function. While usually only a Zeeman field is considered in the theory of transport coefficients, we here investigate the dynamic induced by an additional transverse magnetic field. We find that in this scenario the current-current correlation function exhibits coherent oscillations. In addition a second non-trivial frequency, different from the one expected from the usual Larmor precession, emerges and is studied varying temperature and field. Finally we calculate the frequency-dependent spin and heat conductivity of dimerized spin chains in a magnetic field. Motivated by the recent experimental studies of the phase diagram of C$_5$H$_{12}$N$_2$CuBr$_4$ we take the dimerized chain as a minimal model that exhibits features of the low-temperature region of the observed phase diagram. As a main result, the spin and heat conductivity obtained from linear response theory are enhanced in the field-induced gapless phase. The last application in the field of one-dimensional quantum magnets is the simulation of time-dependent energy-density wave-packets close to as well as far from equilibrium using the time-dependent density renormalization group. The main results are ballistic energy dynamics independently of how far out-of-equilibrium the initial state is and a detailed understanding of the average expansion velocity. The applications in the field of ultra-cold atomic gases focus on the sudden expansion of an initially trapped gas into an empty optical lattice. This setup was recently realized in an experiment performed by U. Schneider {\it et al.} and discussed in the context of electronic transport in the two-dimensional and the three-dimensional Fermi-Hubbard model. Here we investigate the sudden expansion of three different setups: For the expansion of a spin-balanced cloud of fermions, we identify the ballistic regime, and therein investigate the average expansion velocity of the cloud. As a main result the expansion velocity is determined by a small subset of the initial condition over a wide range of parameters. For instance, the Mott-insulating phase of the Hubbard model is characterized by a constant expansion velocity independently of the strength of the interaction. In the case of spinless bosons, we study the expansion from initial states that have a fixed particle number per lattice site and a certain concentration of defects. We study the expansion velocity as a function of interaction strength and investigate whether the time-dependent momentum distribution functions indicate a dynamical quasi-condensation. The last example is the sudden expansion of a spin-polarized gas of fermions in the presence of attractive interactions. This study is motivated by current effort to experimentally detect the Fulde-Ferrell-Larkin-Ovchinnikov state. Our results for the time-dependent momentum distribution functions and the wave-function of the pair condensate suggest that the signatures of the FFLO state vanish quickly, yet a stationary form of the momentum distribution also emerges fast. The latter is shown to be determined by the initial conditions, which might eventually allow for an indirect detection of the FFLO phase.

•Jan 1, 2012•0 citations

Real-time dynamics in spin- 12 chains with adaptive time-dependent density matrix renormalization group

Dominique Gobert, Corinna Kollath, Ulrich Schollwöck +1 more

We investigate the influence of different interaction strengths and dimerizations on the magnetization transport in antiferromagnetic spin 1/2 XXZ chains. We focus on the real-time evolution of the inhomogeneous initial state |upward arrow... upward arrow downward arrow... downward arrow > in using the adaptive time-dependent density-matrix renormalization group (adaptive t-DMRG). Time scales accessible to us are of the order of 100 units of time measured in Planck's/J for almost negligible error in the observables. We find ballistic magnetization transport for small S(z) S(z) interaction and arbitrary dimerization, but almost no transport for stronger S(z) S(z) interaction, with a sharp crossover at J(z) =1 . Additionally, we perform a detailed analysis of the error made by the adaptive time-dependent DMRG using the fact that the evolution in the XX model is known exactly. We find that the error at small times is dominated by the error made by the Trotter decomposition, whereas for longer times the DMRG truncation error becomes the most important, with a very sharp crossover at some "runaway" time. Overall, errors are extremely small before the "runaway" time.

arXiv:cond-mat/0409692•Physical Review E•Mar 3, 2005•279 citations

Energy diffusion in the ergodic phase of a many body localizable spin chain

Vipin Kerala Varma, Alessio Lerose, Francesca Pietracaprina +2 more

Abstract The phenomenon of many-body localization in disordered quantum many-body systems occurs when all transport is suppressed despite the fact that the excitations of the system interact. In this work we report on the numerical simulation of autonomous quantum dynamics for disordered Heisenberg chains when the system is prepared with an initial inhomogeneity in the energy density profile. Using exact diagonalisation and a dynamical code based on Krylov subspaces we are able to simulate dynamics for up to L = 26 spins. We find, surprisingly, the breakdown of energy diffusion even before the many-body localization transition whilst the system is still in the ergodic phase. Moreover, in the ergodic phase we also find a large region in parameter space where the energy dynamics remains diffusive but where spin transport has been previously evidenced to occur only subdiffusively: this is found to be true for initial states composed of infinitely many hydrodynamic modes (square-wave energy profile) or just the single longest mode (sinusoidal profile). This suggestive finding points towards a peculiar ergodic phase where particles are transported slower than energy, reminiscent of the situation in amorphous solids and of the gapped phase of the anisotropic Heisenberg model.

arXiv:1511.09144•Journal of Statistical Mechanics Theory and Experiment•May 1, 2017•41 citations
We study the real-time dynamics of the local energy density in the spin-1/21/21/2 XXZ\mathrm{XXZ}XXZ chain starting from initial states with an inhomogeneous profile of bond energies. Numerical simulations of the dynamics of the initial states are carried out using the adaptive time-dependent density matrix renormalization group method. We analyze the time dependence of the spatial variance associated with the local energy density to classify the dynamics as either ballistic or diffusive. Our results are consistent with ballistic behavior both in the massless and the massive phase. We also study the same problem within Luttinger liquid theory and obtain that energy wave packets propagate with the sound velocity. We recover this behavior in our numerical simulations in the limit of very weakly perturbed initial states.
Using the adaptive time-dependent density matrix renormalization group method, we numerically study the spin dynamics and transport in one-dimensional spin-1/2 systems at zero temperature. Instead of computing transport coefficients from linear-response theory, we study the real-time evolution of the magnetization starting from spatially inhomogeneous initial states. In particular, we are able to analyze systems far away from equilibrium with this setup. By computing the time dependence of the variance of the magnetization, we can distinguish diffusive from ballistic regimes, depending on model parameters. For the example of the anisotropic spin-1/2 chain and at half filling, we find the expected ballistic behavior in the easy-plane phase, while in the massive regime the dynamics of the magnetization is diffusive. Our approach allows us to tune the deviation of the initial state from the ground state and the qualitative behavior of the dynamics turns out to be valid even for highly perturbed initial states in the case of easy-plane exchange anisotropies. We further cover two examples of nonintegrable models, the frustrated chain and the two-leg spin ladder, and we encounter diffusive transport in all massive phases. In the former system, our results indicate ballistic behavior in the critical phase. We propose that the study of the time dependence of the spatial variance of particle densities could be instrumental in the characterization of the expansion of ultracold atoms in optical lattices as well.
We study the spin and energy dynamics in one-dimensional spin-12\frac{1}{2}21​ systems induced by local quantum quenches at finite temperatures using a time-dependent density matrix renormalization group method. System sizes are chosen large enough to ensure that the time-dependent data for the accessible time scales represent the behavior in the thermodynamic limit. As a main result, we observe a ballistic spreading of perturbations of the energy density in the integrable spin-12\frac{1}{2}21​ XXZ chain for all temperatures and exchange anisotropies, related to the divergent thermal conductivity in this model and the exact conservation of the energy current. In contrast, the spin dynamics is ballistic in the massless phase, but shows a diffusive behavior at high temperatures in the easy-axis phase in the case of a vanishing background spin density. We extract a quantitative estimate for the spin-diffusion constant from the time dependence of the spatial variance of the spin density, which agrees well with values obtained from current-current correlation functions using an Einstein relation. Interestingly, the diffusion constant approaches a constant value deep in the easy-axis regime. As an example for nonintegrable models, we consider two-leg ladders, for which we observe indications of diffusive energy and spin dynamics. The relevance of our results for recent experiments with quantum magnets and bosons in optical lattices is discussed.
We study the dynamics of an isotropic spin-12\frac{1}{2}21​ Heisenberg chain starting in a domain-wall initial condition where the spins are initially up on the left half-line and down on the right half-line. We focus on the long-time behavior of the magnetization profile. We perform extensive time-dependent density-matrix renormalization-group simulations (up to t=350t=350t=350) and find that the data are compatible with a diffusive behavior. Subleading corrections decay slowly blurring the emergence of the diffusive behavior. We also compare our results with two alternative scenarios: superdiffusive behavior and enhanced diffusion with a logarithmic correction. We finally discuss the evolution of the entanglement entropy.
The goal of this thesis is to study the transport properties and real-time dynamics of quantum magnets and ultra-cold atomic gases in one spatial dimension using numerical methods. The focus will be on the discussion of diffusive versus ballistic dynamics along with a detailed analysis of characteristic velocities in ballistic regimes. For the simulation of time-dependent density profiles we use the adaptive time-dependent density matrix renormalization group (DMRG). This numerical method allows for the simulation of time-dependent wave functions close to as well as far from equilibrium in a controlled manner. The studies of one-dimensional quantum magnets are partially motivated by the experimental evidence for a highly anisotropic and for insulators comparably high thermal conductivity of certain cuprates. We use linear response theory to study transport coefficients at arbitrary temperatures by diagonalizing small systems exactly and then calculating the current-current correlation functions. As first application we discuss the spin transport in the spin-1/21/21/2 Heisenberg chain with anisotropic exchange interactions (XXZ-chain). The second application of exact diagonalization, here in combination with time-dependent DMRG, is a discussion of the transverse components of the current-current correlation function. While usually only a Zeeman field is considered in the theory of transport coefficients, we here investigate the dynamic induced by an additional transverse magnetic field. We find that in this scenario the current-current correlation function exhibits coherent oscillations. In addition a second non-trivial frequency, different from the one expected from the usual Larmor precession, emerges and is studied varying temperature and field. Finally we calculate the frequency-dependent spin and heat conductivity of dimerized spin chains in a magnetic field. Motivated by the recent experimental studies of the phase diagram of C5_55​H12_{12}12​N2_22​CuBr4_44​ we take the dimerized chain as a minimal model that exhibits features of the low-temperature region of the observed phase diagram. As a main result, the spin and heat conductivity obtained from linear response theory are enhanced in the field-induced gapless phase. The last application in the field of one-dimensional quantum magnets is the simulation of time-dependent energy-density wave-packets close to as well as far from equilibrium using the time-dependent density renormalization group. The main results are ballistic energy dynamics independently of how far out-of-equilibrium the initial state is and a detailed understanding of the average expansion velocity. The applications in the field of ultra-cold atomic gases focus on the sudden expansion of an initially trapped gas into an empty optical lattice. This setup was recently realized in an experiment performed by U. Schneider {\it et al.} and discussed in the context of electronic transport in the two-dimensional and the three-dimensional Fermi-Hubbard model. Here we investigate the sudden expansion of three different setups: For the expansion of a spin-balanced cloud of fermions, we identify the ballistic regime, and therein investigate the average expansion velocity of the cloud. As a main result the expansion velocity is determined by a small subset of the initial condition over a wide range of parameters. For instance, the Mott-insulating phase of the Hubbard model is characterized by a constant expansion velocity independently of the strength of the interaction. In the case of spinless bosons, we study the expansion from initial states that have a fixed particle number per lattice site and a certain concentration of defects. We study the expansion velocity as a function of interaction strength and investigate whether the time-dependent momentum distribution functions indicate a dynamical quasi-condensation. The last example is the sudden expansion of a spin-polarized gas of fermions in the presence of attractive interactions. This study is motivated by current effort to experimentally detect the Fulde-Ferrell-Larkin-Ovchinnikov state. Our results for the time-dependent momentum distribution functions and the wave-function of the pair condensate suggest that the signatures of the FFLO state vanish quickly, yet a stationary form of the momentum distribution also emerges fast. The latter is shown to be determined by the initial conditions, which might eventually allow for an indirect detection of the FFLO phase.
We investigate the influence of different interaction strengths and dimerizations on the magnetization transport in antiferromagnetic spin 1/2 XXZ chains. We focus on the real-time evolution of the inhomogeneous initial state |upward arrow... upward arrow downward arrow... downward arrow > in using the adaptive time-dependent density-matrix renormalization group (adaptive t-DMRG). Time scales accessible to us are of the order of 100 units of time measured in Planck's/J for almost negligible error in the observables. We find ballistic magnetization transport for small S(z) S(z) interaction and arbitrary dimerization, but almost no transport for stronger S(z) S(z) interaction, with a sharp crossover at J(z) =1 . Additionally, we perform a detailed analysis of the error made by the adaptive time-dependent DMRG using the fact that the evolution in the XX model is known exactly. We find that the error at small times is dominated by the error made by the Trotter decomposition, whereas for longer times the DMRG truncation error becomes the most important, with a very sharp crossover at some "runaway" time. Overall, errors are extremely small before the "runaway" time.
Abstract The phenomenon of many-body localization in disordered quantum many-body systems occurs when all transport is suppressed despite the fact that the excitations of the system interact. In this work we report on the numerical simulation of autonomous quantum dynamics for disordered Heisenberg chains when the system is prepared with an initial inhomogeneity in the energy density profile. Using exact diagonalisation and a dynamical code based on Krylov subspaces we are able to simulate dynamics for up to L = 26 spins. We find, surprisingly, the breakdown of energy diffusion even before the many-body localization transition whilst the system is still in the ergodic phase. Moreover, in the ergodic phase we also find a large region in parameter space where the energy dynamics remains diffusive but where spin transport has been previously evidenced to occur only subdiffusively: this is found to be true for initial states composed of infinitely many hydrodynamic modes (square-wave energy profile) or just the single longest mode (sinusoidal profile). This suggestive finding points towards a peculiar ergodic phase where particles are transported slower than energy, reminiscent of the situation in amorphous solids and of the gapped phase of the anisotropic Heisenberg model.