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arXiv:0812.4252 · Physical Review B

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, Ian P. McCulloch, Ulrich Schollwöck

Published Jun 9, 2009• 118 citations
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Abstract

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.

DOI
10.1103/physrevb.79.214409

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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.
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 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.
We study the nonequilibrium spin dynamics in the nonintegrable spin-1 XXZ chains, emerging from inhomogeneous quenches, with two halves initialized in distinct maximally polarized states. Employing the time-dependent variational principle (TDVP) we are able to reach times that are inaccessible by standard time-dependent density matrix renormalization group methods. We identify three distinct phases depending on the anisotropy of the chain, corresponding to diffusive, logarithmic superdiffusive, and insulating. In the easy-axis regime we observe a transient ballistic behavior with a crossover time that diverges as the isotropic point is approached. This leads us to the conclusion that on the intermediate-large timescales the spin dynamics is well described by the integrable Landau-Lifshitz classical differential equation.
In this work we develop a new numerical method, the adaptive time-dependent density-matrix renormalization-group (adaptive t-DMRG), which turns out to be very well suited to investigate time-dependent phenomena in one-dimensional strongly correlated systems. The method is based on the original density-matrix renormalization group method by White and the time evolving block-decimation procedure by Vidal. We further show the applicability of the new method to different physical systems: bosonic, fermionic, and spin chains. We first discuss the consequences of the presence of an external trapping potential on the static properties of trapped cold bosonic atoms in an optical lattice using the Bose-Hubbard model. It is shown that a properly rescaled one-particle density matrix characterizes superfluid versus insulating states just as in the homogeneous system. We note that the superfluid to Mott-insulating transition is seen most directly in the half width of the interference peak. Then we turn to the evolution of density perturbations in these systems. In particular, we discuss the dependence of the velocity and the decay of the amplitude on density, interaction strength, and the extent and height of the perturbation in a numerically exact way, covering a wide range of interaction and perturbation strengths. By comparing our results for the sound velocity to theoretical predictions, we determine the limits of a Gross-Pitaevskii or Bogoliubov type description and the regime where repulsive one-dimensional Bose gases exhibit fermionic behaviour. As a second application of the adaptive t-DMRG, we investigate the phenomenon of spin-charge separation in the Hubbard model and propose an experimental setup for its observation in cold Fermi gases. We show the robustness of this separation beyond the low-energy regime by studying the time evolution of density wave packets of finite strength and at length scales down to a few lattice spacings. A striking signature of spin-charge separation is found in 1D cold Fermi gases in a harmonic trap using the different propagation properties of the liquid and Mott-insulating phases. The third application relates to the area of spintronics. A simplified model for the magnetization transport between two coupled reservoirs with opposite spin polarization is studied. We investigate whether a simple long time limit for the spin transport exists and if so, how it depends on the properties of the system such as integrability and criticality. Time-scales accessible to us are of the order of 100 units of time measured in 1/J while maintaining insignificant error in the observables.
High-temperature spin transport in integrable quantum spin chains exhibits a rich dynamical phase diagram, including ballistic, superdiffusive, and diffusive regimes. While integrability is known to survive in static and periodically driven systems, its fate in the complete absence of time-translation symmetry, particularly in interacting random quantum circuits, has remained unclear. Here we construct integrable random quantum circuits built from inhomogeneous XXZ R-matrices. Remarkably, integrability is preserved for arbitrary sequences of gate layers, ranging from quasiperiodic to fully random, thereby explicitly breaking both continuous and discrete time-translation symmetry. Using large-scale time-dependent density-matrix renormalization group simulations at infinite temperature and half filling, we map out the resulting spin-transport phase diagram and identify ballistic, superdiffusive, and diffusive regimes controlled by the spectral parameters of the R-matrices. The spatiotemporal structure of spin correlations within each regime depends sensitively on the inhomogeneity, exhibiting spatial asymmetry and sharp peak structures tied to near-degenerate quasiparticle velocities. To account for these findings, we develop a generalized hydrodynamics framework adapted to time-dependent integrable circuits, yielding Euler-scale predictions for correlation functions, Drude weights, and diffusion bounds. This approach identifies the quasiparticles governing transport and quantitatively captures both the scaling exponents and fine structures of the correlation profiles observed numerically. Our results demonstrate that exact Yang-Baxter integrability is compatible with stochastic quantum dynamics and establish generalized hydrodynamics as a predictive framework for transport in time-dependent integrable systems.