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arXiv:1303.3942 · New Journal of Physics

Reducing the numerical effort of finite-temperature density matrix renormalization group calculations

Christoph Karrasch, Jens Hjorleifur Bardarson, Joel E. Moore

Published Aug 13, 2013• 96 citations
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Abstract

Finite-temperature transport properties of one-dimensional systems can be studied using the time dependent density matrix renormalization group via the introduction of auxiliary degrees of freedom which purify the thermal statistical operator. We demonstrate how the numerical effort of such calculations is reduced when the physical time evolution is augmented by an additional time evolution within the auxiliary Hilbert space. Specifically, we explore a variety of integrable and non-integrable, gapless and gapped models at temperatures ranging from T = ∞ down to T /bandwidth = 0.05 and study both (i) linear response where (heat and charge) transport coefficients are determined by the current–current correlation function and (ii) non-equilibrium driven by arbitrary large temperature gradients. The modified density matrix renormalization algorithm removes an 'artificial' build-up of entanglement between the auxiliary and physical degrees of freedom. Thus, longer time scales can be reached.

DOI
10.1088/1367-2630/15/8/083031

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Finite-temperature transport properties of one-dimensional systems can be studied using the time dependent density matrix renormalization group via the introduction of auxiliary degrees of freedom which purify the thermal statistical operator. We demonstrate how the numerical effort of such calculations is reduced when the physical time evolution is augmented by an additional time evolution within the auxiliary Hilbert space. Specifically, we explore a variety of integrable and non-integrable, gapless and gapped models at temperatures ranging from T = ∞ down to T /bandwidth = 0.05 and study both (i) linear response where (heat and charge) transport coefficients are determined by the current–current correlation function and (ii) non-equilibrium driven by arbitrary large temperature gradients. The modified density matrix renormalization algorithm removes an 'artificial' build-up of entanglement between the auxiliary and physical degrees of freedom. Thus, longer time scales can be reached.
We propose an easily implemented approach to study time-dependent correlation functions of one-dimensional systems at finite-temperature T using the density matrix renormalization group. The entanglement growth inherent to any time-dependent calculation is significantly reduced if the auxiliary degrees of freedom which purify the statistical operator are time evolved with the physical Hamiltonian but reversed time. We exploit this to investigate the long-time behavior of current correlation functions of the XXZ spin-1/2 Heisenberg chain. This allows a direct extraction of the Drude weight D at intermediate to large T. We find that D is nonzero--and thus transport is dissipationless--everywhere in the gapless phase. At low temperatures we establish an upper bound to D by comparing with bosonization.
This paper provides a study and discussion of earlier as well as novel more efficient schemes for the precise evaluation of finite-temperature response functions of strongly correlated quantum systems in the framework of the time-dependent density matrix renormalization group (tDMRG). The computational costs and bond dimensions as functions of time and temperature are examined for the example of the spin-1/2 XXZ Heisenberg chain in the critical XY phase and the gapped Néel phase. The matrix product state purifications occurring in the algorithms are in a one-to-one relation with the corresponding matrix product operators. This notational simplification elucidates implications of quasi-locality on the computational costs. Based on the observation that there is considerable freedom in designing efficient tDMRG schemes for the calculation of dynamical correlators at finite temperatures, a new class of optimizable schemes, as recently suggested in Barthel, Schollwöck and Sachdev (2012 arXiv:1212.3570), is explained and analyzed numerically. A specific novel near-optimal scheme that requires no additional optimization reaches maximum times that are typically increased by a factor of 2, when compared against earlier approaches. These increased reachable times make many more physical applications accessible. For each of the described tDMRG schemes, one can devise a corresponding transfer matrix renormalization group variant.
One of the most straightforward ways to study thermal properties beyond linear response is to monitor the relaxation of an arbitrarily large left-right temperature gradient ${T}_{L}\ensuremath{-}{T}_{R}$. In one-dimensional systems which support ballistic thermal transport, the local energy currents $\ensuremath{\langle}j(t)\ensuremath{\rangle}$ acquire a nonzero value at long times, and it was recently investigated whether or not this steady state fulfills a simple additive relation $\ensuremath{\langle}j(t\ensuremath{\rightarrow}\ensuremath{\infty})\ensuremath{\rangle}=f({T}_{L})\ensuremath{-}f({T}_{R})$ in integrable models. In this paper, we probe the nonequilibrium dynamics of the Hubbard chain using density matrix renormalization group (DMRG) numerics. We show that the above form provides an effective description of thermal transport in this model; violations are below the finite-time accuracy of the DMRG. As a second setup, we study how an initially equilibrated system radiates into different nonthermal states (such as the vacuum).
We study the real-time dynamics of spin chains driven out of thermal equilibrium by an initial temperature gradient ${T}_{L}\ensuremath{\ne}{T}_{R}$ using density matrix renormalization group methods. We demonstrate that the nonequilibrium energy current saturates fast to a finite value if the linear-response thermal conductivity is infinite, i.e., if the Drude weight DDD is nonzero. Our data suggest that a nonintegrable dimerized chain might support such dissipationless transport (D>0D>0D>0). We show that the steady-state value JE{J}_{E}JE​ of the current for arbitrary ${T}_{L}\ensuremath{\ne}{T}_{R}$ is of the functional form ${J}_{E}=f({T}_{L})\ensuremath{-}f({T}_{R})$, i.e., it is completely determined by the linear conductance. We argue for this functional form, which is essentially a Stefan-Boltzmann law in this integrable model; for the XXX ferromagnet, fff can be computed via the thermodynamic Bethe ansatz in good agreement with the numerics. Inhomogeneous systems exhibiting different bulk parameters as well as Luttinger liquid boundary physics induced by single impurities are discussed briefly.
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.
A major advance in density-matrix renormalization group (DMRG) calculations has been achieved by the invention of highly efficient DMRG techniques for the simulation of real-time dynamics of strongly correlated quantum systems in one dimension. Starting from established linear-response techniques in DMRG and early attempts at real-time dynamics, we go on to review two current methods which both implement the idea of adapting the effective Hilbert space of DMRG to the quantum state evolving in time. We also give an outlook on extensions to finite temperature calculations.