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arXiv:1702.00421 · Physical review. E

Charge diffusion in the one-dimensional Hubbard model

Robin Steinigeweg, Fengping Jin, Hans A. De Raedt, Kristel F. L. Michielsen, Jochen Gemmer B

Published Aug 28, 2017• 18 citations
arXivHTMLPDF

Abstract

We study the real-time and real-space dynamics of charge in the one-dimensional Hubbard model in the limit of high temperatures. To this end, we prepare pure initial states with sharply peaked density profiles and calculate the time evolution of these nonequilibrium states, by using numerical forward-propagation approaches to chains as long as 20 sites. For a class of typical states, we find excellent agreement with linear-response theory and unveil the existence of remarkably clean charge diffusion in the regime of strong particle-particle interactions. We additionally demonstrate that, in the half-filling sector, this diffusive behavior does not depend on certain details of our initial conditions, i.e., it occurs for five different realizations with random and nonrandom internal degrees of freedom, single and double occupation of the central site, and displacement of spin-up and spin-down particles.

DOI
10.1103/physreve.96.020105

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We study the real-time and real-space dynamics of charge in the one-dimensional Hubbard model in the limit of high temperatures. To this end, we prepare pure initial states with sharply peaked density profiles and calculate the time evolution of these nonequilibrium states, by using numerical forward-propagation approaches to chains as long as 20 sites. For a class of typical states, we find excellent agreement with linear-response theory and unveil the existence of remarkably clean charge diffusion in the regime of strong particle-particle interactions. We additionally demonstrate that, in the half-filling sector, this diffusive behavior does not depend on certain details of our initial conditions, i.e., it occurs for five different realizations with random and nonrandom internal degrees of freedom, single and double occupation of the central site, and displacement of spin-up and spin-down particles.
We consider charge and spin transport in the one-dimensional Hubbard model at infinite temperature, half-filling, and zero magnetization. Implementing matrix-product-operator simulations of the nonequilibrium steady states of boundary-driven open Hubbard chains for up to 100 sites we find clear numerical evidence of diffusive transport for weak driving and any (nonzero and finite) value of the interaction UUU. In addition, nonequilibrium steady states are characterized by nonzero spin-up spin-down density-density correlations. For maximal driving, on the other hand, we find an insulating behavior and cosine-shaped density profiles.
We present the exact solution of the one-dimensional extended Hubbard model in the atomic limit within the Green's function and equations of motion formalism. We provide a comprehensive and systematic analysis of the model by considering all the relevant response and correlation functions as well as thermodynamic quantities in the whole parameters space. At zero temperature we identify four phases in the plane (U,n) ( U is the on-site potential and n is the filling) and relative phase transitions as well as different types of charge ordering. These features are endorsed by investigating at T=0 the chemical potential and pertinent local correlators, the particle and double occupancy correlation functions, the entropy, and by studying the behavior in the limit T-->0 of the charge and spin susceptibilities. A detailed study of the thermodynamic quantities is also presented at finite temperature. This study evidences that a finite-range order persists for a wide range of the temperature, as shown by the behavior of the correlation functions and by the two-peak structure exhibited by the charge susceptibility and by the entropy. Moreover, the equations of motion formalism, together with the use of composite operators, allows us to exactly determine the set of elementary excitations. As a result, the density of states can be determined and a detailed analysis of the specific heat allows for identifying the excitations and for ascribing its two-peak structure to a redistribution of the charge density.
We consider two mutually interacting fermionic particle species on a one-dimensional lattice and study how the mass ratio $\ensuremath{\eta}$ between the two species affects the (equilibration) dynamics of the particles. Focusing on the regime of strong interactions and high temperatures, two well-studied points of reference are given by (i) the case of equal masses $\ensuremath{\eta}=1$, i.e., the standard Fermi-Hubbard chain, where initial nonequilibrium density distributions are known to decay, and (ii) the case of one particle species being infinitely heavy, $\ensuremath{\eta}=0$, leading to a localization of the lighter particles in an effective disorder potential. Given these two opposing cases, the dynamics in the case of intermediate mass ratios $0<\ensuremath{\eta}<1$ is of particular interest. To this end, we study the real-time dynamics of pure states featuring a sharp initial nonequilibrium density profile. Relying on the concept of dynamical quantum typicality, the resulting nonequilibrium dynamics can be related to equilibrium correlation functions. Summarizing our main results, we observe that diffusive transport occurs for moderate values of the mass imbalance and manifests itself in a Gaussian spreading of real-space density profiles and an exponential decay of density modes in momentum space. For stronger imbalances, we provide evidence that transport becomes anomalous on intermediate timescales, and in particular, our results are consistent with the absence of strict localization in the long-time limit for any $\ensuremath{\eta}>0$. Based on our numerical analysis, we provide an estimate for the ``lifetime'' of the effective localization as a function of $\ensuremath{\eta}$.
We identify a class of one-dimensional spin and fermionic lattice models that display diverging spin and charge diffusion constants, including several paradigmatic models of exactly solvable, strongly correlated many-body dynamics such as the isotropic Heisenberg spin chains, the Fermi-Hubbard model, and the t-J model at the integrable point. Using the hydrodynamic transport theory, we derive an analytic lower bound on the spin and charge diffusion constants by calculating the curvature of the corresponding Drude weights at half-filling, and demonstrate that for certain lattice models with isotropic interactions some of the Noether charges exhibit superdiffusive transport at finite temperature and half-filling.
We study the real-time evolution of a charge-excited state in a one-dimensional ege_{\rm g}eg​-orbital degenerate Hubbard model, by a time-dependent density-matrix renormalization group method. Considering a chain along the zzz direction, electrons hop between adjacent 3z2−r23z^2-r^23z2−r2 orbitals, while x2−y2x^2-y^2x2−y2 orbitals are localized. For the charge-excited state, a holon-doublon pair is introduced into the ground state at quarter filling. At initial time, there is no electron in a holon site, while a pair of electrons occupies 3z2−r23z^2-r^23z2−r2 orbital in a doublon site. As the time evolves, the holon motion is governed by the nearest-neighbor hopping, but the electron pair can transfer between 3z2−r23z^2-r^23z2−r2 orbital and x2−y2x^2-y^2x2−y2 orbital through the pair hopping in addition to the nearest-neighbor hopping. Thus holon and doublon propagate at different speed due to the pair hopping that is characteristic of multi-orbital systems.
The nonequilibrium dynamics of correlated charge transfer along a one-dimensional chain in the presence of a phonon environment is investigated within a dissipative Hubbard model. For this generalization of the ubiquitous spin-boson model the crucial role of symmetries is analyzed in detail and corresponding invariant subspaces are identified. It is shown that the time evolution typically occurs in each of the disjunct subspaces independently leading, e.g., asymptotically to a non-Boltzmann equilibrium state. Based on these findings explicit results are obtained for two interacting electrons by means of a substantially improved real-time quantum Monte Carlo approach. In the incoherent regime an appropriate mapping of the many-body dynamics onto an isomorphic single particle motion allows for an approximate description of the numerical data in terms of rate equations. These results may lead to control schemes of charge transport in tailored quantum systems as, e.g., molecular chains or quantum dot arrays.