Time-dependent variational principle in matrix-product state manifolds: pitfalls and potential

We study the applicability of the time-dependent variational principle in matrix product state manifolds for the long time description of quantum interacting systems. By studying integrable and nonintegrable systems for which the long time dynamics are known we demonstrate that convergence of long time observables is subtle and needs to be examined carefully. Remarkably, for the disordered nonintegrable system we consider the long time dynamics are in good agreement with the rigorously obtained short time behavior and with previous obtained numerically exact results, suggesting that at least in this case the apparent convergence of this approach is reliable. Our study indicates that while great care must be exercised in establishing the convergence of the method, it may still be asymptotically accurate for a class of disordered nonintegrable quantum systems.Comment: We trade the discussion of a diffusive integrable system in favor of a discussion of diffusive nonintegrable system, which better highlights the outcome of our wor

Numerically exact quantum dynamics of low-dimensional lattice systems

In this thesis I present contributions to the development, analysis and application of tensor network state methods for numerically exact quantum dynamics in one and two-dimensional lattice systems. The setting of numerically exact quantum dynamics is introduced in Chapter 2. This includes a discussion of exact diagonalization approaches and massively parallel implementations thereof as well as a brief introduction of tensor network states. In Chapter 3, I perform a detailed analysis of the performance of n-ary tree tensor network states for simulating the dynamics of two-dimensional lattices. This constitutes the first application of this class of tensor network to dynamics in two spatial dimensions, a long-standing challenge, and the method is found to perform on par with existing state-of-the-art approaches. Chapter 4 showcases the efficacy of a novel tensor network format I developed, tailored to electron-phonon coupled problems in their single-electron sector, through an application to the Holstein model. The applicability of the approach to a broad range of parameters of the model allows to reveal the strong influence of the spread of the electron distribution on the initial state of the phonons at the site where the electron is introduced, for which a simple physical picture is offered. I depart from method development in Chapter 5 and analyse the prospects of using tensor network states evolved using the time-dependent variational principle as an approximate approach to determine asymptotic transport properties with a finite, moderate computational effort. The method is shown to not yield the correct asymptotics in a clean, non-integrable system and can thus not be expected to work in generic systems, outside of finely tuned parameter regimes of certain models. Chapters 6 and 7 are concerned with studies of spin transport in long-range interacting systems using tensor network state methods. For the clean case, discussed in Chapter 6, we find that for sufficiently short-ranged interactions, the spreading of the bulk of the excitation is diffusive and thus dominated by the local part of the interaction, while the tail of the excitation decays with a powerlaw that is twice as large as the powerlaw of the interaction. Similarly, in the disordered case, analysed in Chapter 7, we find subdiffusive transport of spin and sub-linear growth of entanglement entropy. This behaviour is in agreement with the behaviour of systems with local interactions at intermediate disorder strength, but provides evidence against the phenomelogical Griffith picture of rare, strongly disordered insulating regions. We generalize the latter to long-ranged interactions and show that it predicts to diffusion, in contrast to the local case where it results in subdiffusive behaviour

Fragmented superconductivity in the Hubbard model as solitons in Ginzburg-Landau theory

The phenomena of superconductivity and charge density waves are observed in close vicinity in many strongly correlated materials. Increasing evidence from experiments and numerical simulations suggests both phenomena can also occur in an intertwined manner, where the superconducting order parameter is coupled to the electronic density. Employing density matrix renormalization group simulations, we investigate the nature of such an intertwined state of matter stabilized in the phase diagram of the elementary $t$-$t^\prime$-$U$ Hubbard model in the strong coupling regime. Remarkably, the condensate of Cooper pairs is shown to be fragmented in the presence of a charge density wave where more than one pairing wave function is macroscopically occupied. Moreover, we provide conclusive evidence that the macroscopic wave functions of the superconducting fragments are well-described by soliton solutions of a Ginzburg-Landau equation in a periodic potential constituted by the charge density wave. In the presence of an orbital magnetic field, the order parameters are gauge invariant, and superconducting vortices are pinned between the stripes. This intertwined Ginzburg-Landau theory is proposed as an effective low-energy description of the stripe fragmented superconductor.Comment: 12 pages, 9 figure

Phonon-induced disorder in dynamics of optically pumped metals from non-linear electron-phonon coupling

The non-equilibrium dynamics of matter excited by light may produce electronic phases that do not exist in equilibrium, such as laser-induced high-$T_c$ superconductivity. Here we simulate the dynamics of a metal driven at $t=0$ by a pump that excites dipole-active vibrational modes that couple quadratically to electrons, and study the evolution of its electronic and vibrational observables. We provide evidence for enhancement of local electronic correlations, including double occupancy, accompanied by rapid loss of long-range spatial phase coherence. Concurrently, the onsite vibrational reduced density matrix evolves from its initial coherent state to one with a predominantly diagonal structure whose distribution qualitatively resembles the coherent state Poisson character. This rapid loss of coherence controls the electronic dynamics as the system evolves towards a correlated electron-phonon long-time state. We show that a simple model based on an effective disorder potential generated by the oscillator dephasing dynamics for the electrons provides an explanation for the flattening in momentum of electronic correlations. Our results provide a basis within which to understand correlation dynamics of vibrationally coupled electrons in pump-probe experiments.Comment: 7 pages main text + 3 pages appendices, 5 figures main text + 2 figures appendice

Non-equilibrium correlation dynamics in the one-dimensional Fermi-Hubbard model: A testbed for the two-particle reduced density matrix theory

We explore the non-equilibrium dynamics of a one-dimensional Fermi-Hubbard system as a sensitive testbed for the capabilities of the time-dependent two-particle reduced density matrix (TD2RDM) theory to accurately describe time-dependent correlated systems. We follow the time evolution of the out-of-equilibrium finite-size Fermi-Hubbard model initialized by a quench over extended periods of time. By comparison with exact calculations for small systems and with matrix product state (MPS) calculations for larger systems but limited to short times, we demonstrate that the TD2RDM theory can accurately account for the non-equilibrium dynamics in the regime from weak to moderately strong inter-particle correlations. We find that the quality of the approximate reconstruction of the three-particle cumulant (or correlation) required for the closure of the equations of motion for the reduced density matrix is key to the accuracy of the numerical TD2RDM results. We identify the size of the dynamically induced three-particle correlations and the amplitude of cross correlations between the two- and three-particle cumulants as critical parameters that control the accuracy of the TD2RDM theory when current state-of-the art reconstruction functionals are employed.Comment: 18 pages, 13 figure