Session L4: Focus Session: Numerical Methods for Studying Nonequilibrium Many-Body Dynamics; DMRG, DMFT, Truncated Wigner Approximation, Exact Diagonalization

Finite temperature DMRG and the Drude weight of spin 1/2 Heisenberg chains

We propose an easy-to-implement approach to study time-dependent correlation functions of one dimensional systems at finite temperature $T$ using the the density matrix renormalization group (DMRG). If the auxiliary degrees of freedom which purify the statistical operator are time-evolved with the physical Hamiltonian but reversed time, the entanglement blow-up inherent to any time-dependent DMRG calculation is dramatically reduced. The numerical effort of finite temperature DMRG becomes comparable to that at $T=0$, and thus significantly longer timescales can be reached. We exploit this to investigate current correlation functions of the XXZ spin $1/2$ Heisenberg chain. At intermediate to large $T$, we can explicitly extract the Drude weight $D$ from the long-time asymptotics. For the isotropic chain, $D$ is finite. At low temperatures, we establish an upper bound for the Drude weight.   

Universal non-equilibrium quantum dynamics in imaginary time

We propose a method to study the dynamical response of a quantum systems by evolving it with an imaginary-time dependent Hamiltonian. The leading non-adiabatic response of the system driven to a quantum-critical point is universal and characterized by the same exponents in real and imaginary time. For a linear quench protocol, the fidelity susceptibility and the geometric tensor naturally emerge in the response functions. Beyond linear response we extend the finite-size scaling theory of quantum phase transitions to non-equilibrium setups. Imaginary-time dynamics is also amenable to quantum Monte Carlo simulations, which we apply here to quenches of the transverse-field Ising model to quantum critical points in one and two dimensions.   

The Approach To Typicality in Many-Body Quantum Systems

The recent discovery that for large Hilbert spaces, almost all (that is, typical) Hamiltonians have eigenstates that place small subsystems in thermal equilibrium, has shed much light on the origins of irreversibility and thermalization. Here we present numerical evidence that many-body lattice systems generically approach typicality as the number of subsystems is increased, and thus provide further support for the eigenstate thermalization hypothesis. We will present our results that indicate that the deviation of many-body systems from typicality scales as an inverse power of the number of systems, and we compare this with the equivalent scaling for random Hamiltonians.   

Quantum Dynamics of Population-Imbalanced Fermi Mixture in One-dimensional Optical Lattices

We study the (pseudo-)spin dynamics of two-component Fermi mixture with population imbalance in one-dimensional (1D) optical lattices within the framework of 1D Hubbard model, utilizing the time-evolving block decimation (TEBD) algorithm. We consider the situation in which the center region of the system is initially fully occupied and the left-over fully polarized fermions are initially localized on the wings. It is found that for strong interaction the (pseudo-)spin polarization diffuses throughout the system qualitatively in the same way as the diffusion of non-interacting particles in 1D optical lattices. We further look into the time evolution of correlation functions such as pairing correlation and single particle green functions. According to our simulations, the partially polarized inhomogeneous Fermi gas appears to show no tendency of dynamically developing into a FFLO-type of state, contrary to what might have been expected.   

Non-equilibrium dynamics of the driven Hubbard model

We investigate the dynamics of a two-dimensional Hubbard model in a static electric field in order to identify the conditions to reach a non-equilibrium stationary state. For a generic electric field, the convergence to a stationary state requires the coupling to a thermostating bath absorbing the work done by the external force. Following the real-time dynamics of the system, we show that a non-equilibrium stationary state is reached for essentially any value of the coupling to the bath. We map out a phase diagram in terms of dissipation and electric field strengths and identify the dissipation values in which steady current is largest, and correspondingly a suitable entropy function is smallest, for a given field.   

Probing Phases and Quantum Criticality using Deviations from the Local Fluctuation-Dissipation Theorem

One of the major open questions in the study of ultracold atom systems is how to obtain the finite temperature phase diagram of a given Hamiltonian directly from experiments.\footnote{Q. Zhou, et al., {\it Phys. Rev. Lett}. {\bf 103}, 085701 (2009).} Previous work in this direction required quantum Monte Carlo simulations to directly model the experimental situation in order to extract quantitative information, clearly defeating the purpose of an optical lattice emulator. We propose a new method that utilizes deviations from a local fluctuation dissipation theorem to construct a finite temperature phase diagram, for the first time, from local observables accessible by {\it in situ} experimental observations. Our approach extends the utility of the fluctuation-dissipation theorem from thermometry to the identification of quantum phases, associated energy scales and the quantum critical region. We test our ideas using large-scale quantum Monte Carlo simulations of the two-dimensional Bose Hubbard model.\footnote{Y. Kato, et al., {\it Nature Physics} {\bf 4}, 592 - 593 (2008).}   

A new theoretical method to describe nonequilibrium cold atoms in optical lattices

We use perturbation theory in the hopping (strong-coupling expansion) to describe the nonequilibrium dynamics of strongly correlated fermions. Our expansion is a self-consistent expansion for the self-energy which goes beyond the RPA and allows for damping and relaxation effects. We apply this method to solve the homogeneous Fermi - Hubbard model driven by an external field. We investigate the damping of Bloch oscillations (for a uniform dc field) and show results for the current, the nonequilibrium density of states and the momentum distribution. We carefully benchmark the technique using the exact sum rules to determine its accuracy and we discuss regions of parameter space where the method no longer converges. This technique is quite competitive with other methods (such as DMFT) in the regions where it converges.   

A Slave Spin Impurity Solver for Non Equilibrium Dynamical Mean Field Theory

The non equilibrium dynamics of strongly correlated electronic systems represents a challenging theoretical problem in condensed matter physics with applications ranging from pump probe experiments in correlated materials to dynamics in ultracold atomic gases. Dynamical Mean Field Theory (DMFT) has emerged in recent years as a powerful theoretical framework to deal with strong correlations in a non pertubative way. Its extension to the out of equilibrium case requires the solution of an auxiliary quantum impurity model in a non equilibrium bath. Here we present an impurity solver for Non Equilibrium DMFT based on a slave spin representation of the fermionic degrees of freedom. We apply this method to study the quench dynamics in the single band fermionic Hubbard model and compare the results with the time dependent Gutzwiller method.   

A multi-site mean-field theory for cold bosonic atoms in optical lattices

Mean-field theory is one of the most commonly used approximate methods in condensed matter physics. As applied to the Bose-Hubbard model it provides a simple, qualitative explanation of the Mott insulator -- superfluid transition. In its usual form, one invokes a superfluid order parameter to decouple the Bose-Hubbard Hamiltonian into a sum of independent site Hamiltonians. Within this single-site mean-field theory (SSMFT) the equilibrium state is determined by minimizing the grand potential with respect to the order parameter. To improve on this procedure we have developed a multi-site mean-field theory (MSMFT), whereby the lattice is partitioned into small clusters which are decoupled by means of the usual mean-field method. The most general decoupling procedure necessitates the assignment of site-dependent order parameters to the sites bounding the cluster. This leads to a non-trivial topology of the grand potential and one finds, in general, that the equilibrium state is a saddle point. As a result, one cannot use a variational principle to locate the equilibrium states of interest. In this talk, we outline the MSMFT we have developed and give an example of its application.   

Feynman diagrams versus Feynman quantum emulator

Precise understanding of strongly interacting fermions, from electrons in modern materials to nuclear matter, presents a major goal in modern physics. However, the theoretical description of interacting Fermi systems is usually plagued by the intricate quantum statistics at play. Here we present a cross-validation between a new theoretical approach, Bold Diagrammatic Monte Carlo (BDMC), and precision experiments on ultra-cold atoms. Specifically, we compute and measure with unprecedented accuracy the normal-state equation of state of the unitary gas, a prototypical example of a strongly correlated fermionic system. Excellent agreement demonstrates that a series of Feynman diagrams can be controllably resummed in a non-perturbative regime using BDMC. This opens the door to the solution of some of the most challenging problems across many areas of physics.   

Non-equilibrium dynamics, heating, and thermalization of cold atoms in optical lattices

In recent years, out of equilibrium many-body dynamics have become accessible in a controlled way in experiments with ultracold quantum gases. Time-dependent processes in these systems are not only intrinsically interesting, but also extremely important for understanding many-body state preparation, heating, and thermalization as they arise in experiments. They also offer new connections to phenomena studied in solid-state systems. This interest has been complemented by the development of a range of numerical methods, including time-dependent density matrix renormalization group (t-DMRG) methods and matrix product state approaches, which have been applied to study dynamics in 1D lattice systems and spin chains. We have extended and applied these methods to study the non-equilibrium dynamics of cold atoms in optical lattices arising from different heating mechanisms, especially due to spontaneous emissions from incoherent scattering of the lattice light, or via classical noise on the optical potential. Understanding how these heating mechanisms affect the properties of different many-body states is crucial in addressing current experimental challenges in the preparation of interesting quantum phases at low temperatures. The resulting non-equilibrium dynamics typically depend strongly on the properties of the many-body state, with different states being more or less sensitive to different heating mechanisms. Moreover, there is often a separation of timescales between some excitations that thermalize rapidly, and some that do not properly thermalize in the duration of an experimental run, which can strongly modify, and even reduce the overall effects of the heating processes. Part of this work involves the treatment of open many-body quantum systems, where we derive many-body master equations to describe the dynamics and solve these numerically by combining t-DMRG methods with quantum trajectory techniques from quantum optics.   

Pattern formation in time-of-flight images of heavy-light mixtures of atoms undergoing Bloch oscillations

Nonequilibrium dynamical mean-field theory is employed to solve for the response of a light-heavy Fermi-Fermi mixture of atoms to the presence of a uniform electric field (via ``pulling'' the lattice through the cloud of atoms). When we express the (momentum-dependent) light atom distribution functions as functions of the band energy and the projection of the velocity along the direction of the artificial electric field, the system develops characteristic spiral patterns that become more complex as the system evolves, but remain stable for a long period of time. These patterns typically show a contrast of about 10-20\% fluctuations about the mean density, so they are challenging but possible to observe in a time-of-flight measurement. We also show a characteristic change of character of the system between small fields and large fields. The best candidate system for examining these patterns is a Li$^6$-K$^{40}$ mixture on a two-dimensional optical lattice. We expect similar results should occur for light Fermi-heavy Bose mixtures as well, but it is likely this behavior will not be seen in Bose-Bose mixtures.