It is shown that the nonequilibrium self-energy of an interacting lattice-fermion model has a unique Lehmann representation. Based on the construction of a suitable noninteracting effective medium, we provide an explicit and numerically practicable scheme to construct the Lehmann representation for the self-energy, given the Lehmann representation of the single-particle nonequilibrium Green's function. This is of particular importance for an efficient numerical solution of Dyson's equation in the context of approximations where the self-energy is obtained from a reference system with a small Hilbert space. As compared to conventional techniques to solve Dyson's equation on the Keldysh contour, the effective-medium approach allows us to reach a maximum propagation time, which can be several orders of magnitude longer. This is demonstrated explicitly by choosing the nonequilibrium cluster-perturbation theory as a simple approach to study the long-time dynamics of an inhomogeneous initial state after a quantum quench in the Hubbard model on a $10\ifmmode\times\else\texttimes\fi{}10$ square lattice. We demonstrate that the violation of conservation laws is moderate for weak Hubbard interaction and that the cluster approach is able to describe prethermalization physics.
It is shown that the nonequilibrium self-energy of an interacting lattice-fermion model has a unique Lehmann representation. Based on the construction of a suitable noninteracting effective medium, we provide an explicit and numerically practicable scheme to construct the Lehmann representation for the self-energy, given the Lehmann representation of the single-particle nonequilibrium Green's function. This is of particular importance for an efficient numerical solution of Dyson's equation in the context of approximations where the self-energy is obtained from a reference system with a small Hilbert space. As compared to conventional techniques to solve Dyson's equation on the Keldysh contour, the effective-medium approach allows us to reach a maximum propagation time, which can be several orders of magnitude longer. This is demonstrated explicitly by choosing the nonequilibrium cluster-perturbation theory as a simple approach to study the long-time dynamics of an inhomogeneous initial state after a quantum quench in the Hubbard model on a 10\ifmmode×\else\texttimes\fi10 square lattice. We demonstrate that the violation of conservation laws is moderate for weak Hubbard interaction and that the cluster approach is able to describe prethermalization physics.
We rigorously prove that any time-dependent, interacting lattice fermion model can be mapped to a corresponding noninteracting model. By construction, this effective model gives exact access to the one-particle Keldysh-Matsubara Green's function of the system -- at the cost of introducing an exponentially large number of virtual one-particle degrees of freedom. As the physics of the system are invariant under time-dependent unitary transformations involving these virtual degrees of freedom only, we explore two distinct realizations of the effective model: First, we choose the virtual subsystem to be diagonal which, upon tracing out the virtual sites, yields the nonequilibrium self-energy as a superposition of noninteracting isolated modes---the Lehmann representation. This result is highly useful to efficiently solve Dyson's equation numerically in contexts where an approximate self-energy is obtained from a small reference system. Calculating the effective noninteracting model instead of the self-energy, a memory-kernel-free time-propagation algorithm becomes possible.
This is demonstrated explicitly by choosing the nonequilibrium cluster perturbation theory (CPT) as a simple approach to study the long-time dynamics of an inhomogeneous initial state after a quantum quench in the Hubbard model on a 10x10 square lattice. We demonstrate that the violation of conservation laws is moderate for weak Hubbard interaction and that prethermalization physics are contained in the cluster approach. Improving upon plain CPT, we construct a conserving generalization. In form of local constraints on the local spin-dependent particle and the doublon density, we use the macroscopic conservation laws to self-consistently fix the time-dependent hopping in the reference cluster. Using a simple two-site cluster in a proof-of-concept calculation, we consider the dynamics of the two-dimensional, particle-hole-symmetric Hubbard model following a weak interaction quench. Indeed we find the dynamics profoundly altered in the conserving CPT. The doublon density shows a monotonous relaxation while strongly oscillating behavior is found within the plain CPT.
Leaving the topic of cluster-based theories, we also derive a second form of the effective model which is block-tridiagonal. The construction is analog to the Mori-Zwanzig projection technique which we generalize to Keldysh-Matsubara Green's functions. Based on our results we derive a selfconsistent nonequilibrium two-pole approach and explain how to enforce conservation laws. Finally, possible future applications are discussed.
Wir zeigen rigoros, dass jedes zeitabhangige wechselwirkende Gitterfermionenmodell auf ein anderweitig entsprechendes nicht-wechselwirkendes Modell abgebildet werden kann. Dieses effektive Modell ist so konstruiert, dass es exakten Zugriff auf die Einteilchen-Green-Funktion erlaubt --- zum Preis einer exponentiell grosen Anzahl zusatzlicher virtueller Freiheitsgrade. Da die Physik des Systems invariant unter zeitabhangigen unitaren Transformationen bleibt, welche ausschlieslich diese virtuellen Freiheitsgrade involvieren, betrachten wir zwei mogliche Realisationen des effektiven Modells: Zuerst wahlen wir das virtuelle Subsystem in diagonaler Form, wodurch sich bei Ausintegration der virtuellen Platze die Nichtgleichgewichts-Selbstenergie als Uberlagerung nichtwechselwirkender isolierter Moden erweist --- die Lehmann Darstellung. Dieses Resultat is sehr nutzlich, wenn eine numerische Losung der Dyson-Gleichung im Kontext von Approximationen benotigt wird, welche eine genahrte Selbstenergie aus einem kleinen Referenzsystem bestimmen. Indem man anstatt der Selbstenergie das effektive Modell bestimmt, wird ein Markovscher Zeitpropagationsalgorithmus moglich, d.h. ohne Verwendung einer Erinnerungsfunktion.
Wir demonstrieren dies explizit am einfachen Beispiel der zeitabhangigen Cluster-Storungstheorie (CPT), indem wir die Langzeitdynamik eines inhomogenen Anfangszustands nach einem Quanten-Quench, beschrieben durch das Hubbard Modell auf einem 10x10 Quadratgitter, simulieren. Wir zeigen, dass die Verletzung von Erhaltungssatzen im Regime kleiner Hubbard-Wechselwirkungen moderat bleibt, und, dass Prethermalisierungsphysik im Clusteransatz enthalten ist. Daraufhin leiten wir eine erhaltende CPT ab, welche die fundamentalen Erhaltungssatze respektiert. In Form lokaler Zwangsbedingungen fur die spinabhangige Teilchen- und Doublonendichte verwenden wir dabei die Erhaltungssatze, um die zeitabhangige Hupfmatrix im Referenzcluster zu bestimmen. In einer ersten Rechnung betrachten wir die Dynamik eines zweidimensionalen Hubbard-Modells nach einem Quench auf schwache Wechselwirkung. In der Tat ergibt sich in eine starke Veranderung der Dynamik in der erhaltenden CPT. Die Doublonendichte zeigt eine monotone Relaxation, wahrend sich in einer gewohnliche CPT-Rechnung stark oszillierendes Verhalten findet.
Im Folgenden verlassen wir das Thema clusterbasierter Theorien und leiten eine alternative, block-tridiagonale Darstellung des effektiven Modells ab. Die Konstruktion ist analog zum Mori-Zwanzig-Formalismus, welchen wir auf Keldysh-Matsubara-Green-Funktionen verallgemeinern. Basierend auf diesem Resultat leiten wir einen selbstkonsistenten Zweipol-Ansatz ab und zeigen, wie Erhaltungssatze in diesem erzwungen werden konnen. Zuletzt diskutieren wir mogliche zukunftige Anwendungen.
The self-energy functional theory (SFT) is generalized to describe the real-time dynamics of correlated lattice-fermion models far from thermal equilibrium. This is achieved by starting from a reformulation of the original equilibrium theory in terms of double-time Green's functions on the Keldysh-Matsubara contour. With the help of a generalized Luttinger-Ward functional, we construct a functional $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\Omega}}[\ensuremath{\Sigma}]$ which is stationary at the physical (nonequilibrium) self-energy $\ensuremath{\Sigma}$ and which yields the grand potential of the initial thermal state $\ensuremath{\Omega}$ at the physical point. Nonperturbative approximations can be defined by specifying a reference system that serves to generate trial self-energies. These self-energies are varied by varying the reference system's one-particle parameters on the Keldysh-Matsubara contour. In the case of thermal equilibrium, this approach reduces to the conventional SFT. Contrary to the equilibrium theory, however, ``unphysical'' variations, i.e., variations that are different on the upper and the lower branches of the Keldysh contour, must be considered to fix the time dependence of the optimal physical parameters via the variational principle. Functional derivatives in the nonequilibrium SFT Euler equation are carried out analytically to derive conditional equations for the variational parameters that are accessible to a numerical evaluation via a time-propagation scheme. Approximations constructed by means of the nonequilibrium SFT are shown to be inherently causal, internally consistent, and to respect macroscopic conservation laws resulting from gauge symmetries of the Hamiltonian. This comprises the nonequilibrium dynamical mean-field theory but also dynamical-impurity and variational-cluster approximations that are specified by reference systems with a finite number of degrees of freedom. In this way, nonperturbative and consistent approximations can be set up, the numerical evaluation of which is accessible to an exact-diagonalization approach.
The nonequilibrium Dyson (or Kadanoff-Baym) equation, which is an equation of motion with a long-range memory kernel for real-time Green functions, underlies many numerical approaches based on the Keldysh formalism. In this paper we map the problem of solving the Dyson equation in real time onto a noninteracting auxiliary Hamiltonian with additional bath degrees of freedom. The solution of the auxiliary model does not require the evaluation of a memory kernel and can thus be implemented in a very memory efficient way. The mapping is derived for a self-energy which is local in space and is thus directly applicable within nonequilibrium dynamical mean-field theory (DMFT). We apply the method to study the interaction quench in the Hubbard model for an optical lattice with a narrow confinement, using inhomogeneous DMFT in combination with second-order weak-coupling perturbation theory. We find that, although the quench excites pronounced density oscillations, signatures of the two-stage relaxation similar to the homogeneous system can be observed by looking at the time-dependent occupations of natural orbitals.
The framework of nonequilibrium Green functions (NEGFs) is developed to accurately describe interacting quantum systems evolving in the presence of strongly exciting ultra-fast fields.At its core stands the selfconsistent determination of the selfenergy, a functional of the single-particle Green function, which incorporates all relevant many-body physics and, via diagrammatic techniques, guarantees the conservation of the constants of motion.Over the last decades, NEGFs have been successfully applied to a broad range of domains ranging from homogeneous systems such as non-ideal quantum gases and plasmas over condensed matter to highly inhomogeneous systems such as atoms and molecules photo-ionized by laser-fields or colliding nuclear matter.Only recently, the first accurate simulation of the expansion dynamics of clouds of ultra-cold fermionic atoms in a two-dimensional optical lattice could be achieved using NEGFs.The single-particle Green function is the solution of a coupled set of kinetic equations.Its intrinsic dependence on two independent time-arguments, ranging over the complete past of the system, not only enables the determination of all singleparticle observables but also provides access to important two-particle observables and spectral information.With this coupling to the history, though, comes the practical necessity to apply physically motivated approximation techniques to the structure of the selfenergy and the Green functions, to achieve numerical tractability.This thesis is devoted to provide a thorough overview of the possible approximation techniques including perturbative and non-perturbative approaches.Starting from the exact coupled set of equations of motion for the Green functions, the expressions for all selfenergy approximations are derived in the most general basis representation, i.e., for interaction tensor elements w ijkl .From this, the corresponding expressions in a diagonal basis, where the interaction can be described as a matrix w ij , and for the special case of the Hubbard model, where the interaction is just given as a scalar U , are deduced.
We present a nonequilibrium Green's-functional approach to study the dynamics following a quench in weakly interacting Bose-Hubbard model (BHM). The technique is based on the self-consistent solution of a set of equations which represents a particular case of the most general set of Hedin's equations for the interacting single-particle Green's function. We use the ladder approximation as a skeleton diagram for the two-particle scattering amplitude useful, through the self-energy in the Dyson equation, for finding the interacting single-particle Green's function. This scheme is then implemented numerically by a parallelized code. We exploit this approach to study the correlation propagation after a quench in the interaction parameter, for one and two dimensions. In particular, we show how our approach is able to recover the crossover from the ballistic to the diffusive regime by increasing the boson-boson interaction. Finally we also discuss the role of a thermal initial state on the dynamics both for one- and two-dimensional BHMs, finding that, surprisingly, at high temperature a ballistic evolution is restored.
In this article, we present a concise and self-contained introduction to nonequilibrium statistical mechanics with quantum field theory by considering an ensemble of interacting identical bosons or fermions as an example. Readers are assumed to be familiar with the Matsubara formalism of equilibrium statistical mechanics such as Feynman diagrams, the proper self-energy, and Dyson's equation. The aims are threefold: (i) to explain the fundamentals of nonequilibrium quantum field theory as simple as possible on the basis of the knowledge of the equilibrium counterpart; (ii) to elucidate the hierarchy in describing nonequilibrium systems from Dyson's equation on the Keldysh contour to the Navier-Stokes equation in fluid mechanics via quantum transport equations and the Boltzmann equation; (iii) to derive an expression of nonequilibrium entropy that evolves with time. In stage (i), we introduce nonequilibrium Green's function and the self-energy uniquely on the round-trip Keldysh contour, thereby avoiding possible confusions that may arise from defining multiple Green's functions at the very beginning. We try to present the Feynman rules for the perturbation expansion as simple as possible. In particular, we focus on the self-consistent perturbation expansion with the Luttinger-Ward thermodynamic functional, i.e., Baym's Phi-derivable approximation, which has a crucial property for nonequilibrium systems of obeying various conservation laws automatically. We also show how the two-particle correlations can be calculated within the Phi-derivable approximation, i.e., an issue of how to handle the "Bogoliubov-Born-Green-Kirkwood-Yvons (BBGKY) hierarchy".