Using the recently introduced time-local formulation of the nonequilibrium cluster perturbation theory (CPT), we construct a generalization of the approach such that macroscopic conservation laws are respected. This is achieved by exploiting the freedom for the choice of the starting point of the all-order perturbation theory in the intercluster hopping. The proposed conserving CPT is a self-consistent propagation scheme which respects the conservation of energy, particle number, and spin, which treats short-range correlations exactly up to the linear scale of the cluster, and which represents a mean-field-like approach on length scales beyond the cluster size. Using Green's functions, conservation laws are formulated as local constraints on the local spin-dependent particle and the doublon density. We consider them as conditional equations to self-consistently fix the time-dependent intracluster one-particle parameters. Thanks to the intrinsic causality of the CPT, this can be set up as a step-by-step time propagation scheme with a computational effort scaling linearly with the maximum propagation time and exponentially in the cluster size. As a proof of concept, we consider the dynamics of the two-dimensional, particle-hole-symmetric Hubbard model following a weak interaction quench by simply employing two-site clusters only. Conservation laws are satisfied by construction. We demonstrate that enforcing them has strong impact on the dynamics. While the doublon density is strongly oscillating within plain CPT, a monotonic relaxation is observed within the conserving CPT.
Using the recently introduced time-local formulation of the nonequilibrium cluster perturbation theory (CPT), we construct a generalization of the approach such that macroscopic conservation laws are respected. This is achieved by exploiting the freedom for the choice of the starting point of the all-order perturbation theory in the intercluster hopping. The proposed conserving CPT is a self-consistent propagation scheme which respects the conservation of energy, particle number, and spin, which treats short-range correlations exactly up to the linear scale of the cluster, and which represents a mean-field-like approach on length scales beyond the cluster size. Using Green's functions, conservation laws are formulated as local constraints on the local spin-dependent particle and the doublon density. We consider them as conditional equations to self-consistently fix the time-dependent intracluster one-particle parameters. Thanks to the intrinsic causality of the CPT, this can be set up as a step-by-step time propagation scheme with a computational effort scaling linearly with the maximum propagation time and exponentially in the cluster size. As a proof of concept, we consider the dynamics of the two-dimensional, particle-hole-symmetric Hubbard model following a weak interaction quench by simply employing two-site clusters only. Conservation laws are satisfied by construction. We demonstrate that enforcing them has strong impact on the dynamics. While the doublon density is strongly oscillating within plain CPT, a monotonic relaxation is observed within the conserving CPT.
The cluster perturbation theory (CPT) is one of the simplest but systematic quantum cluster approaches to lattice models of strongly correlated electrons with local interactions. By treating the interactions, in addition to the intercluster potential, as a perturbation, it is shown that the CPT can be reformulated as an all-order resummation of diagrams within standard weak-coupling perturbation theory where vertex corrections are neglected. This reformulation is shown to allow for a straightforward generalization of the CPT to the general nonequilibrium case using contour-ordered Green's functions. By solving the resulting generalized CPT equation on the discretized Keldysh-Matsubara time contour, the transient dynamics of an essentially arbitrary initial pure or mixed state can be traced. In this way, the time-dependent expectation values of one-particle observables can be obtained within an approximation that neglects spatial correlations beyond the extension of the reference cluster. The necessary computational effort is very moderate. A detailed discussion and simple test calculations are presented to demonstrate the strengths and the shortcomings of the proposed approach. The nonequilibrium CPT is systematic and is controlled in principle by the inverse cluster size. It interpolates between the noninteracting and the atomic or decoupled-cluster limit which are recovered exactly and is found to predict the correct dynamics at very short times in a general nontrivial case. The effects of initial-state correlations on the subsequent dynamics and the necessity to extend the Keldysh contour by the imaginary Matsubara branch are analyzed carefully and demonstrated numerically. It is further shown that the approach can describe the dissipation of spin and charge to an uncorrelated bath with an essentially arbitrary number of degrees of freedom.
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.
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.
We study the effect of spatially nonlocal correlations on the nonequilibrium dynamics of interacting fermions by constructing the nonequilibrium dynamical cluster theory, a cluster generalization of the nonequilibrium dynamical mean-field theory (DMFT). The formalism is applied to interaction quenches in the Hubbard model in one and two dimensions, and the results are compared with data from single-site DMFT, the time-dependent density matrix renormalization group, and lattice perturbation theory. Both in one and two dimensions the double occupancy quickly thermalizes, while the momentum distribution relaxes only on much longer time scales. For the two-dimensional square lattice we find a strongly momentum-dependent evolution of the momentum distribution around the Fermi energy, with a much faster relaxation near the momenta $(0,\ensuremath{\pi})$ and $(\ensuremath{\pi},0)$ than near $(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$. This result is interpreted as reflecting the momentum-anisotropic quasiparticle lifetime of the marginal Fermi liquid. The method is further applied to the two-dimensional Hubbard model driven by a dc electric field, where the damping of the Bloch oscillation of the current is found to be less effective than predicted by DMFT and lattice perturbation theory.
The Hubbard model is a prototype for strongly correlated many-particle systems, including electrons in condensed matter and molecules, as well as for fermions or bosons in optical lattices. While the equilibrium properties of these systems have been studied in detail, the nonequilibrium dynamics following a strong nonperturbative excitation only recently came into the focus of experiments and theory. It is of particular interest how the dynamics depend on the coupling strength and on the particle number and whether there exist universal features in the time evolution. Here, we present results for the dynamics of finite Hubbard clusters based on a self-consistent nonequilibrium Green functions (NEGF) approach invoking the generalized Kadanoff-Baym ansatz (GKBA). We discuss the conserving properties of the GKBA with Hartree-Fock propagators in detail and present a generalized form of the energy conservation criterion of Baym and Kadanoff for NEGF. Furthermore, we demonstrate that the HF-GKBA cures some artifacts of prior two-time NEGF simulations. Besides, this approach substantially speeds up the numerical calculations and thus presents the capability to study comparatively large systems and to extend the analysis to long times allowing for an accurate computation of the excitation spectrum via time propagation. Our data obtained within the second Born approximation compare favorably with exact diagonalization results (available for up to 13 particles) and are expected to have predictive capability for substantially larger systems in the weak-coupling limit.