Non-equilibrium cluster-perturbation theory

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 inter-cluster potential, in addition to the interactions, as a perturbation, it is shown that the CPT can be reformulated as an all-order re-summation 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 non-equilibrium case using contour-ordered Green's functions. 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 non-equilibrium CPT is systematic and is controlled in principle by the inverse cluster size. It interpolates between the non-interacting 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 non-trivial 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 furthermore 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.Comment: 14 pages, 9 figure

Non-equilibrium Green's function approach to inhomogeneous quantum many-body systems using the Generalized Kadanoff Baym Ansatz

In non-equilibrium Green's function calculations the use of the Generalized Kadanoff-Baym Ansatz (GKBA) allows for a simple approximate reconstruction of the two-time Green's function from its time-diagonal value. With this a drastic reduction of the computational needs is achieved in time-dependent calculations, making longer time propagation possible and more complex systems accessible. This paper gives credit to the GKBA that was introduced 25 years ago. After a detailed derivation of the GKBA, we recall its application to homogeneous systems and show how to extend it to strongly correlated, inhomogeneous systems. As a proof of concept, we present results for a 2-electron quantum well, where the correct treatment of the correlated electron dynamics is crucial for the correct description of the equilibrium and dynamic properties

Efficient grid-based method in nonequilibrium Green's function calculations. Application to model atoms and molecules

We propose and apply the finite-element discrete variable representation to express the nonequilibrium Green's function for strongly inhomogeneous quantum systems. This method is highly favorable against a general basis approach with regard to numerical complexity, memory resources, and computation time. Its flexibility also allows for an accurate representation of spatially extended hamiltonians, and thus opens the way towards a direct solution of the two-time Schwinger/Keldysh/Kadanoff-Baym equations on spatial grids, including e.g. the description of highly excited states in atoms. As first benchmarks, we compute and characterize, in Hartree-Fock and second Born approximation, the ground states of the He atom, the H$_2$ molecule and the LiH molecule in one spatial dimension. Thereby, the ground-state/binding energies, densities and bond-lengths are compared with the direct solution of the time-dependent Schr\"odinger equation.Comment: 11 pages, 5 figures, submitted to Physical Review

A new approach to hierarchical data analysis: Targeted maximum likelihood estimation for the causal effect of a cluster-level exposure

We often seek to estimate the impact of an exposure naturally occurring or randomly assigned at the cluster-level. For example, the literature on neighborhood determinants of health continues to grow. Likewise, community randomized trials are applied to learn about real-world implementation, sustainability, and population effects of interventions with proven individual-level efficacy. In these settings, individual-level outcomes are correlated due to shared cluster-level factors, including the exposure, as well as social or biological interactions between individuals. To flexibly and efficiently estimate the effect of a cluster-level exposure, we present two targeted maximum likelihood estimators (TMLEs). The first TMLE is developed under a non-parametric causal model, which allows for arbitrary interactions between individuals within a cluster. These interactions include direct transmission of the outcome (i.e. contagion) and influence of one individual's covariates on another's outcome (i.e. covariate interference). The second TMLE is developed under a causal sub-model assuming the cluster-level and individual-specific covariates are sufficient to control for confounding. Simulations compare the alternative estimators and illustrate the potential gains from pairing individual-level risk factors and outcomes during estimation, while avoiding unwarranted assumptions. Our results suggest that estimation under the sub-model can result in bias and misleading inference in an observational setting. Incorporating working assumptions during estimation is more robust than assuming they hold in the underlying causal model. We illustrate our approach with an application to HIV prevention and treatment

On the Coulomb-dipole transition in mesoscopic classical and quantum electron-hole bilayers

We study the Coulomb-to-dipole transition which occurs when the separation $d$ of an electron-hole bilayer system is varied with respect to the characteristic in-layer distances. An analysis of the classical ground state configurations for harmonically confined clusters with $N\leq30$ reveals that the energetically most favorable state can differ from that of two-dimensional pure dipole or Coulomb systems. Performing a normal mode analysis for the N=19 cluster it is found that the lowest mode frequencies exhibit drastic changes when $d$ is varied. Furthermore, we present quantum-mechanical ground states for N=6, 10 and 12 spin-polarized electrons and holes. We compute the single-particle energies and orbitals in self-consistent Hartree-Fock approximation over a broad range of layer separations and coupling strengths between the limits of the ideal Fermi gas and the Wigner crystal

Quantum Breathing Mode of Interacting Particles in a One-dimensional Harmonic Trap

Extending our previous work, we explore the breathing mode---the [uniform] radial expansion and contraction of a spatially confined system. We study the breathing mode across the transition from the ideal quantum to the classical regime and confirm that it is not independent of the pair interaction strength (coupling parameter). We present the results of time-dependent Hartree-Fock simulations for 2 to 20 fermions with Coulomb interaction and show how the quantum breathing mode depends on the particle number. We validate the accuracy of our results, comparing them to exact Configuration Interaction results for up to 8 particles

Exploring Millions of 6-State FSSP Solutions: the Formal Notion of Local CA Simulation

In this paper, we come back on the notion of local simulation allowing to transform a cellular automaton into a closely related one with different local encoding of information. This notion is used to explore solutions of the Firing Squad Synchronization Problem that are minimal both in time (2n -- 2 for n cells) and, up to current knowledge, also in states (6 states). While only one such solution was proposed by Mazoyer since 1987, 718 new solutions have been generated by Clergue, Verel and Formenti in 2018 with a cluster of machines. We show here that, starting from existing solutions, it is possible to generate millions of such solutions using local simulations using a single common personal computer

Adaptive Matching in Randomized Trials and Observational Studies

In many randomized and observational studies the allocation of treatment among a sample of n independent and identically distributed units is a function of the covariates of all sampled units. As a result, the treatment labels among the units are possibly dependent, complicating estimation and posing challenges for statistical inference. For example, cluster randomized trials frequently sample communities from some target population, construct matched pairs of communities from those included in the sample based on some metric of similarity in baseline community characteristics, and then randomly allocate a treatment and a control intervention within each matched pair. In this case, the observed data can neither be represented as the realization of n independent random variables, nor, contrary to current practice, as the realization of n/2 independent random variables (treating the matched pair as the independent sampling unit). In this paper we study estimation of the average causal effect of a treatment under experimental designs in which treatment allocation potentially depends on the pre-intervention covariates of all units included in the sample. We define efficient targeted minimum loss based estimators for this general design, present a theorem that establishes the desired asymptotic normality of these estimators and allows for asymptotically valid statistical inference, and discuss implementation of these estimators. We further investigate the relative asymptotic efficiency of this design compared with a design in which unit-specific treatment assignment depends only on the units\u27 covariates. Our findings have practical implications for the optimal design and analysis of pair matched cluster randomized trials, as well as for observational studies in which treatment decisions may depend on characteristics of the entire sample

Mott transition in one dimension: Benchmarking dynamical cluster approaches

The variational cluster approach (VCA) is applied to the one-dimensional Hubbard model at zero temperature using clusters (chains) of up to ten sites with full diagonalization and the Lanczos method as cluster solver. Within the framework of the self-energy-functional theory (SFT), different cluster reference systems with and without bath degrees of freedom, in different topologies and with different sets of variational parameters are considered. Static and one-particle dynamical quantities are calculated for half-filling as a function of U as well as for fixed U as a function of the chemical potential to study the interaction- and filling-dependent metal-insulator (Mott) transition. The recently developed Q-matrix technique is used to compute the SFT grand potential. For benchmarking purposes we compare the VCA results with exact results available from the Bethe ansatz, with essentially exact dynamical DMRG data, with (cellular) dynamical mean-field theory and full diagonalization of isolated Hubbard chains. Several issues are discussed including convergence of the results with cluster size, the ability of cluster approaches to access the critical regime of the Mott transition, efficiency in the optimization of correlated-site vs. bath-site parameters and of multi-dimensional parameter optimization. We also study the role of bath sites for the description of excitation properties and as charge reservoirs for the description of filling dependencies. The VCA turns out to be a computationally cheap method which is competitive with established cluster approaches.Comment: 19 pages, 19 figures, v3 with minor corrections, extended discussio