NESSi
v1.0.1
The NonEquilibrium Systems Simulation Library

NESSi
is an opensource software package for the manipulation of nonequilibrium Green’s functions defined on the KadanoffBaym contour. The Green's function method in its timedependent formulation is a versatile framework for the solution of interacting manybody problems out of equilibrium. NESSi
provides classes representing the various types of Green’s functions, implements the basic operations on these functions and allows to solve the corresponding equations of motion. The library is aimed at the study of transient dynamics from an initial equilibrium state, induced by timedependent model parameters.
NESSi
provides tools for constructing Feynman diagram and solving equations of motion for nonequilibrium Green's functions on the KadanoffBaym contourNESSi
is based on highorder quadrature rules: for \(N\) time slices, the error scales like up to \(\mathcal{O}(N^{p})\) with \(p\) up to \(7\).Please cite the following paper whenever you use parts of NESSi
:
M. Schüler, D. Golež, Y. Murakami, N. Bittner, A. Herrmann, H. U. R. Strand, P. Werner, and M. Eckstein, NESSi: The NonEquilibrium Systems Simulation package, arXiv:1911.01211 (2019)
NESSi
is the shared library libcntr
. It is written in C++ and provides the essential functionalities to treat Green's functions on the KadanoffBaym contour (see Physics background).libcntr
library (see Manual).libcntr
based programs.NESSi
package also contains a number of simple example programs, which demonstrate the usage and functionalities of libcntr
(see Example programs).libcntr
library and the example programs depend on the eigen3
library which implements efficient matrix operations. Furthermore, the hdf5
library and file format can be used for creating binary, machineindependent output such as Green's functions, for instance. The usage of the hdf5
library in the NESSi
package is, however, optional.libcntr
and for reading and postprocessing Green's functions from hdf5
format via the h5py
python package (see Python tools).libcntr
library for solutions of strongly correlated impurity problems and corresponding dynamical mean field theory calculations is in preparation and will be published separately (PseudoParticle Strong coupling (PPSC) library).The libcntr
library provides highly accurate methods for calculating nonequilibrium Green's functions and more. A brief overview of the core routines is presented below. All routines work for fermions and bosons.
Summary of the main routines in libcntr  
Green's function for a constant or timedependent Hamiltonian  Constructs the free Green's functions for a general timedependent Hamiltonian. 
Dyson equation  Solves the Dyson equation along the full KadanoffBaym contour \({\cal C}\): \( [i\partial_t  h(t)] G (t,t')  [\Sigma * G](t,t') = \delta_{\cal C} (t,t') \) for a given selfenergy \(\Sigma(t,t')\). In particular, thermal equilibrium and time evolution are treated on equal footing. 
VIE2  Solves the contour integral equations of the type \( G(t,t^\prime) + [F\ast G](t,t^\prime) = Q(t,t^\prime) \) for given kernel \(F(t,t^\prime)\) and \(Q(t,t^\prime)\). A typical example is the selfconsistent \(GW\) approximation, where the screened interaction obeys the Dyson equation \(W = V + V\ast \Pi \ast W\). Here \(V\) denotes the bare Coulomb interaction, while \(\Pi\) stands for the irreducible polarization. 
Convolution  Computes the convolution \([A\ast B](t,t^\prime) = \int_{\mathcal{C}} d\bar{t} \, A(t,\bar{t}) B(\bar{t},t^\prime)\) of two contour functions \(A(t,t^\prime)\) and \(B(t,t^\prime)\), which is an essential part of solving of the integrodifferential equations. 
Diagram utilities  Constructs the bubble diagrams of the type \( C(t,t^\prime)= i A(t,t^\prime) B(t^\prime, t) \quad \text{or}\quad C(t,t^\prime)= i A(t,t^\prime) B(t, t^\prime)\). 
While the NESSi
package provides a general framework for realtime Green's functions and can be applied to a broad range of nonequilibrium problems, it has so far been mainly used in the context of realtime dynamical meanfield theory (DMFT) simulations. In order to perform DMFT calculations one has to implement a solver for the DMFT effective impurity problem. Two approximate approaches are (i) weak coupling expansions, such as Iterated Perturbation Theory (IPT) and (ii) strong coupling methods.
The strong coupling based methods involve pseudoparticles, one for each state in the local Hilbert space, which obey specific types of Dyson equations. This formulation solves the atomic problem exactly and treats the hybridization with the environment perturbatively. The first and second order dressed expansion of this method is commonly known as the NonCrossing Approximation (NCA) and the OneCrossing Approximation (OCA).
A library implementing these methods called the PseudoParticle Strong Coupling (PPSC) library is currently under development. This library is based on libcntr
and we plan to make it public in the future.
The development of this library has been supported by the Swiss National Science Foundation through SNF Professorship PP0022118866 (ME,PW), Grants 200021140648 and 200021165539 (DG), and NCCR MARVEL (MS,YM), as well as the European Research Council through ERC Starting Grants No. 278023 (AH,HS,PW) and No. 716648 (ME), and ERC Consolidator Grant No. 724103 (MS,NB,PW,YM). The Flatiron Institute as a division of the Simons Foundation. We would like to acknowledge F. Petocchi for help with graphical representation of NESSi logo.
This source code is subject to the terms of the Mozilla Public License, v. 2.0. A copy of the MPL one can obtain at https://mozilla.org/MPL/2.0/.
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