Fermionic Markovian closure

This repository contains the code used for the simulations in [1]. There are some scripts computing the evolution of several types of physical systems, employing the Markovian closure technique as explained in the article.

Installation

1) Clone the repository on your computer. 2) This project depends on some packages that aren't found in Julia's general registry, but in this one, so you need to add it first in order to download those packages automatically (you can skip this step if you already added the registry previously: you need to do it just once per Julia installation). Open a Julia interactive session and enter the Pkg REPL by hitting ] (see the Getting Started with Environments guide) and then run registry add https://github.com/phaerrax/TensorNetworkSimulations.git:

```julia-repl
(@v1.11) pkg> registry add https://github.com/phaerrax/TensorNetworkSimulations.git
     Cloning registry from "https://github.com/phaerrax/TensorNetworkSimulations.git"
       Added registry `TensorNetworkSimulations` to `~/.julia/registries/TensorNetworkSimulations`
```

(If this is the first time using Julia then run `registry add General` too
otherwise the `General` registry won't be automatically added.)

3) Activate the Julia project in this folder with activate .

```julia-repl
(@v1.11) pkg> activate .
  Activating project at `~/markovian_closure_fermions`
```

4) Add the OpenSystemsChainMapping package and download all the dependencies by running dev ./OpenSystemsChainMapping and then instantiate.

Description of the repository

The repository contains a package, called OpenSystemsChainMapping, which defines some "core" functions that allow simulating some open quantum systems in contact with one or two environments.

You will find:

• some Julia scripts which perform the simulation of some physical models such as the SIAM (siam) or a quantum dot impurity (qdot), either using a standard TEDOPA method (pure….jl) or the Markovian closure (mc….jl);
• an example folder which contains the parameter files of some concrete simulations, as well as some frequently used spectral densities.
• some Julia scripts that calculate the chain coefficients, in several ways, starting from the data of a spectral density (chainmapping_….jl);

How it works (WIP)

The parameters for each simulation script must be supplied, in a JSON file, as the first (and only) command-line argument. You can find some examples in the test/spectral_densities directory. Please note that the script that computes the thermofield coefficients currently works only if the chemical potential is zero (this does not affect the generality of our algorithms, since any spectral density can always be shifted so that its chemical potential is zero), so for now care must be taken to write the parameter files according to this specification. All files in test/spectral_densities already follow this convention. In order to run a simulation script, run from the base folder (i.e. the root of the Git repository)

bash julia --project <script.jl> <parameter_file.json>

using the full relative paths of the files, e.g.

bash julia --project siam/spinless/mc.jl test/siam/spinless/mu1/NE8/mc60_NC6.json

Example

Here is an example of a complete workflow, starting from scratch. We want to simulate a spinless SIAM with some given parameter files: test/spectral_densities_semicircle_T4_mu0.5.json representing a semicircle spectral density, and examples/siam_mc.jl with the parameters for the physical simulation of the model with a Markovian closure.

1. Generate the thermofield coefficients from the spectral density, with

bash julia --project chainmapping_thermofield.jl examples/spectral_densities/semicircle_T4_mu0.5.json

The output is a file called test/spectral_densities/semicircle_T4_mu0.5.thermofield, which will be called later in examples/siam_mc.json in the chain_coefficients entry. It is not necessary to generate the coefficient each time, if the file already exists. 2. Run the simulation script with

bash julia --project siam/spinless/mc.jl examples/siam_spinless_mc.json

Note that the parameter file examples/siam_mc.json also specifies some output files which will contain the expectation values of the given observables, an HDF5 file containing the final state, and so on. If one does not need such results, /dev/null or an equivalent destination may be given to avoid creating unnecessary output files.

The examples directory contains other sample parameter files that can be used with other simulation scripts:

bash julia --project siam/spinless/pure.jl examples/siam_spinless_pure.json julia --project siam/spinful/mc.jl examples/siam_spinful_mc.json julia --project qdot/mc_2l.jl examples/qdot_2levels_mc.json

Common parameters and their meaning

• Chain coefficients:

  • chain_coefficients: a file containing the chain coefficients
  • chain_length: an integer specifying how many sites to keep in the environment chains before attaching the closure

• Output files:

  • out_file: expectation values of observables
  • state_file: final MPS (in HDF5 format)
  • ranks_file: bond dimensions of the MPS, step by step
  • times_file: wall-clock time needed for each evolution step

• Integration parameters for the simulation:

  • tmax: final (physical) time of the simulation
  • tstep: integration time step
  • ms_stride: measure observables only each ms_stride steps
  • max_bond: maximum bond dimension for the MPS
  • discarded_w: cutoff for MPS truncation during the evolution

• Observables:

  • observables: a vector specifying the name of the observable (a valid ITensor operator) and a list of sites on which it will be measured, e.g.

    json { "vN": [1,2,3,10,11,16,17], "vX": [1,2,3,4] }

  (note that the v prefix here is used for the "vectorized fermion" systems --- please consult the LindbladVectorizedTensors package documentation for more information)

• Markovian closure parameters:

  • MC_alphas: mc_standard_parameters/alphas_6.dat,
  • MC_betas: mc_standard_parameters/betas_6.dat,
  • MC_coups: mc_standard_parameters/coupls_6.dat

  These are already existing files. You only need to change the number in the file name from 6 to 8 or 10 if you want to use closures with a different number of modes.

• Other system-specific parameters, e.g. when the open system is a two-level system (a fermion, maybe) we need:

  • sys_en: energy of the open system
  • sys_ini: initial state of the open system

References

[1] Ferracin, D., Smirne, A., Huelga, S. F., Plenio M. B. and Tamascelli, D. (2024). Spectral Density Modulation and Universal Markovian Closure of Fermionic Environments. arxiv.org/abs/2407.10017.