NCA Solver Dissipation

A high-performance C++ numerical solver using the Non-Crossing Approximation (NCA) method to solve the dynamics of a fermionic impurity model in the presence of arbitrary Markovian dissipation.

Table of Contents

• Overview
• Physics Features
• Prerequisites
• Compilation
  • Standard Build
  • High-Performance Cluster (HPC) Build
• Code Structure
• Outputs

Overview

This solver handles the dynamics of an Anderson-like impurity coupled to a reservoir under Lindblad master equations. It utilizes the Liouville space representation (Tilde-expansion) to transform super-operators into matrix forms manageable by the Armadillo library.

Physics Features

• NCA Self-Consistent Integration: Iterative resolution of the self-energy equations.
• Dissipation Channels:
  • Standard Dephasing (Up, Down, Spin, Total Charge, Doublon).
  • Two-Body Lossy mechanisms.
• Bath Models: Support for both Flat Band and Semi-elliptic (gapless) fermionic reservoirs.
• Observables: Real-time calculation of:
  • Impurity occupation $n(t)$ and spin magnetization $\sigma_z(t)$.
  • Retarded Green's Functions $G^R(t)$ for Steady State analysis.

Prerequisites

• Compiler: g++ (C++11 or later).
• Library: Armadillo (Linear algebra).
• Dependencies: BLAS, LAPACK, ARPACK, and OpenBLAS.

Compilation

Standard Build

• For local testing with Armadillo installed:

g++ -O3 main.cpp -o nca_solver -larmadillo

High-Performance Cluster (HPC) Build

For deployment on a computation cluster, use the following optimized command to bypass wrappers and link directly against high-performance math libraries:

g++ -O3 -march=native -DNDEBUG -DARMA_DONT_USE_WRAPPER -DARMA_USE_BLAS -DARMA_USE_LAPACK main.cpp -o main -lopenblas -llapack -larpack -lpthread -lm

Compilation notes

• -O3 -march=native
  Enables hardware-specific optimizations.

• -DNDEBUG
  Disables runtime assertions.

• -DARMA_DONT_USE_WRAPPER
  Forces direct linking against external BLAS/LAPACK implementations (recommended on clusters).

Code Structure

• Construction_Matrix()
  Initializes impurity operators in both Hilbert and Liouville (tilde) spaces.

• Local_Lindbladian()
  Constructs the dissipative super-operator associated with the Lindblad terms.

• runDyn_NcaRK2_V2()
  Core solver implementing a second-order Runge–Kutta scheme for self-consistent NCA dynamics.

• Retarded_GreenFunction()
  Computes steady-state retarded Green’s functions.

Outputs

The solver generates .txt output files containing:

• TraceRho
  Time evolution of the density matrix trace.

• RetardedGreenFunction
  Real and imaginary parts of ( G^R ) for spin-up and spin-down channels.

• nt / Sigmaz
  Time evolution of impurity occupation and spin magnetization.

• RhoPolariser
  Full density matrix dynamics for polarized initial states.

⚠️ Warning:
Before execution, ensure that the SavePath variable in main.cpp points to a valid output directory on your system.

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Citation

If you use this solver in your research or project, please cite it as follows:

BibTeX:

@article{Vanhoecke_2025,
   title={Kondo-Zeno crossover in the dynamics of a monitored quantum dot},
   volume={16},
   ISSN={2041-1723},
   url={http://dx.doi.org/10.1038/s41467-025-61287-y},
   DOI={10.1038/s41467-025-61287-y},
   number={1},
   journal={Nature Communications},
   publisher={Springer Science and Business Media LLC},
   author={Vanhoecke, Matthieu and Schirò, Marco},
   year={2025},
   month=jul 
}

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Author: Matthieu Vanhoecke