README.md

Stochastic Series Expansion - SSE

Please cite the code uising the citation file provided in the repository. https://doi.org/10.5281/zenodo.10034200

Implementation of Stochastic Series Expansion (SSE) Monte Carlo method for the spin-S XXZ model. This version uses the Directed Loops [1] method for the loop update. The Hamiltonian of the simulated system is given by

$$ H = J \sum_{\langle i, j \rangle} \left[ \frac{1}{2} (S^+_i S^-_j + S^-_i S^+_j) + \Delta S^z_i S^z_j \right] - h \sum_i S^z_i $$

where $J$ is the coupling constant, $\Delta$ is the magnetic anisotropy along the $z$-direction and $h$ is an external magnetic field. In the code, $J = 1$, then if $\Delta > 0$ the system is antiferromagnetic and if $\Delta < 0$ the system is ferromagnetic. The SSE method is the power series expansion of the partition function

$$ Z = \text{Tr}{e^{-\beta H}} = \sum_{\alpha} \sum_{n=0}^{\infty} \frac{(-\beta)^n}{n!} \langle \alpha |H^n| \alpha \rangle $$

where $\beta \equiv 1 / T$.

The implementation uses binnig for the estimation of the standard deviations of the sampled quantities. It is possible to run each bin in parallel mode, using openMP.

Usage

To use and run the implementation, it is required that you have a C and fortran90 compilers and a version of Python3 installed with numpy.

To compile the code, run

$ source build.sh

in the main directory. This will compile the code and set the enviroment variable $SSE_DIR. To run the code, create a new directory and copy the parameter file (found in scripts_and_parameters/Start/) to the directory and type

$ $SSE_DIR/src/main n_threads $SSE_DIR

This will run the a SSE simulation with the paramertes specified in the parameter file using n_threads threads. To analyse the observables,

$ $SSE_DIR/src/ana obs_1 obs_2 ...

[1] - "Directed loop updates for quantum lattice models", Olav F. Syljuåsen, 2003, Phys. Rev. E 67, 046701, https://doi.org/10.1103/PhysRevE.67.046701