Stochastic Series Expansion - SSE

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 compiler (gcc-12, preferably), and a version of Python3 installed. To install GCC and Python3 you run the following commands if you have a Debian based OS (Ubuntu, Pop_OS, ...)

$ sudo apt install gcc python3

or if you have a MacOS based device with Homebrew

$ brew install gcc python3

You will also need an uptodate version of NumPy. You can do so by running the following command

$ pip3 install numpy

To run a simulation you will need to use the run.sh script and three input files. The first input file read.in contains the information about the system and MC parameters for the simulation.

1, 8, 0.5, 1.0, 0.0, 0.05, PBC
10000, 1000000, 10

The first line is d, L, S, delta, h, epsilon, boundary_cond and the second is therm_cycles, mc_cycles, n_bins. d is the system dimension, 1 and 2, L is the number of unit cells, S is the quantum spin number, delta and h are just the Hamiltonian parameters, epsilon is a parameter for the Directed Loops method and boundary_cond is the boundary condition for the lattice (can be PBC or OBC). epsilon has to be a value larger or equal to 0. Simulations perform best for low values of epsilon.

The second input file is beta.in which has the following structure

2
0.5
1.0

The first line is the number of beta values in the file and the next lines are the beta values for the MC simulation.

The final file matsubara.in is only available for OBC 1-dimensional problems. It contains the information about the measurement of spin conductivity.

5
4, 2

Here the first line is the number of matsubara frequencies to be computed by the system and the second is the x site for measuring the perturbation induced in the y site, respectively.

After setting the simulation parameters, you will need to run it via the run.sh script. You can type

$ ./run.sh -h

to get more information about the arguments. The most important ones are -n <n_threads> which specifies the number of threads used by openMP.

$ ./run.sh -n 5 

[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