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 an input file. This input file contains all of the information about the simulation and simualted system. This is the structure of the input file:

# d, L, S, delta, h, epsilon
1, 8, 0.5, 1.0, 0.0, 0.05

# therm_cycles, mc_cycles, n_bins
10000, 1000000, 10

# test_temps
# len_beta
10
# temp_range
0.05, 4.0

The first line are the system parameters, 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, and epsilon is a parameter for the Directed Loops method. epsilon has to be a value larger or equal to 0. Simulations perform best for low values of epsilon.

To use the test temperatures ( $\beta = {0.5, 1.0, 2.0, 4.0, 8.0, 16.0}$ ), you will need to "uncomment" the line test_tmps. Else, you will have to specify how many temperature point (len_beta) you want to generate, equaly spaced, between the temp_range variables.

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, -i <input_name> and -o <output_name> are used for specifing the input and output filenames. For ease of use, the extensions for the input and output files should be .txt and .csv, respectively. For instance, to run a simulation where the input file is called sim_input.txt and we want an output in the directory outputs/ named sim_output.csv, with 5 threads, you will have to write

$ ./run.sh -n 5 -i input.txt -o outputs/sim_output.csv

The output file has the following structure:

beta,n,n2,n_std,E,E_std,C,C_std,m,m_std,m2,m2_std,m4,m4_std,ms,ms_std,m2s,m2s_std,m4s,m4s_std,sus,sus_std,binder,binder_std

[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