Monte Carlo Simulation of 2D Random Bond Ising Model

Originally Created 3/12/2022 Zhaoyi Li

Technical Details

Hamiltonian: The Hamiltonian is given by $H=J_{ij}\sum_{<i,j>}\sigma_i\sigma_j$, where each spin $\sigma_i$ takes on the value ${-1,1}.$

Assuming constant coupling

$$J_{ij}=J$$

In this case, the thermodynamic variablesare given by: $$\chi\sim\Delta M, C\sim\Delta E$$ Want to compute $\langle E\rangle$, $\langle E^2\rangle$, $\langle M\rangle$, $\langle M^2\rangle$ with thermodynamic average: $$\langle X\rangle = \frac{1}{Z}\sum_{{\sigma}}X({\sigma})e^{-\beta H({\sigma})}$$

Adding randomness in $J_{\langle ij\rangle}$

$$P[J_{ij}] = (1-p)\delta(J_{ij}-J)+p\delta(J_{ij}+J)$$

Here $\langle X\rangle = \mathbb{E}[\frac{1}{Z}\sum_{{\sigma}}X({\sigma})e^{-\beta H({\sigma})}]$

Background

$\cdot$ spin glass
$\cdot$ quantum error correction

Methodology

Metropolis algorithm:

$$w(a\leftarrow b) = \min{(1,e^{\beta(E_a-E_b)})}$$ therefore satisfying the condition: $$\frac{w(a\leftarrow b)}{w(b\leftarrow a)}=\frac{e^{-\beta E_a}}{e^{-\beta E_b}}$$

Variables

This program accepts command-line arguments, which can be used to customize its behavior. The available options for each subprogram are listed in the following table, along with any options that are not available. Hover your mouse over to see more details.

Option	Description	Type	Default Value
-f, --fname	Output filename, the default file name is in the ../data/ folder	string	N/A
--out	output in this directory as an .out file	bool	0
--waitSweep	number of sweeps to wit before sampling	int	10
--rptSweep	number of sweeps sampled	int	10000
-n	Size of Lattice	int	3
-J	interaction strength	float between 0-1	0
-p	randomness	float between 0-1	0
--tmin	minimal temperature	float	0.1
--tmax	maximal temperature	float	2.1
--nt	number of bins (data points -1)	nt	10