Zoo of Classical O(N) Spin Models

This Monte Carlo simulation program evaluates the partition function $\mathcal{Z}$:

for a d-dimensional O(N) spin model, where $d = 1, 2, 3, \ldots$ denotes the spatial dimension and $N = 1, 2, 3, \ldots$ denotes the spin dimension. ${\mathbf{S}_i}$ represents the summation over all possible spin configurations, and $\beta$ is the inverse temperature, defined as $1/T$. Moreover, the Hamiltonian $H$ is given by:

with $J = 1$ (ferromagnetic coupling), leaving $\beta$ (inverse temperature) and $\mathbf{h}$ (external field) as tunable parameters.

The Swendsen–Wang and Wolff algorithms are employed for efficient configuration updates, particularly near critical points. A Metropolis version is also included.

How to use

1. Clone the repository:

   git clone https://github.com/Tensofermi/Zoo_of_Classical_ON_Spin_Model
   cd Zoo_of_Classical_ON_Spin_Model

2. The file named input.txt contains all tunable parameters, so you can modify them as needed. Its basic structure is as follows:

   //----- Model_Parameters
   N           1                         
   D           2                                            
   beta        0.44         
   L           16      
   h           1                    
   
   //----- Simulation_Parameters
   Seed        987654321           
   N_Measure   1                                          
   N_Each      1000    
   N_Therm     10                                          
   N_Total     200                                          
   NBlock      1000                                 
   MaxNBin     1000000                                
   NperBin     1                     
   

3. Once you've determined which parameters you want to use, you can run the program simply by executing:

   ./run.sh
   

   To clear the generated data, use:

   ./clear.sh
   

   To clear everything including compiled files, use:

   ./clear_all.sh
   

4. For more advanced simulations:

   • Use \lsub to run local simulations on your PC.
   • Use \qsub to submit jobs to a server using the PBS system.
   • The output data will be stored in the \data directory.
   • You can then visualize the results using the \plot script.

Features

• Simulates O(N) spin models in arbitrary spatial dimensions d and spin components N.
• Employs cluster algorithms to reduce critical slowing down.
• Measures various observables: energy-like, magnetic-like, correlation length, and cluster-related quantities.

Phase Transitions and Universality Classes

Below is a summary of notable properties for specific values of $d$ and $N$:

$d$	$N$	Phase Transition?	Type	Universality Class
1	Any	No	—	—
2	1	Yes	Continuous	2D Ising
2	2	Yes	BKT transition	2D XY
2	≥3	No	—	—
3	1	Yes	Continuous	3D Ising
3	2	Yes	Continuous	3D XY
3	3	Yes	Continuous	3D Heisenberg
≥4	Any	Yes	Continuous	Mean-field

Notes:

• BKT transition: A topological phase transition occurring in 2D XY models ($N = 2$).
• Mermin–Wagner theorem: Prohibits spontaneous symmetry breaking for $N \geq 2$ in $d = 2$.

Link

Bilayer XY: Two coupled 2D O(2) models, exhibiting a new emergent phase.

Loop-Ising: High-dimensional O(1) model in the loop representation.

LR-O(n): O(n) model with a $1/r^{d+\sigma}$ long-range interaction.

LR-RW: Lévy flights with a step-length distribution $P(r) \sim 1/r^{d+\sigma}$..