Victor Drouin-Touchette, Peter P. Orth, Piers Coleman, Premala Chandra, Tom Lubensky
Addressing the nature of an unexpected smectic-A’ phase in liquid crystal 54COOBC films, we perform large scale Monte Carlo simulations of a coupled hexatic-nematic XY model. The resulting finite-temperature phase diagram reveals a small region with composite Potts Z3 order above the vortex binding transition; this phase is characterized by relative hexatic-nematic ordering though both variables are disordered. The system develops algebraic hexatic and nematic order only at a lower temperature. This multi-step melting scenario agrees well with the experimental observations of a sharp specific heat anomaly that emerges above the onset of hexatic positional order. We therefore propose that the smectic-A’ phase is characterized by composite Potts order and bound- states of fractional vortices
This repository includes the Monte-Carlo code that was used to obtain the data, the jackknife error analysis, as well as information, scripts, and data to generate the figures in the paper.
Details on the specific Monte Carlo routine are presented in the paper. The code can be found in the MC_routine folder. It was written on Python. We use a specifically adapted Wolff algorithm, tailored for coupled XY models. We also use a parallel tempering routine where Nt> temperatures between Tmax and Tmin, on Nc cores. This code was made to run on parallelize cluster environments.
python mcptdoublel.py 10 1.0 2.1 Nt Nc Tmax Tmin
where L=10, Delta=1.0 and lambda = 2.1 are the parameters of the simulation one wants to do. This uses the file functions_mcstep3.py where the functions used for the Monte-Carlo sampling are found.
Then, using
python all_data_process.py 10 1.0 2.1
This processes the obtained raw data, using the Jackknife method to extract errobars, analyze the correlation between different configurations, and overall get the final usable data tables.
Data used for the figures was re-packaged from the full obtained data in the sake of conserving memory space. These new compact files (of .npy format) are found in the data_and_code_for_figures folder. There is also a subfolder with an example of how one of these files was made, including the original data. For more information, contact Victor Drouin-Touchette.
All the codes used to create the figures in the paper are found in the data_and_code_for_figures folder. They are all written in Python (version 3 compatible only, because of the data pickling), and extensively use the matplotlib library.
This is Fig. 3 in the paper. This is obtained using
Phase-Diagram-lambda0.py
This is Fig. 4 in the paper. This is obtained using
Phase-Diagram-full.py
Phase-Diagram-only-inset.py
This is Fig. 5 in the paper. This is obtained using
Plot_low_Delta.py
This is Fig. 6 in the paper. Subfigures (a) and (b) are obtained with
cv_rho_delta1.py
while (c) is obtained with
extrapolation_plot.py
This is Fig. 7 in the paper. This is obtained using
Scaling_Potts.py
This uses previously obtained values for the unbiased scaling fits using the corrections to scaling. Those are done using
pre_Scaling_Potts.ipynb
This is Fig. 8 in the paper. This is obtained using
Energy_Binder.py
This is Fig. 2 of the appendix in the paper. This is obtained using
compare_magnetizations_obs.py
This work was supported by Grant No. DE-SC0020353 (P. Chandra) and Grant No. DE-FG02- 99ER45790 (P. Coleman and V.D.T.), all funded by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, Division of Materials Sciences and Engineering. V.D.T. is thankful for the support of the Fonds de Recherche Quebecois en Nature et Technologie. Part of the research (P.P.O.) was performed at the Ames Laboratory, which is operated for the U.S. DOE by Iowa State University under Contract DE-AC02-07CH11358. Computational resources were provided by the Rutgers University Beowulf cluster.
The data in the Delta1_data.npy file is the most important, and has data for L=[10, 20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 300, 380]. To extract thermodynamical data, one has the following scheme
data = np.load('Delta1_data.npy',allow_pickle = True)
x_data = data[n][0]
y_data = data[n][2*ind + 1]
y_err = data[n][2*ind + 2]
where n is the index for the size studied, and ind is an index corresponding to which observable to look at. These are the following
- Energy
- Specific Heat
- Binder of the energy
- m_theta
- chi_theta
- Binder theta
- m_phi
- chi_phi
- Binder phi
- m_sigma
- chi_sigma
- Binder sigma
- total spin stiffness