Dataset DOI: 10.5061/dryad.s1rn8pkpv

Description of the data and file structure

We include data sets for each figure in the main text. Figure 1 contains directories for each subfigure, which have data for the density profiles and structure factors and optionally the momentum distribution. Figure 2 contains files for each data set plotted, and Figure 3 contains a python script that has arrays storing the data plotted in the phase diagram in Figure 3 of the paper's main text.

Files and variables

File: Figure1.zip

This directory is organized into 5 subdirectories.

Subdirectory "a":

Contains data for the stripe Bose-Einstein condensate (BEC) phase in Figure 1a).

Data represents characterization of a stripe phase found near dipolar parameter strength R = 1.24.

We include a density file where the first two columns represent dimensionless "x" and "y" cartesian coordinate values in the simulation cell. The third column represents the real part of the dimensionless density, and the fourth column represents its corresponding imaginary part. Imaginary parts will be small but non-zero due to finite sample sizes used in Langevin process averaging, but they are accordingly much smaller than the real parts in column 4 and are within statistical error of zero for the data.

The files "rho_k_0.dat", "rho_-k_0.dat", "S_k_00.dat" have the same structure. These are functions of wavevector coordinates in two dimensions. The first two columns of each file represent dimensionless wavevector coordinates. Column's 3 and 4 represent corresponding integer indices for the wavenumber coordinates in columns 1 and 2, respectively. Columns 5 and 6 represents the real and imaginary parts, respectively, of the data.

For "rho_k_0.dat", columns 5 and 6 represent the average density of upspin atoms as a function of wavevector.

"rho_-k_0.dat" represents the average density of upspin atoms as a function of negated wavevector.

"S_k_00.dat" represents the averaged correlator quantity <density(k) density(-k)> for the upspin atoms.

Subdirectory "b":

Contains data for the square-packed Bose-Einstein condensate phase in Figure 1b).

The format of the data files and structure of the data files are precisely the same as in Subdirectory "a". Please see above for descriptions of those files and reported data. Data in this directory represents the square packed Bose-Einstein condensate phase, evaluated near R = 1.26 dipolar parameter strength.

Subdirectory "c":

Contains data for the hexagonal-packed Bose-Einstein condensate phase in Figure 1c).

The format of the data files and structure of the data files are precisely the same as in Subdirectory "a". Please see above for descriptions of those files and reported data. Data in this directory represents the hexagonally-packed Bose-Einstein condensate phase, evaluated near R = 1.267 dipolar parameter strength.

Subdirectory "d_f_g":

Contains data for the octagonal quasicrystal Bose-Einstein condensate phase in Figure 1d) as well as subfigures 1f), 1g).

The format of the data files and structure of the data files are precisely the same as in Subdirectory "a". Please see above for descriptions of those files and reported data. Data in this directory represents the octagonal quasicrystal Bose-Einstein condensate phase, evaluated near R = 1.272 dipolar parameter strength.

This directory includes an additional data set "n_k.dat" which shares the same structure as S_k_00.dat, rho_k_0.dat, rho_-k_0.dat. For "n_k.dat", Columns 5 and 6 represents the real and imaginary parts, respectively, of the momentum distribution N(k) of the Bose-Einstein condensate, where "k" is the wavevector argument.

Subdirectory "e":

Contains data for the isotropic normal fluid phase in Figure 1e).

The format and structure of the data files "S_k_00.dat", "rho_k_0.dat", "rho_-k_0.dat" are precisely the same as in subdirectory "a" (and others). Please see above for descriptions of those files and reported data. Data in this directory represents the isotropic normal "droplet" fluid phase, evaluated near R = 1.272 dipolar parameter strength and higher temperature.

The file "isotropic_density0.dat" represents the density profile in real space of the isotropic fluid in a very large field of view. The numerical values in this file represent the real part of the density profile at a specific point in the simulation cell. The precise image plotted in Figure 1e) is a zoomed-in version of this data file. This data can be imported into a python module by importing numpy as np and then using np.loadtxt('isotropic_density0.dat', unpack=True) and then interpeted as a two-dimensional numpy array of shape 600 x 600, which can then be plotted using plt.imshow() after importing matplotlib.pyplot as plt. The values of the corresponding array represent the real part of the density function of upspin atoms in the simulation cell.

File: Figure2.zip

This folder contains two subdirectories:

Subdirectory "2ab": Contains data for Figure 2a) and Figure 2b).

This directory contains four data files "hex_T_200.0.dat", "octagonal_T_200.dat", "square_T_200.0.dat", and "stripe_T_200.0.dat". Each file corresponds to a particular Bose-Einstein condensate phase (e.g. "stripe" is a particular phase) where simulations were performed at constant temperature (T = 200) but varying dipolar parameter R.

Each file shows the range of "R" values considered in Column 1. Column 2 is the intensive grand free energy and Column 3 is its corresponding standard error of the mean after Langevin sample averaging. Column 4 is the overall superfluid fraction, with its corresponding standard error of the mean in Column 5. Column 6 and 7 are not plotted in the paper but represent the fraction of atoms in a momentum state along the circle |k| = \kappa and its corresponding standard error.

Subdirectory "2cd": Contains data for Figure 2c) and Figure 2d).

This directory contains eight data files.

Four data files are of the form "hex_R_1.272_nX.dat", where "X" is replaced with an integer representing the number of periods of the structure. For example, "hex_R_1.272_n8.dat" correspond with simulations of a hexagonal BEC with 8 periods at R = 1.272. For the hexagonal BEC, the files include n = 6, n = 8, n = 12, and n = 24, corresponding to what is plotted in Figure 2d).

Four data files are of the form "stripe_R_1.25_nX.dat", where "X" is replaced with an integer representing the number of periods of the stripe structure. For example, "stripe_R_1.25_n12.dat" correspond with simulations of a stripe BEC with 12 periods at R = 1.25. For the stripe BEC, the files include n = 4, n = 6, n = 12, and n = 24, corresponding to what is plotted in Figure 2c).

For each, the first column corresponds to temperature data with label "T". Column 2 is the intensive grand free energy and Column 3 is its corresponding standard error of the mean after Langevin sample averaging. Column 4 is the overall superfluid fraction, with its corresponding standard error of the mean in Column 5. Column 6 and 7 correspond to the x-component of the superfluid stiffness tensor and its corresponding standard error of the mean. Column 8 and 9 are not plotted in the paper but represent the fraction of atoms in a momentum state along the circle |k| = \kappa and its corresponding standard error.

File: Figure3.zip

This directory contains a single file "phase_diagram.py", representing a python file for plotting the phase diagram in Figure 3 of the paper. The python file contains numerical data for the points and their error bars in the phase diagram and then plots them.

Code/software

A private repository was used to run the coherent state boson path integral simulations. Python's numpy and matplotlib packages were used to analyze and visualize data.