TOS Heisenberg model on a square lattice
Author Siddhartha Sarkar
We perform a tower of states (TOS) analysis [1] of the Heisenberg model on a square lattice under periodic boundary condition. This model consists of spin \(\frac{1}{2}\) sites at the vertices of square lattice with nearest-neighbor antiferromagnetic Heisenberg couplings such that the Hamiltonian has the form
\[
\mathcal{H} = J\sum_{\langle i,j\rangle} \boldsymbol{S}_i \cdot \boldsymbol{S}_j.
\]
The TOS analysis provides strong evidence for spontaneous symmetry breaking (SSB) in the thermodynamic limit, as the ground state of a finite system is completely symmetric.
The spectrum of this model can be labeled by total magnetization since \([\mathcal{H},S_z]=0\) (in fact the Hamiltonian is \(SU(2)\), and hence the spectrum can be labeled by \(S^2\) and \(S_z\); but for obtaining the eigen-spectrum, we only use \(S_z\)). To perform the TOS analysis, we converged the lowest-lying eigenvalues using the Lanczos algorithm in each symmetry sector. We then determined the total spin quantum number, \(S_{\text{tot}}\), by inspecting, for each energy level, the number of degenerate eigenstates; thus, \(S_{\text{tot}}\) is given by the maximum \(S_z\). Finally, we plotted the energy spectra as a function of \(S_{\text{tot}}\left(S_{\text{tot}} + 1\right)\).
In the figure above, we show the energy spectra as a function of \(S_{\text{tot}}\left(S_{\text{tot}} + 1\right)\) for a \(C_{4}\) symmetric a system of \(N=32\) sites. In this case, the ground state exhibits a Néel order.
#include <xdiag/all.hpp>
int main(int argc, char **argv) {
using namespace xdiag;
using namespace arma;
using fmt::format;
say_hello();
// Parse input arguments
assert(argc == 6);
int n_sites = atoi(argv[1]); // number of sites
int n_up = atoi(argv[2]); // number of upspins
std::string kname = std::string(argv[3]); // momentum k
double J1 = atof(argv[4]);
int seed = atoi(argv[5]);
Log("Diagonalizing H in block nup: {}, k: {}", n_up, kname);
auto lfile = FileToml(format("square.{}.J1.fsl.pbc.toml", n_sites));
std::string ofilename =
format("outfile.square.{}.J1.{:.2f}.nup.{}.k.{}.seed.{}.h5", n_sites, J1,
n_up, kname, seed);
auto ofile = FileH5(ofilename, "w!");
OpSum ops = read_opsum(lfile, "Interactions");
ops["J1"] = J1;
auto irrep = read_representation(lfile, kname);
Log("Creating block ...");
tic();
auto block = Spinhalf(n_sites, n_up, irrep);
toc();
Log("Dimension: {}", block.size());
Log("Running Lanczos ...");
tic();
int n_eig_to_converge = 2;
int max_iterations = 40;
auto tmat = eigvals_lanczos(ops, block, n_eig_to_converge, 1e-12,
max_iterations, 1e-7, seed);
toc();
ofile["Alphas"] = tmat.alphas;
ofile["Betas"] = tmat.betas;
ofile["Eigenvalues"] = tmat.eigenvalues;
ofile["Dimension"] = block.size();
return EXIT_SUCCESS;
}
The interactions terms and the symmetry representation inputs are given in the following TOML file:
# This modelfile was created with the following properties:
# Basis coordinates: (0.0, 0.0)
# Lattice vectors: a1=(1.0, 0.0), a2=(0.0, 1.0)
# Simulation torus vectors: t1=(4, 4), t2=(4, -4)
# Simulation torus matrix: ((4, 4), (4, -4))
# Symmetry center: (0.0, 0.0)
# Lattice Point Group: D4
# Lattice Space Group (infinite Lattice): D4
# K points (K wedge marked with *):
# [3.141592653589793 3.141592653589793] *
# [3.141592653589793 1.5707963267948966] *
# [3.141592653589793 0.0] *
# [3.141592653589793 -1.5707963267948966] *
# [2.356194490192345 2.356194490192345] *
# [2.356194490192345 0.7853981633974483] *
# [2.356194490192345 -0.7853981633974483] *
# [2.356194490192345 -2.356194490192345]
# [1.5707963267948966 3.141592653589793]
# [1.5707963267948966 1.5707963267948966] *
# [1.5707963267948966 0.0] *
# [1.5707963267948966 -1.5707963267948966]
# [0.7853981633974483 2.356194490192345]
# [0.7853981633974483 0.7853981633974483] *
# [0.7853981633974483 -0.7853981633974483]
# [0.7853981633974483 -2.356194490192345]
# [0.0 3.141592653589793]
# [0.0 1.5707963267948966]
# [0.0 0.0] *
# [0.0 -1.5707963267948966]
# [-0.7853981633974483 2.356194490192345]
# [-0.7853981633974483 0.7853981633974483]
# [-0.7853981633974483 -0.7853981633974483]
# [-0.7853981633974483 -2.356194490192345]
# [-1.5707963267948966 3.141592653589793]
# [-1.5707963267948966 1.5707963267948966]
# [-1.5707963267948966 0.0]
# [-1.5707963267948966 -1.5707963267948966]
# [-2.356194490192345 2.356194490192345]
# [-2.356194490192345 0.7853981633974483]
# [-2.356194490192345 -0.7853981633974483]
# [-2.356194490192345 -2.356194490192345]
# High Symmetry Points: M.C4, Z_0.C1, X.C2, Z_1.C1, Sigma_0.C1, None_0.C1, None_1.C1, Sigma_1.C1, Delta.C1, Sigma_2.C1, Gamma.C4,
# Eccentricity: --
Coordinates = [
[0.0, 0.0],
[2.0, -2.0],
[2.0, 0.0],
[2.0, 2.0],
[4.0, -2.0],
[4.0, 0.0],
[4.0, 2.0],
[6.0, 0.0],
[1.0, 0.0],
[3.0, -2.0],
[3.0, 0.0],
[3.0, 2.0],
[5.0, -2.0],
[5.0, 0.0],
[5.0, 2.0],
[7.0, 0.0],
[2.0, -1.0],
[2.0, 1.0],
[4.0, -3.0],
[4.0, -1.0],
[4.0, 1.0],
[4.0, 3.0],
[6.0, -1.0],
[6.0, 1.0],
[1.0, -1.0],
[1.0, 1.0],
[3.0, -3.0],
[3.0, -1.0],
[3.0, 1.0],
[3.0, 3.0],
[5.0, -1.0],
[5.0, 1.0]
]
Interactions = [
['J1', 'SdotS', 0, 18],
['J1', 'SdotS', 8, 25],
['J1', 'SdotS', 25, 12],
['J1', 'SdotS', 24, 8],
['J1', 'SdotS', 2, 17],
['J1', 'SdotS', 17, 3],
['J1', 'SdotS', 3, 22],
['J1', 'SdotS', 1, 16],
['J1', 'SdotS', 16, 2],
['J1', 'SdotS', 10, 28],
['J1', 'SdotS', 28, 11],
['J1', 'SdotS', 11, 29],
['J1', 'SdotS', 29, 15],
['J1', 'SdotS', 26, 9],
['J1', 'SdotS', 9, 27],
['J1', 'SdotS', 27, 10],
['J1', 'SdotS', 5, 20],
['J1', 'SdotS', 20, 6],
['J1', 'SdotS', 6, 21],
['J1', 'SdotS', 21, 0],
['J1', 'SdotS', 18, 4],
['J1', 'SdotS', 4, 19],
['J1', 'SdotS', 19, 5],
['J1', 'SdotS', 13, 31],
['J1', 'SdotS', 31, 14],
['J1', 'SdotS', 14, 24],
['J1', 'SdotS', 12, 30],
['J1', 'SdotS', 30, 13],
['J1', 'SdotS', 7, 23],
['J1', 'SdotS', 23, 1],
['J1', 'SdotS', 22, 7],
['J1', 'SdotS', 15, 26],
['J1', 'SdotS', 0, 8],
['J1', 'SdotS', 8, 2],
['J1', 'SdotS', 25, 17],
['J1', 'SdotS', 24, 16],
['J1', 'SdotS', 2, 10],
['J1', 'SdotS', 17, 28],
['J1', 'SdotS', 3, 11],
['J1', 'SdotS', 1, 9],
['J1', 'SdotS', 16, 27],
['J1', 'SdotS', 10, 5],
['J1', 'SdotS', 28, 20],
['J1', 'SdotS', 11, 6],
['J1', 'SdotS', 29, 21],
['J1', 'SdotS', 26, 18],
['J1', 'SdotS', 9, 4],
['J1', 'SdotS', 27, 19],
['J1', 'SdotS', 5, 13],
['J1', 'SdotS', 20, 31],
['J1', 'SdotS', 6, 14],
['J1', 'SdotS', 21, 24],
['J1', 'SdotS', 18, 25],
['J1', 'SdotS', 4, 12],
['J1', 'SdotS', 19, 30],
['J1', 'SdotS', 13, 7],
['J1', 'SdotS', 31, 23],
['J1', 'SdotS', 14, 1],
['J1', 'SdotS', 12, 3],
['J1', 'SdotS', 30, 22],
['J1', 'SdotS', 7, 15],
['J1', 'SdotS', 23, 26],
['J1', 'SdotS', 22, 29],
['J1', 'SdotS', 15, 0]
]
Symmetries = [
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31],
[1, 0, 4, 5, 2, 3, 7, 6, 9, 8, 12, 13, 10, 11, 15, 14, 18, 19, 16, 17, 22, 23, 20, 21, 26, 27, 24, 25, 30, 31, 28, 29],
[2, 4, 5, 6, 3, 7, 1, 0, 10, 12, 13, 14, 11, 15, 9, 8, 19, 20, 17, 22, 23, 16, 21, 18, 27, 28, 25, 30, 31, 24, 29, 26],
[3, 5, 6, 0, 7, 1, 2, 4, 11, 13, 14, 8, 15, 9, 10, 12, 20, 21, 22, 23, 16, 17, 18, 19, 28, 29, 30, 31, 24, 25, 26, 27],
[4, 2, 3, 7, 5, 6, 0, 1, 12, 10, 11, 15, 13, 14, 8, 9, 17, 22, 19, 20, 21, 18, 23, 16, 25, 30, 27, 28, 29, 26, 31, 24],
[5, 3, 7, 1, 6, 0, 4, 2, 13, 11, 15, 9, 14, 8, 12, 10, 22, 23, 20, 21, 18, 19, 16, 17, 30, 31, 28, 29, 26, 27, 24, 25],
[6, 7, 1, 2, 0, 4, 5, 3, 14, 15, 9, 10, 8, 12, 13, 11, 23, 16, 21, 18, 19, 20, 17, 22, 31, 24, 29, 26, 27, 28, 25, 30],
[7, 6, 0, 4, 1, 2, 3, 5, 15, 14, 8, 12, 9, 10, 11, 13, 21, 18, 23, 16, 17, 22, 19, 20, 29, 26, 31, 24, 25, 30, 27, 28],
[0, 3, 4, 1, 7, 5, 2, 6, 18, 22, 19, 16, 23, 20, 17, 21, 12, 9, 15, 13, 10, 8, 14, 11, 25, 26, 29, 30, 27, 24, 31, 28],
[1, 5, 2, 0, 6, 3, 4, 7, 16, 20, 17, 18, 21, 22, 19, 23, 10, 8, 14, 11, 12, 9, 15, 13, 27, 24, 31, 28, 25, 26, 29, 30],
[2, 6, 3, 4, 0, 7, 5, 1, 17, 21, 22, 19, 18, 23, 20, 16, 11, 12, 8, 15, 13, 10, 9, 14, 28, 25, 24, 29, 30, 27, 26, 31],
[3, 0, 7, 5, 4, 1, 6, 2, 22, 18, 23, 20, 19, 16, 21, 17, 15, 13, 12, 9, 14, 11, 10, 8, 29, 30, 25, 26, 31, 28, 27, 24],
[4, 7, 5, 2, 1, 6, 3, 0, 19, 23, 20, 17, 16, 21, 22, 18, 13, 10, 9, 14, 11, 12, 8, 15, 30, 27, 26, 31, 28, 25, 24, 29],
[5, 1, 6, 3, 2, 0, 7, 4, 20, 16, 21, 22, 17, 18, 23, 19, 14, 11, 10, 8, 15, 13, 12, 9, 31, 28, 27, 24, 29, 30, 25, 26],
[6, 2, 0, 7, 3, 4, 1, 5, 21, 17, 18, 23, 22, 19, 16, 20, 8, 15, 11, 12, 9, 14, 13, 10, 24, 29, 28, 25, 26, 31, 30, 27],
[7, 4, 1, 6, 5, 2, 0, 3, 23, 19, 16, 21, 20, 17, 18, 22, 9, 14, 13, 10, 8, 15, 11, 12, 26, 31, 30, 27, 24, 29, 28, 25],
[0, 1, 7, 3, 6, 5, 4, 2, 15, 14, 13, 12, 11, 10, 9, 8, 23, 22, 21, 20, 19, 18, 17, 16, 26, 29, 24, 31, 30, 25, 28, 27],
[1, 0, 6, 5, 7, 3, 2, 4, 14, 15, 11, 10, 13, 12, 8, 9, 21, 20, 23, 22, 17, 16, 19, 18, 24, 31, 26, 29, 28, 27, 30, 25],
[2, 4, 0, 6, 1, 7, 3, 5, 8, 9, 15, 11, 14, 13, 12, 10, 18, 21, 16, 23, 22, 17, 20, 19, 25, 24, 27, 26, 29, 28, 31, 30],
[3, 5, 4, 0, 2, 1, 7, 6, 12, 10, 9, 15, 8, 14, 13, 11, 19, 18, 17, 16, 23, 22, 21, 20, 30, 25, 28, 27, 26, 29, 24, 31],
[4, 2, 1, 7, 0, 6, 5, 3, 9, 8, 14, 13, 15, 11, 10, 12, 16, 23, 18, 21, 20, 19, 22, 17, 27, 26, 25, 24, 31, 30, 29, 28],
[5, 3, 2, 1, 4, 0, 6, 7, 10, 12, 8, 14, 9, 15, 11, 13, 17, 16, 19, 18, 21, 20, 23, 22, 28, 27, 30, 25, 24, 31, 26, 29],
[6, 7, 3, 2, 5, 4, 0, 1, 11, 13, 12, 8, 10, 9, 15, 14, 22, 17, 20, 19, 18, 21, 16, 23, 29, 28, 31, 30, 25, 24, 27, 26],
[7, 6, 5, 4, 3, 2, 1, 0, 13, 11, 10, 9, 12, 8, 14, 15, 20, 19, 22, 17, 16, 23, 18, 21, 31, 30, 29, 28, 27, 26, 25, 24],
[0, 3, 6, 1, 2, 5, 7, 4, 21, 17, 20, 23, 16, 19, 22, 18, 11, 14, 8, 10, 13, 15, 9, 12, 29, 24, 25, 28, 31, 26, 27, 30],
[1, 5, 7, 0, 4, 3, 6, 2, 23, 19, 22, 21, 18, 17, 20, 16, 13, 15, 9, 12, 11, 14, 8, 10, 31, 26, 27, 30, 29, 24, 25, 28],
[2, 6, 1, 4, 5, 7, 0, 3, 16, 20, 23, 18, 19, 22, 21, 17, 14, 9, 10, 13, 15, 8, 12, 11, 24, 27, 28, 31, 26, 25, 30, 29],
[3, 0, 2, 5, 6, 1, 4, 7, 17, 21, 16, 19, 20, 23, 18, 22, 8, 10, 11, 14, 9, 12, 13, 15, 25, 28, 29, 24, 27, 30, 31, 26],
[4, 7, 0, 2, 3, 6, 1, 5, 18, 22, 21, 16, 17, 20, 23, 19, 15, 8, 12, 11, 14, 9, 10, 13, 26, 25, 30, 29, 24, 27, 28, 31],
[5, 1, 4, 3, 7, 0, 2, 6, 19, 23, 18, 17, 22, 21, 16, 20, 9, 12, 13, 15, 8, 10, 11, 14, 27, 30, 31, 26, 25, 28, 29, 24],
[6, 2, 5, 7, 1, 4, 3, 0, 20, 16, 19, 22, 23, 18, 17, 21, 10, 13, 14, 9, 12, 11, 15, 8, 28, 31, 24, 27, 30, 29, 26, 25],
[7, 4, 3, 6, 0, 2, 5, 1, 22, 18, 17, 20, 21, 16, 19, 23, 12, 11, 15, 8, 10, 13, 14, 9, 30, 29, 26, 25, 28, 31, 24, 27],
[8, 9, 10, 11, 12, 13, 14, 15, 2, 4, 5, 6, 3, 7, 1, 0, 27, 28, 25, 30, 31, 24, 29, 26, 16, 17, 18, 19, 20, 21, 22, 23],
[9, 8, 12, 13, 10, 11, 15, 14, 4, 2, 3, 7, 5, 6, 0, 1, 25, 30, 27, 28, 29, 26, 31, 24, 18, 19, 16, 17, 22, 23, 20, 21],
[10, 12, 13, 14, 11, 15, 9, 8, 5, 3, 7, 1, 6, 0, 4, 2, 30, 31, 28, 29, 26, 27, 24, 25, 19, 20, 17, 22, 23, 16, 21, 18],
[11, 13, 14, 8, 15, 9, 10, 12, 6, 7, 1, 2, 0, 4, 5, 3, 31, 24, 29, 26, 27, 28, 25, 30, 20, 21, 22, 23, 16, 17, 18, 19],
[12, 10, 11, 15, 13, 14, 8, 9, 3, 5, 6, 0, 7, 1, 2, 4, 28, 29, 30, 31, 24, 25, 26, 27, 17, 22, 19, 20, 21, 18, 23, 16],
[13, 11, 15, 9, 14, 8, 12, 10, 7, 6, 0, 4, 1, 2, 3, 5, 29, 26, 31, 24, 25, 30, 27, 28, 22, 23, 20, 21, 18, 19, 16, 17],
[14, 15, 9, 10, 8, 12, 13, 11, 1, 0, 4, 5, 2, 3, 7, 6, 26, 27, 24, 25, 30, 31, 28, 29, 23, 16, 21, 18, 19, 20, 17, 22],
[15, 14, 8, 12, 9, 10, 11, 13, 0, 1, 2, 3, 4, 5, 6, 7, 24, 25, 26, 27, 28, 29, 30, 31, 21, 18, 23, 16, 17, 22, 19, 20],
[16, 20, 17, 18, 21, 22, 19, 23, 2, 6, 3, 4, 0, 7, 5, 1, 28, 25, 24, 29, 30, 27, 26, 31, 10, 8, 14, 11, 12, 9, 15, 13],
[17, 21, 22, 19, 18, 23, 20, 16, 3, 0, 7, 5, 4, 1, 6, 2, 29, 30, 25, 26, 31, 28, 27, 24, 11, 12, 8, 15, 13, 10, 9, 14],
[18, 22, 19, 16, 23, 20, 17, 21, 4, 7, 5, 2, 1, 6, 3, 0, 30, 27, 26, 31, 28, 25, 24, 29, 12, 9, 15, 13, 10, 8, 14, 11],
[19, 23, 20, 17, 16, 21, 22, 18, 5, 1, 6, 3, 2, 0, 7, 4, 31, 28, 27, 24, 29, 30, 25, 26, 13, 10, 9, 14, 11, 12, 8, 15],
[20, 16, 21, 22, 17, 18, 23, 19, 6, 2, 0, 7, 3, 4, 1, 5, 24, 29, 28, 25, 26, 31, 30, 27, 14, 11, 10, 8, 15, 13, 12, 9],
[21, 17, 18, 23, 22, 19, 16, 20, 0, 3, 4, 1, 7, 5, 2, 6, 25, 26, 29, 30, 27, 24, 31, 28, 8, 15, 11, 12, 9, 14, 13, 10],
[22, 18, 23, 20, 19, 16, 21, 17, 7, 4, 1, 6, 5, 2, 0, 3, 26, 31, 30, 27, 24, 29, 28, 25, 15, 13, 12, 9, 14, 11, 10, 8],
[23, 19, 16, 21, 20, 17, 18, 22, 1, 5, 2, 0, 6, 3, 4, 7, 27, 24, 31, 28, 25, 26, 29, 30, 9, 14, 13, 10, 8, 15, 11, 12],
[8, 9, 15, 11, 14, 13, 12, 10, 0, 1, 7, 3, 6, 5, 4, 2, 26, 29, 24, 31, 30, 25, 28, 27, 18, 21, 16, 23, 22, 17, 20, 19],
[9, 8, 14, 13, 15, 11, 10, 12, 1, 0, 6, 5, 7, 3, 2, 4, 24, 31, 26, 29, 28, 27, 30, 25, 16, 23, 18, 21, 20, 19, 22, 17],
[10, 12, 8, 14, 9, 15, 11, 13, 2, 4, 0, 6, 1, 7, 3, 5, 25, 24, 27, 26, 29, 28, 31, 30, 17, 16, 19, 18, 21, 20, 23, 22],
[11, 13, 12, 8, 10, 9, 15, 14, 3, 5, 4, 0, 2, 1, 7, 6, 30, 25, 28, 27, 26, 29, 24, 31, 22, 17, 20, 19, 18, 21, 16, 23],
[12, 10, 9, 15, 8, 14, 13, 11, 4, 2, 1, 7, 0, 6, 5, 3, 27, 26, 25, 24, 31, 30, 29, 28, 19, 18, 17, 16, 23, 22, 21, 20],
[13, 11, 10, 9, 12, 8, 14, 15, 5, 3, 2, 1, 4, 0, 6, 7, 28, 27, 30, 25, 24, 31, 26, 29, 20, 19, 22, 17, 16, 23, 18, 21],
[14, 15, 11, 10, 13, 12, 8, 9, 6, 7, 3, 2, 5, 4, 0, 1, 29, 28, 31, 30, 25, 24, 27, 26, 21, 20, 23, 22, 17, 16, 19, 18],
[15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16],
[16, 20, 23, 18, 19, 22, 21, 17, 1, 5, 7, 0, 4, 3, 6, 2, 31, 26, 27, 30, 29, 24, 25, 28, 14, 9, 10, 13, 15, 8, 12, 11],
[17, 21, 16, 19, 20, 23, 18, 22, 2, 6, 1, 4, 5, 7, 0, 3, 24, 27, 28, 31, 26, 25, 30, 29, 8, 10, 11, 14, 9, 12, 13, 15],
[18, 22, 21, 16, 17, 20, 23, 19, 0, 3, 6, 1, 2, 5, 7, 4, 29, 24, 25, 28, 31, 26, 27, 30, 15, 8, 12, 11, 14, 9, 10, 13],
[19, 23, 18, 17, 22, 21, 16, 20, 4, 7, 0, 2, 3, 6, 1, 5, 26, 25, 30, 29, 24, 27, 28, 31, 9, 12, 13, 15, 8, 10, 11, 14],
[20, 16, 19, 22, 23, 18, 17, 21, 5, 1, 4, 3, 7, 0, 2, 6, 27, 30, 31, 26, 25, 28, 29, 24, 10, 13, 14, 9, 12, 11, 15, 8],
[21, 17, 20, 23, 16, 19, 22, 18, 6, 2, 5, 7, 1, 4, 3, 0, 28, 31, 24, 27, 30, 29, 26, 25, 11, 14, 8, 10, 13, 15, 9, 12],
[22, 18, 17, 20, 21, 16, 19, 23, 3, 0, 2, 5, 6, 1, 4, 7, 25, 28, 29, 24, 27, 30, 31, 26, 12, 11, 15, 8, 10, 13, 14, 9],
[23, 19, 22, 21, 18, 17, 20, 16, 7, 4, 3, 6, 0, 2, 5, 1, 30, 29, 26, 25, 28, 31, 24, 27, 13, 15, 9, 12, 11, 14, 8, 10],
[16, 18, 19, 20, 17, 22, 23, 21, 27, 25, 30, 31, 28, 29, 26, 24, 4, 5, 2, 3, 7, 1, 6, 0, 9, 10, 8, 12, 13, 14, 11, 15],
[17, 19, 20, 21, 22, 23, 16, 18, 28, 30, 31, 24, 29, 26, 27, 25, 5, 6, 3, 7, 1, 2, 0, 4, 10, 11, 12, 13, 14, 8, 15, 9],
[18, 16, 17, 22, 19, 20, 21, 23, 25, 27, 28, 29, 30, 31, 24, 26, 2, 3, 4, 5, 6, 0, 7, 1, 8, 12, 9, 10, 11, 15, 13, 14],
[19, 17, 22, 23, 20, 21, 18, 16, 30, 28, 29, 26, 31, 24, 25, 27, 3, 7, 5, 6, 0, 4, 1, 2, 12, 13, 10, 11, 15, 9, 14, 8],
[20, 22, 23, 16, 21, 18, 19, 17, 31, 29, 26, 27, 24, 25, 30, 28, 7, 1, 6, 0, 4, 5, 2, 3, 13, 14, 11, 15, 9, 10, 8, 12],
[21, 23, 16, 17, 18, 19, 20, 22, 24, 26, 27, 28, 25, 30, 31, 29, 1, 2, 0, 4, 5, 6, 3, 7, 14, 8, 15, 9, 10, 11, 12, 13],
[22, 20, 21, 18, 23, 16, 17, 19, 29, 31, 24, 25, 26, 27, 28, 30, 6, 0, 7, 1, 2, 3, 4, 5, 11, 15, 13, 14, 8, 12, 9, 10],
[23, 21, 18, 19, 16, 17, 22, 20, 26, 24, 25, 30, 27, 28, 29, 31, 0, 4, 1, 2, 3, 7, 5, 6, 15, 9, 14, 8, 12, 13, 10, 11],
[8, 11, 12, 9, 15, 13, 10, 14, 25, 29, 30, 27, 26, 31, 28, 24, 3, 4, 0, 7, 5, 2, 1, 6, 17, 18, 21, 22, 19, 16, 23, 20],
[9, 13, 10, 8, 14, 11, 12, 15, 27, 31, 28, 25, 24, 29, 30, 26, 5, 2, 1, 6, 3, 4, 0, 7, 19, 16, 23, 20, 17, 18, 21, 22],
[10, 14, 11, 12, 8, 15, 13, 9, 28, 24, 29, 30, 25, 26, 31, 27, 6, 3, 2, 0, 7, 5, 4, 1, 20, 17, 16, 21, 22, 19, 18, 23],
[11, 8, 15, 13, 12, 9, 14, 10, 29, 25, 26, 31, 30, 27, 24, 28, 0, 7, 3, 4, 1, 6, 5, 2, 21, 22, 17, 18, 23, 20, 19, 16],
[12, 15, 13, 10, 9, 14, 11, 8, 30, 26, 31, 28, 27, 24, 29, 25, 7, 5, 4, 1, 6, 3, 2, 0, 22, 19, 18, 23, 20, 17, 16, 21],
[13, 9, 14, 11, 10, 8, 15, 12, 31, 27, 24, 29, 28, 25, 26, 30, 1, 6, 5, 2, 0, 7, 3, 4, 23, 20, 19, 16, 21, 22, 17, 18],
[14, 10, 8, 15, 11, 12, 9, 13, 24, 28, 25, 26, 29, 30, 27, 31, 2, 0, 6, 3, 4, 1, 7, 5, 16, 21, 20, 17, 18, 23, 22, 19],
[15, 12, 9, 14, 13, 10, 8, 11, 26, 30, 27, 24, 31, 28, 25, 29, 4, 1, 7, 5, 2, 0, 6, 3, 18, 23, 22, 19, 16, 21, 20, 17],
[16, 18, 21, 20, 23, 22, 17, 19, 24, 26, 29, 28, 31, 30, 25, 27, 0, 6, 1, 7, 3, 2, 5, 4, 8, 14, 9, 15, 11, 10, 13, 12],
[17, 19, 18, 21, 16, 23, 22, 20, 25, 27, 26, 29, 24, 31, 30, 28, 4, 0, 2, 1, 7, 3, 6, 5, 12, 8, 10, 9, 15, 11, 14, 13],
[18, 16, 23, 22, 21, 20, 19, 17, 26, 24, 31, 30, 29, 28, 27, 25, 1, 7, 0, 6, 5, 4, 3, 2, 9, 15, 8, 14, 13, 12, 11, 10],
[19, 17, 16, 23, 18, 21, 20, 22, 27, 25, 24, 31, 26, 29, 28, 30, 2, 1, 4, 0, 6, 5, 7, 3, 10, 9, 12, 8, 14, 13, 15, 11],
[20, 22, 17, 16, 19, 18, 21, 23, 28, 30, 25, 24, 27, 26, 29, 31, 3, 2, 5, 4, 0, 6, 1, 7, 11, 10, 13, 12, 8, 14, 9, 15],
[21, 23, 22, 17, 20, 19, 18, 16, 29, 31, 30, 25, 28, 27, 26, 24, 7, 3, 6, 5, 4, 0, 2, 1, 15, 11, 14, 13, 12, 8, 10, 9],
[22, 20, 19, 18, 17, 16, 23, 21, 30, 28, 27, 26, 25, 24, 31, 29, 5, 4, 3, 2, 1, 7, 0, 6, 13, 12, 11, 10, 9, 15, 8, 14],
[23, 21, 20, 19, 22, 17, 16, 18, 31, 29, 28, 27, 30, 25, 24, 26, 6, 5, 7, 3, 2, 1, 4, 0, 14, 13, 15, 11, 10, 9, 12, 8],
[8, 11, 14, 9, 10, 13, 15, 12, 24, 28, 31, 26, 27, 30, 29, 25, 6, 1, 2, 5, 7, 0, 4, 3, 21, 16, 17, 20, 23, 18, 19, 22],
[9, 13, 15, 8, 12, 11, 14, 10, 26, 30, 29, 24, 25, 28, 31, 27, 7, 0, 4, 3, 6, 1, 2, 5, 23, 18, 19, 22, 21, 16, 17, 20],
[10, 14, 9, 12, 13, 15, 8, 11, 27, 31, 26, 25, 30, 29, 24, 28, 1, 4, 5, 7, 0, 2, 3, 6, 16, 19, 20, 23, 18, 17, 22, 21],
[11, 8, 10, 13, 14, 9, 12, 15, 28, 24, 27, 30, 31, 26, 25, 29, 2, 5, 6, 1, 4, 3, 7, 0, 17, 20, 21, 16, 19, 22, 23, 18],
[12, 15, 8, 10, 11, 14, 9, 13, 25, 29, 24, 27, 28, 31, 26, 30, 0, 2, 3, 6, 1, 4, 5, 7, 18, 17, 22, 21, 16, 19, 20, 23],
[13, 9, 12, 11, 15, 8, 10, 14, 30, 26, 25, 28, 29, 24, 27, 31, 4, 3, 7, 0, 2, 5, 6, 1, 19, 22, 23, 18, 17, 20, 21, 16],
[14, 10, 13, 15, 9, 12, 11, 8, 31, 27, 30, 29, 26, 25, 28, 24, 5, 7, 1, 4, 3, 6, 0, 2, 20, 23, 16, 19, 22, 21, 18, 17],
[15, 12, 11, 14, 8, 10, 13, 9, 29, 25, 28, 31, 24, 27, 30, 26, 3, 6, 0, 2, 5, 7, 1, 4, 22, 21, 18, 17, 20, 23, 16, 19],
[24, 26, 27, 28, 25, 30, 31, 29, 16, 18, 19, 20, 17, 22, 23, 21, 9, 10, 8, 12, 13, 14, 11, 15, 1, 2, 0, 4, 5, 6, 3, 7],
[25, 27, 28, 29, 30, 31, 24, 26, 17, 19, 20, 21, 22, 23, 16, 18, 10, 11, 12, 13, 14, 8, 15, 9, 2, 3, 4, 5, 6, 0, 7, 1],
[26, 24, 25, 30, 27, 28, 29, 31, 18, 16, 17, 22, 19, 20, 21, 23, 8, 12, 9, 10, 11, 15, 13, 14, 0, 4, 1, 2, 3, 7, 5, 6],
[27, 25, 30, 31, 28, 29, 26, 24, 19, 17, 22, 23, 20, 21, 18, 16, 12, 13, 10, 11, 15, 9, 14, 8, 4, 5, 2, 3, 7, 1, 6, 0],
[28, 30, 31, 24, 29, 26, 27, 25, 20, 22, 23, 16, 21, 18, 19, 17, 13, 14, 11, 15, 9, 10, 8, 12, 5, 6, 3, 7, 1, 2, 0, 4],
[29, 31, 24, 25, 26, 27, 28, 30, 21, 23, 16, 17, 18, 19, 20, 22, 14, 8, 15, 9, 10, 11, 12, 13, 6, 0, 7, 1, 2, 3, 4, 5],
[30, 28, 29, 26, 31, 24, 25, 27, 22, 20, 21, 18, 23, 16, 17, 19, 11, 15, 13, 14, 8, 12, 9, 10, 3, 7, 5, 6, 0, 4, 1, 2],
[31, 29, 26, 27, 24, 25, 30, 28, 23, 21, 18, 19, 16, 17, 22, 20, 15, 9, 14, 8, 12, 13, 10, 11, 7, 1, 6, 0, 4, 5, 2, 3],
[24, 28, 25, 26, 29, 30, 27, 31, 8, 11, 12, 9, 15, 13, 10, 14, 17, 18, 21, 22, 19, 16, 23, 20, 2, 0, 6, 3, 4, 1, 7, 5],
[25, 29, 30, 27, 26, 31, 28, 24, 12, 15, 13, 10, 9, 14, 11, 8, 22, 19, 18, 23, 20, 17, 16, 21, 3, 4, 0, 7, 5, 2, 1, 6],
[26, 30, 27, 24, 31, 28, 25, 29, 9, 13, 10, 8, 14, 11, 12, 15, 19, 16, 23, 20, 17, 18, 21, 22, 4, 1, 7, 5, 2, 0, 6, 3],
[27, 31, 28, 25, 24, 29, 30, 26, 10, 14, 11, 12, 8, 15, 13, 9, 20, 17, 16, 21, 22, 19, 18, 23, 5, 2, 1, 6, 3, 4, 0, 7],
[28, 24, 29, 30, 25, 26, 31, 27, 11, 8, 15, 13, 12, 9, 14, 10, 21, 22, 17, 18, 23, 20, 19, 16, 6, 3, 2, 0, 7, 5, 4, 1],
[29, 25, 26, 31, 30, 27, 24, 28, 15, 12, 9, 14, 13, 10, 8, 11, 18, 23, 22, 19, 16, 21, 20, 17, 0, 7, 3, 4, 1, 6, 5, 2],
[30, 26, 31, 28, 27, 24, 29, 25, 13, 9, 14, 11, 10, 8, 15, 12, 23, 20, 19, 16, 21, 22, 17, 18, 7, 5, 4, 1, 6, 3, 2, 0],
[31, 27, 24, 29, 28, 25, 26, 30, 14, 10, 8, 15, 11, 12, 9, 13, 16, 21, 20, 17, 18, 23, 22, 19, 1, 6, 5, 2, 0, 7, 3, 4],
[24, 26, 29, 28, 31, 30, 25, 27, 21, 23, 22, 17, 20, 19, 18, 16, 15, 11, 14, 13, 12, 8, 10, 9, 0, 6, 1, 7, 3, 2, 5, 4],
[25, 27, 26, 29, 24, 31, 30, 28, 18, 16, 23, 22, 21, 20, 19, 17, 9, 15, 8, 14, 13, 12, 11, 10, 4, 0, 2, 1, 7, 3, 6, 5],
[26, 24, 31, 30, 29, 28, 27, 25, 23, 21, 20, 19, 22, 17, 16, 18, 14, 13, 15, 11, 10, 9, 12, 8, 1, 7, 0, 6, 5, 4, 3, 2],
[27, 25, 24, 31, 26, 29, 28, 30, 16, 18, 21, 20, 23, 22, 17, 19, 8, 14, 9, 15, 11, 10, 13, 12, 2, 1, 4, 0, 6, 5, 7, 3],
[28, 30, 25, 24, 27, 26, 29, 31, 17, 19, 18, 21, 16, 23, 22, 20, 12, 8, 10, 9, 15, 11, 14, 13, 3, 2, 5, 4, 0, 6, 1, 7],
[29, 31, 30, 25, 28, 27, 26, 24, 22, 20, 19, 18, 17, 16, 23, 21, 13, 12, 11, 10, 9, 15, 8, 14, 7, 3, 6, 5, 4, 0, 2, 1],
[30, 28, 27, 26, 25, 24, 31, 29, 19, 17, 16, 23, 18, 21, 20, 22, 10, 9, 12, 8, 14, 13, 15, 11, 5, 4, 3, 2, 1, 7, 0, 6],
[31, 29, 28, 27, 30, 25, 24, 26, 20, 22, 17, 16, 19, 18, 21, 23, 11, 10, 13, 12, 8, 14, 9, 15, 6, 5, 7, 3, 2, 1, 4, 0],
[24, 28, 31, 26, 27, 30, 29, 25, 14, 10, 13, 15, 9, 12, 11, 8, 20, 23, 16, 19, 22, 21, 18, 17, 6, 1, 2, 5, 7, 0, 4, 3],
[25, 29, 24, 27, 28, 31, 26, 30, 8, 11, 14, 9, 10, 13, 15, 12, 21, 16, 17, 20, 23, 18, 19, 22, 0, 2, 3, 6, 1, 4, 5, 7],
[26, 30, 29, 24, 25, 28, 31, 27, 15, 12, 11, 14, 8, 10, 13, 9, 22, 21, 18, 17, 20, 23, 16, 19, 7, 0, 4, 3, 6, 1, 2, 5],
[27, 31, 26, 25, 30, 29, 24, 28, 9, 13, 15, 8, 12, 11, 14, 10, 23, 18, 19, 22, 21, 16, 17, 20, 1, 4, 5, 7, 0, 2, 3, 6],
[28, 24, 27, 30, 31, 26, 25, 29, 10, 14, 9, 12, 13, 15, 8, 11, 16, 19, 20, 23, 18, 17, 22, 21, 2, 5, 6, 1, 4, 3, 7, 0],
[29, 25, 28, 31, 24, 27, 30, 26, 11, 8, 10, 13, 14, 9, 12, 15, 17, 20, 21, 16, 19, 22, 23, 18, 3, 6, 0, 2, 5, 7, 1, 4],
[30, 26, 25, 28, 29, 24, 27, 31, 12, 15, 8, 10, 11, 14, 9, 13, 18, 17, 22, 21, 16, 19, 20, 23, 4, 3, 7, 0, 2, 5, 6, 1],
[31, 27, 30, 29, 26, 25, 28, 24, 13, 9, 12, 11, 15, 8, 10, 14, 19, 22, 23, 18, 17, 20, 21, 16, 5, 7, 1, 4, 3, 6, 0, 2]
]
# Irreducible representations
[Gamma.C4.A]
characters = [
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
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[1.0000000000000000, 0.0000000000000000],
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[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
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[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
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[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000]
]
allowed_symmetries = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127]
momentum = [0.0000000000000000, 0.0000000000000000]
[Gamma.C4.B]
characters = [
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000]
]
allowed_symmetries = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127]
momentum = [0.0000000000000000, 0.0000000000000000]
[Gamma.C4.Ea]
characters = [
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[0.0000000000000000, 1.0000000000000000],
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[0.0000000000000000, 1.0000000000000000],
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[0.0000000000000000, 1.0000000000000000],
[0.0000000000000000, 1.0000000000000000],
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[0.0000000000000000, 1.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[0.0000000000000000, -1.0000000000000000],
[0.0000000000000000, -1.0000000000000000],
[0.0000000000000000, -1.0000000000000000],
[0.0000000000000000, -1.0000000000000000],
[0.0000000000000000, -1.0000000000000000],
[0.0000000000000000, -1.0000000000000000],
[0.0000000000000000, -1.0000000000000000],
[0.0000000000000000, -1.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
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[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
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[0.0000000000000000, 1.0000000000000000],
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[0.0000000000000000, 1.0000000000000000],
[0.0000000000000000, 1.0000000000000000],
[0.0000000000000000, 1.0000000000000000],
[0.0000000000000000, 1.0000000000000000],
[0.0000000000000000, 1.0000000000000000],
[0.0000000000000000, 1.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000000],
[0.0000000000000000, -1.0000000000000000],
[0.0000000000000000, -1.0000000000000000],
[0.0000000000000000, -1.0000000000000000],
[0.0000000000000000, -1.0000000000000000],
[0.0000000000000000, -1.0000000000000000],
[0.0000000000000000, -1.0000000000000000],
[0.0000000000000000, -1.0000000000000000],
[0.0000000000000000, -1.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
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[1.0000000000000000, 0.0000000000000000],
[1.0000000000000000, 0.0000000000000000],
[0.0000000000000000, 1.0000000000000000],
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[-1.0000000000000000, 0.0000000000000000],
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[Gamma.C4.Eb]
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[Delta.C1.A]
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[M.C4.A]
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[M.C4.B]
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[M.C4.Ea]
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[M.C4.Eb]
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[None0.C1.A]
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[None1.C1.A]
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momentum = [2.3561944901923448, -0.7853981633974483]
[Sigma0.C1.A]
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allowed_symmetries = [0, 1, 2, 3, 4, 5, 6, 7, 32, 33, 34, 35, 36, 37, 38, 39, 64, 65, 66, 67, 68, 69, 70, 71, 96, 97, 98, 99, 100, 101, 102, 103]
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[Sigma1.C1.A]
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allowed_symmetries = [0, 1, 2, 3, 4, 5, 6, 7, 32, 33, 34, 35, 36, 37, 38, 39, 64, 65, 66, 67, 68, 69, 70, 71, 96, 97, 98, 99, 100, 101, 102, 103]
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[Sigma2.C1.A]
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allowed_symmetries = [0, 1, 2, 3, 4, 5, 6, 7, 32, 33, 34, 35, 36, 37, 38, 39, 64, 65, 66, 67, 68, 69, 70, 71, 96, 97, 98, 99, 100, 101, 102, 103]
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[X.C2.A]
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allowed_symmetries = [0, 1, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23, 32, 33, 34, 35, 36, 37, 38, 39, 48, 49, 50, 51, 52, 53, 54, 55, 64, 65, 66, 67, 68, 69, 70, 71, 80, 81, 82, 83, 84, 85, 86, 87, 96, 97, 98, 99, 100, 101, 102, 103, 112, 113, 114, 115, 116, 117, 118, 119]
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[X.C2.B]
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allowed_symmetries = [0, 1, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23, 32, 33, 34, 35, 36, 37, 38, 39, 48, 49, 50, 51, 52, 53, 54, 55, 64, 65, 66, 67, 68, 69, 70, 71, 80, 81, 82, 83, 84, 85, 86, 87, 96, 97, 98, 99, 100, 101, 102, 103, 112, 113, 114, 115, 116, 117, 118, 119]
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[Z0.C1.A]
characters = [
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[-1.0000000000000000, 0.0000000000000004],
[-1.0000000000000000, 0.0000000000000004],
[1.0000000000000000, -0.0000000000000005],
[-1.0000000000000000, 0.0000000000000006],
[1.0000000000000000, -0.0000000000000007],
[-1.0000000000000000, 0.0000000000000001],
[1.0000000000000000, -0.0000000000000002],
[-1.0000000000000000, 0.0000000000000004],
[1.0000000000000000, -0.0000000000000005],
[1.0000000000000000, -0.0000000000000005],
[-1.0000000000000000, 0.0000000000000006],
[1.0000000000000000, -0.0000000000000007],
[-1.0000000000000000, 0.0000000000000009],
[-0.0000000000000002, -1.0000000000000000],
[0.0000000000000003, 1.0000000000000000],
[0.0000000000000003, 1.0000000000000000],
[-0.0000000000000004, -1.0000000000000000],
[0.0000000000000006, 1.0000000000000000],
[-0.0000000000000024, -1.0000000000000000],
[-0.0000000000000024, -1.0000000000000000],
[-0.0000000000000010, 1.0000000000000000],
[0.0000000000000001, 1.0000000000000000],
[-0.0000000000000002, -1.0000000000000000],
[-0.0000000000000002, -1.0000000000000000],
[0.0000000000000003, 1.0000000000000000],
[-0.0000000000000004, -1.0000000000000000],
[0.0000000000000006, 1.0000000000000000],
[0.0000000000000006, 1.0000000000000000],
[-0.0000000000000024, -1.0000000000000000]
]
allowed_symmetries = [0, 1, 2, 3, 4, 5, 6, 7, 32, 33, 34, 35, 36, 37, 38, 39, 64, 65, 66, 67, 68, 69, 70, 71, 96, 97, 98, 99, 100, 101, 102, 103]
momentum = [3.1415926535897931, 1.5707963267948966]
[Z1.C1.A]
characters = [
[1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000004],
[1.0000000000000000, -0.0000000000000002],
[-1.0000000000000000, 0.0000000000000001],
[-1.0000000000000000, 0.0000000000000006],
[1.0000000000000000, -0.0000000000000005],
[-1.0000000000000000, 0.0000000000000004],
[1.0000000000000000, -0.0000000000000007],
[-1.0000000000000000, 0.0000000000000001],
[1.0000000000000000, -0.0000000000000005],
[-1.0000000000000000, 0.0000000000000004],
[1.0000000000000000, -0.0000000000000002],
[1.0000000000000000, -0.0000000000000007],
[-1.0000000000000000, 0.0000000000000006],
[1.0000000000000000, -0.0000000000000005],
[-1.0000000000000000, 0.0000000000000009],
[0.0000000000000003, 1.0000000000000000],
[-0.0000000000000002, -1.0000000000000000],
[-0.0000000000000024, -1.0000000000000000],
[0.0000000000000006, 1.0000000000000000],
[-0.0000000000000004, -1.0000000000000000],
[0.0000000000000003, 1.0000000000000000],
[-0.0000000000000010, 1.0000000000000000],
[-0.0000000000000024, -1.0000000000000000],
[-0.0000000000000002, -1.0000000000000000],
[0.0000000000000001, 1.0000000000000000],
[0.0000000000000006, 1.0000000000000000],
[-0.0000000000000004, -1.0000000000000000],
[0.0000000000000003, 1.0000000000000000],
[-0.0000000000000002, -1.0000000000000000],
[-0.0000000000000024, -1.0000000000000000],
[0.0000000000000006, 1.0000000000000000]
]
allowed_symmetries = [0, 1, 2, 3, 4, 5, 6, 7, 32, 33, 34, 35, 36, 37, 38, 39, 64, 65, 66, 67, 68, 69, 70, 71, 96, 97, 98, 99, 100, 101, 102, 103]
momentum = [3.1415926535897931, -1.5707963267948966]
To run the above C++ code with the toml file, one needs to execute the following command
where the n_sites, n_up, kname, J1, and seed are to be replaced by their values such as 32, 16, Gamma.C4.A, 1.00, 1, respectively. The Julia code below was used to generate the plot above from the data obtaining running the above code.Plotting Script
using LinearAlgebra
using Plots
using Combinatorics
# using BenchmarkTools
using Kronecker
using LaTeXStrings
using Arpack
# using KernelDensity
using Interpolations
using SparseArrays
# using ArnoldiMethod
using KrylovKit
using JLD2
using HDF5
using Printf
plot_font = "Computer Modern"
default(
fontfamily=plot_font,
linewidth=2,
framestyle=:box,
# xtickfont=font(18),
label=nothing,
left_margin=4Plots.mm,
bottom_margin=2Plots.mm
# ytickfont=font(18),
# legendfont=font(18)
)
n_sites=32
n_seeds = 1
ks=["Gamma.C4.A", "Gamma.C4.B", "Gamma.C4.Ea", "Gamma.C4.Eb", "M.C4.A", "M.C4.B", "M.C4.Ea", "M.C4.Eb", "X.C2.A", "X.C2.B"]#, "Delta.C1.A", "Sigma.C1.A", "Z0.C1.A", "Z1.C1.A"]
seeds = [i for i=1:n_seeds]
nup_start=10
n_ups = [i for i=nup_start:(div(n_sites,2))]
J1=1.00
n_eigs = 10
for seed in seeds
plot()
mineig = 0
eigvs = []
for n_up=n_ups
for k in ks
f = h5open(@sprintf("outfile.square.%d.J1.%.2f.nup.%d.k.%s.seed.%d.h5",n_sites,J1,n_up,k,seed), "r")
eig = read(f["Eigenvalues"])[1:n_eigs]
eigvs = append!(eigvs,eig)
close(f)
end
end
mineig=minimum(eigvs)
eigvs = eigvs-mineig*ones(length(eigvs))
energies2 = round.(eigvs, digits=6)
E0 = unique(energies2)
sort!(E0)
Stot = similar(E0)
for i=1:length(E0)
mask = findall(x -> x == E0[i], energies2)
Sz = abs.(div.(mask,length(ks)*n_eigs).+(nup_start) .- n_sites / 2)
vals = Sz .* (Sz .+ 1)
max_val, arg = findmax(vals)
Stot[i] =abs(max_val)
end
plot(Stot,E0,seriestype=:scatter,mc = :blue,legend=false,xlabel=L"S_\mathrm{tot}(S_{tot}+1)",ylabel=L"E/J_1")
plot!(Stot,0.15*Stot,color = :black,legend=false,ylims=(0,10))
savefig(@sprintf("outfile.square.%d.J1.%.2f.seed.%d-n.pdf",n_sites,J1,seed))
end
references
[1] P. W. Anderson, An Approximate Quantum Theory of the Antiferromagnetic Ground State, Phys. Rev. 86, 694 (1952)