Square lattice \(J_1-J_2\) model Wilson ratio
Author Siddhartha Sarkar
The \(J_1-J_2\) model on a square lattice consists of Heisenberg couplings between nearest neighbors and next nearest neighbors with coupling constants \(J_1\) and \(J_2\), respectively. The Hamiltonian has the form
\[
\mathcal{H} = J_1 \sum_{\langle i,j \rangle} \boldsymbol{S}_i \cdot \boldsymbol{S}_j + J_2 \sum_{\langle\langle i,j \rangle\rangle} \boldsymbol{S}_i \cdot \boldsymbol{S}_j.
\]
The spectrum of this model for a \(C_4\) symmetric \(N=32\) site system is shown below
For small values of \(J_2/J_1 \lessapprox 0.5\), the ground state of this model has Néel order, whereas for large values of \(J_2/J_1 \gtrapprox 0.8\), the ground state has stripe order [2].
The following code was used to obtain the spectrum
#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 == 7);
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]);
double J2 = atof(argv[5]);
int seed = atoi(argv[6]);
Log("Diagonalizing H in block nup: {}, k: {}", n_up, kname);
Log("square.{}.J1J2.fsl.pbc.toml", n_sites);
auto lfile = FileToml(format("square.{}.J1J2.fsl.pbc.toml", n_sites));
std::string ofilename = format(
"outfile.square.{}.J1J2.{:.2f}.nup.{}.k.{}.seed.{}.h5",
n_sites, J1, n_up, kname, seed);
auto ofile = FileH5(ofilename, "w!");
xdiag::OpSum ops = read_opsum(lfile, "Interactions");
ops["J1"] = J1;
ops["J2"] = J2;
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],
['J2', 'SdotS', 0, 25],
['J2', 'SdotS', 8, 17],
['J2', 'SdotS', 25, 3],
['J2', 'SdotS', 24, 2],
['J2', 'SdotS', 2, 28],
['J2', 'SdotS', 17, 11],
['J2', 'SdotS', 3, 29],
['J2', 'SdotS', 1, 27],
['J2', 'SdotS', 16, 10],
['J2', 'SdotS', 10, 20],
['J2', 'SdotS', 28, 6],
['J2', 'SdotS', 11, 21],
['J2', 'SdotS', 29, 0],
['J2', 'SdotS', 26, 4],
['J2', 'SdotS', 9, 19],
['J2', 'SdotS', 27, 5],
['J2', 'SdotS', 5, 31],
['J2', 'SdotS', 20, 14],
['J2', 'SdotS', 6, 24],
['J2', 'SdotS', 21, 8],
['J2', 'SdotS', 18, 12],
['J2', 'SdotS', 4, 30],
['J2', 'SdotS', 19, 13],
['J2', 'SdotS', 13, 23],
['J2', 'SdotS', 31, 1],
['J2', 'SdotS', 14, 16],
['J2', 'SdotS', 12, 22],
['J2', 'SdotS', 30, 7],
['J2', 'SdotS', 7, 26],
['J2', 'SdotS', 23, 9],
['J2', 'SdotS', 22, 15],
['J2', 'SdotS', 15, 18],
['J2', 'SdotS', 0, 24],
['J2', 'SdotS', 8, 16],
['J2', 'SdotS', 25, 2],
['J2', 'SdotS', 24, 1],
['J2', 'SdotS', 2, 27],
['J2', 'SdotS', 17, 10],
['J2', 'SdotS', 3, 28],
['J2', 'SdotS', 1, 26],
['J2', 'SdotS', 16, 9],
['J2', 'SdotS', 10, 19],
['J2', 'SdotS', 28, 5],
['J2', 'SdotS', 11, 20],
['J2', 'SdotS', 29, 6],
['J2', 'SdotS', 26, 0],
['J2', 'SdotS', 9, 18],
['J2', 'SdotS', 27, 4],
['J2', 'SdotS', 5, 30],
['J2', 'SdotS', 20, 13],
['J2', 'SdotS', 6, 31],
['J2', 'SdotS', 21, 14],
['J2', 'SdotS', 18, 8],
['J2', 'SdotS', 4, 25],
['J2', 'SdotS', 19, 12],
['J2', 'SdotS', 13, 22],
['J2', 'SdotS', 31, 7],
['J2', 'SdotS', 14, 23],
['J2', 'SdotS', 12, 17],
['J2', 'SdotS', 30, 3],
['J2', 'SdotS', 7, 29],
['J2', 'SdotS', 23, 15],
['J2', 'SdotS', 22, 11],
['J2', 'SdotS', 15, 21]
]
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],
[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.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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[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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[Sigma0.C1.A]
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[Sigma1.C1.A]
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[Sigma2.C1.A]
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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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[-1.0000000000000000, 0.0000000000000006],
[-1.0000000000000000, 0.0000000000000009],
[1.0000000000000000, -0.0000000000000001],
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[1.0000000000000000, -0.0000000000000002],
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[1.0000000000000000, -0.0000000000000005],
[1.0000000000000000, -0.0000000000000007],
[1.0000000000000000, -0.0000000000000007],
[-1.0000000000000000, 0.0000000000000002],
[-1.0000000000000000, 0.0000000000000002],
[-1.0000000000000000, 0.0000000000000005],
[-1.0000000000000000, 0.0000000000000005],
[-1.0000000000000000, 0.0000000000000005],
[-1.0000000000000000, 0.0000000000000005],
[-1.0000000000000000, 0.0000000000000007],
[-1.0000000000000000, 0.0000000000000007],
[-1.0000000000000000, 0.0000000000000001],
[-1.0000000000000000, 0.0000000000000001],
[-1.0000000000000000, 0.0000000000000004],
[-1.0000000000000000, 0.0000000000000004],
[-1.0000000000000000, 0.0000000000000004],
[-1.0000000000000000, 0.0000000000000004],
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[-1.0000000000000000, 0.0000000000000006],
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[1.0000000000000000, -0.0000000000000004],
[1.0000000000000000, -0.0000000000000004],
[1.0000000000000000, -0.0000000000000004],
[1.0000000000000000, -0.0000000000000006],
[1.0000000000000000, -0.0000000000000006]
]
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]
momentum = [3.1415926535897931, 0.0000000000000000]
[Z0.C1.A]
characters = [
[1.0000000000000000, 0.0000000000000000],
[-1.0000000000000000, 0.0000000000000001],
[1.0000000000000000, -0.0000000000000002],
[-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, 0.1, 1, respectively. Once the code has been run for various values of \(J_2/J_1\), the spectrum can be immediately plotted from the eigenvalues saved in the .h5 files.The expectation value of an observable \(\mathcal{O}\) in the canonical ensemble is
\[
\langle \mathcal{O} \rangle = \frac{\text{Tr}\left(e^{-\beta \mathcal{H}} \mathcal{O}\right)}{\mathcal{Z}},
\]
where \(\beta\) is the inverse temperature and \(\mathcal{Z} = \text{Tr}\left(e^{-\beta \mathcal{H}}\right)\). We approximate the trace stochastically using a set of \(R\) random vectors \(|r\rangle\), leading to
\[
\langle \mathcal{O} \rangle \approx \frac{N_{st}}{\mathcal{Z}R}\sum_{r=1}^R\sum_{j=0}^M e^{-\beta\epsilon_j^r}\langle r|\psi_j^r\rangle \langle \psi_j^r|\mathcal{O}|r\rangle,
\]
and
\[
\mathcal{Z} \approx \frac{N_{st}}{R}\sum_{r=1}^R\sum_{j=0}^M e^{-\beta\epsilon_j^r}|\langle r|\psi_j^r\rangle|^2
\]
where \(|r\rangle\) seves as intial vectors for the Lanczos iteration, resulting in \(M\) eigenvalues \(\epsilon_j^r\) with corresponding \(|\psi_j^r\rangle\) [3], and \(N_{st}\) is the total number of many-body states in the Hilbert space. Once the Lanczos basis \(V\) and the \(M\times M\) tridiagonal matrix \(T = V^\dagger \mathcal{H} V\) is constructed from \(|r\rangle\), \(\langle r|\psi_j^r \rangle\) is just the first component of the \(j\)-th eigenvector of \(T\).
Thus, the algorithm performs the following steps:
-
Generate the Lanczos Basis: Obtain the elements of the tridiagonal matrix \(T\) from each random state \(|r\rangle\).
-
Compute thermodynamic quantities: To obtain the specific heat, $$ C = \beta^2\left[\langle \mathcal{H}^2 \rangle - \langle \mathcal{H} \rangle^2\right], $$ we need to evaluate
\[
\langle \mathcal{H} \rangle \approx \frac{N_{st}}{\mathcal{Z}R}\sum_{r=1}^R\sum_{j=0}^M e^{-\beta\epsilon_j^r}|\langle r|\psi_j^r\rangle|^2 \epsilon_j^r, \\[2exm]
\langle \mathcal{H}^2 \rangle \approx \frac{N_{st}}{\mathcal{Z}R}\sum_{r=1}^R\sum_{j=0}^M e^{-\beta\epsilon_j^r}|\langle r|\psi_j^r\rangle|^2 (\epsilon_j^r)^2.
\]
This part could be done in the post-processing. Similarly, one can obtain magnetic susceptibility $$ \chi = \frac{1}{N}\beta\left[\langle S_z^2 \rangle - \langle S_z \rangle^2\right], $$ where \(N\) is the number of sites, by computing
\[
\langle S_z \rangle \approx \frac{N_{st}}{\mathcal{Z}R}\sum_{r=1}^R\sum_{j=0}^M e^{-\beta\epsilon_j^r}|\langle r|\psi_j^r\rangle|^2 S_z, \\[2exm]
\langle S_z^2 \rangle \approx \frac{N_{st}}{\mathcal{Z}R}\sum_{r=1}^R\sum_{j=0}^M e^{-\beta\epsilon_j^r}|\langle r|\psi_j^r\rangle|^2 S_z^2.
\]
Note that above we used \(\langle \psi_j^r|S_z|r\rangle = \langle \psi_j^r|r\rangle S_z\) because our Hamiltonian is \(S_z\) symmetric, and we use Lanczos algorithm per \(S_z\) sector.
Finally, from the partition function \(\mathcal{Z}\) and \(\langle\mathcal{H}\rangle\), one can directly compute entropy using the formula
\[
s = \frac{1}{N}(\log(\mathcal{Z})+\beta\langle(\mathcal{H}-E_0)\rangle),
\]
where \(N\) is the number of sites and \(E_0\) is the ground state energy. A typical plot of entropy as a funtion of temperature is shown in the figure below
The two subfigures are plotted to show typical behaviors at low and high temperatures. The subplot on the right shows that high temperture entropy satuarates to \(s_{T\rightarrow \infty} =\ln(2)\). The subplot on the left shows that for very low temperature, entropy value obtained using the method mentioned above can be negative [4].
Once we obtain magnetic susceptibility and entropy, we can also plot the Wilson ratio [5]
\[
R(T) = \frac{4\pi^2 T \chi(T)}{3s(T)}.
\]
For antiferromagnets the effective magnon excitations lead to \(s\propto T^2\) for low \(T\), and \(\chi(T\rightarrow 0)>0\) due to contribution of the spin fluctuations transverse to the the magnetic order; hence \(R(T\rightarrow 0) \rightarrow \infty\). Below we show the plot of Wilson ration vs. temperature for several values of \(J_2/J_1\)
Note that each curve is cut off on the left at some \(T\). This is because one can expect that results reach the \(N\rightarrow \infty\) validity only for \(\mathcal{Z}>\mathcal{Z}^*\gg 1\). In reality, we set cut-off of \(\mathcal{Z}^* = 10\), and only plot data for temperatures at which \(\mathcal{Z}>10\) [5]. However, even with this cutoff, we can see that for small \(J_2/J_1 = 0.1, 0.3\) (Néel antiferromagnet) and large \(J_2/J_1 =0.9,1.0\) (stripe antiferromagnet) , the Wilson ratio curve goes upward as temperature is lowered for small \(T\). For \(J_2/J_1 = 0.5, 0.7\), where the transition occurs, the Wilson ratio keeps decreasing as temperature is decreased.
All the thermodynamic quantities mentioned above can be plotted using the Julia script below
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
# 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", "None0.C1.A", "None1.C1.A", "Sigma0.C1.A", "Sigma1.C1.A", "Sigma2.C1.A", "Z0.C1.A"]
kmult=[1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4]
seeds = [i for i=2:2]
n_ups = [i for i=0:div(n_sites,2)]
J1=1.00
J2s=[i/10. for i=0:10]
Jchi = 0.0
temps = collect(range(0.01, 100, length=400))#2.5
mkpath("./data")
mkpath("./plot-therm")
mkpath("./plot-therm/spec-heat/")
mkpath("./plot-therm/susceptibility/")
mkpath("./plot-therm/entropy/")
mkpath("./plot-therm/partition/")
for J2 = J2s
plot(title=L"J_2/J_1 = "*string(J2))
SpecHeat = zeros(n_seeds,length(temps))
partitions = zeros(n_seeds,length(temps))
Susceptibility = zeros(n_seeds,length(temps))
Energy = zeros(n_seeds,length(temps))
Entropy = zeros(n_seeds,length(temps))
n_seed=0
for seed = seeds
n_seed+=1
f = h5open(@sprintf("outfile.square.%d.J1.%.2f.J2.%.2f.Jchi.%.2f.nup.%d.k.%s.seed.%d.h5",n_sites,J1,J2,Jchi,div(n_sites,2),"Gamma.C4.A",seed), "r")
eig0 = read(f["Eigenvalues"])[1] # this is the ground state energy
for ntemp = 1:length(temps)
temp = temps[ntemp]
partition = 0.0
energy = 0.0
energy2 =0.0
sz2 =0.0
# entropy = 0.0
for nk = 1:length(ks)
for n_up = n_ups
factor =2.0
if n_up == div(n_sites,2)
factor=1.0
end
f = h5open(@sprintf("/home/ssarkar/research/ChiralSpinHall/MyThirdCode/outfiles/seed.%d/outfile.square.%d.J1.%.2f.J2.%.2f.Jchi.%.2f.nup.%d.k.%s.seed.%d.h5",seed,n_sites,J1,J2,Jchi,n_up,ks[nk],seed), "r")
alphas = read(f["Alphas"])
betas = read(f["Betas"])
dims = read(f["Dimension"])
tmat = SymTridiagonal(alphas, betas[1:length(alphas)-1])
if length(alphas)==1
partition += exp(-(alphas[1]-eig0)/temp)*kmult[nk]*factor
energy += alphas[1]*exp(-(alphas[1]-eig0)/temp)*kmult[nk]*factor
energy2 += alphas[1]^2*exp(-(alphas[1]-eig0)/temp)*kmult[nk]*factor
elseif length(alphas)>1
F = eigen(tmat)
eig = F.values
vecs = F.vectors
partition += dims*sum([exp(-(eig[m]-eig0)/temp)*vecs[1,m]^2 for m=1:length(eig)])*kmult[nk]*factor
energy += dims*sum([eig[m]*exp(-(eig[m]-eig0)/temp)*vecs[1,m]^2 for m=1:length(eig)])*kmult[nk]*factor
energy2 += dims*sum([eig[m]^2*exp(-(eig[m]-eig0)/temp)*vecs[1,m]^2 for m=1:length(eig)])*kmult[nk]*factor
sz2 += dims*sum([(n_up-n_sites/2)^2*exp(-(eig[m]-eig0)/temp)*vecs[1,m]^2 for m=1:length(eig)])*kmult[nk]*factor
end
end
end
SpecHeat[n_seed,ntemp] = (energy2/partition-(energy/partition)^2)/temp^2
partitions[n_seed,ntemp] = partition
Susceptibility[n_seed,ntemp] = sz2/partition/temp/n_sites
Energy[n_seed,ntemp] = (energy/partition-eig0)
Entropy[n_seed,ntemp] = (log(partition)+(energy/partition-eig0)/temp)/n_sites
end
plot!(temps,SpecHeat[n_seed,:],left_margin=15Plots.mm,bottom_margin=10Plots.mm,xlabel=L"T/J_1",ylabel=L"C_v")
end
savefig(@sprintf("plot-therm/spec-heat/outfile.square.%d.J1.%.2f.J2.%.2f.png",n_sites,J1,J2))
plot(title=L"J_2/J_1 = "*string(J2))
n_seed=0
for seed = seeds
n_seed+=1
plot!(temps,partitions[n_seed,:],left_margin=20Plots.mm,bottom_margin=10Plots.mm,xlabel=L"T/J_1",ylabel=L"Z")
end
savefig(@sprintf("plot-therm/partition/outfile.square.%d.J1.%.2f.J2.%.2f.png",n_sites,J1,J2))
plot(title=L"J_2/J_1 = "*string(J2))
n_seed=0
for seed = seeds
n_seed+=1
plot!(temps[1:200],Susceptibility[n_seed,1:200], xlims=(1e-2, temps[200]), ylims=(1e-5, 1.2e-1),left_margin=20Plots.mm,bottom_margin=10Plots.mm,xlabel=L"T/J_1",ylabel=L"\chi")
end
savefig(@sprintf("plot-therm/susceptibility/outfile.square.%d.J1.%.2f.J2.%.2f..png",n_sites,J1,J2))
plot(title=L"J_2/J_1 = "*string(J2))
n_seed=0
for seed = seeds
n_seed+=1
plot!(temps[1:200],Entropy[n_seed,1:200], xlims=(1e-2, temps[200]),left_margin=20Plots.mm,bottom_margin=10Plots.mm,xlabel=L"T/J_1",ylabel=L"s")
end
savefig(@sprintf("plot-therm/entropy/outfile.square.%d.J1.%.2f.J2.%.2f.png",n_sites,J1,J2))
fid = h5open(@sprintf("data/outfile.square.%d.J1.%.2f.J2.%.2f.h5",n_sites,J1,J2),"w")
fid["SpecHeat"] = SpecHeat
fid["partitions"] = partitions
fid["Susceptibility"] = Susceptibility
fid["Energy"] = Energy
fid["Entropy"] = Entropy
close(fid)
end
plot()
let n_seed = 0
for seed = seeds
n_seed+=1
for J2 = [i/10. for i=[1,3,5,7,9,10]]
fid = h5open(@sprintf("data/outfile.square.%d.J1.%.2f.J2.%.2f.h5",n_sites,J1,J2),"r")
entropy = read(fid["Entropy"]);
susceptibility = read(fid["Susceptibility"])
partitions=read(fid["partitions"])
id=1
j=1
while partitions[n_seed,j]<10
j=j+1
end
id=j
wlratio = 4*pi^2*(temps.*susceptibility[n_seed,:])./(3*entropy[n_seed,:])
plot!(temps[id:200],wlratio[id:200], xlims=(1e-2, temps[200]),ylims=(0.0,3.0),left_margin=20Plots.mm,bottom_margin=10Plots.mm,xlabel=L"T",ylabel=L"\mathrm{Wilson\ ratio\ }R",label=L"J_2 = "*string(J2), dpi=1000)
end
savefig(@sprintf("plot-therm/WR.square.%d.J1.%.2f.Jchi.%.2f.png",n_sites,J1,Jchi))
end
end
References
[1] S. Sugiura and A. Shimizu, Thermal Pure Quantum States at Finite Temperature, Phys. Rev. Lett. 108, 240401 (2012).
[2] Zhang, X. T., Huang, Y., Wu, H. Q., Sheng, D. N., and Gong, S. S. Physical Review B, 109(12), 125146 (2024).
[3] Prelovšek, P., and J. Bonča. "Ground state and finite temperature Lanczos methods." Strongly Correlated Systems: Numerical Methods (2013): 1-30.
[4] Morita, Katsuhiro, and Takami Tohyama. "Finite-temperature properties of the Kitaev-Heisenberg models on kagome and triangular lattices studied by improved finite-temperature Lanczos methods." Phys. Rev. Res. 2, 013205 (2020).
[5] Prelovšek, P., K. Morita, T. Tohyama, and J. Herbrych. "Vanishing Wilson ratio as the hallmark of quantum spin-liquid models." Phys. Rev. Res. 2, no. 2 (2020): 023024.