Extended kagome lattice antiferromagnet
Author Siddhartha Sarkar
We consider kagome lattice with the following Hamiltonian
\[
\mathcal{H} = J_1\sum_{\langle i,j\rangle} \boldsymbol{S}_i \cdot \boldsymbol{S}_j+J_2\sum_{\langle \langle i,j\rangle\rangle} \boldsymbol{S}_i \cdot \boldsymbol{S}_j+J_3\sum_{\langle \langle\langle i,j\rangle\rangle\rangle_h} \boldsymbol{S}_i \cdot \boldsymbol{S}_j.
\]
where \(\boldsymbol{S}_i\) are spin \(1/2\) operators, \(\langle \dots \rangle\) and \(\langle\langle \dots \rangle\rangle\) denote sum over nearest and next-nearest neighbor sites, and \(\langle \langle\langle\dots\rangle\rangle\rangle_h\) denote sum over third-nearest-neighbor sites [2]. This model host plethora of interesting phases for different parameter regimes. Here we we will be interested in the parameter values \(J_2/J_1 =-1.0\) and \(J_3/J_1=-2.0\), for which it was predicted in [2] that the ground state has spin-nematic character, of quadrupolar type. Tower of state analysis [1] provides a strong evidence for this type of 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 (labeled by \(S_z\), momentum, and little group irreducible representations at each momentum) for a \(C_6\) symmetric system with \(N=36\) sites (\(12\) unit cells and \(3\) spins per unit cell). The \(36\)-site kagome lattice and the momentum points in the Brillouin zone are shown in the figure below.
In the left figure above, the twelve triangles show the 12 unit cells. In the right figure above, the different momentum points are shown; there are three \(M\) points (related by \(C_3\) with each other), two \(K\) points (related by \(C_2\)), six \(Z\) points (related by \(C_6\)), and one \(\Gamma\) point. We plot the spectrum vs \(S_z\) in the figure below.
The momenta and the irrep labels of the eigenstates are shown with markers. The degeneracies at different \(S_z\) gives total quantum of number \(S\). Odd-\(S\) sectors are not present in the low-energy tower of states, indicating a quadrupolar spin-nematic phase [2].
The C++ script used to obtain the eigen spectra is given below
#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 == 8);
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]);
double J3 = atof(argv[6]);
int seed = atoi(argv[7]);
Log("Diagonalizing H in block nup: {}, k: {}", n_up, kname);
auto lfile = FileToml(format("kagome.{}.J1J2J3.pbc.toml", n_sites));
std::string ofilename = format(
"outfile.kagome.{}.J1.{:.2f}.J2.{:.2f}.J3.{:.2f}.nup.{}.k.{}.seed.{}.h5",
n_sites, J1, J2, J3, n_up, kname, seed);
auto ofile = FileH5(ofilename, "w!");
xdiag::OpSum ops = read_opsum(lfile, "Interactions");
ops["J1"] = J1;
ops["J2"] = J2;
ops["J3"] = J3;
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 = 100;
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: ((-0.75, -0.4330127018922193), (-0.25, -0.4330127018922193), (-0.5, 0.))
# Lattice vectors: a1=(1.0, 0.0), a2=(-0.5, 0.8660254037844386)
# Simulation torus vectors: t1=(3.0, -1.7320508075688772), t2=(0.0, 3.4641016151377544)
# Simulation torus matrix: ((2, -2), (2, 4))
# Symmetry center: (0.0, 0.0)
# Lattice Point Group: C6
# Lattice Space Group (infinite Lattice): C6
# K points (K wedge marked with *):
# [-2.0943951023931953 0.0]
# [-1.0471975511965976 1.813799364234218]
# [0. 3.627598728468436] *
# [-1.0471975511965976 -1.813799364234218]
# [0. 0.0] *
# [1.0471975511965976 1.813799364234218] *
# [2.094395102393195 3.627598728468436] *
# [1.0471975511965976 -1.813799364234218]
# [2.094395102393195 0.0]
# [3.141592653589793 1.813799364234218]
# [3.141592653589793 -1.813799364234218]
# [4.18879020478639 0.0]
# High Symmetry Points: Gamma.C6, K.C3, M.C2, Z.C1,
# Eccentricity: --
Coordinates = [
[-1.25, -1.299038105676658],
[-0.75, -1.299038105676658],
[-1.0, -0.8660254037844386],
[-0.25, -1.299038105676658],
[0.25, -1.299038105676658],
[0.0, -0.8660254037844386],
[0.75, -1.299038105676658],
[1.25, -1.299038105676658],
[1.0, -0.8660254037844386],
[-1.75, -0.4330127018922193],
[-1.25, -0.4330127018922193],
[-1.5, 0.0],
[-0.75, -0.4330127018922193],
[-0.25, -0.4330127018922193],
[-0.5, 0.0],
[0.25, -0.4330127018922193],
[0.75, -0.4330127018922193],
[0.5, 0.0],
[1.25, -0.4330127018922193],
[1.75, -0.4330127018922193],
[1.5, 0.0],
[-1.25, 0.4330127018922193],
[-0.75, 0.4330127018922193],
[-1.0, 0.8660254037844386],
[-0.25, 0.4330127018922193],
[0.25, 0.4330127018922193],
[0.0, 0.8660254037844386],
[0.75, 0.4330127018922193],
[1.25, 0.4330127018922193],
[1.0, 0.8660254037844386],
[-0.75, 1.299038105676658],
[-0.25, 1.299038105676658],
[-0.5, 1.7320508075688772],
[0.25, 1.299038105676658],
[0.75, 1.299038105676658],
[0.5, 1.7320508075688772]
]
Interactions = [
['J1', 'SdotS', 0, 1],
['J1', 'SdotS', 1, 2],
['J1', 'SdotS', 2, 0],
['J1', 'SdotS', 3, 4],
['J1', 'SdotS', 4, 5],
['J1', 'SdotS', 5, 3],
['J1', 'SdotS', 6, 7],
['J1', 'SdotS', 7, 8],
['J1', 'SdotS', 8, 6],
['J1', 'SdotS', 9, 10],
['J1', 'SdotS', 10, 11],
['J1', 'SdotS', 11, 9],
['J1', 'SdotS', 12, 13],
['J1', 'SdotS', 13, 14],
['J1', 'SdotS', 14, 12],
['J1', 'SdotS', 15, 16],
['J1', 'SdotS', 16, 17],
['J1', 'SdotS', 17, 15],
['J1', 'SdotS', 18, 19],
['J1', 'SdotS', 19, 20],
['J1', 'SdotS', 20, 18],
['J1', 'SdotS', 21, 22],
['J1', 'SdotS', 22, 23],
['J1', 'SdotS', 23, 21],
['J1', 'SdotS', 24, 25],
['J1', 'SdotS', 25, 26],
['J1', 'SdotS', 26, 24],
['J1', 'SdotS', 27, 28],
['J1', 'SdotS', 28, 29],
['J1', 'SdotS', 29, 27],
['J1', 'SdotS', 30, 31],
['J1', 'SdotS', 31, 32],
['J1', 'SdotS', 32, 30],
['J1', 'SdotS', 33, 34],
['J1', 'SdotS', 34, 35],
['J1', 'SdotS', 35, 33],
['J1', 'SdotS', 20, 0],
['J1', 'SdotS', 32, 3],
['J1', 'SdotS', 35, 6],
['J1', 'SdotS', 29, 9],
['J1', 'SdotS', 2, 12],
['J1', 'SdotS', 5, 15],
['J1', 'SdotS', 8, 18],
['J1', 'SdotS', 11, 21],
['J1', 'SdotS', 14, 24],
['J1', 'SdotS', 17, 27],
['J1', 'SdotS', 23, 30],
['J1', 'SdotS', 26, 33],
['J1', 'SdotS', 3, 1],
['J1', 'SdotS', 6, 4],
['J1', 'SdotS', 21, 7],
['J1', 'SdotS', 12, 10],
['J1', 'SdotS', 15, 13],
['J1', 'SdotS', 18, 16],
['J1', 'SdotS', 30, 19],
['J1', 'SdotS', 24, 22],
['J1', 'SdotS', 27, 25],
['J1', 'SdotS', 0, 28],
['J1', 'SdotS', 33, 31],
['J1', 'SdotS', 9, 34],
['J1', 'SdotS', 10, 2],
['J1', 'SdotS', 13, 5],
['J1', 'SdotS', 16, 8],
['J1', 'SdotS', 7, 11],
['J1', 'SdotS', 22, 14],
['J1', 'SdotS', 25, 17],
['J1', 'SdotS', 28, 20],
['J1', 'SdotS', 19, 23],
['J1', 'SdotS', 31, 26],
['J1', 'SdotS', 34, 29],
['J1', 'SdotS', 1, 32],
['J1', 'SdotS', 4, 35],
['J2', 'SdotS', 19, 0],
['J2', 'SdotS', 31, 3],
['J2', 'SdotS', 34, 6],
['J2', 'SdotS', 28, 9],
['J2', 'SdotS', 1, 12],
['J2', 'SdotS', 4, 15],
['J2', 'SdotS', 7, 18],
['J2', 'SdotS', 10, 21],
['J2', 'SdotS', 13, 24],
['J2', 'SdotS', 16, 27],
['J2', 'SdotS', 22, 30],
['J2', 'SdotS', 25, 33],
['J2', 'SdotS', 10, 0],
['J2', 'SdotS', 13, 3],
['J2', 'SdotS', 16, 6],
['J2', 'SdotS', 7, 9],
['J2', 'SdotS', 22, 12],
['J2', 'SdotS', 25, 15],
['J2', 'SdotS', 28, 18],
['J2', 'SdotS', 19, 21],
['J2', 'SdotS', 31, 24],
['J2', 'SdotS', 34, 27],
['J2', 'SdotS', 1, 30],
['J2', 'SdotS', 4, 33],
['J2', 'SdotS', 20, 1],
['J2', 'SdotS', 32, 4],
['J2', 'SdotS', 35, 7],
['J2', 'SdotS', 29, 10],
['J2', 'SdotS', 2, 13],
['J2', 'SdotS', 5, 16],
['J2', 'SdotS', 8, 19],
['J2', 'SdotS', 11, 22],
['J2', 'SdotS', 14, 25],
['J2', 'SdotS', 17, 28],
['J2', 'SdotS', 23, 31],
['J2', 'SdotS', 26, 34],
['J2', 'SdotS', 5, 1],
['J2', 'SdotS', 8, 4],
['J2', 'SdotS', 23, 7],
['J2', 'SdotS', 14, 10],
['J2', 'SdotS', 17, 13],
['J2', 'SdotS', 20, 16],
['J2', 'SdotS', 32, 19],
['J2', 'SdotS', 26, 22],
['J2', 'SdotS', 29, 25],
['J2', 'SdotS', 2, 28],
['J2', 'SdotS', 35, 31],
['J2', 'SdotS', 11, 34],
['J2', 'SdotS', 3, 2],
['J2', 'SdotS', 6, 5],
['J2', 'SdotS', 21, 8],
['J2', 'SdotS', 12, 11],
['J2', 'SdotS', 15, 14],
['J2', 'SdotS', 18, 17],
['J2', 'SdotS', 30, 20],
['J2', 'SdotS', 24, 23],
['J2', 'SdotS', 27, 26],
['J2', 'SdotS', 0, 29],
['J2', 'SdotS', 33, 32],
['J2', 'SdotS', 9, 35],
['J2', 'SdotS', 9, 2],
['J2', 'SdotS', 12, 5],
['J2', 'SdotS', 15, 8],
['J2', 'SdotS', 6, 11],
['J2', 'SdotS', 21, 14],
['J2', 'SdotS', 24, 17],
['J2', 'SdotS', 27, 20],
['J2', 'SdotS', 18, 23],
['J2', 'SdotS', 30, 26],
['J2', 'SdotS', 33, 29],
['J2', 'SdotS', 0, 32],
['J2', 'SdotS', 3, 35],
['J3', 'SdotS', 9, 0],
['J3', 'SdotS', 12, 3],
['J3', 'SdotS', 15, 6],
['J3', 'SdotS', 6, 9],
['J3', 'SdotS', 21, 12],
['J3', 'SdotS', 24, 15],
['J3', 'SdotS', 27, 18],
['J3', 'SdotS', 18, 21],
['J3', 'SdotS', 30, 24],
['J3', 'SdotS', 33, 27],
['J3', 'SdotS', 0, 30],
['J3', 'SdotS', 3, 33],
['J3', 'SdotS', 13, 1],
['J3', 'SdotS', 16, 4],
['J3', 'SdotS', 19, 7],
['J3', 'SdotS', 22, 10],
['J3', 'SdotS', 25, 13],
['J3', 'SdotS', 28, 16],
['J3', 'SdotS', 1, 19],
['J3', 'SdotS', 31, 22],
['J3', 'SdotS', 34, 25],
['J3', 'SdotS', 10, 28],
['J3', 'SdotS', 4, 31],
['J3', 'SdotS', 7, 34],
['J3', 'SdotS', 5, 2],
['J3', 'SdotS', 8, 5],
['J3', 'SdotS', 23, 8],
['J3', 'SdotS', 14, 11],
['J3', 'SdotS', 17, 14],
['J3', 'SdotS', 20, 17],
['J3', 'SdotS', 32, 20],
['J3', 'SdotS', 26, 23],
['J3', 'SdotS', 29, 26],
['J3', 'SdotS', 2, 29],
['J3', 'SdotS', 35, 32],
['J3', 'SdotS', 11, 35]
]
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],
[3, 4, 5, 6, 7, 8, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20, 30, 31, 32, 24, 25, 26, 27, 28, 29, 0, 1, 2, 33, 34, 35, 9, 10, 11],
[6, 7, 8, 21, 22, 23, 24, 25, 26, 15, 16, 17, 18, 19, 20, 30, 31, 32, 33, 34, 35, 27, 28, 29, 0, 1, 2, 3, 4, 5, 9, 10, 11, 12, 13, 14],
[9, 10, 11, 12, 13, 14, 15, 16, 17, 6, 7, 8, 21, 22, 23, 24, 25, 26, 27, 28, 29, 18, 19, 20, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5],
[12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 30, 31, 32, 33, 34, 35, 9, 10, 11, 3, 4, 5, 6, 7, 8],
[15, 16, 17, 18, 19, 20, 30, 31, 32, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 33, 34, 35, 9, 10, 11, 12, 13, 14, 6, 7, 8, 21, 22, 23],
[18, 19, 20, 30, 31, 32, 33, 34, 35, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 21, 22, 23, 24, 25, 26],
[21, 22, 23, 24, 25, 26, 27, 28, 29, 18, 19, 20, 30, 31, 32, 33, 34, 35, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 12, 13, 14, 15, 16, 17],
[24, 25, 26, 27, 28, 29, 0, 1, 2, 30, 31, 32, 33, 34, 35, 9, 10, 11, 12, 13, 14, 3, 4, 5, 6, 7, 8, 21, 22, 23, 15, 16, 17, 18, 19, 20],
[27, 28, 29, 0, 1, 2, 3, 4, 5, 33, 34, 35, 9, 10, 11, 12, 13, 14, 15, 16, 17, 6, 7, 8, 21, 22, 23, 24, 25, 26, 18, 19, 20, 30, 31, 32],
[30, 31, 32, 33, 34, 35, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20, 24, 25, 26, 27, 28, 29],
[33, 34, 35, 9, 10, 11, 12, 13, 14, 3, 4, 5, 6, 7, 8, 21, 22, 23, 24, 25, 26, 15, 16, 17, 18, 19, 20, 30, 31, 32, 27, 28, 29, 0, 1, 2],
[35, 6, 4, 8, 18, 16, 20, 0, 28, 32, 3, 1, 5, 15, 13, 17, 27, 25, 29, 9, 34, 2, 12, 10, 14, 24, 22, 26, 33, 31, 11, 21, 7, 23, 30, 19],
[11, 21, 7, 23, 30, 19, 32, 3, 1, 35, 6, 4, 8, 18, 16, 20, 0, 28, 2, 12, 10, 5, 15, 13, 17, 27, 25, 29, 9, 34, 14, 24, 22, 26, 33, 31],
[14, 24, 22, 26, 33, 31, 35, 6, 4, 11, 21, 7, 23, 30, 19, 32, 3, 1, 5, 15, 13, 8, 18, 16, 20, 0, 28, 2, 12, 10, 17, 27, 25, 29, 9, 34],
[5, 15, 13, 17, 27, 25, 29, 9, 34, 2, 12, 10, 14, 24, 22, 26, 33, 31, 35, 6, 4, 11, 21, 7, 23, 30, 19, 32, 3, 1, 8, 18, 16, 20, 0, 28],
[8, 18, 16, 20, 0, 28, 2, 12, 10, 5, 15, 13, 17, 27, 25, 29, 9, 34, 11, 21, 7, 14, 24, 22, 26, 33, 31, 35, 6, 4, 23, 30, 19, 32, 3, 1],
[23, 30, 19, 32, 3, 1, 5, 15, 13, 8, 18, 16, 20, 0, 28, 2, 12, 10, 14, 24, 22, 17, 27, 25, 29, 9, 34, 11, 21, 7, 26, 33, 31, 35, 6, 4],
[26, 33, 31, 35, 6, 4, 8, 18, 16, 23, 30, 19, 32, 3, 1, 5, 15, 13, 17, 27, 25, 20, 0, 28, 2, 12, 10, 14, 24, 22, 29, 9, 34, 11, 21, 7],
[17, 27, 25, 29, 9, 34, 11, 21, 7, 14, 24, 22, 26, 33, 31, 35, 6, 4, 8, 18, 16, 23, 30, 19, 32, 3, 1, 5, 15, 13, 20, 0, 28, 2, 12, 10],
[20, 0, 28, 2, 12, 10, 14, 24, 22, 17, 27, 25, 29, 9, 34, 11, 21, 7, 23, 30, 19, 26, 33, 31, 35, 6, 4, 8, 18, 16, 32, 3, 1, 5, 15, 13],
[32, 3, 1, 5, 15, 13, 17, 27, 25, 20, 0, 28, 2, 12, 10, 14, 24, 22, 26, 33, 31, 29, 9, 34, 11, 21, 7, 23, 30, 19, 35, 6, 4, 8, 18, 16],
[29, 9, 34, 11, 21, 7, 23, 30, 19, 26, 33, 31, 35, 6, 4, 8, 18, 16, 20, 0, 28, 32, 3, 1, 5, 15, 13, 17, 27, 25, 2, 12, 10, 14, 24, 22],
[2, 12, 10, 14, 24, 22, 26, 33, 31, 29, 9, 34, 11, 21, 7, 23, 30, 19, 32, 3, 1, 35, 6, 4, 8, 18, 16, 20, 0, 28, 5, 15, 13, 17, 27, 25],
[19, 20, 18, 28, 29, 27, 34, 35, 33, 7, 8, 6, 16, 17, 15, 25, 26, 24, 31, 32, 30, 4, 5, 3, 13, 14, 12, 22, 23, 21, 1, 2, 0, 10, 11, 9],
[31, 32, 30, 1, 2, 0, 10, 11, 9, 22, 23, 21, 19, 20, 18, 28, 29, 27, 34, 35, 33, 7, 8, 6, 16, 17, 15, 25, 26, 24, 4, 5, 3, 13, 14, 12],
[34, 35, 33, 4, 5, 3, 13, 14, 12, 25, 26, 24, 31, 32, 30, 1, 2, 0, 10, 11, 9, 22, 23, 21, 19, 20, 18, 28, 29, 27, 7, 8, 6, 16, 17, 15],
[28, 29, 27, 34, 35, 33, 4, 5, 3, 16, 17, 15, 25, 26, 24, 31, 32, 30, 1, 2, 0, 13, 14, 12, 22, 23, 21, 19, 20, 18, 10, 11, 9, 7, 8, 6],
[1, 2, 0, 10, 11, 9, 7, 8, 6, 19, 20, 18, 28, 29, 27, 34, 35, 33, 4, 5, 3, 16, 17, 15, 25, 26, 24, 31, 32, 30, 13, 14, 12, 22, 23, 21],
[4, 5, 3, 13, 14, 12, 22, 23, 21, 31, 32, 30, 1, 2, 0, 10, 11, 9, 7, 8, 6, 19, 20, 18, 28, 29, 27, 34, 35, 33, 16, 17, 15, 25, 26, 24],
[7, 8, 6, 16, 17, 15, 25, 26, 24, 34, 35, 33, 4, 5, 3, 13, 14, 12, 22, 23, 21, 31, 32, 30, 1, 2, 0, 10, 11, 9, 19, 20, 18, 28, 29, 27],
[10, 11, 9, 7, 8, 6, 16, 17, 15, 28, 29, 27, 34, 35, 33, 4, 5, 3, 13, 14, 12, 25, 26, 24, 31, 32, 30, 1, 2, 0, 22, 23, 21, 19, 20, 18],
[13, 14, 12, 22, 23, 21, 19, 20, 18, 1, 2, 0, 10, 11, 9, 7, 8, 6, 16, 17, 15, 28, 29, 27, 34, 35, 33, 4, 5, 3, 25, 26, 24, 31, 32, 30],
[16, 17, 15, 25, 26, 24, 31, 32, 30, 4, 5, 3, 13, 14, 12, 22, 23, 21, 19, 20, 18, 1, 2, 0, 10, 11, 9, 7, 8, 6, 28, 29, 27, 34, 35, 33],
[22, 23, 21, 19, 20, 18, 28, 29, 27, 10, 11, 9, 7, 8, 6, 16, 17, 15, 25, 26, 24, 34, 35, 33, 4, 5, 3, 13, 14, 12, 31, 32, 30, 1, 2, 0],
[25, 26, 24, 31, 32, 30, 1, 2, 0, 13, 14, 12, 22, 23, 21, 19, 20, 18, 28, 29, 27, 10, 11, 9, 7, 8, 6, 16, 17, 15, 34, 35, 33, 4, 5, 3],
[9, 34, 29, 33, 31, 26, 30, 19, 23, 0, 28, 20, 27, 25, 17, 24, 22, 14, 21, 7, 11, 18, 16, 8, 15, 13, 5, 12, 10, 2, 6, 4, 35, 3, 1, 32],
[12, 10, 2, 9, 34, 29, 33, 31, 26, 3, 1, 32, 0, 28, 20, 27, 25, 17, 24, 22, 14, 30, 19, 23, 18, 16, 8, 15, 13, 5, 21, 7, 11, 6, 4, 35],
[15, 13, 5, 12, 10, 2, 9, 34, 29, 6, 4, 35, 3, 1, 32, 0, 28, 20, 27, 25, 17, 33, 31, 26, 30, 19, 23, 18, 16, 8, 24, 22, 14, 21, 7, 11],
[6, 4, 35, 3, 1, 32, 0, 28, 20, 9, 34, 29, 33, 31, 26, 30, 19, 23, 18, 16, 8, 27, 25, 17, 24, 22, 14, 21, 7, 11, 15, 13, 5, 12, 10, 2],
[21, 7, 11, 6, 4, 35, 3, 1, 32, 12, 10, 2, 9, 34, 29, 33, 31, 26, 30, 19, 23, 0, 28, 20, 27, 25, 17, 24, 22, 14, 18, 16, 8, 15, 13, 5],
[24, 22, 14, 21, 7, 11, 6, 4, 35, 15, 13, 5, 12, 10, 2, 9, 34, 29, 33, 31, 26, 3, 1, 32, 0, 28, 20, 27, 25, 17, 30, 19, 23, 18, 16, 8],
[27, 25, 17, 24, 22, 14, 21, 7, 11, 18, 16, 8, 15, 13, 5, 12, 10, 2, 9, 34, 29, 6, 4, 35, 3, 1, 32, 0, 28, 20, 33, 31, 26, 30, 19, 23],
[18, 16, 8, 15, 13, 5, 12, 10, 2, 21, 7, 11, 6, 4, 35, 3, 1, 32, 0, 28, 20, 9, 34, 29, 33, 31, 26, 30, 19, 23, 27, 25, 17, 24, 22, 14],
[30, 19, 23, 18, 16, 8, 15, 13, 5, 24, 22, 14, 21, 7, 11, 6, 4, 35, 3, 1, 32, 12, 10, 2, 9, 34, 29, 33, 31, 26, 0, 28, 20, 27, 25, 17],
[33, 31, 26, 30, 19, 23, 18, 16, 8, 27, 25, 17, 24, 22, 14, 21, 7, 11, 6, 4, 35, 15, 13, 5, 12, 10, 2, 9, 34, 29, 3, 1, 32, 0, 28, 20],
[0, 28, 20, 27, 25, 17, 24, 22, 14, 30, 19, 23, 18, 16, 8, 15, 13, 5, 12, 10, 2, 21, 7, 11, 6, 4, 35, 3, 1, 32, 9, 34, 29, 33, 31, 26],
[3, 1, 32, 0, 28, 20, 27, 25, 17, 33, 31, 26, 30, 19, 23, 18, 16, 8, 15, 13, 5, 24, 22, 14, 21, 7, 11, 6, 4, 35, 12, 10, 2, 9, 34, 29],
[32, 30, 31, 23, 21, 22, 11, 9, 10, 35, 33, 34, 26, 24, 25, 14, 12, 13, 2, 0, 1, 29, 27, 28, 17, 15, 16, 5, 3, 4, 20, 18, 19, 8, 6, 7],
[35, 33, 34, 26, 24, 25, 14, 12, 13, 11, 9, 10, 29, 27, 28, 17, 15, 16, 5, 3, 4, 2, 0, 1, 20, 18, 19, 8, 6, 7, 32, 30, 31, 23, 21, 22],
[11, 9, 10, 29, 27, 28, 17, 15, 16, 14, 12, 13, 2, 0, 1, 20, 18, 19, 8, 6, 7, 5, 3, 4, 32, 30, 31, 23, 21, 22, 35, 33, 34, 26, 24, 25],
[2, 0, 1, 20, 18, 19, 8, 6, 7, 5, 3, 4, 32, 30, 31, 23, 21, 22, 11, 9, 10, 35, 33, 34, 26, 24, 25, 14, 12, 13, 29, 27, 28, 17, 15, 16],
[5, 3, 4, 32, 30, 31, 23, 21, 22, 8, 6, 7, 35, 33, 34, 26, 24, 25, 14, 12, 13, 11, 9, 10, 29, 27, 28, 17, 15, 16, 2, 0, 1, 20, 18, 19],
[8, 6, 7, 35, 33, 34, 26, 24, 25, 23, 21, 22, 11, 9, 10, 29, 27, 28, 17, 15, 16, 14, 12, 13, 2, 0, 1, 20, 18, 19, 5, 3, 4, 32, 30, 31],
[23, 21, 22, 11, 9, 10, 29, 27, 28, 26, 24, 25, 14, 12, 13, 2, 0, 1, 20, 18, 19, 17, 15, 16, 5, 3, 4, 32, 30, 31, 8, 6, 7, 35, 33, 34],
[14, 12, 13, 2, 0, 1, 20, 18, 19, 17, 15, 16, 5, 3, 4, 32, 30, 31, 23, 21, 22, 8, 6, 7, 35, 33, 34, 26, 24, 25, 11, 9, 10, 29, 27, 28],
[17, 15, 16, 5, 3, 4, 32, 30, 31, 20, 18, 19, 8, 6, 7, 35, 33, 34, 26, 24, 25, 23, 21, 22, 11, 9, 10, 29, 27, 28, 14, 12, 13, 2, 0, 1],
[20, 18, 19, 8, 6, 7, 35, 33, 34, 32, 30, 31, 23, 21, 22, 11, 9, 10, 29, 27, 28, 26, 24, 25, 14, 12, 13, 2, 0, 1, 17, 15, 16, 5, 3, 4],
[26, 24, 25, 14, 12, 13, 2, 0, 1, 29, 27, 28, 17, 15, 16, 5, 3, 4, 32, 30, 31, 20, 18, 19, 8, 6, 7, 35, 33, 34, 23, 21, 22, 11, 9, 10],
[29, 27, 28, 17, 15, 16, 5, 3, 4, 2, 0, 1, 20, 18, 19, 8, 6, 7, 35, 33, 34, 32, 30, 31, 23, 21, 22, 11, 9, 10, 26, 24, 25, 14, 12, 13],
[7, 11, 21, 10, 2, 12, 1, 32, 3, 19, 23, 30, 22, 14, 24, 13, 5, 15, 4, 35, 6, 31, 26, 33, 25, 17, 27, 16, 8, 18, 34, 29, 9, 28, 20, 0],
[22, 14, 24, 13, 5, 15, 4, 35, 6, 31, 26, 33, 25, 17, 27, 16, 8, 18, 7, 11, 21, 34, 29, 9, 28, 20, 0, 19, 23, 30, 10, 2, 12, 1, 32, 3],
[25, 17, 27, 16, 8, 18, 7, 11, 21, 34, 29, 9, 28, 20, 0, 19, 23, 30, 22, 14, 24, 10, 2, 12, 1, 32, 3, 31, 26, 33, 13, 5, 15, 4, 35, 6],
[16, 8, 18, 7, 11, 21, 10, 2, 12, 28, 20, 0, 19, 23, 30, 22, 14, 24, 13, 5, 15, 1, 32, 3, 31, 26, 33, 25, 17, 27, 4, 35, 6, 34, 29, 9],
[19, 23, 30, 22, 14, 24, 13, 5, 15, 1, 32, 3, 31, 26, 33, 25, 17, 27, 16, 8, 18, 4, 35, 6, 34, 29, 9, 28, 20, 0, 7, 11, 21, 10, 2, 12],
[31, 26, 33, 25, 17, 27, 16, 8, 18, 4, 35, 6, 34, 29, 9, 28, 20, 0, 19, 23, 30, 7, 11, 21, 10, 2, 12, 1, 32, 3, 22, 14, 24, 13, 5, 15],
[34, 29, 9, 28, 20, 0, 19, 23, 30, 7, 11, 21, 10, 2, 12, 1, 32, 3, 31, 26, 33, 22, 14, 24, 13, 5, 15, 4, 35, 6, 25, 17, 27, 16, 8, 18],
[28, 20, 0, 19, 23, 30, 22, 14, 24, 10, 2, 12, 1, 32, 3, 31, 26, 33, 25, 17, 27, 13, 5, 15, 4, 35, 6, 34, 29, 9, 16, 8, 18, 7, 11, 21],
[1, 32, 3, 31, 26, 33, 25, 17, 27, 13, 5, 15, 4, 35, 6, 34, 29, 9, 28, 20, 0, 16, 8, 18, 7, 11, 21, 10, 2, 12, 19, 23, 30, 22, 14, 24],
[4, 35, 6, 34, 29, 9, 28, 20, 0, 16, 8, 18, 7, 11, 21, 10, 2, 12, 1, 32, 3, 19, 23, 30, 22, 14, 24, 13, 5, 15, 31, 26, 33, 25, 17, 27],
[10, 2, 12, 1, 32, 3, 31, 26, 33, 22, 14, 24, 13, 5, 15, 4, 35, 6, 34, 29, 9, 25, 17, 27, 16, 8, 18, 7, 11, 21, 28, 20, 0, 19, 23, 30],
[13, 5, 15, 4, 35, 6, 34, 29, 9, 25, 17, 27, 16, 8, 18, 7, 11, 21, 10, 2, 12, 28, 20, 0, 19, 23, 30, 22, 14, 24, 1, 32, 3, 31, 26, 33]
]
# Irreducible representations
[Gamma.C6.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]
]
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]
momentum = [0.0000000000000000, 0.0000000000000000]
[Gamma.C6.B]
characters = [
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0]
]
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]
momentum = [0.0000000000000000, 0.0000000000000000]
[Gamma.C6.E1a]
characters = [
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386]
]
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]
momentum = [0.0000000000000000, 0.0000000000000000]
[Gamma.C6.E1b]
characters = [
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386]
]
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]
momentum = [0.0000000000000000, 0.0000000000000000]
[Gamma.C6.E2a]
characters = [
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[0.5000000000000001, 0.8660254037844386],
[0.5000000000000001, 0.8660254037844386],
[0.5000000000000001, 0.8660254037844386],
[0.5000000000000001, 0.8660254037844386],
[0.5000000000000001, 0.8660254037844386],
[0.5000000000000001, 0.8660254037844386],
[0.5000000000000001, 0.8660254037844386],
[0.5000000000000001, 0.8660254037844386],
[0.5000000000000001, 0.8660254037844386],
[0.5000000000000001, 0.8660254037844386],
[0.5000000000000001, 0.8660254037844386],
[0.5000000000000001, 0.8660254037844386],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[0.49999999999999983, -0.8660254037844387],
[0.49999999999999983, -0.8660254037844387],
[0.49999999999999983, -0.8660254037844387],
[0.49999999999999983, -0.8660254037844387],
[0.49999999999999983, -0.8660254037844387],
[0.49999999999999983, -0.8660254037844387],
[0.49999999999999983, -0.8660254037844387],
[0.49999999999999983, -0.8660254037844387],
[0.49999999999999983, -0.8660254037844387],
[0.49999999999999983, -0.8660254037844387],
[0.49999999999999983, -0.8660254037844387],
[0.49999999999999983, -0.8660254037844387]
]
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]
momentum = [0.0000000000000000, 0.0000000000000000]
[Gamma.C6.E2b]
characters = [
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[0.5000000000000001, -0.8660254037844386],
[0.5000000000000001, -0.8660254037844386],
[0.5000000000000001, -0.8660254037844386],
[0.5000000000000001, -0.8660254037844386],
[0.5000000000000001, -0.8660254037844386],
[0.5000000000000001, -0.8660254037844386],
[0.5000000000000001, -0.8660254037844386],
[0.5000000000000001, -0.8660254037844386],
[0.5000000000000001, -0.8660254037844386],
[0.5000000000000001, -0.8660254037844386],
[0.5000000000000001, -0.8660254037844386],
[0.5000000000000001, -0.8660254037844386],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-0.49999999999999983, -0.8660254037844387],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[-0.5000000000000001, 0.8660254037844386],
[0.49999999999999983, 0.8660254037844387],
[0.49999999999999983, 0.8660254037844387],
[0.49999999999999983, 0.8660254037844387],
[0.49999999999999983, 0.8660254037844387],
[0.49999999999999983, 0.8660254037844387],
[0.49999999999999983, 0.8660254037844387],
[0.49999999999999983, 0.8660254037844387],
[0.49999999999999983, 0.8660254037844387],
[0.49999999999999983, 0.8660254037844387],
[0.49999999999999983, 0.8660254037844387],
[0.49999999999999983, 0.8660254037844387],
[0.49999999999999983, 0.8660254037844387]
]
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]
momentum = [0.0000000000000000, 0.0000000000000000]
[K.C3.A]
characters = [
[1.0, 0.0],
[-0.5, 0.8660254037844386],
[-0.5, -0.8660254037844386],
[-0.5, 0.8660254037844386],
[-0.5, -0.8660254037844386],
[1.0, 0.0],
[-0.5, 0.8660254037844386],
[1.0, 0.0],
[-0.5, 0.8660254037844386],
[-0.5, -0.8660254037844386],
[-0.5, -0.8660254037844386],
[1.0, 0.0],
[1.0, 0.0],
[-0.5, 0.8660254037844386],
[-0.5, -0.8660254037844386],
[-0.5, 0.8660254037844386],
[-0.5, -0.8660254037844386],
[1.0, 0.0],
[-0.5, 0.8660254037844386],
[1.0, 0.0],
[-0.5, 0.8660254037844386],
[-0.5, -0.8660254037844386],
[-0.5, -0.8660254037844386],
[1.0, 0.0],
[1.0, 0.0],
[-0.5, 0.8660254037844386],
[-0.5, -0.8660254037844386],
[-0.5, 0.8660254037844386],
[-0.5, -0.8660254037844386],
[1.0, 0.0],
[-0.5, 0.8660254037844386],
[1.0, 0.0],
[-0.5, 0.8660254037844386],
[-0.5, -0.8660254037844386],
[-0.5, -0.8660254037844386],
[1.0, 0.0]
]
allowed_symmetries = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59]
momentum = [2.0943951023931953, 3.627598728468436]
[K.C3.Ea]
characters = [
[1.0, 0.0],
[-0.49999999999999983, 0.8660254037844387],
[-0.5000000000000001, -0.8660254037844386],
[-0.49999999999999983, 0.8660254037844387],
[-0.5000000000000001, -0.8660254037844386],
[1.0, 0.0],
[-0.49999999999999983, 0.8660254037844387],
[1.0, 0.0],
[-0.49999999999999983, 0.8660254037844387],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[1.0, 0.0],
[-0.49999999999999983, 0.8660254037844387],
[-0.5000000000000001, -0.8660254037844386],
[1.0, 0.0],
[-0.5000000000000001, -0.8660254037844386],
[1.0, 0.0],
[-0.49999999999999983, 0.8660254037844387],
[-0.5000000000000001, -0.8660254037844386],
[-0.49999999999999983, 0.8660254037844387],
[-0.5000000000000001, -0.8660254037844386],
[1.0, 0.0],
[1.0, 0.0],
[-0.49999999999999983, 0.8660254037844387],
[-0.5000000000000001, -0.8660254037844386],
[1.0, 0.0],
[-0.49999999999999983, 0.8660254037844387],
[1.0, 0.0],
[-0.49999999999999983, 0.8660254037844387],
[-0.5000000000000001, -0.8660254037844386],
[1.0, 0.0],
[-0.5000000000000001, -0.8660254037844386],
[1.0, 0.0],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.5000000000000001, -0.8660254037844386]
]
allowed_symmetries = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59]
momentum = [2.0943951023931953, 3.627598728468436]
[K.C3.Eb]
characters = [
[1.0, 0.0],
[-0.49999999999999983, 0.8660254037844387],
[-0.5000000000000001, -0.8660254037844386],
[-0.49999999999999983, 0.8660254037844387],
[-0.5000000000000001, -0.8660254037844386],
[1.0, 0.0],
[-0.49999999999999983, 0.8660254037844387],
[1.0, 0.0],
[-0.49999999999999983, 0.8660254037844387],
[-0.5000000000000001, -0.8660254037844386],
[-0.5000000000000001, -0.8660254037844386],
[1.0, 0.0],
[-0.5000000000000001, -0.8660254037844386],
[1.0, 0.0],
[-0.49999999999999983, 0.8660254037844387],
[1.0, 0.0],
[-0.49999999999999983, 0.8660254037844387],
[-0.5000000000000001, -0.8660254037844386],
[1.0, 0.0],
[-0.5000000000000001, -0.8660254037844386],
[1.0, 0.0],
[-0.49999999999999983, 0.8660254037844387],
[-0.49999999999999983, 0.8660254037844387],
[-0.5000000000000001, -0.8660254037844386],
[-0.49999999999999983, 0.8660254037844387],
[-0.5000000000000001, -0.8660254037844386],
[1.0, 0.0],
[-0.5000000000000001, -0.8660254037844386],
[1.0, 0.0],
[-0.49999999999999983, 0.8660254037844387],
[-0.5000000000000001, -0.8660254037844386],
[-0.49999999999999983, 0.8660254037844387],
[-0.5000000000000001, -0.8660254037844386],
[1.0, 0.0],
[1.0, 0.0],
[-0.49999999999999983, 0.8660254037844387]
]
allowed_symmetries = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59]
momentum = [2.0943951023931953, 3.627598728468436]
[M.C2.A]
characters = [
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0]
]
allowed_symmetries = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47]
momentum = [0.0, 3.627598728468436]
[M.C2.B]
characters = [
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[-1.0, 0.0],
[1.0, 0.0],
[1.0, 0.0]
]
allowed_symmetries = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47]
momentum = [0.0, 3.627598728468436]
[Z.C1.A]
characters = [
[1.0, 0.0],
[0.5000000000000001, 0.8660254037844386],
[-0.49999999999999983, 0.8660254037844387],
[0.5000000000000001, 0.8660254037844386],
[-0.49999999999999983, 0.8660254037844387],
[-1.0, 0.0],
[-0.5000000000000001, -0.8660254037844386],
[-1.0, 0.0],
[-0.5000000000000001, -0.8660254037844386],
[0.49999999999999983, -0.8660254037844387],
[0.49999999999999983, -0.8660254037844387],
[1.0, 0.0]
]
allowed_symmetries = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]
momentum = [1.0471975511965976, 1.813799364234218]
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.C6.A, 1.00, -1.00, -2.00, 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=8Plots.mm,
bottom_margin=2Plots.mm
# ytickfont=font(18),
# legendfont=font(18)
)
colors = palette(:default)#palette(:thermal,length(14:div(n_sites,2))+1)
markers = filter((m->begin
m in Plots.supported_markers()
end), Plots._shape_keys)
n_sites=36
n_seeds = 1
ks=["Gamma.C6.A", "Gamma.C6.B", "Gamma.C6.E1a", "Gamma.C6.E1b", "Gamma.C6.E2a", "Gamma.C6.E2b", "K.C3.A", "K.C3.Ea", "K.C3.Eb", "M.C2.A", "M.C2.B"]#, "Z.C1.A", "Delta.C1.A", "Sigma.C1.A", "Z0.C1.A", "Z1.C1.A"]
ksl=["Γ.C6.A", "Γ.C6.B", "Γ.C6.E1a", "Γ.C6.E1b", "Γ.C6.E2a", "Γ.C6.E2b", "K.C3.A", "K.C3.Ea", "K.C3.Eb", "M.C2.A", "M.C2.B"]
seeds = [i for i=1:n_seeds]
nup_start=10
n_ups = [i for i=nup_start:(div(n_sites,2))]#
J1=1.00
J2=-1.0
J3=-2.0
n_eigs = 10
for seed in seeds
plot(xlabel=L"S_z",ylabel=L"E/J_1",ylims=(-0.02,1),xlims=(-0.1,10),dpi=400,xticks=(0:2:10,[string(i) for i=0:2:10]))
mineig = 0
eigvs = []
Sz = []
f = h5open(@sprintf("outfile.kagome.%d.J1.%.2f.J2.%.2f.J3.%.2f.nup.%d.k.%s.seed.%d.h5",n_sites,J1,J2,J3,div(n_sites,2),"Gamma.C6.A",seed), "r")
mineig = read(f["Eigenvalues"])[1]
for n_up=n_ups
c=0
for k in ks
c=c+1
f = h5open(@sprintf("outfile.kagome.%d.J1.%.2f.J2.%.2f.J3.%.2f.nup.%d.k.%s.seed.%d.h5",n_sites,J1,J2,J3,n_up,k,seed), "r")
eig = read(f["Eigenvalues"])[1:n_eigs].-mineig
sz=abs(n_up- n_sites / 2)*[1 for i=1:n_eigs]
if n_up==nup_start
plot!(sz,eig,seriestype=:scatter,m=markers[c],mc=colors[c],label=ksl[c])
else
plot!(sz,eig,seriestype=:scatter,m=markers[c],mc=colors[c],primary=false)
end
close(f)
end
end
savefig(@sprintf("outfile.kagome.%d.J1.%.2f.J2.%.2f.J3.%.2f.seed.%d-n.png",n_sites,J1,J2,J3,seed))
end
references
[1] P. W. Anderson, An Approximate Quantum Theory of the Antiferromagnetic Ground State, Phys. Rev. 86, 694 (1952)
[2] Wietek, Alexander, and Andreas M. Läuchli. "Valence bond solid and possible deconfined quantum criticality in an extended kagome lattice Heisenberg antiferromagnet." Physical Review B 102.2 (2020): 020411