"""Hardcore boson Haldane model - minimal FCI example
Based on Eq.1 of [Grushin2015]_ with:
- bosons instead of fermions
- mu=0, V=0 (only infinite onsite repulsion, via hardcore constraint)
- 1/2 filling of the lowest band (i.e. 1/4 total filling)
"""
# Copyright (C) TeNPy Developers, Apache license
import warnings
import numpy as np
from tenpy.algorithms import dmrg
from tenpy.models.haldane import BosonicHaldaneModel
from tenpy.networks.mps import MPS
def plot_model(model_params, phi_ext=0.1):
model_params['phi_ext'] = phi_ext
M = BosonicHaldaneModel(model_params)
import matplotlib.pyplot as plt
plt.figure()
ax = plt.gca()
M.lat.plot_sites(ax)
M.coupling_terms['t1 Bd_i B_j'].plot_coupling_terms(ax, M.lat)
M.coupling_terms['t2 Bd_i B_j'].plot_coupling_terms(ax, M.lat, text='{op_j!s} {strength_angle:.2f}', text_pos=0.9)
print(M.coupling_terms['t1 Bd_i B_j'].to_TermList())
ax.set_aspect(1.0)
plt.show()
def run(model_params, phi_ext=np.linspace(0, 2.0, 7)):
data = dict(phi_ext=phi_ext, QL=[], ent_spectrum=[])
dmrg_params = {
'mixer': True, # setting this to True helps to escape local minima
'mixer_params': {'amplitude': 1.0e-5, 'decay': 1.2, 'disable_after': 30},
'trunc_params': {
'svd_min': 1.0e-10,
},
'lanczos_params': {'N_min': 5, 'N_max': 20},
'chi_list': {0: 9, 10: 49, 20: 100},
'max_E_err': 1.0e-10,
'max_S_err': 1.0e-6,
'max_sweeps': 150,
}
prod_state = [1]
if 2 * model_params['Lx'] * model_params['Ly'] % 4 != 0:
warnings.warn('Total filling factor = 1/4 cannot be achieved with this unit cell geometry.')
for i in range(1, 2 * model_params['Lx'] * model_params['Ly']):
if i % 4 == 0:
prod_state.append(1)
else:
prod_state.append(0)
print(prod_state)
eng = None
for phi in phi_ext:
print('=' * 100)
print('phi_ext = ', phi)
model_params['phi_ext'] = phi
if eng is None: # first time in the loop
M = BosonicHaldaneModel(model_params)
psi = MPS.from_product_state(
M.lat.mps_sites(), prod_state, bc=M.lat.bc_MPS, unit_cell_width=M.lat.mps_unit_cell_width
)
eng = dmrg.TwoSiteDMRGEngine(psi, M, dmrg_params)
else:
del eng.options['chi_list']
M = BosonicHaldaneModel(model_params)
eng.init_env(model=M)
E, psi = eng.run()
data['QL'].append(psi.average_charge(bond=0)[0])
data['ent_spectrum'].append(psi.entanglement_spectrum(by_charge=True)[0])
return data
def plot_results(data):
import matplotlib.pyplot as plt
plt.figure()
ax = plt.gca()
ax.plot(data['phi_ext'], data['QL'], marker='o')
ax.set_xlabel(r'$\Phi_y / 2 \pi$')
ax.set_ylabel(r'$ \langle Q^L(\Phi_y) \rangle$')
plt.savefig('haldane_FCI_charge_pump.pdf')
plt.figure()
ax = plt.gca()
colors = ['b', 'g', 'r', 'c', 'm', 'y', 'k']
color_by_charge = {}
for phi_ext, spectrum in zip(data['phi_ext'], data['ent_spectrum']):
for q, s in spectrum:
q = q[0]
label = ''
if q not in color_by_charge:
label = f'{q:d}'
color_by_charge[q] = colors[len(color_by_charge) % len(colors)]
color = color_by_charge[q]
ax.plot(phi_ext * np.ones(s.shape), s, linestyle='', marker='_', color=color, label=label)
ax.set_xlabel(r'$\Phi_y / 2 \pi$')
ax.set_ylabel(r'$ \epsilon_\alpha $')
ax.set_ylim(0.0, 8.0)
ax.legend(loc='upper right')
plt.savefig('haldane_FCI_ent_spec_flow.pdf')
if __name__ == '__main__':
t1_value = -1
phi = np.arccos(3 * np.sqrt(3 / 43))
t2_value = (np.sqrt(129) / 36) * t1_value * np.exp(1j * phi) # optimal band flatness
model_params = dict(
conserve='N',
t1=t1_value,
t2=t2_value,
mu=0,
V=0,
bc_MPS='infinite',
order='default',
Lx=1,
Ly=4,
bc_y='cylinder',
)
# plot_model(model_params)
data = run(model_params)
plot_results(data)