"""Chiral pi flux model - Chern insulator example
Based on the model in [Neupert2011]_
"""
# Copyright (C) TeNPy Developers, Apache license
import numpy as np
from tenpy.algorithms import dmrg
from tenpy.models import lattice
from tenpy.models.model import CouplingMPOModel
from tenpy.networks import site
from tenpy.networks.mps import MPS
from tenpy.networks.site import FermionSite
class BipartiteSquare(lattice.Lattice):
def __init__(self, Lx, Ly, siteA, **kwargs):
basis = np.array(([2, 0.0], [0, 2]))
pos = np.array(([0, 0], [1, 0], [0, 1], [1, 1]))
kwargs.setdefault('order', 'default')
kwargs.setdefault('bc', 'periodic')
kwargs.setdefault('bc_MPS', 'infinite')
kwargs.setdefault('basis', basis)
kwargs.setdefault('positions', pos)
super().__init__([Lx, Ly], [siteA, siteA, siteA, siteA], **kwargs)
self.NN = [
(0, 1, np.array([0, 0])),
(1, 3, np.array([0, 0])),
(3, 2, np.array([0, 0])),
(2, 0, np.array([0, 0])),
(2, 0, np.array([0, 1])),
(1, 3, np.array([0, -1])),
(0, 1, np.array([-1, 0])),
(3, 2, np.array([1, 0])),
]
self.nNNdashed = [
(0, 3, np.array([0, 0])),
(2, 1, np.array([0, 0])),
(3, 0, np.array([1, 1])),
(1, 2, np.array([1, -1])),
]
self.nNNdotted = [
(1, 2, np.array([1, 0])),
(3, 0, np.array([1, 0])),
(2, 1, np.array([0, 1])),
(3, 0, np.array([0, 1])),
]
class FermionicPiFluxModel(CouplingMPOModel):
def init_sites(self, model_params):
conserve = model_params.get('conserve', 'N')
site = FermionSite(conserve=conserve)
return site
def init_lattice(self, model_params):
Lx = model_params.get('Lx', 1)
Ly = model_params.get('Ly', 3)
fs = self.init_sites(model_params)
lat = BipartiteSquare(Lx, Ly, fs)
return lat
def init_terms(self, model_params):
t = np.asarray(model_params.get('t', -1.0))
V = np.asarray(model_params.get('V', 0.0))
mu = np.asarray(model_params.get('mu', 0.0))
phi_ext = 2 * np.pi * model_params.get('phi_ext', 0.0)
t1 = t * np.exp(1j * np.pi / 4)
t2 = t / np.sqrt(2)
self.add_onsite(mu, 0, 'N', category='mu N')
self.add_onsite(-mu, 1, 'N', category='mu N')
for u1, u2, dx in self.lat.NN:
t1_phi = self.coupling_strength_add_ext_flux(t1, dx, [0, phi_ext])
self.add_coupling(t1_phi, u1, 'Cd', u2, 'C', dx, 'JW', category='t1 Cd_i C_j', plus_hc=True)
self.add_coupling(V, u1, 'N', u2, 'N', dx, category='V N_i N_j')
for u1, u2, dx in self.lat.nNNdashed:
t2_phi = self.coupling_strength_add_ext_flux(t2, dx, [0, phi_ext])
self.add_coupling(t2_phi, u1, 'Cd', u2, 'C', dx, 'JW', category='t2 Cd_i C_j', plus_hc=True)
for u1, u2, dx in self.lat.nNNdotted:
t2_phi = self.coupling_strength_add_ext_flux(t2, dx, [0, phi_ext])
self.add_coupling(-t2_phi, u1, 'Cd', u2, 'C', dx, 'JW', category='-t2 Cd_i C_j', plus_hc=True)
def plot_lattice():
import matplotlib.pyplot as plt
ax = plt.gca()
fs = site.FermionSite()
lat = BipartiteSquare(3, 3, fs, basis=[[2, 0], [0, 2]])
lat.plot_sites(ax)
lat.plot_coupling(ax, lat.NN, linestyle='-', color='green')
lat.plot_coupling(ax, lat.nNNdashed, linestyle='--', color='black')
lat.plot_coupling(ax, lat.nNNdotted, linestyle='--', color='red')
ax.set_aspect('equal')
plt.show()
def run(phi_ext=np.linspace(0, 1.0, 7)):
data = dict(phi_ext=phi_ext, QL=[], ent_spectrum=[])
model_params = dict(conserve='N', t=-1, V=0, mu=0, Lx=1, Ly=3)
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 = ['empty', 'full'] * (model_params['Lx'] * model_params['Ly'] * 2)
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 = FermionicPiFluxModel(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 = FermionicPiFluxModel(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$')
plt.savefig('chiral_pi_flux_charge_pump.pdf')
plt.figure()
ax = plt.gca()
colors = ['b', 'g', 'r', 'c', 'm', 'y', 'k']
color_by_charge = {}
for phi_ext, spectrum in zip(data['phi_ext'], data['ent_spectrum']):
for q, s in spectrum:
q = q[0]
label = ''
if q not in color_by_charge:
label = f'{q:d}'
color_by_charge[q] = colors[len(color_by_charge) % len(colors)]
color = color_by_charge[q]
ax.plot(phi_ext * np.ones(s.shape), s, linestyle='', marker='_', color=color, label=label)
ax.set_xlabel(r'$\Phi_y / 2 \pi$')
ax.set_ylabel(r'$ \epsilon_\alpha $')
ax.set_ylim(0.0, 8.0)
ax.legend(loc='upper right')
plt.savefig('chiral_pi_flux_ent_spec_flow.pdf')
if __name__ == '__main__':
# plot_lattice()
data = run()
plot_results(data)