def H_tracking_evolution_equation(current_time, gamma, fermihubbard,
observables, J_target):
D = fermihubbard.operator_dict['hop_left_op'].expt_value(gamma[:-1])
phi = fixed_point(H_tracking_implicit_phi_function,
observables.phi[-1],
args=(gamma, J_target(current_time), fermihubbard,
observables))
# phi = 0
psi_dot = -expiphi(phi) * fermihubbard.perimeter_params.t \
* fermihubbard.operator_dict['hop_left_op'].dot(gamma[:-1])
psi_dot -= expiphiconj(phi) * fermihubbard.perimeter_params.t \
* fermihubbard.operator_dict['hop_right_op'].dot(gamma[:-1])
psi_dot += fermihubbard.operator_dict['H_onsite'].dot(gamma[:-1])
y_dot = -(-fermihubbard.perimeter_params.t * (expiphi(phi) * D
+ expiphiconj(phi) * D.conj())
+ fermihubbard.operator_dict['H_onsite'].expt_value(gamma[:-1]))\
* (J_target.derivative())(current_time)/(J_target(current_time)**2)
gamma_dot = np.append(psi_dot, y_dot)
return gamma_dot
def R_tracking_evolution_equation(current_time, gamma, fermihubbard,
observables, J_target):
D = fermihubbard.operator_dict['hop_left_op'].expt_value(gamma[:-1])
comm = fermihubbard.operator_dict['commutator_HK'].expt_value(gamma[:-1])
R = np.abs(D)
theta = np.angle(D)
a = fermihubbard.perimeter_params.a
t = fermihubbard.perimeter_params.t
J_dot_target = J_target.derivative()
R_dot = (comm.real * D.imag - comm.imag * D.real) / R
# theta_dot = (comm.imag * D.imag + comm.real * D.real)/R**2
phi = observables.phi_init + theta - gamma[-1]
psi_dot = -expiphi(
phi) * t * fermihubbard.operator_dict['hop_left_op'].dot(gamma[:-1])
psi_dot -= expiphiconj(
phi) * t * fermihubbard.operator_dict['hop_right_op'].dot(gamma[:-1])
psi_dot += fermihubbard.operator_dict['H_onsite'].dot(gamma[:-1])
psi_dot = -1j * psi_dot
y_dot = (J_dot_target(current_time) / (2 * a * t) +
R_dot * np.sin(phi - theta)) / (R * np.cos(phi - theta))
gamma_dot = np.append(psi_dot, y_dot)
return gamma_dot
def append_observables(self, psi, phi):
self.neighbour.append(
self.fermihubbard.operator_dict['hop_left_op'].expt_value(psi))
self.current.append(-1j * self.fermihubbard.perimeter_params.a *
self.fermihubbard.perimeter_params.t *
(expiphi(phi) * self.neighbour[-1] -
expiphiconj(phi) * self.neighbour[-1].conj()))
self.energy.append(
-self.fermihubbard.perimeter_params.t *
(expiphi(phi) * self.neighbour[-1] +
expiphiconj(phi) * self.neighbour[-1].conj()) +
self.fermihubbard.operator_dict['H_onsite'].expt_value(psi))
while abs(phi.real - self.phi[-1].real) > PI:
if phi.real - self.phi[-1].real > PI:
phi -= 2 * PI
elif phi.real - self.phi[-1].real < -PI:
phi += 2 * PI
self.phi.append(phi)
if self.continuity:
self.number.append(
self.fermihubbard.operator_dict['n'].expt_value(psi))
for _ in range(self.fermihubbard.perimeter_params.nx - 1):
self.numbersite[_].append(
self.fermihubbard.operator_dict["num" +
str(_)].expt_value(psi))
K_t = expiphi(phi) * self.fermihubbard.operator_dict[
"K" + str(_)].expt_value(psi)
self.currentsite[_].append(
-1j * self.fermihubbard.perimeter_params.t *
self.fermihubbard.perimeter_params.a * (K_t - K_t.conj()))
if self.fermihubbard.perimeter_params.pbc:
self.numbersite[self.fermihubbard.perimeter_params.nx -
1].append(
self.fermihubbard.operator_dict["num5"].
expt_value(psi))
K_t = expiphi(phi) * self.fermihubbard.operator_dict[
"K5"].expt_value(psi)
self.currentsite[self.fermihubbard.perimeter_params.nx -
1].append(
-1j *
self.fermihubbard.perimeter_params.t *
self.fermihubbard.perimeter_params.a *
(K_t - K_t.conj()))
def H_tracking_implicit_phi_function(phi_j0, gamma, J_target, fermihubbard,
observables):
D = fermihubbard.operator_dict['hop_left_op'].expt_value(gamma[:-1])
a = fermihubbard.perimeter_params.a
t = fermihubbard.perimeter_params.t
expi = expiphi(phi_j0)
doub = fermihubbard.operator_dict['H_onsite'].expt_value(gamma[:-1])
phi_0 = observables.phi_init
bt = observables.boundary_term
phi_j1 = -a * (-t * (expi * D + (expi * D).conj()) +
doub) / J_target + phi_0 + a * gamma[-1] + bt
return phi_j1.real
def switch_tracking_methods(self, method, fermihubbard, psi, J_target):
if method == "R2H":
D = fermihubbard.operator_dict['hop_left_op'].expt_value(psi)
self.phi_init = self.phi[-1]
self.boundary_term = fermihubbard.perimeter_params.a \
* (-fermihubbard.perimeter_params.t * (expiphi(self.phi_init) * D
+ expiphiconj(self.phi_init)
* fermihubbard.perimeter_params.t * D.conj())
+ fermihubbard.operator_dict['H_onsite'].expt_value(psi))/J_target
elif method == "H2R":
self.phi_init = self.phi[-1]
self.boundary_term = 0.0
return 0.0
def __init__(self, psi_init, J_init, phi_init, fermihubbard, continuity=0):
self.fermihubbard = fermihubbard
self.psi = [
psi_init,
]
self.neighbour = [
fermihubbard.operator_dict['hop_left_op'].expt_value(psi_init),
]
self.current = [
J_init,
]
self.phi = [
phi_init,
]
D = self.neighbour[-1]
expi = expiphi(phi_init)
doub = fermihubbard.operator_dict['H_onsite'].expt_value(psi_init)
a = fermihubbard.perimeter_params.a
t = fermihubbard.perimeter_params.t
self.energy = [
-t * (D * expi + (D * expi).conj()) + doub,
]
# self.y = 0
# self.psi = psi_init
self.phi_init = phi_init
if np.isclose(J_init, 0):
self.boundary_term = 0
else:
self.boundary_term = a * self.energy[-1] / J_init
self.number = [
self.fermihubbard.perimeter_params.nup +
self.fermihubbard.perimeter_params.ndown,
]
self.numbersite = [
[],
]
self.currentsite = [
[],
]
self.continuity = continuity
if self.continuity:
for _ in range(self.fermihubbard.perimeter_params.nx - 1):
self.numbersite.append([])
self.currentsite.append([])
for _ in range(self.fermihubbard.perimeter_params.nx - 1):
self.numbersite[_] = [
self.fermihubbard.operator_dict["num" + str(_)].expt_value(
psi_init),
]
K_t = expiphi(phi_init) * self.fermihubbard.operator_dict[
"K" + str(_)].expt_value(psi_init)
self.currentsite[_] = [
-1j * self.fermihubbard.perimeter_params.t *
self.fermihubbard.perimeter_params.a * (K_t - K_t.conj()),
]
if self.fermihubbard.perimeter_params.pbc:
self.numbersite[self.fermihubbard.perimeter_params.nx -
1].append(
self.fermihubbard.operator_dict["num5"].
expt_value(psi_init))
K_t = expiphi(phi_init) * self.fermihubbard.operator_dict[
"K5"].expt_value(psi_init)
self.currentsite[self.fermihubbard.perimeter_params.nx -
1].append(
-1j *
self.fermihubbard.perimeter_params.t *
self.fermihubbard.perimeter_params.a *
(K_t - K_t.conj()))
self.add_var = dict()