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Josephson_ED_varied_number.py
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Josephson_ED_varied_number.py
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import os
os.environ['OMP_NUM_THREADS']='4' # set number of OpenMP threads to run in parallel
from quspin.operators import hamiltonian # Hamiltonians and operators
from quspin.basis import spinful_fermion_basis_1d # bosonic Hilbert space
from quspin.tools.measurements import diag_ensemble
from scipy.sparse.linalg import expm_multiply
from quspin.tools.lanczos import lanczos_full,lanczos_iter,lin_comb_Q_T,expm_lanczos
import numpy as np # general math functions
import matplotlib.pyplot as plt # plotting library
import math
import cmath
import itertools as it
import scipy.linalg as sla
#from time import time # timing package
#ti = time() # start timer
no_checks = dict(check_herm=False,check_symm=False,check_pcon=False)
#
##### define model parameters
# initial seed for random number generator
np.random.seed(0) # seed is 0 to produce plots from QuSpin2 paper
# setting up parameters of simulation
L = 2 # length of chain
Ns = 2*L+1 # number of sites
N = Ns+2 # number of sites with leads
evens = [i for i in range(0,N+1,2)]
odds = [i for i in range(1,N+1,2)]
Nfs_even = [(i,j) for i in evens for j in evens]
Nfs_even.extend(list((i,j) for i in odds for j in odds))
Nfs_odd = [(i,j) for i in odds for j in evens]
Nfs_odd.extend(list((i,j) for i in evens for j in odds))
print(Nfs_even)
print(Nfs_odd)
sps = 2 # number of states per site
t_AA = -1.0 # AA hopping
t_AB = np.sqrt(2.0)*t_AA # AB hopping
t_SL = 5.0
t_SR = 5.0
#mu = 0.7
mu = 0.0
#DeltaL = 100.0*cmath.exp(0.5*3.14159265*1.0j)
DeltaL = 100.0*cmath.exp(0.99*3.14159265*1.0j)
DeltaR = 100.0
U = -2.5 # interaction strength
VB = 1.0 # boundary potential
##### set up Hamiltonian and observables
# define site-coupling lists
E_list = [[VB-mu,1,1],[VB-mu,N-2,N-2]] # boundary potential
E_list.extend([[-mu,i,i] for i in range(2,N-2)])
E_list.extend(([0.0,0,0],[0.0,N-1,N-1]))
print(E_list)
Deltaup_list = [[np.conj(DeltaL),0,0],[np.conj(DeltaR),N-1,N-1]]
Deltadown_list = [[-DeltaL,0,0],[-DeltaR,N-1,N-1]]
int_list = [[U,i,i] for i in range(1,N-1,1)] # on-site interactions within lattice
int_list.extend(([0.0,0,0],[0.0,N-1,N-1]))
print(int_list)
# setting up hopping lists
hop_left = [[t_AA,i,(i+2)] for i in range(1,N-2,2)] # AA hopping left
hop_right = [[-t_AA,i,(i+2)] for i in range(1,N-2,2)] # AA hopping right
hop_left.extend([[t_AB,i,(i+1)] for i in range(1,N-2,1)]) # AB hopping left
hop_right.extend([[-t_AB,i,(i+1)] for i in range(1,N-2,1)]) # AB hopping right
hop_left.extend([[np.conj(t_SL),0,1],[np.conj(t_SR),N-2,N-1]]) # lead hopping 1
hop_right.extend([[-t_SL,0,1],[-t_SR,N-2,N-1]]) # lead hopping 2
#hop_list_hc = [[J.conjugate(),i,j] for J,i,j in hop_list] # add h.c. terms
# setting up local Fock basis
# there are even and odd sectors
basis_even = spinful_fermion_basis_1d(L=N,Nf = Nfs_even)
basis_odd = spinful_fermion_basis_1d(L=N,Nf = Nfs_odd)
# non-interacting Hamiltonian
static = [
["+-|",E_list], # on_site energies up
["|+-",E_list], # on_site energies down
["+-|",hop_left], # hopping up
["-+|",hop_right], # hopping up h.c.
["|+-",hop_left], # hopping down
["|-+",hop_right], # hopping down h.c.
["+|+",Deltaup_list], # lead Delta up
["-|-",Deltadown_list], # lead Delta down
["n|n",int_list] # interaction
]
H_even = hamiltonian(static,[],basis=basis_even,dtype=np.complex128,check_pcon = False)
H_odd = hamiltonian(static,[],basis=basis_odd,dtype=np.complex128,check_pcon = False)
print(basis_even.Ns)
print(basis_odd.Ns)
#Lanczos algorithm for ground state
m_evo = 30 # Krylov subspace dimension
#initial state
v0_even = np.random.normal(0,1,size=basis_even.Ns)
v0_odd = np.random.normal(0,1,size=basis_odd.Ns)
E_full_even,V_full_even,Q_full_even = lanczos_full(H_even,v0_even,m_evo)
E_full_odd,V_full_odd,Q_full_odd = lanczos_full(H_odd,v0_odd,m_evo)
# unitary evolution time step
dt = 0.3
v_even = expm_lanczos(E_full_even,V_full_even,Q_full_even,a=-1j*dt)
v_odd = expm_lanczos(E_full_odd,V_full_odd,Q_full_odd,a=-1j*dt)
print(E_full_even[0])
print(E_full_odd[0])
E_GS_even,psi_GS_even = H_even.eigsh(k=1,which="SA")
psi_GS_even = psi_GS_even.ravel()
E_GS_odd,psi_GS_odd = H_odd.eigsh(k=1,which="SA")
psi_GS_odd = psi_GS_odd.ravel()
print(E_GS_even)
print(E_GS_odd)
psi_GS_even_lanczos = lin_comb_Q_T(V_full_even[:,0],Q_full_even)
print(np.abs(np.log(np.abs(np.vdot(psi_GS_even_lanczos,psi_GS_even)))))
I_SL = hamiltonian([["+-|",[[1.0j*t_SL,1,0]]], ["+-|",[[-1.0j*t_SL,0,1]]],["|+-",[[1.0j*t_SL,1,0]]], ["|+-",[[-1.0j*t_SL,0,1]]]],[],basis=basis_even,dtype=np.complex128,**no_checks)
n_op = hamiltonian([["+-|",[[1.0,i+1,i+1] for i in range(Ns)]],["|+-",[[1.0,i+1,i+1] for i in range(Ns)]]],[],basis=basis_even,dtype=np.complex128,**no_checks)
Delta_A2 = hamiltonian([["-|-",[[U,3,3]]]],[],basis=basis_even,dtype=np.complex128,**no_checks)
#expt_I_SL = I_SL.expt_value(psi_GS_even).real
#print(expt_I_SL)
#for i in range(Ns):
# Delta_A1 = hamiltonian([["-|-",[[U,i+1,i+1]]]],[],basis=basis_even,dtype=np.complex128,**no_checks)
# expt_Delta_A1 = Delta_A1.expt_value(psi_GS_even)
# print([abs(expt_Delta_A1),cmath.phase(expt_Delta_A1)])
n_mus = 100
n_phis = 100
mus = np.linspace(-4.2,2.0,n_mus)
phis = np.linspace(0,2*np.pi,n_phis)
nvsmu = np.zeros(n_mus)
Ivsmu = np.zeros(n_mus)
DeltaA2vsmu = np.zeros(n_mus)
Ivsphi = np.zeros(n_phis)
for i in range(n_mus):
#mu = 0.7
mu = mus[i]
# define site-coupling lists
E_list = [[VB-mu,1,1],[VB-mu,N-2,N-2]] # boundary potential
E_list.extend([[-mu,i,i] for i in range(2,N-2)])
E_list.extend(([0.0,0,0],[0.0,N-1,N-1]))
#DeltaL = 100.0*cmath.exp(1.0j*phis[i])
#Deltaup_list = [[np.conj(DeltaL),0,0],[np.conj(DeltaR),N-1,N-1]]
#Deltadown_list = [[-DeltaL,0,0],[-DeltaR,N-1,N-1]]
static = [
["+-|",E_list], # on_site energies up
["|+-",E_list], # on_site energies down
["+-|",hop_left], # hopping up
["-+|",hop_right], # hopping up h.c.
["|+-",hop_left], # hopping down
["|-+",hop_right], # hopping down h.c.
["+|+",Deltaup_list], # lead Delta up
["-|-",Deltadown_list], # lead Delta down
["n|n",int_list] # interaction
]
H_even = hamiltonian(static,[],basis=basis_even,dtype=np.complex128,check_pcon = False)
H_odd = hamiltonian(static,[],basis=basis_odd,dtype=np.complex128,check_pcon = False)
E_GS_even,psi_GS_even = H_even.eigsh(k=1,which="SA")
E_GS_odd,psi_GS_odd = H_odd.eigsh(k=1,which="SA")
print(E_GS_even)
print(E_GS_odd)
psi_GS_even = psi_GS_even.ravel()
expt_I_SL = I_SL.expt_value(psi_GS_even).real
n_expt = n_op.expt_value(psi_GS_even).real
DeltaA2_expt = abs(Delta_A2.expt_value(psi_GS_even))
Ivsmu[i] = expt_I_SL
nvsmu[i] = n_expt
DeltaA2vsmu[i] = DeltaA2_expt
#Ivsphi[i] = expt_I_SL
plt.figure()
plt.plot(mus,Ivsmu)
plt.xlabel("Chemical potential")
plt.ylabel("Current")
plt.figure()
plt.plot(mus,nvsmu)
plt.xlabel("Chemical potential")
plt.ylabel("Particle number")
plt.figure()
plt.plot(mus,DeltaA2vsmu)
plt.xlabel("Chemical potential")
plt.ylabel("Delta at A2")
plt.show()
#plt.figure()
#plt.plot(mus,Ivsphi)
#plt.xlabel("Phase twist")
#plt.ylabel("Current")
#plt.show()
# diagonalise Hamiltonian
#E_even,V_even = H_even.eigh()
#E_odd,V_odd = H_odd.eigh()
#print("Eigenenergies", E)
## initial state
#psi = V[:,-1]
#
## setting up observables
#n_list = [hamiltonian([["n",[[1.0,i]]]],[],basis=basis,dtype=np.complex128,**no_checks) for i in range(N)]
##I = commutator(H, n_list[4])
#it_AA = 1.0j*t_AA
#it_AB = 1.0j*t_AB
#
#
##I_A2A3 = hamiltonian([["+-",[[1.0j*t_AA,N-1,N-3]]], ["+-",[[-1.0j*t_AA,N-3,N-1]]]],[],basis=basis,dtype=np.complex128,**no_checks)
##I_B2A3 = hamiltonian([["+-",[[1.0j*t_AB,N-1,N-2]]], ["+-",[[-1.0j*t_AB,N-2,N-1]]]],[],basis=basis,dtype=np.complex128,**no_checks)
##I_A2A3 = hamiltonian([["+-",[[t_AA,4,2]]], ["+-",[[-t_AA,2,4]]]],[],basis=basis,dtype=np.complex128,**no_checks)
##I_B2A3 = hamiltonian([["+-",[[t_AB,4,3]]], ["+-",[[-t_AB,3,4]]]],[],basis=basis,dtype=np.complex128,**no_checks)
#
#
## setting up parameters for evolution
##start,stop,num = 0,100,2501 # 0.1 equally spaced points
#start,stop,num = 0,400,5001 # 0.1 equally spaced points
#times = np.linspace(start,stop,num)
#n_VBs = 50
#VBs = np.linspace(0.0, 2.0, n_VBs)
#n_Vstate_A3 = np.zeros(n_VBs)
#curs_A2A3_Vstate = np.zeros(n_VBs)
#curs_B2A3_Vstate = np.zeros(n_VBs)
#curs_A2A3_Vstate_diag = np.zeros(n_VBs)
#curs_B2A3_Vstate_diag = np.zeros(n_VBs)
#n_A3_Vstate_diag = np.zeros(n_VBs)
#curs_total_Vstate = np.zeros(n_VBs)
#
#for i in range(n_VBs):
# VB = VBs[i]
# print(VB)
# E_list = [[VB,0],[VB,N-1]] # boundary potential
# # set up static and dynamic lists
# static = [
# ["+-",hop_list], # hopping
# ["-+",hop_list_hc], # hopping h.c.
# ["++--",int_list], # two-particle interaction
# ["n",E_list]
# ]
# dynamic = [] # no dynamic operators
# # build real-space Hamiltonian
# H = hamiltonian(static,dynamic,basis=basis,dtype=np.complex128,**no_checks)
# Es,Vs = H.eigh()
# #print("Eigenvalue",Es)
#
# ##### do time evolution
# # calculating the evolved states
# psi_t = H.evolve(psi,t0=times[0],times=times)
#
# """
# expt_I_A2A3 = np.zeros(num)
# expt_I_B2A3 = np.zeros(num)
# for j,psi in enumerate(psi_t):
# expt_I_A2A3[j] = I_A2A3.expt_value(psi,time=times[j]).imag
# expt_I_B2A3[j] = I_B2A3.expt_value(psi,time=times[j]).imag
# """
#
# # calculating the local densities as a function of time
# expt_n_t = n_list[N-1].expt_value(psi_t).real
# expt_I_A2A3 = I_A2A3.expt_value(psi_t).real
# expt_I_B2A3 = I_B2A3.expt_value(psi_t).real
#
# # Diagonal ensemble averages
# Diag_ens_I_A2A3 = diag_ensemble(N,psi,Es,Vs,Obs=I_A2A3,delta_t_Obs=True)
# Diag_ens_I_B2A3 = diag_ensemble(N,psi,Es,Vs,Obs=I_B2A3,delta_t_Obs=True)
# Diag_ens_n_A3 = diag_ensemble(N,psi,Es,Vs,Obs=n_list[N-1],delta_t_Obs=True)
#
#
# # mean particle number at A3
# n_Vstate_A3[i] = np.mean(expt_n_t[math.floor(num/2):1:-1])
# n_A3_Vstate_diag[i] = Diag_ens_n_A3['Obs_pure']
# # RMS particle current
# curs_A2A3_Vstate_diag[i] = Diag_ens_I_A2A3['Obs_pure']
# curs_B2A3_Vstate_diag[i] = Diag_ens_I_B2A3['Obs_pure']
#
# print(curs_A2A3_Vstate_diag[i])
# print(curs_B2A3_Vstate_diag[i])
# #curs_A2A3_Vstate[i] = np.sqrt(np.mean(expt_I_A2A3[math.floor(num/2):1:-1]**2))
# #curs_B2A3_Vstate[i] = np.sqrt(np.mean(expt_I_B2A3[math.floor(num/2):1:-1]**2))
# #curs_total_Vstate[i] = np.sqrt(np.mean((expt_I_A2A3[math.floor(num/2):1:-1]+expt_I_B2A3[math.floor(num/2):1:-1])**2))
#
# curs_A2A3_Vstate[i] = np.abs(expt_I_A2A3[1])
# curs_B2A3_Vstate[i] = np.abs(expt_I_B2A3[1])
# curs_total_Vstate[i] =np.abs(expt_I_A2A3[1]+expt_I_B2A3[1])
#
# #curs_A2A3_Vstate[Ui] = np.mean(np.abs(expt_I_A2A3))
# #curs_B2A3_Vstate[Ui] = np.mean(np.abs(expt_I_B2A3))
#
#print("simulation took {0:.4f} sec".format(time()-ti))
#
## plotting static figures
#print("Plotting")
#
#plt.plot(VBs, n_Vstate_A3)
#plt.xlim(VBs[0],VBs[-1])
#plt.ylim(ymin=0)
#plt.xlabel("Boundary energy",fontsize=20)
#plt.ylabel("$n_\\mathrm{A3}$",fontsize=20)
#plt.savefig("particle_number_"+str(Nb)+"_"+str(VBs[0])+"-"+str(VBs[-1])+".png")
#
#plt.figure()
#
#plt.plot(VBs, curs_A2A3_Vstate,label="A2->A3")
#plt.plot(VBs, curs_B2A3_Vstate,label="B2->A3")
#plt.plot(VBs, curs_total_Vstate,label="total")
#plt.xlim(VBs[0],VBs[-1])
#plt.ylim(ymin=0)
#plt.xlabel("Boundary energy",fontsize=20)
#plt.ylabel("Particle current",fontsize=20)
#plt.legend()
#plt.grid()
#plt.savefig("current_"+str(Nb)+"_"+str(VBs[0])+"-"+str(VBs[-1])+".png")
#
#plt.figure()
#
#plt.plot(VBs, curs_A2A3_Vstate_diag,label="A2->A3")
#plt.plot(VBs, curs_B2A3_Vstate_diag,label="B2->A3")
#plt.xlim(VBs[0],VBs[-1])
#plt.ylim(ymin=0)
#plt.xlabel("Boundary potential",fontsize=20)
#plt.ylabel("Particle current",fontsize=20)
#plt.legend()
#plt.grid()
#
#plt.figure()
#
#plt.plot(VBs, n_A3_Vstate_diag)
#plt.xlim(VBs[0],VBs[-1])
#plt.ylim(ymin=0)
#plt.xlabel("Boundary potential",fontsize=20)
#plt.ylabel("Particle numbet at A_3",fontsize=20)
#
#plt.show()
#plt.close()