def CalculateBerryConnectGraphene(k, params, n0, n1):
h = 0.0001;
H = HaldaneHamiltonianPaulis(k, params)
d0,v0 = GetEvalsAndEvecsGen(H)
#first eigenvector
u0=v0[:,n0]
u1=v0[:,n1]
#dx direction
kxx = k + np.array([h,0])
H = HaldaneHamiltonianPaulis(kxx, params)
dx,vx = GetEvalsAndEvecsGen(H)
ux1 = vx[:,n1]
#dy direction
kyy = k+np.array([0,h])
H = HaldaneHamiltonianPaulis(kyy, params)
dy,vy = GetEvalsAndEvecsGen(H)
uy1=vy[:,n1]
xder = (ux1-u1)/h
yder = (uy1-u1)/h
berryconnect = 1j*np.array([np.dot(np.conj(u0),xder),np.dot(np.conj(u0),yder)])
return berryconnect
def BerryCurvature(Hamiltonian, k, n0, n1, params):
h = 0.0001
H = Hamiltonian(k,params)
d0,v0 = GetEvalsAndEvecsGen(H)
#first eigenvector
u0bx=v0[:,n0]
u0by=v0[:,n1]
#eigenvalues
lowerband = d0[0]
upperband = d0[1]
#dx direction
kxx = k + np.array([h,0])
H = Hamiltonian(kxx, params)
dx,vx = GetEvalsAndEvecsGen(H)
ux = vx[:,n0] # first eigenvector
ux = AlignGaugeBetweenVecs(u0bx, ux)
#dy direction
kyy = k+np.array([0,h])
H = Hamiltonian(kyy, params)
dy,vy = GetEvalsAndEvecsGen(H)
uy=vy[:,n1] # first eigenvector
uy = AlignGaugeBetweenVecs(u0by, uy)
xder = (ux-u0bx)/h
yder = (uy-u0by)/h
berrycurve = 2*np.imag(np.dot(np.conj(xder), yder))
return berrycurve, lowerband, upperband
# centres = [24, 25]
centres = [24]
# funcs = [Cosine, Cosine]
funcs = [Cosine]
# paramss = [[a, omega1, phi1, onsite1], [a, omega2, phi2, onsite2]]
paramss = [[a, omega1, phi1, onsite1]]
# form = "SS-p"; hamiltonianString="$H(t)=H_0 + a \> \hat{n}_b \cos (\omega t + \phi) $"; paramsString = r"$a="+str(a)+r", \omega = "+"{:.2f}".format(omega)+", \phi = "+PhiString(phi)+r"$"
# form = "DS-p"; hamiltonianString = "$H(t)=H_0 + a \> \hat{n}_b \cos (\omega_1 t + \phi_1) + a \> \hat{n}_{b+1} \cos (\omega_2 t + \phi_2)]$"; paramsString = r"$a=$"+str(a)+", "+r"$\omega_1="+"{:.2f}".format(omega1)+", \omega_2 = "+str(omegaMultiplier)+" \omega_1, \phi_1 = "+PhiString(phi1)+", \phi_2 = \phi_1 + \pi/2, N = "+str(N)+", b = "+str(centre)+"$ "
# form = "SSDF-p"; hamiltonianString = "$H(t)=H_0 + a \> \hat{n}_b [\cos (\omega_1 t + \phi_1) + \cos (\omega_2 t + \phi_2)]$"; paramsString = r"$a=$"+str(a)+", "+r"$\omega_1="+ "{:.2f}".format(omega1)+", \omega_2 = "+str(omegaMultiplier)+" \omega_1, \phi_1 ="+PhiString(phi1)+", \phi_2 = \phi_1 + \pi/2, N = "+str(N)+", b = "+str(centre)+"$ "
# form = "H0_PhasesNNHop"; hamiltonianString = "$H_0$"; paramsString=""
UT, HF = CreateHFGeneral(N, centres, funcs, paramss, T, circleBoundary)
# HF = H0_PhasesNNHop(N, centres, phases )
evals, evecs = GetEvalsAndEvecsGen(HF)
# evecs = OrderEvecs(evecs, N)
func = np.real
colour = "dodgerblue"
# title = ("real(evecs); "
# + form +r"; "+hamiltonianString+"\n"
# +paramsString)
title = ""
sz=8
fig, ax = plt.subplots(figsize=(sz*1.4,sz))
ax.plot(range(N), func(evals), '.', color=colour, markersize=20)
ax.set_ylabel("e-value")
ax.set_xlabel("eval index")
# fig.suptitle(r"evals; "+ form +r"; "+hamiltonianString+"\n"
omega12 = random()
omega23 = random()
omega31 = random()
print(omega12, omega23, omega31)
N = 100
evals = np.empty((100, 3))
evec0s = np.empty((100, 3))
evec1s = np.empty((100, 3))
evec2s = np.empty((100, 3))
delta3s = np.linspace(-0.1, 0.1, 100)
for i, delta3 in enumerate(delta3s):
H = Gen3Ham(delta1, delta2, delta3, omega12, omega23, omega31)
dx, vx = GetEvalsAndEvecsGen(H)
evals[i] = dx
evec0s[i] = vx[:, 0]
evec1s[i] = vx[:, 1]
evec2s[i] = vx[:, 2]
fig, ax = plt.subplots()
ax.plot(delta3s, evals[:, 0], '.', ms=10, color="#090446", label="e0")
ax.plot(delta3s, evals[:, 1], '.', ms=6, color="#F71735", label="e1")
ax.plot(delta3s, evals[:, 2], '.', ms=2, color="#FA9F42", label="e2")
plt.legend()
plt.show()
fig, ax = plt.subplots()
ax.plot(delta3s, evec0s[:, 0], '.', ms=10, label="a")
ax.plot(delta3s, evec0s[:, 1], '.', ms=6, label="b")
_, evecs0 = GetEvalsAndEvecsEuler(H)
evecsOld = evecs0
# go through and align vecs until the end
for k_vec in kline[1:]:
H = Euler2Hamiltonian(k_vec)
_, evecs = GetEvalsAndEvecsEuler(H)
evecs = SetGaugeSVG(evecs, evecsOld)
# evecs = SetGaugeByBand( evecsOld, evecs)
evecsOld = evecs
#calculate overlap
WilsonLine = CalcOverlapEvecs(evecsOld, evecs0)
WLevals, _ = GetEvalsAndEvecsGen(WilsonLine)
wilsonLineEvalsByPathWidth[i] = WLevals
wilsonevals0 = np.angle(wilsonLineEvalsByPathWidth[:, 0])
wilsonevals0newphase = np.where(wilsonevals0 < 0, np.abs(wilsonevals0),
wilsonevals0)
plt.plot(np.linspace(0, 1, Nw), wilsonevals0)
plt.show()
#%%
"""
Calculate Wilson Line -
Calculate half baked wilson lines around a circle
omega = a / jn_zeros(0, 1)[0]
phi = 0
form = "SS-p"
hamiltonianString = "$H(t)=H_0 + a \> \hat{n}_b \cos (\omega t + \phi) $"
paramsString = r"$a=" + str(a) + r", \omega = " + "{:.2f}".format(
omega) + ", \phi = " + PhiString(phi) + r"$"
# form = "DS-p"; hamiltonianString = "$H(t)=H_0 + a \> \hat{n}_b \cos (\omega_1 t + \phi_1) + a \> \hat{n}_{b+1} \cos (\omega_2 t + \phi_2)]$"; paramsString = r"$a=$"+str(a)+", "+r"$\omega_1="+"{:.2f}".format(omega1)+", \omega_2 = "+str(omegaMultiplier)+" \omega_1, \phi_1 = "+PhiString(phi1)+", \phi_2 = \phi_1 + \pi/2, N = "+str(N)+", b = "+str(centre)+"$ "
# form = "SSDF-p"; hamiltonianString = "$H(t)=H_0 + a \> \hat{n}_b [\cos (\omega_1 t + \phi_1) + \cos (\omega_2 t + \phi_2)]$"; paramsString = r"$a=$"+str(a)+", "+r"$\omega_1="+ "{:.2f}".format(omega1)+", \omega_2 = "+str(omegaMultiplier)+" \omega_1, \phi_1 ="+PhiString(phi1)+", \phi_2 = \phi_1 + \pi/2, N = "+str(N)+", b = "+str(centre)+"$ "
# form = "H0_PhasesNNHop"; hamiltonianString = "$H_0$"; paramsString=""
# UT, HF = CreateHF(form, rtol, N, centre, a, omega, phi, onsite)
UT, HF = CreateHFGeneral(N, centre, funcs, paramss, T, circleBoundary)
# HF = H0_PhasesNNHop(N, centres, phases )
evals, evecs = GetEvalsAndEvecsGen(HF)
# evecs = OrderEvecs(evecs, N)
func = np.real
colour = "dodgerblue"
# title = ("real(evecs); "
# + form +r"; "+hamiltonianString+"\n"
# +paramsString)
title = ""
plotevecs(evecs, N, func, colour, title, ypos=0.955)
sz = 5
fig, ax = plt.subplots(figsize=(sz * 1.4, sz))
ax.plot(range(N), func(evals), 'x', color=colour)
phases2 = [0, 0, 0, 0, 0, 0] # phases with non trivial loops and NNN hopping aSiteStart1 = 20; # aSiteStart2 = 110 psi01 = np.zeros(N, dtype=np.complex_); psi01[aSiteStart1] = 1; # psi02 = np.zeros(N, dtype=np.complex_); psi02[aSiteStart2] = 1; # p0 = pi/7; p1 = pi/3; p2 = pi/2; p3 = pi/4 HPhases1 = H0_PhasesNNHop(N, centres, phases1) HPhases2 = H0_PhasesNNHop(N, centres, phases2) evals1, evecs1 = GetEvalsAndEvecsGen(HPhases1) evals2, evecs2 = GetEvalsAndEvecsGen(HPhases1) # psi01 = evecs2[:,0] # HPhasesFlip = np.flip(np.flip(HPhases, 0).T, 0) #parameters for ODE solver nOscillations = 10 #how many steps we want. NB, this means we will solve for nTimesteps+1 times (edges) nTimesteps = nOscillations*100 n_osc_divisions = 2 tspan = (0,nOscillations*T) t_eval = np.linspace(tspan[0], tspan[1], nTimesteps) #solve differential equation psi1 = SolveSchrodingerTimeIndependent(HPhases1, tspan, nTimesteps, psi01)