def wfnPropagation(iMPS, HMPO, nsteps, dt, ephtable, thresh=0, \
cleanexciton=None, prop_method="C_RK4", compress_method="svd", QNargs=None):
'''
simple wavefunction propagation through Runge-Kutta methods
'''
tableau = RK.runge_kutta_explicit_tableau(prop_method)
propagation_c = RK.runge_kutta_explicit_coefficient(tableau)
ketMPS = mpslib.add(iMPS, None, QNargs=QNargs)
Hset = [] # energy
Vset = [] # overlap
for isteps in xrange(nsteps):
if isteps != 0:
ketMPS = tMPS(ketMPS, HMPO, dt, ephtable, propagation_c, thresh=thresh, \
cleanexciton=cleanexciton, compress_method=compress_method, \
QNargs=QNargs)
Hset.append(mpslib.dot(mpslib.conj(ketMPS,QNargs=QNargs), \
mpslib.mapply(HMPO, ketMPS, QNargs=QNargs), QNargs=QNargs))
Vset.append(mpslib.dot(mpslib.conj(ketMPS,QNargs=QNargs), \
ketMPS, QNargs=QNargs))
return Hset, Vset
def ApproxPropagatorMPO(HMPO, dt, ephtable, propagation_c, thresh=0, \
compress_method="svd", QNargs=None):
'''
e^-iHdt : approximate propagator MPO from Runge-Kutta methods
'''
# Identity operator
if QNargs is not None:
nmpo = len(HMPO[0])
else:
nmpo = len(HMPO)
MPOdim = [1] * (nmpo + 1)
MPOQN = [[0]] * (nmpo + 1)
MPOQNidx = nmpo - 1
MPOQNtot = 0
IMPO = []
for impo in xrange(nmpo):
if QNargs is not None:
mpo = np.ones([1, HMPO[0][impo].shape[1], 1], dtype=np.complex128)
else:
mpo = np.ones([1, HMPO[impo].shape[1], 1], dtype=np.complex128)
IMPO.append(mpo)
IMPO = hilbert_to_liouville(IMPO)
QNargslocal = copy.deepcopy(QNargs)
if QNargs is not None:
IMPO = [IMPO, MPOQN, MPOQNidx, MPOQNtot]
# a real MPO compression
QNargslocal[1] = True
approxMPO = tMPS(IMPO, HMPO, dt, ephtable, propagation_c, thresh=thresh, \
compress_method=compress_method, QNargs=QNargslocal)
print "approx propagator thresh:", thresh
if QNargs is not None:
print "approx propagator dim:", [mpo.shape[0] for mpo in approxMPO[0]]
else:
print "approx propagator dim:", [mpo.shape[0] for mpo in approxMPO]
chkIden = mpslib.mapply(mpslib.conj(approxMPO, QNargs=QNargs),
approxMPO,
QNargs=QNargs)
print "approx propagator Identity error", np.sqrt(mpslib.distance(chkIden, IMPO, QNargs=QNargs) /\
mpslib.dot(IMPO, IMPO, QNargs=QNargs))
return approxMPO
def FiniteT_emi(mol, pbond, iMPO, HMPO, dipoleMPO, nsteps, dt, \
ephtable, insteps, thresh=0, temperature=298, prop_method="C_RK4", compress_method="svd",
QNargs=None):
'''
Finite temperature emission, already included in FiniteT_spectra
'''
tableau = RK.runge_kutta_explicit_tableau(prop_method)
propagation_c = RK.runge_kutta_explicit_coefficient(tableau)
beta = constant.T2beta(temperature)
ketMPO = thermal_prop(iMPO,
HMPO,
insteps,
ephtable,
prop_method=prop_method,
thresh=thresh,
temperature=temperature,
compress_method=compress_method,
QNargs=QNargs)
braMPO = mpslib.add(ketMPO, None, QNargs=QNargs)
#\Psi e^{\-beta H} \Psi
Z = mpslib.dot(mpslib.conj(braMPO, QNargs=QNargs), ketMPO, QNargs=QNargs)
print "partition function Z(beta)/Z(0)", Z
AketMPO = mpslib.mapply(dipoleMPO, ketMPO, QNargs=QNargs)
autocorr = []
t = 0.0
ketpropMPO, ketpropMPOdim = ExactPropagatorMPO(mol,
pbond,
-1.0j * dt,
QNargs=QNargs)
dipoleMPOdagger = mpslib.conjtrans(dipoleMPO, QNargs=QNargs)
if compress_method == "variational":
braMPO = mpslib.canonicalise(braMPO, 'l', QNargs=QNargs)
for istep in xrange(nsteps):
if istep != 0:
t += dt
AketMPO = mpslib.mapply(ketpropMPO, AketMPO, QNargs=QNargs)
braMPO = tMPS(braMPO,
HMPO,
dt,
ephtable,
propagation_c,
thresh=thresh,
cleanexciton=1,
compress_method=compress_method,
QNargs=QNargs)
AAketMPO = mpslib.mapply(dipoleMPOdagger, AketMPO, QNargs=QNargs)
ft = mpslib.dot(mpslib.conj(braMPO, QNargs=QNargs),
AAketMPO,
QNargs=QNargs)
autocorr.append(ft / Z)
autocorr_store(autocorr, istep)
return autocorr
def FiniteT_spectra(spectratype, mol, pbond, iMPO, HMPO, dipoleMPO, nsteps, dt,\
ephtable, insteps=0, thresh=0, temperature=298,\
algorithm=2, prop_method="C_RK4", compress_method="svd", QNargs=None, \
approxeiHt=None, GSshift=0.0, cleanexciton=None, scheme="P&C"):
'''
finite temperature propagation
only has algorithm 2, two way propagator
'''
assert algorithm == 2
assert spectratype in ["abs", "emi"]
tableau = RK.runge_kutta_explicit_tableau(prop_method)
propagation_c = RK.runge_kutta_explicit_coefficient(tableau)
beta = constant.T2beta(temperature)
print "beta=", beta
# e^{\-beta H/2} \Psi
if spectratype == "emi":
ketMPO = thermal_prop(iMPO, HMPO, insteps, ephtable,\
prop_method=prop_method, thresh=thresh,\
temperature=temperature, compress_method=compress_method,\
QNargs=QNargs, approxeiHt=approxeiHt)
elif spectratype == "abs":
thermalMPO, thermalMPOdim = ExactPropagatorMPO(mol, pbond, -beta/2.0,\
QNargs=QNargs, shift=GSshift)
ketMPO = mpslib.mapply(thermalMPO, iMPO, QNargs=QNargs)
#\Psi e^{\-beta H} \Psi
Z = mpslib.dot(mpslib.conj(ketMPO, QNargs=QNargs), ketMPO, QNargs=QNargs)
print "partition function Z(beta)/Z(0)", Z
autocorr = []
t = 0.0
exacteiHpt, exacteiHptdim = ExactPropagatorMPO(mol, pbond, -1.0j*dt,\
QNargs=QNargs, shift=GSshift)
exacteiHmt, exacteiHmtdim = ExactPropagatorMPO(mol, pbond, 1.0j*dt,\
QNargs=QNargs, shift=GSshift)
if spectratype == "abs":
ketMPO = mpslib.mapply(dipoleMPO, ketMPO, QNargs=QNargs)
else:
dipoleMPOdagger = mpslib.conjtrans(dipoleMPO, QNargs=QNargs)
if QNargs is not None:
dipoleMPOdagger[1] = [[0] * len(impsdim)
for impsdim in dipoleMPO[1]]
dipoleMPOdagger[3] = 0
ketMPO = mpslib.mapply(ketMPO, dipoleMPOdagger, QNargs=QNargs)
braMPO = mpslib.add(ketMPO, None, QNargs=QNargs)
if compress_method == "variational":
ketMPO = mpslib.canonicalise(ketMPO, 'l', QNargs=QNargs)
braMPO = mpslib.canonicalise(braMPO, 'l', QNargs=QNargs)
if approxeiHt is not None:
approxeiHpt = ApproxPropagatorMPO(HMPO, dt, ephtable, propagation_c,\
thresh=approxeiHt, compress_method=compress_method, QNargs=QNargs)
approxeiHmt = ApproxPropagatorMPO(HMPO, -dt, ephtable, propagation_c,\
thresh=approxeiHt, compress_method=compress_method, QNargs=QNargs)
else:
approxeiHpt = None
approxeiHmt = None
for istep in xrange(nsteps):
if istep != 0:
t += dt
# for emi bra and ket is conjugated
if istep % 2 == 0:
braMPO = mpslib.mapply(braMPO, exacteiHpt, QNargs=QNargs)
braMPO = tMPS(braMPO, HMPO, -dt, ephtable, propagation_c,\
thresh=thresh, cleanexciton=1, compress_method=compress_method, \
QNargs=QNargs, approxeiHt=approxeiHmt, scheme=scheme,\
prefix=scheme+"2")
else:
ketMPO = mpslib.mapply(ketMPO, exacteiHmt, QNargs=QNargs)
ketMPO = tMPS(ketMPO, HMPO, dt, ephtable, propagation_c, \
thresh=thresh, cleanexciton=1, compress_method=compress_method, \
QNargs=QNargs, approxeiHt=approxeiHpt, scheme=scheme,\
prefix=scheme+"1")
ft = mpslib.dot(mpslib.conj(braMPO, QNargs=QNargs),
ketMPO,
QNargs=QNargs)
if spectratype == "emi":
ft = np.conj(ft)
wfn_store(braMPO, istep, "braMPO.pkl")
wfn_store(ketMPO, istep, "ketMPO.pkl")
autocorr.append(ft / Z)
autocorr_store(autocorr, istep)
return autocorr
def ZeroTCorr(iMPS, HMPO, dipoleMPO, nsteps, dt, ephtable, thresh=0, \
cleanexciton=None, algorithm=1, prop_method="C_RK4",\
compress_method="svd", QNargs=None, approxeiHt=None, scheme="P&C"):
'''
the bra part e^iEt is negected to reduce the oscillation
algorithm:
algorithm 1 is the only propagte ket in 0, dt, 2dt
algorithm 2 is propagte bra and ket in 0, dt, 2dt (in principle, with
same calculation cost, more accurate, because the bra is also entangled,
the entanglement is not only in ket)
compress_method: svd or variational
cleanexciton: every time step propagation clean the good quantum number to
discard the numerical error
thresh: the svd threshold in svd or variational compress
'''
AketMPS = mpslib.mapply(dipoleMPO, iMPS, QNargs=QNargs)
# store the factor and normalize the AketMPS, factor is the length of AketMPS
factor = mpslib.dot(mpslib.conj(AketMPS, QNargs=QNargs),
AketMPS,
QNargs=QNargs)
factor = np.sqrt(np.absolute(factor))
print "factor", factor
AketMPS = mpslib.scale(AketMPS, 1. / factor, QNargs=QNargs)
if compress_method == "variational":
AketMPS = mpslib.canonicalise(AketMPS, 'l', QNargs=QNargs)
AbraMPS = mpslib.add(AketMPS, None, QNargs=QNargs)
autocorr = []
t = 0.0
tableau = RK.runge_kutta_explicit_tableau(prop_method)
propagation_c = RK.runge_kutta_explicit_coefficient(tableau)
if approxeiHt is not None:
approxeiHpt = ApproxPropagatorMPO(HMPO, dt, ephtable, propagation_c,\
thresh=approxeiHt, compress_method=compress_method, QNargs=QNargs)
approxeiHmt = ApproxPropagatorMPO(HMPO, -dt, ephtable, propagation_c,\
thresh=approxeiHt, compress_method=compress_method, QNargs=QNargs)
else:
approxeiHpt = None
approxeiHmt = None
for istep in xrange(nsteps):
if istep != 0:
t += dt
if algorithm == 1:
AketMPS = tMPS(AketMPS, HMPO, dt, ephtable, propagation_c, thresh=thresh, \
cleanexciton=cleanexciton, compress_method=compress_method, \
QNargs=QNargs, approxeiHt=approxeiHpt, normalize=1., \
scheme=scheme, prefix=scheme)
if algorithm == 2:
if istep % 2 == 1:
AketMPS = tMPS(AketMPS, HMPO, dt, ephtable, propagation_c, thresh=thresh, \
cleanexciton=cleanexciton, compress_method=compress_method, QNargs=QNargs,\
approxeiHt=approxeiHpt, normalize=1., scheme=scheme, \
prefix=scheme+"1")
else:
AbraMPS = tMPS(AbraMPS, HMPO, -dt, ephtable, propagation_c, thresh=thresh, \
cleanexciton=cleanexciton, compress_method=compress_method, QNargs=QNargs,\
approxeiHt=approxeiHmt, normalize=1., scheme=scheme,\
prefix=scheme+"2")
ft = mpslib.dot(mpslib.conj(AbraMPS, QNargs=QNargs),
AketMPS,
QNargs=QNargs) * factor**2
wfn_store(AbraMPS, istep, str(dt) + str(thresh) + "AbraMPS.pkl")
wfn_store(AketMPS, istep, str(dt) + str(thresh) + "AketMPS.pkl")
autocorr.append(ft)
autocorr_store(autocorr, istep)
return autocorr
def Exact_Spectra(spectratype, mol, pbond, iMPS, dipoleMPO, nsteps, dt,\
temperature, GSshift=0.0, EXshift=0.0):
'''
0T emission spectra exact propagator
the bra part e^iEt is negected to reduce the osillation
and
for single molecule, the EX space propagator e^iHt is local, and so exact
GS/EXshift is the ground/excited state space energy shift
the aim is to reduce the oscillation of the correlation fucntion
support:
all cases: 0Temi
1mol case: 0Temi, TTemi, 0Tabs, TTabs
'''
assert spectratype in ["emi", "abs"]
if spectratype == "emi":
space1 = "EX"
space2 = "GS"
shift1 = EXshift
shift2 = GSshift
if temperature != 0:
assert len(mol) == 1
else:
assert len(mol) == 1
space1 = "GS"
space2 = "EX"
shift1 = GSshift
shift2 = EXshift
if temperature != 0:
beta = constant.T2beta(temperature)
print "beta=", beta
thermalMPO, thermalMPOdim = ExactPropagatorMPO(mol,
pbond,
-beta / 2.0,
space=space1,
shift=shift1)
ketMPS = mpslib.mapply(thermalMPO, iMPS)
Z = mpslib.dot(mpslib.conj(ketMPS), ketMPS)
print "partition function Z(beta)/Z(0)", Z
else:
ketMPS = iMPS
Z = 1.0
AketMPS = mpslib.mapply(dipoleMPO, ketMPS)
if temperature != 0:
braMPS = mpslib.add(ketMPS, None)
else:
AbraMPS = mpslib.add(AketMPS, None)
t = 0.0
autocorr = []
propMPO1, propMPOdim1 = ExactPropagatorMPO(mol,
pbond,
-1.0j * dt,
space=space1,
shift=shift1)
propMPO2, propMPOdim2 = ExactPropagatorMPO(mol,
pbond,
-1.0j * dt,
space=space2,
shift=shift2)
# we can reconstruct the propagator each time if there is accumulated error
for istep in xrange(nsteps):
if istep != 0:
AketMPS = mpslib.mapply(propMPO2, AketMPS)
if temperature != 0:
braMPS = mpslib.mapply(propMPO1, braMPS)
if temperature != 0:
AbraMPS = mpslib.mapply(dipoleMPO, braMPS)
ft = mpslib.dot(mpslib.conj(AbraMPS), AketMPS)
autocorr.append(ft / Z)
autocorr_store(autocorr, istep)
return autocorr
def Quasi_Boson_MPO(opera, nqb, trunc, base=2, C1=1.0, C2=1.0):
'''
nqb : # of quasi boson sites
opera : operator to be decomposed
"b + b^\dagger"
'''
assert opera in ["b + b^\dagger","b^\dagger b", "b", "b^\dagger", \
"C1(b + b^\dagger) + C2(b + b^\dagger)^2"]
# the structure is [bra_highest_bit, ket_highest_bit,..., bra_lowest_bit,
# ket_lowest_bit]
mat = np.zeros([
base,
] * nqb * 2)
if opera == "b + b^\dagger" or opera == "b^\dagger" or opera == "b":
if opera == "b + b^\dagger" or opera == "b^\dagger":
for i in xrange(1, base**nqb):
# b^+
lstring = np.array(map(int, baseConvert(i, base).zfill(nqb)))
rstring = np.array(
map(int,
baseConvert(i - 1, base).zfill(nqb)))
pos = tuple(roundrobin(lstring, rstring))
mat[pos] = np.sqrt(i)
if opera == "b + b^\dagger" or opera == "b":
for i in xrange(0, base**nqb - 1):
# b
lstring = np.array(map(int, baseConvert(i, base).zfill(nqb)))
rstring = np.array(
map(int,
baseConvert(i + 1, base).zfill(nqb)))
pos = tuple(roundrobin(lstring, rstring))
mat[pos] = np.sqrt(i + 1)
elif opera == "C1(b + b^\dagger) + C2(b + b^\dagger)^2":
# b^+
for i in xrange(1, base**nqb):
lstring = np.array(map(int, baseConvert(i, base).zfill(nqb)))
rstring = np.array(map(int, baseConvert(i - 1, base).zfill(nqb)))
pos = tuple(roundrobin(lstring, rstring))
mat[pos] = C1 * np.sqrt(i)
# b
for i in xrange(0, base**nqb - 1):
lstring = np.array(map(int, baseConvert(i, base).zfill(nqb)))
rstring = np.array(map(int, baseConvert(i + 1, base).zfill(nqb)))
pos = tuple(roundrobin(lstring, rstring))
mat[pos] = C1 * np.sqrt(i + 1)
# bb
for i in xrange(0, base**nqb - 2):
lstring = np.array(map(int, baseConvert(i, base).zfill(nqb)))
rstring = np.array(map(int, baseConvert(i + 2, base).zfill(nqb)))
pos = tuple(roundrobin(lstring, rstring))
mat[pos] = C2 * np.sqrt(i + 2) * np.sqrt(i + 1)
# b^\dagger b^\dagger
for i in xrange(2, base**nqb):
lstring = np.array(map(int, baseConvert(i, base).zfill(nqb)))
rstring = np.array(map(int, baseConvert(i - 2, base).zfill(nqb)))
pos = tuple(roundrobin(lstring, rstring))
mat[pos] = C2 * np.sqrt(i) * np.sqrt(i - 1)
# b^\dagger b + b b^\dagger
for i in xrange(0, base**nqb):
lstring = np.array(map(int, baseConvert(i, base).zfill(nqb)))
rstring = np.array(map(int, baseConvert(i, base).zfill(nqb)))
pos = tuple(roundrobin(lstring, rstring))
mat[pos] = C2 * float(i * 2 + 1)
elif opera == "b^\dagger b":
# actually Identity operator can be constructed directly
for i in xrange(0, base**nqb):
# I
lstring = np.array(map(int, baseConvert(i, base).zfill(nqb)))
rstring = np.array(map(int, baseConvert(i, base).zfill(nqb)))
pos = tuple(roundrobin(lstring, rstring))
mat[pos] = float(i)
# check the original mat
# mat = np.moveaxis(mat,range(1,nqb*2,2),range(nqb,nqb*2))
# print mat.reshape(base**nqb,base**nqb)
# decompose canonicalise
MPO = []
mat = mat.reshape(1, -1)
for idx in xrange(nqb - 1):
U, S, Vt = scipy.linalg.svd(mat.reshape(mat.shape[0]*base**2,-1), \
full_matrices=False)
U = U.reshape(mat.shape[0], base, base, -1)
MPO.append(U)
mat = np.einsum("i, ij -> ij", S, Vt)
MPO.append(mat.reshape(-1, base, base, 1))
print "original MPO shape:", [i.shape[0] for i in MPO] + [1]
# compress
MPOnew = mpslib.compress(MPO, 'l', trunc=trunc)
print "trunc", trunc, "distance", mpslib.distance(MPO, MPOnew)
fidelity = mpslib.dot(mpslib.conj(MPOnew), MPO) / mpslib.dot(
mpslib.conj(MPO), MPO)
print "compression fidelity:: ", fidelity
print "compressed MPO shape", [i.shape[0] for i in MPOnew] + [1]
return MPOnew