def calculate_pairing_matrix(u, v, norm):
assert is_square_matrix(u)
assert is_square_matrix(v)
assert u.shape[0] == v.shape[0]
# Warn if u is singular, but don't abort cuz sometimes this is OK..
(signdet, logdet) = numpy.linalg.slogdet(u)
logger.info('|det(u)| = %g', numpy.abs(numpy.exp(logdet))) # actually, just log it for now..
phiMat = numpy.dot( numpy.linalg.inv(u.conjugate().transpose()), v.conjugate().transpose() )
# logger.debug('Pairing matrix = ...')
# logger.debug(phiMat)
# logger.debug("Pairing matrix's main diagonal = ...")
# logger.debug(phiMat.diagonal())
# Check that phiMat is symmetric:
if numpy.max(numpy.abs(phiMat - phiMat.transpose())) > 1e-8:
raise RuntimeError('"Symmetricness" of pairing matrix = %.9g', numpy.max(numpy.abs(phiMat - phiMat.transpose())))
# Check against TI code (must first run *_Fourier.py with same parameters):
# phiMat_TI = numpy.loadtxt('phiMat_TI.dat').view(complex)
# logger.debug('max|phiMat - phiMat_TI| = %.9g ', numpy.max(numpy.abs(phiMat - phiMat_TI)))
phiNorm = norm_of_pairing_matrix(phiMat)
logger.info('Norm of pairing matrix before normalization = %f', phiNorm)
if norm is not None:
phiMat = norm * (phiMat / phiNorm)
phiNorm = norm_of_pairing_matrix(phiMat)
logger.info('Norm of pairing matrix after normalization = %f', phiNorm)
return phiMat
def calculate_free_fermion_pairing_matrix(evecs, evals, Nparticles, norm):
assert is_square_matrix(evecs)
Nsites = evecs.shape[0]
assert Nsites == len(evals)
M = Nparticles/2 # ( = N_up = N_down )
assert M <= Nsites
phiMat = numpy.zeros((Nsites, Nsites), dtype=complex)
for ind_r in range(0, Nsites):
for ind_rp in range(0, Nsites):
phiMat[ind_r, ind_rp] = numpy.dot( evecs[ind_r, 0:M], evecs[ind_rp, 0:M] )
# logger.debug('Pairing matrix = ...')
# logger.debug(phiMat)
# logger.debug("Pairing matrix's main diagonal = ...")
# logger.debug(phiMat.diagonal())
# Check that phiMat is symmetric:
if numpy.max(numpy.abs(phiMat - phiMat.transpose())) > 1e-8:
raise RuntimeError('"Symmetricness" of pairing matrix = %.9g', numpy.max(numpy.abs(phiMat - phiMat.transpose())))
phiNorm = norm_of_pairing_matrix(phiMat)
logger.info('Norm of pairing matrix before normalization = %f', phiNorm)
if norm is not None:
phiMat = norm * (phiMat / phiNorm)
phiNorm = norm_of_pairing_matrix(phiMat)
logger.info('Norm of pairing matrix after normalization = %f', phiNorm)
return phiMat
def add_noise_to_matrix_diagonal(mat, noiselevel=.001):
# only adds noise to the diagonal
assert 0 <= noiselevel < 1
assert is_square_matrix(mat)
assert numpy.max(numpy.abs(mat - mat.transpose())) < 1e-12
mat = mat.copy()
if noiselevel == 0:
return mat
for i in six.moves.xrange(mat.shape[0]):
mat[i, i] *= 1 + random.uniform(-noiselevel, noiselevel)
return mat
def add_noise_to_symmetric_matrix(mat, noiselevel=.01):
# currently does not add noise to the diagonal
assert 0 < noiselevel < 1
assert is_square_matrix(mat)
assert numpy.max(numpy.abs(mat - mat.transpose())) < 1e-12
mat = mat.copy()
for i in six.moves.xrange(mat.shape[0]):
for j in six.moves.xrange(i + 1, mat.shape[1]):
z = random.uniform(1 - noiselevel, 1 + noiselevel)
mat[(i, j)] *= z
mat[(j, i)] *= z
return mat
def bcs_stats(u, v):
assert is_square_matrix(u)
assert is_square_matrix(v)
assert u.shape[0] == v.shape[0]
Nsites = u.shape[0]
fdagf = numpy.zeros((Nsites), dtype=float)
fdownfup = numpy.zeros((Nsites), dtype=complex)
Tvec = numpy.zeros((Nsites, 3), dtype=float)
Ttot = numpy.zeros((Nsites), dtype=float)
for ind_r in range(0, Nsites):
fdagf[ind_r] = 2.0 * numpy.dot( v[ind_r, :], v[ind_r, :].conjugate() ).real
fdownfup[ind_r] = -numpy.dot( u[ind_r, :], v[ind_r, :].conjugate() )
for ind_r in range(0, Nsites):
Tvec[ind_r, 0] = fdownfup[ind_r].real
Tvec[ind_r, 1] = -fdownfup[ind_r].imag
Tvec[ind_r, 2] = 0.5 * (fdagf[ind_r] - 1.0)
Tvec[ind_r, :] = 2.0 * Tvec[ind_r, :] # 2.0 to be consistent with Lesik.
Ttot[ind_r] = numpy.sqrt( numpy.dot( Tvec[ind_r, :], Tvec[ind_r, :].conjugate() ) ).real
return {'fdagf':fdagf, 'fdownfup':fdownfup, 'Tvec':Tvec, 'Ttot':Ttot}
def add_onsite_terms(offsite, onsite):
assert is_square_matrix(offsite)
check_offsite_for_onsite(offsite) # since we're singling out the on-site terms with this function
Nsites = offsite.shape[0]
# total_{ij} = offsite_{ij} + onsite_i * krondelta_{ij}
if isinstance(onsite, numpy.ndarray):
assert onsite.ndim == 1
assert onsite.shape[0] == 1 or onsite.shape[0] == Nsites
if onsite.shape[0] == 1:
total = offsite + onsite * numpy.identity(Nsites)
elif onsite.shape[0] == Nsites:
total = offsite + numpy.diag(onsite)
else:
total = offsite + onsite * numpy.identity(Nsites)
return total
def free_fermion_soln(t):
assert is_square_matrix(t)
Nsites = t.shape[0]
# t_{ij} = ttilde_{ij} = t_{ij} + mu_i * krondelta_{ij} from notes (FIXME: take t and mu as separate parameters?)
H = -t
assert numpy.max(numpy.abs(H - H.conjugate().transpose())) < 1e-12
eigsys = numpy.linalg.eigh(H)
idx = eigsys[0].argsort()
evals = eigsys[0][idx]
evecs = eigsys[1][:,idx]
logger.debug('Eigenvalues = ...')
logger.debug(evals)
return {'evecs':evecs, 'evals':evals}
def plot_pairing_matrix(phiMat, callshow=True):
assert is_square_matrix(phiMat)
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
fig = plt.figure()
ax = fig.gca(projection='3d')
X = numpy.arange(0, phiMat.shape[0], 1)
Y = numpy.arange(0, phiMat.shape[0], 1)
X, Y = numpy.meshgrid(X, Y)
surf = ax.plot_surface(X, Y, numpy.abs(phiMat), rstride=1, cstride=1, cmap=cm.coolwarm, linewidth=0, antialiased=False)
#ax.zaxis.set_major_locator(LinearLocator(10))
ax.zaxis.set_major_formatter(FormatStrFormatter('%.02f'))
fig.colorbar(surf, shrink=0.5, aspect=5)
if callshow:
plt.show()
def bcs_soln(t, delta):
# both t and delta should include on-site terms, e.g., t_{ij} = ttilde_{ij} = t_{ij} + mu_i * krondelta_{ij} from notes
assert is_square_matrix(t)
assert is_square_matrix(delta)
assert t.shape[0] == delta.shape[0]
Nsites = t.shape[0]
H = numpy.zeros((Nsites, Nsites, 2, 2), dtype=complex)
for ind_r in range(0, Nsites):
for ind_rp in range(0, Nsites):
H[ind_r, ind_rp] = -numpy.array([[t[ind_r, ind_rp], delta[ind_r, ind_rp]],
[delta[ind_r, ind_rp].conjugate(), -t[ind_r, ind_rp].conjugate()]])
Hflat = numpy.zeros((2*Nsites, 2*Nsites), dtype=complex)
for i in range(0, 2*Nsites):
for j in range(0, 2*Nsites):
Hflat[i,j] = H[i//2, j//2, numpy.mod(i,2), numpy.mod(j,2)]
assert numpy.max(numpy.abs(Hflat - Hflat.conjugate().transpose())) < 1e-12
eigsys = numpy.linalg.eigh(Hflat)
idx = eigsys[0].argsort()
evals = eigsys[0][idx]
evecs = eigsys[1][:,idx]
# Check that eigenvectors are orthonormal:
Icheck = numpy.dot( evecs.conjugate().transpose(), evecs )
assert numpy.max(numpy.abs(Icheck - numpy.identity(Icheck.shape[0], dtype=complex))) < 1e-12
evalsNeg = evals[0:Nsites]
evecsNeg = evecs[:, 0:Nsites]
evalsPos = evals[Nsites:2*Nsites]
evecsPos = evecs[:, Nsites:2*Nsites]
logger.debug('Positive eigenvalues = ...')
logger.debug(evalsPos)
# logger.debug('Negative eigenvalues = ...')
# logger.debug(evalsNeg[::-1])
# Check that positive eigenvalues are positive and that eigenvalues have come in plus-minus pairs:
# FIXME: probably shouldn't be so strict, since this gets called when trying to tune the ansatz,
# and we could potentially run into degenerate mean field solutions (i.e., those with zero modes) in that process;
# if we don't assert evalsPos > 0, then we treat zero modes as positive and don't populate them in the the BCS solution,
# which I think is OK; for now, we'll just warn instead of assert
# assert numpy.min(evalsPos) > 1e-12
if not numpy.min(evalsPos) > 1e-12:
logger.warning('Some of evalsPos are not greater than 1e-12')
assert numpy.max(numpy.abs(evalsNeg[::-1] + evalsPos)) < 1e-12
u = evecsPos[0:evecsPos.shape[0]:2, :]
v = evecsPos[1:evecsPos.shape[0]:2, :]
# Check that (v, -u)^* from evecsPos are indeed negative eigenvectors:
errr = []
for n in range(0, Nsites):
evecTest = numpy.zeros(2*Nsites, dtype=complex)
for i in range(0, Nsites):
evecTest[2*i] = v[i,n].conjugate()
evecTest[2*i+1] = -u[i,n].conjugate()
errr.append(numpy.max(numpy.abs( numpy.dot( Hflat, evecTest ) - (-evalsPos[n] * evecTest ))))
assert numpy.max(errr) < 1e-12
return {'u':u, 'v':v, 'evalsPos':evalsPos}
