def covDeriv(self, scalarField, gaugeField, cpt):
scalarFieldShifted = self.shiftScalarField(scalarField, cpt, +1)
lieAlgField = scalarField * FieldTools.pauliMatrix(3)
lieAlgFieldShifted = scalarFieldShifted * FieldTools.pauliMatrix(3)
covDeriv = gaugeField[:,:,:,cpt,:,:] @ lieAlgFieldShifted @\
tf.linalg.adjoint(gaugeField[:,:,:,cpt,:,:]) - lieAlgField
return covDeriv
def shiftGaugeField(self, gaugeField, cpt, sign):
gaugeFieldShifted = tf.roll(gaugeField, -sign, cpt)
pauliMatNum = self.boundaryConditions[cpt]
if pauliMatNum == 0:
return gaugeFieldShifted
latShape = tf.shape(gaugeField)[0:3]
indices = FieldTools.boundaryIndices(latShape, cpt, sign)
updates = tf.gather_nd(gaugeFieldShifted, indices)
updates = FieldTools.pauliMatrix(pauliMatNum) @\
updates @ FieldTools.pauliMatrix(pauliMatNum)
gaugeFieldShifted = tf.tensor_scatter_nd_update(
gaugeFieldShifted, indices, updates)
return gaugeFieldShifted
def covDerivTerm(self, higgsField, isospinField, hyperchargeField):
energyDensity = tf.zeros(tf.shape(higgsField)[0:3], dtype=tf.float64)
higgsMagnitude = tf.linalg.trace(
self.higgsMagnitude(tf.math.real(higgsField)))
numDims = 3
# This is only valid in the unitary gauge, but it keeps the isospin
# gradients symmetric which is good for convergence to the saddle
for ii in range(numDims):
higgsFieldShifted = self.shiftHiggsField(higgsField, ii, +1)
higgsMagnitudeShifted = tf.linalg.trace(
self.higgsMagnitude(tf.math.real(higgsFieldShifted)))
energyDensity += higgsMagnitude**2
energyDensity += higgsMagnitudeShifted**2
energyDensity -= higgsMagnitude * higgsMagnitudeShifted *\
tf.math.real(tf.linalg.trace(isospinField[:,:,:,ii,:,:])) *\
tf.math.real(tf.linalg.trace(hyperchargeField[:,:,:,ii,:,:]))
energyDensity += higgsMagnitude * higgsMagnitudeShifted *\
tf.math.imag(tf.linalg.trace(isospinField[:,:,:,ii,:,:] @\
FieldTools.pauliMatrix(3))) *\
tf.math.imag(tf.linalg.trace(hyperchargeField[:,:,:,ii,:,:]))
return energyDensity