def calc_Vsh(self, n, sqrt_r):
"""Generates m.H(V[n][s]) for a given n, used for generating B[n][s]
This is described on p. 14 of arXiv:1103.0936v2 [cond-mat.str-el] for left
gauge fixing. Here, we are using right gauge fixing.
Array slicing and reshaping is used to manipulate the indices as necessary.
Each V[n] directly depends only on A[n] and r[n].
We return the conjugate m.H(V) because we use it in more places than V.
"""
R = sp.zeros((self.D[n], self.q[n], self.D[n-1]), dtype=self.typ, order='C')
for s in xrange(self.q[n]):
R[:,s,:] = m.mmul(sqrt_r, m.H(self.A[n][s]))
R = R.reshape((self.q[n] * self.D[n], self.D[n-1]))
V = m.H(ns.nullspace_qr(m.H(R)))
#print (q[n]*D[n] - D[n-1], q[n]*D[n])
#print V.shape
#print sp.allclose(mat(V) * mat(V).H, sp.eye(q[n]*D[n] - D[n-1]))
#print sp.allclose(mat(V) * mat(Rh).H, 0)
V = V.reshape((self.q[n] * self.D[n] - self.D[n - 1], self.D[n], self.q[n])) #this works with the above form for R
#prepare for using V[s] and already take the adjoint, since we use it more often
Vsh = sp.empty((self.q[n], self.D[n], self.q[n] * self.D[n] - self.D[n - 1]), dtype=self.typ, order=self.odr)
for s in xrange(self.q[n]):
Vsh[s] = m.H(V[:,:,s])
return Vsh
def calc_Vsh_l(A, lm1_sqrt, sanity_checks=False):
D = A.shape[2]
Dm1 = A.shape[1]
q = A.shape[0]
if q * Dm1 - D <= 0:
return None
L = sp.zeros((D, q, Dm1), dtype=A.dtype, order='C')
for s in xrange(q):
L[:, s, :] = lm1_sqrt.dot(A[s]).conj().T
L = L.reshape((D, q * Dm1))
V = ns.nullspace_qr(L)
if sanity_checks:
if not sp.allclose(L.dot(V), 0):
log.warning("Sanity Fail in calc_Vsh_l!: LV != 0")
if not sp.allclose(V.conj().T.dot(V), sp.eye(V.shape[1])):
log.warning("Sanity Fail in calc_Vsh_l!: V H(V) != eye")
V = V.reshape((q, Dm1, q * Dm1 - D))
Vsh = sp.transpose(V.conj(), axes=(0, 2, 1))
Vsh = sp.asarray(Vsh, order='C')
if sanity_checks:
M = eps_l_noop(lm1_sqrt, A, V)
if not sp.allclose(M, 0):
log.warning("Sanity Fail in calc_Vsh_l!: Bad Vsh")
return Vsh
def calc_Vsh(A, r_s, sanity_checks=False):
D = A.shape[2]
Dm1 = A.shape[1]
q = A.shape[0]
if q * D - Dm1 <= 0:
return None
R = sp.zeros((D, q, Dm1), dtype=A.dtype, order='C')
for s in xrange(q):
R[:, s, :] = r_s.dot(A[s].conj().T)
R = R.reshape((q * D, Dm1))
Vconj = ns.nullspace_qr(R.conj().T).T
if sanity_checks:
if not sp.allclose(mm.mmul(Vconj.conj(), R), 0):
log.warning("Sanity Fail in calc_Vsh!: VR != 0")
if not sp.allclose(mm.mmul(Vconj,
Vconj.conj().T), sp.eye(Vconj.shape[0])):
log.warning("Sanity Fail in calc_Vsh!: V H(V) != eye")
Vconj = Vconj.reshape((q * D - Dm1, D, q))
Vsh = Vconj.T
Vsh = sp.asarray(Vsh, order='C')
if sanity_checks:
Vs = sp.transpose(Vsh, axes=(0, 2, 1)).conj()
M = eps_r_noop(r_s, Vs, A)
if not sp.allclose(M, 0):
log.warning("Sanity Fail in calc_Vsh!: Bad Vsh")
return Vsh
def calc_Vsh_l(A, lm1_sqrt, sanity_checks=False):
D = A.shape[2]
Dm1 = A.shape[1]
q = A.shape[0]
if q * Dm1 - D <= 0:
return None
L = sp.zeros((D, q, Dm1), dtype=A.dtype, order='C')
for s in xrange(q):
L[:,s,:] = lm1_sqrt.dot(A[s]).conj().T
L = L.reshape((D, q * Dm1))
V = ns.nullspace_qr(L)
if sanity_checks:
if not sp.allclose(L.dot(V), 0):
log.warning("Sanity Fail in calc_Vsh_l!: LV != 0")
if not sp.allclose(V.conj().T.dot(V), sp.eye(V.shape[1])):
log.warning("Sanity Fail in calc_Vsh_l!: V H(V) != eye")
V = V.reshape((q, Dm1, q * Dm1 - D))
Vsh = sp.transpose(V.conj(), axes=(0, 2, 1))
Vsh = sp.asarray(Vsh, order='C')
if sanity_checks:
M = eps_l_noop(lm1_sqrt, A, V)
if not sp.allclose(M, 0):
log.warning("Sanity Fail in calc_Vsh_l!: Bad Vsh")
return Vsh
def calc_Vsh(A, r_s, sanity_checks=False):
D = A.shape[2]
Dm1 = A.shape[1]
q = A.shape[0]
if q * D - Dm1 <= 0:
return None
R = sp.zeros((D, q, Dm1), dtype=A.dtype, order='C')
for s in xrange(q):
R[:,s,:] = r_s.dot(A[s].conj().T)
R = R.reshape((q * D, Dm1))
Vconj = ns.nullspace_qr(R.conj().T).T
if sanity_checks:
if not sp.allclose(mm.mmul(Vconj.conj(), R), 0):
log.warning("Sanity Fail in calc_Vsh!: VR != 0")
if not sp.allclose(mm.mmul(Vconj, Vconj.conj().T), sp.eye(Vconj.shape[0])):
log.warning("Sanity Fail in calc_Vsh!: V H(V) != eye")
Vconj = Vconj.reshape((q * D - Dm1, D, q))
Vsh = Vconj.T
Vsh = sp.asarray(Vsh, order='C')
if sanity_checks:
Vs = sp.transpose(Vsh, axes=(0, 2, 1)).conj()
M = eps_r_noop(r_s, Vs, A)
if not sp.allclose(M, 0):
log.warning("Sanity Fail in calc_Vsh!: Bad Vsh")
return Vsh
def calc_Vsh(self, n, sqrt_r):
"""Generates mm.H(V[n][s]) for a given n, used for generating B[n][s]
This is described on p. 14 of arXiv:1103.0936v2 [cond-mat.str-el] for left
gauge fixing. Here, we are using right gauge fixing.
Array slicing and reshaping is used to manipulate the indices as necessary.
Each V[n] directly depends only on A[n] and r[n].
We return the conjugate mm.H(V) because we use it in more places than V.
"""
R = sp.zeros((self.D[n], self.q[n], self.D[n-1]), dtype=self.typ, order='C')
for s in xrange(self.q[n]):
R[:,s,:] = mm.mmul(sqrt_r, mm.H(self.A[n][s]))
R = R.reshape((self.q[n] * self.D[n], self.D[n-1]))
Vconj = ns.nullspace_qr(mm.H(R)).T
if self.sanity_checks:
if not sp.allclose(mm.mmul(Vconj.conj(), R), 0):
print "Sanity Fail in calc_Vsh!: VR_%u != 0" % (n)
if not sp.allclose(mm.mmul(Vconj, mm.H(Vconj)), sp.eye(Vconj.shape[0])):
print "Sanity Fail in calc_Vsh!: V H(V)_%u != eye" % (n)
Vconj = Vconj.reshape((self.q[n] * self.D[n] - self.D[n - 1], self.D[n], self.q[n]))
Vsh = Vconj.T
Vsh = sp.asarray(Vsh, order='C')
if self.sanity_checks:
M = sp.zeros((self.q[n] * self.D[n] - self.D[n - 1], self.D[n]), dtype=self.typ)
for s in xrange(self.q[n]):
M += mm.mmul(mm.H(Vsh[s]), sqrt_r, mm.H(self.A[n][s]))
if not sp.allclose(M, 0):
print "Sanity Fail in calc_Vsh!: Bad Vsh_%u" % (n)
return Vsh
def calc_Vsh(self, r_sqrt):
R = np.zeros((self.D, self.q, self.D), dtype=self.typ, order='C')
for s in xrange(self.q):
R[:,s,:] = m.mmul(r_sqrt, m.H(self.A[s]))
R = R.reshape((self.q * self.D, self.D))
Vconj = ns.nullspace_qr(m.H(R)).T
#R can be pretty huge for large q and D. The decomp. can take a long time...
if self.sanity_checks:
if not np.allclose(np.dot(Vconj, m.H(Vconj)), np.eye(self.q*self.D - self.D)):
print "Sanity check failed: V . H(V) not eye!"
if not np.allclose(np.dot(Vconj.conj(), R), 0):
print "Sanity check failed: V . R not zero!"
Vconj = Vconj.reshape(((self.q - 1) * self.D, self.D, self.q))
#prepare for using V[s] and already take the adjoint, since we use it more often
Vsh = Vconj.T
Vsh = np.asarray(Vsh, order='C')
return Vsh
