def calc_K_l(self, n_low=-1, n_high=-1):
"""Generates the K matrices used to calculate the B's.
For the left gauge-fixing case.
"""
if n_low < 2:
n_low = 2
if n_high < 1:
n_high = self.N
self.K[1] = sp.zeros((self.D[1], self.D[1]), dtype=self.typ)
for n in xrange(n_low, n_high + 1):
#if n <= self.N - self.ham_sites + 1:
if self.ham_sites == 2:
self.K[n], ex = tm.calc_K_l(self.K[n - 1], self.C[n - 1], self.l[n - 2],
self.r[n], self.A[n], self.AA[n - 1])
else:
assert False, 'left gauge fixing not yet supported for three-site Hamiltonians'
self.h_expect[n - 1] = ex
#else:
# self.K[n].fill(0)
if n_high == self.N:
self.H_expect = sp.asscalar(self.K[self.N])
def calc_K_l(self, n_low=-1, n_high=-1):
"""Generates the K matrices used to calculate the B's.
For the left gauge-fixing case.
"""
if n_low < 2:
n_low = 2
if n_high < 1:
n_high = self.N
self.K[1] = sp.zeros((self.D[1], self.D[1]), dtype=self.typ)
for n in xrange(n_low, n_high + 1):
#if n <= self.N - self.ham_sites + 1:
if self.ham_sites == 2:
self.K[n], ex = tm.calc_K_l(self.K[n - 1], self.C[n - 1], self.l[n - 2],
self.r[n], self.A[n], self.A[n - 1],
sanity_checks=self.sanity_checks)
else:
assert False, 'left gauge fixing not yet supported for three-site Hamiltonians'
self.h_expect[n - 1] = ex
#else:
# self.K[n].fill(0)
if n_high == self.N:
self.H_expect = sp.asscalar(self.K[self.N])
def calc_K(self):
"""Generates the right K matrices used to calculate the B's
K[n] contains 'column-vectors' such that <l[n]|K[n]> = trace(l[n].dot(K[n])).
K_l[n] contains 'bra-vectors' such that <K_l[n]|r[n]> = trace(K_l[n].dot(r[n])).
"""
self.h_expect = sp.zeros((self.N + 1), dtype=self.typ)
self.uni_r.calc_AA()
self.uni_r.calc_C()
self.uni_r.calc_K()
self.K[self.N + 1][:] = self.uni_r.K[0]
self.uni_l.calc_AA()
self.uni_l.calc_C()
K_left, h_left_uni = self.uni_l.calc_K_l()
self.K_l[0][:] = K_left[-1]
for n in xrange(self.N, self.N_centre - 1, -1):
self.K[n], he = tm.calc_K(self.K[n + 1], self.C[n], self.get_l(n - 1),
self.r[n + 1], self.A[n], self.get_AA(n))
self.h_expect[n] = he
for n in xrange(1, self.N_centre + 1):
self.K_l[n], he = tm.calc_K_l(self.K_l[n - 1], self.C[n - 1], self.get_l(n - 2),
self.r[n], self.A[n], self.get_AA(n - 1))
self.h_expect[n - 1] = he
self.dH_expect = (mm.adot_noconj(self.K_l[self.N_centre], self.r[self.N_centre])
+ mm.adot(self.l[self.N_centre - 1], self.K[self.N_centre])
- (self.N + 1) * self.uni_r.h_expect)
def calc_K(self):
"""Generates the right K matrices used to calculate the B's
K[n] contains 'column-vectors' such that <l[n]|K[n]> = trace(l[n].dot(K[n])).
K_l[n] contains 'bra-vectors' such that <K_l[n]|r[n]> = trace(K_l[n].dot(r[n])).
"""
self.h_expect = sp.zeros((self.N + 1), dtype=self.typ)
self.uni_r.calc_AA()
self.uni_r.calc_C()
self.uni_r.calc_K()
self.K[self.N + 1][:] = self.uni_r.K
self.uni_l.calc_AA()
self.uni_l.calc_C()
K_left, h_left_uni = self.uni_l.calc_K_l()
self.K_l[0][:] = K_left
for n in xrange(self.N, self.N_centre - 1, -1):
self.K[n], he = tm.calc_K(self.K[n + 1],
self.C[n],
self.get_l(n - 1),
self.r[n + 1],
self.A[n],
self.A[n + 1],
sanity_checks=self.sanity_checks)
self.h_expect[n] = he
for n in xrange(1, self.N_centre + 1):
self.K_l[n], he = tm.calc_K_l(self.K_l[n - 1],
self.C[n - 1],
self.get_l(n - 2),
self.r[n],
self.A[n],
self.A[n - 1],
sanity_checks=self.sanity_checks)
self.h_expect[n - 1] = he
self.dH_expect = (
mm.adot_noconj(self.K_l[self.N_centre], self.r[self.N_centre]) +
mm.adot(self.l[self.N_centre - 1], self.K[self.N_centre]) -
(self.N + 1) * self.uni_r.h_expect)
def take_step_split(self, dtau, ham_is_Herm=True):
"""Take a time-step dtau using the split-step integrator.
This is the one-site version of a DMRG-like time integrator described
at:
http://arxiv.org/abs/1408.5056
It has a fourth-order local error and is symmetric. It requires
iteratively computing two matrix exponentials per site, and thus
has less predictable CPU time requirements than the Euler or RK4
methods.
NOTE:
This requires the expokit extension, which is included in evoMPS but
must be compiled during, e.g. using setup.py to build all extensions.
Parameters
----------
dtau : complex
The (imaginary or real) amount of imaginary time (tau) to step.
ham_is_Herm : bool
Whether the Hamiltonian is really Hermitian. If so, the lanczos
method will be used for imaginary time evolution.
"""
#TODO: Compute eta.
self.eta_sq.fill(0)
self.eta = 0
assert self.canonical_form == 'right', 'take_step_split only implemented for right canonical form'
assert self.ham_sites == 2, 'take_step_split only implemented for nearest neighbour Hamiltonians'
dtau *= -1
from expokit_expmv import zexpmv, zexpmvh
if sp.iscomplex(dtau) or not ham_is_Herm:
expmv = zexpmv
fac = 1.j
dtau = sp.imag(dtau)
else:
expmv = zexpmvh
fac = 1
norm_est = abs(self.H_expect.real)
KL = [None] * (self.N + 1)
KL[1] = sp.zeros((self.D[1], self.D[1]), dtype=self.typ)
for n in xrange(1, self.N + 1):
lop = Vari_Opt_Single_Site_Op(self, n, KL[n - 1], tau=fac)
#print "Befor A", n, sp.inner(self.A[n].ravel().conj(), lop.matvec(self.A[n].ravel())).real
An = expmv(lop, self.A[n].ravel(), dtau/2., norm_est=norm_est)
self.A[n] = An.reshape((self.q[n], self.D[n - 1], self.D[n]))
self.l[n] = tm.eps_l_noop(self.l[n - 1], self.A[n], self.A[n])
norm = m.adot(self.l[n], self.r[n])
self.A[n] /= sp.sqrt(norm)
#print "After A", n, sp.inner(self.A[n].ravel().conj(), lop.matvec(self.A[n].ravel())).real, norm.real
#shift centre matrix right (RCF is like having a centre "matrix" at "1")
G = tm.restore_LCF_l_seq(self.A[n - 1:n + 1], self.l[n - 1:n + 1],
sanity_checks=self.sanity_checks)
if n > 1:
self.AA[n - 1] = tm.calc_AA(self.A[n - 1], self.A[n])
self.C[n - 1] = tm.calc_C_mat_op_AA(self.ham[n - 1], self.AA[n - 1])
KL[n], ex = tm.calc_K_l(KL[n - 1], self.C[n - 1], self.l[n - 2],
self.r[n], self.A[n], self.AA[n - 1])
if n < self.N:
lop2 = Vari_Opt_SC_op(self, n, KL[n], tau=fac)
#print "Befor G", n, sp.inner(G.ravel().conj(), lop2.matvec(G.ravel())).real
G = expmv(lop2, G.ravel(), -dtau/2., norm_est=norm_est)
G = G.reshape((self.D[n], self.D[n]))
norm = sp.trace(self.l[n].dot(G).dot(self.r[n].dot(G.conj().T)))
G /= sp.sqrt(norm)
#print "After G", n, sp.inner(G.ravel().conj(), lop2.matvec(G.ravel())).real, norm.real
for s in xrange(self.q[n + 1]):
self.A[n + 1][s] = G.dot(self.A[n + 1][s])
self.AA[n] = tm.calc_AA(self.A[n], self.A[n + 1])
self.C[n] = tm.calc_C_mat_op_AA(self.ham[n], self.AA[n])
for n in xrange(self.N, 0, -1):
lop = Vari_Opt_Single_Site_Op(self, n, KL[n - 1], tau=fac, sanity_checks=self.sanity_checks)
#print "Before A", n, sp.inner(self.A[n].ravel().conj(), lop.matvec(self.A[n].ravel())).real
An = expmv(lop, self.A[n].ravel(), dtau/2., norm_est=norm_est)
self.A[n] = An.reshape((self.q[n], self.D[n - 1], self.D[n]))
self.l[n] = tm.eps_l_noop(self.l[n - 1], self.A[n], self.A[n])
norm = m.adot(self.l[n], self.r[n])
self.A[n] /= sp.sqrt(norm)
#print "After A", n, sp.inner(self.A[n].ravel().conj(), lop.matvec(self.A[n].ravel())).real, norm.real
#shift centre matrix left (LCF is like having a centre "matrix" at "N")
Gi = tm.restore_RCF_r_seq(self.A[n - 1:n + 1], self.r[n - 1:n + 1],
sanity_checks=self.sanity_checks)
if n < self.N:
self.AA[n] = tm.calc_AA(self.A[n], self.A[n + 1])
self.C[n] = tm.calc_C_mat_op_AA(self.ham[n], self.AA[n])
self.calc_K(n_low=n, n_high=n)
if n > 1:
lop2 = Vari_Opt_SC_op(self, n - 1, KL[n - 1], tau=fac, sanity_checks=self.sanity_checks)
Gi = expmv(lop2, Gi.ravel(), -dtau/2., norm_est=norm_est)
Gi = Gi.reshape((self.D[n - 1], self.D[n - 1]))
norm = sp.trace(self.l[n - 1].dot(Gi).dot(self.r[n - 1].dot(Gi.conj().T)))
G /= sp.sqrt(norm)
#print "After G", n, sp.inner(Gi.ravel().conj(), lop2.matvec(Gi.ravel())).real, norm.real
for s in xrange(self.q[n - 1]):
self.A[n - 1][s] = self.A[n - 1][s].dot(Gi)
self.AA[n - 1] = tm.calc_AA(self.A[n - 1], self.A[n])
self.C[n - 1] = tm.calc_C_mat_op_AA(self.ham[n - 1], self.AA[n - 1])
def vari_opt_ss_sweep(self, ncv=None):
"""Perform a DMRG-style optimizing sweep to reduce the energy.
This carries out the MPS version of the one-site DMRG algorithm.
Combined with imaginary time evolution, this can dramatically improve
convergence speed.
"""
assert self.canonical_form == 'right', 'vari_opt_ss_sweep only implemented for right canonical form'
KL = [None] * (self.N + 1)
KL[1] = sp.zeros((self.D[1], self.D[1]), dtype=self.typ)
for n in xrange(1, self.N + 1):
if n > 2:
k = n - 1
KL[k], ex = tm.calc_K_l(KL[k - 1], self.C[k - 1], self.l[k - 2],
self.r[k], self.A[k], self.AA[k - 1])
lop = Vari_Opt_Single_Site_Op(self, n, KL[n - 1], sanity_checks=self.sanity_checks)
evs, eVs = las.eigsh(lop, k=1, which='SA', v0=self.A[n].ravel(), ncv=ncv)
self.A[n] = eVs[:, 0].reshape((self.q[n], self.D[n - 1], self.D[n]))
#shift centre matrix right (RCF is like having a centre "matrix" at "1")
G = tm.restore_LCF_l_seq(self.A[n - 1:n + 1], self.l[n - 1:n + 1],
sanity_checks=self.sanity_checks)
#This is not strictly necessary, since r[n] is not used again,
self.r[n] = G.dot(self.r[n].dot(G.conj().T))
if n < self.N:
for s in xrange(self.q[n + 1]):
self.A[n + 1][s] = G.dot(self.A[n + 1][s])
#All needed l and r should now be up to date
#All needed KR should be valid still
#AA and C must be updated
if n > 1:
self.AA[n - 1] = tm.calc_AA(self.A[n - 1], self.A[n])
self.C[n - 1] = tm.calc_C_mat_op_AA(self.ham[n - 1], self.AA[n - 1])
if n < self.N:
self.AA[n] = tm.calc_AA(self.A[n], self.A[n + 1])
self.C[n] = tm.calc_C_mat_op_AA(self.ham[n], self.AA[n])
for n in xrange(self.N, 0, -1):
if n < self.N:
self.calc_K(n_low=n + 1, n_high=n + 1)
lop = Vari_Opt_Single_Site_Op(self, n, KL[n - 1], sanity_checks=self.sanity_checks)
evs, eVs = las.eigsh(lop, k=1, which='SA', v0=self.A[n].ravel(), ncv=ncv)
self.A[n] = eVs[:, 0].reshape((self.q[n], self.D[n - 1], self.D[n]))
#shift centre matrix left (LCF is like having a centre "matrix" at "N")
Gi = tm.restore_RCF_r_seq(self.A[n - 1:n + 1], self.r[n - 1:n + 1],
sanity_checks=self.sanity_checks)
#This is not strictly necessary, since l[n - 1] is not used again
self.l[n - 1] = Gi.conj().T.dot(self.l[n - 1].dot(Gi))
if n > 1:
for s in xrange(self.q[n - 1]):
self.A[n - 1][s] = self.A[n - 1][s].dot(Gi)
#All needed l and r should now be up to date
#All needed KL should be valid still
#AA and C must be updated
if n > 1:
self.AA[n - 1] = tm.calc_AA(self.A[n - 1], self.A[n])
self.C[n - 1] = tm.calc_C_mat_op_AA(self.ham[n - 1], self.AA[n - 1])
if n < self.N:
self.AA[n] = tm.calc_AA(self.A[n], self.A[n + 1])
self.C[n] = tm.calc_C_mat_op_AA(self.ham[n], self.AA[n])