def testAKLTStateHasCorrectCorrelations(self):
phys_d = 3
bond_d = 2
# Follow Annals of Physics Volume 326, Issue 1, Pages 96-192.
# AKLT correlations appear between Eqs. (115) and (116).
# The tensors below correspond to not normalized state in the thermodynamic limit.
# They should all be multiplied by sqrt(4/3) to get a normalized state.
# One can also normalize the final result with the dominant eigenvalue.
Aplus = np.array([[0, 1 / np.sqrt(2)], [0, 0]])
Aminus = np.array([[0, 0], [-1 / np.sqrt(2), 0]])
A0 = np.array([[-1 / 2, 0], [0, 1 / 2]])
A_matrices = np.array([Aplus, A0, Aminus])
aklt = tfimps.Tfimps(phys_d, bond_d, A_matrices, symmetrize=False)
# Spin 1 operators.
X = tf.constant([[0, 1, 0], [1, 0, 1], [0, 1, 0]],
dtype=tf.float64) / np.sqrt(2)
iY = tf.constant([[0, -1, 0], [1, 0, -1], [0, 1, 0]],
dtype=tf.float64) / np.sqrt(2)
Z = tf.constant([[1, 0, 0], [0, 0, 0], [0, 0, -1]], dtype=tf.float64)
# Range of of values j-i
range = 6
with self.test_session() as sess:
sess.run(tf.global_variables_initializer())
xx_eval = sess.run(aklt.correlator(Z, range))
xx_exact = 12 / 9 * (-1 / 3)**np.arange(1, range)
self.assertAllClose(xx_eval, xx_exact)
def testDominantEigenvectorIsEigenvector(self):
phys_d = 3
bond_d = 5
imps = tfimps.Tfimps(phys_d, bond_d)
with self.test_session() as sess:
sess.run(tf.global_variables_initializer())
T = sess.run(imps.transfer_matrix)
val, vec = sess.run(imps.dominant_eig)
self.assertAllClose(T @ vec, val * vec)
def testMPSInLeftCanonicalForm(self):
phys_d = 2
bond_d = 3
imps = tfimps.Tfimps(phys_d, bond_d, symmetrize=False)
with self.test_session() as sess:
sess.run(tf.global_variables_initializer())
A = sess.run(imps.A)
self.assertAllClose(np.tensordot(A, A, axes=([0, 1], [0, 1])),
np.identity(bond_d))
def testRightEigenvectorHasUnitEigenvalue(self):
phys_d = 2
bond_d = 4
imps = tfimps.Tfimps(phys_d, bond_d, symmetrize=False)
with self.test_session() as sess:
sess.run(tf.global_variables_initializer())
T = sess.run(imps.transfer_matrix)
vec = sess.run(imps.right_eigenvector)
self.assertAllClose(T @ vec, vec)
def testIdentityHamiltonianHasEnergyOneRandomMPS(self):
phys_d = 3
bond_d = 5
I = tf.eye(phys_d, dtype=tf.float64)
h = tf.einsum('ij,kl->ikjl', I, I)
imps = tfimps.Tfimps(phys_d, bond_d, hamiltonian=h)
with self.test_session() as sess:
sess.run(tf.global_variables_initializer())
actual = sess.run(imps.variational_energy)
self.assertAllClose(1, actual)
def testTransferMatrixForIdentity(self):
phys_d = 2
bond_d = 2
A1 = A0 = np.identity(phys_d)
A_matrices = np.array([A0, A1])
imps = tfimps.Tfimps(phys_d, bond_d, A_matrices)
with self.test_session() as sess:
sess.run(tf.global_variables_initializer())
actual = sess.run(imps.transfer_matrix)
self.assertAllClose(phys_d * np.identity(4), actual)
def testIdentityHamiltonianHasEnergyOneDiagonalMPS(self):
phys_d = 2
bond_d = 5
A0 = np.diag(np.random.rand(bond_d))
A1 = np.diag(np.random.rand(bond_d))
A_matrices = np.array([A0, A1])
I = tf.eye(phys_d, dtype=tf.float64)
h = tf.einsum('ij,kl->ikjl', I, I)
imps = tfimps.Tfimps(phys_d, bond_d, A_matrices, hamiltonian=h)
with self.test_session() as sess:
sess.run(tf.global_variables_initializer())
actual = sess.run(imps.variational_energy)
self.assertAllClose(1, actual)
def testAKLTStateHasCorrectEnergy(self):
phys_d = 3
bond_d = 2
# Follow Annals of Physics Volume 326, Issue 1, Pages 96-192.
# Note that even though the As are not symmetric, the transfer matrix is.
# We normalize these to be in left (and right) canonical form
Aplus = np.array([[0, 1 / np.sqrt(2)], [0, 0]])
Aminus = np.array([[0, 0], [-1 / np.sqrt(2), 0]])
A0 = np.array([[-1 / 2, 0], [0, 1 / 2]])
A_matrices = np.array([Aplus, A0, Aminus]) * np.sqrt(4 / 3)
# Spin 1 operators.
X = tf.constant([[0, 1, 0], [1, 0, 1], [0, 1, 0]],
dtype=tf.float64) / np.sqrt(2)
iY = tf.constant([[0, -1, 0], [1, 0, -1], [0, 1, 0]],
dtype=tf.float64) / np.sqrt(2)
Z = tf.constant([[1, 0, 0], [0, 0, 0], [0, 0, -1]], dtype=tf.float64)
XX = tf.einsum('ij,kl->ikjl', X, X)
YY = -tf.einsum('ij,kl->ikjl', iY, iY)
ZZ = tf.einsum('ij,kl->ikjl', Z, Z)
hberg = XX + YY + ZZ
h_aklt = hberg + tf.einsum('abcd,cdef->abef', hberg, hberg) / 3
aklt = tfimps.Tfimps(phys_d,
bond_d,
A_matrices,
symmetrize=False,
hamiltonian=h_aklt)
with self.test_session() as sess:
sess.run(tf.global_variables_initializer())
aklt_energy = sess.run(aklt.variational_energy)
self.assertAllClose(-2 / 3, aklt_energy)