def AKLT(L, pbc=False):
''' S1 * S1 + 1/3 (S1 * S1)^2 '''
print "\n--> Construct the Hamiltonian for the AKLT model of size L=%d ..." % L
store = [[None, None, None, None, None, None, None] for i in range(L + 1)]
store[1][0] = op.spin1_x().times(0)
store[1][1] = op.spin1_x()
store[1][2] = op.spin1_x()
store[1][3] = op.spin1_y()
store[1][4] = op.spin1_y()
store[1][5] = op.spin1_z()
store[1][6] = op.spin1_z()
def iterate(l):
if store[l] != store[0]: return store[l]
le = int(l / 2)
lH, llx, lrx, lly, lry, llz, lrz = iterate(le)
rH, rlx, rrx, rly, rry, rlz, rrz = iterate(l - le)
lid = lH.identity()
rid = rH.identity()
store[l][0] = lH.kron(rid)
temp = lid.kron(rH)
store[l][0].plus(temp)
temp = lrx.kron(rlx)
temp.plus(lry.kron(rly))
temp.plus(lrz.kron(rlz))
store[l][0].plus(temp)
store[l][0].plus(temp.mult(temp).times(1 / 3.))
store[l][1] = llx.kron(rid)
store[l][2] = lid.kron(rrx)
store[l][3] = lly.kron(rid)
store[l][4] = lid.kron(rry)
store[l][5] = llz.kron(rid)
store[l][6] = lid.kron(rrz)
return store[l]
H, lx, rx, ly, ry, lz, rz = iterate(L)
if pbc:
edge = lx.mult(rx)
edge.plus(ly.mult(ry))
edge.plus(lz.mult(rz))
H.plus(edge)
H.plus(edge.mult(edge).times(1. / 3))
print "--> Hamiltonian constructed."
H.basis = op.basis(3)
H.L = L
return H
def Potts_n(L, n=3, J=1., h=1., theta=0., phi=0., pbc=False):
''' - J * Sz * Sz * e^{i theta} + h.c. - h * Sx * e^{i phi}'''
print "\n--> Construct the Hamiltonian for the %d-Potts model of size L=%d ..." % (
n, L)
store = [[None, None, None] for i in range(L + 1)]
store[1][0] = op.tau_n(n).times(-h / 2 * (np.cos(phi) + 1j * np.sin(phi)))
store[1][1] = op.sigma_n(n)
store[1][2] = op.sigma_n(n).hermitian().times(
-J * (np.cos(theta) + 1j * np.sin(theta)))
def iterate(l):
if store[l] != store[0]: return store[l]
le = int(l / 2)
lH, llo, lro = iterate(le)
rH, rlo, rro = iterate(l - le)
lid = lH.identity()
rid = rH.identity()
store[l][0] = lH.kron(rid)
temp = lid.kron(rH)
store[l][0].plus(temp)
store[l][0].plus(lro.kron(rlo))
store[l][1] = llo.kron(rid)
store[l][2] = lid.kron(rro)
return store[l]
H, lo, ro = iterate(L)
if pbc:
edge = operator(L, [store[1][1], store[1][2]], [0, L - 1])
edge.times(-J)
H.plus(edge)
temp = H.hermitian()
H.plus(temp)
print "--> Hamiltonian constructed."
H.basis = op.basis(n)
H.L = L
return H
def Kitaev(L, t=1., u=1., d=1., pbc=False):
''' - t * c^d * c + 2u * n + d * c * c
= ( -t * c^d + d * c) * c + h.c. + 2u * n '''
print "\n--> Construct the Hamiltonian for the Kitaev model of size L=%d ..." % (
L)
store = [[None, None, None] for i in range(L + 1)]
store[1][0] = op.fermion_c().hermitian().mult(op.fermion_c()).times(u)
store[1][1] = op.fermion_c().hermitian().times(-t).plus(
op.fermion_c().times(d))
store[1][2] = op.fermion_c()
def iterate(l):
if store[l] != store[0]: return store[l]
le = int(l / 2)
lH, llo, lro = iterate(le)
rH, rlo, rro = iterate(l - le)
lid = lH.identity()
rid = rH.identity()
store[l][0] = lH.kron(rid)
temp = lid.kron(rH)
store[l][0].plus(temp)
store[l][0].plus(lro.kron(rlo))
store[l][1] = llo.kron(rid)
store[l][2] = lid.kron(rro)
return store[l]
H, lo, ro = iterate(L)
if pbc:
edge = operator(L, [store[1][1], store[1][2]], [0, L - 1])
edge.times(-d)
H.plus(edge)
temp = H.hermitian()
H.plus(temp)
print "--> Hamiltonian constructed."
H.basis = op.basis(2)
H.L = L
return H
def SSH(L, t=1., d=1., pbc=False):
''' - (t-d) * c^d * c - (t+d) * c^d * c '''
print "\n--> Construct the Hamiltonian for the SSH model of size L=%d ..." % (
L)
ope_z = op.fermion_c().times(0.0)
ope_c = op.fermion_c()
ope_cd = op.fermion_c().hermitian()
def iterate(a, b):
if a == b: return [ope_z, ope_c, ope_cd]
ab = a + int((b - a + 1) / 2)
lH, llo, lro = iterate(a, ab - 1)
rH, rlo, rro = iterate(ab, b)
lid = lH.identity()
rid = rH.identity()
H = lH.kron(rid)
temp = lid.kron(rH)
H.plus(temp)
td = -t + d
if ab % 2: td = -t - d
H.plus(lro.kron(rlo).times(td))
return [H, llo.kron(rid), lid.kron(rro)]
H, lo, ro = iterate(1, L)
if pbc:
edge = operator(L, [ope_cd, ope_c], [0, L - 1])
edge.times(t + d)
H.plus(edge)
temp = H.hermitian()
H.plus(temp)
print "--> Hamiltonian constructed."
H.basis = op.basis(2)
H.L = L
return H
nphot0 = 0 tmax = 20 dt = 0.1 # setup basis with ntls spin, each of Hilbert space dimentsion # 2 and photon with dimension nphot setup_basis(ntls, 2, nphot) #run other setup routines list_equivalent_elements() setup_convert_rho() from basis import nspins, ldim_p, ldim_s #setup inital state and calculate Liouvillian L = setup_Dicke(omega, omega0, U, g, gp, kappa, gam_phi, gam_dn, 3) initial = setup_rho(basis(ldim_p, nphot0), basis(ldim_s, 1)) print "setup L" #operators to calculate expectation values for na = tensor(create(ldim_p) * destroy(ldim_p), qeye(ldim_s)) sz = tensor(qeye(ldim_p), sigmaz()) #propagate t0 = time() resultscomp = time_evolve(L, initial, tmax, dt, [na, sz]) tf = time() - t0 print "Time evollution complete" print tf #plot time evolution
def FK(L=8, sec=0, t1=1., t2=1.0, f0=1., f1=1., pbc=False):
''' see the paper for Hamiltonian
hopping t: i c^d c
f0: -(2n-1)(2n-1)
f1: -4(n+n-1)(n+n-1) + 8(cccc+h.c.) '''
if L not in [1,2,4,8]:
raise Exception("Illegal L value")
print "\n--> Construct the Hamiltonian for the FK model of size L=%d ..." % (L)
n_up = op.fermion_up4().hermitian().mult( op.fermion_up4() )
n_down = op.fermion_down4().hermitian().mult( op.fermion_down4() )
pp = op.fermion_up4().mult( op.fermion_down4() ) # cc
ide = n_up.identity()
n2_up = n_up.copy().times(-2).plus( ide ) # 1-2n
n2_down = n_down.copy().times(-2).plus( ide ) # 1-2n
nn = n2_up.copy().plus(n2_down) # 2(1-n-n)
mH = n2_up.mult( n2_down ).times(-f0/2.)
if L==1:
H = mH.hermitian().plus(mH)
H.L = 1
H.basis = op.basis(4)
return H
# the 2-stie ham with the hopping t1 and onsite-interaction f0
H2 = op.fermion_up4().hermitian().kron( op.fermion_up4() )
H2.plus( op.fermion_down4().hermitian().kron( op.fermion_down4() ) )
H2.times( 2.j*float(t1) )
H2.plus( mH.kron(ide).plus( ide.kron(mH) ) )
if L==2:
H = H2.hermitian().plus(H2)
H.L = 1
H.basis = op.basis(4)
return H
# the 4 site ham with interaction f1 and hopping t2
H4 = ham.operator(4, [op.fermion_up4().hermitian(), op.fermion_up4()], [1,3])
H4.plus( ham.operator(4, [op.fermion_down4().hermitian(), op.fermion_down4()], [1,3]) )
H4.times( 2.j*float(t2) )
ide = H2.identity()
H4.plus( H2.kron(ide).plus( ide.kron(H2) ) )
H4.plus( ham.operator(4, [nn, nn], [1,2]).times(-f1/2.) )
H4.plus( ham.operator(4, [pp, pp], [1,2]).times(f1*8.) )
if L==4:
H = H4.hermitian().plus(H4)
H.L = 1
H.basis = op.basis(4)
return H
def kron_s(A, B, sec):
fp = A.fparity
ss = A.sym_sum
sym = A.sym
dim = sym * A.dim * B.dim
ret = csr((dim, dim), dtype=np.complex)
for row1 in range(sym):
for col1 in range(sym):
row2 = ss(sec, -row1)
col2 = ss(sec, -col1)
if not A._empty_(row1, col1) and not B._empty_(row2, col2):
sign = 1 - 2*( fp[col1] * fp[ss(col2, -row2)] )
temp = coo( sparse.kron(A.val[row1][col1], B.val[row2][col2]) * sign )
block = A.dim * B.dim
add = coo((temp.data, (temp.row + row1*block, temp.col + col1*block)),(dim, dim), dtype=np.complex)
ret += add
return ret
# the 8 site ham
lup = ham.operator(4, [op.fermion_up4().hermitian()], [2])
rup = ham.operator(4, [op.fermion_up4()], [0])
ldown = ham.operator(4, [op.fermion_down4().hermitian()], [2])
rdown = ham.operator(4, [op.fermion_down4()], [0])
ide = H4.identity()
ln = ham.operator(4, [nn], [3])
rn = ham.operator(4, [nn], [0])
lp = ham.operator(4, [pp], [3])
rp = ham.operator(4, [pp], [0])
Hs = coo((16384, 16384), dtype=np.complex)
Hs += kron_s(lup, rup, sec)
Hs += kron_s(ldown, rdown, sec)
Hs *= 2.j*float(t2)
Hs += kron_s(H4, ide, sec)
Hs += kron_s(ide, H4, sec)
Hs += kron_s(ln, rn, sec)*(-f1/2.)
Hs += kron_s(lp, rp, sec)*(f1*8.)
if pbc:
edge = kron_s(rup,lup, sec)
edge += kron_s(rdown, ldown, sec)
edge *= -2.j*float(t2)
Hs += edge
Hs += kron_s(rn, ln, sec)*(-f1/2.)
Hs += kron_s(rp, lp, sec)*(f1*8.)
Hs += Hs.conjugate().transpose()
print "--> Hamiltonian constructed."
return Hs
H = FK(L=8, sec=sec, t1=t1, t2=t2, f0=f0, f1=f1, pbc=pbc) evals, evecs = eigsh(H, which='SA') arg = np.argsort(evals) evals_all.append( evals[arg] ) evecs_all.append( evecs[:,arg] ) return np.array(evals_all), np.array(evecs_all) H = FK(L=L, sec=0, t1=t1, t2=t2, f0=f0, f1=f1, pbc=pbc) for sec in range(0,4): D, U = eigh(H.val[sec][sec].todense()) evals_all.append(D) evecs_all.append(U) return np.array(evals_all), np.array(evecs_all) L = 8 basis = ham.basis(L, op.basis(4), op.fermion_up4().sym_sum) evals, evecs = FK_eig(L=L, t1=1.0, t2=1., f0=0.25, f1=0.25, pbc=True) print evals[:,:5] sec = 0 rho = func.reduced_density_matrix(evecs[sec,:,0], basis[sec], 4, 4) u, d, v = svd(rho) ee = -np.sum(d**2 * np.log( d**2) ) print "Entanglement entropy:", ee, "or", ee/np.log(2), "ln2" rho = func.reduced_density_matrix(evecs[sec,:,1], basis[sec], 4, 4) u, d, v = svd(rho) ee = -np.sum(d**2 * np.log( d**2) ) print "Entanglement entropy:", ee, "or", ee/np.log(2), "ln2" sec = 3 rho = func.reduced_density_matrix(evecs[sec,:,0], basis[sec], 4, 4) u, d, v = svd(rho)
def FK(L=8, sec=0, t1=1., t2=1.0, f0=1., f1=1., pbc=False):
''' see the paper for Hamiltonian
hopping t: i c^d c
f0: -(2n-1)(2n-1)
f1: -4(n+n-1)(n+n-1) + 8(cccc+h.c.) '''
if L not in [1, 2, 4, 8]:
raise Exception("Illegal L value")
print "\n--> Construct the Hamiltonian for the FK model of size L=%d ..." % (
L)
n_up = op.fermion_up4().hermitian().mult(op.fermion_up4())
n_down = op.fermion_down4().hermitian().mult(op.fermion_down4())
pp = op.fermion_up4().mult(op.fermion_down4()) # cc
ide = n_up.identity()
n2_up = n_up.copy().times(-2).plus(ide) # 1-2n
n2_down = n_down.copy().times(-2).plus(ide) # 1-2n
nn = n2_up.copy().plus(n2_down) # 2(1-n-n)
mH = n2_up.mult(n2_down).times(-f0 / 2.)
if L == 1:
H = mH.hermitian().plus(mH)
H.L = 1
H.basis = op.basis(4)
return H
# the 2-stie ham with the hopping t1 and onsite-interaction f0
H2 = op.fermion_up4().hermitian().kron(op.fermion_up4())
H2.plus(op.fermion_down4().hermitian().kron(op.fermion_down4()))
H2.times(2.j * float(t1))
H2.plus(mH.kron(ide).plus(ide.kron(mH)))
if L == 2:
H = H2.hermitian().plus(H2)
H.L = 1
H.basis = op.basis(4)
return H
# the 4 site ham with interaction f1 and hopping t2
H4 = ham.operator(4, [op.fermion_up4().hermitian(),
op.fermion_up4()], [1, 3])
H4.plus(
ham.operator(4, [op.fermion_down4().hermitian(),
op.fermion_down4()], [1, 3]))
H4.times(2.j * float(t2))
ide = H2.identity()
H4.plus(H2.kron(ide).plus(ide.kron(H2)))
H4.plus(ham.operator(4, [nn, nn], [1, 2]).times(-f1 / 2.))
H4.plus(ham.operator(4, [pp, pp], [1, 2]).times(f1 * 8.))
if L == 4:
H = H4.hermitian().plus(H4)
H.L = 1
H.basis = op.basis(4)
return H
def kron_s(A, B, sec):
fp = A.fparity
ss = A.sym_sum
sym = A.sym
dim = sym * A.dim * B.dim
ret = csr((dim, dim), dtype=np.complex)
for row1 in range(sym):
for col1 in range(sym):
row2 = ss(sec, -row1)
col2 = ss(sec, -col1)
if not A._empty_(row1, col1) and not B._empty_(row2, col2):
sign = 1 - 2 * (fp[col1] * fp[ss(col2, -row2)])
temp = coo(
sparse.kron(A.val[row1][col1], B.val[row2][col2]) *
sign)
block = A.dim * B.dim
add = coo(
(temp.data,
(temp.row + row1 * block, temp.col + col1 * block)),
(dim, dim),
dtype=np.complex)
ret += add
return ret
# the 8 site ham
lup = ham.operator(4, [op.fermion_up4().hermitian()], [2])
rup = ham.operator(4, [op.fermion_up4()], [0])
ldown = ham.operator(4, [op.fermion_down4().hermitian()], [2])
rdown = ham.operator(4, [op.fermion_down4()], [0])
ide = H4.identity()
ln = ham.operator(4, [nn], [3])
rn = ham.operator(4, [nn], [0])
lp = ham.operator(4, [pp], [3])
rp = ham.operator(4, [pp], [0])
Hs = coo((16384, 16384), dtype=np.complex)
Hs += kron_s(lup, rup, sec)
Hs += kron_s(ldown, rdown, sec)
Hs *= 2.j * float(t2)
Hs += kron_s(H4, ide, sec)
Hs += kron_s(ide, H4, sec)
Hs += kron_s(ln, rn, sec) * (-f1 / 2.)
Hs += kron_s(lp, rp, sec) * (f1 * 8.)
if pbc:
edge = kron_s(rup, lup, sec)
edge += kron_s(rdown, ldown, sec)
edge *= -2.j * float(t2)
Hs += edge
Hs += kron_s(rn, ln, sec) * (-f1 / 2.)
Hs += kron_s(rp, lp, sec) * (f1 * 8.)
Hs += Hs.conjugate().transpose()
print "--> Hamiltonian constructed."
return Hs
evals, evecs = eigsh(H, which='SA')
arg = np.argsort(evals)
evals_all.append(evals[arg])
evecs_all.append(evecs[:, arg])
return np.array(evals_all), np.array(evecs_all)
H = FK(L=L, sec=0, t1=t1, t2=t2, f0=f0, f1=f1, pbc=pbc)
for sec in range(0, 4):
D, U = eigh(H.val[sec][sec].todense())
evals_all.append(D)
evecs_all.append(U)
return np.array(evals_all), np.array(evecs_all)
L = 8
basis = ham.basis(L, op.basis(4), op.fermion_up4().sym_sum)
evals, evecs = FK_eig(L=L, t1=1.0, t2=1., f0=0.25, f1=0.25, pbc=True)
print evals[:, :5]
sec = 0
rho = func.reduced_density_matrix(evecs[sec, :, 0], basis[sec], 4, 4)
u, d, v = svd(rho)
ee = -np.sum(d**2 * np.log(d**2))
print "Entanglement entropy:", ee, "or", ee / np.log(2), "ln2"
rho = func.reduced_density_matrix(evecs[sec, :, 1], basis[sec], 4, 4)
u, d, v = svd(rho)
ee = -np.sum(d**2 * np.log(d**2))
print "Entanglement entropy:", ee, "or", ee / np.log(2), "ln2"
sec = 3
rho = func.reduced_density_matrix(evecs[sec, :, 0], basis[sec], 4, 4)
u, d, v = svd(rho)