# define model params L=100 # system size J=1.0 #uniform hopping deltaJ=0.2 # hopping difference Delta=0.2 # staggered potential # define site-coupling lists hopping=[[J+deltaJ*(-1)**i,i,(i+1)%L] for i in range(L)] # PBC stagg_pot=[[Delta,i] for i in range(L)] # define basis basis=spin_basis_1d(L,Nup=1,pauli=False) basis_args = (L,) blocks=(dict(Nup=1,pauli=False,kblock=i,a=2) for i in range(L//2)) # define static and dynamic lists static=[["+-",hopping],["-+",hopping],['z',stagg_pot]] dynamic=[] #### calculate Hamiltonian H=hamiltonian(static,dynamic,basis=basis,dtype=np.float64) E=H.eigvalsh() _,Hblock = block_diag_hamiltonian(blocks,static,dynamic,spin_basis_1d,basis_args,np.complex128) Eblock=Hblock.eigvalsh() #print(_.shape) print(E) print(Eblock) plt.scatter(np.arange(L),E) plt.show()
stagg_pot=[[Delta*(-1)**i,i] for i in range(L)]
# define static and dynamic lists
static=[["+-",hop_pm],["-+",hop_mp],['n',stagg_pot]]
dynamic=[]
# build real-space Hamiltonian
H=hamiltonian(static,dynamic,basis=basis,dtype=np.float64)
print(H.toarray())
# diagonalise real-space Hamiltonian
E,V=H.eigh()
#
##### compute Fourier transform and momentum-space Hamiltonian #####
# define basis blocks and arguments
blocks=[dict(Nf=1,kblock=i,a=2) for i in range(L//2)] # only L//2 distinct momenta
basis_args = (L,)
# construct block-diagonal Hamiltonian
FT,Hblock = block_diag_hamiltonian(blocks,static,dynamic,spinless_fermion_basis_1d,
basis_args,np.complex128,get_proj_kwargs=dict(pcon=True))
print(np.around(Hblock.toarray(),2))
# diagonalise momentum-space Hamiltonian
Eblock,Vblock=Hblock.eigh()
#
##### plot spectra
import matplotlib.pyplot as plt # plotting library
plt.plot(np.arange(H.Ns),E/L,
marker='o',color='b',label='real space')
plt.plot(np.arange(Hblock.Ns),Eblock/L,
marker='x',color='r',markersize=2,label='momentum space')
plt.xlabel('state number',fontsize=16)
plt.ylabel('energy',fontsize=16)
plt.xticks(fontsize=16)
plt.yticks(fontsize=16)
plt.legend(fontsize=16)
def test():
L = 5
start = 0
stop = 10
num = 10
J = [[1.0, i, (i + 1) % L] for i in range(L)]
h = [[1.0, i] for i in range(L)]
static = [["xx", J], ["yy", J], ["zz", J]]
dynamic = [["z", h, np.sin, ()]]
blocks = []
for Nup in range(L + 1):
blocks.append({"Nup": Nup})
H_block = block_diag_hamiltonian(blocks, static, dynamic, spin_basis_1d,
(L, ), np.float64)
H = hamiltonian(static, dynamic, N=L)
for t in np.linspace(0, 2 * np.pi):
E = H.eigvalsh(time=t)
E_blocks = H.eigvalsh(time=t)
np.testing.assert_allclose(E, E_blocks)
static = [["zz", J]]
dynamic = [["x", h, f, []]]
blocks = []
for kblock in range(L):
blocks.append({"kblock": kblock})
H = hamiltonian(static, dynamic, N=L)
[E0] = H.eigsh(time=0.3, k=1, which='SA', return_eigenvectors=False)
block_op = block_ops(blocks,
static,
dynamic,
spin_basis_1d, (L, ),
np.complex128,
compute_all_blocks=True)
# real time.
expH = exp_op(H,
a=-1j,
start=start,
stop=stop,
iterate=True,
num=num,
endpoint=True)
times = np.linspace(start, stop, num=num, endpoint=True)
psi0 = np.random.ranf(H.Ns)
psi0 /= np.linalg.norm(psi0)
psi_exact_1 = H.evolve(psi0,
0,
times,
iterate=True,
atol=1e-15,
rtol=1e-15)
psi_block_1 = block_op.evolve(psi0,
0,
times,
iterate=True,
atol=1e-15,
rtol=1e-15,
n_jobs=4)
psi_exact_2 = expH.dot(psi0, time=0.3)
psi_block_2 = block_op.expm(psi0,
H_time_eval=0.3,
start=start,
stop=stop,
iterate=True,
num=num,
endpoint=True,
n_jobs=2,
block_diag=True)
for psi_e_1, psi_e_2, psi_b_1, psi_b_2 in izip(psi_exact_1, psi_exact_2,
psi_block_1, psi_block_2):
np.testing.assert_allclose(psi_b_1, psi_e_1, atol=1e-7)
np.testing.assert_allclose(psi_b_2, psi_e_2, atol=1e-7)
expH.set_iterate(False)
psi_exact_1 = H.evolve(psi0,
0,
times,
iterate=False,
atol=1e-15,
rtol=1e-15)
psi_block_1 = block_op.evolve(psi0,
0,
times,
iterate=False,
atol=1e-15,
rtol=1e-15,
n_jobs=4)
psi_exact_2 = expH.dot(psi0, time=0.3)
psi_block_2 = block_op.expm(psi0,
H_time_eval=0.3,
start=start,
stop=stop,
iterate=False,
num=num,
endpoint=True,
block_diag=True)
for psi_e_1, psi_b_1 in izip(psi_exact_1, psi_block_1):
np.testing.assert_allclose(psi_b_1, psi_e_1, atol=1e-7)
for psi_e_2, psi_b_2 in izip(psi_exact_2, psi_block_2):
np.testing.assert_allclose(psi_b_2, psi_e_2, atol=1e-7)
# imaginary time
expH = exp_op(H,
a=-1,
start=start,
stop=stop,
iterate=True,
num=num,
endpoint=True)
times = np.linspace(start, stop, num=num, endpoint=True)
psi0 = np.random.ranf(H.Ns)
psi0 /= np.linalg.norm(psi0)
psi_exact_2 = expH.dot(psi0, time=0.3, shift=-E0)
psi_block_2 = block_op.expm(psi0,
a=-1,
shift=-E0,
H_time_eval=0.3,
start=start,
stop=stop,
iterate=True,
num=num,
endpoint=True,
block_diag=True)
for psi_e_2, psi_b_2 in izip(psi_exact_2, psi_block_2):
np.testing.assert_allclose(psi_b_2, psi_e_2, atol=1e-7)
# same for iterate=False
expH.set_iterate(False)
psi_exact_2 = expH.dot(psi0, time=0.3, shift=-E0)
psi_block_2 = block_op.expm(psi0,
a=-1,
shift=-E0,
H_time_eval=0.3,
start=start,
stop=stop,
iterate=False,
num=num,
endpoint=True,
block_diag=True)
for psi_e_2, psi_b_2 in izip(psi_exact_2, psi_block_2):
np.testing.assert_allclose(psi_b_2, psi_e_2, atol=1e-7)