def test(Lx, Ly):
N = Lx * Ly
tr = square_lattice_trans(Lx, Ly)
basis_dict = {}
Nfs = range(N + 1)
for Nf in Nfs:
basis_blocks = []
pcon_basis = spinful_fermion_basis_general(N, Nf=(Nf, N - Nf))
Ns_block = 0
for blocks in tr.allowed_blocks_iter_parity():
print(N, Nf, N - Nf)
print(blocks)
basis = spinful_fermion_basis_general(N, Nf=(Nf, N - Nf), **blocks)
Ns_block += basis.Ns
basis_blocks.append(basis)
assert (Ns_block == pcon_basis.Ns)
basis_dict[Nf] = (pcon_basis, basis_blocks)
J = [[np.sqrt(2.0), i, tr.T_x[i]] for i in range(N)]
J.extend([[np.sqrt(2.0), i, tr.T_y[i]] for i in range(N)])
Jp = [[1.0, i, tr.T_x[i]] for i in range(N)]
Jp.extend([[1.0, i, tr.T_y[i]] for i in range(N)])
Jm = [[-1.0, i, tr.T_x[i]] for i in range(N)]
Jm.extend([[-1.0, i, tr.T_y[i]] for i in range(N)])
static = [["n|n", J], ["+-|", Jp], ["-+|", Jm], ["|+-", Jp], ["|-+", Jm]]
E_symm = {}
for Nf, (pcon_basis, basis_blocks) in basis_dict.items():
H_pcon = hamiltonian(static, [], basis=pcon_basis, dtype=np.float64)
if H_pcon.Ns > 0:
E_pcon = np.linalg.eigvalsh(H_pcon.todense())
else:
E_pcon = np.array([])
E_block = []
for basis in basis_blocks:
H = hamiltonian(static, [], basis=basis, dtype=np.complex128)
if H.Ns > 0:
E_block.append(np.linalg.eigvalsh(H.todense()))
E_block = np.hstack(E_block)
E_block.sort()
np.testing.assert_allclose(E_pcon, E_block, atol=1e-13)
print("passed Nf={} sector".format(Nf))
basis_fermion = spinless_fermion_basis_general(N_2d, make_basis=False, Nf=N_2d//2, kxblock=(T_x,0),kyblock=(T_y,0), rblock=(R,0), pxblock=(P_x,0),pyblock=(P_y,0), ) basis_fermion_full = spinless_fermion_basis_general(N_2d, make_basis=True, Nf=N_2d//2, ) basis_spinful_fermion = spinful_fermion_basis_general(N_2d, make_basis=False, Nf=(N_2d//8,N_2d//8), kxblock=(T_x,0),kyblock=(T_y,0), rblock=(R,0), pxblock=(P_x,0),pyblock=(P_y,0), ) basis_spinful_fermion_full = spinful_fermion_basis_general(N_2d, make_basis=True, Nf=(N_2d//8,N_2d//8), ) bases_2d=[basis_boson,basis_spin,basis_fermion,basis_spinful_fermion] bases_2d_full=[basis_boson_full,basis_spin_full,basis_fermion_full,basis_spinful_fermion_full] for i,(basis_2d,basis_2d_full) in enumerate(zip(bases_2d,bases_2d_full)): # grab states of full basis states=basis_2d_full.states
# visualise graph
pos = nx.spring_layout(hex_graph, seed=42, iterations=100)
nx.draw(hex_graph, pos=pos, with_labels=True)
plt.show(block=False)
#
###### model parameters
#
N_up = 2 # number of spin-up fermions
N_down = 2 # number of spin-down fermions
t = 1.0 # tunnelling matrix element
U = 2.0 # on-site fermion interaction strength
#
##### set up Fermi-Hubbard Hubbard Hamiltonian with quspin #####
#
### compute basis
basis = spinful_fermion_basis_general(N, Nf=(N_up, N_down))
print('Hilbert space size: {0:d}.\n'.format(basis.Ns))
#
# define site-coupling lists
tunnelling = [[-t, i, j] for i in range(N) for j in hex_graph.adj[i]]
interactions = [[U, i, i] for i in range(N)]
#
# define site-coupling lists [hermitian conjugates "-+|" and "|-+" contained in tunnelling list]
static = [["n|n", interactions], ["+-|", tunnelling], ["|+-", tunnelling]]
dynamic = []
#
### construct Hamiltonian
H = hamiltonian(static, dynamic, basis=basis, dtype=np.float64)
#
# compute eigensystem
E, V = H.eigsh(k=4, which='SA', maxiter=1E4)
)
basis_fermion = spinless_fermion_basis_general(N_2d,
make_basis=False,
Nf=N_2d // 2,
**basis_dict)
basis_fermion_full = spinless_fermion_basis_general(
N_2d,
make_basis=False,
Nf=N_2d // 2,
)
basis_spinful_fermion = spinful_fermion_basis_general(N_2d,
make_basis=False,
Nf=(N_2d // 8,
N_2d // 8),
**basis_dict)
basis_spinful_fermion_full = spinful_fermion_basis_general(
N_2d,
make_basis=False,
Nf=(N_2d // 8, N_2d // 8),
)
bases_2d = [basis_boson, basis_spin, basis_fermion, basis_spinful_fermion]
bases_2d_full = [
basis_boson_full, basis_spin_full, basis_fermion_full,
basis_spinful_fermion_full
]
compare(static_list,basis,basis_op)
print('passed bosons')
if Np==None:
Nf=Np
elif type(Np) is list:
Nf=list(zip(Np,Np))
else:
Nf=(Np,Np)
basis=spinful_fermion_basis_general(N_2d, make_basis=False,
Nf=Nf,
kxblock=(T_x,0),kyblock=(T_y,0),
pxblock=(P_x,0),pyblock=(P_y,0),
)
basis_op=spinful_fermion_basis_general(N_2d, make_basis=True,
Nf=Nf,
kxblock=(T_x,0),kyblock=(T_y,0),
pxblock=(P_x,0),pyblock=(P_y,0),
)
compare(static_list_spfs,basis,basis_op)
print('passed spinful fermios')
t_y = x + Lx * (
(y + 1) % Ly) # translation along y-direction for one spin species
# create the spin-up spin-down combined transformations
s = np.arange(
2 * N_2d) # sites [0,1,2,...,N_2d-1,...,2*N_2d-1] in advanced notation
T_x = np.hstack(
(t_x, t_x + N_2d)) # translation along x-direction for both spin species
T_y = np.hstack(
(t_y, t_y + N_2d)) # translation along y-direction for both spin species
PH = -(s + 1) # particle-hole in the advanced case
#
###### setting up bases ###### (note optional argument simple_symm=False)
#basis_2d=spinful_fermion_basis_general(N_2d,simple_symm=False,Nf=(2,2),kxblock=(T_x,0),kyblock=(T_y,0))
basis_2d = spinful_fermion_basis_general(N_2d,
simple_symm=False,
Nf=(6, 6),
kxblock=(T_x, 0),
kyblock=(T_y, 0),
phblock=(PH, 0))
#
###### setting up hamiltonian ######
# setting up site-coupling lists for advanced case
hopping_left = [[-J, i, T_x[i]]
for i in range(2 * N_2d)] + [[-J, i, T_y[i]]
for i in range(2 * N_2d)]
hopping_right = [[+J, i, T_x[i]]
for i in range(2 * N_2d)] + [[+J, i, T_y[i]]
for i in range(2 * N_2d)]
interaction = [[U, i, i + N_2d] for i in range(N_2d)]
#
static = [
["+-", hopping_left], # spin-up and spin-down hop to left
def test(Lx,Ly):
N = Lx*Ly
nmax = int(eval("2*1/2"))
sps = nmax+1
tr = square_lattice_trans(Lx,Ly)
basis_dict = {}
basis_dict_f = {}
basis_dict_combined = {}
Nups=range(nmax*N+1)
for Nup in Nups:
basis_blocks=[]
basis_blocks_f=[]
pcon_basis = spin_basis_general(N,Nup=Nup,pauli=False)
pcon_basis_f = spinful_fermion_basis_general(N,Nf=(Nup,N-Nup),double_occupancy=False)
Ns_block = 0
for blocks in tr.allowed_blocks_spin_inversion_iter(Nup,sps):
basis = spin_basis_general(N,Nup=Nup,pauli=False,**blocks)
Ns_block += basis.Ns
basis_blocks.append(basis)
Ns_block_f = 0
for blocks_f in tr.allowed_blocks_spin_inversion_iter(Nup,sps): # requires simple symmetry definition
basis_f = spinful_fermion_basis_general(N,Nf=(Nup,N-Nup),double_occupancy=False,**blocks_f)
Ns_block_f += basis_f.Ns
basis_blocks_f.append(basis_f)
try:
assert(Ns_block == pcon_basis.Ns)
except AssertionError:
print(Nup,Ns_block,pcon_basis.Ns)
raise AssertionError("reduced blocks don't sum to particle sector.")
try:
assert(Ns_block_f == pcon_basis_f.Ns)
except AssertionError:
print(Nup,Ns_block_f,pcon_basis_f.Ns)
raise AssertionError("fermion reduced blocks don't sum to particle sector.")
try:
assert(Ns_block == pcon_basis_f.Ns)
except AssertionError:
print(Nup,Ns_block_f,pcon_basis_f.Ns)
raise AssertionError("fermion reduced blocks don't match spin blocks.")
basis_dict[Nup] = (pcon_basis,basis_blocks)
basis_dict_f[Nup] = (pcon_basis_f,basis_blocks_f)
basis_dict_combined[Nup] = (pcon_basis,basis_blocks,pcon_basis_f,basis_blocks_f)
J = [[1.0,i,tr.T_x[i]] for i in range(N)] + [[1.0,i,tr.T_y[i]] for i in range(N)]
J_nn_ij = [[-0.25,i,tr.T_x[i]] for i in range(N)] + [[-0.25,i,tr.T_y[i]] for i in range(N)]
J_nn_ji = [[-0.25,tr.T_x[i],i] for i in range(N)] + [[-0.25,tr.T_y[i],i] for i in range(N)]
J_nn_ij_p = [[0.25,i,tr.T_x[i]] for i in range(N)] + [[0.25,i,tr.T_y[i]] for i in range(N)]
J_nn_ji_p = [[0.25,tr.T_x[i],i] for i in range(N)] + [[0.25,tr.T_y[i],i] for i in range(N)]
J_cccc_ij = [[-1.0,i,tr.T_x[i],tr.T_x[i],i] for i in range(N)] + [[-1.0,i,tr.T_y[i],tr.T_y[i],i] for i in range(N)]
J_cccc_ji = [[-1.0,tr.T_x[i],i,i,tr.T_x[i]] for i in range(N)] + [[-1.0,tr.T_y[i],i,i,tr.T_y[i]] for i in range(N)]
static = [["zz",J],["+-",J],["-+",J]]
static_f = [["nn|",J_nn_ij_p],["|nn",J_nn_ji_p],["n|n",J_nn_ij],["n|n",J_nn_ji],["+-|+-",J_cccc_ij],["+-|+-",J_cccc_ji]]
E_symm = {}
E_symm_f = {}
#'''
for N,(pcon_basis,basis_blocks,pcon_basis_f,basis_blocks_f) in basis_dict_combined.items():
H_pcon = hamiltonian(static,[],basis=pcon_basis,dtype=np.float64)
H_pcon_f = hamiltonian(static_f,[],basis=pcon_basis_f,dtype=np.float64)
if H_pcon.Ns>0:
E_pcon = np.linalg.eigvalsh(H_pcon.todense())
else:
E_pcon = np.array([])
if H_pcon_f.Ns>0:
E_pcon_f = np.linalg.eigvalsh(H_pcon_f.todense())
else:
E_pcon_f = np.array([])
E_block = []
E_block_f = []
for basis, basis_f in zip(basis_blocks,basis_blocks_f):
H = hamiltonian(static,[],basis=basis,dtype=np.complex128)
H_f = hamiltonian(static_f,[],basis=basis_f,dtype=np.complex128)
if H.Ns>0:
E_block.append(np.linalg.eigvalsh(H.todense()))
if H_f.Ns>0:
E_block_f.append(np.linalg.eigvalsh(H_f.todense()))
E_block = np.hstack(E_block)
E_block.sort()
E_block_f = np.hstack(E_block_f)
E_block_f.sort()
np.testing.assert_allclose(E_pcon,E_block,atol=1e-13)
np.testing.assert_allclose(E_pcon_f,E_block_f,atol=1e-13)
np.testing.assert_allclose(E_pcon,E_pcon_f,atol=1e-13)
print("passed N={} sector".format(N))
U_onsite = [[1.0, i, i] for i in range(N)]
operator_list_0 = [["+-|", Jp], ["-+|", Jm], ["|+-", Jp], ["|-+", Jm]]
operator_list_1 = [["n|n", U_onsite]]
operator_dict = dict(H0=operator_list_0, H1=operator_list_1)
basis_f = spinless_fermion_basis_1d(L=N, Nf=2)
basis_1 = tensor_basis(basis_f, basis_f)
basis_f = spinless_fermion_basis_general(N, Nf=2)
basis_2 = tensor_basis(basis_f, basis_f)
basis_3 = spinful_fermion_basis_1d(L=N, Nf=(2, 2))
basis_4 = spinful_fermion_basis_general(N, Nf=(2, 2))
basis_dict = dict(tensored_spinless_fermion_basis_1d=basis_1,
spinful_fermion_basis_1d=basis_3,
tensored_spinless_fermion_basis_general=basis_2,
spinful_fermion_basis_general=basis_4)
for basis_name, basis in basis_dict.items():
H_U = quantum_operator(operator_dict,
basis=basis,
dtype=np.float64,
check_pcon=False,
check_symm=False,
check_herm=False)
E_quspin = np.zeros(E_paper.shape, )
# symmetry-free
basis_1 = spin_basis_1d(L=L, Nup=range(0, L, 2))
basis_1g = spin_basis_general(N=L, Nup=range(0, L, 2))
basis_2 = boson_basis_1d(L=L, Nb=range(0, L, 2))
basis_2g = boson_basis_general(N=L, Nb=range(0, L, 2))
basis_3 = spinless_fermion_basis_1d(L=L, Nf=range(0, L, 2))
basis_3g = spinless_fermion_basis_general(N=L, Nf=range(0, L, 2))
basis_4 = spinful_fermion_basis_1d(L=L,
Nf=product(range(0, L, 2),
range(0, L, 2)))
basis_4g = spinful_fermion_basis_general(N=L,
Nf=product(
range(0, L, 2),
range(0, L, 2)))
# symmetry-ful
t = (np.arange(L) + 1) % L
basis_1 = spin_basis_1d(L=L, Nup=range(0, L, 2), kblock=0)
basis_1g = spin_basis_general(N=L, Nup=range(0, L, 2), kblock=(t, 0))
basis_2 = boson_basis_1d(L=L, Nb=range(0, L, 2), kblock=0)
basis_2g = boson_basis_general(N=L, Nb=range(0, L, 2), kblock=(t, 0))
basis_3 = spinless_fermion_basis_1d(L=L, Nf=range(0, L, 2), kblock=0)
basis_3g = spinless_fermion_basis_general(N=L,
Nf=range(0, L, 2),
###### setting up user-defined BASIC symmetry transformations for 2d lattice ######
s = np.arange(N_2d) # sites [0,1,2,...,N_2d-1] in simple notation
x = s % Lx # x positions for sites
y = s // Lx # y positions for sites
T_x = (x + 1) % Lx + Lx * y # translation along x-direction
T_y = x + Lx * ((y + 1) % Ly) # translation along y-direction
P_x = x + Lx * (Ly - y - 1) # reflection about x-axis
P_y = (Lx - x - 1) + Lx * y # reflection about y-axis
S = -(s + 1) # fermion spin inversion in the simple case
#
###### setting up bases ######
basis_2d = spinful_fermion_basis_general(
N_2d,
Nf=(3, 3),
double_occupancy=False,
kxblock=(T_x, 0),
kyblock=(T_y, 0),
pxblock=(P_x, 1),
pyblock=(P_y, 0), # contains GS
sblock=(S, 0))
print(basis_2d)
#
###### setting up hamiltonian ######
# setting up site-coupling lists for simple case
hopping_left = [[-J, i, T_x[i]] for i in range(N_2d)] + [[-J, i, T_y[i]]
for i in range(N_2d)]
hopping_right = [[+J, i, T_x[i]] for i in range(N_2d)] + [[+J, i, T_y[i]]
for i in range(N_2d)]
potential = [[-mu, i] for i in range(N_2d)]
interaction = [[U, i, i] for i in range(N_2d)]
#
Jp = [[1.0, i, T_x[i]] for i in range(2 * N)]
Jp.extend([[1.0, i, T_y[i]] for i in range(2 * N)])
Jm = [[-1.0, i, T_x[i]] for i in range(2 * N)]
Jm.extend([[-1.0, i, T_y[i]] for i in range(2 * N)])
U_onsite = [[1.0, i, i + N] for i in range(N)]
operator_list_0 = [["+-", Jp], ["-+", Jm]]
operator_list_1 = [["nn", U_onsite]]
operator_dict = dict(H0=operator_list_0, H1=operator_list_1)
basis = spinful_fermion_basis_general(N,
Nf=(2, 2),
ky=(T_y, 0),
kx=(T_x, 0),
simple_symm=False)
basis_name = "general spinful fermions advanced"
H_U = quantum_operator(operator_dict,
basis=basis,
dtype=np.float64,
check_pcon=False,
check_symm=False,
check_herm=False)
E_quspin = np.zeros(E_paper.shape)
for j, U in enumerate(np.linspace(0.0, 4.0, 9)):
def build_basis(Lx, Ly, U=4.0, μ=0):
N_2d = Lx * Ly
Nup = N_2d
Ndown = N_2d
J = 1.0
s = np.arange(N_2d) # sites [0,1,2,....]
x = s % Lx # x positions for sites
y = s // Lx # y positions for sites
T_x = (x + 1) % Lx + Lx * y # translation along x-direction
P_x = x + Lx * (Ly - y - 1) # reflection about x-axis
T_y = x + Lx * ((y + 1) % Ly) # translation along y-direction
P_y = (Lx - x - 1) + Lx * y # reflection about y-axis
S = -(s + 1) # fermion spin inversion in the simple case
P_xy = (Lx - x - 1) + Lx * (Ly - y - 1) # point reflection about origin
#basis = spinful_fermion_basis_general(
# N_2d, pxyblock=(P_xy, 1)
#)
basis = spinful_fermion_basis_general(N_2d, double_occupancy=False)
#print(basis)
hop_right = ([[+0.5, i, T_x[i]]
for i in range(N_2d)] + [[+0.5, i, T_y[i]]
for i in range(N_2d)])
hop_left = ([[-0.5, i, T_x[i]]
for i in range(N_2d)] + [[-0.5, i, T_y[i]]
for i in range(N_2d)])
print(hop_right)
coupling_1 = ([[-J / 2, i, T_x[i]]
for i in range(N_2d)] + [[-J / 2, i, T_y[i]]
for i in range(N_2d)] +
[[-J / 2, T_x[i], i]
for i in range(N_2d)] + [[-J / 2, T_y[i], i]
for i in range(N_2d)])
coupling_2 = ([[J / 2, i, i, T_x[i]]
for i in range(N_2d)] + [[J / 2, i, i, T_y[i]]
for i in range(N_2d)] +
[[J / 2, T_x[i], i, T_x[i]]
for i in range(N_2d)] + [[J / 2, T_y[i], i, T_y[i]]
for i in range(N_2d)])
coupling_3 = ([[J / 2, i, T_x[i], i]
for i in range(N_2d)] + [[J / 2, i, T_y[i], i]
for i in range(N_2d)] +
[[J / 2, i, T_x[i], T_x[i]]
for i in range(N_2d)] + [[J / 2, i, T_y[i], T_y[i]]
for i in range(N_2d)])
coupling_4 = ([[-J, i, T_x[i], i, T_x[i]]
for i in range(N_2d)] + [[-J, i, T_y[i], i, T_y[i]]
for i in range(N_2d)])
coupling_5 = ([[-J / 2, i, T_x[i], T_x[i], i]
for i in range(N_2d)] + [[-J / 2, i, T_y[i], T_y[i], i]
for i in range(N_2d)] +
[[-J / 2, T_x[i], i, i, T_x[i]]
for i in range(N_2d)] + [[-J / 2, T_y[i], i, i, T_y[i]]
for i in range(N_2d)])
pot = [[-U / 2, i] for i in range(N_2d)] # -\mu \sum_j n_{j \sigma}I
static_free = [
["+-|", hop_left], # up hops left
["|+-", hop_left], # down hops left
["-+|", hop_right], # up hops right
["|-+", hop_right], # down hops right
]
# Interaction
interact = [[U, i, i]
for i in range(N_2d)] # U/2 \sm_j n_{j,up} n_{j,down}
mass1 = [[(-1)**(i % 2 + math.floor(i / 2)) * 1e-5, i]
for i in range(N_2d)]
if U <= 0:
mass2 = [[(-1)**(i % 2 + math.floor(i / 2)) * 1e-5, i]
for i in range(N_2d)]
else:
mass2 = [[-1 * (-1)**(i % 2 + math.floor(i / 2)) * 1e-5, i]
for i in range(N_2d)]
static_int = [
["n|n", interact], # up-down interaction
["n|", pot], # up on-site potention
["|n", pot], # down on-site potent ion
]
# Proper slater state
slater_free = [
["+-|", hop_left], # up hops left
["|+-", hop_left], # down hops left
["-+|", hop_right], # up hops right
["|-+", hop_right], # down hops right
["n|", mass1],
["|n", mass2],
]
num = [[1, i] for i in range(N_2d)]
particle_number = [
["n|", num],
["|n", num],
]
particle_up = [
["n|", num],
]
tJ = [["n|n", coupling_1], ["n|nn", coupling_2], ["nn|n", coupling_3],
["nn|nn", coupling_4], ["+-|+-", coupling_5]]
return basis, static_free, static_int, slater_free, particle_number, particle_up, tJ
# J=1.0 # hopping matrix element U=2.0 # onsite interaction mu=0.5 # chemical potential # ###### setting up user-defined symmetry transformations for 2d lattice ###### s = np.arange(N_2d) # sites [0,1,2,....] x = s%Lx # x positions for sites y = s//Lx # y positions for sites T_x = (x+1)%Lx + Lx*y # translation along x-direction T_y = x +Lx*((y+1)%Ly) # translation along y-direction P_x = x + Lx*(Ly-y-1) # reflection about x-axis P_y = (Lx-x-1) + Lx*y # reflection about y-axis # ###### setting up bases ###### basis_2d=spinful_fermion_basis_general(N_2d,kxblock=(T_x,0),kyblock=(T_y,0),pxblock=(P_x,0),pyblock=(P_y,0)) # ###### setting up hamiltonian ###### # setting up site-coupling lists hopping_left=[[-J,i,T_x[i]] for i in range(N_2d)] + [[-J,i,T_y[i]] for i in range(N_2d)] hopping_right=[[+J,i,T_x[i]] for i in range(N_2d)] + [[+J,i,T_y[i]] for i in range(N_2d)] potential=[[-mu,i] for i in range(N_2d)] interaction=[[U,i,T_x[i]] for i in range(N_2d)] + [[U,i,T_y[i]] for i in range(N_2d)] # static=[["+-|",hopping_left], # spin up hops to left ["-+|",hopping_right], # spin up hops to right ["|+-",hopping_left], # spin down hopes to left ["|-+",hopping_right], # spin up hops to right ["n|",potential], # onsite potenial, spin up ["|n",potential], # onsite potential, spin down ["n|n",interaction]] # spin up-spin down interaction
N_2d = Lx*Ly # number of sites for spin 1 # J=1.0 # hopping matrix element U=2.0 # onsite interaction mu=0.5 # chemical potential # ###### setting up user-defined BASIC symmetry transformations for 2d lattice ###### s = np.arange(N_2d) # sites [0,1,2,...,N_2d-1] in simple notation x = s%Lx # x positions for sites y = s//Lx # y positions for sites T_x = (x+1)%Lx + Lx*y # translation along x-direction T_y = x + Lx*((y+1)%Ly) # translation along y-direction S = -(s+1) # fermion spin inversion in the simple case # ###### setting up bases ###### basis_2d=spinful_fermion_basis_general(N_2d,Nf=(2,2),kxblock=(T_x,0),kyblock=(T_y,0),sblock=(S,0)) # ###### setting up hamiltonian ###### # setting up site-coupling lists for simple case hopping_left =[[-J,i,T_x[i]] for i in range(N_2d)] + [[-J,i,T_y[i]] for i in range(N_2d)] hopping_right=[[+J,i,T_x[i]] for i in range(N_2d)] + [[+J,i,T_y[i]] for i in range(N_2d)] potential=[[-mu,i] for i in range(N_2d)] interaction=[[U,i,i] for i in range(N_2d)] # static=[["+-|",hopping_left], # spin up hops to left ["-+|",hopping_right], # spin up hops to right ["|+-",hopping_left], # spin down hopes to left ["|-+",hopping_right], # spin up hops to right ["n|",potential], # onsite potenial, spin up ["|n",potential], # onsite potential, spin down ["n|n",interaction]] # spin up-spin down interaction
1) % Lx + Lx * y # translation along x-direction for one spin species
t_y = x + Lx * (
(y + 1) % Ly) # translation along y-direction for one spin species
# create the spin-up spin-down combined transformations
s = np.arange(
2 * N_2d) # sites [0,1,2,...,N_2d-1,...,2*N_2d-1] in advanced notation
T_x = np.hstack(
(t_x, t_x + N_2d)) # translation along x-direction for both spin species
T_y = np.hstack(
(t_y, t_y + N_2d)) # translation along y-direction for both spin species
S = np.roll(s, N_2d) # fermion spin inversion in the advanced case
#
###### setting up bases ###### (note optional argument simple_symm=False)
basis_2d = spinful_fermion_basis_general(N_2d,
simple_symm=False,
Nf=(2, 2),
kxblock=(T_x, 0),
kyblock=(T_y, 0),
sblock=(S, 0))
#
###### setting up hamiltonian ######
# setting up site-coupling lists for advanced case
hopping_left = [[-J, i, T_x[i]]
for i in range(2 * N_2d)] + [[-J, i, T_y[i]]
for i in range(2 * N_2d)]
hopping_right = [[+J, i, T_x[i]]
for i in range(2 * N_2d)] + [[+J, i, T_y[i]]
for i in range(2 * N_2d)]
potential = [[-mu, i] for i in range(2 * N_2d)]
interaction = [[U, i, i + N_2d] for i in range(N_2d)]
#
static = [
