This is a small python library to generate the basis and many-body Hamiltonian for the Fermi-Hubbard model.
The Fermi-Hubbard model we are concerned with here is defined on some lattice is described by the Hamiltonian,
Here the sum runs over the lattice indices and spin components (up and down), and the operator ci annihilates a particle on site i. Here the ni operator—appearing in the definition of the interaction operator—counts the total number of particles on site i.
This library requires numpy and scipy installed on your system.
As an example consider a ring with four sites.
Construct the model from a onsite enegies, hopping amplitudes and interaction energy.
import numpy as np
import lintable as lin
import fermihubbard
# The model parameters:
# The onsite energies (all zero)
H1E = np.zeros((4, 4))
# Hopping on a ring (amplitude t = -1)
H1T = np.zeros((4, 4))
for i in xrange(4):
H1T[i, (i+1) % 4] = -1.
H1T[(i+1) % 4, i] = -1.
# Interaction (her onsite only at U = 2)
H1U = 2*np.eye(4)
# Construc the model
m = fermihubbard.Model(H1E, H1T, H1U)Because the Fermi-Hubbard Hamiltonian commutes with the total number operator, we can investigate each particle number sector separately,
# Construct a specific chargestate with eight electrons:
ns4 = m.numbersector(4)Assuming that our Fermi-Hubbard model also commutes with the spin component along e.g. the z-direction, we can investigate each of the z-projections separately,
# Consider the sector with net zero spin in the z direction, meaning that
# 4 electrons = 2 spin-up electrons + 2 spin-down electrons
s0 = ns4.szsector(0)Our basis is managed in a clever way referred to informally as Lin tables. You do not need to consider this here, but let us look at the many-body basis used for spin-up electrons:
# Let us then take a look at the basis. Generate the Lin-table:
lita = lin.Table(4, 2, 2)
# Print the basis
print('Spin-up basis:')
print(lita.basisu)
# Spin-up basis:
# [[0 0 1 1]
# [0 1 0 1]
# [0 1 1 0]
# [1 0 0 1]
# [1 0 1 0]
# [1 1 0 0]]In our simple case the spin-down many-body basis looks exactly the same. We are now ready to look at the many-body Hamiltonian:
# Compute the Hamiltonian
(HTu, HTd, HUd) = s0.hamiltonian
# Print the hopping Hamiltonian for the spin-up electronic systems
print('Spin-up hopping:')
print(HTu)
# Spin-up hopping:
# (0, 1) -1.0
# (0, 4) 1.0
# (1, 0) -1.0
# (1, 2) -1.0
# (1, 3) -1.0
# (1, 5) 1.0
# (2, 1) -1.0
# (2, 4) -1.0
# (3, 1) -1.0
# (3, 4) -1.0
# (4, 0) 1.0
# (4, 2) -1.0
# (4, 3) -1.0
# (4, 5) -1.0
# (5, 1) 1.0
# (5, 4) -1.0We can explicitly construct the many-body Hamiltonian from these different components,
import scipy.sparse as sparse
# The Total hamiltonian can be generated from these parts.
Nu, Nd = lita.Ns
# The interaction part in sparse format
HU = sparse.coo_matrix((HUd, (np.arange(Nu*Nd), np.arange(Nu*Nd))), shape=(Nu*Nd, Nu*Nd)).tocsr()
# The kronecker product of the two hopping sectors.
HT = sparse.kron(np.eye(Nd), HTu, 'dok') + sparse.kron(HTd, np.eye(Nu), 'dok')
print('The total Hamiltonian:')
print(HU + HT)
# The total Hamiltonian:
# (0, 0) 2.0
# (0, 1) -1.0
# (0, 4) 1.0
# (0, 6) -1.0
# (0, 24) 1.0
# (1, 0) -1.0
# (1, 2) -1.0
# (1, 3) -1.0
# (1, 5) 1.0
# (1, 7) -1.0
# : :
# (33, 34) -1.0
# (34, 10) 1.0
# (34, 28) -1.0
# (34, 30) 1.0
# (34, 32) -1.0
# (34, 33) -1.0
# (34, 35) -1.0
# (35, 11) 1.0
# (35, 29) -1.0
# (35, 31) 1.0
# (35, 34) -1.0
# (35, 35) 2.0Also check out example.py. Where you can run and modify the above example code. The library code can easily be extended for your own pleasure. Remember though that it is released under a MIT license!