bosehubbard

This is a lite python library to generate the basis and many-body Hamiltonian for the Bose-Hubbard model.

The Bose-Hubbard model on some lattice is described by the Hamiltonian,

Here the sum runs over the lattice indices, and as is normal the operator b(i) annihilates a particle on site i.

This library requires numpy and scipy installed on your system.

How to use it?

As an example consider a ring with 8 sites.

Construct the model from a list of onsite-energies omegas, a list of links and hopping strengths, and the interaction energy U.

import bosehubbard

# The model parameters
omegas = [0]*8
links = [[i, (i+1) % 8, .1] for i in xrange(8)]
U = 2

# Construct the model
m = bosehubbard.Model(omegas, links, U)

# Look at the single-particle hopping hamiltonian
print(m.hopping)

#  [[ 0.   0.1  0.   0.   0.   0.   0.   0.1]
#   [ 0.1  0.   0.1  0.   0.   0.   0.   0. ]
#   [ 0.   0.1  0.   0.1  0.   0.   0.   0. ]
#   [ 0.   0.   0.1  0.   0.1  0.   0.   0. ]
#   [ 0.   0.   0.   0.1  0.   0.1  0.   0. ]
#   [ 0.   0.   0.   0.   0.1  0.   0.1  0. ]
#   [ 0.   0.   0.   0.   0.   0.1  0.   0.1]
#   [ 0.1  0.   0.   0.   0.   0.   0.1  0. ]]

Because the Bose-Hubbard Hamiltonian commutes with the total number operator, we can investigate each particle number sector separately,

# Investigate the model with two bosons in it.
m2 = m.numbersector(2)

# Construct the many-body Hamiltonian (in sparse format)
H = m2.hamiltonian

print(H)

#	( 0,  0)  2.0
#	( 0,  1)  0.141421356237
#	( 0,  7)  0.141421356237
#	( 1,  0)  0.141421356237
#	( 1,  2)  0.1
#	( 1,  8)  0.141421356237
#	( 1, 14)  0.1
#	  :	  :
#	(34,  6)  0.1
#	(34, 32)  0.1
#	(34, 33)  0.141421356237
#	(34, 35)  0.141421356237
#	(35,  7)  0.141421356237
#	(35, 34)  0.141421356237
#	(35, 35)  2.0

One can access the many-body basis in the two boson sector by looking at the Basis object,

basis = m2.basis
print(basis.vs)

#  [[2 0 0 0 0 0 0 0]
#   [1 1 0 0 0 0 0 0]
#   [1 0 1 0 0 0 0 0]
#   [1 0 0 1 0 0 0 0]
#   [1 0 0 0 1 0 0 0]
#   [1 0 0 0 0 1 0 0]
#   [1 0 0 0 0 0 1 0]
#   [1 0 0 0 0 0 0 1]
#   [0 2 0 0 0 0 0 0]
#          ...
#   [0 0 0 0 0 0 1 1]
#   [0 0 0 0 0 0 0 2]]