Matrix Product Toolkit | NCTS / BoseHubbard

In this tutorial, we will examine a more realistic example, of the Bose-Hubbard model in a 1D harmonic trap.

The Bose-Hubbard model is

Note that the Hamiltonian preserves particle-number symmetry, and as we are working in a finite 1D system it is not spontaneously broken.

We define the trapping potential as

The homogeneous system with no trap has a phase diagram (from Freericks and Monien 1994),

This model can be constructed using the program bosehubbard-spinless-u1. Try a 60-site lattice. It is necessary to choose a cutoff in the number of bosons possible per site. The default is 5 bosons maximum, which is good enough except for small U.

Next, make a random wavefunction with N=60 particles.

The hopping, coulomb and trap terms in the Hamiltonian are implemented as the operators H_J, H_U and H_trap repsectively. We can also get a quartic trap or a simple ramp with H_trap2 and H_ramp.

To see some paramters in the superfluid region, try the Hamiltonian "H_J + 1*H_U + 50*H_trap".

To see some fringes of Mott insulator, try increasing U, to 10 or 20. m=50 states is probably sufficient. It will take a long time to converge because of the non-uniform groundstate! You will probably need up to a few hundred sweeps. You could try starting off with a smaller number of states so that the program runs faster intially, and then increase the number of states towards the end of the calculation.

What is the difference in the local density profile as U is increased?

Try the mp-localexpectation program to obtain quickly local quantities.

Try mp-localexpectation lattice psi N 1 60 | xmgrace -pipe& to get a quick way of viewing the local density.

Is a maximum of 5 bosons sufficient for small U? The lattice file defines projectors onto the particle number states, P_1, P_2, ...., P_N which can be used to calculate the occupations.