README.md

GutzwillerDynamics

Important

This code is in principle complete and ready to use, but there might be glitches, or edge cases where it fails. Feel free to open a github issue if you need support, or if you see any unepxected behavior of the program.

What is this?

This program implements the Gutzwiller variational wave function for the Bose-Hubbard model. The Gutzwiller coefficients are site-dependent (so that, for instance, one can add an additional confining potential) and complex-valued (so that this code can be used both for imaginary- and real-time evolution). Note: time-evolution is implemented with a sequential updates over all sites, as suggested in this work - see discussion below Eq. (2).

How to use it?

This is a python code, which requires the numpy, scipy and cython libraries. You may also need the future library (if you are on Python 2) and matplotlib or gnuplot (for some of the examples).

Before being imported in a python script, the module lib_inhomogeneous_gutzwiller.pyx has to be compiled through the command

$ python setup_cython.py build_ext --inplace

to be issued in the Code folder. After this step, lib_inhomogeneous_gutzwiller can be imported in ordinary python scripts (have a look at the Examples folder).

Elementary tests are available in the Code/Tests folder, and they are performed at each commit - see the current status on https://travis-ci.org/tcompa/GutzwillerDynamics.

Relevant parameters

Here are the relevant parameters of the Gutzwiller ansatz and of the Bose-Hubbard Hamiltonian:

Gutzwiller ansatz for a Bose-Hubbard model.

Parameters:
    D               Lattice dimensionality (allowed values: 1, 2)
    L               Linear size of lattice
    OBC             Boundary conditions (OBC=1 for OBC, OBC=0 for PBC)
    J               Nearest-neighbor hopping parameter (see Hamiltonian)
    U               On-site interaction parameter (see Hamiltonian)
    mu              Chemical potential (see Hamiltonian)
    Vnn             Nearest-neighbor interaction parameter (see Hamiltonian)
    VT              Trap prefactor (see Hamiltonian)
    alphaT          Trap exponent (see Hamiltonian)
    trap_center     Trap center (see Hamiltonian)
    nmax            Cutoff on the local occupation number (state indices go from 0 to nmax)

The Hamiltonian includes several terms.
Here we denote the sum over the neighbors of site i as sum_{j~i}.
    - J * sum_{i} sum_{j~i} b_i^dagger b_j
    + U * sum_{i} n_i * (n_i - 1)
    + sum_{i} n_i * ( -mu + VT * |x_i - trap_center|^alpha )
    + (Vnn/2) sum_{i} sum_{j~i} n_i * n_j

NOTE:
1) In the nearest-neighbor-interaction term, I use (Vnn/2) because each pair of neighbors is counted twice.
2) In the hopping term, I use J (and not J/2) because pairs (i,j) and (j,i) are not equivalent (due to the Hermitian conjugate).