DWA-02 Density Profile

Density profile

Attention: this implementation of the directed worm algorithm is deprecated and will be removed in a future version of ALPS. It should currently only be used for Bose-Hubbard models with on-site interactions only.

As a second example of the directed worm algorithm QMC code, we will study the density profile of the Bose-Hubbard model subject to an harmonic trap.

Column integrated density

In this subsection, we compute the density profile of a superfluid in a harmonic trap. To visualize, we will sum the density over the third direction.

Preparing and running the simulation from the command line

We create the parameter file parm2a to set up a Monte Carlo simulation of a $21^3$ optical lattice trap:

LATTICE="inhomogeneous simple cubic lattice"
L=21

MODEL='boson Hubbard"
Nmax=5

t=1.
U=8.11
mu="4.05 - (0.2*(x-(L-1)/2.)*(x-(L-1)/2.) + 0.2*(y-(L-1)/2.)*(y-(L-1)/2.) + 0.2*(z-(L-1)/2.)*(z-(L-1)/2.))"

THERMALIZATION=2000
SWEEPS=20000
SKIP=10

MEASURE[Local Density]=1

{ T=1. }

We load the local density measurements from all output files starting with parm2a.

import pyalps
import pyalps.plot as aplt;
data = pyalps.loadMeasurements(pyalps.getResultFiles(prefix='parm2a'), 'Local Density');

and visualize the column integrated density:

aplt.plot3D(data, centeredAtOrigin=True)

Cross section density

We want to observe a Mott plateau in the trapand therefore create the parameter file parm2b to set up a Monte Carlo simulation of the Bose-Hubbard model with a large interaction strength:

LATTICE="inhomogeneous simple cubic lattice"
L=21

MODEL="boson Hubbard"
Nmax=5

t=1.
U=60.
mu="30. - (0.2*(x-(L-1)/2.)*(x-(L-1)/2.) + 0.2*(y-(L-1)/2.)*(y-(L-1)/2.) + 0.2*(z-(L-1)/2.)*(z-(L-1)/2.))"

THERMALIZATION=100000
SWEEPS=2000000
SKIP=1000

MEASURE[Local Density]=1

{ T=1. }

We run the same code as last time on parm2b to prepare the plots, except this time, we want to visualize the cross-section density at the center. Therefore, we pass layer="center" to aplt.plot3D.

import pyalps
import pyalps.plot as aplt;
data = pyalps.loadMeasurements(pyalps.getResultFiles(prefix='parm2b'), 'Local Density');
aplt.plot3D(data, centeredAtOrigin=True, layer="center")

By rotating the figure, a clear Mott plateau can be seen in the center of the trap.

Contributors

• Matthias Troyer
• Ping Nang Ma

Revisions

• Lode Pollet