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Monte Carlo Simulations
MC-05 Bosons

MC-05 Bosons

Quantum phase transitions in the Bose-Hubbard model

As an example of the worm QMC code, we will study a quantum phase transition in the Bose-Hubbard mode.

Superfluid density in the Bose Hubbard model

Preparing and running the simulation from the command line

The parameter file parm5a with the following contents sets up Monte Carlo simulations of the quantum Bose Hubbard model on a square lattice with 4x4 sites for a couple of hopping parameters (t=0.01, 0.02, …, 0.1) using the worm code.

LATTICE="square lattice";
L=4;
MODEL="boson Hubbard";
NONLOCAL=0;
U    = 1.0;
mu   = 0.5;
Nmax = 2;
T = 0.1;
SWEEPS=500000;
THERMALIZATION=10000;
{ t=0.01; }
{ t=0.02; }
{ t=0.03; }
{ t=0.04; }
{ t=0.05; }
{ t=0.06; }
{ t=0.07; }
{ t=0.08; }
{ t=0.09; }
{ t=0.1; }

The corresponding Python script is found at tutorial5a.py.

Evaluating the simulation and preparing plots using Python

To load the results and prepare plots we load the results from the output files and collect the magntization density as a function of magnetic field from all output files starting with parm5a.

data = pyalps.loadMeasurements(pyalps.getResultFiles(prefix='parm5a'),'Stiffness')
magnetization = pyalps.collectXY(data,x='h',y='Stiffness')

To make plots we call the pyalps.plot.plot and then set some nice labels, a title, and a range of y-values:

plt.figure()
pyalps.plot.plot(rhos)
plt.xlabel('Hopping $t/U$')
plt.ylabel('Superfluid density $\\rho _s$')
plt.show()

Questions

What is the signature of the phase transition?

The transition from the Mott insulator to the superfluid

We next want to pin down the location of the phase transition more accurately. For this we simulate a two-dimensional square lattice for various system sizes and look for a crossing of the quantity $\rho_s L$.

Preparing and running the simulation from the command line

In the parameter file parm5b we focus on the region around the critical point for three system sizes L=4, 6, and 8:

LATTICE="square lattice";
MODEL="boson Hubbard";
NONLOCAL=0;
U    = 1.0;
mu   = 0.5;
Nmax = 2;
T = 0.05;
SWEEPS=600000;
THERMALIZATION=150000;
{ L=4; t=0.045; }
{ L=4; t=0.05; }
{ L=4; t=0.0525; }
{ L=4; t=0.055; }
{ L=4; t=0.0575; }
{ L=4; t=0.06; }
{ L=4; t=0.065; }
{ L=6; t=0.045; }
{ L=6; t=0.05; }
{ L=6; t=0.0525; }
{ L=6; t=0.055; }
{ L=6; t=0.0575; }
{ L=6; t=0.06; }
{ L=6; t=0.065; }
{ L=8; t=0.045; }
{ L=8; t=0.05; }
{ L=8; t=0.0525; }
{ L=8; t=0.055; }
{ L=8; t=0.0575; }
{ L=8; t=0.06; }
{ L=8; t=0.065; }

The corresponding Python script is found at tutorial5b.py.

Evaluating the simulation using Python

We first load the superfluid density (stiffness) into three different data sets, one for each system size L:

data = pyalps.loadMeasurements(pyalps.getResultFiles(prefix='parm5b'),'Stiffness')
rhos = pyalps.collectXY(data,x='t',y='Stiffness',foreach=['L'])

Next we multiply each data set by the size L:

for s in rhos:
s.y = s.y * float(s.props['L'])

And finally we make a plot in the usual way:

plt.figure()
pyalps.pyplot.plot(rhos)
plt.xlabel('Hopping $t/U$')
plt.ylabel('$\\rho _sL$')
plt.legend()
plt.title('Scaling plot for Bose-Hubbard model')
plt.show()

Note the legend and labels that are nicely set up.

Questions

  • How can you determine the location of the quantum phase transition in the thermodynamic limit?
  • Tip: Multiply your results for the superfluid stiffness by the respective linear system size L.
  • Compare your result to the exact result (t/U)c = 0.05974…
  • Why does the Monte Carlo simulation overestimate the critical point of the transition?

Contributors

  • Simon Trebst
  • Synge Todo
  • Matthias Troye
  • Abdullah “Amina” Al-Harbi
MC-04 MeasurementsMC-06 QWL