# Difference between revisions of "Tutorial:WormAlgorithm"

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## Quantum phase transitions in the Bose-Hubbard model

### The Bose-Hubbard model

The parameter file quantum3/parm5a 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_LIBRARY="../lattices.xml";
LATTICE="square lattice";
L=4;

MODEL_LIBRARY="../models.xml";
MODEL="boson";
U    = 1.0;
mu   = 0.5;
Nmax = 2;

T = 0.1;

SWEEPS=500000;
THERMALIZATION=100000;

{ 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; }
```

Using the following standard sequence of commands you can run the simulation using the quantum worm code and extract the calculated energy from the XML output files

```parameter2xml parm5a
worm -Tmin 10 parm5a.in.xml
extractxmgr stiffnessplot.xml parm5a.task*.out.xml > plot5a.xmgr
xmgrace plot5a.xmgr
```

where the plot is specified in the file stiffnessplot.xml like

```<?xml version="1.0" encoding="UTF-8"?>
<?xml-stylesheet type="text/xsl" href="http://xml.comp-phys.org/2003/4/plot2html.xsl"?>

<plot name="Stiffness versus temperature for the 2D BHM">

<legend show="true"/>
<xaxis label="Temperature"    type="PARAMETER" name="T"/>
<yaxis label="Stiffness"      type="SCALAR_AVERAGE"/>

<set label="rho vs. T"/>

</plot>
```

### Questions

• What is the signature of the phase transition?

### The transition from the Mott insulator to suprafluidity

The parameter file quantum3/parm5b sets up Monte Carlo simulations of the quantum Bose Hubbard model on a two-dimensional square lattice for various system sizes.

``` LATTICE_LIBRARY="../lattices.xml";
LATTICE="square lattice";

MODEL_LIBRARY="../models.xml";
MODEL="boson";
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; }

```

Using the following standard sequence of commands you can run the simulation using the quantum worm code and extract the calculated energy from the XML output files

```parameter2xml parm5b
worm -Tmin 10 parm5b.in.xml
extractxmgr stiffnessplot2.xml parm5a.task*.out.xml > plot5b.xmgr
xmgrace plot5b.xmgr
```

### 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 = 0.0549...
• Why does the Monte Carlo simulation overestimate the critical point of the transition?

(c) 2003-2005 by Simon Trebst and Synge Todo