Tutorial:WormAlgorithm

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Contents

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Quantum phase transitions in the Bose-Hubbard model

The following examples can be found in tutorial/quantum3 in the applications source directory.

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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>
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Questions

  • What is the signature of the phase transition?


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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
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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?


© 2003-2005 by Simon Trebst and Synge Todo

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