Difference between revisions of "Tutorial:Particle in a box"
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Revision as of 13:23, 8 May 2005
Density Matrix Renormalization Group for a particle in a box (noninteracting DMRG)
Particle in a box
The onedimensional tightbinding chain
The parameter file dmrg/particle_in_box/parm1 sets up a DMRG simulation of a single particle tightbinding model on a onedimensional chain with 10 sites and open boundary conditions, thus simulating a quantummechanical particle in a box.
LATTICE = "chain lattice" LATTICE_LIBRARY = "lattices_dmrg.xml" L = 10 t = 1 V = 0 SWEEPS = 10 WAVEFUNCTION_FILE = "psi.dat" OUTPUT_LEVEL = 1
You can run the simulation using the noninteracting DMRG code by typing
simple_dmrg < parm1
The output file "psi.dat" contains (for the default OUTPUT_LEVEL = 1) the groundstate wavefunction psi(x) obtained in the last sweep. The wavefunction can be plotted e.g. using xmgrace or gnuplot. Observe how the ground state energy changes at the end of every sweep. Increase the system size and adjust the number of sweeps needed to obtain convergence.
Questions
 What is the maximum system size you can calculate within 5 minutes of playing around?
 Depending on the system size: How many sweeps do you need for convergence?
The onedimensional tightbinding chain with nextnearestneighbor hopping
Adjust the parameter file dmrg/particle_in_box/parm1 in order to be able to run a simulation with nextnearestneighbour hopping t1=0.7. In order to do this, you need to change the LATTICE to "nextnearest chain lattice" and specify the additional parameter t1 which represents the strength of the nextnearestneighbour hopping. The "nextnearest chain lattice" describes a onedimensional chain with a unitcell that also has a hopping term to a nextnearestneighbour site (you may want to have a look at the latticelibrary file "lattices_dmrg.xml" to see how this is done). Perform the same runs as for the plain tightbinding chain.
Questions
 How does the convergence behavior change?
 What is now the maximum system size you can reach within 5 minutes? How many sweeps do you need?
Onedimensional tightbinding chain in an external potential
Onedimensional tightbinding chain with an harmonic trap
The parameter file dmrg/particle_in_box/parm2 sets up a DMRG simulation of a onedimensional tightbinding chain with 20 sites and a parabolic external potential.
LATTICE = "chain lattice" LATTICE_LIBRARY = "lattices_dmrg.xml" L = 20 t = 1 V = 0.5 * (x/L  0.5) * (x/L  0.5) SWEEPS = 100 WAVEFUNCTION_FILE = "psi_harmonic_potential.dat" OUTPUT_LEVEL = 1
Questions
 What does the wavefunction now look like?
 What happens if you switch on the nextnearestneighbour hopping?
 What happens if you have only nextnearestneighbour hopping?
Onedimensional tightbinding chain with a general potential
The parameter file dmrg/particle_in_box/parm3 sets up a DMRG simulation of a onedimensional tightbinding chain with 20 sites with nextnearestneighbour hopping t1=0.01 in a zigzag potential.
LATTICE = "nextnearest chain lattice" LATTICE_LIBRARY = "lattices_dmrg.xml" L = 20 t = 1 t1 = 0.01 K = 2*3.1415927*2/L V = cos(K*x) + cos(3*K*x) / 9 + cos(5*K*x) / 25 + cos(7*K*x) / 36  2 SWEEPS = 100 WAVEFUNCTION_FILE = "psi.dat" OUTPUT_LEVEL = 1
It is possible to specify a general (periodic) potential by specifying its Fourierseries expansion. Try different potentials of your own choice and describe the results and the convergence behavior of the DMRG run. Play around with the parameters (system size, strength of the potential, value of the hopping terms). Which of the potentials you tried do you think is the most difficult potential for DMRG? Why?
Questions
 Play around with the parameters (system size, strength of the potential, value of the hopping terms)
 Which of the potentials you tried do you think is the most difficult potential for the DMRG method? Why?
 Bonus: You may redo some of the runs and answer some of the questions for a chain with periodic boundary conditions (i.e. one particle on a ring). This is done by changing the BOUNDARY tag in the lattice library file "lattices_dmrg.xml".
(c) 20032005 by Salvatore Manmana and Ian McCulloch