Tutorial VI: Density Matrix Renormalization Group for a particle in a box (non-interacting DMRG)

Tutorial I: Running and evaluating MC simulations using ALPS
Tutorial II: Classical Monte Carlo Simulations
Tutorial III: Quantum Monte Carlo Simulations (SSE)
Tutorial IV: Quantum Monte Carlo Simulations (worm code)
Tutorial V: Quantum Wang-Landau Monte Carlo Simulations
Tutorial VI: Full Diagonalization of a Quantum Hamiltonian
Tutorial VII: Density Matrix Renormalization Group for a particle in a box (non-interacting DMRG)

Part I: particle in a box

The one-dimensional tight-binding chain

The parameter file dmrg/particle_in_box/parm1 sets up a DMRG simulation of a single particle tight-binding model on a one-dimensional chain with 10 sites and open boundary conditions, thus simulating a quantum-mechanical 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 non-interacting DMRG code by typing

simple_dmrg < parm1

The output file "psi.dat" contains (for the default OUTPUT_LEVEL = 1) the ground-state 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.

Part II: One-dimensional tight-binding chain in an external potential

One-dimensional tight-binding chain with an harmonic trap

The parameter file dmrg/particle_in_box/parm2 sets up a DMRG simulation of a one-dimensional tight-binding 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 next-nearest-neighbour hopping?

What happens if you have only next-nearest-neighbour hopping?

One-dimensional tight-binding chain with a general potential

The parameter file dmrg/particle_in_box/parm3 sets up a DMRG simulation of a one-dimensional tight-binding chain with 20 sites with next-nearest-neighbour hopping t1=0.01 in a zigzag potential.

LATTICE = "next-nearest 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 Fourier-series 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".