NCCU /
## IntroductionFinite## Introduction to using the Matrix Product Toolkit## In this tutorial, we will obtain the groundstate of a spin-1/2 Heisenberg chain and calculate some observables.## 1. Constructing the lattice fileThe finite-system toolkit uses two types of file. Firstly the
All of the programs in the toolkit will display a help message when they are run without any arguments. You should try this for all of the tools introduced in these tutorials. For
This will make a file named ## 2. Constructing an initial wavefunctionNow, we need a wavefunction. The finite-system DMRG program does a variational optimization towards the groundstate, but we need an initial wavefunction to start from. We will use a random wavefunction.
This makes a random wavefunction named ## 3. Some basic information about the wavefunctionWe now have everything we need to start the DMRG calculation. However, we will first introduce some simple tools for finding out some information about the wavefunction. The first tool is
which produces Symmetry list is Sz:U(1) State transforms as 0 Number of sites = 40 This is what we already knew; the wavefunction is defined on a 40-site lattice and has U(1) symmetry with good quantum number
This shows the reduced density matrix eigenvalues at each partition of the wavefunction. Can you guess from the form of the density matrix, some clues as to how ## 4. running the DMRG programThe main tool for running DMRG optimizations is
will perform 2 half-sweeps with m=50 states kept using the operator H defined by the spinchain-u1 program. Two sweeps will not be enough to obtain a converged wavefunction, so lets run 4 more sweeps by using the
The DMRG program will show the variance of the energy (H-E)^2, which should be about 1E-6 for this number of states kept. Now lets calculate some observables. ## 4. ObservablesThe main tool for calculating observables is
The z-component of the spin at site 1 (the left-hand edge). This isn't so exciting because reflection symmetry demands that this expectation value is zero. Note that we enclosed the operator expression in double quotes " to prevent the shell from interpretting the brackets and other symbols.
The spin-spin correlation between sites 1 and 2.
The square of the spin-spin correlation between sites 1 and 2. This might be a good time to revisit the ## 5. SymmetriesThe previous calculation used U(1) symmetry of the z-component of the total spin. The actual symmetry group of the Hamiltonian is SU(2). This group is non-abelian, and the representations labelled by a total spin It is also possible to use SU(2) symmetry in the Matrix Product Toolkit. Try repeating the calculation with the model |

Page last modified on August 09, 2010, at 11:25 PM