Computational Physics

Exercises 365

lll+ 1 l+ 2 + 3

S E

S E

Figure 11.6. The third step in the finite-size DMRG algorithm.

Each time the sweep has reached one of the ends of the chain, the representation of
the corresponding end is exact, as it contains only one (or a few) spins. This exact
representation is responsible for the improvements in successive sweeps. Step 3 is
shown in Figure 11.6. In the particular example of finite-size scaling, the matrices
stored for sizeLcan be used for the next sizeL−2. So part of the work is used
later again. Interestingly, the dynamics of polymer chains, which is governed by a
Master equation, can also be treated within the DMRG approach, and for reptation
and electrophoresis, useful results have been obtained [28, 29].
The major effort in constructing a program for the finite-size chain consists of
storing the relevant operator matricesH,Szetc. for the various sizes in memory.
The algorithm can then be implemented straightforwardly. As a check, you can
verify whether the ground state energy found corresponds to that found for finite
Heisenberg chains in Section 11.5.1.
A major point is whether the DMRG method is applicable to systems in
more than one dimension. This turns out to be difficult. Trials have been made
in two directions. The first is to consider a two-dimensional system as a one-
dimensional chain, ‘wrapped up’ to fill the plane[ 30 ]. The second is to formulate
the Hubbard Hamiltonian ink-space, which renders the hopping term diagonal and
the Coulomb interaction no longer local. Reviews concerning DMRG can be found
inRefs.[ 31 – 33 ].

Exercises

11.1 The Pauli matrixσxis given as

σx=

(

01

10

)

.

Usingσx^2 =1, show that

eβLσx=cosh(βL)+σxsinh(βL).