Chemistry Reference
In-Depth Information
eigenstates of a large many-particle system iteratively from the low-energy
states of smaller blocks, using the density matrix to decide which states to
keep and which to discard. In one space dimension, this procedure works
very well and gives accuracies comparable to exact diagonalization for
much larger system sizes. Since its introduction, the DMRG has quickly
become a method of choice for many one-dimensional quantum many-particle
problems including various spin chains and spin ladders with and without
frustration. Electronic systems such as Hubbard chains and Hubbard ladders
can be studied efficiently as well because the DMRG is free of the fermionic
sign problem. An extensive review of the DMRG method and its applications
can be found in Ref. 125.
In the context of our interest in quantum phase transitions, however, we
note that the accuracy of the DMRG method suffers greatly in the vicinity of
quantum critical points. This was shown explicitly in two studies of the one-
dimensional Ising model in a transverse field, as given by the Hamiltonian of
Eq. [14].
126,127
Legaza and Fath
127
studied chains of up to 300 sites and found
that the relative error of the ground-state energy at the quantum critical point
is several orders of magnitude larger than off criticality. (This is caused by the
fact that the quantum critical system is gapless; it thus has many low-energy
excitations that must be retained in the procedure.) Andersson, Boman and
O
ยจ
stlund
128
studied the behavior of the correlation function in a DMRG study
of gapless free fermions (or equivalently, a spin-
2
XX model). They found that
the DMRG result reproduces the correct power law at small distances, but it
always drops exponentially at large distances. The fake correlation length
grows as
M
1:3
with the number of states
M
retained in each DMRG step.
When studying a critical point, this fake correlation length should be larger
than the physical correlation length, which increases greatly the numerical
effort. While the standard DMRG method does not work very well in dimen-
sions larger than one, an interesting generalization
129,130
has arisen recently in
the quantum information community. It is based on so-called projected
entangled-pair states (PEPS). First applications to quantum many-particle
systems look promising (e.g., Ref. 131 reports a study of bosons in a two-
dimensional optical lattice), but the true power of the method has not been
explored fully.
Another useful technique for studying one-dimensional spin systems
involves mapping the system onto
noninteracting
fermions. This method
was developed by Lieb, Schultz and Mattis
76
in the 1960's and was applied
to the nonrandom transverse-field Ising model of Eq. [14] by Katsura
132
and
Pfeuty.
133
The resulting fermionic Hamiltonian can be solved analytically by
Fourier transformation in the nonrandom case. Young and Rieger
74,75
applied
the same method to the random transverse-field Ising chain, Eq. [30]. The
mapping onto fermions now results in a disordered system; the fermionic
Hamiltonian must therefore be diagonalized numerically. However, since
one is simulating a
noninteracting
system, the numerical effort is still much
