Computational Physics

11.7 The density matrix renormalisation group method 361

S E

lll+ 1 l+ 2 + 3

Figure 11.5. Schematic representation of the DMRG procedure. A spin is added

to the ‘old’ system S, which before this addition represented a block oflspins. An

environment E is created by reflecting the new S (including the extra spin). Both

states are now perfectly known.

over the environment. We can, however, use the information that we have generated
on S to create anartificialenvironment E of which all is known. More specifically,
we take for E thereflectionof system S as shown inFigure 11.5. As stated above,
we have a basis|m,sl+ 1 〉for S. Note that the two indices together belong to S; they
can together be viewed as the state|σ〉of our general discussion of the density
matrix above. The reflected system E has a basis|sl+ 2 ,n〉, where we have reversed
the indices (and replacedmbyn) to represent the structure of E. The combined
system U, described by the basis|m,sl+ 1 ,sl+ 2 ,n〉should be in the ground state. If
we can find this state, then we can trace out the degrees of freedom of E in order to
find the density matrixρSdescribing S. Remember the aim was to find theMmost
representative basis states of system S, whose Hilbert space has dimensionM·MS.
We should therefore find theMstates which best represent the density matrixρS.
For this we use the idea behind the singular value decomposition, well known from
numerical linear algebra [27]: we simply take theMeigenstates ofρSwith the
highest eigenvalues (all eigenvalues of the density matrix are positive). In practice,
these eigenvalues rapidly decrease so that the errors made in this procedure are very
small. The new density matrix is then given as a truncated expansion

ρS(M)=

1

∑M

m= 1 λm

∑M

m= 1

λm|m〉〈m| (11.59)

where|m〉are the eigenstates ofρS. The states|m〉are the ones which are carried
over to the next stage, analogous to the quantum renormalisation method of the
previous section: the Hamiltonian of S is transformed according to

HS(M)=V†HS(M·MS)V, (11.60)

whereVcontains theM‘most important’ eigenstates ofρS(M·MS). The fact
that this procedure is indeed the best we can follow is addressed in Problems 11.4
and 11.5. The fact that the states kept are derived from an estimate of the density
matrix of the system S is reflected in the namedensity matrix renormalisation group
(DMRG) for this method.