Computational Physics

368 Transfer matrix and diagonalisation of spin chains

(b) Show thatSz=

∑L

l= 1 Sz,lcommutes withH. Also show that

S^2 =

(L

∑

l= 1

Sl

) 2

commutes withH.

11.4 In the DMRG, we take the set of eigenvectors|m〉with the highest eigenvalues ofρS
as representatives of the system S. To show that this is indeed the best procedure, we
consider the representation of an arbitrary state|ψ〉, of the universe U, which can be
expanded as
|ψ〉=

∑

mn

Cmn|mn〉,

where, as usual,mlabels a basis vector of S andnone of E.

We now replace the basis|m〉of S by a smaller basis|uα〉,α=1,...,Mwhere

we choose the orthonormal basis vectors|uα〉such that the state|ψ〉can be

optimally represented by them. This means that the norm of the difference between

the exact and the expanded state

∣∣

∣∣

∣

∑

mn

Cmn|mn〉−

∑

αn

Dαn|uαn〉

∣∣

∣∣

∣

2

is minimal.

We can expand the basis vectors|uα〉in terms of the basis|m〉:

|uα〉=

∑

m

uαm|m〉.

(a) Formulate the error (i.e. the norm of the difference) in terms of the coefficients

Dαnanduαm. Minimise this error with respect to both theDαnand theuαmand

show that this leads to the equation

−

∑

m

Cnmuαm+

∑

mα′

Dαnuαmuα′m= 0

and

−

∑

n

CmnDαn+

∑

nα′

DαnDα′nuα′m=0.

(b) Use the orthonormality of the|uα〉to infer from the first equation:

∑

m

Cmnuαm=Dαn.

(c) Substitute this into the second equation to find

∑

nm′

CmnCm′nuαm′=

∑

nm′m′′

Cm′nCm′′nuαm′uα′m′′uα′m.

Using

ρmmS ′=

∑

n

CmnCm′n,