The Lanczos algorithm and the many-body problem

One of the most powerful methods in large-scale matrix computing is theLanczos

method, which is an iterative scheme suitable for large, especially sparse (most

elements are zero) matrices. The advantage of the Lanczos method is extremely noticeable when we only need a few eigenvalues and eigenvectors, or when the system is extremely large and sparse. Here we just sketch a very basic Lanczos algorithm. More elaborate discussions on various Lanczos methods can be found in several specialized books, for example, Wilkinson (1965), Cullum and Willoughby (1985), and Hackbusch (1994).

Assuming that the matrixHis ann×nreal symmetric matrix, we can tridi- agonalize anm×msubset ofHwith

OTHO=H˜, (5.144)

whereOis ann×mmatrix with itskth column given by

vk=

uk

Nk

, (5.145)

for k=1,2, . . . ,m, where Nk=uT_kuk is the normalization constant and the

vectorsukare generated recursively from an arbitrary vectoru1as

uk+1=Hvk−αkvk−βkvk−1, (5.146)

withβk=H˜k−1k=vTk−1Hvkandαk=H˜kk=vTkHvk. The recursion is started

atβ1=0 withu1being a selected vector. Note thatu1can be a normalized vec- tor with each element generated from a uniform random-number generator, for example. In principle, the vectorsvk, fork=1,2, . . . ,m, form an orthonormal

set, but in practice we still need to carry out the Gram–Schmidt orthogonaliza- tion procedure, at every step of the recursion, to remove the effects of rounding errors. We can show that the eigenvalues of the tridiagonal submatrix ˜Hare the approximations of the ones ofHwith the largest magnitudes. We can use the standard methods discussed earlier to diagonalize ˜Hto obtain its eigenvectors and eigenvalues from ˜Hx˜k=λkx˜k. The eigenvectors ˜xkcan be used to obtain the

approximate eigenvectors ofHwithxkOx˜k.

The approximation is improved if we construct a new initial stateu1from the eigenvectors ˜xkwithk=1,2, . . . ,m, for example,

u1=

m

k=1

5.9 The Lanczos algorithm and the many-body problem 157

and then the recursion is repeated again and again. We can show that this iterative scheme will eventually lead to them eigenvectors of H, corresponding to the eigenvalues with the largest magnitudes. In practice, the selection of the coeffi- cientsckis rather important in order to have a fast and accurate algorithm. Later

in this section we will introduce one of the choices made by Dagotto and Moreo (1985) in the study of the ground state of a quantum many-body system.

We can work out the eigenvalue problem for a specified region of the spectrum ofHwith the introduction of the matrix

G=(H−µI)−1_{. (5.148)}

We can solve Gwith the Lanczos algorithm to obtain the eigenvectors with eigenvalues nearµ. Note that

Gxk=

1

λk−µ

xk (5.149)

if Hxk =λkxk. This is useful if one wants to know about the spectrum of a

particular region.

At the beginning of this chapter, we used a many-body Hamiltonian (the Hubbard model) in Eq. (5.17) to describe the electronic behavior of H+3. It is generally believed that the Hubbard model and its variants can describe the ma- jority of highly correlated quantum systems, for example, transition metals, rare earth compounds, conducting polymers, and oxide superconducting materials. There are good reviews of the Hubbard model in Rasetti (1991). Usually, we want to know the properties of the ground state and the low-lying excited states. For example, if we want to know the ground state and the first excited state of a cluster ofN sites withN0<N electrons, we can solve the problem using the Lanczos method by takingm10 and iterating the result a few times. Note that the number of many-body states increases exponentially with both N0 andN. The iteration converges to the ground state and the low-lying excited state only if they have the largest eigenvalue magnitudes. The ground state and low-lying excited state energies carry the largest magnitudes if the chemical potential of the system is set to be zero. We also have to come up with a construction for the next guess ofu1. We can, for example, use

u(1l+1)= 5

k=1

v(kl). (5.150)

A special choice of the iteration scheme for the ground-state properties is given by Dagotto and Moreo (1985). The (l+1)th iteration ofv1is taken as

v(₁l+1)= √ 1 1+a2 v(₁l)+av(₂l) , (5.151)

whereais determined from the minimization of the expectation value ofHunder v(₁l+1), which gives

wherebis expressed in terms of the expectation values of thelth iteration as b= d3−3d1d2+2d 3 1 2(d2−d12)3/2 (5.153) withd1=vT1Hv1,d2=vT1H2v1, andd3=vT1H3v1. The second vector

v2=

1

d2−d12

(Hv1−d1v1) (5.154)

is also normalized under such a choice ofv1. The advantage of this algorithm is that we only need to store three vectorsv1,Hv1, andH2v1during each iteration. The eigenvalue with the largest magnitude is also given iteratively from

λ1=vT1Hv1+

a

d2−d12

. (5.155)

This algorithm is very efficient for calculating the ground-state properties of many-body quantum systems. For more discussions on the method and its appli- cations, see Dagotto (1994).