Al Wd Journal of Magnetism and Magnetic Materials 104-107 (1992) X61-862 North-Holland M F

A ZYE

Real space renormalisation group study of Heisenberg spin chain

Tao Xiang ‘9’ and G.A. Gehring ’

‘I Department of Theoretical Physics, Unkersity of Oxford, UK ’ Department of Physics, Unicersity of Sheffield, Sheffield S3 7RH, UK

A new method for determining the ground state and first excitation state energy of a one-dimensional quantum

Hamiltonian with short range interaction is proposed. This method, based on the Wilson renormalisation group technique, is

illustrated by applying it to the one-dimensional quantum antiferromagnetic systems. Long spin chain calculations are

performed iteratively by diagonalising a smaller system, retaining the lowest rn states, and constructing from these a longer

spin chain basis and corresponding Hamiltonian matrices. The computation can be easily and quickly done for spin S = l/2

and 1 system on a chain up to 60 sites. By extraploting with respect to both M and the chain length L, the ground state

energy of the spin l/2 Heisenberg model is estimated to be -0.44313. It agrees to one part in 10’ to the exact result. The

energy gap between the first excitation state and the ground state for the spin 1 Heisenberg model with both open and

closed boundary conditions is calculated. As m increases, it converges to the exact result rapidly.

The number of basis states for a quantum spin chain of length L is (2s + lIL. This limits the size of chain that can be diagonalised exactly to 30 sites for S = 1, 18 sites for S = 1, 14 sites for S = i, and 12 sites for S = 2 [l]. The result for an infinite chain is ob- tained by extrapolation using finite size scaling. For a longer spin chains Monte Carlo simulations have been employed [2]. However, the accuracy of the results obtained from Monte Carlo calculations is always im- paired by various statistical and systematic errors.

In this paper we investigate the possibility that one might do better by taking a restricted basis set for longer chains by choosing the lowest eigenstates sys- tematically. This method is only feasible provided that the ground state is not highly degenerate - hence it works best for a singlet ground state and is inapplica- ble to a ferromagnetic chain.

We note that we do not assume a broken symmetry; our basic assumption is that there is a significant overlap between the ground state and low lying excited states for a L spin chain and ground state wavefunc- tion for an infinite chain projected on to L spins.

The method is straightforward. Consider a quantum spin chain where each spin has q degrees of freedom (4 = 2s + 1). A cluster of r spins is diagonalised ex- actly: this has qr basis states. We retain m states and a further spin added to the cluster to give qm states. The r + 1 cluster is diagonalised with in the qm basis states and again m states are retained and the procedure is repeated out to a cluster of length L. The ground state energy and gap to the first excited state are given by E(m,L) and AE(m,L). The m states which are re- tained may be the m lowest states for each cluster;

’ Present address: Physics Department, Warwick University, Coventry CV4 7AL, UK.

however we found better convergence if we took the lowest p states of the z component of the total angular momentum up to a value of J where p = m/(2J + 1). For example for the S = 1 chain we had J = 5 and so kept p states of M = 0, k I,. . . I.5. The procedure may be used for both open and closed chains. This method of adding single spin to the chain is different from the method of Kovarik [3] who doubled the chain length at each iteration.

We first consider how to evaluate the ground state energy, lim, _z,m ,yl E(m,L) = E,,, which is an ap- proximation to the ground state energy. In this calcula- tion we use the full Hamiltonian to evaluate the ener- gies of the ground state and first excited state using a restricted basis set. Thus estimated our ground state energy is always too high. We have evaluated the ground state energy of the Heisenberg S = l/2 chain as a check on our method. In fig. 1 we show the ground state energy per bond E(p,L)/(L - 1) as a function of L at p = 10 and its linear extrapolation with respect to l/L (inset). A plot of E(p) = lim,,,E(p,L) is shown in fig. 2. At first sight it is surprising that the function E(p) is not a monotonic function of p. The reason is that although the ground energy keeping p + 1 basis states has involved diagonalising a q(p + 1) dimen- sional matrix compared with the 9p dimensional matrix for the p basis states all the states in the manifold may be slightly different. By extrapolating E(p) with re- spect to p we get E” = -0.44313 which agrees to one part in lo5 to the exact value l/2 - ln2 = -0.443147. We also confirm that the spectrum is gapless.

We now look at the S = 1 antiferromagnetic chain with both Free Boundary Conditions (FBC) and Peri- odic Boundary Conditions (PBC). We find good values for the ground first excited state are radically different from open and closed chains as is well known. We confirm that open chains are gapless. Our result for

0312.8853/92/$05.00 0 1992 - Elsevier Science Publishers B.V. All rights reserved

862

-0.46

- -0.48

3 -0.50

\ ,..-._ -0.52

1

Q CL

-0.54

w -0.56

-0.58

-0.60

T. Xiang, G.A. Gehring / Heisenherg spin chain

h’..: :.. :.:-.: .“k. :...

5 IO 15 25 30 35 20

L

\

-0.44

=

d -0.46 :: A

;” -0.48

-0.50

0.00 0.04 0.08 0.12 0.16

t/L

40

Fig. 1. Spin l/2 Heisenberg model. E( p,L)/(L - I) vs. L for

p = 10. The straight line is the exact ground state energy of

the spin l/2 Heisenberg model at thermodynamic limit. The

inset is a linear extrapolation of E(p,L) with respect to l/L

for the same p.

6 8 IO 12 14 16 I8 20 22 24 26 28 30

P Fig. 2. Spin l/2 Heisenberg model.,E(p) vs. p. The dash line

is an exponential extrapolation of E(p) to p. The solid line is

the exact ground state energy at thermodynamic limit.

1.0

. . 0.9

p=20

+ + p=50 0.8 :

0.7 L 0 0 exact

: *.

7 0.6 ‘n.. PBC

2 0.5 ‘+._

-+-~g.,~:+:.+--+- +..-+S4’:

0.4

y 0.3

0.2

0.1

0.0

4 6 8 10 I2 14 I6 I8 20 22 24 26 28 30

L Fig. 3. Spin 1 Heisenberg model. AE(p,L) vs. I, for p = 20

and SO. The data for the exact result are from ref. [4].

the Haldane gap is shown in fig. 3. We note that we are getting good convergence to the accepted result without applying any additional perturbation theory as Kovarik [3] did.

This new method is time and computer-memory intensive. The amount of computer time involved in our scaling approach is less by a large factor compared with the exact diagonalisations or Monte Carlo calcula- tions.

This work was supported by Basic Research Action 3041 and partly by SERC grant No. GR/E/79798.

References

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Kubo, Nucl. Phys. B 300 (1988) 301. S. Liang, Phys. Rev.

Lett. 64 (1990) 1597.

[3] M.D. Kovarik, Phys. Rev. B 41 (1990) 6889. [4] D.C. Mattis and C.Y. Pan, Phys. Rev. Lett. 61 (1988) 463.

[5] R. Sakai and M. Takahashi, Phys. Rev. B 42 (1990) 1090.

T. Kennedy, J. Phys.: Condens. Matter. 2 (1990) 5737.