Some exact excited states in a linear antiferromagnetic spin system

Volume 88A, number 2

PHYSICS LETTERS

22 February 1982

SOME EXACT EXCITED STATES IN A LINEAR ANTIFERROMAGNETIC SPIN SYSTEM W.J. CASPERS and W. MAGNUS’ Twente University of Technology, Enschede, The Netherlands Received 20 October 1981 Revised manuscript received 12 November 1981

Exact expressions are derived for some excited states in a linear quantum spin system for which the exact ground state has been studied in the last decade.

1. Hamiltonian and notation. We consider a linear antiferromagnetic lattice which contains a next nearest neighbour interaction (NNN) with half the strength of the nearest neighbour interaction (NN). The hamiltonian is of the Heisenberg type: —

~,. ~=

i

‘

~=

i

~

I

describes the coupling between the kth and the (k + 1)-st cell, which contains I nn interaction and 2 nnn interactions. 2. Eigenstates and eige,ivalues ofHOk. The eigenstates of Hok are chosen to be simultaneous eigenstates of (S2k 1 +S2k)2 and ~~kl +S~k,and can thus be labeled by the total spin Sk and its z-component Mk. In the two-spin representation of the kth cell,

2N is the number of spins (S 1 = 1/2). Assuming periodic boundary conditions, we put ~ = Si for every lattice site i. For convenience, the chain is divided into N cells of 2 spins, and H can be split up into two parts [1—41:

the spectrum of Hok splits up into a singlet and a triplet, as is shown in table 1.

N HOIiHOk

V~~o>0>1

N H’=~H,~,k+l

and

,

(2)

3. The ground state of H. Consider the product N

10>k.

(5)

® 0>2 ®~® ‘°>N = k1 “

turns out to be not only the ground state of H where

52k —2 [(S2k1+S2k) — ~] (3) 2 3 Hok —— ‘~2k—1 describes the internal interaction in the kth cell, contaming the spin pair (52k_1 S2k)~and —

=

+ =

~2k

(

2k—i

4. Shorthand notations. Let I~k(M)>be a many-

S2k+l

21~S2k—1 S 2k+ 1 2 ~ +~ 2k)

0, but also an eigenstate of H H0 +H’ with the lowest . states of eigenvalue H, corresponding withThere this lowest possible E0 = —3N. are twopossible eigeneigenvalue. The other one is found by translation over one lattice spacing [1—41.

+ S 2k

•~

2k+ 2)~

~~ ~4j

~ 2k+1 +

2k~ 2k+1 +S2k+2)

Instituut voor Theoretische Universiteit Leuven, B-3030 Leuven, Belgium. On leave of absence from

Fysika,

0 03l-9163/82/0000—0000/$ 02.75 © 1982 North-Holland

cell state in which onlyM). the In kth is inway a triplet state (with quantum number thecell same I ~kl(M,M’)> denotes a state with only two cells (k, 1) in a triplet state (MrespM’). In other words, if a particular cell / does not appear in the notation of a many-cell state, it is assumed to be in the cell ground state, i.e. in the 103

Volume 88A, number 2

PHYSICS LEJT[RS

22 lebruary 1982

Table I

1k

~ IO)k=2~2(+ ~)k_I_+)k)a) IM)k

0

0

“okI°~k= ~3~°~k

=

I

I

11ok I

=2~2(i+~~)k+~

1

0

1

1

I

=

a) +_-)k denotes a cell state with spin up and down on sites 2k

—

1. 2k respectively, and so on.

singlet state I0)~.Finally, ISM>kl =

lM>k =

is the Kronecker delta symbol. With the help of these equations, the following set

(lM~1M

~

2 I 1 1SM>I ~kl~Ml, M2)>,

M= --S, ~S + I

S = 0,1,2,

S

—

l,S,

of eigenstates has been constructed for the full harniltonian:

(6)

(1) A triplet (S

=

1) ofeigenstates with energyE=

—3N + 4, containing only states with one excited cell:

describes the addition of the cell spins of the cells k and!, when both cells are in a triplet state. K1M1 1M21 X 1 ISM> is a Clebsch—Gordan coefficient [5] 5. Some excited states. Excited states can be constructed exactly, by taking appropriate linear combinations of some basic states 1CIk(M)> and ISM)kl. In

order to prove that those are exact eigenstates ofH, one has to figure out how H’ acts on the basic states. Using conventional Clebsch—Gordan technology, and applying the Wigner—Eckart theorem [5]for irredu-

cible tensor operators, one finds:

N

I ~~)

=

(~)k I ~k(M)>,

~ ~k=1

N even.

(8)

(2) A singlet (S = 0), with energy E = —3N ÷ 4, containing only states with two excited cells: N

I

~

~

(—l)”~i00>k,k+l ,

N even.

(9)

(3) A triplet (S = 1), with energy E = —3N + 8, containing only states with one or two excited cells:

Hkk+lkl)o>=0, ~

N

/Ikk+1I~k~Ml)>_~llM1>kk+I ‘

L

k=1

~ H~k+lI~k+1(M2)>=~2I1M2>k,k+1 ‘

1k k+1(Ml ,M k+1 V

(_l)k+1IlM>kl]

,

Nodd.

(10)

l~k
2)>

6. Concluding remarks. The presented excited states =

_4~

~

(lF 1

—~

1100>

l~ 1M1 Ill

1M~)

correspond to exact eigenstates of the hamiltonian H

pM~M~ (7)

X K 1—u 1M2 Ill 1M~)l~k, k÷ 1 (M~ , Mi)> +

M’ ~ KiM1 1M2 Ill lM’>(I ~k(M)>

hlk,k+1ISM>k,k+1

=

[S(S

+

1)

4IISM>kk+1

+ V’2öSl II~k(M)>+ I ~‘~k+1 (RI)] .

104

1~k+1(M’))), +

(1), and are characterized by a fixed maximum number of cell excitations. Recently, Shastry and Sutherland [3] investigated, for the same system, extwo system represented singlet pairs citedphases states of of the a different nature. Theybyconsidered the given in

(2k

—

1,2k) and (2!, 21 + 1) respectively, separated

by defects, i.e. single spins. To meet periodic boundary conditions the number of defects should be even. In particular they analyzed states with two defects and

Volume 88A, number 2

PHYSICS LETTERS

solved for these states an approximate secular prob-

22 February 1982

References

lem, resulting in elementary excitations with the same

wave number, energy and spin as those represented by the states I ~ w> and I ~) of our formulas (8) and (9)

[1] P.M. van den Broek, Phys. Lett. 77A (1980) 261. [2j P.M. van den Broek, W.J. Caspers, M.W.M. Willemse,

respectively. This agreement merits further detailed analysis. For further exact results in linear quantum spin

[31 B.S. Shastry (1981) 964. and B. Sutherland, Phys. Rev. Lett. 47

systems we may refer to refs. [6,71

.

Physica 104A (1980) 298.

[41C.K. Majumdar and D.K. Ghosh,J. Phys. C3 (1970) 911; J. Math. Phys. 10 (1969) 1388,1399. [51AR. Edmonds, Angular momentum in quantum mechan-

7. Two-dimensional systems. Using the same method, it is also possible to construct exact eigenstates

ics (Princeton U.P., 1960). [6j J.C. Bonner and M.E. Fisher, Phys. Rev. 135A (1964)

for some two-dimensional versions of the presented hamiltonian. Details wifi be published in a separate paper.

[7] C.J. Thompson, in: Phase transitions and critical phenomena, Vol. 1, eds. C. Domband and M.S. Green (Academic

640.

Press, New York, 1976) p. 425.

The authors are very much indebted to Dr. F.W. Wiegel for critical reading of the manuscript.

105