Volume 94B, number 2 PHYSICS LETTERS 28 July 1980

THE QUANTUM INVERSE SCATTERING METHOD AND THE JORDAN-WIGNER

TRANSFORMATION FOR THE HEISENBERG-ISING SPIN CHAIN

Michael FOWLER Department o f Physics, University o f Virginia, Charlottesville, VA 22901, USA

Received 31 March 1980

The quantum inverse scattering method for the spin one-half chain is presented in a formulation closely analogous to the standard inverse scattering treatment for the nonlinear Sehrodinger equation. It is shown that for the particular case of the isotropic X Y spin chain, the quantum inverse scattering transformation is equivalent to the Jordan-Wigner transformation.

Recently, Kulish and Sklyanin [1] formulated the quantum inverse scattering method (ISM) for the Heisenberg ferromagnet, and demonstrated that in a

continuum limit it gave the quantum nonlinear Schr6dinger equation. Their work was based in part on Baxter's [2] solution of the eight-vertex model. In this note, we analyze Baxter's algebra from somewhat different point of view, making more explicit the rela- tionship to the familiar ISM representation of scattering matrix elements by series expansions in fields or field operators [3]. Applying this technique to the isotropic spin one-halfXY chain, it is not difficult to see that in this case the quantum inverse scattering transforma-

tion is equivalent to the Jordan-Wigner transformation. To set up the notation, we briefly recapitulate

Baxter's construction of the eight-vertex model trans- fer matrix. The model has a square lattice and directed bonds with an even number pointing into each vertex. The vertex energy depends on the configuration of bonds at that vertex, but is invariant with respect to reversing all four bonds. This implies there are four distinct possible vertex energies, in general. Labelling the north, south, east and west bonds at a vertex by t~',/l, ~', ~. (each variable being + 1 or - 1 ) Baxter wrote the weight at a single vertex as

4

R(U,U'IX, X') =:--~i w ,~/ ,~/ i v.~,, vxx, , (1)

where e 1 , o 2, o 3 are the Pauli matrices, 04 is the unit

matrix, and the wi ' s are simple linear combinations of

the weights e x p ( - e i / k T ) , e i being the vertex energies. The free energy is found by multiplying together one R for each vertex in the system, and summing over all

~'s and/ fs . The transfer matrix is given by summing over all ~.'s in one row - this is accomplished by writing

R as a 2 X 2 matrix, the rows and columns labelled by = +, - ; ~' = +, - , and finding the trace:

Tr T N = T r R 1 . . . R N ,

where 1, . . . , N label the vertices along the row. Explicit- ly,

71 61 T2 62 TN 6N

where the a's, etc., are themselves 2 X 2 matrices with rows and columns labelled by/1 = +, - ; / ~ ' = +, - . For these matrices, a + + + + n = 6n , f3n = 3'n, A N = D N , CN = B N [4].

Expanding out the matrix product on the right- hand side of eq. (2), we find that the elements of T N

are given by

A N = a l a 2 ... a N + aa/36~i"/c~ ... a N + . . . . (3)

Bfv = 6 1 6 2 . . . 6 7 a . . . a N + . . . .

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Volume 94B, number 2 PHYSICS LETTERS 28 July 1980

The elements are sums over all products which have the form of runs of a's, 8's with/3, 7 signalling the changes from c~ to 6 and 8 to a, respectively. The a, 8 runs may have zero length. A N begins and ends with an ~ run, B~¢ begins with a/5 run and ends with an

• + c~ run. Since each ~, etc. is a 2 × 2 matr lX ,AN,B N act on a direct product of N spin one-half spaces. Baxter constructed a one-parameter family of transfer ma- trices TN(O), whose traces commuted with each other and with the spin one-hal fXYZ chain hamiltonian

H = ~ (Jx °x °x+ 1 + Jy oF °Y+ 1 + Jz Oz °z+ 1 ) ' (4) i

for a particular Jx : Jy : Jz related to the allowed sets of wi's in the family. It has recently been shown that the elements of TN(O ) are closely related to creation operators for the generalized Bethe Ansatz excitations for this spin chain [4].

For the Heisenberg-Ising chain, Jx: Jy : Jz = 1 : 1 : A with A = -cos /a , 0 </a < 7r. Following Yang and Yang, we introduce the variable p defined by [5]

[- sink ½ (a ~ i/a)7,

p = - i in sinh ½ (a + i/a)-] (5)

1". v = -~1(0l + 17 0 , c~ is real.

To exhibit the structural similarity between the products (3) and the ISM as developed by Thacker and Wilkinson [3], we let N-+ oo and denote by [g2) the (unphysical) vacuum state having all spins down.

The matrices c~, 13, 7, 8, are

a = ( ~ lp 0),1

1 0

/a ]1/2(0 1) (6) 7 = e i p / 2 [ 2 ( c o s p + c o s ) 0 0 '

> In the above, phase factors e -+ ip/2 have been removed from each matrix so that each term in A N is well de- fined when applied to a state tending to I~ ) at infinity. The leading term of B~v is just the Fourier transform of o +. We can write

A = - - + - - o - ... o+ - - + e t c . (7)

B + = ... cr + - - + ... a + - - a - ... o + - - + e tc . ,

where - - denotes a run of a 's . . . . a run of 8's. These series are very similar to those given by Thacker and Wilkinson [3] for the quantum nonlinear Schr6dinger equation.

In the quantum ISM, the creation operator for a Bethe Ansatz excitation of momentum p is [4]

R+ (p) = B+ ( p ) A - l ( p ) . (8)

For the isotropic X Y chain, the BA excitations are free fermions given by the Jordan-Wigner transformation

S+(p) = ~ eipno+ n I-I<n ( - s z , ) . (9)

To establish the equivalence of R +, S + we need to prove that

S + ( p ) A ( p ) = B + ( p ) . (10)

Here S + raises a single spin. From eqs. (7) we see that A, B + can raise and lower many spins. The validity of eq. (10) can be proved with straightforward but tedious algebra. The main nontrivial point is that one must in- clude terms in which S + raises a spin lowered by A. Thus different orders in the expansion of A contribute to the same term in B +. There are extensive cancella- tions between terms, which occur differently depending on the particular configuration of up spins in the state acted upon. For nonzero but small A, these cancellations almost occur - evidently, R + has only small multiparti- cle components. These presumably correspond, at least in leading order, to the weak coupling bosonization results [6].

Since we have an unbounded spin chain, we cannot extend this method to derive, for example, the distri- bution of momenta in the physical ground state, which follows from periodic boundary conditions in the usual BA analysis. However, the algebraic structure o f the operators is simpler in the unbounded case. (See, for example, Lieb et al. [7] for the X Y chain, Fadeev [4] for a general discussion.) A possible way around this is sug- gested by the work of Creamer et al. [8] on a closely related problem - the equivalence of the Jo rdan - Wigner transformation to ISM for the non-linear. Schr6dinger equation in the infinitely repulsive limit. In their work, the momentum distribution derived with periodic boundary conditions is inserted into a quantum

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Volume 94B, number 2 PHYSICS LETTERS 28 July 1980

Gelfand-Levitan equation constructed on the infinite line.

Appendix . In this appendix we give algebraic details of the argument leading to eq. (10) of the text.

We define B + as the series

B+(p) = ... a + - + ... a + - a - ... o + - + etc.

where ... represents a product of matrices (1 0 eip), - - represents a product of matrices ~0 zp 0), a+ here represents eip/2(2 cosp) 1/2 ~ 1) and there is an over- . O U

all multiplicative factor e lp/2 (2 cos p)- l /2 ,

A+(p) = + - - o - ... o + - - + e t c .

S+(P) = ~-'n eipn a+ Iln' < n ( - ° n ' ) , where the o's

here are just the Pauli matrices.

Theorem: B+(p) = S+(p)A(p) . We indicate the method of proof of this theorem by considering how

the two operators behave on successively more com-

plicated states. First we take the all-spins-down state. A then gives

unity, S + a linear superposition of terms, the nth having the nth spin raised and a multiplying factor e ipn. Only the first term of the B + series gives a non-zero result on the all-spin-down state, and it is easy to see that B + = S+A on this state.

We consider now B +, S+A operating on the state having a single up spin, which is at site m. The only non-zero matrix elements are to states with two up spins. We check those to states having the up spins at

m, n. First, take n < m . T h e n A , B + both give extra phase factors e ip, and otherwise the argument is the same as for the vacuum state. Supposing now m < n, A gives a phase factor e ip , but B + gives e -ip , and S + gives ( -1) . However, another term in the A series con- tributes to this matrix element - beginning with a single up spin at m, the second term in A can lower m, raise n, then S + raises m. The appropriate multiplying factor for this sequence is

(2e ip cos p) eiP (n - m - 1)eipm

Adding this to the contribution from the first term in the A series leaves e ip(n- 1), the same amplitude as

B + between these states. Consider now the final state having up spins at

only n l , n 2 where n 1 4: m, n 2 g: m. I f m is between n l , n 2 the argument is straightforward. Otherwise, we

notice from the structure o f B + above that the matrix element must be zero. That it is zero is not so evident from the S+A series. For m < n, < n~, the oZa + from the second term of A , together with an2 from

S +, give a matrix element

(2e ip cos p)e ip (n l - m - 1) eipn= (_ 1).

This is exactly cancelled by the term where A contri-

butes a m an2, + S + gives a +nl - in which case ther is no ( - 1 ) . Notice, though, that this cancellation does not occur if the initial state has a second up spin at m', between n l , n2, for in this case another ( - 1 ) ap- pears in S + and the two terms add. However, in this case we have to go to the next term in the A series

-oTn an +, am , an+2, with S + giving O~n,. It is straight- forward to check that this term cancels the sum of the two just discussed. Without the initial up spin at m', of course, this term gives zero.

In fact, going to higher order terms or more com- plicated initial states, the arguments are nor more in- volved than those given above, for two basic reasons. First, S + only raises a single spin, so a particular term in the B + series only corresponds to two terms in theA series. Second, the structure ofB + ensures that the only non-zero matrix elements are between states in which the changed spins alternate in sign going along the chain - and we can check as above that this holds true for S+A as well. Hence the proof for the general case can be constructed in the manner described above

I wish to thank V.J. Emery for his hospitality at Brookhaven National Laboratory where part of this work was done.

References

[1] P.P. Kulish and E.K. Sklyanin, Phys. Lett. 70A (1979) 461.

[2] R.J. Baxter, Ann. Phys. 70 (1972) 193,323. [3] H.B. Thacker and D. Wilkinson, Phys. Rev. D19 (1979)

3660. [4] L.D. Fadeev, Steklov Mathematical Institute Preprint P-

2-79 (Leningrad, 1979). [5] C.N. Yang and C.P. Yang, Phys. Rev. 150 (1966) 321. [6] A. Luther and I. Peschel, Phys. Rev. B12 (1975) 3908. [7] E. Lieb, T. Schultz and D. Mattis, Ann. Phys. 16 (1961)

407. [8] D.B. Creamer, H.B. Thacker and D. Wilkinson, Fermilab

preprint 80/17 (1980).

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