Long Time Behaviour of an Integrable Heisenberg Spin Chain

Long Time Behaviour of an Integrable Heisenberg Spin ChainAuthor(s): Spyridon KamvissisSource: Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 452, No. 1955 (Dec.8, 1996), pp. 2629-2638Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/52863 .

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Long time behaviour of an integrable Heisenberg spin chain

BY SPYRIDON KAMVISSISt

Ecole Normale Superieure, 61 avenue du President Wilson, 94235 cachan, Paris, France

We consider an integrable modification of the standard Heisenberg spin chain which is known to be a discretization of the continuous Heisenberg magnet model and indeed equivalent to a discretization of the nonlinear Schrodinger equation. In particular we investigate the long time behaviour of the system; we are mostly interested in the 'non-soliton' part, thus completing the discussion in Papanicolaou (1987) of the solitonic solutions.

Our method makes use of the related inverse scattering problem and its formulation as a Riemann-Hilbert factorization problem. We show that the problem can be reduced to the analogous one for the Toda lattice and provide explicit asymptotic formulae.

1. Introduction

In this paper we consider the following system suggested by Haldane (1982) (and also Papanicolaou 1987; Fadeev & Takhtajan 1987; Nijhoff et al. 1982; Ishimori 1982):

1+ Sn Sn+l

Here Sn(t) = (S (t), S2(t), S3(t)) is a three-dimensional vector function of time and the dot signifies differentiation with respect to time. More specifically we consider the long time asymptotic behaviour of (1.1) under initial data such that S(0), S2(0) and S3 (O)-1 decay rapidly (faster than polynomially will do) as n -* ?oo. In particular, we are mostly interested in the effect of the continuous spectrum, as opposed to that of the discrete spectrum which corresponds to solitons (see Papanicolaou 1987).

It is well known (see Papanicolaou 1987; Fadeev & Takhtajan 1987; Nijhoff et al. 1982; Ishimori 1982) that the above system is completely integrable and hence is easier to handle than the standard Heisenberg chain:

Sn - Jn -Jn-1i Jn = 2Sn x Sn+l, Sn * Sn = 1 (1.2)

More precisely, (1.1) can be treated using the machinery of inverse scattering the-

ory. In Papanicolaou (1987) that theory is introduced and used to study the pure soliton solutions of (1.1). In the present work, we actually ignore the contribution of the 'soliton' part to the solution and we concentrate on the 'background radiation'

t Present address: Institut Henri Poincar6, 11 rue Pierre et Marie Curie, 75005 Paris, France.

Proc. R. Soc. Lond. A (1996) 452, 2629-2638 ) 1996 The Royal Society Printed in Great Britain 2629 TEX Paper



part. In other words, we make the assumption that the linear 'Lax operator' associ- ated to (1.1) has no bound states. However, it is very easy to add solitons and we will indicate how to do this at the end of ? 3.

Our treatment relies on the fact that the relevant inverse scattering problem can be restated as a Riemann-Hilbert factorization problem. It is now established that this is indeed the most fruitful formulation. As far as the long time behaviour of integrable equations is concerned, there has recently appeared a series of papers attacking such

problems, beginning with the pioneering paper of Deift & Zhou (1993; see also Deift et al. 1993, 1994, 1996; Deift & Zhou 1994; Kamvissis 1993). In this work, we make

heavy use of results stated in Kamvissis (1993), where a similar problem for the Toda lattice is treated.

As in Papanicolaou (1987), we use the convenient stereographic parametrization:

Sn + iS2 n= + = tan(On/2)e'O. (1.3)

The angles On and q)n are the spherical variables parametrizing the unit vector Sn. Auxiliary potentials a, and bn can be defined as follows (see (3.6), (3.10) and

(3.11) of Papanicolaou (1987)):

Qn = (1 + ,n) 1/2 1

) Un = Qn+lQn+2) Un -b* an b

Our main result is expressed in terms of both the auxiliary potentials and the Ons.

Theorem 1.1. (i) In the physically interesting region jn/2tl < 1, we have the uniform estimates

an(t) 1 + O(t-1/2),

(4t2 _ n2)1/4q(t +

O(t),(1.4)

[2t]

(t) (4t2 n2)1/4 qn(t), as t - oo,

where qn(t) is given by formulae (3.9) and (0) of 3. Indeed, for all n, i(t) -? 0, as t -- oo.

(ii) In the region 1n/2tl > 1, we have the uniform estimates

an(t) = 1 + O(t-), bn(t) = O(t-), ) n(t) = O(t-1), (1.5)

for any positive constant 1.

The plan of the rest of this paper is the following. In ?2 we state the inverse scattering problem (in terms of the auxiliary potentials an and bn) and reduce it to a Riemann-Hilbert factorization problem. In ? 3, we reduce that problem to a similar one used in Kamvissis (1993) for the Toda lattice and thus recover the answer. For a more detailed discussion of the inverse scattering problem and the physical motivation of this work one should consult Papanicolaou (1987), to which our work should be considered as the sequel and whose notation we follow.

We finally note that in Fadeev & Takhtajan (1987) the authors show that the Heisenberg spin chain is a discretization of the continuous Heisenberg model (and

Proc. R. Soc. Lond. A (1996)

2630 S. Kamvissis



Integrable Heisenberg spin chains

indeed call it 'lattice isotropic Heisenberg magnet model') and also equivalent to a discretization of the nonlinear Schr6dinger equation. Also, in Ishimori (1982) it is shown that the Heisenberg spin chain is gauge-equivalent to a different discretization of the nonlinear Schrodinger equation.

It is interesting that we have here a discretization of NLS that has different features in its long time behaviour than NLS itself. This is an immediate consequence of the connection with the Toda lattice. For example, there is no 'Painleve' region for NLS while there is one for the Heisenberg chain. On the contrary, such difference is not observed between Toda and (its continuum limit) KdV (see Kamvissis 1993; Deift et al. 1994).

Remarks. An open problem is to find the long time asymptotics for the autocorrelation function for the Heisenberg chain. It is known (Its & Korepin, personal communication) that such a problem can be reduced to solving an operator-valued Riemann-Hilbert factorization problem and that more sophisticated methods are needed to recover the solution. For a precedent in the (much simpler) case of the

Ising chain see Deift & Zhou (1994) and the references quoted there. A more ambitious project is to calculate long time asymptotics for the autocor-

relation function of the non-integrable Heisenberg chain. Although numerical exper- iments point to a similarity between the two cases (integrable and non-integrable) the analysis of the non-integrable case is much harder.

2. Inverse scattering as a factorization problem

In this section, we rewrite the inverse scattering problem as a Riemann-Hilbert factorization problem on the unit circle.

We first recall a few facts concerning the scattering data for the Heisenberg- Haldane chain (see Papanicolaou 1987).

Jost functions

(n Z) = (d(n,z)) ,(n ,) (1(n,z))

satisfying the eigenvalue equations

'i(n + 1, z) = zanl(n, z) + -bn2(n, z), z1 }(2.1)

$02(n + 1, z) - --zbnl (n, z) + -a 2(n, z) z

(and similarly for b(n, z)) are defined on the unit circle by their asymptotics:t

(0 z as n--oo, ( as n-o, (2.2)

with normalization

11,(n,z) 2 + 12(n,)12 = , 1ll(n, z)12 + 102(n, z)12 = 1.

Here, the star denotes conjugation. Now, functions 0 and 0 can be meromorphically

t Not quite uniquely; see Venakides et al. (1991) for more rigorous definitions, in the context of the Toda lattice.


2631



extended in the exterior of the unit disc. In fact, under the assumption of no solitons, i.e. no bound states for the Lax operator, the only possible poles are at infinity. Indeed, we can write

z-il (n, ) = Ll(n)z-21, 1=0 00

z lni(n, z) = C (n)z21, 1=0

for Iz I 1, with

00

z-'2(n, z) - E DI (n)z-2 1=0

00

Zn?42(n, z) = E K, (n) 1=0-- I

Lo(n) - 1, as n - -oo,

Do(n) --0, as n - -oo,

Co(n) -0, as n--oo, Ko(n) - 1, as n- oo.

Let

4)(n,z) 1 = ( q(n,1/z*) , )I '(n,z)= ( b-)(n, 1/z*) ^ - {-/(n,1l/z*)

-

Then q and 0L are holomorphic in 0 < [zI < 1, with possible poles at z- 0. On Izi = 1, (0, b) and (), b) are linearly independent pairs of Jost functions. We

thus obtain the 'scattering relations', defining the scattering coefficients T(z), p(z), T(z) and p(z):

T(z)q(n, z) = (n, z) + p(z) (n, z), It follows that

1

T() = (51'2 - 21/1)(n, z),

T(z)

T(z)b(n, z) = V((n, z) + p(z)b(n, z). (2.5)

1

T(-) (021 - 1'2)*(n, z),

which enable us to extend T as a holomorphic function outside the unit circle. Sim- ilarly, T can be extended as a holomorphic and bounded function inside the unit circle. Also, one has

T(z) =-T*(z) and p(z) =-p*(z), for lzl 1,

and 1

T(oo) -)- -- I

) Lo(n)Ko(n) - Co(n)Do(n)' T() Co (n)DO(n)- L*(n)Ko(n) (26)

Next, define

fi(n, ) = z-nT(z))(n, z), for Izl > 1, = (z)-n (n,z), for lzl < 1,

f2(n, ) z= zn(n, z), for IZ1 > 1, = (z)nT(z)0(n,z), for Izl < 1.

(2.7)

The 2 x 2-matrix F = (fi, f2) is analytic and bounded in the complement of the unit circle. Furthermore,

F(z = ) = T( )Do(n) Proc. R. Soc. Lo nd. A (1996)

Proc. R?. Soc. Lond. A (1996)

Co(n) Ko(n)) (2.8)

(2.3)

(2.4)

2632 S. Kamvissis

1




and

F(z = 0) - K*() Dn)T(0) (2.9) -C;(n) -L;(n)T(0)) Letting F+ and F_ denote the limits of F on the unit circle, from the inside and the outside respectively, we have (after a few calculations)

I -p*(z)z2n F+ (z) = F_(z) _p(z) -2n + Ip(Z)12 ) on I I 1. (2.10)

The coefficients CI and K1 satisfy a set of algebraic equations from which an and bn can be recovered. For example (cf. relations (3.26) of Papanicolaou 1987),

Ko(n) Lo(n + 1) _ (n) -D(n + 1) an b_ - _ C D Co((n) = 0.

Ko(n + 1) Lo(n) Ko(n + 1) L (n) Cn (2.11)

Keeping in mind the asymptotics of a, and bn as n -? ?oo, we derive the following expression for K0:

00 j=n-1 j=n-2

Ko(n)=Haj, Lo(n)= U7 aj, Do(n)=-b_ 1 J aj. (2.12) j=n -00oo -00

Taking into account the evolution of p

p(z, t) = p(z, 0)ei(-1/z)2t

we end up with a Riemann-Hilbert factorization problem across the unit circle:

F is analytic inside and outside of the unit disc, with: ( F+(z; n,t) F_(z; n, t)un,t(z), on Izl = J

where

() ( 1 -p* (z, O)z2ne-i(z-1/z)2t on I) 1 Un'(Z -(Z, O)e(z-/)2t--2n 1 + p(z, 0)12 on 1,

(2.14) with

F(oo)-( 1/Ko(n;t) 0 ) (215) F(~) =

Do(n;t)/Lo(n;t)Ko(n;t) K(n;t) (2.1

Consider now the standard Riemann-Hilbert problem: to find a Q(z) analytic in the complement of the unit circle, such that

Q+(z) = Q-(z)u,t(z), on zl = 1, Q(oo) = I. (2.16)

Note that as det Un,t = 1, det Q has no jump, hence it is entire and by Liouville's theorem det Q = 1. In particular, Q (and similarly F) is invertible.

As Q and F satisfy the same jump condition, again by a Liouville's theorem-type argument, we must have

Q(z) = AF(z), (2.17) where A is a constant matrix. From (2.16), we get

( Ko(n) 0

A -Do (n) 1 (2.18)

Ko(n)Lo(n) Ko(n) Proc. R. Soc. Lond. A (1996)

2633



Hence / ^2 -O~-D(n)Ko(n) N TKo(n) 12 0 * (n)

0(0) - Oo* 1 (2n19) Q(0) Do( n)() Do(n) 2 + Lo(nr) 2 (2.

L Lo(n)Ko(n) IKo(n)Lo(n)l2 1

It follows, that once problem (2.17) is solved, then

Ko(n) n b*(t) I IDo(n + 1)[ lan(t)l = KO(n + 1) and (l _ l +1) (2.20)

lao((t)I- Lo(n + 1)I are readily recovered.

Indeed, as we shall see in the next section, we will even recover an(t), bn(t) and n (t) asymptotically.

3. Reduction to the Toda lattice

In this section, we consider the Riemann-Hilbert problem (2.17) with jump matrix given by (2.15).

Letting

= iz2 (3.1) the 'phase'

(z) - i) - - log (3.2)

becomes 1 n win

() =- - log ( + - 2i. (3.3) t 2t

Defining

r(n,) = -inp(z, t 0), (3.4) the jump matrix becomes

V ((n f /I 2i) r*(n, ()(-(C /C2i) on (- 1. (3.5) ^^ -(r(n,

-n C eK "/--2-i)t l+l IrIn

( ^2

Setting

Y() = Q(z), (3.6) we end up with the following factorization problem:

Y+(() = Y_(()vn,t((), on I- 1, Y(oo) = , (3.7) with Vn,t(() given by (3.5).

Remark. The reduction above is due to the symmetry Q(z) Q(-z) which follows immediately from the symmetry un,t(z) = un,t(-z) of the jump matrix, which in turn follows from the symmetry of the reflection coefficient p(z) = p(-z): an immediate consequence of the definitions (2.5) and the expansions (2.3).

At this point, we note that the above problem is very similar to the problem corresponding to the Toda lattice (see Kamvissis 1993). There are, in fact, three main differences. First, the new 'reflection coefficient' r(n, z) depends on n. This, however, is a trivial complication: as one can see from (3.4), one can consider separately the


2634 S. Kamvissis




four distinct cases where n = j(mod4), j = 1,2,3,4, since for each of this cases

r(n, z) is independent of n. It also seems as if the new reflection coefficient depends on time through a factor e-2it. However, one can incorporate this term in the new

phase (see (3.3)) and just note that the stationary phase points z0 (see (3.10)) are not affected.

Second, the jump matrix in the Toda case has 1 - Irl2 in the (2,2) position. The

important point, however, is that an obvious upper-lower factorization is still possible. The analysis pursued in the Toda case can then be immitated here, without

any changes. In fact, there is an extra 'bonus'; as is clear from the treatment in Kamvissis (1993), complications arise when a zero appears in the (2,2) position of the jump matrix. In the present case, since 1 + Ir2 can never be zero, we do not need a special analysis of that possibiliy. On the contrary, in the Toda (or the KdV) case, we need a special argument when Ir(z)l = 1, and in fact the answer of the problem is different: physically, one observes the so-called 'collisionless shock' phenomenon (cf. Ablowitz & Segur 1977; Deift et al. 1994; Kamvissis 1996), that is an extra region appears, where the solution is asymptotically identified with an 'algebro-geometric' type solution, connected with the theory of Riemann surfaces and the Abel map.

Finally, in our case, as opposed to the Toda case, one does not in general have the extra symmetry r(z*) = r*(z). Again, however, this does not result in any essential

complications in the related computations. With the discussion above in mind, we use the results of Kamvissis (1993):

Theorem 3.1. If Y solves problem (3.7), then

(i) In the region In/2tl < 1 - O(t-1/3) we have

I (-t\O q(t) +

Ologt) Y(O) - I+ ( ( 2t iqnt)) 0(/ t3/ (3.8)

uniformly in n, where

q (t) -= v 1/2e2it(sin 0o-0o cos o)+ivllog(2t)e-2G(zo) -3ivl log zo-2iv2 log(zo--z)

1/2e-2it(sin o0-00 cos 0o)+iV2 log(2t)e-2G(z )-3iv2 log(zO )-2ivl log(zO -z0)

(3.9)

and

zo - e

n cos 0o= - 0 < 0 < <r/2,

2t'

1 /1 log(1 + Ir(zo)l2) > 0, 2ir (3.10)

v2 -= log(1 + r(z*)12) > 0,

() I log( +r(w)12 dw

C being the arc of the unit circle joining z and counterclockwise.--

C being the arc of the unit circle joining zo and z counterclockwise.

(ii) In the region 11 - n/2t = O(t-1/3), we have the uniform estimate

Y(O) = I + (t-/3).


2635



(iii) In the region In/2tl > 1 + O(t1/3), we have the uniform estimate

Y(O) = I + C(n)O(t-l),

for any positive number 1, where C(n) decays rapidly as Inl -+ oo.

Proof. Follows from the discussion in ??5 and 6 of Kamvissis (1993). Two main differences arise because of the lack of symmetry of the reflection coefficient (see the preceeding paragraph). First, the solution of the auxiliary scalar Riemann-Hilbert problem (see Kamvissis 1993, ? 2) is slightly different. Second, the contribution to the final formula (3.8) due to the first stationary phase point is not equal to the conjugate of the contribution due to the second stationary phase point. The fast decay of the term C(n)O(t-l) in part (iii) follows from the proofs in Kamvissis (1993), although not stated explicitly there (but see, however, the explicit analogous statements in ? 1 of Deift & Zhou (1993)). U

It follows immediately from (2.21) that

la(t)l =l+ t(2 ), (3.11)

since from (3.8), (3.6) and (2.20),

IKo(n)l - l + 0o t3/2)

Similarly b (t) Ko(n) -q (t) +Ologt (12)

an(t) KO(n) (2tsino)1/2 +

t3/2 (3.12)

since by (2.12) bn Do(n + 1) an Lo(n + 1)'

From the definition of an, bn (Papanicolaou 1987, eqn (3.11)) 1 + !+l *n+2

an(t) = (1 + +21 1 + In+22) (3.13) I~ (1+P112)1/2(1 +1 ^+212)1/2 ^

b*(t) +1- n+2 (3.14 = (3.14) an(t) 1 + O 1n+l *n+2

and

_ + 2 Re(?P+i!P*+2) + I ?p+ l' 1 +2 12 lan (t)l 1+l

- (P +l) + I pP l 2 + n121 2 (3.15)

l-? 1 5n+ I +12 + In+212 + Ikn+2121i + 12' We next show that for each n, n (t) -- 0, as t -- oo. Set n = 0 and assume that

So(tj) -- So (S, S2 , S3 ) for some sequence of times tj -0 oo. As

1- Sg(tj) I f0 (tj) 12 1 S3(tj)

1 + Sg(t)'

it follows that oo = limj,oo o(tj) exists, where we include the possibility that 00 = oo. Now, from (3.11), (3.12), (3.13) and (3.14) we have

Pn+l(tj) -

,n+2(tj) 0 (1 + I,n+1(tj) 12)1/2(1 + In +2(tj) 12)1/2

s (3.16)


2636 S. Kamvissis




and from a simple induction argument using (3.16) it follows that On(t) --+ oo, for all integers n. But then limj,, Sn(tj) = SoO for all n. On the other hand we have from (1.1) the conservation law

00

E (1-S3(tj))=c, (3.17) n= -oo

a positive constant, and hence as I - S3(tj) > 0 for all n and j,

N

c > (1 - S) (3.18) -N

for any fixed integer N. As c < oo, we must have S3 =- 1 and hence S1 = S2 = 0. This shows that limn,oo Sn(t) = (0,0, 1) for all n and hence limn1nO Pn(t) = 0 for all n, as desired.

Now solving (3.14) for On+l, substituting in (3.13) and using (3.12) we see that an(t) = -n(t) + Cn(t) where yn(t) > 0 and ec(t) = O(t-1/2) and hence an(t) 1 + O(t-1/2). Substituting in (3.12) yields the estimate for bn(t). To complete the proof of estimates (1.4), note that we have

[2t] (t) n(t) (1 + 3+l^(t)m+2(t))

(t

(3.19) ~~+n

2 3()aj(t)'

by the fast decay of the C(n)/t' term referred to earlier, hence

[2t] [2t]

P(t) E(1 + o( )) 2- E (3.20) n aj n aj

and hence the estimate (1.4) follows, since Ko(n) ~ I (note that the estimate o(1) is uniform).

Remark. Our paper concentrates on the effect of the continuous spectrum on the long time behaviour of (1.1). Indeed, the above results are strictly true only when no bound states exist. However, it is easy to recover the most general (generic) asymptotics to add solitons. The easiest way to do so is to follow the procedure described in Fadeev & Takhtajan (1987, ch. II, ? 2). If problem (2.14) now includes poles for F, we can factor away 'Blaschke-Potapov' factors responsible for the poles and end up with a problem similar to (2.14) and without poles. Solving that as in ? 3, we then factor back the Blaschke-Potapov factors thus recovering the most general solutions.

We thank N. Papanicolaou for pointing our attention to this problem and for useful discussions. We also thank the Physics Department of the University of Crete in Greece, where most of this work was conducted, for its hospitality. Finally, we thank one of the referees for several important comments, especially concerning the proof of theorem 3.1.

References

Ablowitz, M. J. & Segur, H. 1977 Asymptotic solutions of the Korteweg-de Vries equation. Stud. Appl. Math. 57, 13-44.

Deift, P. & Zhou, X. 1993 A steepest descent method for oscillatory Riemann-Hilbert problems. Ann. Math. March.


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Deift, P., Its, A. R. & Zhou, X. 1993 Important developments in soliton theory, 1980-1990 (ed. A. Fokas & V. Zakharov). Springer.

Deift, P. & Zhou, X. 1994 Long time asymptotics for the autocorrelation function of the trans- verse Ising chain at the critical magnetic field. In Singular limits of dispersive waves (ed. N. Ercolani et al.). New York: Plenum Press.

Deift, P., Venakides, S. & Zhou, X. 1994 The collisionless shock region for the long-time behavior of solutions of the KdV equation. Commun. Pure Appl. Math. 47, 199-206.

Deift, P., Kamvissis, S., Kriecherbauer, T. & Zhou, X. 1996 The Toda rarefaction problem. Commun. Pure Appl. Math. 49, 35-84.

Fadeev, L. D. & Takhtajan, L. A. 1987 Hamiltonian methods in the theory of solitons. Springer. Haldane, F. D. M. 1982 J. Phys. C 15, L1309.

Ishimori, Y. 1982 An integrable classical spin chain. J. Phys. Soc. Jap. 51, 3417-3418.

Kamvissis, S. 1993 On the long time behavior of the Toda lattice under initial data decaying at infinity. Commun. Math. Phys. 153, 479-519.

Kamvissis, S. 1996 Long time behavior of the focusing nonlinear Schr6dinger equation with real spectral singularities. Commun. Math. Phys. 180, 325-342.

Nijhoff, F., van der Linden, J., Quispel, G. R. W., Capel, H. W. & Velthuizen, J. 1982 Physica 116A, 1-33.

Papanicolaou, N. 1987 Complete integrability for the discrete Heisenberg chain. J. Phys. A 20, 3637-3652.

Venakides, S., Deift, P. & Oba, R. 1991 The Toda shock problem. Commun. Pure Appl. Math. 44, 1171-1242.

Received 22 May 1995; revised 10 July 1996; accepted 26 July 1996


2638 S. Kamvissis



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