Following Sutherland's work on one-dimensional integrable systems we
formulate and study its two-dimensional version. Physically it expresses the
absence of true 3-body forces among an assembly of N particles leaving
exclusively effective 2-body interactions. This criterion may be a suitable
candidate for an integrability condition.Comment: 13 pages,uses LATeX2e and epsf. File prepared using the uufiles
  packag

Azhari, A.

Truong, T. T.

English

arXiv

Following Sutherland's work on one-dimensional integrable systems we formulate and study its two-dimensional version. Physically it expresses the absence of true 3-body forces among an assembly of N particles leaving exclusively effective 2-body interactions. This criterion may be a suitable candidate for an integrability condition

Azhari, A

Truong, T T

CERN Document Server

ON THE SUTHERLAND'S
INTEGRABILITY CONDITION
FOR TWO-DIMENSIONAL N-PARTICLE
SYSTEMS.
A. AZHARI
?;1
and T.T. TRUONG
y;2
?) Laboratoire de Modeles de Physique Mathematique,
Departement de Physique, Universite de Tours,
Parc de Grandmont, F-37200 Tours, France.
y) Groupe de Physique Statistique. Departement de Physique,
Universite de Cergy-Pontoise B.P.8428,
F-95806 Cergy-Pontoise Cedex France.
July 19, 1995
Abstract
Following Sutherland's work on one-dimensional integrable sys-
tems we formulate and study its two-dimensional version.Physically
it expresses the absence of true 3-body forces among an assembly of
N particles leaving exclusively eective 2-body interactions.This cri-
terion may be a suitable candidate for an integrability condition.
1
e-mail: azhari@cel.phys.univ-tours.fr
2
e-mail: truong@u.cergy.fr
1
1 Introduction
In the last twenty years, the quantum N-body problem has been chiey
studied in one space dimension with great success thanks to the discovery
of integrable systems. Among those, one can count systems soluble with the
Bethe ansatz wave function as well as others systems discovered originally
by Calogero and Sutherland at the beginning of the 70's. The last topic has
evolved considerably and connections to many algebraic structure have been
established [1].
One main ingredient to the solubility is the fact that the dynamics of
the N-particle system is only due to pair potential and eventually external
applied elds. This abscence of other type of N-body forces renders the prob-
lem completly soluble whether it is relativistic or not.
Few breakthroughs have been made in 2 or 3 space dimensions. Due to
more \room" available, particles can go around each other instead of being
conned on the line where they must scatter inevitably. The resulting eect
is that it is dicult to give a concept of \integrable" systems. Yet Calogero
and Machioro [5] have discovered that if one includes some type of 3-body
forces which are highly dependent on direction, one may obtain exactly sol-
uble multiparticle dynamics in three space dimensions.
The interest in recent years in two-dimensional quantum systems stems
from the discovery of the fractional quantum Hall eect which indirectly re-
vived an older work of Leinass and Myrheim [2] on quantum theory of many-
particles in two-dimensions as well as quantum dots and electronic plasmas
[19]. At the center of this topic are new objects, the anyons which seems to
be responsible for some physical phenomenona (such as fractional quantum
Hall eect)[15] or realizations of \exotic" statistics which are not excluded
in two-dimensions. Hence dierent many-particle wavefunctions have been
proposed in this spirit such as the known Laughlin wavefunction for the frac-
tional quantum Hall eect [16, 17].
In this paper we shall not discuss the physical phenomena, but instead
pose the following problem: under which condition, in two-dimensions can
one have only pair interactions? And more generally when does one obtain
2
an integrable system? We shall see then that the emergence of fractional
statistics appears to be natural and consistent with the existence of pair-
potentials among the N-particles, only for a specic class of pair-potentials.
Let us rst recall that in non-relativistic quantum mechanics of a single
particle the wavefunction depends on two real variables x and y i.e.  (x; y).
One may equivalently use the complex combinations:
z = x+ iy and z = x  iy (1)
and consider instead  (z; z); this last wave function is in fact a restriction of
a fonction of two complex variables  (z; z
0
) , such that z
0
= z. To account for
manifest correlations among particles of the systems it is natural to postulate
the Bijl-Dingle-Jastrow wavefunction [8, 18] which is generally proposed in
one-dimension to describe the ground state of a system of N particles.
	 = 	(z
1
; z
1
; : : : ; z
N
; z
N
) =
Y
i<j
 (z
ij
; z
ij
) (2)
where z
ij
= z
i
 z
j
. Excited states may be constructed from the ground state
using a standard method [9].
In this wavefunction, the order of pairs of particles is single out by the
pair wavefunction  (z
ij
; z
ij
). This is a general feature in all soluble N-particle
systems in one-dimension, and becomes thus a valuable starting point for the
study of two-dimensional systems.
In section I, we review the situation in one-dimension to establish the
grounds for such procedure. There, the central object is the Sutherland's
condition for the solubility of the problem. This condition merely states
that the 3-body potential arising from a state described by the Bijl-Dingle-
Jastrow wavefunction, may be recast into a sum of pair-potentials so that the
whole system behaves pratically under pure eective pair potentials. This is
our main argument and this may be related to what is known in integrable
quantum eld theory in one-space dimension. In fact when the S-matrix of
such a quantum eld theory is factorizable or can be written as product of
S-matrices for pairs of particles, subjected to the usual conditions of unitarity
and analyticity, does one have an innite number of concerved quantities in
3
the theory. What is more interesting is that A.B. Zamolodchikov and Al.B.
Zamolodchikov [4] have shown that in the non-relativistic limit, these pair
S-matrices reproduce the scattering phase shifts of the soluble pair potentials
in one-dimension. It is then tempting to identify the Sutherland's condition
as the integrability condition in one-dimension. In fact this is really so be-
cause such a condition contains all the known soluble pair potentials.
Our objective is thus to explore the Sutherland's condition in two-dimensions.
In section II, we shall derive the condition of Sutherland in two dimensions
and study its properties in details. As we shall see, many new aspects emerge
as compared to those in one-dimension: the inclusion of fractional statistics,
the nature of the eective pair-potentials, and the triviality of the attractive
harmonic potential in two-dimensions.
In section III, we shall raise the question whether there exists a repul-
sive interaction among pairs which might insure stability against the collapse
due to the harmonic attractive potential . In fact we shall present a two-
dimensional version of the Sutherland elliptic potential as a concrete exam-
ple. The model has the merit of providing an example of Wigner solid in
two dimensions. In the scaling limit of this potential we shall see how the
fact that particles can never \over take" each other in two-dimensions is the
key to the solubility of the problem. In fact the particles are locked inside
two-dimensional rectangular cells and remain \impenetrable".
We conclude by presenting some ideas on how one may generalize the
previous ideas to nd non-trivial integrable systems in two-dimensions as well
as to seek limits to construct integrable relativistic quantum eld theories in
two-space dimensions.
2 The one-dimensionnal case revisited
.
This has been worked out in the seventies by B.Sutherland [8, 9] and
F.Calogero [12, 13]. The N-particule wavefunction is assumed to be of the
Dingle-Bijl-Jastrow form:
4
	 =
Y
i<j
j (x
i
  x
j
)j

(3)
where  (x) is the pair wavefunction and  a real constant.
Introducing the associated functions  and ' by:
 = exp ' =
d
dx
(4)
and applying the kinetic energy operator on 	, one obtains:
 
N
X
i=1
@
2
@x
2
i
	 = f 2
X
j<i
['
0
(x
i
  x
j
) + '
2
(x
i
  x
j
)]
+2
2
P
i;j;k
['(x
k
  x
i
)'(x
i
  x
j
) + '(x
i
  x
j
)'(x
j
  x
k
)
+'(x
j
  x
k
)'(x
k
  x
i
)]g	:
(5)
The rst term of the r.h.s. of (5) represents the pair potentiel between
pairs of particles due to the choice of  (x). the second term is an induced
3-body potential beetwen any triplet i,j,k of the N-particles.
Sutherland proposed then to choose  such that this 3-body potential
breaks up into additional pair potentials, namely that:
'(x)'(y) + '(y)'(z) + '(z)'(x) = f(x) + f(y) + f(z) (6)
for x+ y + z = 0
The form of f(x) being essentially due to the choice of  (x). If such a
choice is made, then there exists among the N-particles only an eective pair
potential:
V (x) = ['
0
(x) + '
2
(x)  f(x)]: (7)
The philosophy of this statement is analoguous to the one adopted in the
theory of integrable quantized elds in 1+1-dimensions. There it is stated
that the N-body S-matrices are factorized into two-body S-matrices. In fact
Zamolodchikov and Zamolodchikov [4] have shown that in one of these theo-
ries, the non-relativistic limit of such 2-body S-matrix is precisely the phase-
shift in a relative pair potential of the type (7). In this sense, one may say
5
that the Sutherland's condition (6), is in fact an integrability condition for
an N-body problem in one-dimension.
To get more insight we may advantageously replaced the condition (6) by
the following one, using elementary algebra:
('(x) + '(y) + '(z))
2
= g(x) + g(y) + g(z) (8)
with x+ y + z = 0 and g(x) = '
2
(x) + 2f(x).
The form of (8) turns out to be exactly a relation satised by the Weier-
strassian elliptic functions (x) and P(x) namely:
((x) + (y) + (z))
2
= P (x) + P (y) + P (z) (9)
with x+ y + z = 0
Sutherland who discovered this connection identied then  (x) to (x), the
Weierstrassian -function . The periodicity of (x) is instrumental in ex-
hibiting an example of Wigner solid in one-dimension [6, 7].
Some remarks on the properties of '(x) are now in order:
a) '(x) is in fact dened up to a linear term: the substitution '(x) !
'(x) + ax+ b leaves relation (6) invariant.The linear term ax+ b induces a
pair potential:
V (x) = fa+

2
(ax+ b)
2
  xg (10)
which is essentially a shifted harmonic oscillator potential. In this sense, the
harmonic oscillator pair potential is simply a trivial one.
b) As shown by Sutherland [8], particular limits of the (x) potential re-
produce all the non-singular pair-potential known in one-dimension.
c) We note also that the pair -function is also contained in (6) which in
this case the r.h.s of [7] is simply constant and the 	 take up the form of a
Bethe ansatz wavefunction.
6
3 Two-dimensional case
As pointed out before, the pair wavefunction is in fact a restricted wavefunc-
tion of 2 complex variables  (z; z
0
) with z
0
= z. In analogy to section 2 we
shall introduce the notations:
 = exp
' =
d
dz
 = (z; z)
' =
d
dz
(11)
The application of the kinetic energy operator, assuming that particles are
of unit mass.
N
X
i=1
@
2
@z
j
@z
i
=  
1
2
N
X
i=1
(
@
2
@x
2
i
+
@
2
@y
2
i
)
on a N-particle Dingle-Bijl-Jastrow wavefunction yields a sum of pair-potentials
and an induced 3-body potential as in one-dimension. Therefore following
Sutherland we are led to the generalized condition:
f '(x; x)'(y; y) + '(y; y)'(x; x)
+ '(y; y)'(z; z) + '(z; z)'(y; y)
+ '(z; z)'(x; x) + '(x; x)'(z:z) g
= [f(x; x) + f(y; y) + f(z; z)]
(12)
with:
x+ y + z = 0 and x+ y + z = 0
which evidently states that the 3-body potential between any triplet of par-
ticles will be break up into a sum of 2-body potentials.
Unfortunatly, there is up to now no theory of factorized S-matrices in
(2+1) dimensions. Yet, we may call this condition an integrability condition
if non-trivial interesting solutions can be found. In the sequel we shall seek
to construct some solutions based on experience in one-dimension.
But before doing so let us mention that an alternative way for formulating
(12) would be, in analogy to eq (8)
f'(x; x) + '(y; y) + '(z; z)gf'(x; x) + '(y; y) + '(z; z)g
= g(x; x) + g(y; y) + g(z; z)
(13)
7
whenever:
x+ y + z = 0 and x+ y + z = 0
Under this form, eq.(13) remains obviously invariant under the double
substitution:
'(x; x)! '(x; x) + ax+ a
0
x+ b (14)
'(x; x)! '(x; x) + a
0
x+ ax+ b (15)
Again this linear part is responsible for the shifted harmonic oscillator pair-
potential between particles. Thus such a potential is of trivial nature and has
tendency to cause a collapse to the center of mass of the system of N-particles.
Morever,
! + a
0
zz +
1
2
(az
2
+ az
2
) + bz + bz + const:
and the new potential is:
V ! (V + a
0
) + '(x; x) + '(x; x) + (b+ b): (16)
This transfomation may be used to generate new pair potential from a known
one.The two-dimensional eective pair-potential in general has the form:
V = (   1)
@
@z
@
@z
+ 
2
@
2

@z@z
 

2
2
f(z; z)
We are now in a position to study some particular situations:
a) If  is a solution of the Laplace operator
@
2
@z@z
 = 0
Then  = f(z)+ g(z), here f and g are independent functions not related to
those of eq.(12) or (13). Sutherland's condition becomes
ff
0
(x)+f
0
(y)+f
0
(z)gfg
0
(x)+g
0
(y)+g
0
(z)g = h(x; x)+h(y; y)+h(z; z) (17)
for
x+ y + z = 0:
8
We note that if g
0
=const ( or resp. f
0
=const ) the condition (17) is auto-
matically satised:  = exp f(z) or  = exp g(z) is a solution.
b) If  = (zz), function of the distance zz, this represents a physically
reasonable situation for which the Sutherland 's condition is:

0
(yy)
0
(xx)(xy + yx) + 
0
(yy)
0
(zz)(yz + yz) + 
0
(zz)
0
(xx)(xz + zx)
= h(x; x) + h(y; y) + h(z; z)
(18)
Note that for 
0
=const one recovers the harmonic oscillator pair-potential,
which has been recently investigated by Mushkevich et al [11]. We observe
that if the pair wavefunction is taken az  (z; z)

=
z

exp ( zz) for example,
the eective pair-potential is
V = (
3
2

2
  )zz    + (

2
2
  )
Thus the anyonic factor z

simply shifts V by a constant amount. In fact
this is the only known instance where exotic statistics seem to be consistent
with the Sutherland condition.
Since:
xy + yx = 2~x:~y
represents the scalar product of the vector ~x and ~y in the plane, the Suther-
land's condition takes a new vector form as a scalar product:
[
0
(j~xj)~x+ 
0
(j~yj)~y + 
0
(j~zj)~z]
2
= h(~x) + h(~y) + h(~z) (19)
with
~x+ ~y + ~z =
~
0
This has the same structure as eq.(6). Calogero and Machioro [5] have
treated in three-dimensions, in the same spirit, the problem of N-particles
only with potential dependent on the interparticle distance. In other words
the wave function is of the Dingle-Bijl-Jastrow type. However they kept the
induced three-body potential and have not look at the possibility it may
decompose into pair-potentials.
9
4 Two dimensional Sutherland's model
Inspired by the work of Sutherland wemay extend his model in two-dimensions
obtaining a Wigner solid in the plane. The wavefunction 	 is taken as :
	 = c
Y
i<j

1
(

L
(x
i
  x
j
))
1
(

L
0
(y
i
  y
j
)) (20)
where 
1
is the Jacobian odd-theta function and L, L
0
are lengths in the x
and y directions.
Applying the kinetic energy operator on the wavefunction 	 yields :
1
	
(
N
X
i=1
(
@
2
@x
2
i
+
@
2
@y
2
i
)	) =
X
i<j
[N

00
1
(

L
(x
i
  x
j
);
ir
L
)

1
(

L
(x
i
  x
j
);
ir
L
)
+ 2(
L
2
)(N   2)
1
L
]
+
X
i<j
[N

00
1
(

L
(y
i
  y
j
);
ir
0
L
0
)

1
(

L
0
(y
i
  y
j
);
ir
0
L
0
)
+ 2(
L
0
2
)(N   2)
1
L
0
]
(21)
In this case the integrability condition (13) will appears as being derived
by a combination of (9).In fact we have :
 = ln
1
(

L
x) + ln
1
(

L
0
x) (22)
consequently
' =
d
dz
=
1
2
d
dx
+
1
2i
d
dy
(23)
or
' =
1
2
((x)  (w)
x
w
) 
i
2
((y)  (w
0
)
y
w
0
) (24)
where w and w
0
are the periods of the (z)-function of Weierstrass.
Similarly we obtain
'(z; z) =
1
2
((x)  (w)
x
w
) +
i
2
((y)  (w
0
)
y
w
0
) (25)
10
As observed before the parts in ' and ' linear in x, y and constants will
not aect the integrability condition the l.h.s. of (13) which in fact becomes
f[(x
1
) + (x
2
) + (x
3
)]  i[(y
1
) + (y
2
) + (y
3
)]g
f[(x
1
) + (x
2
) + (x
3
)] + i[(y
1
) + (y
2
) + (y
3
)]g
= [(x
1
) + (x
2
) + (x
3
)]
2
+ [(y
1
) + (y
2
) + (y
3
)]
2
(26)
But this is precisely, because of the identity [10]
[P (x
1
) + P (x
2
) + P (x
3
)] + [P (y
1
) + P (y
2
) + P (y
3
)]
if x
1
+ x
2
+ x
3
= 0 y
1
+ y
2
+ y
3
= 0
One may thus says that the double Sutherland's model fullls an in-
tegrability condition which is merely the sum of the separate integrability
conditions in the x and the y directions.
To get a further insight on the integrability of this two dimensional prob-
lem one may consider one of its limiting case where the pair potential is
essentially
V (~x 
~
x
0
) =
g
(x  x
0
)
2
+
g
(y   y
0
)
2
(27)
Here g is a coupling constant.
There one see that, relative to particle ~x, particle
~
x
0
remains constantly
in one of the quadrants centred at point ~x, and conversely, this is the two-
dimensional form of the non- overtaking aspect of the dynamics a particle
can never get out of the quadrant in which it is located with respect to its
neighbor.(see g.1)
Thus after a scattering the two particles will y away from each other
but continue to remain in these quadrant sectors for ever.
We can generalise this picture to an ensemble of N-particles and under-
stand why these conservation laws are behind the integrability of the model.
Although the periodical aspect leads to a two dimensional Wigner crystal,
the particle dynamics in the particular limit of the pair- potential is rather
articial due to the presence of "forbidden lines" parallel to the axis attached
to each particle.
11
x’
.
.
x
Figure 1: Particle ~x is in third quadrant of particle
~
x
0
and particle
~
x
0
is in
the rst quadrant of particle ~x.
5 Conclusion and outlook.
In this article we have tried to nd a generalized version of the condition
found by Sutherland,for one dimensional integrable systems. Such systems
admits for ground state wavefunction as an N-particle wavefunction of the
Dingle-Bijl-Jastrow form.
The Sutherland's condition merely states that for such a system there
exists no true three-body potential but only eective pair-potentials. Suther-
land was able to nd the most general solution in one dimension, it is a peri-
odic one and its describes a Wigner crystal. In two dimensions the general-
ized Sutherland's condition does not admit obvious repulsive pair-potentials
as in one dimension nor local -function pair-potentials. But it does not ex-
clude also other type of pair-potentials which remain to be discovered. We
have constructed a simple example of two-dimensional solution fullling the
generalized Sutherland's condition, this study seems to suggest that a more
general solution should be given by a 2-variable theta functions depending on
three modulu. Work is in progres in this direction. It is expected that special
limits would lead to new form of soluble pair-potentials in two-dimensions.
12
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13


On the Sutherland's Integrability Condition for Two-Dimensional N-Particle Systems

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