Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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(all other brackets of the coordinate functions
vanish), and the system [11] is Hamiltonian with
respect to this bracket, with the Ham ilton function
H
2
¼
1
2
X
b
2
n
þ
X
a
n
However, one can define also a different Poisson
bracket for the variables a
n
, b
n
:
fb
n
; a
n
g
2
¼b
n
a
n
fa
n
; b
nþ1
g
2
¼a
n
b
nþ1
fb
n
; b
nþ1
g
2
¼a
n
fa
n
; a
nþ1
g
2
¼a
n
a
nþ1
½20
with the following properties: it is compatible with
the first one (i.e., their linear combinations are again
Poisson brackets), and the system [11] is Hamilto-
nian with respect to this bracket, with the Hamilton
function H
1
=
P
b
n
. So, the Toda lattice in the form
[11] is a bi-Hamiltonian system. This result is due
to M Adler (1979). The bi-Hamiltonian property,
introduced by F Magri in 1978 on the example of
the Korteweg–de Vries equation, has been estab-
lished since then as an alternative (and highly
effective and infor mative) definition of integrability.
Actually, the Toda lattice [11] is even tri-Hamiltonian,
since there exists one more local Poisson bracket for
the variables a
n
, b
n
with similar properties, discovered
by B Kupershmidt in 1985.
Darboux–Ba¨ cklund Transformations
and Discretization
A further indispensable attribute of integrable
systems are the so-called Darboux–Ba¨cklund trans-
formations. For the Toda lattice they were first
found by M Toda and M Wadati in 1975.A
Ba¨cklund transformation (q
n
, p
n
) 7!(
e
q
n
,
e
p
n
) with the
parameter h can be written as
1 þ hp
n
¼ e
~
q
n
q
n
þ h
2
e
q
n
~
q
n1
1 þ h
e
p
n
¼ e
~
q
n
q
n
þ h
2
e
q
nþ1
~
q
n
½21
This is a canonical transformation, possessing a
classical generating function. These formulas can be
given a fundamentally important interpretation in
terms of the matrices
U
þ
¼
X
n2Z
e
~
q
n
q
n
E
n;n
þ h
X
n2Z
E
nþ1;n
½22
U
¼ I þ h
X
n2Z
e
q
nþ1
~
q
n
E
n;nþ1
½23
The first formula in [21] is equivalent to the
factorization I þ hL = U
þ
U
, while the second one
is equivalent to the factorization I þ h
e
L = U
U
þ
with the flipped factors. The Ba¨ cklund trans forma-
tion [21] serves also as an integrable discretization
of the Toda flow [2] with the time step h.
Finite Open-End Toda Lattice
Model
The infinite Toda lattice [1] can be reduced to finite-
dimensional systems by imposing suitable boundary
conditions, different from the rap idly decaying ones.
Particularly important are ‘‘open-end boundary
conditions,’’ which correspond to placing the parti-
cles 0 and N þ 1atq
0
= þ1 and q
Nþ1
= 1,
respectively. In terms of the Flaschka–Manakov
variables, this means that a
0
= a
N
= 0 and b
0
=
b
N þ1
= 0. The Hamilton function of the resulting
system with N degrees of freedom is
H
2
ðp; qÞ¼
1
2
X
N
n¼1
p
2
n
þ
X
N1
n¼1
e
q
nþ1
q
n
½24
This system consists of N particles subject to
repulsive forces between nearest neighbors, and
exhibits a scattering behavior both as t !1and
t !þ1. It admits a Lax representation of the same
form [12] or [15] as in the infinite case, but with all
the matrices being now of finite size N N, so that
[13]–[14] and [16]–[17] are replaced by
L ¼
X
N
n¼1
b
n
E
n;n
þ
X
N1
n¼1
a
n
E
n;nþ1
þ
X
N1
n¼1
E
nþ1;n
½25
A
þ
¼
X
N
n¼1
b
n
E
n;n
þ
X
N1
n¼1
E
nþ1;n
A
¼
X
N1
n¼1
a
n
E
n;nþ1
½26
and
L
0
¼
X
N
n¼1
b
n
E
n;n
þ
X
N1
n¼1
a
1=2
n
E
nþ1;n
þ E
n;nþ1
½27
A
0
¼
1
2
X
N1
n¼1
a
1=2
n
E
nþ1;n
E
n;nþ1
½28
The qualitative behavior of the solutions is easily
understood: as a consequence of repulsive interac-
tions, the pairwi se distances between particles grow
infinitely, a
n
(t) = e
q
nþ1
(t)q
n
(t)
!0ast !1, so that
the matrix L
0
becomes asymptotically diagonal,
with the limit velocities b
n
(1) =
_
q
n
(1) as the
diagonal entries. Due to the isospectral evolution of
L
0
, these limit velocities have to coincide with the
eigenvalues
j
of L
0
, which are integrals of motion.
238 Toda Lattices
As t !1, they appear on the diagonal in the
increasing order (the rightmost particle q
1
being the
slowest, and the leftmost q
N
being the fastest), while
as t !þ1, their order on the diagonal changes to
the decreasing one (the particle q
1
becoming the
fastest and q
N
becoming the slowest).
Moser’s Solution
Integration of this syst em has been first perfor med by
J Moser in 1975. His solution can be interpreted
within the general scheme of the IST (see Figure 1).
The spectral data in this case consist, for example, of
the eigenvalues
j
(j = 1, ..., N) of the matrix L
0
and
the first components r
j
of the corresponding ortho-
normal eigenvectors. The evolution of these data
induced by the Toda flow [2] turns out to be simple:
j
¼ const:; r
2
j
ðtÞ¼
r
2
j
ð0Þe
j
t
P
N
i¼1
r
2
i
ð0Þe
i
t
½29
The IST is expressed by the identity
X
N
j¼1
r
2
j
j
¼
1
b
1
a
1
b
2
.
.
.
a
N1
b
N
½30
both parts of which represent the entry (1, 1) of the
matrix (I L
0
)
1
. It im plies that all variables
a
n
(t), b
n
(t) are rational functions of
j
and e
j
t
;in
particular, one finds:
a
n
ðtÞ¼e
q
nþ1
ðtÞq
n
ðtÞ
¼
n1
ðtÞ
nþ1
ðtÞ
2
n
ðtÞ
½31
where
n
(t) can be represented as an n n Hankel
determinant
n
ðtÞ¼det c
jþk
ðtÞ
0j;kn1
c
j
ðtÞ¼
X
N
i¼1
j
i
r
2
i
ðtÞ
½32
Factorization Solution
The Lax representation [12] is a particular instance of
a general construction, known under the name of
Adler–Kostant–Symes (AKS) method and found
around 1980. The ingredients of this construction are:
a Lie algebra g , equipped with a nondegenerate
scalar product which is used to identify g with its
dual space g
;
a splitting of g into a direct sum of its two
subspaces g
which are also Lie subalg ebras, with
: g !g
being the corresponding projections;
the Lie group G of the Lie algebra g , and its Lie
subgroups G
with the Lie algebras g
; and
a function : g !g covariant with respect to the
adjoint action of G (in the case of matrix Lie
algebras and groups, one can take, e.g.,
(L) = L
s
).
The AKS method provides a formula for the solution
of the initial-value problem for Lax equations of the
form [12] with the Lax matrix L 2 g and
A
=
((L)). The solution is given by
LðtÞ¼U
1
þ
ðtÞLð0ÞU
þ
ðtÞ¼U
ðtÞLð 0ÞU
1
ðtÞ½33
where the elements U
(t) 2 G
solve the factoriza-
tion problem
exp
tðLð0ÞÞ
¼ U
þ
ðtÞU
ðtÞ½34
For the open-end Toda lattice g = gl(N), the Lie
algebra of all N N matrices, g
consistofall
lower-triangular, resp., s trictly upper-triangular,
matrices. Accordingly, G = GL(N), the Lie group
of all nondegenerate N N matrices, and G
consist of all nondegenerate lower-triangular
matrices, resp., of upper-triangular matrices with
units on the diagonal. The corresponding factor-
ization problem in G is well known in the linear
algebra under the name of LR factorization, and is
related t o the Gaussian elimination. From [33] and
the well-known expression of the diagonal ele-
ments of the lower-triangular factor in the LR
factorization through the minors of the factorized
matrix, we find:
a
n
ðtÞ¼
nþ1
ðtÞ
n1
ðtÞ
2
n
ðtÞ
a
n
ð0Þ½35
where
n
(t) is the upper-left n n minor of
the matrix exp(tL(0)). If L(t) is the Lax matrix
along the solution of the Toda flow ((L) = L), then
the sampling of the matrix exp(L(t)) at the integer
times t 2 Z coincides with the result of application
of the Rutishauser’s LR algor ithm to the matrix
exp(L(0)). The LR algorithm applied to the matrix
I þ hL(0) is nothing other but the Ba¨ cklund trans-
formation [21] in the open-end situation.
Finite Periodic Toda Lattice
Model
A different reduction of the infinite Toda lattice to a
finite-dimensional system appears by imposing peri-
odic boundary conditions, q
nþN
(t) q
n
(t) for all
n 2 Z (of course, such relations hold also for the
Flaschka–Manakov variables a
n
, b
n
). The Hamilton
Toda Lattices 239
function of the resul ting system with N degrees of
freedom is
H
2
ðp; q Þ¼
1
2
X
n2Z=NZ
p
2
n
þ
X
n2Z=NZ
e
q
nþ1
q
n
½36
This system consists of N particles q
n
(n = 1, ..., N),
and it is always assumed that q
Nþ1
q
1
and q
0
q
N
.
Thus, the potential energy in [36] differs from the
potential energy in [24] by one additional term e
q
1
q
N
.
However, this modest difference leads to much more
complicated dynamics of the system (quasiperiodic
instead of scattering). It is convenient to replace
infinite matrices in the Lax representation [12] by
finite ones, of size N N, but depending on an
additional parameter (called the spectral parameter):
L ¼
X
n2Z=NZ
b
n
E
n;n
þ
1
X
n2Z=NZ
a
n
E
n;nþ1
þ
X
n2Z=NZ
E
nþ1;n
½37
A
þ
¼
X
n2Z=NZ
b
n
E
n;n
þ
X
n2Z=NZ
E
nþ1;n
½38
A
¼
1
X
n2Z=NZ
a
n
E
n;nþ1
½39
The Lax representation [12] holds identically in ,
so that the spectral parameter drops out of the
equations of motion. Note that, unlike the open-end
case, L is no more a tridiagonal matrix, because of
the nonvanishing entries in the positions (N,1)
and (1, N).
Inverse-Spectral Transformation
Solution of the periodic lattice in terms of multi-
dimensional theta functions has been given indepen-
dently by E Date and S Tanaka, and by I Krichever
in 1976. In this case, the set of the spectral data is
more complicated; it includes:
a hyperelliptic Riemann surface R of genus N 1
determined by the eigenvalues of the periodic
boundary-value proble m for the operator L, or,
in other words , by the equation R(, ) =
det(L() I) = 0; and
N 1 points P
k
on R, which correspo nd to the
eigenvalues of L with vanishing boundary
conditions.
Due to [12], the Riemann surface R itself is an
integral of motion, and the evolution of points P
k
is
such that the image of the divisor P
1
þþP
N1
under the Abel map moves along a straight line in
the Jacobi variety of R. Solution of the inverse-
spectral problem is given in terms of
multidimensional theta-functions by formula [35]
with
n
(t) = (nU tV þ D), where U, V, D are
certain vectors on the Jacobian of R (the first two
of them depending on the spectrum R only).
Loop Algebras
The periodic Toda lattice can be included into the
general AKS scheme, if one interprets the Lax
matrix L as an element of the loop algebra g
which consists of Laurent polynomials (in ) with
coefficients from gl(N), singled out by the additional
condition
g ¼ LðÞ2glðNÞ½;
1
:LðÞ
1
¼ Lð!Þ
where =diag(1,!,...,!
N 1
), ! = exp(2i=N). Sub-
algebras g
consist of Laurent polynomials with
respect to non-negative, resp., strictly negative
powers of . The Lie group G corresponding to the
Lie algebra g consists of GL(N)-valued functions
U() of the complex parameter , regular in
CP
1
n{0,1} and satisfying U()
1
=U(!). Its
subgroups G
corresponding to the Lie algebras g
are singled out by the following conditions: elements
of G
þ
are regular in the neighborhood of = 0,
while elements of G
are regular in the neighbor-
hood of =1 and take at = 1 the value I. The
corresponding factorization is called the generalized
LR factorization. As opposed to the open-end case,
finding such a factorization is a problem of the
Riemann–Hilbert type which is solved in terms of
algebraic geometry and theta-functions rather than in
terms of linear algebra and exponential functions. This
approach to the periodic Toda lattice is due to Reyman
and Semenov-Tian-Shansky (1979) and, indepen-
dently, to M Adler and P van Moerbeke (1980).
Generalizations: Lie-Algebraic Systems
The AKS interpretation of the finite Toda lattices
leads directly to their generalizations by replaci ng
the algebra gl(N), resp., the loop algebra over gl(N),
by simple Lie algebras, resp. affine Lie algebras.
These generalized Toda systems were introduced in
1976 by O Bogoyavlensky and solved in 1979
independently by M Olshanetsky, A Perelomov,
and by B Kostant.
Simple Lie Algebras
Let g be a simple Lie algebra (complex or real split),
and h its Cartan subalgebra. Let further =
þ
[
be the root system of g , decomposed into the sets of
positive roots
þ
and the set of negative roots
.
One has a direct vector space g = g
þ
g
,whereg
þ
is
spanned by t he root spaces for positive roots and by h,
240 Toda Lattices
while g
is spanned by the root spaces for negative
roots (Borel decomposition). For 2 let E
be a
corresponding root vector. So, [H, E
] = (H)E
for all
H 2 h. The root 2 h
may be identified with H
2 h
defined by hH
, Hi= (H) for all H 2 h. It is easy to
deduce that [E
, E
] = c
H
, where c
= hE
, E
i.
The system of simple roots will be denoted by
þ
.
The generalized Toda lattice for the Lie algebra g
is the following system of differential equations on
h h:
_
Q ¼ P
_
P ¼
X
2
e
ðQÞ
½E
; E
¼
X
2
c
e
ðQÞ
H
½40
This system can be given a Hamiltonian formula-
tion, with the Hamilton function
H
g
¼
1
2
hP; Piþ
X
2
c
e
ðQÞ
½41
It is completely integrable, and has a Lax represen-
tation [12] with
L ¼ P þ
X
2
E
þ
X
2
e
ðQÞ
E
½42
A
þ
¼ P þ
X
2
E
; A
¼
X
2
e
ðQÞ
E
½43
The usual open-end Toda lattice corresponds to the
algebra sl(N) (series A
N1
), so that the Hamilton
function [24] can be denoted by H
A
N1
. The
Hamilton functions of the generalized lattices
corresponding to other classical algebras so(2N þ
1) (series B
N
), sp(N) (series C
N
), and so(2N) (series
D
N
) can be written in the canonically con jugate
variables q
n
, p
n
(n = 1, ..., N)as
H
g
ðp;qÞ¼H
A
N1
ðp;qÞþ
e
q
N
; g ¼B
N
e
2q
N
; g ¼C
N
e
q
N
q
N1
; g ¼D
N
8
>
<
>
:
½44
Affine Lie Algebras
Turning to the generalizations of the periodic Toda
lattice, let be a Coxeter automorphism of a simple
complex algebra g , the order of being m. Introduce
the loop algebra g as the Lie algebra of Laurent
polynomials
g ¼ LðÞ2g ½;
1
: ðLðÞÞ ¼ Lð!Þ
where ! = exp(2i=m). Denote by g
j
the eigenspaces
of corresponding to the eigenvalues !
j
(j 2 Z=mZ).
Set a = g
0
, and let s denote the dimension of a .By
definition of the Coxeter automorphism, a is an
abelian subalgebra of g .Denoteby the set of 2
a
for which there exist nonzero elements E
2 g
1
with
[H, E
] = (H)E
for all H 2 a . The elements E
2
g
1
are defined similarly. It can be shown that
contains s þ 1 elements, so that between them there
exists exactly one linear relation. The elements of
are called simple weights of the loop algebra g.TheLie
algebra g is a direct sum of its two subspaces g
consisting of Laurent polynomials with non-negative,
resp., with strictly negative powers of ;these
subspaces are also Lie subalgebras.
Now the generalized Toda lattice related to the loop
algebra g can be introduced as the system of differential
equations on a a , which looks formally exactly as
[40], and has the Hamilton function which looks
exactly as [41], but with the set of simple roots of g
being replaced by the set of simple weights of g.The
matrices participating in the Lax representation [12]
belong now to the loop algebra g:
LðÞ¼P þ
X
2
E
þ
1
X
2
e
ðQÞ
E
½45
A
þ
ðÞ¼P þ
X
2
E
A
ðÞ¼
1
X
2
e
ðQÞ
E
½46
For the classical series of loop algebras, the
Hamilton functions H
g
in the canonically conjugate
variables q
n
, p
n
(n = 1, ..., N) can be presented as
H
g
ðp;qÞ¼H
A
N1
ðp;qÞ
þ
e
q
N
þe
q
1
þq
2
; g ¼B
ð1Þ
N
e
2q
N
þe
2q
1
; g ¼C
ð1Þ
N
e
q
N
q
N1
þe
q
1
þq
2
; g ¼D
ð1Þ
N
e
2q
N
þe
q
1
þq
2
; g ¼A
ð2Þ
2N1
e
q
N
þe
2q
1
; g ¼A
ð2Þ
2N
e
q
N
þe
q
1
; g ¼D
ð2Þ
Nþ1
8
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
:
½47
Actually, one can find even more general integrable
systems of the Toda type: one can add to H
A
N1
(p,q)
any of the two potentials e
q
N
q
N1
or e
q
N
þe
2q
N
on one end combined with any of the two potentials
e
q
1
þq
2
or e
q
1
þe
2q
1
on the other end, where
,,, are arbitrary constants. This result is due
to E Sklyanin (1987).
Generalizations: Lattices with
Nearest-Neighbor Interactions
There exist further integrable lattice systems with
the nearest-neighbor interaction apart from the
classical exponential Toda lattice [1]. Those of the
Toda Lattices 241
type
€
q
n
= r(
_
q
n
)(g(q
nþ1
q
n
) g(q
n
q
n1
)) have
been classified by R Yamilov in 1982, and the list
contains, apart from the usual Toda lattice [1], the
following ones:
€
q
n
¼
_
q
n
e
q
nþ1
q
n
e
q
n
q
n1
ðÞ½48
€
q
n
¼
_
q
n
ðq
nþ1
2q
n
þ q
n1
Þ½49
€
q
n
¼ð
_
q
2
n
2
Þ
1
q
nþ1
q
n
1
q
n
q
n1
½50
€
q
n
¼ð
_
q
2
n
2
Þ cothðq
nþ1
q
n
Þð
cothð q
n
q
n1
ÞÞ ½51
Equations [48] are known as the ‘ ‘modified Toda
lattice.’’ Equations [49] describe the ‘‘dual Toda lattice’ ’
which was instrumental in the original discovery by
Toda (see Toda (1989)). All systems [49]–[51] can be
obtained from [11] via suitable parametrizations of the
variables a
n
, b
n
by canonically conjugate ones q
n
, p
n
,
similar to [10] for [1],seeSuris (2003).
A remarkable discovery of the integrable relati-
vistic Toda lattice is due to S Ruijsenaars (1990).
This lattice with the equations of motion
€
q
n
¼ð1 þ
_
q
n
Þð1 þ
_
q
nþ1
Þ
e
q
nþ1
q
n
1 þ
2
e
q
nþ1
q
k
ð1 þ
_
q
n1
Þ
e
q
n
q
n1
1 þ
2
e
q
n
q
n1
½52
can be considered as the perturbation of the usual
Toda lattice with the small parameter (the inverse
speed of light).
A class of integrable lattice systems of the relativistic
Toda type
€
q
n
= r(
_
q
n
)(
_
q
nþ1
f (q
nþ1
q
n
)
_
q
n1
f (q
n
q
n1
) þ g(q
nþ1
q
n
) g(q
n
q
n1
)) is richer than
that of the Toda type, and has been isolated by Yu
B Suris and by V Adler and A Shabat in 1997. The list
contains, apart from the relativistic Toda lattice [52],
two more -perturbations of the usual Toda lattice [1]:
€
q
n
¼ð1 þ
_
q
nþ1
Þe
q
nþ1
q
n
ð1 þ
_
q
n1
Þe
q
n
q
n1
2
e
2ðq
nþ1
q
n
Þ
e
2ðq
n
q
n1
Þ
½53
€
q
n
¼ð1
_
q
n
Þ
2
ð1
_
q
nþ1
Þe
q
nþ1
q
n
ð1
_
q
n1
Þe
q
n
q
n1
½54
two -perturbations of the modified Toda lattice [48]:
€
q
k
¼
_
q
n
e
q
nþ1
q
n
e
q
n
q
n1
þ
_
q
nþ1
e
q
nþ1
q
n
1 þ e
q
nþ1
q
n
_
q
n1
e
q
n
q
n1
1 þ e
q
n
q
n1
½55
€
q
n
¼
_
q
n
ð1
_
q
n
Þð1
_
q
nþ1
Þ
e
q
nþ1
q
n
1 þ e
q
nþ1
q
n
ð1
_
q
n1
Þ
e
q
n
q
n1
1 þ e
q
n
q
n1
½56
two -perturbations of the dual Toda lattice [49]:
€
q
n
¼
_
q
n
ðq
nþ1
2q
n
þ q
n1
Þþ
_
q
nþ1
_
q
n
1 þ ðq
nþ1
q
n
Þ
_
q
n
_
q
n1
1 þ ðq
n
q
n1
Þ
½57
€
q
n
¼
_
q
n
ð1 þ
2
_
q
n
Þ
q
nþ1
q
n
_
q
nþ1
1 þ ðq
nþ1
q
n
Þ
q
n
q
n1
_
q
n1
1 þ ðq
n
q
n1
Þ
½58
and one -perturbation of each of the systems [50]
and [51]:
€
q
n
¼
_
q
2
n
2
q
nþ1
q
n
_
q
nþ1
ðq
nþ1
q
n
Þ
2
ðÞ
2
q
n
q
n1
_
q
n1
ðq
n
q
n1
Þ
2
ðÞ
2
!
½59
€
q
n
¼
1
2
_
q
2
n
2
sinh 2ðq
nþ1
q
n
Þ
1
sinhð2Þ
_
q
nþ1
sinh
2
ðq
nþ1
q
n
Þsinh
2
ðÞ
sinh 2ðq
n
q
n1
Þ
1
sinhð2Þ
_
q
n1
sinh
2
ðq
n
q
n1
Þsinh
2
ðÞ
!
½60
A detailed study of all these systems, their interrelations,
and time discretizations can be found in Suris (2003).
There exist also lattices with more complicated
nearest-neighbor interactions, involving elliptic
functions. They were discovered by A Shabat and
R Yamilov (1990), and by I Krichever (2000). For
example, the nonrelativistic elliptic Toda lattice is
governed by the equations
€
q
n
¼
_
q
2
n
1
Vðq
n
; q
nþ1
ÞþVðq
n
; q
n1
ÞðÞ½61
where V(q, q
0
) = (q þ q
0
) þ (q q
0
) (2q)isan
elliptic function in both arguments q, q
0
(here (q)is
the Weierstrass -function).
Further Developments
and Generalizations
Sato’s Theory
Formulas [6], [31], and [35] have the same structure,
with the case-dependent functions
n
(t) given by the
determinants [7] for the multisoliton solution in the
242 Toda Lattices
infinite case, by the Hankel determinants [32] or by the
minors of the matrix exp(L(0)) in the open case, and
by the multidimensional theta functions in the periodic
case. All these seemingly different objects are actually
particular cases of a beautiful construction due to M
Sato (1981), developed by E Date, M Jimbo, M
Kashiwara, T Miwa (1981–83), and by G Segal and G
Wilson (1985), which provides one of the major
unifying schemes for the theory of integrable
systems. In this construction, integrable systems are
interpreted as simple dynamical systems on an infinite-
dimensional Grassmannian. The -function (first
invented by R Hirota in 1971) receives in this theory
a representation-theoretical interpretation in terms of
the determinant bundle over the Grassmannian.
Band Matrices
The Lax matrices [13] and [16] in the Manakov–
Flaschka variables can be easily generalized: in the
symmetric matrix L
0
one can admit nonvanishing
elements in the band of the width 2s þ 1> 3 around
the main diagonal, in the Heisenberg matrix L one
can admit more nonvanishing diagonals in the
upper-triangle part. A systematic presentation of
a large body of relevant results is given in
Kupershmidt (1985). In the setting of finite lattices,
the integrability of such systems becomes a non-
trivial problem (as opposed to the tridiagonal
situation), because the number of independent
conjugation-invariant functions tr(L
s
) becomes
less than the number of degrees of freedom. An
effective approach to this problem based on the
semi-invariant functions ha s been found by P Deift,
L-Ch Li, T Nanda, and C Tomei in 1986.
Two-Dimensional Toda Lattices
Up to now, we considered integrable lattices with
one continuous and one discrete independent vari-
ables. This allows for a further generalization.
Integrable systems with two continuous and one
discrete independent variables are well known and
widely used as models of the field theory. For
instance, the Toda field theory deals with the system
ðq
n
Þ
xy
¼ e
q
nþ1
q
n
e
q
n
q
n1
½62
introduced in the soliton theory by AMikhailovin
1979. This two-dimensional system admits all possi-
ble kinds of reductions and generalizations mentioned
above for the usual Toda lattice. In particular, the
periodic two-dimensional Toda lattice is referred to
as the affine Toda field theory (with the prominent
example of the sine-Gordon field which corresponds
to the period 2). Later, it was realized that the
equivalent equation ( log v
n
)
xy
= v
nþ1
2v
n
þ v
n1
,
which is obtained from [62] by setting v
n
=
exp(q
nþ1
q
n
), already appeared in studies by
G Darboux in the 1880s, as the equation satisfied
by the Laplace invariants of the chain of Laplace
transformations of a given conjugate net. This
relation to the classical differential geometry was
extensively studied by G Darboux, G Tzitze´ica, and
others long before the advent of the theory of
integrable systems. Another link to the differential
geometry is a more recent observation, and relates the
two-dimensional Toda lattice, with the d’Alembert
operator ()
xy
on the left-hand side of [62] replaced by
the Laplace operator ()
z
z
, to harmonic maps. For
instance, the sinh-Gordon equation u
z
z
= sinh u gov-
erns harmonic maps from C into the unit sphere S
2
,
which can be interpreted also as Gauss maps of the
constant mean curvature surfaces in R
3
. A review of
this topic can be found in Guest (1997).
Discretization of Toda lattices, nonabelian Toda
Lattices, quantization of Toda lattices, dispersionless
limit of Toda lattices, etc., are only some of the
further relevant topics, which cannot be discussed in
any detail in the restricted frame of this article, and
the same holds, unfortunately, for such fascinating
applications of the Toda lattice as the Frobenius
manifolds, Laplacian growth problem, quantum
cohomology, random matrix theory, two-dimensional
gravity, etc.
See also: Ba
¨
cklund Transformations; Bi-Hamiltonian
Methods in Soliton Theory; Classical r-Matrices,
Lie Bialgebras, and Poisson Lie Groups; Current Algebra;
Dynamical Systems and Thermodynamics; Functional
Equations and Integrable Systems; Integrable Discrete
Systems; Integrable Systems and Discrete Geometry;
Integrable Systems and the Inverse Scattering Method;
Integrable Systems: Overview; Lie Groups: General
Theory; Multi-Hamiltonian Systems; Quantum
Calogero–Moser Systems; Separation of Variables for
Differential Equations; Solitons and Kac–Moody Lie
Algebras; WDVV Equations and Frobenius Manifolds.
Further Reading
Adler M (1979) On a trace functional for formal pseudo–differential
operators and the symplectic structure for Korteweg–de Vries
type equations. Invent. Mathematics 50: 219–248.
Adler VE and Shabat AB (1997) On a class of Toda chains. Teor.
Mat. Phys. 111: 323–334 (in Russian; English translation:
Theor. Math. Phys. 111: 647–657); Generalized Legendre
transformations. Teor. Mat. Phys. 112: 179–194 (in Russian;
English translation: Theor. Math. Phys. 112: 935–948).
Adler M and van Moerbeke P (1980) Completely integrable systems,
Kac–Moody algebras and curves; Advances in Mathematics 38:
267–317. Linearization of Hamiltonian systems, Jacobi varieties
and representation theory. Advances in Mathematics 38: 318–379.
Toda Lattices 243
Bogoyavlensky OI (1976) On perturbations of the periodic Toda
lattice. Communications in Mathematical Physics 51: 201–209.
Date F, Jimbo M, and Miwa T (1982–83) Method for generating
discrete soliton equations. I–V. Journal of the Physical Society
of Japan 51: 4116–4124, 4125–4131; 52: 388–393, 761–765,
766–771.
Date E and Tanaka S (1976) Analogue of inverse scattering theory for
the discrete Hill’s equation and exact solutions for the periodic
Toda lattice. Progress in Theoretical Physics 55: 457–465.
Deift P, Li L-C, Nanda T, and Tomei C (1986) The Toda flow on
a generic orbit is integrable. Communications in Pure and
Applied Mathematics 39: 183–232.
Flaschka H (1974) On the Toda lattice II. Inverse scattering
solution. Progress in Theoretical Physics 51: 703–716.
Guest MA (1997) Harmonic Maps, Loop Groups, and Integrable
Systems. Cambridge: Cambridge University Press.
He´non M (1974) Integrals of the Toda lattice. Physical Review B
9: 1921–1923.
Krichever IM (1978) Algebraic curves and nonlinear difference
equations. Uspekhi Mat. Nauk 33: 215–216 (in Russian).
Krichever I (2000) Elliptic analog of the Toda Lattice. Int. Math.
Res. Notes 8: 383–412.
Kupershmidt BA (1985) Discrete Lax equations and differential-
difference calculus. Asterisque 123: 212.
Magri F (1978) A simple model of the integrable Hamiltonian
equation. Journal of Mathematical Physics 19: 1156–1162.
Manakov SV (1974) On the complete integrability and stochas-
tization in discrete dynamical systems. Zh. Exp. Theoretical
Physics 67: 543–555 (in Russian; English translation: Soviet
Physics JETP 40: 269–274).
Mikhailov AV (1979) Integrability of a two–dimensional general-
ization of the Toda chain. Pis’ma Zh. Eksp. Teor. Fiz. 30: 443–448
(in Russian; English translation: JETP Letters 30: 414–418).
Moser J (1975) Finitely many mass points on the line under the
influence of an exponential potential – an integrable system.
Lecture Notes Physics 38: 467–497.
Olshanetsky MA and Perelomov AM (1979) Explicit solutions of
classical generalized Toda models. Invent. Math. 54: 261–269.
Reyman AG and Semenov-Tian-Shansky MA (1979–81) Reduc-
tion of Hamiltonian systems, affine Lie algebras and Lax
equations. I, II. Invent. Math. 54: 81–100, 63: 423–432.
Reyman AG and Semenov-Tian-Shansky MA (1994) Group
theoretical methods in the theory of finite dimensional
integrable systems. In: Arnold VI and Novikov SP (eds.)
Encyclopaedia of Mathematical Science, vol. 16. Dynamical
Systems VII, pp. 116–225. Berlin: Springer.
Ruijsenaars SNM (1990) Relativistic Toda systems. Communica-
tions in Mathematical Physics 133: 217–247.
Sato M and Sato Y (1983) Soliton equations as dynamical systems
on infinite-dimensional Grassmann manifold. In: Nonlinear
Partial Differential Equations in Applied Science (Tokyo,
1982), pp. 259–271. North-Holland: Amsterdam.
Segal G and Wilson G (1985) Loop groups and equations of KdV
type. Inst. Hautes E
´
tudes Sci. Publ. Math. 61: 5–65.
Shabat AB and Yamilov RI (1990) Symmetries on nonlinear
chains. Algebra i Analiz 2: 183–208 (in Russian; English
translation: Leningrad Mathematical Journal 2: 377–400).
Sklyanin EK (1987) Boundary conditions for integrable equations.
Funkts. Anal. Prilozh. 21: 86–87 (in Russian; English
translation: Funct. Anal. Appl. 21: 164–166).
Suris YB (1997) New integrable systems related to the relativistic Toda
lattice. Journal of Physics A: Math. and Gen. 30: 1745–1761.
Suris YuB (2003) The Problem of Integrable Discretization:
Hamiltonian Approach. Basel: Birkha¨user.
Toda M (1967) Vibration of a chain with nonlinear interaction.
Journal of Physical Society Japan 22: 431–436.
Toda M (1989) Theory of Nonlinear Lattices. Berlin: Springer.
Toda M and Wadati M (1975) A canonical transformation for the
exponential lattice. Journal of the Physical Society of Japan
39: 1204–1211.
Yamilov RI (1982) On the classification of discrete equations.
In: Integrable Systems, Ufa, 95–114 (in Russian).
Toeplitz Determinants and Statistical Mechanics
E L Basor, California Polytechnic State University,
San Luis Obispo, CA, USA
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
A finite Toeplitz matrix is an n n matrix with the
following structure:
a
0
a
1
a
2
a
nþ1
a
1
a
0
a
1
a
nþ2
a
2
a
1
a
0
a
nþ3
.
.
.
.
.
.
.
.
.
.
.
.
a
n1
a
n2
a
n3
a
0
0
B
B
B
B
B
@
1
C
C
C
C
C
A
½1
The entries depend on the difference i j and hence
they are constant down all the diagonals. There are
two cases when the determinant is easy to compute.
One is when the matrix is upper- or lower-triangular
and the determinant is a
n
0
. The other case is when
the matrix is of the form
a
0
a
n1
a
n2
a
1
a
1
a
0
a
n1
a
2
a
2
a
1
a
0
a
3
.
.
.
.
.
.
.
.
.
.
.
.
a
n1
a
n2
a
n3
a
0
0
B
B
B
B
B
@
1
C
C
C
C
C
A
½2
In this latter case, the matrix is called a circulant
matrix and the eigenvalues are given by the formula
n
ðe
i2k=n
Þ; 0 k n 1
where
n
ðe
i
Þ¼
X
n1
j¼0
a
i
e
ij
244 Toeplitz Determinants and Statistical Mechanics
The corresponding eigenvector for eigenvalue
n
(e
i2k=n
)is
½1; e
i2k=n
; ...; e
i2kðn1Þ=n
This can be verified by direct comput ation. The role
of circulant matrices will not be emphasized in this
article, although they are used in the computation of
the generating function for certain dimer configura-
tions and also in applications using the discrete
Fourier transform.
The most common way to generate a finite
Toeplitz matrix is with the Fourier coefficients of
an integrable function. Let : T ! C be a function
defined on the unit circle with Fourie r coefficients
k
¼
1
2
Z
ðe
i
Þe
ik
d ½3
We define T
n
() to be the Toeplitz matrix:
T
n
ðÞ¼
ij
n1
i;j¼0
A basic problem that in large part has been
motivated by statistical mechanics is to determine
the behavior of the asymptotics of the determinant
of T
n
()asn !1. The determinant will be
referred to as D
n
(), where is called the genera ting
function of the determinant. If the generating
function has the property that its Fourier coefficients
vanish for negative index (positive index) then the
corresponding matrix is lower-triangular (upper-
triangular) and hence the determinant is
n
0
. For
other cases, the determinant is not easy to determine
and requires additional mathematical machinery.
Some of the primary motivation to study the
determinant of these matrices comes from the two-
dimensional Ising model. We consider the Onsager
lattice in the absence of a magnetic field with sites
labeled by
ði; jÞ; 0 i M ; 0; j N
and with a value
i, j
= 1 assigned to each site. In
the Ising model,
i, j
signifies the state of the spin at
the site (i, j). To each possible configuration of spins,
we define an energy
EðÞ¼E
1
X
i;j
i;j
iþ1;j
E
2
X
i;j
i;j
i;jþ1
Let
Z ¼
X
e
EðÞ
be the partition funct ion. Then the probability of a
given configuration is
1
Z
e
EðÞ
Here E
1
, E
2
, and = 1=kT are, without loss of
generality, assumed to be positive constants, T is the
temperature, and k is the Boltzmann constant. If X
is a random variable defined on the space of
configurations, the expectation is given by
EðXÞ¼
1
Z
X
¼1
XðÞe
EðÞ
Let n be fixed for the moment and assume toroidal
boundary conditions for the lattice and then let
N, M !1. It is known that the random variable
XðÞ¼
0;0
0;n
has expectation h
0,0
0,n
i given by D
n
(), where
ðe
i
Þ¼
ð1
1
e
i
Þð1
2
e
i
Þ
ð1
1
e
i
Þð1
2
e
i
Þ
1=2
1
¼ z
1
1 z
2
1 þ z
2
;
2
¼ z
1
1
1 z
2
1 þ z
2
and
z
1
¼ tanh E
1
; z
2
¼ tanh E
2
The square root is taken so that (e
i
) = 1. This
formula was first stated by Onsager and later
verified in a difficult computation by Montroll,
Potts, and Ward.
The spontaneous magnetization M for the Ising
model is defined by
M
2
¼ lim
n!1
h
0;0
0;n
i¼lim
n!1
D
n
ðÞ
Note that it is the square root of the correlation
between two distant sites. Hence, the asymp totics of
the Toeplitz determinants will determine whether
the magnetization is positive or tends to zero as
n !1.
Strong Szego¨ Limit Theorem
To determine the behavior of the determinants, we
need to analyze the genera ting function . Let us
first consider the case where
2
< 1. (It is always the
case that 0 <
1
< 1.) This generating function is
differentiable, nonzero and has winding number
zero, and it is for functions of this type that a
second-order expansion of the Toeplitz determinants
can be described. The expansion first formulated by
Szego¨ , in response to the question concerning the
Toeplitz Determinants and Statistical Mechanics 245
spontaneous magnetization, is called the ‘‘strong
Szego¨ limit theorem.’’
Before proving the Szego¨ theorem, it should be
remarked that we can view the finite Toeplitz matrix
as a truncation of an infinite array,
0
1
2
1
0
1
.
.
.
2
1
0
.
.
.
.
.
.
.
.
.
.
.
.
0
B
B
B
B
B
@
1
C
C
C
C
C
A
½4
The above infinite array is the matrix representation
for the Toeplitz operator
TðÞ : H
2
! H
2
defined by
TðÞf ¼ Pðf Þ
where H
2
is the Hardy space
ff 2 L
2
ðTÞjf
k
¼ 0; k < 0g
the function 2 L
1
(T), and P is the orthogonal
projection of L
2
(T) onto H
2
. The matrix representa-
tion given in [4] is with respect to the Hilbert space
basis of H
2
,
fe
ik
j0 k < 1g
and is called the symbol of the operator. Now
define P
n
: H
2
! H
2
by
P
n
ðf
0
; f
1
; f
2
; ...Þ¼ðf
0
; f
1
; f
2
; ...; f
n1
; 0; 0; ...Þ
The finite Toeplitz matrix can be thought of as the
upper-left corner of the array given in [4] or as
P
n
T()P
n
.
To prove the Strong Szego¨ limit theorem, we
introduce the Banach algebra B of bounded func-
tions f satisfying
P
1
k = 1
jkkf
k
j
2
< 1.
Theorem 1 (Strong Szego¨ limit theorem). Assume
=
þ
, where
have logarithms in B. Suppose
log
, log
þ
2 H
2
. Then
lim
n!1
D
n
ðÞ=GðÞ
n
¼ EðÞ¼exp
X
1
k¼1
ks
k
s
k
!
where G() = exp (( log )
0
) and s
k
= log
k
.
Since B is a Banach algebra, it follows that if
log
belong to B so do
;
þ
;
1
þ
;
1
;;
1
and hence they are bounded. Since
þ
is in H
2
as
well, its Fourier coefficients vanish for negative
index and the Toeplitz operator has a corresponding
infinite array that is lower-triangular. The Fourier
coefficients vanish for positive index for
and
hence the infinite array is upper triangular. From
this, it follows that
Tð
þ
ÞTð
1
þ
Þ¼Tð
1
ÞTð
Þ¼I ½5
Tð
ÞTð
þ
Þ¼TðÞ½6
and
P
n
Tð
þ
Þ¼P
n
Tð
þ
ÞP
n
P
n
Tð
ÞP
n
¼ Tð
ÞP
n
½7
This yields
D
n
ðÞ¼det T
n
ðÞ¼det P
n
TðÞP
n
½8
¼ det P
n
Tð
þ
ÞTð
1
þ
ÞTðÞTð
1
ÞTð
ÞP
n
½9
¼detP
n
Tð
þ
ÞP
n
Tð
1
þ
ÞTðÞTð
1
ÞP
n
Tð
ÞP
n
½10
z ¼ det P
n
Tð
þ
ÞP
n
detðP
n
Tð
1
þ
ÞTðÞTð
1
ÞP
n
Þ
det P
n
Tð
ÞP
n
½11
The determinants of the right-hand side and the left-
hand side of the above expression are ((
)
0
)
n
,
respectively. Now given the Banach algebra condi-
tions imposed on the symbol , it follows that the
operator
Tð
1
þ
ÞTðÞTð
1
Þ
is of the form I þ K, where K is trace class. Hence,
the eigenvalues
i
of K satisfy
X
j
i
j < 1
and the infinite (Fredholm) determinant of I þ K is
defined. To verify the claim that the operator
Tð
1
þ
ÞTðÞTð
1
Þ¼Tð
1
þ
ÞTð
ÞTð
þ
ÞTð
1
Þ
is I plus a trace class operator, we use the identity
TðfgÞTðf ÞTðgÞ¼Hðf ÞHð
~
gÞ½12
where H(f) has matrix form (f
iþjþ1
)
1
i,j = 0
,and
˜
g(e
i
) = g(e
i
). Our Banach algebra conditions
show that if f is in B then the operator H(f) satisfies
P
i, j
ja
ij
j
2
< 1, where the a
ij
are the matrix entries
of the operator. Any operator satisfying this is called
a Hilbert–Schmidt operator, and it is known that the
product of two Hilbert–Schmidt is trace class.
Applying the identity to
T
1
þ
Tð
Þ
shows that this operator is T(
1
þ
) plus trace class.
The operator
Tð
þ
ÞT
1
246 Toeplitz Determinants and Statistical Mechanics
is thus T(
þ
1
) plus trace class and one more
application of the identity combined with the fact
that trace class operators form an ideal yield the
desired result.
From the theory of infinite determinants, as
n !1,
det P
n
T
1
þ
TðÞT
1
P
n
½13
converges to
det T
1
þ
TðÞT
1
½14
At this point, we have proved that
lim
n!1
D
n
ðÞ=ðð
Þ
0
Þ
n
ðð
þ
Þ
0
Þ
n
¼ lim
n!1
det P
n
T
1
þ
TðÞT
1
P
n
¼ det T
1
þ
TðÞT
1
½15
It only remains to identify the constants. To see that
GðÞ¼ðð
Þ
0
Þ
n
ðð
þ
Þ
0
Þ
n
we note that
GðÞ¼expððlog Þ
0
Þ¼exp
1
2
Z
2
0
log ðe
i
Þd
¼ exp
1
2
Z
2
0
ðlog
ðe
i
Þþlog ðe
i
ÞÞd
¼ expð log
Þ
0
expðlog
þ
Þ
0
¼ð
Þ
0
ð
þ
Þ
0
To compute the determinant of
T
1
þ
TðÞT
1
we write
det T
1
þ
TðÞT
1
¼ det T
1
þ
Tð
þ
ÞT
1
¼ det T
1
þ
Tð
ÞTð
þ
ÞT
1
This last expression is the form
e
A
e
B
e
A
e
B
where
A ¼Tðlog
þ
Þ and B ¼ Tðlog
Þ
If AB BA is trace class then
det e
A
e
B
e
A
e
B
¼ e
trðABBAÞ
The operator AB BA is
Tðlog
þ
ÞTðlog
ÞþTðlog
ÞTðlog
þ
Þ
which equals
Tðlog
þ
ÞTðlog
ÞþTððlog
Þðlog
þ
ÞÞ
and, by the identity from eqn [12], becomes
Hðlog
ÞHðlog
þ
Þ
It can be directly computed that
trðHðlog
ÞHðlog
þ
ÞÞ
equals
X
1
k¼1
ks
k
s
k
and the theorem is proved.
Returning to the Ising model, one needs to
compute the asymp totics of the determinants for
the generating function
ðe
i
Þ¼
ð1
1
e
i
Þð1
2
e
i
Þ
ð1
1
e
i
Þð1
2
e
i
Þ
1=2
The term G() = 1 and for k > 0
ks
k
s
k
¼
1
4
2k
1
k
þ
2k
2
k
þ 2
k
1
k
2
k
from which it follows that
lim
n!1
D
n
ðÞ¼
1
2
1
1
2
2
ð1
1
2
Þ
2
"#
1=4
Recalling the definition of
1
and
2
yields
lim
n!1
h
0;0
0;n
i¼ 1
1
ðsinh 2E
1
sinh 2E
2
Þ
2
"#
1=4
or the spontaneous magnetization M as
M ¼ 1
1
ðsinh 2E
1
sinh 2 E
2
Þ
2
!
1=8
In order for this computation to be valid, it was
necessary for 0 <
2
< 1, and by elementary com-
putations one can show that this is equivalent to the
inequality
sinh 2E
1
sinh 2E
2
> 1
Nonsmooth Symbols or T = T
c
A problem occurs in the analysis just outlined when
the inequality 0 <
2
< 1 does not hold. There are
two separate possibilities,
2
> 1or
2
= 1. First, we
consider the latter case. For fixed E
1
and E
2
, this
happens for exactly one fixed value of the constant
c
= 1=kT
c
and the corresponding temperature T
c
is
called the critical temperature. The ‘‘strong Szego¨
Toeplitz Determinants and Statistical Mechanics 247