Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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(L
1
)
J
,
(L
2
)
J
, ...,
(L
N
)
J
has weight N
1
. The ensemble
N
converges to a metastate
J
as N !1, in the
following sense: for every (nice) function g on states
(e.g., a function of finitely many correlations),
lim
N!1
N
1
X
N
‘¼1
gð
ðL
‘
Þ
Þ¼
Z
gðÞ d
J
ðÞ½5
The information contained in
J
effectively specifies
the fraction of cube sizes L
‘
which the system spends
in differe nt (possibly mixed) thermodynamic states
as ‘ !1.
A different, but in the end equivalent, approach
based on J-randomness is due to Aizenman and
Wehr (1990). Here one considers the random pair
(J,
(L)
J
), defined on the underlying probability space
of J, and takes the limit
y
(with conditional
distribution
y
J
, given J), via finite-dimensional
distributions along some subsequence. The details
are omitted here, and the reader is referred to the
work by Aizenman and Wehr (1990) and Newm an
and Stein (1998a). We note, however, the important
result that a ‘‘deterministic’’ subsequence of volumes
can be found on which [5] is valid and also (J,
(L)
J
)
converges, with
y
J
=
J
(Newman and Stein
1998a).
In what follows we use the term ‘‘metastate’’ as
shorthand for the
J
constructed using periodic
boundary conditions on a sequence of volumes
chosen independently of the couplings, and along
which
J
=
y
J
. We choose periodic boundary
conditions for specificity; the results and claims
discussed are expected to be independent of the
boundary conditions used, as long as they are
chosen independently of the co uplings.
Low-Temperature Structure
of the EA Model
There have been several scenarios proposed for the
spin-glass phase of the Edwards–Anderson model at
sufficiently low temperature and high dimension.
These remain speculative, because it has not even
been proved that a phase trans ition from the high-
temperature phase exists at positive temperature in
any finite dimension.
As noted earlier, at sufficiently high temperature
in any dimension (and at all nonzero temperatures in
one and presumably two dimensions, although the
latter assertion has not been proved), there is a
unique Gibbs state. It is conceivable that this
remains the case in all dimensions and at all nonzero
temperatures, in which case the metastate
J
is, for
a.e. J, supported on a single, pure Gibbs state
J
.
(It is important to note, however, that in principle
such a trivial metastate cou ld occur even if N > 1;
indeed, just such a situation of ‘‘weak uniqueness’’
(van Enter and Fro¨ hlich 1985, Campanino et al.
1987) happens in very long range spin glasses at
high temperatures (Fro¨ hlich and Zegarlinski 1987,
Gandolfi et al. 1993).)
A phase transition has been proved to exist
(Aizenman et al. 1987) in the Sherrington–
Kirkpatrick (SK) model (Sherrington and Kirkpa-
trick 1975), which is the infinite-range version of
the EA model. Numerical (Ogielski 1985, Ogielski
and Morgenstern 1985, Binder and Young 1986,
Kawashima and Young 1996) and some analytical
(Fisher and Singh 1990, Thill and Hilhorst 1996) work
has led to a general consensus that above some
dimension (typically around three or four) there does
exist a positive-temperature phase transition below
which spin-flip symmetry is broken, that is, in which
pure states come in pairs, as discussed below eqn [4].
Because much of the literature has focused on this
possibility, we assume it in what follows, and the
metastate approach turns out to be highly useful in
restricting the scenarios that can occur. The simplest
such scenario is a two-state picture in which, below the
transition temperature T
c
, there exists a single pair of
global flip-related pure states
J
and
J
.Inthiscase,
there is no CSD for periodic boundary conditions and
the metastate can be written as
J
¼
1
2
J
þ
1
2
J
½6
That is, the metastate is supported on a single
(mixed) thermodynamic state.
The two-state scenario that has received the most
attention in the literature is the ‘‘droplet/scaling’’
picture (McMillan 1984, Fisher and Huse 1986,
1988, Bray and Moore 1985). In this picture a low-
energy excitation above the ground state in
L
is a
droplet whose surface area scales as l
d
s
, with l
O(L) and d
s
< d, and whose surface energy scales as
l
, with >0 (in dimensions where T
c
> 0). More
recently, an alternative picture has arisen (Krzakala
and Martin 2000, Palassini and Young 2000) in
which the low-energy excitations differ from those
of droplet/scaling, in that their energies scale as l
0
,
with
0
= 0.
The low-temperature picture that has perhaps
generated the most attention in the literature is
the replica symmetry breaking (RSB) scenario
(Binder and Young 1986, Marinari et al. 1994,
1997, Franz et al. 1998, Marinari et al. 2000,
Marinari and Parisi 2000, 2001, Dotsenko 2001),
which assumes a rather complicated pure-state
structure, inspired by Parisi’s solution of the SK
model (Parisi 1979, 1983, Me´zard et al. 1984,
1987). This is a many-state picture (N = 1 for a.e.
572 Short-Range Spin Glasses: The Metastate Approach
J) in which the ordering is described in terms of the
‘‘overlaps’’ between states. There has been some
ambiguity in how to describe such a picture for
short-range models; the prevailing, or standard,
view. Consider any reasonably constructed thermo-
dynamic state
J
(see Newman and Stein (1998a)
for more details) – e.g., the ‘‘average’’ over the
metastate
J
J
¼
Z
d
J
ðÞ½7
Now choose and
0
from the product distribu-
tion
J
()
J
(
0
). The overlap Q is defined as
Q ¼ lim
L!1
j
L
j
1
X
x2
L
x
0
x
½8
and P
J
(q) is defined to be its probability
distribution.
In the standard RSB picture,
J
is a mixture of
infinitely many pure states, each with a specific
J-dependent weight W:
J
ðÞ¼
X
W
J
J
ðÞ½9
If is drawn from
J
and
0
from
J
, then the
expression in eqn [8] equals its thermal mean,
q
J
¼ lim
L!1
j
L
j
1
X
x2
L
h
x
i
h
x
i
½10
and hence P
J
is given by
P
J
ðqÞ¼
X
;
W
J
W
J
ð q q
J
Þ½11
The ‘‘self-overlap,’’ or EA order parameter, is given
by q
EA
= q
J
and (at fixed T) is thought to be
independent of both and J (with probability 1).
According to the standard RSB scenario, the W
J
’s
and q
J
’s are non-self-averaging (i.e., J-dependent)
quantities, except for = or its global flip, where
q
J
= q
EA
. The average P
s
(q)ofP
J
(q) over the
disorder distribution of J is predicted to be a
mixture of two delta-fun ction components at q
EA
and a continuous part between them. However, it
was proved by Newman and Stein (1996c) that this
scenario cannot occur, because of the translation
invariance of P
J
(q) and the translation ergod icity of
the disorder distribution. Nevertheless, the metastate
approach suggests an alternative, nonstandard, RSB
scenario, which is described next.
The idea behind the nonstandard RSB picture
(referred to by us as the nons tandard SK picture in
earlier papers) is to produce the finite-volume
behavior of the SK model to the maximum extent
possible. We therefore assume in this picture that in
each
L
, the finite-volume Gibbs state
(L)
J
is well
approximated deep in the interior by a mixed
thermodynamic state
(L)
, decomposable into many
pure states
L
(explicit dependence on J is
suppressed for ease of notation). More precisely,
each in
J
satisfies
¼
X
W
½12
and is presumed to have a nontrivial overlap
distribution for ,
0
from ()(
0
):
P
ðqÞ¼
X
;
W
W
q q
½13
as did
J
in the standard RSB picture.
Because
J
, like its counterpart
J
in the standard
SK picture, is translation covariant, the resulting
ensemble of overlap distributions P
is independent
of J. Because of the CSD present in this scenario,
the overlap distribution for
(L)
J
varies with L,no
matter how large L becomes. So, instead of
averaging the overlap distribution over J, the
averaging must now be done over the states
within the metastate
J
, all at fixed J:
P
ns
ðqÞ¼
Z
P
ðqÞ
J
ðÞd ½14
The P
ns
(q) is the same for a.e. J, and has a form
analogous to the P
s
(q) in the standard RSB picture.
However, the nonstandard RSB scenario seems
rather unlikely to occur in any natural setting,
because of the following result:
Theorem Newman and Stein 1998b).(Consider
two metastates constructed along (the same) deter-
ministic sequence of
L
’s, using two different
sequences of flip-related, coupling-independent
boundary conditions (such as periodic and antiper-
iodic). Then with probability one, these two
metastates are the same.
The proof is given by Newman and Stein (1998b),
but the essential idea can be easily described here.
As discussed earlier,
J
=
y
J
; but
y
J
is constructed
by a limit of finite-dimensional distributions, which
means averaging over other couplings including the
ones near the system boundary, and hence gives the
same metastate for two flip-related boundary
conditions.
This invariance with respect to differe nt sequences
of periodic and antiperiodic boundary conditions
means essentially that the frequency of appearance
of various thermodynamic states
(L)
in finite
volumes
L
is independent of the choice of
boundary conditions. Moreover, this same
Short-Range Spin Glasses: The Metastate Approach 573
invariance property holds among any two sequences
of fixed boundary conditions (and the fixed bound-
ary condition of choice may even be allowed to vary
arbitrarily along any single sequence of volumes)! It
follows that, with respect to changes of boundary
conditions, the metastate is extraordinarily robust.
This should rule out all but the simplest overlap
structures, and in particular the nonstandard RSB
and related pictures (for a full discussion,
see Newman and Stein 1998b). It is therefore
natural to ask whether the property of metastate
invariance allows any many-state picture.
There is one such picture, namely the ‘‘chaotic pairs’’
picture, which is fully consistent with metastate
invariance (our belief is that it is the only many-state
picture that fits naturally and easily into results
obtained about the metastate.)
Here the periodic bounda ry condition metastate is
supported on infinitely many pairs of pure states,
but instead of eqn [12] one has
¼ð1=2Þ
þð1=2Þ
½15
with overlap
P
¼ð1=2Þ q q
EA
Þþð1=2Þðq þ q
EA
ðÞ½16
So there is CSD in the states but not in the overlaps,
which have the same form as a two-state picture in
every volume. The difference is that, while in the latter
case, one has the ‘‘same’’ pair of states in every volume,
in chaotic pairs the pure-state pair varies chaotically as
volume changes. If the chaotic pairs picture is to be
consistent with metastate invariance in a natural way,
then the number of pure-state pairs should be
‘‘uncountable.’’ This allows for a ‘‘uniform’’ distribu-
tion (within the metastate) over all of the pure states,
and invariance of the metastate with respect to
boundary conditions could follow naturally.
Open Questions
We have discussed how the metastate approach to the
EA spin glass has narrowed considerably the set of
possible scenarios for low-temperature ordering in any
finite dimension, should broken spin-flip symmetry
occur. The remaining possibilities are either a two-state
scenario, such as droplet/scaling, or the chaotic-pairs
picture if there exist many pure states at some (, d).
Both have simple overlap structures. The metastate
approach appears to rule out more complicated
scenarios such as RSB, in which the approximate
pure-state decomposition in a typical large, finite
volume is a nontrivial mixture of many pure-state pairs.
Of course, this does not answer the question of
which, if either, of the remaining pictures actually
does occur in real spin glasses. In this section we list
a number of open questions relevant to the above
discussion.
Open Question 1 Determine whether a phase
transition occurs in any finite dimension greater
than one. If it does, find the lower critical dimension.
Existence of a phase transition does not necessa-
rily imply two or more pure states below T
c
. It could
happen, for example, that in some dimension there
exists a single pure state at all nonzero temperatures,
with two-point spin correlations decaying exponen-
tially above T
c
and more slowly (e.g., as a power
law) below T
c
. This leads to:
Open Question 2 If there does exist a phase
transition above some lower critical dimension,
determine whether the low-tempe rature spin-glass
phase exhibits broken spin-flip symmetry.
If broken symmetry does occur in some dimen-
sion, then of course an obvious open question is to
determine the number of pure-state pairs, and hence
the nature of ordering at low tem perature. A
(possibly) easier question (but still very difficult),
and one which does not rely on knowing whether a
phase transition occurs, is to determine the zero-
temperature – i.e., ground state – properties of spin
glasses as a function of dimensionality. A ground
state is an infinite-volume spin configuration whose
energy (governed by eqn [1]) cannot be lowered by
flipping any finite subset of spins. That is, all ground
state spin configurations must satisfy the constraint
X
x;yhi2C
J
xy
x
y
0 ½17
along any closed loop C in the dual lattice.
Open Question 3 How many ground state pairs is
the T = 0 periodic boundary condition metastate
supported on, as a function of d?
The answer is known to be one for 1D, and a partial
result (Newman and Stein 2000, 2001a) points
towards the answer being one for 2D as well. There
are no rigorous, or even heuristic (except based on
underlying ‘‘ansa¨ tze’ ’) arguments in higher dimension.
An interesting – but unrealistic – spin-glass model
in which the ground state structure can be exactly
solved (although not yet completely rigorously) was
proposed by the authors (Newman and Stein 1994,
1996a) (see also Banavar 1994). This ‘‘highly
disordered’’ spin glass is one in which the coupling
magnitudes scale nonlinearly with the volume (and so
are no longer distributed independently of the
volume, although they remain independent and
identically distributed for each volume). The model
574 Short-Range Spin Glasses: The Metastate Approach
displays a transition in ground state multiplicity:
below eight dimensions, it has only a single pair of
ground states, while above eight it has uncountably
many such pairs. The mechanism behind the transi-
tion arises from a mapping to invasion percolation
and minimal spanning trees (Lenormand and Bories
1980, Chandler et al. 1982, Wilkinson and Will-
emsen 1983): the number of ground state pairs can be
shown to equal 2
N
, where N = N(d) is the number of
distinct global components in the ‘‘minimal spanning
forest.’’ The zero-temperature free boundary con di-
tion metastate above eight dimensions is supported
on a uniform distribution (in a natural sense) on
uncountably many ground state pairs.
Interestingly, the high-dimensional ground state
multiplicity in this model can be shown to be
unaffected by the presence of frustration, although
frustration still plays an interesting role: it leads to
the appearance of chaotic size dependence when free
boundary conditions are used.
Returning to the more difficult problem of ground
state multiplicity in the EA model, we note as a final
remark that there could, in principle, exis t ground
state pairs that are not in the support of metastates
generated through the use of coupling-independent
boundary conditions. If such states exist, they may
be of some interest mathematically, but are not
expected to play any significant physical role. A
discussion of these putative ‘‘invisible states’’ is
given by Newman and Stein (2003).
Open Question 4 If there exists bro ken spin-flip
symmetry at a range of positive temperatures in
some dimensions, then what is the number of pure-
state pairs as a function of (, d)?
Again, the answer to this is not known above one
dimension; indeed, the prerequisite existence of
spontaneously broken spin-flip symmetry has not
been proved in any dimension. A speculative paper
by the authors (Newman and Stein 2001b), using a
variant of the highly disordered model, suggests that
there is at most one pair of pure states in the EA
model below eight dimensions; but no rigorous
arguments are known at this time.
Acknowledgments
This work was supported in part by NSF Grants
DMS-01-02587 and DMS-01-02541.
See also: Glassy Disordered Systems: Dynamical
Evolution; Mean Field Spin Glasses and Neural
Networks; Spin Glasses.
Further Reading
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Binder K and Young AP (1986) Spin glasses: experimental facts,
theoretical concepts, and open questions. Review of Modern
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Bray AJ and Moore MA (1985) Critical behavior of the three-
dimensional Ising spin glass. Physical Review B 31: 631–633.
Campanino M, Olivieri E, and van Enter ACD (1987) One
dimensional spin glasses with potential decay 1=r
1þ
. Absence
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Chandler R, Koplick J, Lerman K, and Willemsen JF (1982)
Capillary displacement and percolation in porous media.
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Chowdhury D (1986) Spin Glasses and Other Frustrated Systems.
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Dotsenko V (2001) Introduction to the Replica Theory of
Disordered Statistical Systems. Cambridge: Cambridge
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Fisher DS and Huse DA (1986) Ordered phase of short-range
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Fisher DS and Huse DA (1988) Equilibrium behavior of the spin-
glass ordered phase. Physical Review B 38: 386–411.
Fisher ME and Singh RRP (1990) Critical points, large-dimensionality
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Georgii HO (1988) Gibbs Measures and Phase Transitions.
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Short-Range Spin Glasses: The Metastate Approach 575
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Sine-Gordon Equation
S N M Ruijsenaars, Centre for Mathematics and
Computer Science, Amsterdam, The Netherlands
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
The sine-Gordon equation
@
2
@x
2
@
2
@t
2
¼ sin ½1
may be viewed as a prototype for a nonlinear
integrable field theory. It is manifestly invariant
under spacetime translations and Lorentz boosts,
ðx; tÞ7!ðx ; t Þ
ðx; tÞ7!ðx cosh t sinh ; t cosh x sinh Þ
½2
It shares this relativistic invariance property with the
linear Klein–Gordon equation, which is obtained
upon replacing sin by . (The name sine-Gordon
equation is derived from this relation, and was
introduced by Kruskal.) The sine-Gordon equation
can also be defined and studied in the form
@
2
@u @v
~
¼ sin
~
;
~
ðu; vÞ¼ðt; xÞ½3
where
u ¼ðx þ tÞ=2; v ¼ðx tÞ=2 ½4
are the so-called light-cone variables.
There are two interpretations of the field (t, x)
that are quite different, both from a physic al and
from a mathematical viewpoint. The first one
consists in viewing it as a real-valued function, so
576 Sine-Gordon Equation
that [1] is simply a nonlinear PDE in two variables.
In the second version, one views (t, x)asan
operator-valued distribution on a Hilbert space.
(Thus, one should smear (t, x) with a test function
f (t, x) in Schwartz space to obtain a genuine
operator on the Hilbert space.) In spite of their
different character, the classical and quantum field
theory versions have several striking features in
common, including the presence of an infinite
number of conservation laws and the occurrence of
solitonic excitations.
The classical sine-Gordon equation has been used
as a model for various wave phenomena, including
the propagation of dislocations in crystals, phase
differences across Josephson junctions, torsion
waves in strings and pendula, and waves along
lipid membranes. It was already studied in the
nineteenth century in connection with the theory of
pseudospherical surfaces. The quantum version is
used as a simple model for solid-state excitations.
The desi gnation ‘‘sine-Gordon’’ is also used for
various equations that generalize [1] or bear
resemblance to it. These include the so-called
homogeneous and symmetric space sine-Gordon
models, discrete and supersymmetric versions, and
generalizations to higher-dimensional spacetimes
(i.e., in [1] the spatial derivative is replaced by the
Laplace operator in several variables). In this
contribution we focus on [1], however.
Our main goal is to discuss the integrability and
solitonic properties, both at the classical and at the
quantum level. First, we sketch the inver se-scattering
transform (IST) solution to the Cauchy problem for
[1]. Following Faddeev and Takhtajan, we emphasize
the interpretation of the IST as an action-angle
transformation for an infinite-dimensional Hamilto-
nian system. Next, the particle-like solutions are
surveyed by using a description in terms of variables
that may be viewed as relativistic action-angle
coordinates. This is followed by a section on the
quantum field theory version, paying special atten-
tion to the factorized scattering that is the quantum
analog of the solitonic classical scattering. Finally, we
sketch the intimate relation between the N-particle
subspaces of the classical and quantum sine-Gordon
field theory and certain integrable relativistic syst ems
of N point particles on the line.
The Classical Version: An Integrable
Hamiltonian System
In order to tie in the hyperbolic evolution equation
[1] with the notion of infinite-dimensional integrable
system, it is necessary to restrict attention to initial
data (0, x) = ( x)and@
t
(0, x ) = (x) with special
properties. First of all, the energy functional
H ¼
Z
1
1
1
2
ðxÞ
2
þ
1
2
@
x
ðxÞ
2
þð1 cos ðxÞÞ
dx
½5
and symplectic form
! ¼
Z
1
1
dðxÞ^dðxÞdx ½6
should be well defined on the phase space of initial
data. Indeed, in that case [1] amounts to the
Hamilton equation associated to [5] via [6].
Second, there exists a sequence of functionals
I
2lþ1
ð; Þ; l 2 Z ½7
that formally Poisson-commute with H and among
themselves.
In particular, H equals 2(I
1
þ I
1
), whereas
2(I
1
I
1
) equals the momentum functional
P ¼
Z
1
1
ðxÞ@
x
ðxÞdx ½8
The functional I
2lþ1
contains x-derivatives of order
up to j2l þ 1j, so one needs to require that the
functions @
x
(x)and(x) be smooth and that all of
their derivatives have suff icient decrease for
x !1.
A natural choice guaranteeing the latter require-
ments is
@
x
ðxÞ;ðxÞ2S
R
ðRÞ½9
where S
R
(R) denotes the Schwartz space of
real-valued functions on the line. To render the integral
over 1 cos (x) (and similar integrals occurring for
the sequence [7]) finite, one also needs to require
ðxÞ!2k
; x !1; k
2Z ½10
On this phase space of initial data, the Cauchy
problem for the evolution equation [1] is not only
well posed, but can be solved in explicit form by
using the IST. More generally, the Hamiltonians
I
2lþ1
give rise to evolution equations that are
simultaneously solved via the IST, yielding an
infinite sequence of commuting Hamiltonian flows
on .
Before sketching the overall picture resulting from
the IST, it should be mentioned at this point that [1]
admits explicit solutions of interest that do not
belong to . First, there is a class of algebro-
geometric solutions that have no limits as x !1.
These solutions can be obtained via finite-gap
integration methods, yielding formulas involving
Sine-Gordon Equation 577
the Riemann theta functions associated to compact
Riemann surfa ces. Second, there are the tachyon
solutions. They arise from the particle-like solutions
that do belong to by the transformation
ðt; xÞ!ðx; tÞþ ½11
(Observe that the equation of motion [1] is invariant
under [11], whereas due to the finite-energy require-
ment [10] this is not true for solutions evolving in .)
The IST via which the above Cauchy problem can
be solved starts from an auxiliary system of two
linear ordinary differential equations involving
(0, x) and @
t
(0, x ). It is beyond our scope to
describe the system in detail. The results derived
from it, however, are to a large extent the same as
those obtained via a simpler auxiliary linear opera-
tor that is associated to the light-cone Cauchy
problem. The latter operator is of the Ablowitz–
Kaup–Newell–Segur (AKN S) form. That is, the
linear operator is an ordinary differential operator
of Dirac type given by
L ¼
i
d
dx
iq
ir i
d
dx
!
½12
where the external potentials r(u)andq(u) depend
on the evolution equation at hand. For the light-
cone sine-Gordon equation [3] , one needs to choose
r ¼q ¼ð@
u
~
Þðu; 0 Þ=2 ½13
In both settings, the associated spectral features
are invariant under the sine-Gordon evolution and
all of the evolutions generated by the Hamiltonians
I
2lþ1
, yielding the so-called isospectral flows. More
specifically, if the initial data give rise to bound-state
solutions of the linear problem (square-integrable
wave functions), then the corresponding eigenvalues
are time independent. Furthermore, due to the decay
requirements on the potential in the linear system,
there exist scattering solutions with plane-wave
asymptotics for all initial data in . A suitable
normalization leads to the so-called Jost solutions
(x, ). (Here is the spectral parameter, which
varies over the real line for scattering solutions.)
Their x !1asymptotics is encoded in transition
coefficients a()andb(), with a() and jb()j being
time independent, whereas arg b() has a linear
dependence on time when the potential evolves
according to the sine-Gordon equation. The bound
states correspond to special -values
1
, ...,
N
with
positive imaginary part (namely the zeros of the
coefficient a( ), which is analytic in the upper-half
-plane); their normalization coefficients
1
, ...,
N
have an essentially linear time evolution, just
as b().
The crux of the IST is now that the potentials can
be reconstructed from the spectral data
fbðÞ;
1
; ...;
N
;
1
; ...;
N
g½14
by solving a linear integral equation of Gelfa nd–
Levitan–Marchenko (GLM) type. (Alternatively,
Riemann–Hilbert problem techniques can be used.)
Hence, the nonlinear Cauchy problem can be
replaced by the far simpler linear problems of
determining the spectral data [14] of a linear
operator (the direct problem) and then solving the
linear GLM equation for the time-evolved scattering
data (the inverse problem).
From the Hamiltonian perspective, the IST may
be reinterpreted as a transformation to action-angle
variables. The action variables are defined in terms
of jb()j and
1
, ...,
N
. They are time independent
under the sine-Gordon and higher Hamiltonian
flows. The angle variables are arg b() and suitable
functions of the normalization coefficients. They
depend linearly on the evolution times of the flows.
The Hamiltonians can be explicitly expressed in
action variables.
Next, we point out that there is a large subspace
of Cauchy data ((x), (x)) that do not give rise to
bound states in the auxiliary linear problem. The
associated solutions are the so-called radiation
solutions: they decrease to 0 for large times. These
solutions can be obtained from the inverse transform
involving the GLM equation by only taking b()
into account.
The other extreme is to choose b() = 0 and
arbitrary bound states and normalization coeffi-
cients in the GLM equation. This special case of
vanishing reflection leads to the particle-like solu-
tions that are studied in the next section. For general
Cauchy data, one has both b() 6¼ 0 and a finite
number of bound states. These so-called mixed
solutions have a radiation component (encoded in
b()) which decays for asymptotic times, whereas
the bound states show up for t !1as isolated
solitons, antisolitons, and breathers.
Classical Solitons, Antisolitons,
and Breathers
Just as for other classical soliton equations, the case
of reflectionless data can be handled in complete
detail, since the GLM equation reduces to an N N
system of linear equations. The case N = 1 yields the
1-soliton and 1-antisoliton solutions. Resting at the
origin, these one-particle solutions are given by
4 arctanðe
x
Þ½15
578 Sine-Gordon Equation
and have energy 8 (cf. [5]). (We normalize all
solutions by requiring
lim
x !1
ðt; xÞ¼0 ½16
Note that one can add arbitrary multiples of 2
without changing the energy H [5].) A spatial
translation and Lorentz boost then yields the general
solutions
ðt; xÞ
¼4 arctanðexpð q x cosh þ t sinh ÞÞ ½17
with energy 8 cosh and momentum 8 sinh (cf. [8]).
Defining the topological charge of a solution
(with normalization [16])by
Q ¼
1
2
lim
x !1
ðt; xÞ½18
the different ch arges Q = 1 and Q = 1 of the
soliton and antisoliton reflec t a signature associated
to the special value of the spectral parameter on the
imaginary axis for which a bound state in the linear
problem occurs. More generally, for bound-state
eigenvalues on the imaginary axis these signatures
must be specified in the IST setting, a point gloss ed
over in the previous section.
Bound states in the linear problem can also arise
from -values off the imaginary axis, which come in
pairs ia b,witha, b > 0. Such pairs give rise to
solutions containing breathers, which can be viewed as
bound states of a soliton and an antisoliton. The one-
breather solution breathing at the origin is given by
4 arctan cot
sinðt sin Þ
coshðx cos Þ
;2ð0;=2Þ½19
and has energy 16 cos . A spacetime translation and
Lorentz boost then yields the genera l solution
b
ðt; xÞ
¼4 arctan cot
sin½=2 þsinðt cosh xsinhÞ
cosh½y=2 cosðxcosh tsinhÞ
½20
which has energy 16coshcos and momentum
16sinhcos. It may be obtained by analytic
continuation from the solution describing a collision
between a soliton with velocity tanh
1
and an
antisoliton with veloc ity tanh
2
, taking
2
<
1
.
The latter is given by
þ
ðt; xÞ
¼4 arctan cothðð
1
2
Þ=2Þ
sinhðð
1
2
Þ=2Þ
coshðð
1
þ
2
Þ=2Þ
2
<
1
½21
where
j
¼ q
j
x cosh
j
þ t sinh
j
; q
j
;
j
2 R ½22
and
b
results from
þ
by substituting
1
2
! i; q
1
2
!ðy iÞ=2 ½23
(For the case
1
<
2
, one needs an extra minus sign
on the right-hand side of [21].)
There is yet another possibility for an eigenvalue
on the imaginary axis we have not mentioned thus
far: it may have an arbit rary multiplicity, giving rise
to the so-called multipole solutions. This is illu-
strated by the breather solution
b
: when one sets
= 2q
0
and lets tend to 0, one obtains a
solution
sep
ðt; xÞ
¼ 4 arctan
q
0
þ t cosh x sinh
cosh½y=2 x cosh þ t sinh
½24
From a physical viewpoint, the soliton and anti-
soliton have just enough energy to prevent a bound
state from being formed. Notice that in this case the
distance between soliton and antisoliton diverges
logarithmically in jtj as t !1, whereas for
þ
one obtains linear increase.
The 2-soliton and 2-antisoliton solutions can also be
obtained by analytic continuation of
þ
.Theyread
¼4 arctan
cothðð
1
2
Þ=2Þ
coshðð
1
2
Þ=2Þ
sinhðð
1
þ
2
Þ=2Þ
;
2
<
1
½25
where
j
is given by [22]. Thus, they arise by
taking q
2
!q
2
þ i and q
1
!q
1
þ i in [21],resp.
The e qual-signature eigenvalues corresponding to
these two solutions cannot collide and move off
the imaginary axis; physically speaking, equal-
charge particles repel each other. The energy and
momentum of the solutions [25] and [21] are given
by 8 cosh
1
þ 8cosh
2
and 8 sinh
1
þ 8sinh
2
,
respectively.
Up to scale factors, the above variables
1
,
2
and
, are the action variables resulting from the IST,
whereas q
1
, q
2
and y, are the canonically con-
jugated angle variables. Accordingly, the time and
space translation flows (generated by H [5] and P
[8], resp.) shift the angles linearly in the evolution
parameters t and x.
We conclude this section with a description of the
N-soliton solution and its large time asymptotics. It
can be expressed in terms of the N N matrix
L
jk
¼ expð
j
Þ
Q
l6¼j
jcothðð
j
l
Þ=2Þj
coshðð
j
k
Þ=2Þ
½26
Sine-Gordon Equation 579
where
j
is given by [22] with
q
1
; ...; q
N
2 R;
N
< <
1
½27
Specifically, one has
Nþ
ðt; xÞ¼4 tr arctan ðLÞ
¼2i lnðj1
N
þ iLj=j 1
N
iLjÞ
¼2i ln 1 þ
X
N
l¼1
i
l
S
l
ðLÞ
!
=c:c:
!
½28
where S
l
is the lth symmetric function of L. Using
Cauchy’s identity, one obtains the explicit formula
S
l
¼
X
If1;...;Ng
jIj¼l
exp
X
j2I
j
!
Y
j2I
k=2 I
jcothðð
j
k
Þ=2Þj ½29
In order to specify the t !1asymptotics of
Nþ
,
we introduce the 1-soliton solutions
j
ðt; xÞ¼4 arctanðexpð
j
j
=2ÞÞ ½30
where
j
¼
X
k<j
X
k>j
0
@
1
A
ð
j
k
Þ½31
ð Þ¼ln cothð=2Þ
2
½32
Then, one has
sup
x2R
Nþ
ðt; xÞ
X
N
j¼1
j
ðt; xÞ
¼ OðexpðjtjrÞÞ
t !1 ½33
where the decay rate is given by
r ¼ min
j6¼k
ðcoshð
j
Þjtanh
j
tanh
k
jÞ ½34
Thus, the soliton profile with velocity tanh
j
incurs
a shift
j
= cosh
j
as a result of the collision. The
factor 1= cosh
j
may be viewed as a Lorentz
contraction factor.
The Quantum Version: A Soliton
Quantum Field Theory
From a perturbation-theoretic viewpoint, the quan-
tum sine-Gordon Hamiltonian is given by
H ¼
Z
1
1
:
1
2
ð@
t
Þ
2
þ
1
2
ð@
x
Þ
2
þ
m
2
2
ð1 cos Þ
: dx; m; >0 ½35
Here, (0, x) is a neutral Klein–Gordon field with
mass m and the double dots denote a suitable
ordering prescription. The associated equation of
motion
xx
tt
¼
m
2
sin ½ 36
is equivalent to [1] on the classical level, but not on
the quantum level. (If ( t, x) is a classical solution to
[36], then (t=m, x=m) solves [1].) This difference
is due to the extremely singular character of
interacting relativistic quantum field theory, a
context in which ‘‘solving’’ the field theory has
slowly acquired a meaning that is vastly different
from the classical notion. Indeed, one can at best
hope to verify [36] in the sense of expectation values
in suitable quantum states, and this is precisely what
has been achieved within the form-factor program
sketched later on.
From the perspective of functional analysis, the
existence of a well-defined Wightman field theory with
all of the features mentioned below is wide open. More
precisely, beginning with pioneering work by Fro¨ hlich
some 30 years ago, various authors have contributed
to a mathematically rigorous construction of a sine-
Gordon quantum field theory version, but to date it
seems not feasible to verify that the resulting Wight-
man field theory has any of the explicit features we are
going to sketch. (For example, not even the free
character of the field theory for
2
= 4 has been
established; cf. below.)
That said, we proceed to sketch some highlights
of the impressive, but partly heuristic lore that has
been assembled in a great many theoretical physics
papers. A key result we begin with is the equivalence
to a field theory that looks very different at face
value. This is the massive Thirring model, formally
given by the Hamiltonian
H
T
¼
Z
1
1
:
ði
5
@
x
þ
0
MÞ þ
g
2
ðJ
2
0
J
2
1
Þ
: dx
M 2ð0; 1Þ; g 2 R ½37
Here, (0, x) is the charged Dirac field with mass M
and the double dots stand for normal ordering. For
the -algebra, one may choose
0
¼
01
10
;
1
¼
0 1
10
5
¼
0
1
¼
10
0 1
½38
and J
is the Dirac current,
J
0
¼
; J
1
¼
10
01
½39
580 Sine-Gordon Equation
The equivalence argument (due to Coleman) consists
in showing that the quantities
2
@
;
m
2
2
: cos : ½40
in the sine-Gordon theory have the same vacuum
expectation values (in perturbation theory) as the
massive Thirring quantities
: J
:; M :
0
: ½41
resp., provided the parameters are related by
4
2
¼ 1 þ
g
½42
This yields an equivalence between the charge-0
sector of the massive Thirring model and the sector
of the sine-Gordon theory obtained by the action of
the fields [40] on the vacuum vector. But the
charged sectors of the Thirring model can also be
viewed as new sectors in the sine -Gordon theory,
obtained by a solitonic field construction (first
performed by Mandelstam).
In this picture, the fermions and antifermions in
the massive Thirring model correspond to new
excitations in the sine-Gordon theory, the quantum
solitons and antisolitons. The latter are viewed as
coherent states of the sine-Gordon ‘‘mesons’’ in the
vacuum sector, the rest masses being related by
M ¼
8m
2
1
2
8
½43
in the semiclassical limit
2
!0.
Even at the formal level involved in the corre-
spondence, the theories are not believed to exist for
2
> 8 and g < =2, since there is positivity
breakdown for this range of couplings. The free
Dirac case g = 0 corresponds to
2
= 4. In parti-
cular, there is no interaction between the sine-
Gordon solitons and antisolitons for this -value.
In the range
2
2 (4,8) there is interaction, but
bound soliton–antisoliton pairs (quantum breathers,
alias sine-Gordon mesons) do not occur.
By contrast, for
2
< 4 there exist breathers with
rest masses
m
n
¼ 2M sinðn þ 1Þ ; m=2M;
n þ 1 ¼ 1; 2; ...; L <=2 ½44
Thus, the ‘‘particle spectrum’’ consists of solitons
and antisolitons with mass M and mesons C
1
, ...,C
L
with masses m
1
, ...,m
L
given by [44]. The latter
formula was first established by semiclassical quan-
tization of the classi cal breathers (Dashen–
Hasslacher–Neveu), and ever since is usually called
the DHN formula. Notice that for near zero m
1
and
m are nearly equal, and that for
2
4 there are no
longer any sine-Gordon mesons present in the theory.
A priori, the existence of infinitely many classical
conserved Hamiltonians does not even formally
imply the same feature for the quantum field theory,
as anomalies may occur. For the sine-Gordon and
massive Thirring cases, anomalies have been shown
to be absent, however. This entails not only that the
number of solitons, antisolitons, and breathers in a
scattering process is conserved, but also that the set
of incoming rapidities equals the set of outgoing
rapidities.
The latter stability features and the DHN formula
[44] are corroborated by the S-matrix, which is
known in complete detail. The two-body amplitudes
involving solitons and antisolitons can be written in
terms of the function
uðzÞ
¼ exp i
Z
0
1
dx
x
sinhð =2Þx
sinh x cosh x=2
sin 2xz
½45
They are given by
ðu
þþ
; t
þ
; r
þ
; u
ÞðÞ
¼ uð=2Þ 1;
sinhð=2Þ
sinhðði Þ=2Þ
;
i sinð
2
=2Þ
sinhðði Þ=2Þ
; 1
½46
where denotes the rapidity difference. (Due to
fermion statistics, one gets only one amplitude for a
soliton or antisoli ton pair. But a soliton and an
antisoliton have opposite charge, so they can be
distinguished. In that case, therefore, the notion of
reflection and transmission coefficients makes sense.)
The S-matrix involving an arbitrary number of
solitons, antisolitons, and their bound states is also
explicitly known. The amplitudes involving no
breathers are readily described in terms of the above
two-body amplitudes. Indeed, the S-matrix factorizes
as a sum of products of the amplitudes [46], yielding a
picture of particles scattering independently in pairs,
just as at the classical level. The factorization can be
performed irrespective of the temporal ordering
assumed for the pair scattering processes, since the
four functions occurring inside the parentheses of
[46] satisfy the Yang–Baxter equations.
Roughly speaking, the S-matrix for processes invol-
ving breathers can be calculated by analytic continua-
tion from the soliton–antisoliton S-matrix. The details
are however quite substantial. We only add that
scattering amplitudes involving solely breathers can
be expressed using only hyperbolic functions.
Sine-Gordon Equation 581