-viour, in

Physics Letters A 334 (2005) 173–179

www.elsevier.com/locate/pla

Chaos in the thermodynamic Bethe ansatz

Olalla Castro-Alvaredoa , Andreas Fringb,∗

a Laboratoire de Physique, Ecole Normale Supérieurede Lyon, 46 Allée d’Italie, 69364 Lyon cedex, Franceb Centre for Mathematical Science, City University, Northampton Square, London EC1V 0HB, UK

Received 16 June 2004; accepted 8 November 2004

Available online 24 November 2004

Communicated by A.P. Fordy

Abstract

We investigate the discretized version of the thermodynamic Bethe ansatz equation for a variety of(1+1)-dimensional quantum field theories. By computing Lyapunov exponents we establish that many systems of this type exhibit chaotic behathe sense that their orbits through fixed points are extremely sensitive with regard to the initial conditions. 2004 Elsevier B.V. All rights reserved.

PACS:05.45.+b

Keywords:Chaos; Thermodynamic Bethe ansatz

a-f

es.ch

tra-nn,-

ions e-

es

ith

1. Introduction

The thermodynamic Bethe ansatz (TBA) eqution [1,2] is an important tool in the context o(1+ 1)-dimensional integrable quantum field theoriIt serves to extract various types of informations, suas the Virasoro central charge of the underlying ulviolet conformal field theory[3], vacuum expectatiovalues[1,4], etc. As it is a nonlinear integral equatioit can be solved analytically only in very few circumstances. In general, one relies on numerical solut

* Corresponding author.E-mail addresses:ocastroa@ens-lyon.fr

(O. Castro-Alvaredo),a.fring@city.ac.uk(A. Fring).

0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserveddoi:10.1016/j.physleta.2004.11.009

of its discretised version

εn+1A (θ) = rmA coshθ

−∑B

∞∫−∞

dθ ′ ϕAB(θ − θ ′)

(1.1)× ln(1+ e−εn

B(θ ′)).Herer is the inverse temperature,mA is the mass of aparticle of typeA, θ is the rapidity andϕAB denotesthe logarithmic derivative of the scattering matrix btween the particles of typeA and B. The unknownquantities in these equationsare the pseudo-energiεA(θ). The standard solution procedure for(1.1)con-sists of a consecutive iteration of the equation w

.

174 O. Castro-Alvaredo, A. Fring / Physics Letters A 334 (2005) 173–179

s,tivess.andd-enqueys-ave

isd ise

lesmeysingionsjustati-

se,n.

hiser-om-ondersarecanure

ewedon

ed

-

on-

can

f

, if

mele

ed

initial valuesε0A(θ) = rmA coshθ . At the heart of this

procedure lie theassumptionsthat the exact solutionis reached forn → ∞, i.e., the sequence convergeand furthermore that the final answer is non-sensiwith regard to the initial values, that is its uniqueneIn general these assumptions are poorly justifiedonly few rigorous investigations for some simple moels exist[5,2,6]. So far the outcome has always bethat these assumptions on the existence and uniness of the solution indeed hold, albeit for certain stems convergence problems in certain regimes hbeen noted[7,8]. The main purpose of this Letterto show that this believe has to be challenged anin fact unjustified for certain well defined theories. Wnote that our findings do neither effect the principof the TBA itself nor the consistency of the quantufield theory it is meant to investigate. However, thindicate that one needs to be very cautious when uthe above solution procedure and making deductabout the physics for such theories as one mightbe mislead by the non-convergence of the mathemcal procedure used to solve the TBA-equations.

Here we will not analyze the full TBA-equation(1.1), but rather concentrate on the ultraviolet regimthat isr ≈ 0, in which it possess some approximatioClearly, the occurrence of chaotic behaviour in tregime will have consequences for the finite tempature regime. We encounter the interesting phenenon that the iterative procedure is convergent beythe ultraviolet (or for a certain choice of parametin some theories), but that unstable fixed pointspresent in the ultraviolet, meaning that this regimenever be reached by the iterative solution procedfor (1.1).

2. Stability of fixed points and Lyapunovexponents

For completeness, let us first briefly recall a fwell-known basic facts concerning the nature of fixpoints which may be found in standard textbooksdynamical systems, see e.g.[9]. The objects of our in-vestigations are difference equations of the type

(2.1)�xn+1 = �F(�xn),

where n ∈ N0, �xn ∈ R� and �F :R� → R

� is a vec-tor function. We are especially interested in the fix

-

points�xf of this system being defined as

(2.2)�F(�xf ) = �xf .

The fixed point is reached by iterating(2.1), if for aperturbation of it, defined as�yn = �xn − �xf , we havelimn→∞ �yn = 0. From(2.1)we find

�yn+1 + �xf = �F(�yn + �xf )

= �F(�xf ) + J · �yn +O(|�yn|2

)(2.3)for �yn → 0

whereJ is the�� Jacobian matrix of the vector function �F(�x)

(2.4)Jij = ∂Fi

∂xj

∣∣∣∣�xf

for 1 � i, j � �.

The linearized system which arises from(2.3) for�yn → 0

(2.5)�yn+1 = J · �yn

governs the nature of the fixed point under certain cditions[9]. Evidently, it is solved by

(2.6)�yn = qni �vi with J · �vi = qn

i �vi for 1 � i � �.

Excluding the case when the eigenvectors�vi of theJacobian matrix are not linearly independent, weexpand the initial value uniquely

(2.7)�y0 =�∑

i=1

ςi �vi

such that

(2.8)�yn =�∑

i=1

ςiqni �vi .

It is now obvious from(2.8) that the perturbation othe fixed point�yn will grow for increasingn if |qi | > 1for somei ∈ {1, . . . , �}. In that case the fixed point�xf

is said to be linearly unstable. On the other hand|qi | < 1 for all i ∈ {1, . . . , �} the perturbation will tendto zero for increasingn and the fixed point�xf is saidto be linearly stable. It can be shown that under soconditions[9] the fixed points are nonlinearly stabwhen they are linearly stable.

In general, that is for any point�x rather than just thefixed points�xf , stability properties are easily encodin the Lyapunov exponentsλi . Roughly speaking the

O. Castro-Alvaredo, A. Fring / Physics Letters A 334 (2005) 173–179 175

ntiaun-

e

ianyn

int

d

ete

canasales-i-

n-

-

ct

na-

-

heyndl

-hell

them-olu-lhibit

edeeate

t

Lyapunov exponents are a measure for the exponeseparation of neighbouring orbits. One speaks ofstable (chaotic) orbits ifλi > 0 for somei ∈ {1, . . . , �}and stable orbits ifλi < 0 for all i ∈ {1, . . . , �}. Foran arbitrary point�x the� Lyapunov exponents for thabove mentioned system(2.1)are defined as

λi = limn→∞

[1

nln

∣∣qi

[ �Fn(�x)]∣∣]

(2.9)= limn→∞

[1

n

n−1∑k=0

ln∣∣qi

[ �Fk(�x)]∣∣],

where theqi(�x) are the eigenvalues of the Jacobmatrix as defined in(2.6), but now at some arbitrarpoint �x. Taking the point to be a fixed point, we carelate(2.9) to the above statements. At the fixed powe have of course�Fk(�xf ) = �xf , such that

(2.10)λi = limn→∞

[1

n

n−1∑k=0

ln∣∣qi(�xf )

∣∣] = ln |qi|.

Therefore, a stable fixed point is characterized byλi <

0 or |qi| < 1 for all i ∈ {1, . . . , �} and an unstable fixepoint by λi > 0 or |qi | > 1 for somei ∈ {1, . . . , �}.We can now employ this criterion for some concrsystems.

3. Unstable fixed points in constant TBAequations

We adopt here the notation of[10–12], by whicha large class of integrable quantum field theoriesbe referred to in a general Lie algebraic formg|g-theories. Their underlying ultraviolet conformfield theories can be described by the theories invtigated in[13–15](and special cases thereof) with Vrasoro central chargec = ��h/(h + h). Here� (�) andh (h) are the rank and the Coxeter number ofg (g),respectively. In particular,g|A1 is identical to the min-imal affine Toda theories (ATFT)[16,17] and An|gcorresponds to thegn+1-homogeneous sine-Gordo(HSG) models[18,19]. In this formulation each particle is labelled by two quantum numbers(a, i), whichtake their values in 1� a � � and 1� i � �. Hence,in total we have� × � different particle types. It is astandard procedure in this context[1,2] to approximatethe pseudo-energies in(1.1) by εi

a(θ) = εia = const

lin a large region forθ when r is small. For convenience one then introduces further the quantityxi

a =exp(−εi

a) such that(1.1)can be cast into the compaform

xia =

�∏b=1

�∏j=1

(1+ x

jb

)Nijab =: F i

a(�x)

(3.1)with Nijab = δabδij − K−1

ab Kij .

The matrix Nij

ab in (3.1) encodes the informatioon the asymptotic behaviour of the scattering mtrix. As stated in(3.1) it is specific to each of theg|g-theories withK and K being the Cartan matrix of g and g, respectively. The equations(3.1)are referred to as the constant TBA-equations. Tgovern the ultraviolet behaviour of the system atheir solutions yield directly the effective centracharge

(3.2)ceff = 6

π2

�∑a=1

�∑i=1

L(

xia

1+ xia

)

with L(x) = ∑∞n=1 xn/n2 + lnx ln(1 − x)/2 denot-

ing Rogers dilogarithm (see, e.g.,[20] for proper-ties).

Let us now discretise(3.1) and analyze it with regard to the nature of its fixed points. According to targument of Section2, we have to compute first of athe Jacobian matrices of�F

(3.3)Jijab = ∂F i

a

∂xjb

∣∣∣∣�xf

= Nijab

(xia)f

1+ (xjb )f

.

Next we need to determine the eigensystem ofJacobian matrix. This is not possible to do in a copletely generic way at present, since not even the stions, i.e. fixed points, of(3.1)are known in a generafashion. Instead, we present some examples to exthe possible types of behaviour.

3.1. Stable fixed points

We start with a simple example of a stable fixpoint. We present theA2|A1 case, which after the frefermion (A1|A1) is the next non-trivial example in thseries of the minimal ATFTs, the scaling three-stPotts model with Virasoro central chargec = 4/5.The TBA has been investigated in[1,2]. The constan

176 O. Castro-Alvaredo, A. Fring / Physics Letters A 334 (2005) 173–179

ix

en--the

hisitive

ra-

lyti-BA

eiesct,uchples

s a. a

thealis

co-

e.,val-see

ill

ence

TBA-equations(3.1)

x1 = (1+ x1)−1/3(1+ x2)

−2/3 = F1(�x),

(3.4)x2 = (1+ x1)−2/3(1+ x2)

−1/3 = F2(�x),

can be solved analytically by the golden ratioτ :=(√

5−1)/2= x1 = x2. Using this solution for the fixedpoint �xf = (x1, x2), we compute the Jacobian matrof �F(�x)

(3.5)J (�xf ) = −1

3

(τ2 2τ2

2τ2 τ2

),

with eigensystem

q1 = −τ2 ≈ −0.38197, �v1 = (1,1),

q2 = τ2/3≈ 0.12732, �v2 = (−1,1).

As the eigenvectors are obviously linearly indepdent and|q1| < 1, |q2| < 1, we deduce that all Lyapunov exponents are negative and therefore thatfixed point is stable. Indeed, numerical studies of tsystem exhibit a fast convergence and a non-sensbehaviour with regard to the initial values of the itetive procedure.

For some theories the solutions are known anacally in a closed form. For instance, the constant Tequations for theA1|A�-theories (≡ SU(�+1)2-HSG-model) are solved by

(3.6)xi1 =

[sin[π(1+ i)λ]

sin(πλ)

]2

− 1, for 1� i � �

with λ = 1/(3+�) [21–23,12]. Taking this solution forthe fixed point we compute the Jacobian matrix(3.3)with Nij = (δi,j+1 + δi,j−1)/2 to

Jij

11(�xf ) = 1

2

[sin[π(2+ i)λ]

sin(iπλ)δi,j+1

(3.7)+ sin(iπλ)

sin[π(2+ i)λ]δi,j−1

]

and the eigenvalues toqi = cos[π(i+1)λ]. As |qi | < 1for all i ∈ {1, . . . , �} the fixed points are stable. Winvestigated various minimal affine Toda field theorwhich all posses fixed points of this nature. In fathe general assumption is that all systems exhibit sa behaviour. We present now some counter examwhich refute this believe.

3.2. Stable two-cycles

We start with a system which does not possesstable fixed point, but rather a stable two-cycle, i.esolution for

(3.8)�G(�x) := �F ( �F(�x)) = �x.

We consider theA2|A2-theories (≡ SU(3)3-HSGmodel) studied already previously by means ofTBA in [24]. Its extreme ultraviolet Virasoro centrcharge isc = 2. The constant TBA-equations for thcase read

x11 = (1+ x2

1)2/3(1+ x22)1/3

(1+ x11)1/3(1+ x1

2)2/3= F1(�x),

(3.9)x12 = (1+ x2

1)1/3(1+ x22)2/3

(1+ x11)2/3(1+ x1

2)1/3= F2(�x),

x21 = (1+ x1

1)2/3(1+ x12)1/3

(1+ x21)1/3(1+ x2

2)2/3= F3(�x),

(3.10)x22 = (1+ x1

1)1/3(1+ x12)2/3

(1+ x21)2/3(1+ x2

2)1/3= F4(�x),

with analytic solutionx11 = x1

2 = x21 = x2

2 = 1. Takingthis solution as the fixed point, we compute the Jabian matrix for �F(�x)

(3.11)J (�xf ) = 1

6

−1 −2 2 1

−2 −1 1 2

2 1 −1 −2

1 2 −2 −1

with eigensystem

q1 = −1, �v1 = (−1,−1,1,1),

q2 = 1/3, �v2 = (−1,1,−1,1),

q3 = 0, �v3 = (1,0,0,1),

(3.12)q4 = 0, �v4 = (0,1,1,0).

We observe that there is one eigenvalue with|q1| = 1,which is generally called a marginal behaviour, i.the stability properties depend on the other eigenues and on the next leading order. In fact, we canfrom (2.8) that the perturbation of the fixed point wremain the same even for large values ofn, flippingbetween two values and thus suggesting the existof a stable two cycle(3.8). We find that(3.8) can be

O. Castro-Alvaredo, A. Fring / Physics Letters A 334 (2005) 173–179 177

-ian

sta-ionheotr of

ute

g-eies,of

fort aolu-tial

e

ns

,

a-

e-

ibiton

icheansply

solved by

(3.13)x11 = x1

2 = 1/x21 = 1/x2

2 = κ,

for any arbitrary value ofκ . To determine the stability of the two-cycle we have to compute the Jacobmatrix forG(�x)

(3.14)

J (�xf ) = 1

9+ 9κ

5κ 4κ −4κ2 −5κ2

4κ 5κ −5κ2 −4κ2

−4/κ −5/κ 5 4

−5/κ −4/κ 4 5

which has eigensystem

q1 = 1, �v1 = (−κ2,−κ2,1,1),

q2 = 1/9, �v2 = (−κ2, κ2,−1,1),

q3 = 0, �v3 = (κ,0,0,1),

(3.15)q4 = 0, �v4 = (0, κ,1,0).

We conclude from this that one approaches able two-cycle when iterating the discretised versof (3.8). Thus, we note that the TBA-system for tSU(3)3-HSG model in the ultraviolet regime does nposses a stable fixed point but an infinite numbestable two-cycles of the type(3.13). It is now intrigu-ing to note that when using this solution to compthe effective Virasoro central charge(3.2), one alwaysobtains the expected valuec = 2 for any value ofκ ∈ R, simply due to an identity for the Rogers diloarithmL(1 − x) + L(x) = π2/6. Hence, despite thfact, that one is using entirely wrong pseudo-energone obtains by pure luck an apparent confirmationthe theories consistency.

3.3. Unstable fixed points, chaotic behaviour

In this section we present some TBA-systemswell-defined quantum field theories, which exhibichaotic behaviour in the sense that their iterative stions are extremely sensitive with regard to the inivalues.

3.3.1. A4|A4This model is theSU(5)5-HSG model with extreme

ultraviolet Virasoro central chargec = 8. To reduce thecomplexity of the model, we exploit already from thvery beginning theZ2-symmetries in theA4-Dynkindiagrams and identifyx1

1 = x41 = x1

4 = x44, x2

2 = x32 =

x23 = x3

3, x12 = x1

3 = x42 = x4

3, x21 = x2

4 = x31 = x3

4.With these identifications the constant TBA-equatiocan be brought into the form

x11 = (1+ x2

1)(1+ x22)

(1+ x11)(1+ x1

2)2= F1(�x),

(3.16)x12 = (1+ x2

1)(1+ x22)2

(1+ x11)2(1+ x1

2)3= F2(�x),

x21 = (1+ x1

1)(1+ x12)

(1+ x22)

= F3(�x),

(3.17)x22 = (1+ x1

1)2(1+ x12)2

(1+ x21)(1+ x2

2)= F4(�x).

We can solve these equations analytically by oneτ

andτ := 1/τ = (√

5+ 1)/2

x11 = x4

1 = x14 = x4

4 = x22 = x3

2 = x23 = x3

3 = 1,

x21 = x2

4 = x31 = x3

4 = τ ,

(3.18)x12 = x1

3 = x42 = x4

3 = τ.

With this solution for�xf at hand we compute the Jcobian matrix of�F(�x) in (3.16), (3.17)

(3.19)J (�xf ) =

−1/2 −2τ τ2 1/2

−τ −3τ2 2τ − 1 τ

τ /2 1 0 −τ /2

1/2 2τ −τ2 −1/2

.

Now we find the eigensystem

q1 = −τ2 ≈ −2.6180, �v1 = (−1,−1,1,1),

q2 = 4τ − 2 ≈ 0.47214, �v2 = (−1, τ2,−τ2,1),

q3 = 0, �v3 = (1,0,0,1),

(3.20)q4 = 0, �v4 = (−2τ2, τ,1,0).

As |q1| > 1 and the eigenvectors are linearly indpendent, we deduce that the Lyapunov exponentλ1 ispositive and therefore that the fixed point(3.18)is un-stable. Indeed, numerical studies of this system exhthat any small perturbation away from the soluti(3.18)will lead to a divergent iterative procedure.

Nonetheless, by some manipulations of(3.16),(3.17)one can find equivalent sets of equations whposses stable fixed points and can be solved by mof an iterative procedure. For example, when sim

178 O. Castro-Alvaredo, A. Fring / Physics Letters A 334 (2005) 173–179

ts

w-

full-

G-eirom

-andeseethedl-

intof

lethe

tobe

ms,

G-

T.

.,

in

Gely

m-d-e-BA-

nt

substitutingx11 in F2(�x) we obtain the equations

x11 = F1(�x) = F ′

1(�x),

(3.21)x12 = x1

1(1+ x22)

(1+ x11)(1+ x1

2)= F ′

2(�x),

x21 = F3(�x) = F ′

3(�x),

(3.22)x22 = F4(�x) = F ′

4(�x),

which are of this kind. Now all Lyapunov exponenresulting from the Jacobian matrix for�F ′(�x) at thefixed point(3.18)are negative. One should note, hoever, that even though�F ′(�x) it is easily constructedby trial and error from �F(�x) for the constant TBA-equations, the equivalent manipulations on theTBA-equations(1.1)are quite unnatural, albeit not impossible to perform once(3.1)is analyzed.

One of the distinguishing features of the HSmodels is that they contain unstable particles in thspectrum, whose masses are characterized by sresonance parametersσij with 1 � i, j � �. We cannow interpret these parameters as bifurcation parameters as common in the study of chaotic systemsinvestigate the nature of the fixed points when thparameters are varied. In[7] a precise decoupling rulwas provided, which describes the behaviour oftheories when some of theσ ’s become large and tento infinity. For SU(5)5 we have for instance the folowing possibilities

limσ12→∞ SU(5)5 = SU(2)5 ⊗ SU(4)5 or

(3.23)limσ23→∞ SU(5)5 = SU(3)5 ⊗ SU(3)5.

For the algebras involved we found that the fixed poof SU(4)5 is unstable, whereas the fixed pointsSU(3)5 andSU(2)5 are stable. For ourSU(5)5 exam-ple this implies that the fixed point inSU(3)5⊗SU(3)5will be stable, whereas the fixed point inSU(2)5 ⊗SU(4)5 will be unstable. In general, we find that, whiapproaching the ultraviolet from the infrared, oncenature of the fixed point has changed from stableunstable it remains that way. This behaviour canencoded naturally in standard bifurcation diagrawhich we present elsewhere.

We also found unstable fixed points for other HSmodels related to simply laced algebras andg|g-theories which are neither HSG nor minimal ATF

e

A priori the behaviour is difficult to predict, e.gwhereasD4|D4 (see[10] for the solution of(3.1)) andD4|A4 have unstable fixed points, the fixed pointD4|A2 is stable.

3.3.2. A1|C2This model is the simplest example of an HS

model related to a non-simply laced algebra, namthe Sp(4)2-HSG model with central chargec = 2.In the TBA analysis carried out in[7] convergenceproblems in the ultraviolet regime were already comented upon. In fact, we find here that all HSG moels which are related to non-simply laced Lie algbras posses unstable fixed points. The constant Tequations forA1|C2 read

x11 =

√(1+ x2

1

)(1+ x2

3

)(1+ x2

2

) = F1(�x),

(3.24)x21 = 1

(1+ x22)

√(1+ x1

1)

(1+ x21)(1+ x2

3)= F2(�x),

x22 = (1+ x1

1)

(1+ x21)(1+ x2

2)(1+ x23)

= F3(�x),

(3.25)x23 = 1

(1+ x22)

√(1+ x1

1)

(1+ x21)(1+ x2

3)= F4(�x),

with solutions

(3.26)x11 = 3, x2

1 = x23 = 2/3 and x2

1 = 4/5.

The corresponding Jacobian matrix for�F(�x) reads

(3.27)

J (�xf ) =

0 9/10 5/3 9/10

1/12 −1/5 −10/27 −1/5

1/5 −12/25 −4/9 −12/25

1/12 −1/5 −10/27 −1/5

with eigensystem

q1 ≈ −1.3647, �v1 = (−3.53,1,1.8104,1),

q2 ≈ 0.3973, �v2 = (−80.2656,1,−20.2124,1),

q3 ≈ 0.1229, �v3 = (2.1956,1,−0.9180,1),

(3.28)q4 = 0, �v4 = (0,−1,0,1).

Since|q1| > 0 we find a positive Lyapunov exponeand therefore the fixed point�xf in (3.26)is unstable.

O. Castro-Alvaredo, A. Fring / Physics Letters A 334 (2005) 173–179 179

onBA-

nsbitex-

.G-orallntlyedyetin

b-ean-n-veould

yblese

delsed

be

rkm.

.

.

9.35

1)

am-

9)

ucl.

-

Mi-

8

s,

We also checked explicitlyA1|C3,A1|C4,A1|G2,

A2|B2 and found a similar behaviour. Basedthese examples we conjecture that the constant Tequations(3.1) related tog-HSG-models withg non-simply laced have unstable fixed points.

4. Conclusions

We showed that the discretised TBA-equatiofor many well-defined quantum field theories exhichaotic behaviour in the sense that their orbits aretremely sensitive with regard to the initial conditionsIn particular, we found several examples for HSmodels andg|g-theories which are neither HSG nminimal ATFT. Apart from the statements, thatA1|A�-theories have stable fixed points and appareall HSG-models which are related to non-simply lacmodels have unstable fixed points, we did not finda general pattern which characterizes such theoriesa more concise way.

Our findings clearly explain the convergence prolems reported upon earlier in[7] and we stress herthat they do neither effect the consistency of the qutum field theories nor the validity of the principles uderlying the TBA, but only point out the need to solthese theories by alternative means. The closest wbe to alter the iterative procedure for(1.1)as indicatedin Section3.3.1 for the constant TBA-equations, bdefining equivalent sets of equations which have stafixed points. Unfortunately, we cannot settle with thearguments the convergence problems for the mostudied in[8], as for those the fixed points are situatat infinity.

Our results clearly indicate that one can onlyconfident about results obtained from iterating(1.1)ifthe nature of the fixed points is clarified.

Acknowledgements

This work is supported in part by the EU netwo“EUCLID, Integrable models and applications: frostrings to condensed matter”, HPRN-CT-2002-00325

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