The ODE/IM correspondence in Toda field...

The ODE/IMcorrespon-

S. Negro

Introduction

Integrability:some basicfacts

The ODE/IMCorrespon-dence for theBullough-DoddModel

Conclusionsandperspectives

The ODE/IM correspondence in Toda fieldtheories

Stefano NegroAdvisor: Roberto Tateo

Dipartimento di fisicaUniversita degli Studi di Torino

19 february 2013

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Introduction

Table of Contents

1 Introduction

2 Integrability: some basic facts

3 The ODE/IM Correspondence for the Bullough-Dodd Model

4 Conclusions and perspectives

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Introduction

Table of Contents

1 Introduction

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Introduction

The ODE/IM Correspondence

• The ODE/IM correspondence is a newly found linkbetween two apparently separated areas of mathematicalphysics

• It is based on an equivalence between functional relationsappearing in the study of eigenvalue problems of ODEs onone side and emerging from the analysis of the partitionfunction in 2D quantum integrable theories on the otherside

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Introduction

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Introduction

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Introduction

The first instance of this equivalence arose in a work ofP.Dorey and R.Tateo in 1998

• ODE: d2

dx2ψ(x) + x2Mψ(x) = −Eψ(x)

• IM: The six-vertex model

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Introduction

The first instance of this equivalence arose in a work ofP.Dorey and R.Tateo in 1998

• ODE: d2

dx2ψ(x) + x2Mψ(x) = −Eψ(x)

• IM: The six-vertex model

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Introduction

The Six-Vertex Model

• To each local configuration is associated a Boltzmannweight:

↑→ →

↓← ←

= a(ν, η)

↓→ →

↑← ←

= b(ν, η)

↑→ ←

↓← →

= c(ν, η)

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Introduction

• To each local configuration is associated a Boltzmannweight:

↑→ →

↓← ←

= a(ν, η)

↓→ →

↑← ←

= b(ν, η)

↑→ ←

↓← →

= c(ν, η)

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Introduction

• The physics of the model is encoded in the PartitionFunction:

Z =∑C

∏sites

·· ··

• Which can be written in terms of the Transfer Matrix:Z = TrTN

• The free energy density assume a simple form in thethermodynamic limit

f = − 1

NN ′lnZ = − 1

NN ′lnTrTN ∼ − 1

N ′ln t0

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Introduction

Z =∑C

∏sites

·· ··

f = − 1

NN ′lnZ = − 1

N ′ln t0

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Introduction

Z =∑C

∏sites

·· ··

f = − 1

NN ′lnZ = − 1

N ′ln t0

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Introduction

Z =∑C

∏sites

·· ··

f = − 1

NN ′lnZ = − 1

N ′ln t0

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Introduction

• The six-vertex model is integrable, this means that in thethermodynamic limit, the eigenvalues of T satisfy theBethe Ansatz equations:

t0(E )q0(E ) = q0(ω2E ) + q0(ω−2E )

q0(E ) =∞∏`=1

(1− E

)∞∏`=1

(E` − ω2Ei

E` − ω−2Ei

)= −1

• Where E = e2ν and ω = −e−2iη

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Introduction

t0(E )q0(E ) = q0(ω2E ) + q0(ω−2E )

q0(E ) =∞∏`=1

(1− E

)∞∏`=1

(E` − ω2Ei

E` − ω−2Ei

)= −1

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Introduction

t0(E )q0(E ) = q0(ω2E ) + q0(ω−2E )

q0(E ) =∞∏`=1

(1− E

)∞∏`=1

(E` − ω2Ei

E` − ω−2Ei

)= −1

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Introduction

Table of Contents

1 Introduction

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Introduction

Integrable Hamiltonian Systems

Definition (Liouville 1855)

A competely integrable Hamiltonian system is a dynamicalsystem admitting a Hamiltonian description and possessing amaximal set of independent conserved quantities, so that it canbe solved by quadratures.

n-Dimensional Systems

• the phase space M is a 2n-dimensional symplecticmanifold equipped with some Poisson structure {·, ·}

• the conserved quantities are functions Fi on M (F1 = H)Poisson commuting with the Hamiltonian: {Fi ,H} = 0

• the system is completely integrable if there exist nconserved quantities {Fi}ni=1 such that{Fi ,Fj} = 0 ; ∀i , j = 1, . . . , n

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Introduction

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Introduction

∞-Dimensional Systems

• the phase space M is ∞-dimensional

• thus we need an infinite set of integrals of motion {Fi}∞i=1

in involution

Example: the Korteweg-de Vries (KdV) equation

∂tφ+ ∂3xφ− 6φ∂xφ = 0

L = 12∂xψ∂tψ + (∂xψ)3 − 1

(∂2xψ)2

; φ = ∂xψ

• Integrals of motion (Miura, Gardner & Kruskal, 1969):F2n−1 =

∫∞−∞ P2n−1

(φ, ∂φ, ∂

2xφ, . . .

)dx ; F2n = 0

Pn = − ddxPn−1 +

∑n−2i=1 PiPn−1−i ; P1 = φ

• Soliton solutions: φ(x , t) = c2sech

c x−ct−a2

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Introduction

in involution

∂tφ+ ∂3xφ− 6φ∂xφ = 0

L = 12∂xψ∂tψ + (∂xψ)3 − 1

(∂2xψ)2

; φ = ∂xψ

∫∞−∞ P2n−1

(φ, ∂φ, ∂

2xφ, . . .

)dx ; F2n = 0

∑n−2i=1 PiPn−1−i ; P1 = φ

c x−ct−a2

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Introduction

in involution

∂tφ+ ∂3xφ− 6φ∂xφ = 0

L = 12∂xψ∂tψ + (∂xψ)3 − 1

(∂2xψ)2

; φ = ∂xψ

∫∞−∞ P2n−1

(φ, ∂φ, ∂

2xφ, . . .

)dx ; F2n = 0

∑n−2i=1 PiPn−1−i ; P1 = φ

c x−ct−a2

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Introduction

in involution

∂tφ+ ∂3xφ− 6φ∂xφ = 0

L = 12∂xψ∂tψ + (∂xψ)3 − 1

(∂2xψ)2

; φ = ∂xψ

∫∞−∞ P2n−1

(φ, ∂φ, ∂

2xφ, . . .

)dx ; F2n = 0

∑n−2i=1 PiPn−1−i ; P1 = φ

c x−ct−a2

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Introduction

in involution

∂tφ+ ∂3xφ− 6φ∂xφ = 0

L = 12∂xψ∂tψ + (∂xψ)3 − 1

(∂2xψ)2

; φ = ∂xψ

∫∞−∞ P2n−1

(φ, ∂φ, ∂

2xφ, . . .

)dx ; F2n = 0

∑n−2i=1 PiPn−1−i ; P1 = φ

c x−ct−a2

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Introduction

in involution

∂tφ+ ∂3xφ− 6φ∂xφ = 0

L = 12∂xψ∂tψ + (∂xψ)3 − 1

(∂2xψ)2

; φ = ∂xψ

∫∞−∞ P2n−1

(φ, ∂φ, ∂

2xφ, . . .

)dx ; F2n = 0

∑n−2i=1 PiPn−1−i ; P1 = φ

c x−ct−a2

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Introduction

Inverse Scattering Transform

Lax Representation

• in 1968 P.D.Lax observed that the KdV equation can berealized as the commutation condition[L, ∂t − A] = 0 ⇒ ∂tL = [A, L]for the auxiliary linear differential operatorsL = −∂2

x +φ(x , t) ; A = 4∂3x −3 [2φ(x , t)∂x + φx(x , t)]

• he further showed that it’s possible to recover the solitonsolution of KdV equation through the Inverse ScatteringTransform (IST); schematically:

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Introduction

Lax Representation

x +φ(x , t) ; A = 4∂3x −3 [2φ(x , t)∂x + φx(x , t)]

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Introduction

Lax Representation

x +φ(x , t) ; A = 4∂3x −3 [2φ(x , t)∂x + φx(x , t)]

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Introduction

Zero Curvature Representation

More generally the IST method is based on the observationthat some 2D PDEs appear as consistency conditions for linearsystems

∂xΨ = U(x , t;λ)Ψ ; ∂tΨ = V (x , t;λ)Ψ

which takes the form of a Zero Curvature Condition (ZCC)

∂tU(x , t;λ)− ∂xV (x , t;λ) + [U(x , t;λ),V (x , t;λ)] = 0

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Introduction

Table of Contents

1 Introduction

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Introduction

The Bullough-Dodd Model

• The modified EOMs are

∂∂η(z , z) + e−η(z,z) − p(z)p(z)e2η(z,z) = 0

where p(x) = x3M − E .

• We search for periodic solutions η(ρ, φ+ 2π3M ) = η(ρ, φ),

real-valued and finite for real ρ 6= 0 and φ, satisfying theasymptotics:

η(ρ, φ) ∼ρ→∞

−2M ln ρ ; η(ρ, φ) ∼ρ→0−2g ln ρ

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Introduction

η(ρ, φ) ∼ρ→∞

−2M ln ρ ; η(ρ, φ) ∼ρ→0−2g ln ρ

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Introduction

η(ρ, φ) ∼ρ→∞

−2M ln ρ ; η(ρ, φ) ∼ρ→0−2g ln ρ

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Introduction

The Associated Linear Problem

The Differential Equations

• By appropriately defining the components of Ψ we canwrite the linear problem LΨ = 0 ; LΨ = 0 as a pair ofdifferential equations:

∂3ψ −[(∂η)2 + 2∂2η

]∂ψ +

[λ3p(z)− ∂η∂2η − ∂3η

]ψ = 0

∂3ψ −[(∂η)2 + 2∂2η

]∂ψ +

[λ−3p(z)− ∂η∂2η − ∂3η

]ψ = 0

• The functions are defined asΨ1 = λ−

η2 ∂[eη∂(e−ηψ)] = λ−

32 e−

η2 ψ

• The asymptotics for ρ→ 0 is

∂3ψ − g(g + 2)

z2∂ψ +

g(g + 2)

z3ψ = 0

which has solutions {χn ∼ zµn}1n=−1 with

µn = 1 + n(g + 1)

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Introduction

∂3ψ −[(∂η)2 + 2∂2η

]∂ψ +

[λ3p(z)− ∂η∂2η − ∂3η

]ψ = 0

∂3ψ −[(∂η)2 + 2∂2η

]∂ψ +

[λ−3p(z)− ∂η∂2η − ∂3η

]ψ = 0

η2 ∂[eη∂(e−ηψ)] = λ−

32 e−

η2 ψ

∂3ψ − g(g + 2)

z2∂ψ +

g(g + 2)

z3ψ = 0

µn = 1 + n(g + 1)

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Introduction

∂3ψ −[(∂η)2 + 2∂2η

]∂ψ +

[λ3p(z)− ∂η∂2η − ∂3η

]ψ = 0

∂3ψ −[(∂η)2 + 2∂2η

]∂ψ +

[λ−3p(z)− ∂η∂2η − ∂3η

]ψ = 0

η2 ∂[eη∂(e−ηψ)] = λ−

32 e−

η2 ψ

∂3ψ − g(g + 2)

z2∂ψ +

g(g + 2)

z3ψ = 0

µn = 1 + n(g + 1)

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Introduction

∂3ψ −[(∂η)2 + 2∂2η

]∂ψ +

[λ3p(z)− ∂η∂2η − ∂3η

]ψ = 0

∂3ψ −[(∂η)2 + 2∂2η

]∂ψ +

[λ−3p(z)− ∂η∂2η − ∂3η

]ψ = 0

η2 ∂[eη∂(e−ηψ)] = λ−

32 e−

η2 ψ

∂3ψ − g(g + 2)

z2∂ψ +

g(g + 2)

z3ψ = 0

µn = 1 + n(g + 1)

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Introduction

The WKB approximation

• In the limit ρ→∞ the equation becomes (λ = eθ)(∂3 + e3θz3M

)ψ = e3θEψ

• From WKB approximation we find that on the real axis

ψ ∼ρ→∞

ρ−Me−2 ρM+1

is the unique subdominant solution.

• We define ψk(z , z ; θ) = ψ(z , z ; θ − 2πi k3 )

• It can be shown that W [ψk , ψk+1, ψk+2] = −3i√

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Introduction

)ψ = e3θEψ

ψ ∼ρ→∞

ρ−Me−2 ρM+1

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Introduction

)ψ = e3θEψ

ψ ∼ρ→∞

ρ−Me−2 ρM+1

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Introduction

)ψ = e3θEψ

ψ ∼ρ→∞

ρ−Me−2 ρM+1

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Introduction

)ψ = e3θEψ

ψ ∼ρ→∞

ρ−Me−2 ρM+1

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Introduction

From the Linear Problem to the Bethe Ansatz

The ψ-system

• Direct calculation shows that u−k,k = W [ψ−k , ψk ] satisfythe equation

∂3u−k,k −[(∂η)2 + 2∂2η

]∂u−k,k+

−(e−2πikp(z) + ∂η∂2η + ∂3η

)u−k,k = 0

which coincides with the original one for k ∈ 12Z

• Evaluating the WKB expansion for one can show thatu− 1

2= i√

3ψ0, obtaining our ψ-system:

W [ψ− 12, ψ 1

2] = i√

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Introduction

The ψ-system

∂3u−k,k −[(∂η)2 + 2∂2η

]∂u−k,k+

−(e−2πikp(z) + ∂η∂2η + ∂3η

)u−k,k = 0

2= i√

W [ψ− 12, ψ 1

2] = i√

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Introduction

The ψ-system

∂3u−k,k −[(∂η)2 + 2∂2η

]∂u−k,k+

−(e−2πikp(z) + ∂η∂2η + ∂3η

)u−k,k = 0

2= i√

W [ψ− 12, ψ 1

2] = i√

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Introduction

Towards the BAEWe now set χn;k(z , z ; θ) = ωn(g+1)χn(z , z ; θ) where

ω = e2πi

3(M+1) and expand the ψk functions in terms of the χbasis:

ψk(z , z ; θ) =1∑

n=−1

Qn(θ − 2πik

3)χn;k(z , z ; θ)

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Introduction

The BAENow, using the last equation and the ψ-system and consideringterms proportional to z−g , we obtain

3Q1(θ) =(g + 1)ω−g+1

2 Q1(θ + iπ

3)Q0(θ − i

−(g + 1)ωg+1

2 Q1(θ − iπ

3)Q0(θ + i

which, considered at θ = θn ± i π3 with Q1(θn) = 0 yelds

Q1(θn + i 2π3 )Q1(θn − i π3 )

Q1(θn − i 2π3 )Q1(θn + i π3 )

= −ωg+1

This equation, along with the corresponding for Q−1(θ),coincides with the BAE for the ground state of theIzergin-Korepin model.

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Introduction

Table of Contents

1 Introduction

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Introduction

Conclusions

• A first work published: Phil. Trans A371 (2013) 23,[arXiv:1209.5517]

• The work of these years focused on the general TodaModels, of which the BD model is a representative

• The result are being collected and ordered in view of apublication

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Introduction

Conclusions

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Conclusions

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Conclusions

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Introduction

Perspectives

• The ODE/IM is a surprising and interesting mathematicalstructure which is not yet well understood; it might beworthy to investigate more deeply.

• It has also proven to be a powerful tool in variousbranches of the physics, such as condensed matter physics,boundary CFT and PT-symmetric QM

• It can be applied also to the amplidude calculations inN = 4 Super Yang-Mills theory, via the AdS/CFTcorrespondence

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Introduction

Perspectives

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Perspectives

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Introduction

Perspectives

The ODE/IM correspondence in Toda field...

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