An introduction to the IM/ODE correspondanceAn introduction to the IM/ODE correspondance Clare...

An introduction to the IM/ODE correspondance

Clare Dunning

University of Kent

Integrability in Low Dimensional Systems, MATRIX, July 2017

ODE/IM Correspondence

The same functional relations are satisfied by

I spectral determinants of Ordinary Differential Equations

I T and Q functions in Integrable Models

Consequently

eigenvalues of the ODE = the Bethe roots of the quantum IM

Apply IM techniques to ODEs and vice versa





Consequently







Consequently



Which IMs and ODEs?

Simplest example

XXZ spin chain

6-vertex model ←→(− d2

dx2 + x2M)ψ(x) = Eψ(x)

c ≤ 1 conformal field theory

Outline

I An integrable model: the 6-vertex model

I A differential equation: the cubic oscillator

I The IM/ODE correspondence (massless)

I The IM/ODE correspondence (massive)

I Summary

Square ice in nature?

Square ice in Nature

One atom thin layer of water between two layers of graphene

Square ice in graphene nanocapillariesG. Algara-Siller, O. Lehtinen, F. C. Wang, R. R. Nair, U. Kaiser,H. A. Wu, A. K. Geim I. V. Grigorieva, Nature 519 (2015) 443

Square ice in Nature. Perhaphs not?

One atom thin layer of water between layers of graphene

The observation of square ice in graphene questionedW. Zhou, K. Yin, C. Wang, Y. Zhang, T. Xu, A. Borisevich, L.Sun, J C Idrobo, M.F. Chisholm, S.T. Pantelides, R.F Klie andA.R. Lupini, Nature 528 (2015) E1

Square ice rules

Arrange water molecules on an N by N ′ lattice

I Oxygen atom at each site

I Two hydrogen ions bind to O by strong covalent bonds

I Two hydrogen ions bind to O by weak hydrogen bonds

Tetrahedral ice Square ice

Square ice

I Replace covalent bonds with down/left arrows

I Replace hydrogen bonds with up/right arrows

I Assume periodic boundary conditions

Six possible configurations of arrows at each vertex

6-vertex modelAllow probabilities of each configuration to vary:

W

[→↑↑→

]= W

[←↓↓←

]= sin(η + iν)

W

[→↓↓→

]= W

[←↑↑←

]= sin(η − iν)

W

[→↑↓←

]= W

[←↓↑→

]= sin(2η)

Pictorial representation of Boltzmann weight

��

α

α

νββW

αβ β

α(ν )

where anisotropy η is fixed and spectral parameter ν varies

Partition function Z

Z =∑

configs

∏sites

W[· ·· ·

]

Transfer matrix

Tαα′αα (ν) =

∑{βi}

W

[β1α′1α1β2

](ν)W

[β2α′2α2β3

](ν) . . .W

[βNα′NαNβ1

](ν)

or representing this 2N by 2N matrix pictorially

ν

α α αα

β1β2 β3 4β βNβ1

ν ν ν(ν)T

α 1 α α

1 2

3

N

N

3

2

{βi }Σ

α

��

��

��

��

ThenZ = Trace

[TN′]

Setting ∑αα′

Tαα′αα Ψ

(j)αα′ = tj(ν)Ψ(j)

αα .

Free energy per site in the limit N ′ →∞ can be obtained as

f = − 1NN′ logZ = − 1

NN′ log Trace[TN′]∼ − 1

N log t0

Baxter’s TQ approach

Since [T(ν),T(ν ′)] = 0 we solve for each tj(ν) independently

Baxter introduced an auxiliary function q(ν) such that

t(ν)q(ν) = sin(η + iν)Nq(ν + 2iη) + sin(η − iν)Nq(ν − 2iη)

Properties

I t(ν) and q(ν) are entire functions

I t(ν) and q(ν) are iπ-periodic functions

Suppose q(ν1) = q(ν2) · · · = q(νn) = 0. Then

q(ν) =n∏

l=1

sinh(ν − νl)





Properties




q(ν) =n∏

l=1

sinh(ν − νl)





Properties




q(ν) =n∏

l=1

sinh(ν − νl)





Properties




q(ν) =n∏

l=1

sinh(ν − νl)

TQ relation → BAE

Set ν = νi in


Rearranging we have

q(νi − 2iη)

q(νi + 2iη)= −sin(η + iν)N(νi , η)

sin(η − iν)N

The Bethe ansatz equations

(−1)nn∏

l=1

sinh(2iη − νi + νl)

sinh(2iη − νl + νi )= −sin(η + iν)N

sin(η − iν)N, i = 1 . . . n

TQ relation → BAE

Set ν = νi in


Rearranging we have

q(νi − 2iη)


sin(η − iν)N


(−1)nn∏

l=1



sin(η − iν)N, i = 1 . . . n

TQ relation → BAE

Set ν = νi in


Rearranging we have

q(νi − 2iη)


sin(η − iν)N


(−1)nn∏

l=1



sin(η − iν)N, i = 1 . . . n

Which solution of BAE?

The ground state eigenvalue t0(ν) has N/2 distinct, real roots

ν ν ν ν ν ν ν ν0 1 2 3 4 5 6 7

ν

6-vertex model and the XXZ model

The transfer matrix eigenvectors of the XXZ model

HXXZ = −1

2

N∑j=1

(σxj σ

xj+1 + σyj σ

yj+1 − cos 2η σzj σ

zj+1

)coincide with those of the 6-vertex model

HXXZ = −i sin 2ηd

dνlnT(ν)

∣∣∣ν=−iη

− 1

2cos 2η I⊗N .

Continuum limit of the 6-vertex model

Take N →∞, the lattice spacing d → 0 with Nd finite and scale νappropriately to find

ln t0(N) = −f N +πceff

6N+ . . .

TQ relation becomes

t(E )q(E ) = e iφq(ω2E ) + e−iφq(ω−2E )

where ω = −e−2iη and q(El) = 0 implies

∞∏l=1

(El − ω2Ei

El − ω−2Ei

)= −e−2iφ , i = 1 . . .∞

Have also sneaked in a twist φ in the periodic boundary conditions




6N+ . . .

TQ relation becomes



∞∏l=1

(El − ω2Ei

El − ω−2Ei

)= −e−2iφ , i = 1 . . .∞





6N+ . . .

TQ relation becomes



∞∏l=1

(El − ω2Ei

El − ω−2Ei

)= −e−2iφ , i = 1 . . .∞


Ordinary differential equations

We shall study several eigenproblems associated with the ODE[− d2

dx2+ x2M

]y(x ,E ) = E y(x ,E ) , M ≥ 1

Simple example: P(x) = x3 − E

Consider the WKB approximation for large-|x | when M = 3/2

ψ(x) ∼ 1

P1/4(x)exp

(±∫ x√

P(t) dt

), |x | → ∞

Thus there are two asymptotic behaviours as x →∞ :

ψ±(x) ∼ x−3/4 exp

(±2

5x

52

)

Call solution that grows for large-x as dominant and the solutionthat decays as subdominant

E belongs to the spectrum iff the subdominant solution asx → −∞ is also subdominant as x →∞



ψ(x) ∼ 1

P1/4(x)exp

(±∫ x√

P(t) dt

), |x | → ∞


ψ±(x) ∼ x−3/4 exp

(±2

5x

52

)





ψ(x) ∼ 1

P1/4(x)exp

(±∫ x√

P(t) dt

), |x | → ∞


ψ±(x) ∼ x−3/4 exp

(±2

5x

52

)



WKB expansion does not hold near x such that P(x) = 0 socannot simply continue x from −∞ to ∞

Instead we can take x to be complex and continue x throughcomplex values from large negative real x to large positive real x

We need to take into account the Stokes phenomenon where thedominant component of an asymptotic solution, if nonzero, canhide a ‘discontinuous’ change in the size of its subdominantcomponent as x varies past a Stokes line

Stokes phenomenon

Set x = −iρe iθ with ρ, θ ∈ R then

ψ±(x) ∼ ρ−3/4 exp

(±2

5e i

5θ2 ρ

52

)ρ→∞

The dominant and subdominant solutions can be distinguished forall θ except when

<e(e i52θ) = 0

As θ sweeps past

θ =π

5± 2πn

5, n = 0, 1, 2

previously subdominant solutions swap to be dominant and viceversa.

Stokes phenomenon


ψ±(x) ∼ ρ−3/4 exp

(±2

5e i

5θ2 ρ

52

)ρ→∞


<e(e i52θ) = 0

As θ sweeps past

θ =π

5± 2πn

5, n = 0, 1, 2


Stokes phenomenon


ψ±(x) ∼ ρ−3/4 exp

(±2

5e i

5θ2 ρ

52

)ρ→∞


<e(e i52θ) = 0

As θ sweeps past

θ =π

5± 2πn

5, n = 0, 1, 2


Stokes sectorsThe complex plane is thus split into five Stokes sectors

Sk =

∣∣∣∣arg(x)− 2πk

5

∣∣∣∣ < π

5

separated by the anti-Stokes lines along which both WKB solutionsoscillate

.

x

S2

S1

0S

S−1

S−2

TQ relations from ODEs

Consider [− d2

dx2+ x3

]y(x ,E ) = E y(x ,E )

The ODE has a unique solution [Sibuya (1970s)]

I y is an entire function of x and E

I For |x | → ∞ with |arg x | < 3π/5

y ∼ 1√2ix−3/4 exp

[−2

5x5/2

]

Family of solutions

There is a family of solutions

yk(x ,E , l) = ωk/2y(ω−kx , ω2kE , l) ω = exp(2πi/5)

such that

I yk exists and is an entire function of x and E

I yk is subdominant in the Stokes sector Sk , and is dominant inSk−1 and Sk+1 .

MoreoverW [yk , yk+1] := yk y

′k+1 − y ′k yk+1 = 1

implies yk , yk+1 are linearly independent solutions

Family of solutions



such that




′k+1 − y ′k yk+1 = 1


Family of solutions



such that




′k+1 − y ′k yk+1 = 1


Expand y−1 in basis of y0 ≡ y , y1:

y−1(x ,E ) = C (E ) y0(x ,E ) + C (E ) y1(x ,E )

The Stokes multipliers C (E ) and C (E ) are

and

C (E ) =W [y−1, y0]

W [y1, y0]= −W [y−1, y0]

W [y0, y1]= −1

Thus the Stokes relation is

C (E )y(x ,E ) = ω−1/2 y(ωx , ω−2E ) + ω1/2 y(ω−2x , ω2E )


y−1(x ,E ) = C (E ) y0(x ,E ) + C (E ) y1(x ,E )


and

C (E ) =W [y−1, y0]

W [y1, y0]= −W [y−1, y0]

W [y0, y1]= −1




y−1(x ,E ) = C (E ) y0(x ,E ) + C (E ) y1(x ,E )


and

C (E ) =W [y−1, y0]

W [y1, y0]= −W [y−1, y0]

W [y0, y1]= −1



TQ relations

If we x = 0 and D(E ) = y(0,E ) we have

C (E )D(E ) = ω−1/2D(ω−2E ) + ω1/2D(ω2E )

which we can compare with

t(E )q(E ) = e−iφq(ω−2E ) + e iφq(ω2E )

TQ relations

If we x = 0 and D(E ) = y(0,E ) we have

C (E )D(E ) = ω−1/2D(ω−2E ) + ω1/2D(ω2E )

which we can compare with

t(E )q(E ) = e−iφq(ω−2E ) + e iφq(ω2E )

What are C and D?

They are spectral determinants–functions that vanish at theeigenvalues of an eigenvalue problem

We see thatD(Ek) = 0 ⇐⇒ y(0,Ek) = 0

and by definition

y(x ,E )→ 0 as |x | → ∞ in S0

Therefore zeros of D(E ) are the eigenvalues of the ODE with

I Dirichlet boundary condition at x = 0

I ψ(x) ∈ L2(R+)

What are C and D?



and by definition

y(x ,E )→ 0 as |x | → ∞ in S0



I ψ(x) ∈ L2(R+)

What are C and D?



and by definition

y(x ,E )→ 0 as |x | → ∞ in S0



I ψ(x) ∈ L2(R+)

What are C and D?



and by definition

y(x ,E )→ 0 as |x | → ∞ in S0



I ψ(x) ∈ L2(R+)

Could also set D(E ) = y ′(0,E ) to obtain

C (E )D(E ) = ω1/2D(ω−2E ) + ω−1/2D(ω2E )

We see thatD(Ek) = 0 ⇐⇒ y ′(0,Ek) = 0

and by definition

y(x ,E )→ 0 as |x | → ∞ in S0


I Neumann boundary condition at x = 0

I ψ(x) ∈ L2(R+)


C (E )D(E ) = ω1/2D(ω−2E ) + ω−1/2D(ω2E )


and by definition

y(x ,E )→ 0 as |x | → ∞ in S0



I ψ(x) ∈ L2(R+)


C (E )D(E ) = ω1/2D(ω−2E ) + ω−1/2D(ω2E )


and by definition

y(x ,E )→ 0 as |x | → ∞ in S0



I ψ(x) ∈ L2(R+)

C is also a spectral determinant

SinceC (ek) = 0 ⇐⇒ W [y−1(x , ek), y1(x , ek)] = 0

which impliesy−1(x , ek) ∝ y1(x , ek)

This means there is a solution of the ODE with E = ek that decayssimultaneously in S−1 and S1

This is equivalent to a spectral problem in PT -symmetric quantummechanics:[

− d2

dx2+ ix3

]y(x ,E ) = −e y(x ,E ) , y ∈ L2(R)






− d2

dx2+ ix3

]y(x ,E ) = −e y(x ,E ) , y ∈ L2(R)






− d2

dx2+ ix3

]y(x ,E ) = −e y(x ,E ) , y ∈ L2(R)

Three eigenproblems

Solve Hψ = Eψ for H = p2 + x3 subject to

1. ψ(0) = 0 and ψ ∈ L2(R+)

2. ψ′(0) = 0 and ψ ∈ L2(R+)

3. ψ ∈ L2(C) where C is complex contour from S−1 to S1

.

x

S2

S1

0S

S−1

S−2

Bethe ansatz equations

Setting E = Ej where

D(E ) = D(0)∞∏i=0

(1− E

Ej

)

Then evaluating the TQ relation at E = Ej

C (Ej)D(Ej) = ω−1/2D(ω−2Ej) + ω1/2D(ω2Ej)

implies the Bethe ansatz equations

∞∏k=1

Ek − ω2Ej

Ek − ω−2Ej= −e

−2iπ5 j = 1, 2, . . .

Bethe ansatz equations

Setting E = Ej where

D(E ) = D(0)∞∏i=0

(1− E

Ej

)

Then evaluating the TQ relation at E = Ej

C (Ej)D(Ej) = ω−1/2D(ω−2Ej) + ω1/2D(ω2Ej)

implies the Bethe ansatz equations

∞∏k=1

Ek − ω2Ej

Ek − ω−2Ej= −e

−2iπ5 j = 1, 2, . . .

The generalised problem

The eigenvalues {Ej}(− d2

dx2+ x2M +

l(l + 1)

x2

)ψ(x ,E , l) = Eψ(x ,E , l) ψ ∈ L2(R+)

satisfy the Bethe ansatz equations

∞∏k=1

Ek − q2Ej

Ek − q−2Ej= −e

iπ(2l+1)M+1 j = 1, 2, . . .

Many eigenvalue problems

T1 1/2

C

T

x

IM/ODE dictionary

6-vertex model with twistφ = −(2l + 1)π/(2M+2)

Schrodinger equation withpotential x2M + l(l + 1)/x2

Spectral parameter ↔ Energy

Anisotropy ↔ Degree of potential

Twist parameter ↔ Angular momentum

Transfer matrix ↔ The Stokes multiplier C (E )

(Fused) transfermatrices

↔ Lateral spectral problems

defined at |x |=∞

q functions ↔ Radial spectral problems

linking |x |=∞ and |x |=0

Many generalisations

I Perk-Schultz model/hairpin model of boundary interaction

I Spin-j su(2) quantum chains in thermodynamic limit and theboundary parafermionic sinh-Gordon model

I vertex models with Lie algebra symmetry (simply andnon-simply laced)

I finite spin-j XXZ quantum chains at ∆ = ±1/2

I Coqblin-Schrieffer model

I Circular Brane model

I Paperclip models

I Finite spin-1/2 XYZ quantum chain

I...

Bazhanov, Dorey, Dunning, Hibberd, Khoroshkin, Lukyanov, Mangazeev, Masoero, Raimondo, Suzuki, Tateo,Tsvelik, Valeri, Vitchev, Zamolodchikov,Zamolodchikov . . .

Massive ODE/IM correspondance

Following work of Gaiotto, Moore & Neitzke, Lukyanov &Zamolodchikov established the correspondence for the quantumsine-Gordon field theory on a finite cylinder

Instead of an ODE, start with a classical integrable model, amodification of the sinh-Gordon model, and study its linear problem

∂z∂zη − e2η + p(z)p(z)e−2η = 0

where p(z) = z2M − s2M

Several of the people above + Adamopoulou, Faldella, Ito, Locke, Negro have generalised this idea

Summary

Date post:	28-Mar-2021
Category:	Documents
Upload:	others
View:	12 times
Download:	0 times

An introduction to the IM/ODE correspondanceAn introduction to the IM/ODE correspondance Clare...

Documents