Hirota solutions of TBA and NLIE Francesco Ravanini Cortona 2010 A.D. 1088.

Hirota solutions of TBA

and NLIE

Francesco Ravanini

Cortona 2010

A.D. 1088

Exact tools for FSE 2Les Houches, 1 Mar 2010

Finite size effects

Luscher 1980-84

Finite size effects S-matrix

term F-ter( ) ..s .msMLe

M LL ML


Scaling functions

( )

6i

i

c lE l ML

L

M

l

E

Cardy, Blote, Nightingale 1984 link with CFT at UV

for the i-th excited state (0=vacuum)

(0) 12( )ii ic c


T.B.A. (Thermodynamic Bethe Ansatz)

1

space time

finite size effect t

sp

hermod

ace time

Ha

ynamics

miltoni a n:

R R TL L

0

0 0

( ) (

( )

)

Hilbert space:

Partition f

u

n

ction [ ] [ ]

Tr Tr

(

)

LR

R

H

R

ll rr

L R

RE R L

L

R

H

Rf

R

HT dr T dl

e e

e

H R

e

LR

R L

Al. Zamolodchikov 1990

Integrability (2D)

Thermodynamic Bethe Ansatz R

L


0

6( ) ( ) ( ) ( ) ,E R Rf R c r f r r MR

Scaling function of the vacuum (Casimir effect)

Compute f(r): Dynamics dictated by Bethe-Yang eqs. in the thermodynamic limit

1

( ) 1N

ipRj

j

e S

Form of S-matrix

,0, ( ) dressing fact( )) ( ) ( )R-matro xr( io j jjj jS R

sinh

0 0,1

·

Tr (

( |{ })

( |{ }) color transfer matr

(

i

)

x)N

j jj

jj

jir

j

e

R

T

T

Diagonalize color transfer matrix by Bethe ansatz (Al. Zamolodchikov, 1991)


Example: Sine-Gordon – S-matrix (Zam-Zam, 1979) has a dressing factor

2

2

sinh ( 1)2

exp ,2 8 12sinh cos

(

2

)h

2

kp

dkp

k kk pp

and a matrix part coinciding with the XXZ spin ½ R-matrix

sinh(

( )

)

sinh sin1

sin sinhsinh( )

sinh( )

i

i i

i

i

Ri i


1/2 1

{

1

1/2

1

}1

Bethe equations s ( ) s (

sinh ( )where s ( )

s

)

s ( )

inh ( )

Eigenvalues ( |{ })

terms vanishing for

j

N M

r k

M

j j k

jj

r

r

x ix

x i

N

The color transfer matrix is diagonalized by the fully inhomogeneous XXZ spin ½ Bethe ansatz

Full set of Bethe-Yang equations for Sine-Gordon given by eigenvalues coupled to the exp(iPL) term and Bethe equations for the Bethe roots (magnon excitations)


sinh1/2

1 1

11/2 1

1 1

(

particle

) ( )

1

rapidities , magno

)

, ns

( ) (

n

N Mir

n r n jr j

N M

j

n r

k

k

j kr

j

r

e s u

s u s u u

u u

String hypotesis for the Bethe roots: in the thermodynamic limit the Bethe roots tend to organize as follows:

( ) ( ),

( ) ( )

( ) ( )

( 1 2 ) 1,...,2

string centres (not necessarily roots)

( 1) (strings of the second kind)2

n nj a j

n nj j

n nj j

iu u n a a n

u u n

iu u p


In the thermodynamic limit the number of roots tends to infinity and they become dense.

Intorduce density for each type of n-string and for the corresponding holes

) density of centres of strings of type

( ) density of centres of holes of type

(n

n

n

n

Logs of Bethe-Yang equations give coupled integral eqs. for the densities

,

,0

) ( ) ) * )

where

( ( (

) cosh driving term(

nn n n m mm

n n

K

r


From density: compute energy, entropy and free energy of the system. Minimum of free energy gives the conditions (TBA equations)

11) log ( ) ( *log(1 ( ))( )

2

( ) 1( ) ( )

( ) cosh

adiacency matrix of a (magnon) graph

(

)n n nm nm

nn nm nm

n

y K y

y K H

H

H

If more particles, each type is coupled to an equation with its mass term

1

,

, ,

1 1log ( *log(1 )) ( *log(1 ( ) )

2 2

const.( ) ( ) log ( )

cosh

cosh if the model is massive( )

) if the model is massl(2

es

i i i ja a ab b ij a

b j

ab ab

i Ma

ia i L i Ra

y K y H K y

dK K i S

d

m

m Re e

R

s


,

, ,

1log ( ) * [ log(1 ] log 1 ( )

kernel: ( ) 4 cosh

2

cosh (massive)driving terms: )

( ) (massl

(

es(

)

)

)

s2

i i ia ab b ij

b j

i M

ia i L R

ia b j

a

i

K

gK

g

MR

M

y

Re

y

e

y

G

G

G H

Through Fourier transform, one can prove a useful identity valid for all ADE S-matrices, that brings TBA in the so-called Universal Form

(Al. Zamolodchikov, 1991 without magnons)

(FR, Tateo, Valleriani, 1992 and Quattrini, FR, Tateo, 1993 with magnons)


Dynkin TBA or… the Y-system1) ( ) (1 ))( (1 ijabi i i j

a a b ajb

iq y iqy y y HG_

Equivalent to TBA if complemented by the asymptotic conditions

cosh if node ( , ) is massive)

const. if node ( , ) is mag(

nonic

am L ijia Ly

e a i

a i

Draw diagrams with adjacency G and H


(Dynkin) masses Perron-Frobeniusaam M GG

(magnons)H

1For AG

13(Vir)p M2

Sine-Gordon at ,8 1

pp

p

H diagram can be Dynkin or extended Dynkin (or something else ??? ) (Quattrini, FR, Tateo, 1993)

FR, Tateo, Valleriani, 1993 (Dynkin TBA’s)

FR, 1992

Hollowood, 1994

Al.Zam. 1991-92


Sine-Gordon for general p rational → Continued Fraction

Takahashi, Suzuki (1972) – Mezincescu, Nepomechie (1990) – Tateo (1994)

Tateo snakes


Bazhanov, Lukyanov, A.Zamolodchikov (1995) suggest that nodes on H direction are linked with internal symmetry algebra: j labels IRREPS

Kuniba, Nakanishi, Suzuki (1996): a,j related to Young tableau of IRREPS

But then what about diagrams D or D(1) ?

Truncation of Q-group reps. when

22

with integer8 1

i pq e p

p

IRREPS are only up to j=p-1 plus two type II reps labelled by p, p+1

Adjacency:

General rational p: Mezincescu, Nepomechie: it gives the same continued fraction decomposition as strings → Tateo snakes


1 1( ) ( ) ( ) ( ) where 1k k k ky u i y u i u Y u Y yY

SU(2)xSU(2) Principal chiral model: Y-system y+y- = YY

T-system

TT = 1 + T+T-

1 1( )( ) where

( ) is an arbitrary function

( ) ( )kk k u

u ki u ki

T u Ty u

*1 1( ) ( ) ( ) ( )( ) ( )k kk k u kT u i T u i u ki TT ui u

Finite difference equation of Hirota type, 2nd order. Solution by Lax pair, known as TQ-relation: TQ = Q+ + Q-

1

1

( ) ( ) ( ) ( ( 2) ) ) ( ( 2) )

( ) ( ( 2) ) (

(

( ) () )) (

k k

k k

T u Q u ki T u i Q u k i u ki Q u k i

T u Q u k i T u i Q u ki u ki Q u ki

2u


Baxter Q-operator

Entire function with zeroes coincident with the Bethe roots.

Up to prefactor not containing zeroes

( ) rational (non periodic)

( ) ) trigonometric (Im periodicity 2

) elliptic (double periodicity Re & Im)

where ( ) is an entire function a

sinh ( )

(

is a cn onsta td n

j

j

j

j

j

j

x x

Q x

f x

x x

x x

Im-Periodicity related to relevant perturbing field of UV CFT

Al. Zamolodchikov (1991)


Solution Hirota (1981) expresses all the Tk ’s in terms of

1

0

( ( 1) )( )

( ( 1) )

( ( 1) ) ( ( 1) )( ( 1 2 )

( ( 1 2 )

) ( ( 1 2

(

)

)

)k

k

j

Q u k iT u

Q u k i

Q u k i Q u k iQ u k j

T u k

u k j

i Q u

i

k i

i

j

0 and (( ) )T u u

“Gauge” symmetry *

( ) ( ) ( ) (

( )

)

( ) ( ) ( )

( ) ( ) ( )k k

u g

T u g u ki g u ki

u i g u i u

Q u g u i Q u

T u

Leaves the T-system invariant. Y functions are invariant

Auxiliary functions (gauge invariant)

( )( ( 2) )( ) ( ) 1 ( )

( ( 2) ) ( )k

k k k

T u iQ u k ib u B u b u

Q u u kk i i


1( ) ( ) ( ) ( ) ( ) ( )kk k kk kb u Y u B u i B u i Yu ub

log ( ) cosh sources 2 Im ( ) log ( )2 i

b u ML u dxG u x B x

ò

Satisfy

Use logarithmic index theorem, Fourier transform and some algebraic manipulation to show that

0( )) (b b u iu

satisfies the NLIE

( ) log ( )h u i b utaking the DdV NLIE for Sine-Gordon is recovered


Conclusions• TBA applied to relativistic integrable QFT has given very good results on Finite Size Effects

• NLIE can be seen as a powerful tool, sort of Bethe ansatz for QFT in the continuum

• The two methods are related thanks to their functional forms: Y-systems and T-systems, and the Baxter Q-operator

•Applications to recent problems (AdS/CFT): resummation is still an issue (sum of massive nodes is difficult…)

• Other models can be dealt with this methods, like sausages, surely interesting for a group based in Bologna…

Thank you

Hirota solutions of TBA and NLIE Francesco Ravanini Cortona 2010 A.D. 1088.

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