hirota solutions of tba and nlie francesco ravanini cortona 2010 a.d. 1088
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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
Exact tools for FSE 3Les Houches, 1 Mar 2010
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
Exact tools for FSE 4Les Houches, 1 Mar 2010
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
Exact tools for FSE 5Les Houches, 1 Mar 2010
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)
Exact tools for FSE 6Les Houches, 1 Mar 2010
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
Exact tools for FSE 7Les Houches, 1 Mar 2010
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)
Exact tools for FSE 8Les Houches, 1 Mar 2010
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
Exact tools for FSE 9Les Houches, 1 Mar 2010
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
Exact tools for FSE 10Les Houches, 1 Mar 2010
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
Exact tools for FSE 11Les Houches, 1 Mar 2010
,
, ,
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)
Exact tools for FSE 12Les Houches, 1 Mar 2010
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
Exact tools for FSE 13Les Houches, 1 Mar 2010
(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
Exact tools for FSE 14Les Houches, 1 Mar 2010
Sine-Gordon for general p rational → Continued Fraction
Takahashi, Suzuki (1972) – Mezincescu, Nepomechie (1990) – Tateo (1994)
Tateo snakes
Exact tools for FSE 15Les Houches, 1 Mar 2010
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
Exact tools for FSE 16Les Houches, 1 Mar 2010
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
Exact tools for FSE 17Les Houches, 1 Mar 2010
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)
Exact tools for FSE 18Les Houches, 1 Mar 2010
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
Exact tools for FSE 19Les Houches, 1 Mar 2010
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
Exact tools for FSE 20Les Houches, 1 Mar 2010
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
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