Field-Theoretical Formulation of the
Thermodynamical Bethe Ansatz
Ivan Kostov
Institute de Physique Théorique, CEA - Saclay

I.K., arXiv[hep-th]1909xxxx
I.K., Didina Serban, D. L. Vu, arXiv[hep-th]1805.02591, 1809.05705, 1906.01909

Workshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan!1

Thermodynamic Bethe Ansatz(TBA) [Yang&Yang, 1969]3— thermodynamics of 1-dim.
Since [Al. ns
ZamolodchikovTBA(1990)and tree expansionniopa
integrable
systems at finiteex
BA and tree
⇤
(1)
⇥ RH
TBA became the main tool3⇤
temperature.
e
r
⇥
T
=
)
R
,
(L
Z
toecompute
RH
expansion finite size
and treTBA and tree expansion
Z(L,
R)
=
Tr
e
ading
le(1)
e
th
⇥ RH ⇤
tealu
aeﬀects in 1+1 dim.
ev
to
is
l
a
o
g
r
u
o
at R ⌧ L,thgin⇥ RH ⇤
m
u
ss
e
A
.
Tr
e
=
c
a
R)
L,
sp
Z(
rt
e
b
il
H
ll
fu
e
n in thtaketheories
Z(L, R)= Tr e that R ⌧the trace isfield
whererelativistic
where
the
trace
is
taken
in
the
full
Hilbertspace.
Assuming
luate the leadingeva
e free energy
to
is
al
go
r
ou
L,
erm of thce
⌧
R
t
tha
ing
sum
As
ce.
spa
rt
lbe
l Hi of the free energy
re the tra is taken in the fulterm(2)⌧
where the trace1is taken in the full Hilbert space. Assuming that R
log ZL (R).
limof the free energy(R)of=the
F
term
free
energyL! 1 LCut the cylinder and1 glue it
1
type
le
g
n
F
(R)
=
lim
log
Z
(R).
si
e
(2)
n
o
d
n
a
L
)
v
back
by
insertingacomplete
,
(u
S
ix
tr
).
a
(R
Zit
log
Lh diagonal S-m=se olim)
Fle(R
L!1 L1
w
ry
o
e
th
a
f
a
c
ises
st
tr
e
p
m
1
m
L
ra
L!
si
a
e
p
h
th
F
(R)
=
lim
log
Z
(R).
ic
to
h
s
w
e
,
lv
set
of
virtualstates(wrapping
le
e
L
b
rs
a
u
ri
o
a
t
v
c
y
We restrie rapidit
th
is
u
re
e
L!1 L
H
l.
ra
e
n
e
g
is
t
n
e
e
m
th
More recently
TBA
relatedmethods
are
used
in
u
,
g
rg
a
n
ri
e
e
th
tt
t
a
u
sc
b
l
,
a
n
n
o
o
ti
g
a
particles)
it
ia
e
c
d
typ
h
ex
gle
it
sin
le
d one with diagonryv)aantheoryu,ofS(wtheixocase
a atr
uasipartic
r
S-m
o
l
F
of qcomputationnaWe
restrictourselvestothesimplest
go
y.
dia
a
th
w
wi
t
ory
n
the
ie
a
n
of
e
e
v
cas
n
st
o
ple
c
sim
a
the
to
in
s
lve
le e.g.
of pcorrelation
functions,t ourseasd, inparticWesire
We restric
a
h
u
q
ic
h
a
w
f
o
s
)
n
o
(u
ti
a
u
q
=
ise
e
etr
p
e
ram
restrictourselvestothesimplest
caseofatheorywithdiagona
pa
m
th
e
ich
tu
B
n
wh
,
e
e
ble
th varia is general. Here u is th
y ty
mom
idi
n bargument
erap
the hexagonalization
the
iv
is
u
g
re
He
.
ral
R
ne
e
ge
of
quasiparticle
excitation,
but
the
c
is
t
n
en
re
um
fe
arg
the
m
t
u
bu
c
,
on
ir
ati
c
cit
h
ex
method
in
N=4
SYM
e
it
icl
w
uasipart tion condition on a cylinder of quasiparticle excitation, but the argument
theis thring, uis na
general.
l scatteHere
go
dia
th
wi
ntisa
ory
the
a
r
qua(see
Fo
y.
wa
t
ien
en
nv
co
a
in
le
rtic
pa
asi momentum p = p(u) of a quasiparticle in a convenient way. Fo) of a quthep(up=Didina’stalk).
m lemomentur,
tod, in Focsewhich reaway.
themomentum
p
=
p(u)
of
a
quasiparticle
inaconvenient
ns
ic
tio
ua
eq
the
Be
the
by
the n-partco
en
giv
is
R
e
nc
ere
mf
cu
cir
th
(3
der wiconditiononacylinderwithcircumference
R
is
given
ntisation ndition on a cylinquantisation.
M,.
1, . .
quantisation
conditionona cylinder
with circumference R is given
j=
r,
e
g
te
in
n
h
it
w
j
n
⇡
2
j
Need
eﬃciently
the sum overtor, how to evaluate
seclearn
rticleto
n-pa
j =
the
n-particle
sector,the n-particle
sector,the virtual particles in diﬀerent problems.
(3)
n. .o.n,, M .
g
a
m
r
o
,
le
1,
ic
=
j
rt
a
p
er,
eg
h
int
-t
n
j
th
e
wi
th
n
r
2⇡
j
j fo
phase
g
j n=
ri
e
tt
a
sc
l
ta
to
e
th
is
re
He
j
njninteger,
1,
= 2⇡n
= 1,
j withj integer, jj =j = j2⇡nj withIs it possible to replace the original TBA arguments
M on,
X
gn
ma
or
(4
le,
rtic
pa
h
j-t
the
for
1
ase
ph
g
rin
tte
sca
al
tot
the
is
e by
).
u
,
j a more refined QFT/statistical formulation?
(uj fork the j-th particle, or magnon
log Sphase
+
Here
is
the
total
scattering
L
p
⌘
)
j
j
u
,
.
.
.
Here
is
the
total
scattering
phase
for
the
j-th particle, or magnon,Mj (u1 jiMj)
1 Xk(6=
M (4) th
).
u
,
u
S(
log
k
j
+
X
L
p
⌘
)
u
,
.
.
.
tor !2in,ra(u1September 2019,ep
MOsaka CityjUniversity, Japan
o
M
1
ty
ti
n
e
Workshop New Trends in Integrable Systems,j9-20
id
e
th
d
n
a
)
X
(3
i
numbers

The question was posed decades ago and the answer is in principle yes
although the eﬀective QFT has not been yet formulated
Balog’94, Saleur 1999
Woynarovich, 2004: gaussian fluctuations aroundthe saddle point of the Y-Y potential.
Pozsgay, 2010: showed that there is another O(1)contribution from the measure.
Kato&Wadati, 2004: exact cluster expansion.
I.K., Serban, Vu 2018 graph expansion for the freeenergy with periodic and open b.c.
In this talk we construct from scratch an eﬀective QFT
generating the exact cluster expansion for TBA— For simplicity we take an integrable theory with one single neutral particle.
— In view of applications to N=4 SYM the scattering matrix is not supposed
to be of diﬀerence type and no relativistic symmetry is assumed, only amirror transformation.

Workshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan!3

Euclidean 1+1 dimensional integrable field
theory with factorized scattering==(c.f. Roberto’s talk)
Rapidity variable:(θ ≡ πu)p = p(u), E = E(u)unitarity: S(u, v) S(v, u) = 1
Two-particle S-matrix: uv

crossing

S(u, v)S(u, v 2γ ) = 1===

S(u, u) = − 1

E → ip̃, p → iẼ
PhysicaltheoryMirrortheoryWorkshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japantime— no Lorentz invariance assumed, only a mirror
transformation exchanging space and time

mirror particles

physical particles

space

γ

Physical theory → Mirror theoryγγ

S(u, v) → S̃(u, v) ≡ S(u , v )p(u) → p̃(u) = − iE(u γ )E(u) → Ẽ(u) = − ip(u γ )uuγγ : u → uγ

Example 1: Lorentz-invariant
massive integrable QFT

E2 − p2 = m2
E = m cosh(πu)
p = m sinh(πu)
Phys=Mirror

u γ ≡ u + i/2uu −γ ≡ u − i/2

Workshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japantime

For an observer living in the physical space,a mirror particle looks as physical particle
with complex rapidity

physical particles

mirror particles

Mirror transformation as analytical continuation

space

Example 2: N=4 SYM−2g + i /2

2g + i /2u−2g − i /2

2g − i /2

u −γ
uγ

!5

I. PERIODIC BOUNDARY CONDITIONS

A

B
Physical theory defined on the A-cycle of length L,
Mirror theory defined on the B-cycle of length R
Degrees of freedom of the eﬀective QFT:
particles winding around the A and B cycles

Workshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan!6

1. Wrapping operators
The Hilbert space of the eﬀective QFT is spanned on the elementary
excitations on a torus with asymptotically large space and time circles.
Two types of them: time-wrapping particles in the physical theory and andspace-wrapping particles in the mirror theory

Operators creating wrapping particles:

LR
physical space= mirror time

mirror space=physical time

mirror particles

physical particles

A(u) = time-wrapping operator
B(u) = space-wrapping operator
Particles wrapping the same cycle do not scatterbut particles wrapping diﬀerent cycles do:=

Workshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan

!7

A(u)

creates real particle in the physical theory/virtual particle in the mirror theory

B(u)creates real particle in the mirror theory/virtual particle in the physical theoryγ

creates space-wrapping particle in the mirror theory−γ

creates space-wrapping particle in the physical theory

A(u )
B(u )

Workshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan!8

2. Algebra of the wrapping operators
...

u1 u2

Expectation value of N time-wrappingand M space-wrapping operators:

⟨

M

∏
j=1

B(vj)

N

∏
k=1

A(wk)⟩ =MN∏∏
j=1 k=1

S(vjγ, wk)

uM

wN
w2
w1
M

∏

e −RẼ(vj)

j=1

N

∏

e −LE(wk)

k=1

Fock-space realisation:

B(v)A(u) = S(v γ, u) A(u)B(v),⟨L | A(u) = e −LE(u) ⟨L | ,[B(u), B(v)] = [A(u), A(v)] = 0

B(u) | R⟩ = e −RẼ(u) | R⟩

For any operator define

Fock-space expectation value:
⟨

M

∏
j=1

B(vj)

N

∏
k=1

A(wk)⟩ ≡ ⟨L |

M

∏
j=1

B(vj)

⟨L | R⟩ = 1

N

∏
k=1

A(wk) | R⟩

Workshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan⟨𝒪⟩ = ⟨L |∙∙ 𝒪 ∙∙ | R⟩

where ∙∙ ∙∙ is the anti-normal product:
all B’s are on the left of all A’s!93. Operator form of Bethe-Yang equations
M+N equations for M+N rapidities

...

u1 u2

uM

wN
w2
w1

B-Y equations in the mirror channel

1 + e −iLp̃(uj)

M

∏
k=1

S̃(uk, uj) = 0,

j = 1,…, M

ujOperator form:
M

⟨∏
j=1

B(uj)

N

A(wk) (1 + A(ujγ )) = 0,∏⟩
k=1

j = 1,...,Mujγ

Similarly, in the physical channel
⟨

−γ
1
+
B(w
)
(
k )

M

∏
j=1

B(uj)

N

A(wk) = 0,
⟩
k=1

∏

k = 1,...,NWorkshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japanwkuj
u1 u2 ujuM

u1 u2 uj

uM

.
.

wk−γ
!10

4. Partition function at finite volume R
Finite volume partition function in the physical theory= thermal partition function in the mirror theory— can be computed by the techniques of the
Thermodynamical Bethe
Ansatz (justified in the
TBA and tree expansion
thermodynamical limit when the number of the⇥ RH ⇤
TBA and tree expansion
Z(L, R) = Tr ewrapping particles is large).

physical3(1)⇥ RH ⇤
mirror thatZ(L,where the trace is taken in the full Hilbert space. Assuming
R ⌧R)
L, =
ourTrgoale is to evaluate the leading
term of the free energy
where the trace is taken in the full Hilbert space. Assuming that R ⌧ L, our goal i
1termof theover
free energy
Instead, we will evaluate exactly thesumparticles
using TBA:
F(R) = lim without
log ZL (R).
(2)
L!1 L
1
F
(R)
=
lim
log ZL (R).
∞We restrict ourselvesto the simplest case of a theory with diagonal L!1
S-matrix−R H̃
L S(u, v) and one single type
|
|
⟨ψ
e
ψ⟩
Ẽ(urapidity
quasiparticle
excitation, but the=argument is general. Hereeu−R(
is thevariable,1)+...+Ẽ(uM )) which parametrises
𝒵(L, R) = Tr[e −R H̃of]=the momentum= p(u)
a quasiparticlethe∑ p ⟨ψ
∑ intoa convenient∑
Weofrestrict
ourselvesthesimplestway.
caseForof aa theory
theorywith
withdiagonal
diagonalscattering,
S-matrix S(u|ψ⟩ψ condition
M=0circumferencen1<nbut
quantisationa cylinder withR
the Betheequations
read,
in
of on
quasiparticle
excitation,
the argument
general.
Hereu is thewhich
rapidityvaria
2<...<nM is givenisby
the n-particle sector,the momentum p = p(u) of a quasiparticle in a convenient way. For a theory withquantisation condition on a cylinder with circumference R is given by the Bethe enj integer, j = 1, . . . , M .
(3)
j = 2⇡nj with
The sum goes over the Bethe quantum
numbers
the n-particle
sector, whichappear in the logarithmicof theBethe-Yangequations:
Here form
scattering
phase for the j-th
particle, or magnon,
j is the total= 2⇡n with n integer,
M

j

j

j

j = 1, . . . , M .

M

1 X
Here
is
the
total
scattering
phase
for thej(u1 , . . . , uM ) ⌘ pj L +log j-thS(ujparticle,, uk ). or magnon,(4)−Lp̃(uj) + i
log S̃(uk, uj) = 2πnj, j = 1,…,j Mi∑k(6=j)
k=1
M
1 X(u1 , . . . , numbers
uM ) ⌘ p(3)+ the identity
log S(uj L andj , uk ).in the
The states in finite volume are labeled by discrete jquantum
operatori
Our aim is to sum overthe solutionsof theBethe-Yangequationsj)
M -particle
sector of the Hilbert
space
can be decomposed
as a sum of products k(6of=normalised
states(with no approximation) without solving
them. X
The statesInin=finite volume
discrete quantum numbers (3) and the|n1are, . . labeled. , nM ihnby(5)
1 , . . . , nn |.
M -particle sectorof theMHilbert space can be decomposed as a sum of products of
n1 <...<nWorkshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan

!11

.

4. Fock-space representation of the partition function
Relax the constraint n1 < n2 < … < nM by introducing multi-wrapping particles
as explained yesterday by Dinh-Long
∑

M=0

M

∑

∏

n1<n2<…<nM j=1

e −RẼ(uj)m1(−1)rj−1 −rj RẼ(uj)→e∑ m! ∑∑ ∏rj
m=0
r ,...,r ≥1 n ,...,n j=1
∞

1

1

m

TBA and tree expansion
Z(L, R) (1)= Tr e

RH

3

n
Z(L,
R) = Tr e

RH⇤physicalTBA and tree expansion

⇥

m

⇥

⇤

3

physical𝒵(L, R) =∞⇥ RH ⇤⇥ RH ⇤
mirror thatZ(L,
where
the
trace
taken
Hilbertthe
space.
Assuming
R ⌧R)
L, =
ourTr
goale is to evaluate theR)
Trgoaline isthetofull(1)
mirrorspace. Assuming
thatZ(L,
R⌧
L,is=
our
evaluate
leadingterm of the free energy
where thatthe trace
taken
the isfull
space.
Assuming that R ⌧ L, ou
ken in the full Hilbert space. Assuming
R ⌧isL,ouringoal
toHilbert
evaluatethe leading
1
scatters (r-1)
term
of
the
free
energy
1
F
(R)
=
lim
log
Z
(R).
rgy = limLtimes with itself,R)
log ZL (R).
(2) L
L!1
S(u,u)=-1
L!1 L
1
1 to the simplest case of a theory withFdiagonal(R) = S-matrixlimlogZLv)(R).
We
restrictourselvesS(u,and one singr-wrapping
particle:
r−1
Fdiagonal(R) = S-matrixlimlogZLv)(R).
(2)
L!1 L
se of a theory with
S(u,
and
one
singletype(−1)
wraps lengthr Rparam
L!1of quasiparticle
excitation,
but the argument is general. Hereeu−rR
is Ẽ(u)the rapidity variable,whichL
ent is general. Hereu is the rapidity
variable,which parametrisesrthe momentum
p = p(u)a quasiparticle
into
a convenientway.
Weofrestrict
ourselvesthe simplest
caseForof aa theory
theorywith
withdiagonal
diagonalscatteri
S-mat
le intoa convenient
way.
with
scattering,
the thev)argumentlves
the simplest
caseForof aconditiona theory
theorydiagonalwith
S-matrixS(u,one issinglequantisationona diagonal
cylindercircumference
Rand
is given
bygeneral.
the type
Betheequationsof withquasiparticle
excitation,butHereu is thewhich
rapidir
circumferenceRgivenisbygeneral.
the Betheequations
read,tation,but the argument
u is thepwhich
rapidity
variable,which parametrises
theisn-particle
sector,theHeremomentum= p(u)
ofainquasiparticle
in a convenient way. For a theo
cyclic symmetryp(u) of a quasiparticle in a convenient
way. Fora theory
diagonal
scattering,
the R is given by theof orderrquantisation
condition
on witha cylinder
withcircumference
withnj integer,
1, . . .read,, M . in
j = 2⇡non on a cylinder with circumference
R is given
byj theBetheequationsj =which
the n-particle
sector,with n integer, j = 1, . . . , M .
(3), Workshop jNew Trends in Integrable
9-20total
September
2019, Osakaphase
City University,
Here Systems,
is thescattering
for Japan
the j-th particle, or magnon,
j

!12

.
.

Instead of solving the B-Y equations we can impose them by the operator constraint∏j(A(uj) + 1) = 0 - selects both the physical and the unphysical solutions

Express the discrete sum over Bethe and wrapping numbers as a contour integral
around the real axis

.

M

∑ ∑ ∏

e −rj RẼ(uj) =r1,...,rm n1,...,nm j=1

⟨∮
ℝ

⋯

m

γ
∂
log
1
+
A(u
j ))
du
(

∮ ∏ 2πi
ℝ

j

j=1

∂uj

(−1)rj−1 m
B(uj)rj∑∏r⟩jr ,...,rj=1
1m

perform the sum over all wrapping numbers:𝒵(L, R) = ⟨Ω⟩du∂Ω ≡ exp
log (1 + B(u))
log (1 + A(u γ))∂u[ ∮ℝ 2πi]Ω — operator creating the physicalvacuum (finite R) out of the barevacuum (asymptotically large R)

Workshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan

!13

5. Excitedstates in the physical channel
TBA and tree expansion
physical⇥ RH ⇤
Partition function forTBA
anand
excitedtree expansion
Z(L, R) = Tr e
state with rapidities w = {w1, . . . , wN}⇥ RH ⇤
mirror thatZ(L,where the trace is taken in the full Hilbert space. Assuming
R ⌧R)
L, =
ourTrgoale is to evaluate th

term of the free energy N
𝒵(L, R, w) = whereA(wΩ is taken in the full Hilbert space. Assuming that R ⌧ L, othek)trace
(1)
∏
⟨ k=1
⟩
1
term of the free energy
F (R) = lim
log ZL (R).
L!1 L
The rapidities of the excited state are determined by the exact Bethe equations
1(R) = S-matrixlimlogZLv)(R).
We restrict ourselvesN to the simplest case of a theory withFdiagonalS(u,and one si
L!1
L
−γ
1
+
B(w
A(w
Ω argument= 0, j =is1,2,...,Nof quasiparticle
excitation,but
general. Here u is the rapidity variable, which par(2)k ) the
(
j )∏
⟨
⟩
k=1
the momentum
p = p(u)ofaquasiparticle
into
a convenient
We restrict ourselves
the simplestway.
caseForof aa theory
theorywith
withdiagonal
diagonalscatteS-mquantisation conditiona cylinder withcircumference
R is givenisbygeneral.
the Betheequationsof onquasiparticle
excitation,
but the argument
Here
u is thewhich
rapid
the n-particle sector,the momentum p = p(u) of a quasiparticle in a convenient way. For a theRelation to Dorey-Tateo: quantisation condition on a cylinder with circumference R is given by th
j = 1, . . . , M .
j = 2⇡nj with nj integer,
the n-particle
sector,Write
a contour
integral:Here(1)jas
is thetotal scattering
phase for the j-th particle, or magnon,j = 2⇡nj with nj integer,j = 1, . . . , M

M
du
γX ∂
1
𝒵(L, R, w) = (−1)−, .totallog
A(u )forlog+
B(u)= ⟨Ωw⟩
Here jexp
isthescattering
j-th(j1L+phase) the(1jparticle,) orΩmagnon,
(u
.
.
,
u
)
⌘
p
+
log
S(u
,
u
).
∮
j
1
M
k
∂u
⟨
[ w−γ 2πi
] ⟩iNk(6=j)MX
1(u1 , . . . , numbers
uM ) ⌘∂ p(3)+ the identity
log S(uj L andj, u
The states in finite volume are labeled by discrete
opera
du jquantum
i+ A(u γ )
Ω
≡
exp
log
1
+
B(u)
log
1
j) )
(
)
(
w
M -particle sector of the Hilbert space∮can be decomposed as a sum of products k(6of=normalised
sta
2πi
∂u
[ ℝ+w−γ]X
The statesInin=finite volume
discrete quantum numbers (3)|n1are, . . labeled. , nM ihnby
1 , . . . , nn |.
M -particle sectorof theMHilbert space can be decomposed as a sum of pro
n1 <...<nWorkshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan

!14

5. Free-field representation
Wrapping operators as vertex operators:

B(u) = e −φ(u),A(u γ) = e −iφ̄(u)[φ(u), φ̄(v)] = i log S̃(u, v)[φ̄(u), φ̄(v)] = [φ(u), φ(v)] = 0⟨L | φ̄(u) = φ̄∘(u)⟨L | ,φ(u) | R⟩ = φ ∘ | R⟩

Expectation value:⟨⋯⟩ = ⟨L |∙∙ ⋯ ∙∙ | R⟩φ ∘(u) = R Ẽ(u),φ̄∘(u) = Lp̃(u)with anti-normal product ∙∙ ∙∙(all φ′s on the left of all φ′s)⟨φ̄(u) φ(v)⟩ = iS̃(u, v) + φ̄(u)∘ φ ∘(v)⟨e−iφ̄(u) −φ(v)e⟩ = S̃(v, u) e

𝒵(L, R) = ⟨Ω⟩−iφ̄∘(u)−φ ∘(v)du
Ω = exp
log⟨1 + e −φ(u)⟩ ∂u log (1 + e −iφ̄(u))[ ∮𝒞 2πi]Workshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan!15

5. Continuum spectrum approximation
Now expand the contour slightly away the real axis:∮ℝd log (1 + e
2πi−iφ̄(u)

)

∞

(⋯) →

d φ̄(u)∫ 2π+ exponentially small in L−∞dφ̄(u) = φ̄′du + . . . (?)
i log S̃(v, u)dv
φ̄(u) −∂vφ̄(v)
Ω =0
by the Ward identity
φ(v)
∫ℝ 2π
1+e
⟨(
) ⟩

dv ̂
dφ̄(u) → φ̄′(u) −K(u, v) φ̄′(v) du
[
]
∫ 2π

1
̂
K(u, v) = −i∂u log S(v, u)
1 + e φ(v)=> Jacobian = (functional) Gaudin determinant

Workshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan!16

Introduce by hand fermions which generate the non-diagonal terms in the
Gaudin determinant:𝒵(L, R) = ⟨Ω̌⟩∞du
ψ̄(u)ψ′(u)−φ(u)Ω̌ = explog (1 + eφ̄′(u) +)∫ 2π [
1 + e φ(u) ]−∞⟨φ̄(u) φ(v)⟩c = i log S̃(u, v),⟨ψ(u)ψ̄(v)⟩ = ilogS̃(u, v)

Workshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan!17

5. Path integral
Impose the correlators by introducing a pair of auxiliary fields
φ, φ̄ ρ, ρ̄
commutative

ρ, ρ̄ θ, θ̄ψ̄, ψ ϑ, ϑ̄grassmanian𝒵(L, R) = 𝒟[fields] e −𝒜[fields]
∫

du
ψ̄ ∂ψ
−φ
∘
∘
−𝒜[fields] =log(1 + e ) ∂φ̄ +
+
(
φ̄
−
φ̄
)ρ+(φ
−
φ
)ρ̄ + ϑ̄ψ + ψ̄ϑφ)∫ 2π (
1+e
du dv−ilog S̃(u, v)(ρ̄(u)ρ(v) − ϑ(u)ϑ̄(v))∫ 2π 2π.

the dependence on R and L through the classical fields:

φ ∘(u) = R Ẽ(u), φ̄∘(u) = Lp̃(u)The bosonic part of the path integral was obtained using diﬀerent
arguments by Jiang, Komatsu and Veskovi [ARXIV:1906.07733]

Workshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan

!18

5. Localisationdu𝒜=𝒜 +Q(u) ℬ
∫ 2π
∘

du ∘
𝒜 =−
φ̄ ρ
∫ 2π
∘

Q-exact localisation termδδδδQ = ψ̄+ φ̄ + ρ̄ + ϑ̄
δφ
δψ
δϑ
δρ

Q2 = 0

du
du dv
−φ
∘
ℬ=
− log(1 + e ) ∂ψ + ψρ + θ(φ − φ )) − iθ(u) log S̃(u, v) ρ(v)(∫ 2π
∫ 2π 2π
By standard localisation argument integral localises to the critical point:

𝒜 → 𝒜t = 𝒜 + tQℬ,
𝒵 → 𝒵t = e∫∂𝒵t−𝒜∘−tQℬ−𝒜∘−tQℬ= e(−Qℬ) = Q (eℬ) = 0∫∫∂t−𝒜t

Take the limit of
infinite perturbation: t → ∞ :

∘

𝒵=e

du
𝒜 ≡−L
p′(u)ρ(u) .
∫ 2π
∘

−𝒜∘

Workshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan

Qℬ=0
!19

The critical point:dv
φ(u) = R Ẽ(u) − i
log S̃(u, v) ρ(v)∫ 2π
ρ(u) = ∂u log(1 + e −φ(u))

equation for thecritical point identical to
the TBA integral equation:

The partition function:ϵ(u) = R Ẽ(u) − K̃(v, u) log(1 + e −ϵ(v))∫ϵ(u) = φ crit(u) - pseudo energydp̃(u)
𝒵(L, R) = exp L
log [1 + e −ϵ(u)]( ∫ 2π)As a consequence of localisation : the theory is one-loop exact and thegaussian fluctuations of the bosons and the fermions cancel => no quantum
corrections to the critical action at allWorkshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan

!20

6. Feynman graphs = exact cluster expansion
Vertices:

I.K., Didina Serban, D. L. Vu,
arXiv[hep-th]1805.02591,
1809.05705, 1906.01909

∞

n−1
(−1)
φ̄′ log(1 + e −φ) =φ̄′ φ n∑nn=1Propagators:⟨φ̄′(u)φ(v)⟩ = − K̃(u, v),⟨ψ̄′(u)ψ(v)⟩ = K̃(u, v)

1
K̃(u, v) = ∂u log S̃(u, v)iFeynman rules (c.f. Dinh-Long’s talk)

trees

n

n

(u , r)bosonic loops(u , r)nu1(u , r)

u2

u1

u2

fermionic loops

The bosonic loops and the fermionic loops cancel and the freeenergy is given by the sum over tree graphs.
Workshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan!21

The meaning of the tree diagrams in the cluster expansion:

TBA and tree expansion

⇥

nsion
Z(L, R) = Tr e

RH

3

3

⇥

⇤

⇤

RH
TBA and tree expansion
3(1) Z(L, R) = Tr e

(1)

⇥ RH ⇤⇥ RH ⇤
trace
taken in theHilbert space. Assuming thatR ⌧ R)L, ourgoaleis to evaluate the leading
Z(L,
= Tr
bert space. Assuming thatZ(L,R ⌧ whereL, ourgoal
the fullleadingR)= theTr
eis tois evaluate(1)term of the free energy
in the full
Hilbertspace. Assuming that R ⌧ L, our goal is to
is taken in the full Hilbert space. Assuming that Rwhere⌧ L,theourtracegoalisistaken
to evaluatetheleading1
1term of the free energy
F
(R)
=
lim
log ZL (R).
(2)=
energyF (R) = lim
log ZL (R).
(2)
L!1 L
L!1 L
1
1
F
(R)
=
lim
log
We
restrictourselvestothesimplest
caseofatheorywithdiagonal
S-matrixS(u,Zv)and one single type
L (R).
(R) = S-matrix
lim
log
(2)
st case of a theory with F
diagonal
S(u,Zv)and one single type
L (R).
L!1 L
L!1
L
of
quasiparticle
excitation, but the argument is general. Here u is the rapidity variable, which parametrises
gument is general.
Here
u
is
the
rapidity
variable,which parametrises
Wrapping particles
the momentum p = p(u)Weof restrict
a quasiparticle
in atoconvenientway.
Forofaatheory
withscattering,ourselvesthesimplest
case
theorywithdiagonal
diagonal
S-matrix theS(u, v)
Non-interacting
clustersofarticleinaconvenient
way.
For
a
theorywithdiagonal
scattering,theurselves to the simplest
case
of
a
theorywithdiagonal
S-matrix
S(u,
v)
and
one
singletype
weakly interacting
quantisationafter
conditiona cylinder withcircumferenceis given isbygeneral.
the Betheequations
read,
in
ofon
quasiparticle
excitation,
but the R
argumentHere
u is thewhich
rapidity
variablewithcircumference
R
is
given
by
the
Bethe
equations
which
read,
in
wrapping
excitation, but being
the argument
isthe
general.
u is the rapidity variable, whichparametrisesparticles: behave as freesector,put inan-particle
largeHerebox
the momentum p = p(u) of a quasiparticle in a convenient way. For a theory with dia
p = p(u) of a quasiparticle in a convenient way. For a theory with diagonalfermionsscattering, therenormalized
quantisation condition on a cylinder withwith
circumference
R is givenenergy
by the Bethe equa
=
2⇡n
with
n
integer,
j
=
1,
.
.
.
,
M
.
(3)
ndition on a cylinder with circumference R is given by thej Bethe equationsjj which read, inthe n-particle (3)sector,nj with nj integer, j = 1, . . . , M .

ector,
Here j is the total scattering phase for the j-th particle, or magnon,or the j-th particle, or magnon,
j = 1, . . . , M .
j = 2⇡nj with nj integer,j = 1, . . . , M .
(3)
j = 2⇡nj with nj integer,
M
X
1
M
Here
is
the
total
scattering
phase
for log
the S(uj-thj ,particle,
or magnon,
X
(u
,
.
.
.
,
u
)
⌘
p
L
+
uk ).
(4)
jj 1
M
j
1
i
otal
scattering
phase for log
the S(uj-thj ,particle,
or magnon,. , uM) ⌘ pj L +
uk ).
(4)
k(6=j)iMk(6=j)X
1M(u1 , . . . ,numbers
uM ) ⌘ p(3)+ the identity
log S(uThe states1 inX
finite volume are labeled by discrete jquantum
operatorj Landj , uk ).in thei(u1 , . . . ,numbers
uM ) ⌘Mp(3)
+ the
log
S(u
ed by discrete jquantum
identity
operator
the can be decomposed as a(4)j Landj , uk ).in
-particle
sectorof theHilbertspace
sum of products ofnormalised
statesk(6=j)
ie can be decomposed as a sum of products of normalised statesk(6=j)X
The statesInin=finite volume
the ide|n1are, . . .labeled, nM ihnby. . , nn |. quantum numbers (3) and (5)X
1 , . discrete
nite
volumenumbers(3)University,andtheidentity
operator in theWorkshopNew
TrendsIntegrable
9-20quantum
September 2019,
Osaka
City
M -particle
sectorof theof2 nor|n1are, . . .labeled, innMihnby. discrete. . , nn |.
(5)nJapan1 <...<n
M Hilbert space can be decomposed as a sum of products !2
1 , Systems,

II. OPEN B. C. IN THE MIRROR CHANNEL
Thermal partition function with open b.c.
time

⟨Ba |

= Tr[e −H̃ab(L)R]a

Parity conserving bulk scattering matrix

| Bb⟩

b

time

𝒵ab(R, L) = ⟨Ba | e −H(R)L | Bb⟩

a,b - integrable boundary
conditions

S(u, − v)S(−u, v) = 1

boundary reflection matrix [Ghoshal-Zamolodchikov]

R̃a(u)R̃a(−u) = 1==

R̃a(−u) = S̃(u, − u)R̃a(u)Boundary entropyR̃a(u) ≡ Ra(u γ )

=

=

=

[Aﬄeck-Ludvig’ 91]

R-finite, L-asymptotically largeℱab(R, L) ≡ log 𝒵ab(R, L) − log 𝒵(R, L) = log ga(R) + log gb(R) + O(e −mL)
“g-functions”
Workshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan!23

Asymptotically large cylinder= [circle of length R]x[interval of length L]
Excitations in the eﬀective QFT:
In the physical theory: parity
invariant states propagating
between the two boundaries

a

b

B(u)

time

u

−u

a

b

Algebra of the wrapping operators:

b

a
u

time

Aab(u)aIn the mirror theory: particles
wrapping the cylinderb

B(v)Aab(u) = S(v γ, u)S(−v γ, u) Aab(u)B(v)[Aab(u), Aab(u)] = 0, [B(u), B(v)] = 0
Fock space vacua:

Expectation value:⟨L | Aab(u) = ⟨L | e −2LE(u)Ra(u)Rb(u)B(u) | R⟩ = e −RẼ(u) | R⟩⟨𝒪⟩ ≡ ⟨L |∙∙ 𝒪 ∙∙ | R⟩

Workshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan!24

Boundary Bethe-Yang equations
b

b = a

a

= a

b

uj

uj

uj

Operator representation:
M

⟨∏
k=1

⟩

B(uk) (1 + Aab(ujγ ))= 0,

j = 1,...,M .

ujγ
a

uj

b

Fock-space representation of the cylinder partition function

𝒵ab(L, R) = ⟨Ωab⟩duΩab ≡ explog (1 + B(u)) ∂u log [1 + Aab(u γ)]∮ℝ 2πi+Workshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan!25

Free-field representation:

Aab(u γ) = e −iφ̄ab(u),B(u) = e −φ(u)[φ(v), φ̄ab(u)] = − i log[S̃(v, u)S̃(−v, u)]⟨L | φ̄ab(u) = φ̄∘ab(u) ⟨L | ,φ(u) | R⟩ = φ ∘ | R⟩φ̄∘ab(u) = 2Lp̃(u) + i log Ra(u)Rb(u) − 2π sign(u)φ ∘(u) = R Ẽ(u)Continuum spectrum approximation: add fermionic partners for the measure[ψab(v)ψ̄(u)] = − i log[S̃(v, u)S̃(−v, u)]𝒵ab(R, L) = ⟨Ω̌ab⟩Ω̌ab = exp∞∫0

du φ̄ab∂uφ − ψ̄ab∂uψ
2π
1 + eφWorkshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan!26

Path integral for open boundary fonditions:

𝒵ab(L, R) = 𝒟[fields] e −𝒜[fields]∫
du φ̄ab ∂φ − ψ̄ab ∂ψ
∘
∘
−𝒜 =
+
(
φ̄
−
φ̄
)ρ+(φ
−
φ
)ρ̄ + θ̄ψ + ψ̄abθ
ab
ab
φ
)
∫ 2π (
1+e
du dv+ilog[S̃(u, v)S̃(u, − v)](− ρ̄(u)ρ(v) + θ(u)θ̄(v))∫ 2π 2πφ̄∘ab(u) = 2Lp̃(u) + i log R̃a(u) + i log R̃b(u) − 2π sign(u), φ ∘(u) = R Ẽ(u)The theory is
one-loop exact:∞
Det(1 − K̂−)
du
𝒵ab =
exp
∂uφ̄∘ab log[1 + e −φ]( ∫0 2π)Det(1 − K̂+)11±
̂± log S̃(u, − v))
K (u, v) =
∂
log
S̃(u,
v)
(
u
i 1 + e ϵ(u)Dorey, Fioravanti, Rim, Tateo’04; Pozsgay’2010
Workshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan!27

III. EXAMPLE: SINH-GORDON21
2μ
𝒜 = d2x(∂μϕ)2 +cosh(bϕ)2∫
sin πb[ 4π]1
ν = 1 + 2,b

One particle, no bound states;
relativistic theory: mirror=physicalp(u) = m sinh πu,S(u, v) =1
K0(u, v) = ∂u log S0(u, v)i(0 < b ≤ 1)

tanh ( 2 − 2ν )πu

E(u) = m cosh πulog S(u) = (𝔻a + 𝔻−a)log S0(u)π(u − i/2)
S0(u) = tanh
2

2
a =1−
ν

iπ

tanh ( 2 + 2ν )
πu

iπ

i ∂
Shift operator: 𝔻 = exp( 2 ∂u )- “universal kernel”(𝔻 + 𝔻−1)K0(u) = 2πδ(u)KWorkshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan→ 𝕂 =

𝔻 a + 𝔻− a
𝔻 + 𝔻 −1

K (u ) = K 0 (u + i a / 2) + K 0 (u − i a / 2) = (𝔻 a + 𝔻− a ) K 0

K0(u , v ) =πcosh π ( u − v )K (u , v ) =1∂ log S ( u , v )i u

𝔻 ≡ e i∂ u / 2!28

(all expression below need to be rigorously defined)
du φ̄ φ′ − ψ̄ ψ′∘∘𝒜[fields] =
+
(
φ̄
−
φ̄
)ρ+(φ
−
φ
)ρ̄ + θ̄ψ + ψ̄θ)∫ 2π ( 1 + e φ
du dv
a
−a
+i
ρ̄(u)ρ(v)
+
θ(u)
θ̄(v)
(𝔻
+
𝔻
)log S0(u, v))∫ 2π 2π (The action can be cast into a quasi-local form by a field redefinitionφ̄ → Lp(u) + (𝔻 + 𝔻−1)φ̄ψ̄ → (𝔻 + 𝔻−1)ψ̄

a
−a
du
(𝔻
+
𝔻
)∂φ̄
−1
𝒜=
φ(𝔻 + 𝔻 )∂φ̄ −∫ 2π [
1 + e −φ ]
a
−a
du
ψ̄(𝔻
+
𝔻
)∂ψ
−1 ¯
+
ψ(𝔻 + 𝔻 )∂ψ −]∫ 2π [
1 + eφCritical point:

−1
a
−a
−φ
𝔻
+
𝔻
φ
=
−
(𝔻
+
𝔻
)log(1+e)[]“Discrete Liouville equation”[Zamolodchikov,
Lukyanov]φ → φ + R E (u )Workshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan!29

Summary
Path integral formulation of the Thermodynamic Bethe Ansatz
The theory is one-loop exact. Explains why there are only
exponential corrections to the free energyWorks also for scattering matrices not of diﬀerence type, as in AdS/CFTCan be generalised to the case of non-diagonal scattering (nested Bethe
Ansatz) and bound states
Hopefully can be adapted to other geometries with application to
AdS/CFTWorkshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan!30

Thank you!

Workshop New Trends in Integrable Systems, 9-20 September 2019, Osaka City University, Japan!31