We give a recipe for for exact evaluation of the sum over a complete - - PowerPoint PPT Presentation

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Workshop on higher-point correlation functions and integrable AdS/CFT, 2018, HMI Dublin

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Ivan Kostov Institut de Physique Théorique, Saclay

w/D. Serban and D.L. Vu, in progress

TBA and tree expansion

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Workshop on higher-point correlation functions and integrable AdS/CFT, 2018, HMI Dublin

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We give a recipe for for exact evaluation of the sum over a complete set of states in an integrable 1+1 dimensional field theory, based on a graph expansion of the Gaudin measure. Motivation: to regularise the sum over wrapping particles in the hexagon bootstrap proposal. The method allows to sum up the cluster expansion by transforming it into a set of diagrammatic rules. A statistical alternative of the TBA. The method is tested against the TBA for

• 1. Partition function on a cylinder
• 2. Exact energy of a physical state in finite volume
• 3. LeClair-Mussardo formula for the one-point correlators

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Workshop on higher-point correlation functions and integrable AdS/CFT, 2018, HMI Dublin

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Computation of the thermal partition function in a theory with diagonal scattering

Thermal partition function in the mirror theory

momentum p = p(u) of condition on

gy E = E(u) quantisation condition on

Dispersion relation: u = rapidity

S-matrix S(u, v) and variable, which

Factorised S-matrix: u v

Z(L, R) = Tr

phys[eLHphys] = Tr mir [eRHmir].

phys

mirror

L, R) = T Z ( L ,

L, R) = T (L,

[Al. Zamolodchikov]

transformation x = i˜ t, t = i˜ x

mirror transformation

transformation γ : u ! ˜ u of

! E(˜ u) = i˜ p(u), p(˜ u) = i ˜ E(u).

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Workshop on higher-point correlation functions and integrable AdS/CFT, 2018, HMI Dublin

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In = X

n1<...<nM

|n1, . . . , nMihn1, . . . , nn|.

Resolution of the identity: The Hilbert space is spanned by the on-shell states with energies X |n1, . . . , nMi (n1 < ... < nM)

| i E(n1, . . . , nM) = E(u1) + · · · + E(uM)

The free energy can be computed by taking into account the excluded volume via cumulant expansion

˜ φj = 2πnj with nj integer, j = 1, . . . , M

˜ φj(u1, . . . , uM) ⌘ ˜ p(uj)L + 1 i

M

X

k(6=j)

log ˜ S(uj, uk).

— For large L, the quantisation of the momenta of a M-particle state in the mirror theory is determined by the Bethe-Yang equations for the scattering phases φj ~

Z(L, R) =

1

X

M=0

X

n1<n2<···<nM

eR ˜

E(n1,...,nM).

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Workshop on higher-point correlation functions and integrable AdS/CFT, 2018, HMI Dublin

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• 1. Excluded volume => multi-wrapping particles

Introduce a factor imposing that all mode numbers are different: and expand The expansion is a sum of physical and unphysical solutions of the Bethe-Yang eqs for which some mode numbers can coincide: with ri = 1, 2, . . . . = solution of the BY equations for M magnons with m distinct mode numbers rapidities, r1 + · · · + rm = M.

X |nr1

1 , . . . nrM m i

This state is a linear combination of plane waves with momenta . The energy of such a state is

momenta rjp(uj),

with rapidities determined by the Bethe-Yang equations

Z(L, R) =

1

X

M=0

1 M! X

n1,...,nM M

Y

j<k

• 1 δnj,nk
• eR ˜

E(n1,...,nM).

Z(L, R) = 1 + X

n

eR ˜

E(n) + 1

2! X

n1,n2

eR ˜

E(n1,n2) 1

2 X

n

eR ˜

E(n,n) + . . .

˜ E(nr1

1 , . . . ; nrm m ) = r1 ˜

E(u1) + · · · + rm ˜ E(um).

˜ φj ⌘ ˜ p(uj)L + 1 i

m

X

k(6=j)

rk log ˜ S(uj, uk) + π(rj 1) = 2πnj (j = 1, . . . , m)

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Workshop on higher-point correlation functions and integrable AdS/CFT, 2018, HMI Dublin

Fixing the combinatorics of non-restricted mode numbers

We have to fix the coefficients in the expansion The numbers are purely combinatorial and can be computed by comparing with free fermions

,...,r

Cr1,...,rme

Zfree fermions(L, R) = Y

n2Z

⇣ 1 + eRE(n)⌘ = exp X

n2Z 1

X

r=1

(−1)r1 r erRE(n) = 1 +

1

X

m=1

(−1)m m! X

n1,...,nm

X

r1,...,rm

(−1)r1+···+rm r1 . . . rm er1RE(n1)···rmRE(nm). (12)

,...,r

Cr1,...,rme By analogy with the free fermions, one expects that the solutions with multiplicities describe particles wrapping several times the time circle. multiplicity r = wrapping number

Z(L, R) =

1

X

m=0

(−1)m m! X

n1,...,nm

X

r1,...,rm

(−1)r1+···+rm Cr1...rm eR ˜

E(nr1

1 , ... , nrm m ),

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• 2. From sum over mode numbers to integral over rapidities

The Jacobian = generalized Gaudin determinant depends both on the rapidities and on the wrapping numbers

• scattering kernel

X

n1,...,nm

= Z d˜ φ1 2π . . . d˜ φm 2π .

Z(L, R) =

1

X

m=0

(−1)m m! X

r1,...,rm

(−1)r1+···+rm r1 . . . rm Z du1 2π . . . dum 2π × ˜ G(ur1

1 , . . . , urm m ) er1E(u1) . . . ermE(um).

˜ G = det

m⇥m

˜ Gkj,

˜ Gkj = ∂ ∂uk ˜ φj

kj =

L˜ p0(uj) +

m

X

l=1

rlK(uj, ul) ! δjk − rkK(uk, uj),

where K(u, v) = 1

i ∂u log ˜

S(u, v) =

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• 2. From sum over mode numbers to integral over rapidities

˜ p0

j ≡ ˜

p0(uj) Kjk ≡ K(uj, uk).

˜ G(ur) = L˜ p0 , ˜ G(ur1

1 , ur2 2 ) = L2˜

p0

1˜

p0

2 + L˜

p0

1r1K21 + L˜

p0

2r2K12,

˜ G(ur1

1 ; u2 2, ur3 3 ) = L3˜

p0

1˜

p0

2˜

p0

3

+ L2˜ p0

2˜

p0

3r2K12 + L2˜

p0

2˜

p0

3r3K13 + L2˜

p0

1˜

p0

3r1K21

+ L2˜ p0

1˜

p0

3r3K23 + L2˜

p0

1˜

p0

2r1K31 + L2˜

p0

1˜

p0

2r2K32

+ ˜ p0

3Lr1r3K13K21 + ˜

p0

3Lr2r3K12K23 + ˜

p0

3Lr2 3K13K23

+ ˜ p0

2Lr1r2K12K31 + ˜

p0

1Lr2 1K21K31 + ˜

p0

1Lr3r1K23K31

+ ˜ p0

1Lr2r1K21K32 + ˜

p0

2Lr2 2K12K32 + ˜

p0

2Lr2r3K13K32,

No cycles type K12K21 the Gaudin determinant

• r K12K23K31.

determinant for general

• r

How to compute the Gaudin measure for a state with arbitrary many particles? m=1: m=2: m=3:

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Gaudin determinant as a sum over spanning trees

ˆ Gkj = ˆ Dk δkj − ˆ Kkj with ˆ Dj = Lrj ˜ p0(uj) and ˆ Kk,j = rkrjKkj − δkj

m

X

l=1

rjrlKjl. diagonal matrix Laplacian matrix = zero row sums

det

m⇥m

⇣ ˆ Djδjk − ˆ Kjk ⌘ = X

F

Y

vj2roots

ˆ Dj Y

`jk2F

ˆ Kkj.

Kirchhoff’s Matrix-Tree theorem G is equal to the sum of all “forests” of trees spanning the fully connected graph with vertices at the points 1,2,…, m

spanning forest

Introduce a slightly modified matrix with interesting properties: ˆ Gkj ≡ ˜ Gkjrj

˜ G = det ˆ Gjk Qm

j=1 rj

,

Gaudin measure in terms of the modified matrix:

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Gaudin determinant as a sum over spanning trees

1 2 3 1 2 3 1 2 3 1 2 3 1 2 1 2 3

+ + + …+ + …+

1 2

+

1 2

+

1

m = 1: m = 2: m = 3:

= 2:

the general term:

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Workshop on higher-point correlation functions and integrable AdS/CFT, 2018, HMI Dublin

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Tree expansion of the thermal partition function

= sum over all connected tree diagrams with root at the point (u,r ) Reverse the order of summation in the partition sum => => Gas of non-interacting tree “Feynman graphs” embedded in

) R ⇥ N

Feynman rules: vertices propagators a vertex can have at most one incoming and any number of outgoing lines The partition function exponentiates: Rapidity u and wrapping number r assigned to each vertex of the tree

(u,r) … (u,r) …

1

(u , r )

1

(u , r )

2 2

= (1)r1 r2 erR ˜

E(u),

= r1r2K(u2, u1)

log Z(L, R) = L Z du 2⇡ ˜ p0(u)

1

X

r=1

r ˜ Yr(u),

where ˜ Yr(u)

• Eq. (30) gi

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Summing up the connected trees

satisfies a non-linear equation The sum of connected trees rooted at (u,r)

( u , r ) (u,r) =

Solution:

˜ Yr(u) = (−1)r1 r2 [ ˜ Y1(u)]r, r = 1, 2, 3, . . . .

˜ Y1(u) = eR ˜

E(u) e P

r

R

dv 2π rK(v,u) ˜

Yr(v).

X

r

r ˜ Yr(v) = log h 1 + ˜ Y1(v) i .

=>

˜ Y1(u) = eR ˜

E(u)+ R

dv 2π K(v,u) log[1+ ˜

Y1(v)]

function Yr(u) ertices and

=

(u,r)

=

( u , r ) ( u , r ) ( u , r ) ( u , r )

+ + 1/2! + 1/3! + …

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Workshop on higher-point correlation functions and integrable AdS/CFT, 2018, HMI Dublin

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Energy of an an excited state in the physical channel

Hamilton evolution of a Bethe rapidities u = {u1, . . . , uM}. circle is sufficiently lar

Bethe state in the physical channel

˜ j(v1, . . . , vM) ⌘ ˜ p(vj)L + 1 i

N

X

k=1

log S(˜ vj, uk) + 1 i

M

X

l(6=j)

log S(˜ vj, ˜ vl), j = 1, . . . , M.

• processes should be indistinguishable.

namely ˜ φj(v1, . . . , vN) = 2π˜ nj, where Bethe-Yang equations in the mirror channel: This leads to a minor redefinition of the Feynman rules

Y1(v) = Y

1 (v) e R du

2π log(1+Y1(u))K(u,v).

vertices propagators

(u,r) … (u,r) …

1

(u , r )

1

(u , r )

2 2

= r1r2K(u2, u1)

˜ Y

1 (v) ⌘ e ˜ E(v) M

Y

k=1

S(˜ v, uk)

X (1)r1 r2 [Y

1 (u)]r

Y1(uj) = 1, j = 1, . . . , N.

TBA equation Exact Y-B equations

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One-point functions of local operators at finite temperature

• 1. Diagonal form factors in infinite volume:

The connected diagonal form factors are defined as the limit

F O

n (u1, . . . , un) = h0|O|u1, . . . , uni∞.

F c

2n(¯

un + iεn, . . . , ¯ u1 + iε1, u1, . . . , un) = F c

2n(u1, . . . , un) + ε-dependent terms.

hurM

M , . . . , ur1 1 |O|ur1 1 , . . . , urM M iL =

P

α∪¯ α={u1,...,uM}

Q

j∈α rj F c 2|α|(α) ⇥ detj,k∈¯ α Gjk

detj,k∈{u1,...,uM} Gjk .

huM, . . . , u1|O|u1, . . . , uMiL = P

α∪¯ α={u1,...,uM} F c 2|α|(α) ⇥ detj,k∈¯ α Gjk

detj,k∈{u1,...,uM} Gjk + o(e−L),

minors of the Gaudin determinant

We need to generalise this formula to the multi-wrapping states: The only effect of the multi-wrapping is a multiplication by the wrapping number r.

Balog 1994, Saleur 1999, Pozsgay-Takacs 2007 Bajnok, Balog, Wu, 2017

14

The finite volume matrix elements are expressed in terms of the connected form factors and the minors of the Gaudin determinant

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One-point functions of local operators at finite temperature

1|O|

hOiL,R =

• 2. Finite temperature matrix elements

We want to compute the thermal trace with the operator O inserted using the tree expansion.

L

R

R = ∞

X

M=0

X

n1<n2<···<nM

e−RE(n1,...,nM)hn1, . . . , nM|O|nM, . . . , n1i insert a complete set of states

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One-point functions of local operators at finite temperature

• 3. Finite temperature matrix elements in terms of the connected diagonalform factors

hOiR = 1 Z(R, L)

∞

X

m=0

(1)m m! X

r1,...,rm

Z du1 2π . . . dum 2π e−Lr1E(u1)−LrmE(um) r1 . . . rm ⇥ X

↵∪¯ ↵={u1,...,um}

Y

j∈↵

rj F c

2|↵|(α) detj,k∈¯ ↵ ˆ

Gjk Q

i∈¯ ↵ ri

,

jk

K(u, v) = 1

i ∂u log S(u, v).

1 r2

1 . . . r2 m

Y

j∈↵

r2

j.

the wrapping numbers give a total factor

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• 4. Minors of the Gaudin determinant in terms of trees

One-point functions of local operators at finite temperature

det

j,k∈¯ ↵

ˆ Gjk = X

F∈Fα, ¯

α

Y

roots in ¯ ↵

ˆ Dj Y

`jk∈F

ˆ Kkj.

1 2

+…+

3 4 1 2 3 4

+

1 2 3 4 1 2 3 4

+…+ +…

The minor obtained by deleting the rows and the columns in the subset is equal to the sum of spanning forests such that all points of this subset are roots. .

X

Fα

ˆ Dj

X

Fα u,r)

α, ¯ α

The roots in have weight 1 and the roots in have weight

the complementary subset

(α = {1, 2}, ¯ α = {3, 4}.)

E.g. for

with ˆ Dj = Lrj ˜ p0(uj)

and ˆ Kk,j = rkrjK(uk, uj)

(u,r)

=

Σ

u

r

2

F

c 2 n

hOiR =

The final formula:

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R = ∞

X

n=1

1 n! Z

n

Y

j=1

duj 2π f(uj) F c

2n(u1, . . . , un),

f(u) = Y1(u) 1 + Y1(u).

F c

2 n

hOiR =

( u , r )

=

Σ

r

2

u

X

r

r2Yr(u) = X

r

(1)r−1[Y1(u)]r = Y1(u) 1 + Y1(v) = f(u).

= The sum over tree Feynman graphs is identical to the LeClair-Mussardo series:

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The case of operator type conserved charge

Saleur 1999

O = L1 Z dxO(x), hun, . . . , u1|O|u1, . . . , uni hun, . . . , u1|u1, . . . , uni = 1 L

n

X

j=1

O(uj).

hOiR =

1

X

n=1

Z

n

Y

j=1

duj 2⇡ p0(u1)f(u1)K(u2, u1)f(u2)K(u3, u2) . . . K(un, un1)f(un)O(un)

+ + … = +

The tree expansion is over trees with a marked vertex 1 2 1=2 1 2 2 1

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We proposed a statistical alternative to the TBA based on an exact treatment of the sum

• ver a complete set of Bethe eigenstates using a graph expansion of the Gaudin measure.

The free energy and the observables are expressed in terms of tree Feynman graphs. The trees stand for clusters of vacuum loops winding around the compactified space dimension. The method seems to generalisable to theories with non-diagonal scattering. Next tasks: — Try to resolve the divergencies in the hexagon proposal for the correlators

• f trace operators in AdS/CFT.

— Try to generalise the method to a theory with open boundaries with integrable boundary conditions. In such theories the graph expansion of the Gaudin measure contains

• cycles. Summing up these cycles is expected to give the exact g-function, to be compared

with Pozsgay 2010. — Non-perturbative integrability for the one-point functions in AdS/dCFT?

CONCLUSION

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thank you!