TBA and tree expansion Ivan Kostov Institut de Physique Théorique, Saclay w/D. Serban and D.L. Vu, in progress Workshop on higher-point correlation functions and integrable AdS/CFT, 2018, HMI Dublin 1
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 Workshop on higher-point correlation functions and integrable AdS/CFT, 2018, HMI Dublin 2
Computation of the thermal partition function in a theory with diagonal scattering Dispersion relation: momentum p = p ( u ) of u = rapidity condition on gy E = E ( u ) quantisation condition on Factorised S-matrix: S-matrix S ( u, v ) and v u variable, which ! u ) = i ˜ transformation γ : u ! ˜ u of E (˜ u ) = i ˜ p ( u ) , p (˜ E ( u ) . mirror transformation L, R ) = T transformation x = � i ˜ mirror t, t = � i ˜ x [Al. Zamolodchikov] phys Z ( L , L, R ) = T ( L, Thermal partition function phys [ e � LH phys ] = Tr mir [ e � RH mir ] . Z ( L, R ) = Tr in the mirror theory 3 Workshop on higher-point correlation functions and integrable AdS/CFT, 2018, HMI Dublin
— 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 φ j ˜ φ j = 2 π n j with n j integer , j = 1 , . . . , M M p ( u j ) L + 1 ˜ X log ˜ X φ j ( u 1 , . . . , u M ) ⌘ ˜ S ( u j , u k ) . i k ( 6 = j ) | i The Hilbert space is spanned by the on-shell states | n 1 , . . . , n M i ( n 1 < ... < n M ) with energies E ( n 1 , . . . , n M ) = E ( u 1 ) + · · · + E ( u M ) Resolution of the identity: X I n = | n 1 , . . . , n M ih n 1 , . . . , n n | . n 1 <...<n M 1 e � R ˜ X X E ( n 1 ,...,n M ) . Z ( L, R ) = M =0 n 1 <n 2 < ··· <n M The free energy can be computed by taking into account the excluded volume via cumulant expansion Workshop on higher-point correlation functions and integrable AdS/CFT, 2018, HMI Dublin 4
1. Excluded volume => multi-wrapping particles Introduce a factor imposing that all mode numbers are different: 1 M 1 e � R ˜ X X Y E ( n 1 ,...,n M ) . � � Z ( L, R ) = 1 � δ n j ,n k M ! M =0 n 1 ,...,n M j<k and expand E ( n ) + 1 E ( n 1 ,n 2 ) � 1 E ( n,n ) + . . . e � R ˜ e � R ˜ e � R ˜ X X X Z ( L, R ) = 1 + 2! 2 n n 1 ,n 2 n X The expansion is a sum of physical and unphysical solutions of the Bethe-Yang eqs for which some mode numbers can coincide: | n r 1 1 , . . . n r M = solution of the BY equations for M magnons m i rapidities, r 1 + · · · + r m = M . with m distinct mode numbers with r i = 1 , 2 , . . . . This state is a linear combination of plane waves with momenta . momenta r j p ( u j ) , The energy of such a state is ˜ m ) = r 1 ˜ E ( u 1 ) + · · · + r m ˜ E ( n r 1 1 , . . . ; n r m E ( u m ) . with rapidities determined by the Bethe-Yang equations m p ( u j ) L + 1 ˜ X r k log ˜ φ j ⌘ ˜ S ( u j , u k ) + π ( r j � 1) = 2 π n j ( j = 1 , . . . , m ) i k ( 6 = j ) 5 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 1 ( − 1) m ( − 1) r 1 + ··· + r m C r 1 ...r m e � R ˜ E ( n r 1 1 , ... , n rm X X X m ) , Z ( L, R ) = m ! m =0 n 1 ,...,n m r 1 ,...,r m The numbers are purely combinatorial and can be computed C r 1 ,...,r m e by comparing with free fermions ,...,r 1 ( − 1) r � 1 ⇣ 1 + e � RE ( n ) ⌘ Y X X e � rRE ( n ) Z free fermions ( L, R ) = = exp r r =1 n 2 Z n 2 Z (12) 1 ( − 1) r 1 + ··· + r m ( − 1) m X X X e � r 1 RE ( n 1 ) � ··· � r m RE ( n m ) . = 1 + m ! r 1 . . . r m m =1 n 1 ,...,n m r 1 ,...,r m C r 1 ,...,r m e ,...,r 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 Workshop on higher-point correlation functions and integrable AdS/CFT, 2018, HMI Dublin
2. From sum over mode numbers to integral over rapidities Z d ˜ 2 π . . . d ˜ φ 1 φ m X = 2 π . n 1 ,...,n m Z du 1 1 ( − 1) r 1 + ··· + r m ( − 1) m 2 π . . . du m X X Z ( L, R ) = m ! r 1 . . . r m 2 π m =0 r 1 ,...,r m m ) e � r 1 E ( u 1 ) . . . e � r m E ( u m ) . × ˜ G ( u r 1 1 , . . . , u r m ˜ ˜ G = det G kj , The Jacobian = generalized Gaudin determinant m ⇥ m depends both on the rapidities and on the wrapping numbers m ! ∂ ˜ ˜ X p 0 ( u j ) + G kj = kj = L ˜ r l K ( u j , u l ) δ jk − r k K ( u k , u j ) , φ j ∂ u k l =1 i ∂ u log ˜ where K ( u, v ) = 1 S ( u, v ) = - scattering kernel Workshop on higher-point correlation functions and integrable AdS/CFT, 2018, HMI Dublin 7
2. From sum over mode numbers to integral over rapidities How to compute the Gaudin measure for a state with arbitrary many particles? p 0 p 0 ( u j ) ˜ j ≡ ˜ K jk ≡ K ( u j , u k ) . p 0 , ˜ m =1: G ( u r ) = L ˜ m =2: ˜ G ( u r 1 1 , u r 2 2 ) = L 2 ˜ p 0 p 0 p 0 p 0 1 ˜ 2 + L ˜ 1 r 1 K 21 + L ˜ 2 r 2 K 12 , m =3: ˜ G ( u r 1 2 , u r 3 3 ) = L 3 ˜ p 0 p 0 p 0 1 ; u 2 1 ˜ 2 ˜ 3 + L 2 ˜ 3 r 2 K 12 + L 2 ˜ 3 r 3 K 13 + L 2 ˜ p 0 p 0 p 0 p 0 p 0 p 0 2 ˜ 2 ˜ 1 ˜ 3 r 1 K 21 + L 2 ˜ 3 r 3 K 23 + L 2 ˜ 2 r 1 K 31 + L 2 ˜ p 0 p 0 p 0 p 0 p 0 p 0 1 ˜ 1 ˜ 1 ˜ 2 r 2 K 32 3 Lr 2 p 0 p 0 p 0 + ˜ 3 Lr 1 r 3 K 13 K 21 + ˜ 3 Lr 2 r 3 K 12 K 23 + ˜ 3 K 13 K 23 1 Lr 2 p 0 p 0 p 0 + ˜ 2 Lr 1 r 2 K 12 K 31 + ˜ 1 K 21 K 31 + ˜ 1 Lr 3 r 1 K 23 K 31 2 Lr 2 p 0 p 0 p 0 + ˜ 1 Lr 2 r 1 K 21 K 32 + ˜ 2 K 12 K 32 + ˜ 2 Lr 2 r 3 K 13 K 32 , No cycles or type K 12 K 21 or K 12 K 23 K 31 . the Gaudin determinant determinant for general Workshop on higher-point correlation functions and integrable AdS/CFT, 2018, HMI Dublin 8
Gaudin determinant as a sum over spanning trees G kj ≡ ˜ ˆ Introduce a slightly modified matrix with interesting properties: G kj r j G kj = ˆ ˆ D k δ kj − ˆ K kj m with ˆ p 0 ( u j ) and ˆ X D j = Lr j ˜ K k,j = r k r j K kj − δ kj r j r l K jl . l =1 diagonal Laplacian matrix matrix = zero row sums 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 ⇣ ⌘ D j δ jk − ˆ ˆ ˆ ˆ X Y Y det = K jk D j K kj . m ⇥ m v j 2 roots F ` jk 2 F spanning forest Gaudin measure in terms of the modified matrix: G = det ˆ G jk ˜ , Q m 9 j =1 r j Workshop on higher-point correlation functions and integrable AdS/CFT, 2018, HMI Dublin
Gaudin determinant as a sum over spanning trees m = 1: 1 + + m = 2: 1 2 1 2 1 2 3 3 3 3 3 + + + …+ + …+ m = 3: 1 2 1 2 1 2 1 2 1 2 the general term: Workshop on higher-point correlation functions and integrable AdS/CFT, 2018, HMI Dublin 10 = 2:
Tree expansion of the thermal partition function Reverse the order of summation in the partition sum => => Gas of non-interacting tree “Feynman graphs” embedded in ) R ⇥ N Rapidity u and wrapping number r assigned to each vertex of the tree Feynman rules: vertices = ( � 1) r � 1 … … e � rR ˜ E ( u ) , ( u,r ) ( u,r ) r 2 propagators ( u , r ) ( u , r ) = r 1 r 2 K ( u 2 , u 1 ) 1 1 2 2 a vertex can have at most one incoming and any number of outgoing lines Z du 1 r ˜ X The partition function exponentiates: p 0 ( u ) log Z ( L, R ) = L 2 ⇡ ˜ Y r ( u ) , r =1 = sum over all connected tree diagrams with root at the point ( u,r ) where ˜ Y r ( u ) Eq. (30) gi Workshop on higher-point correlation functions and integrable AdS/CFT, 2018, HMI Dublin 11
Summing up the connected trees ( u,r ) = = The sum of connected trees rooted at ( u,r ) function Y r ( u ) ( u , r ) satisfies a non-linear equation ertices and = + 1/3! + + 1/2! + … ( u,r ) ( u , r ) ( u , r ) ( u , r ) ( u , r ) Y r ( u ) = ( − 1) r � 1 ˜ [ ˜ Solution: Y 1 ( u )] r , r = 1 , 2 , 3 , . . . . r 2 Y 1 ( u ) = e � R ˜ 2 π rK ( v,u ) ˜ E ( u ) e ˜ R dv P Y r ( v ) . r Y 1 ( u ) = e � R ˜ 2 π K ( v,u ) log [ 1+ ˜ Y 1 ( v ) ] h i dv R ˜ X r ˜ 1 + ˜ E ( u )+ Y r ( v ) = log Y 1 ( v ) => . r Workshop on higher-point correlation functions and integrable AdS/CFT, 2018, HMI Dublin 12
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