Octagon with finite bridge: free fermions and determinant identities
PPrepared for submission to JHEP
Octagon with finite bridge: free fermions anddeterminant identities
Ivan Kostov a and Valentina B. Petkova b a Universit´e Paris-Saclay, CNRS, CEA, Institut de physique th´eorique, 91191, Gif-sur-Yvette,France b Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bul-garia
E-mail: [email protected] , [email protected] Abstract:
We continue the study of the octagon form factor which helps to evaluatea class of four-point correlation functions in N = 4 SYM theory. The octagon is charac-terised, besides the kinematical parameters, by a “bridge” of (cid:96) propagators connecting twonon-adjacent operators. In this paper we construct an operator representation of the oc-tagon with finite bridge as an expectation value in the Fock space of free complex fermions.The bridge (cid:96) appears as the level of filling of the Dirac sea. We obtain determinant iden-tities relating octagons with different bridges, which we derive from the expression of theoctagon in terms of discrete fermionic oscillators. The derivation is based on the existenceof a previously conjectured similarity transformation, which we find here explicitly. Corresponding author. a r X i v : . [ h e p - t h ] F e b ontents (cid:96) = 0 14 (cid:96) = 0 A new non-perturbative approach for the computation of the correlation functions of single-trace operators in the N = 4 supersymmetric Yang-Mills theory has been developed inthe past five years [1–6]. The method, mostly referred to as hexagonalisation, is basedon the world-sheet integrability of N = 4 SYM [7]. The hexagonalisation prescribes todecompose the correlation function into elementary blocks called hexagon form factors,or shortly hexagons, which are almost uniquely determined by the huge symmetry of thetheory. Being formulated in terms if infinite-volume form factors, the prescription involves– 1 –ivergencies and, in spite of some important progress [8], it still awaits an appropriateregularisation procedure.Remarkably, a class of four-point functions of half-BPS operators with large R-chargesand specially tuned polarisations, discovered in [9, 10], are free of divergencies and canbe evaluated exactly for any value of the ’t Hooft coupling. In these correlation functionsthe hexagons couple only pairwise. The composite form factors representing two pairedhexagons, named octagons , completely factorise. The factorisation was shown to take placein all orders of the 1 /N c expansion [11]. If there are (cid:96) propagators sandwiched between thetwo hexagons, one speaks of octagon with bridge (cid:96) .The two constituent hexagons are bound by exchanging virtual particles in the mirrorchannel. In [12, 13], the octagon was represented as a Fredholm pfaffian and was also givena more tractable representation as the pfaffian of a discrete kernel K representing a complexsemi-infinite anti-symmetric matrix K m,n with m, n ≥
0. It was also conjectured that theoctagon kernel can be rotated to a simplified kernel ◦ K which is a real half-sparse matrixand as such can be split into two equivalent diagonal blocks. Based on this conjecture,the pfaffian was expressed as the determinant of one of the blocks. The simplified kernel ◦ K was defined in [13] by its perturbative series and then non-perturbatively in [14–16].This second representation of the octagon allowed the authors of [14–16] to reformulatethe latter as Fredholm determinant of a generalised Bessel kernel, for which powerful math-ematical methods have been developed previously. However the existence of a similaritytransformation turning K into ◦ K has not been established. One of the goals of this paperis to construct explicitly such a transformation. The octagon is the simplest of a family of computable observables in N = 4 SYM, suchas the cusp anomalous dimension [17] and the MHV six-gluon amplitude in the collinear[18] and close-to-the origin [19] limits. As emphasised in [19], these objects exhibit similarmathematical structures involving semi-infinite matrices.In this paper we propose an operator description for the octagon based on a pairof complex fermionic fields, ψ ( x ) and ψ ∗ ( x ), with the holomorphic variable x being theZhukovsky parametrisation of the rapidities of the virtual particles. Similar descriptionsexist for all observables mentioned above. Below we present, for reader’s convenience, ashort summary of our main results.The operator formalism proposed here is a Fock space realisation of the descriptionwith real fermions presented in [13]. The Fock space for the complex fermions is a directsum of sectors characterised by the U (1) charge of the vacuum or, in other words, by thelevel of filling of the Dirac sea. The octagon with bridge (cid:96) is constructed as the expectationvalue of a product of exponential operators in the sector of charge (cid:96) , O (cid:96) = (cid:104) (cid:96) | exp[ ψ K ψ ] exp[ − ψ ∗ C ψ ∗ ] | (cid:96) (cid:105) , (1.1) While we were working on this manuscript, we learned that Andrey Belitsky and Gregory Korchemskyfound another solution for the similarity transformation, to be published as appendix to v2 of [16]. Wecomment on their solution in our appendix B. The two solutions are related by a transformation whichleaves the kernel K invariant. – 2 – is the octagon kernel and C is a standard quasi-diagonal symplectic matrix. The rightexponential imposes non-trivial correlation of the modes of ψ and resembles the operatorsof boundary states in CFT, hence the notation | (cid:96) (cid:105)(cid:105) def = exp[ − ψ ∗ C ψ ∗ ] | (cid:96) (cid:105) . (1.2)As the two exponents contain either two creation operators, or two annihilation operators,the U (1) charge is not preserved and the expectation value is given by a Fredholm pfaffian[13].The Fock-space realisation (1.1) gives a nice interpretation of the bridge as an operatorcomposed of the (cid:96) lowest fermion oscillator modes. Based on this we show that the octagonwith non-zero bridge (cid:96) is obtained by multiplying the octagon with (cid:96) = 0 by a pfaffian ofa 2 (cid:96) × (cid:96) matrix of fermionic correlators.We give an explicit solution for the similarity transformation mentioned above andexplore its consequences for the fermionic oscillator model. For any (cid:96) ≥
0, the similaritytransformation acts only on the oscillators ψ n , ψ ∗ n above the Fermi level, n ≥ (cid:96) , by asemi-infinite matrix U (cid:96) , ˜ ψ j = (cid:88) k ≥ (cid:96) [ U (cid:96) ] jk ψ k , ˜ ψ ∗ j = (cid:88) k ≥ (cid:96) [ U − ] kj ψ ∗ k . (1.3)The canonical transformation (1.3) preserves the matrix C and transforms the octagonkernel K into the simplified kernel ◦ K . We will give its explicit formula for any (cid:96) , but whatis important is the very fact of its existence. The operator expression for the octagon thentakes the form O (cid:96) = (cid:104) (cid:96) | exp[ ψ ◦ K ψ ] exp[ − ψ ∗ C ψ ∗ ] | (cid:96) (cid:105) . (1.4)Here we replaced { ˜ ψ, ˜ ψ ∗ } by { ψ, ψ ∗ } , as the existence of a transformation (1.3) for any (cid:96) then guarantees that the vacuum states have the same form for the original and thetransformed fermions.Both the simplified kernel ◦ K and the matrix C relate only modes of different parity.Thanks to this property, half of the modes in (1.4) can be eliminated and the resultingoperator expression is exponential of a fermion bilinear which, unlike the exponential oper-ators in (1.4), preserves the U (1) charge. This leads to the Fredholm determinant formulafor the octagon and to finite determinant relations between octagons with different bridges.For even/odd bridge we expressed the octagon as an expectation value in the Fockspace built on the odd/even oscillators, ψ e j = ψ j , ψ ∗ e j = ψ ∗ j and ψ o j = ψ j +1 , ψ ∗ o = ψ ∗ j +1 , O (cid:96) = (cid:40) (cid:104) m, o | e − ψ o K oo ψ ∗ o | m, o (cid:105) = det[(1 − K oo ) ≥ m ] if (cid:96) = 2 m, (cid:104) m, e | e − ψ e K ee ψ ∗ e | m, e (cid:105) = det[(1 − K ee ) ≥ m ] if (cid:96) = 2 m − . (1.5)The vacuum states | m, o (cid:105) and | m, e (cid:105) in (1.5) are the standard vacuum vectors of charge m respectively for the ensembles of odd and the even oscillator modes. By K ee and K oo wedenoted respectively the even-even and the odd-odd blocks of the block diagonal product– 3 – KC = K ee ⊕ K oo and 1 stays for the identity matrix. Finally, for any semi-infinite matrix A = { A i,j } i,j ≥ , the symbol ( A ) ≥ m denotes the semi-infinite matrix obtained by delet-ing the first m roes and columns, ( A ) ≥ m = { A i,j } i,j ≥ m . The determinants in (1.5) areequivalent to those formulated in [13, 14], only the matrix elements are indexed differently.The operator representations in the form (1.5) give rise to m × m determinant identities,presented in section 4, which relate the octagons with finite bridge (cid:96) = 2 m − (cid:96) = 2 m to the octagon with zero bridge. Hence the ratio O m and O as an m × m determinant, O m O = det [(1 + R oo ) The bi-local weights are defined in terms of a single function W ( u, v ) = x ( u ) − x ( v ) x ( u ) x ( v ) − x ( u ) is defined by the Zhukovsky map u/g = x + 1 /x (2.5)transforming the physical sheet in the rapidity plane into the exterior of the unit circle.Namely W a,b ( u, v ) = W ( u + i a, v + i b ) W ( u + i a, v − i b ) W ( u − i a, v + i b ) W ( u − i a, v − i b ) . (2.6) • Local weights. The one-particle factors are W ± a ( u, ξ ) = 1 g ( − a χ ± a Ω (cid:96) ( u + i a, ξ ) Ω (cid:96) ( u − i a, ξ ) × W ( u + i a, u − i a ) . (2.7)– 5 –here Ω (cid:96) ( u, ξ ) = 1 x (cid:96) e igξ [ x − /x ] x − /x = g e igξ [ x − /x ] x (cid:96) d log xdu , (2.8) χ ± a is essentially the character of the a -th antisymmetric representation of psu (2 | χ ± a ( φ, ϕ, θ ) = ( − a sin( aφ )sin φ [2 cos φ − ϕ ± iθ )] . (2.9)For simplicity we will assume that θ = 0. The function Ω (cid:96) ( u, ξ ) reflects the form of themomentum and the energy of the mirror magnons as functions of the rapidity u ,˜ p a ( u ) = g ( x − x ) u + ia/ + g ( x − x ) u − ia/ , ˜ E a ( u ) = log x | u + ia/ + log x | u − ia/ . (2.10)The psu (2 | 2) characters are determined by the generating function W ( t ) = 1 + ∞ (cid:88) a =1 ( − a χ a e − at = 1 − cosh ϕ − cos φ cosh t − cos φ . (2.11) The fermionic representation we give here was sketched in [20]. Let us first give ourconventions, mostly following the conventions of [21], with ψ here = ψ ∗ there , ψ ∗ here = ψ there .The pair of fermionic fields is defined as ψ ( x ) = (cid:88) n ∈ Z ψ n x − n , ψ ∗ ( x ) = (cid:88) n ∈ Z ψ ∗ n x n , [ ψ m , ψ ∗ n ] + = δ m,n , m, n ∈ Z . (2.12)The operators ψ n , ψ ∗ n act in the standard fermionic Fock space H , which splits as a sum ofFock spaces with given U (1) charge (cid:96) , H = ⊕ (cid:96) ∈ Z H (cid:96) . (2.13)The Fock space H (cid:96) is built on the highest-weight state | (cid:96) (cid:105) and its dual (cid:104) (cid:96) | , constructed for (cid:96) ≥ (cid:104) (cid:96) | = (cid:104) | (cid:96) − (cid:89) n =0 ψ n , | (cid:96) (cid:105) = (cid:96) − (cid:89) n =0 ψ ∗ n | (cid:105) . (2.14)The two vacua satisfy ψ ∗ n | (cid:96) (cid:105) = 0 , (cid:104) (cid:96) | ψ n = 0 ( n < (cid:96) ) , (cid:104) (cid:96) | ψ ∗ n = 0 , ψ n | (cid:96) (cid:105) = 0 ( n ≥ (cid:96) ) . (2.15)The non-vanishing correlators are (cid:104) (cid:96) | ψ m ψ ∗ n | (cid:96) (cid:105) = (cid:40) δ m,n if m ≥ (cid:96), m < (cid:96) (2.16)– 6 –nd the two-point function is G ( x, y ) ≡ (cid:104) (cid:96) | ψ ( x ) ψ ∗ ( y ) | (cid:96) (cid:105) | y | < | x | = ( y/x ) (cid:96) − y/x . (2.17)The correlation function of a product of fermions is given by the determinant of the two-point correlators.In [13], the bi-local weights in the expansion (2.3) of section 1 were expressed in termsof the two-point function of the field ψ ( x ) whose form was postulated. On the presentinterpretation the two-point function of the field ψ results from replacing the right vacuumby a coherent state | (cid:96) (cid:105)(cid:105) def = e − ψ ∗ C ψ ∗ | (cid:96) (cid:105) , ψ ∗ C ψ ∗ = (cid:88) m,n ≥ ψ ∗ m C mn ψ ∗ n (2.18)where C is the skew-symmetric matrix with elements C m,n = δ m +1 ,n − δ m,n +1 ( m, n ≥ . (2.19)For the action of the fermionic oscillators ψ n on the coherent state one obtains( ψ m + [ C ψ ∗ ] m ) | (cid:96) (cid:105)(cid:105) = 0 , m ≥ (cid:96) . (2.20)With the ket vacuum replaced by the coherent state, the ψ -oscillators have a non-vanishingcorrelation (cid:104) (cid:96) | ψ m ψ n | (cid:96) (cid:105)(cid:105) = C m,n (2.21)and their two-point function takes the desired form in the x -representation (cid:104) (cid:96) | ψ ( x ) ψ ( y ) | (cid:96) (cid:105)(cid:105) = (cid:88) m,n ≥ (cid:96) C mn x − m y − n = ( xy ) − (cid:96) x − yxy − . (2.22)As in any ensemble of fermions, the 2 n -point correlator is the pfaffian of the matrix of thetwo-point correlators: (cid:104) (cid:96) | ψ ( x ) ...ψ ( x n ) | (cid:96) (cid:105)(cid:105) = Pf (cid:32)(cid:20) x j x k ) (cid:96) x j − x k x j x k − (cid:21) ni,j =1 (cid:33) = n (cid:89) i =1 x (cid:96)i n (cid:89) j 12 1(2 πi ) (cid:73) dxx (cid:73) dyy ψ ( x ) K ( x, y ) ψ ( y ) (cid:19) | (cid:96) (cid:105)(cid:105) ,K ( x, y ) = 2 e igξ ( x − x + y − y ) g (cid:90) ∞| ξ | dt sin (cid:104) gt ( x + x − y − y ) (cid:105) X( t ) ,X ( t ) = cos φ − cosh ξ cos φ − cosh t . (2.26)In terms of the fermionic oscillators the quadratic form is represented by the semi-infinitematrix K = { K m,n } m,n ≥ . Using the integration formula12 πi (cid:73) dxx x − n e igξ ( x − /x ) ± igt ( x +1 /x ) = (cid:32) i (cid:115) t + ξt − ξ (cid:33) ± n J n (2 g (cid:112) t − ξ ) θ ( t ± ξ ) . (2.27)where the contour integration goes along the unit circle, the discrete kernel K can beexpressed in terms of Bessel finctions, O (cid:96) = (cid:104) (cid:96) | exp (cid:88) m,n ≥ ψ m K m,n ψ n | (cid:96) (cid:105)(cid:105) , (2.28) K m,n = gi (cid:90) ∞| ξ | dt X ( t ) (cid:32) i (cid:115) t + ξt − ξ (cid:33) m − n − (cid:32) i (cid:115) t + ξt − ξ (cid:33) n − m × J m (2 g (cid:112) t − ξ ) J n (2 g (cid:112) t − ξ ) , (2.29)– 8 –or vacuum states of charge (cid:96) , the sum in the exponential in (2.28) is effectively restrictedto m, n ≥ (cid:96) . The computation of the expectation value (2.28) is straightworward and reproduces thepfaffian formula of [13], O (cid:96) = Pf (cid:32) C ≥ (cid:96) − − K ≥ (cid:96) (cid:33) = exp (cid:18) 12 Tr log (cid:2) − C ≥ (cid:96) K ≥ (cid:96) (cid:3)(cid:19) . (2.30)In this expression the semi-infinite matrices C ≥ (cid:96) and K ≥ (cid:96) are obtained from C and K by deleting the first (cid:96) rows and columns. For example, K ≥ (cid:96) = { K m,n } m,n ≥ (cid:96) . The rhsof (2.30) is defined rigorously by first truncating the semi-infinite matrices C ≥ (cid:96) and , K ≥ (cid:96) to N × N matrices and then taking the limit N → ∞ . The limit is convergent for anyfinite g because K m,n decay exponentially when m, n → ∞ . A more direct derivation ofthe pfaffian is based on the formulation of the expectation value as an integral over theGrassmann variables [22], O (cid:96) = (cid:90) (cid:89) m ≥ (cid:96) dζ m dζ ∗ m e S ( ζ,ζ ∗ ) , S ( ζ, ζ ∗ ) = − (cid:88) m,n ≥ (cid:96) ζ m C m,n ζ n + (cid:88) n ≥ (cid:96) ζ ∗ n ζ n + 12 (cid:88) m,n ≥ (cid:96) ζ ∗ m K m,n ζ ∗ n . (2.31) Take the operator representation of the octagon with bridge (cid:96) , eq. (2.28) and consider theright and left vacua as the result of the action of the (cid:96) lowest fermion oscillators as in eq.(2.14), O (cid:96) = (cid:104) | ψ ...ψ (cid:96) − e ψ K ψ e − ψ ∗ C ψ ∗ ψ ∗ (cid:96) − ...ψ ∗ | (cid:105) . (2.32)Hence one can obtain O (cid:96) by inserting in the expectation value for O an operator creating (cid:96) pairs of fermions, O (cid:96) = (cid:104) | e ψ K ψ B (cid:96) | (cid:105)(cid:105) , B (cid:96) = ψ ...ψ (cid:96) − ψ ∗ (cid:96) − ...ψ ∗ . (2.33)This can be used to derive an expression for the octagon with bridge (cid:96) in terms of theexpectation value of the operator B (cid:96) , O (cid:96) O = (cid:104) B (cid:96) (cid:105) , (2.34) Of course C ≥ (cid:96) and C are identical as matrices, but considered as functions of two discrete variablesthey are related by a shift by (cid:96) in both arguments. If the semi-infinite matrix is truncated to a N × N -dimensional matrix, there will be an extra signfactor ( − N ( N − / multiplying the pfaffian. – 9 –here the expectation value of an operator O is defined as (cid:104) O (cid:105) def = (cid:104) | e ψ K ψ O | (cid:105)(cid:105)(cid:104) | e ψ K ψ | (cid:105)(cid:105) . (2.35)As any expectation value of free fermions, (cid:104) B (cid:96) (cid:105) is equal to the pfaffian of the two-point correlation functions of the fermions involved. A direct calculation gives, for j, k =0 , , , ... , (cid:104) ψ ∗ j ψ ∗ k (cid:105) = − [ K (1 + R )] j,k , (cid:104) ψ j ψ ∗ k (cid:105) = [1 + R ] j,k , (cid:104) ψ ∗ j ψ k (cid:105) = − [1 + R ] k,j , (cid:104) ψ j ψ k (cid:105) = [(1 + R ) C ] j,k (2.36)where R = CK − CK . (2.37)The matrix of all correlators is the inverse of the quadratic form in the representation asintegral over grassman variables, as it should, (cid:32) (1 + R ) C R − (1 + R ) T − K (1 + R ) (cid:33) = (cid:32) − K − II C (cid:33) − . (2.38)Now we can express the ratio O (cid:96) / O as an 2 (cid:96) × (cid:96) pfaffian O (cid:96) O = ( − (cid:96) ( (cid:96) − Pf (cid:34)(cid:32) (1 + R ) C R − (1 + R ) T − K (1 + R ) (cid:33) <(cid:96) (cid:35) . (2.39)Here we introduced the symbol X <(cid:96) which represents the truncation of the semi-infinitematrix X to an (cid:96) × (cid:96) matrix { X m,n } ≤ m,n<(cid:96) . The truncation is applied to all four blocksof the matrix.We have checked the finite pfaffian relation (2.39) for (cid:96) ≤ g . As another-consistency check let us consider the limit (cid:96) → ∞ where O (cid:96) → 1. Then after taking intoaccount (2.38), the identity (2.39) reproduces the original pfaffian formula (2.30) for (cid:96) = 0.An obvious generalisation of (2.39) relates two octagons with bridges (cid:96) < (cid:96) , O (cid:96) O (cid:96) = ( − ( (cid:96) − (cid:96) )( (cid:96) − (cid:96) − Pf (1 + R ≥ (cid:96) ) C ≥ (cid:96) R ≥ (cid:96) − − R T ≥ (cid:96) K ≥ (cid:96) (1 + R ≥ (cid:96) ) <(cid:96) . (2.40)In particular, the octagons with bridges (cid:96) and (cid:96) + 1 are related as O (cid:96) +1 O (cid:96) = (cid:20) − C ≥ (cid:96) K ≥ (cid:96) (cid:21) (cid:96),(cid:96) (2.41)which provides a factorised form for the relation (2.39).– 10 – The similarity transformation In this section we give explicit expression for the similarity transformation relating theoriginal and the simplified octagon kernels, which corresponds to the canonical transfor-mation (1.3) of the fermion oscillators. The operator representation based on the new setof oscillators has the advantage that it preserves the U (1) charge and therefore leads to adeterminant instead of a pfaffian.In this subsection we remind the definition of the two kernels. It is convenient tochange the variables in (2.29) as ξ ≡ σg , t ≡ g (cid:112) τ + σ . (3.1)so that the integration now spreads on the whole positive real axis and the dependence onthe ’t Hooft coupling is carried only by the weight function X. In the new variables, theweight function takes the form χ ( τ, σ ) ≡ X ( √ τ + σ g ) = cos φ − cosh ϕ cos φ − cosh √ τ + σ g (3.2)and the integral formula for the matrix elements (2.29) becomes K m,n = 2 (cid:90) ∞ dτ χ ( τ, σ ) K m,n ( τ ; σ ) ( m, n ≥ , (3.3)with K m,n ( σ, τ ) = Π m − n ( σ/τ ) J m (2 τ ) J n (2 τ ) , (3.4)Π n ( z ) def = i n (cid:16) √ z + 1 + z (cid:17) n − i − n (cid:16) √ z + 1 − z (cid:17) n i √ z + 1 = − Π − n ( z ) . (3.5)Importantly, Π n is a polynomial,Π ( z ) = 0 , Π ( z ) = 1 , Π ( z ) = 2 iz, Π ( z ) = − − z , etc . (3.6)It equals the ( n − n ( z ) = U n − ( iz ) = n − (cid:88) p =0 sin π n − p A ( p ) n ( − iz ) p p ! ( n ≥ 1) (3.7)with A ( p ) n = ( − p Γ (cid:2) ( n + 1 + p ) (cid:3) Γ (cid:2) ( n + 1 − p ) (cid:3) . (3.8)– 11 –ummarising, the integrand in (3.3) is given by a sum of products of Bessel functions, K m,n ( τ ; σ ) = | m − n |− (cid:88) j =0 − ( − m − n − j A ( j ) m − n ( iσ ) j j ! i m − n − j J m (2 τ ) J n (2 τ ) τ j . (3.9)The octagon kernel (3.3) depends on the cross ratios of the spacetime coordinates (theparameters σ and φ ) through the weight function χ and also through the polynomialsΠ m − n ( σ/τ ). It was noticed [13] that the second dependence is redundant in the sense thatonly the constant terms Π m − n (0) = sin( m − n π ) of these polynomials contribute. Based onthis observation, it was conjectured that in the pfaffian formula (2.30), the kernel K can bereplaced with a simplified kernel whose matrix elements ◦ K m,n are real and vanish if m and n have the same parity. The last property implies that the pfaffian (2.30) can be writtenas a determinant. The simplified kernel was found as a perturbative series in [13] and inintegral form in [15], ◦ K m,n = 2 (cid:90) ∞ dτ χ ( τ, σ ) ◦ K m,n ( τ ) ( m, n ≥ ◦ K m,n ( τ ) = sin (cid:0) m − n π (cid:1) J m (2 τ ) J n (2 τ ) , (3.10)where χ ( τ, σ ) is the weight function defined in (3.2).The conjecture of [13] states, with the interpretation of the bridge we adopted here, thatfor any (cid:96) ≥ 0, the matrices C ≥ (cid:96) K ≥ (cid:96) and C ≥ (cid:96) ◦ K ≥ (cid:96) are related by a similarity transformation.(We remind that X ≥ (cid:96) denotes the matrix X with its first (cid:96) rows and columns deleted.) Thisis equivalent to claiming that there exists a symplectic transformation preserving C andrelating K ≥ (cid:96) and ◦ K ≥ (cid:96) , C ≥ (cid:96) K ≥ (cid:96) = U − (cid:96) C ≥ (cid:96) ◦ K ≥ (cid:96) U (cid:96) ⇔ (cid:40) K ≥ (cid:96) = U T (cid:96) ◦ K ≥ (cid:96) U (cid:96) , C ≥ (cid:96) = U (cid:96) C ≥ (cid:96) U T (cid:96) . (3.11)In terms of the ensemble of fermions, the above statements mean, first, that the operatorsin the expectation values (1.1) and (1.4) are related by the canonical transformation (1.3),and second, that the canonical transformation in question leaves the bra and ket vacua ofcharge (cid:96) invariant. The solution for the matrix U (cid:96) in (3.11) is not unique. We found a particular solution ofthe first equation (3.11) in the form of a power series in σ , U (cid:96) = ∞ (cid:88) p =0 ( − iσ ) p p ! (cid:0) C ≥ (cid:96) M ≥ (cid:96) (cid:1) p Q ( p ) (cid:96) (3.12)– 12 –here the diagonal matrices M ≥ (cid:96) and Q ( p ) (cid:96) are defined as M ≥ (cid:96) = diag (cid:26) n (cid:27) n ≥ (cid:96) = diag (cid:26) (cid:96) , (cid:96) + 1 , (cid:96) + 2 , ... (cid:27) (3.13) Q ( p ) (cid:96) = diag (cid:110) θ n +1 − p − (cid:96) (cid:16) α n − (cid:96) − p A ( p ) n − (cid:96) + α n − − (cid:96) − p ( − p B ( p ) n − (cid:96) (cid:17)(cid:111) n ≥ (cid:96) . (3.14)Here α k ≡ (1 − ( − k ) , θ k = (cid:40) k ≤ , k > A ( p ) n are defined above in (3.8), and the coefficients B ( p ) m are given by B ( p ) m = m ( − p − A ( p − m = 2 p − m Γ (cid:0) ( m + p ) (cid:1) Γ (cid:0) ( m + 2 − p ) (cid:1) . (3.16)These coefficients appear in the Taylor expansions (cid:16) √ z + 1 − z (cid:17) m √ z + 1 = (cid:88) p ≥ A ( p ) m z p p ! , (cid:16)(cid:112) z + 1 + z (cid:17) m = (cid:88) p ≥ B ( p ) m z p p ! . (3.17)For fixed m, n ≥ (cid:96) , the matrix element ( U (cid:96) ) m,n is a polynomial in σ of degree n − (cid:96) for m − (cid:96) even, or of degree n − − (cid:96) for m − (cid:96) odd. The coefficients of this polynomialdepend explicitly on the bridge length (cid:96) . The lowest matrix elements (cid:96) ≤ m, n ≤ (cid:96) + 3 are U (cid:96) = iσ(cid:96) +1 − σ ( (cid:96) +1)( (cid:96) +2) − iσ ( (cid:96) +1)( (cid:96) +2)( (cid:96) +3) ∗ iσ(cid:96) +2 − σ ( (cid:96) +2)( (cid:96) +3) ∗ − iσ(cid:96) +1 σ ( (cid:96) +1)( (cid:96) +3) 3 iσ ( (cid:96) +1)( (cid:96) +4)+12 iσ ( (cid:96) +1)( (cid:96) +3)( (cid:96) +4) ∗ − iσ(cid:96) +2 σ ( (cid:96) +2)( (cid:96) +4) ∗ − σ ( (cid:96) +2)( (cid:96) +3) − iσ ( σ + (cid:96) ( (cid:96) +7)+10 ) ( (cid:96) +2)( (cid:96) +3)( (cid:96) +5) ∗∗ ∗ ∗ ∗ ∗ . (3.18)We give the idea of the derivation of the symplectic transformation in appendix C. Theproof is based on a linear relation between K and ◦ K , K m,n = m − n − (cid:88) k =0 ( iσ ) k k ! α m − n − k A ( k ) m − n [( MC ) k K ◦ ] m,n ( m > n ) K m,n = − n − m − (cid:88) k =0 ( − iσ ) k k ! α m − n − k A ( k ) n − m [( K ◦ ( CM ) k ] m,n ( m < n ) , (3.19) The lowest matrix element [ M ≥ (cid:96) ] (cid:96),(cid:96) is singular for (cid:96) = 0, but it does not appear neither in (3.12) norin the matrix relations further on. It can be set to any finite number but we prefer to keep it as it is, sothat formally the matrix M ≥ (cid:96) is the inverse of N ≡ { n } n ≥ (cid:96) . – 13 –hich follows from the expansion (3.9) and the recurrence relation for the Bessel functions J m +1 (2 τ ) + J m − (2 τ ) = m J m (2 τ ) τ , (3.20)see appendix A. Since eq. (3.19) is consistent with restricting the matrix elements to m, n ≥ (cid:96) , it also implies a linear relation between K ≥ (cid:96) and ◦ K ≥ (cid:96) . Concerning the secondrelation in (3.11), we checked that it is satisfied by the series (3.12) for the first severalorders in σ , but we do not know how to prove it analytically in general. In the nextsubsection we give another form of the solution (3.12) for (cid:96) = 0, for which this propertycomes out naturally.As we mentioned before, the similarity transformation is not unique, and anothersolution was independently obtained by Belitsky and Korchemsky [16]. In appendix Bwe re-derive their result as a solution of an ordinary differential equation describing theoperator flow connecting K and ◦ K . (cid:96) = 0When (cid:96) = 0, the solution (3.12) for the similarity transformation can be written in a quasiexponential form, U (cid:96) =0 = ∞ (cid:88) j =0 (cid:104)(cid:16) P e e − σ CMS + P o e − σ SCM (cid:17) e iσ C (cid:105) j P ( j ) . (3.21)Here [ ... ] j denotes the coefficient of the power σ j in the expansion of the expression in thebrackets, M ≡ M (cid:96) =0 is given by (3.13), the matrix S is defined as S m,n = δ m +1 ,n + δ m,n +1 ( m, n ≥ , (3.22) P ( j ) is as in (3.14) the projector to the matrices with the first j columns vanishing, P ( j ) = diag { θ n +1 − j } n ≥ , (3.23)and P e and P o are the projectors respectively to the even and odd subsets, P e = diag { α m +1 } m ≥ , P o = diag { α m } m ≥ (3.24)with α k given by (3.15).To get some intuition on the origin of the two exponential factors in (3.21), let us writethe simplified kernel for (cid:96) = 0 in x -representation, ◦ K ( x, y ) = (cid:88) m,n ∈ Z x m y n ◦ K m,n = 2 (cid:90) ∞ dτ χ ( τ, σ ) sin (cid:104)(cid:16) x + x − y − y (cid:17) τ (cid:105) , (3.25)and compare it with the original kernel (2.26), written in terms of the variables (3.1), K ( x, y ) = 2 e iσ ( x − x + y − y ) (cid:90) ∞ dτ χ ( τ, σ ) τ √ τ + σ sin (cid:104)(cid:16) x + x − y − y (cid:17)(cid:112) τ + σ (cid:105) . (3.26)The first expression is obtained from the second by setting σ = 0. In (3.21), the rightexponential factor e iσ C accounts for the factor e iσ ( x − /x + y − /y ) in (3.26). Indeed, in x -representation, the operator C acts as a multiplication by x − /x . The second factor in(3.26) originates from the σ -dependence of the integrand of (3.26). The latter expands asa series in σ , with the constant term given by the integrand of (3.25).– 14 – Determinant identities Since in the simplified kernel (3.10) the matrix elements with the same parity vanish, the2 (cid:96) × (cid:96) pfaffians in the finite pfaffian formulas obtained in section 2.4 can now be writtenas (cid:96) × (cid:96) determinants. It turns out that these determinants can be simplified further andwritten as determinants of approximately twice less size. More precisely, for (cid:96) = 2 m − (cid:96) = 2 m , the ratio O (cid:96) / O is an m × m determinant.To obtain the reduced determinant identities, we first notice that in the Fock spacerepresentation (1.4) the exponents are bilinear forms of the even and odd modes, O (cid:96) = (cid:104) (cid:96) | exp (cid:88) j,k ≥ ψ j +1 ◦ K j +1 , k ψ k exp − (cid:88) j,k ≥ ψ ∗ j C j, k +1 ψ ∗ k +1 | (cid:96) (cid:105) . (4.1)We will show that, depending on the parity of (cid:96) , one can eliminate either the even or the oddmodes from the expectation value by replacing the right exponent with the identification(2.20). Let us assume that the length of the bridge is even, (cid:96) = 2 m . In the operator expression(4.1), we can commute all the even modes of ψ ∗ to the left and all the even modes of ψ to the right until they both are annihilated by the corresponding vacua. As a result weobtain an operator expression only in terms of the odd modes, O (cid:96) =2 m = ∞ (cid:88) n =0 ( − n n ! (cid:88) j ,...,jn ≥ mk ,...,nn ≥ m (cid:104) (cid:96) | n (cid:89) a =1 ψ j a +1 n (cid:89) a,b =1 K oo j a ,k b n (cid:89) b =1 ψ ∗ k b +1 | (cid:96) (cid:105) = (cid:104) (cid:96) | ◦◦ exp − (cid:88) k,j ≥ ψ j +1 K oo j,k ψ ∗ k +1 ◦◦ | (cid:96) (cid:105) . (4.2)In the last line ◦◦ ◦◦ denotes the anti-normal ordering where all ψ ∗ are on the right of all ψ .By K oo we denoted odd-odd diagonal block of the matrix ◦ KC , K oo j,k def = [ ◦ KC ] j +1 , k +1 , (4.3)whose matrix elements are given explicitly by K oo i,j = 2(2 j + 1) ( − i − j (cid:90) ∞ dτ χ ( τ ) J i +1 (2 τ ) J j +1 (2 τ ) τ ( i, j ≥ . (4.4)To obtain the rhs of (4.2) we used the identity( ◦ K ≥ (cid:96) C ≥ (cid:96) ) j +1 , k +1 = ([ ◦ KC ] ≥ (cid:96) ) j +1 , k +1 ( (cid:96) = 2 m ) (4.5)which follows from the fact that the matrix C is quasi-diagonal. Evaluating the expectationvalue with the correlators (2.16), we obtain the determinant formula for the octagon O (cid:96) =2 m = det[(1 − K oo ) ≥ m ] . (4.6)– 15 –ow we would like to evaluate the ratio of the octagons with (cid:96) = 2 m and (cid:96) = 0 asan expectation value, as in section 2.4. The identity (4.5) also guarantees that the evenfermion modes in the vacuum states can be removed without altering the result, | (cid:96) (cid:105) → ψ ∗ m − ψ ∗ m − · · · ψ ∗ | (cid:105) ≡ | m, o (cid:105) , (cid:104) (cid:96) | → (cid:104) | ψ ψ · · · ψ m − ≡ (cid:104) m, o | ( (cid:96) = 2 m, m ≥ . (4.7)We express the octagon with bridge (cid:96) = 2 m as the result of the insertion of m pairs ofodd fermionic modes, and divide by the octagon with bridge zero, O m O = (cid:104) | ψ ψ · · · ψ m − ◦◦ e ψ K oo ψ ∗◦◦ ψ ∗ m − ψ ∗ m − · · · ψ ∗ | (cid:105)(cid:104) | ◦◦ e ψ K oo ψ ∗◦◦ | (cid:105)≡ (cid:104) m − (cid:89) j =0 ψ j +1 m − (cid:89) j =0 ψ ∗ j +1 (cid:105) . (4.8)The expectation value is equal to the determinant of the two-point correlators (cid:104) ψ j +1 ψ ∗ k +1 (cid:105) = δ j,k + R oo j,k , (4.9)where the semi-infinite matrix R oo is related to K oo by(1 + R oo )(1 − K oo ) = 1 . (4.10)Hence the ratio O m and O is an m × m determinant, O m O = det [(1 + R oo ) In a similar way, in the case (cid:96) = 2 m − O (cid:96) =2 m − = ∞ (cid:88) n =0 ( − n n ! (cid:88) j ,...,jn ≥ mk ,...,nn ≥ m (cid:104) (cid:96) | n (cid:89) a =1 ψ j a n (cid:89) a,b =1 K ee j a ,k b n (cid:89) b =1 ψ ∗ k b | (cid:96) (cid:105) = (cid:104) (cid:96) | ◦◦ exp − (cid:88) k,j ≥ ψ j K ee j,k ψ ∗ k ◦◦ | (cid:96) (cid:105) . (4.15)The matrix elements of K ee i,j ≡ [ ◦ KC ] j, k (4.16)are given explicitly by ( j, k ≥ K ee i,j = 2 (cid:90) ∞ dτ χ ( τ ) (cid:20) (1 − δ j, )( − i − j j J i (2 τ ) J j (2 τ ) τ + δ j, ( − j J i J (cid:21) . (4.17)Thanks to the identity [ ◦ K ≥ (cid:96) C ≥ (cid:96) ] j, k = [[ ◦ KC ] ≥ (cid:96) ] j, k for (cid:96) odd, we can eliminate the oddmodes also from the left and the right vacuum, and formulate the octagon as an expectationvalue in the ensemble of the even oscillators, O (cid:96) =2 m − = (cid:104) m, e | ◦◦ e − ψ e K ee ψ ∗ e ◦◦ | m, e (cid:105) = det (cid:104) (1 − K ee ) ≥ m (cid:105) ( m ≥ . (4.18)The bra and ket vacuum states here are the m -charged vacua for the even modes, | m, e (cid:105) = ψ ∗ m − ψ ∗ m − · · · ψ ∗ | (cid:105) , (cid:104) m, e | = (cid:104) | ψ ψ · · · ψ m − . (4.19)Again, the ratio O m − / O can be computed as an expectation value of m fermionpairs which gives an m × m determinant O m − O = (cid:104) m − (cid:89) j =0 ψ e2 j ψ ∗ e2 j (cid:105) = det (cid:2) (1 + R ee ) 2] replaced by sin [( m − n )( π/ − α )].The fermionic representation and the subsequent analysis (except for section 4 which isrelevant only to the case α = 0) can be generalised to a generic angle α . In particular, wechecked that the flow equation obtained in appendix B holds for the tilted kernel as well. Acknowledgments We thank A. Belitsky and G. Korchemsky for useful discussions and for sharing theirunpublished notes and D. Serban for critical remarks on the manuscript. A Proof of the linear relation (3.19) between the original and the sim-plified kernels We will show that the linear relation (3.9) holding at the level of the integrands (3.3),(3.10), leads to the relation (3.19) for the integral kernels. Consider the bilinear of Besselfunctions J m,n ( τ ) = i m − n − J m (2 τ ) J n (2 τ ) . (A.1)With this normalization the functional relation (3.20) for the Bessel function is rewrittenwith the help of the matrix C in (2.19) and M m,n = m δ m,n as i J m,n ( τ ) τ = ( MC ) m,m (cid:48) J m (cid:48) ,n ( τ ) = J m,n (cid:48) ( τ )( CM ) n (cid:48) ,n , m, n (cid:54) = 0 . (A.2)Repeated for an arbitrary power of τ this gives, in matrix notations, (cid:18) iτ (cid:19) j J ( τ ) = ( MC ) j J ( τ ) , (A.3)where the explicit expressions for the matrix powers of MC are[( MC ) s ] m,m (cid:48) = s (cid:88) r = − s (cid:18) ss − | r | (cid:19) Γ( m − s + r )( m + 2 r )Γ( m + s + r + 1) ( − r − s δ m +2 r,m (cid:48) , [( MC ) s +1 ] m,m (cid:48) = s (cid:88) r = − s (cid:18) ss − | r | (cid:19) Γ( m − s + r )Γ( m + s + r + 1) ( − r − s ( δ m +2 r +1 ,m (cid:48) − δ m +2 r − ,m (cid:48) ) . (A.4)– 21 –he expression (A.4) is defined for positive integer m with the power j of the matrix ( MC )restricted to j ≤ m . (A.5)The second m (cid:48) index in (A.4) runs between m − j and m + j (mod 2) . For the even power j = 2 s , the formula (A.4) has sense for j = 0, reproducing the identity [( MC ) ] m,m (cid:48) = δ m,m (cid:48) . Furthermore in this case the formula extends for m = 0, taking into account (A.5),i.e., [( MC ) s ] ,m (cid:48) = δ s, δ ,m (cid:48) . The powers of CM are obtained by transposition,[( CM ) j ] n (cid:48) ,n = ( − j [( MC ) j ] n,n (cid:48) . (A.6)Next we observe that for odd values j + m − n , eq. (A.3) turns into (cid:18) iτ (cid:19) j J m,n ( τ ) = [( MC ) j ] m,m (cid:48) ◦ K ( τ )] m (cid:48) ,n ( j + m = n − , (A.7)where ◦ K ( τ ) m,n = − ( − m − n J m,n (3.10) . Inserting (A.7) in (3.9) we obtain after integration(3.19) K m,n = (cid:88) j =0 ,...,m − n − j + m + n =odd ( iσ ) j j ! A ( j ) m − n [( MC ) j ◦ K ] m,n ( m > n )= (cid:88) j =0 ,...,n − m − j + m + n =odd ( − iσ ) j j ! A ( j ) n − m [ ◦ K ( CM ) j ] m,n ( n > m ) (A.8)The inequality (A.5) is fulfilled in (A.8). In our problem the indices of the Bessel functions J m take values m ≥ (cid:96) . This implies that the power j of the matrices ( M ≥ (cid:96) C ≥ (cid:96) ) (cf. (3.13))is restricted to j ≤ m − (cid:96) , (A.9)[( M ≥ (cid:96) C ≥ (cid:96) ) s ] (cid:96),m (cid:48) = δ s, δ (cid:96),m (cid:48) . The relation (A.8) then holds with M , C replaced by M ≥ (cid:96) , C ≥ (cid:96) .The restriction of the indices of the kernel K m,n given by (A.8) to m, n ≥ (cid:96) projects, takinginto account the upper bound (A.9), the second index of [( MC ) j ] m,m (cid:48) in the rhs to m (cid:48) ≥ (cid:96) . B Flow equation The matrix elements K m,n satisfy the following differential equation and its conjugate, m∂ σ K m,n − iσ∂ σ ( K m +1 ,n + K m − ,n ) + i ( m − n ) ( K m +1 ,n − K m − ,n ) = 0 , (B.1)for any m, n ≥ 0. Here the weight function χ is treated as a functional parameter and thederivative in σ does not act on it. The equations follow straightforwardly from (3.19) usingthe relations for the coefficients A j +1 m − n = − ( m − n ∓ j ) A jm ± − n , A m − nm +1 − n = 0 . (B.2)– 22 –ntroducing the diagonal matrix N m,n = n δ m,n , m, n ≥ N − iσ S ) ∂ σ K + i [ N , CK ] = 0 ∂ σ K ( N − iσ S ) + i [ N , KC ] = 0 . (B.3)The flow equation (B.3) determines the evolution of the full kernel K which char-acterises the octagon with zero bridge. The equation for non-zero bridge is obtained byreplacing the kernel and the matrices involved with K ≥ (cid:96) , N ≥ (cid:96) , S ≥ (cid:96) , C ≥ (cid:96) . Remark. After substituting K ≥ (cid:96) → U T (cid:96) ◦ K ≥ (cid:96) U (cid:96) , the equation (B.1) turns into an equationfor the similarity operator U (cid:96) . In general, it is not straightforward to integrate it. Sincethere is a continuum of solutions, one can impose additional conditions on the solution.Belitsky and Korchemsky imposed [16] the condition that the semi-infinite similarity matrix Ω = { Ω i,k } k,j ≥ acts trivially on the first two columns, Ω k, = δ k, , Ω k, = δ k, . In ourconventions their matrix Ω corresponds to a matrix ˜ U (cid:96) with elements[ ˜ U (cid:96) ] k + (cid:96),j + (cid:96) = [ Ω ] k,j . (B.4)Under these conditions [ ˜ U (cid:96) ] k + (cid:96),(cid:96) = δ k, and [ ˜ U (cid:96) ] k + (cid:96), (cid:96) = δ k, one obtains K m + (cid:96),(cid:96) = [ ˜ U T (cid:96) ◦ K ] m + (cid:96),(cid:96) , K m + (cid:96), (cid:96) = [ ˜ U T (cid:96) ◦ K ] m + (cid:96), (cid:96) ( m ≥ . (B.5)In this way the differential equation (B.1) reduces to an equation for ˜ U (cid:96) . Accordingly thefirst/second relation (B.5) determines the matrix elements of Ω T with odd/even first indexdirectly from (3.19) Ω k +1 ,n = (cid:88) j =0 ,...,n − j − n = odd ( − iσ ) j j ! [( CM ) j ] k +1+ (cid:96),n + (cid:96) A ( j ) n = [ U (cid:96) ] k +1+ (cid:96),n + (cid:96) Ω k,n = δ n, δ k, + (cid:88) j =0 ,...,n − j − n = even ( − iσ ) j j ! [( CM ) j ] k + (cid:96),n + (cid:96) A ( j ) | n − | = δ n, δ k, + [ U (cid:96) +1 ] k − (cid:96) +1) ,n − (cid:96) +1) . (B.6)The last equality relates the matrix elements of Ω to a different projection of the solution[ U (cid:96) ] n (cid:48) ,n (3.12) to odd n (cid:48) − (cid:96) . C Proof of the similarity transformation (3.11) - (3.12) In this appendix we will omit the index (cid:96) in U (cid:96) and M (cid:96) in order to avoid ugly formulas.We will show that the linear transformation (3.19) can be written as adjoint action matrixrelation (3.11) K m + (cid:96),n + (cid:96) = (cid:88) m (cid:48) ,n (cid:48)≥ n (cid:48)− m (cid:48) =odd U T m + (cid:96),m (cid:48) + (cid:96) ◦ K m (cid:48) + (cid:96),n (cid:48) + (cid:96) U n (cid:48) + (cid:96),n + (cid:96) . (C.1)– 23 –he matrix elements of U = U (cid:96) in (3.12) read more explicitly U k +1+ (cid:96),n + (cid:96) = (cid:88) j =0 ,...,n − j − n =odd ( − iσ ) j j ! A ( j ) n [( CM ) j ] k +1+ (cid:96),n + (cid:96) U k + (cid:96),n + (cid:96) = (cid:88) j =0 ,...,nj − n =even ( iσ ) j j ! B ( j ) n [( CM ) j ] k + (cid:96),n + (cid:96) . (C.2)They satisfy the relations U k + (cid:96),(cid:96) = δ k, , U k +1+ (cid:96), (cid:96) = δ k, , (C.3)which differ from (B.5) since U k + (cid:96), (cid:96) = iσ ( δ k, − δ k, ). Note that the zeros of thecoefficients A ( j ) n and B ( j ) n (3.17) compensate the poles in the expressions (A.4) for the matrixpowers ( CM ) jn (cid:48) + (cid:96),n + (cid:96) and one can write regularised closed expressions for the operators(C.2). For example U m + (cid:96), n + (cid:96) = δ m, σ n (2 n − ( − n (1 − δ n, ) + δ n, ) Γ(1 + (cid:96) )Γ(2 n + (cid:96) + 1)+ θ m n (cid:88) s =0 (2 σ ) s ( − m + n (2 m + (cid:96) ) n Γ( n + s ) (cid:81) m + (cid:96) − k =1 ( n − s + k )Γ( s − m + n + 1)Γ( s + m − n + 1)Γ( n + m + (cid:96) + s + 1) . (C.4)We want to prove (C.1) with the operators given in (C.2). The key ingredient of theproof is the intertwining relation[( MC ) j ◦ K ] m + (cid:96),n + (cid:96) = [( MC ) j − p ◦ K ( CM ) p ] m + (cid:96),n + (cid:96) ( j ≤ m , p ≤ n ) . (C.5)The latter allows to redistribute the matrix powers in (A.8) on both sides of ◦ K . E.g., − A (1) m − n [ MC ◦ K ] m,n = ( m − n )[ MC ◦ K ] m,n = [ m MC ◦ K ) − n ◦ K CM ] m,n A (2) m − n [( MC ) ◦ K ] m,n = [ m ( MC ) ◦ K − m ( MC ) ◦ K ( CM ) n + ( n − ◦ K ( CM ) ] m,n . (C.6)In what follows we assume, for the sake of simplicity of the presentation, that (cid:96) = 0.To illustrate the general procedure, consider the matrix element K ,n . We can split thecoefficient in (3.15) as A ( j ) n − = j B (1)1 A ( j − n + ( − j B ( j ) n A (0)1 (C.7)and distribute accordingly the matrix powers in agreement with the inequality (A.5) K ,n = ◦ K ,n (cid:48) (cid:88) j =0 ,...,n − j − n =even ( − iσ ) j j ! [( CM ) j ] n (cid:48) ,n A ( j ) | n − | = (cid:88) j =0 ,...,n − j − n =even ( − iσ ) j j ! (cid:16) j B (1)1 [( MC ) ◦ K ( CM ) ( j − ] ,n A ( j − n + ( − j [ ◦ K ( CM ) ( j ) ] ,n B ( j ) n (cid:17) . (C.8) This expression does not change if the upper bound of the summation is extended to s ≤ n + m + (cid:96) − – 24 –his is almost (C.1) in this particular case, with odd intermediate summation index n (cid:48) inthe first term in the rhs of (C.8) and n (cid:48) - even in the second. The two terms reproducethe corresponding expressions for the operators in (C.2), but up to the upper bounds. Infact the upper bound in (C.8) extends from n − n . In the first line this is so duethe vanishing of A ( n ) n − = 0. Equivalently for j = n the two terms in (C.7) and theircontributions to (C.8) compensate each other due to the relation (3.17). Hence, takinginto account U T1 , k +1 = δ k, , we reproduce (C.1) for this particular example K ,n = [ U T ◦ KU ] ,n = (cid:88) n (cid:48) − odd U T1 ,m (cid:48) ◦ K m (cid:48) ,n (cid:48) U n (cid:48) ,n + (cid:88) n (cid:48) − even U T1 ,m (cid:48) ◦ K m (cid:48) ,n (cid:48) U n (cid:48) ,n . (C.9)The generalization of (C.7) for m < n reads A ( j ) n − m = (cid:88) s =0 ,...,ms − m =even (cid:18) js (cid:19) B ( s ) m A ( j − s ) n + ( − j (cid:88) s =0 ,...,m − s − m =odd (cid:18) js (cid:19) A ( s ) m B ( j − s ) n . (C.10)For general K m,n one proceeds as in the example K ,n considered above, distributingaccordingly the matrix powers to the left and right of ◦ K . The last step is to ensure theupper bounds as in (C.2). E.g., for odd m the upper bound in the initial expression (A.8)can be lifted to n adding m + 1 terms without violating (A.5) exploiting the vanishing ofthe coefficients A ( j ) n − m for j ≥ n − m + 1 + 2 k , k ∈ Z + . The upper bound in (3.19) can beextended to n adding m + 1 (for odd m ) terms without violating (A.5) using the fact thatthe coefficients A ( j ) n − m for j = n − m + 1 + 2 k, k ≥ 0. Moreover, the upper bound can beextended further to n + m − CM ), sothat to comply with the inequalities (A.5). In the process, some zeros may appear also ineach of the two terms in (C.10). Altogether this ensures the correct upper bounds of theoperators U in the transformed expresssion obtained using (C.10). D Proof of the exponential representation for (cid:96) = 0 Here we show that the matrix U defined in (3.21) is given by the series (3.12) for (cid:96) = 0.The matrix U and its transposed U T factorise as U = ˆ UP , ˆ U = (cid:16) P e e − σ CMS + P o e − σ SCM (cid:17) e iσ C = (cid:88) j ≥ [ ˆ U ] j σ j ; (D.1) U T = P ˆ U T , ˆ U T = e − iσ C (cid:16) e σ SMC P e + e σ MCS P o (cid:17) = (cid:88) j ≥ [ ˆ U T ] j σ j , (D.2)where P is the projector restricting the power j of σ of the matrix element [ ˆ U ] n (cid:48) ,n to n or n − U T , restricting ourselvesto the piece acting in the even sector. The coefficients X ( j ) = [ ˆ U T ] j P e in the expansion ofthe first term in (D.2) read j ! X ( j ) = j ! [ j ] (cid:88) k =0 β jk C j − k ( SMC ) k , β jk = ( − k k ( j − k )! k ! . (D.3)– 25 –n the other hand, the piece of the transposed matrix (3.12) restricted to the even sector, U Te = U T P e , is expanded as[ U Te ] m,m (cid:48) = (cid:88) j =0 ,...,mj − m =even ( − iσ ) j j ! B ( j ) m [( MC ) j ] m,m (cid:48) , m (cid:48) − even . (D.4)We have to show that B ( j ) m [( MC ) j ] m,m (cid:48) = j ! X ( j ) m,m (cid:48) . (D.5)Let us see how this works with the lowest coefficients j = 1 , 2. By the expression (3.17) forthe coefficients B ( j ) m , for j = 1 , B (1) m ( MC ) m,m (cid:48) = m ( MC ) m,m (cid:48) = C m,m (cid:48) ,B (2) m [( MC ) ] m,m (cid:48) = m [( MC ) ] m,m (cid:48) = m (cid:88) ± ± [ MC ] m ± ,m (cid:48) = (cid:88) ± ± (( m ± ∓ 1) [ MC ] m ± ,m (cid:48) = [ C − SMC ] m,m (cid:48) . (D.6)We see that the computation for j = 2 in the last line of (D.6) is reduced to that for j = 1.The general proof of (D.5) can be done by induction. For that we will use the followingrecursive formula for B ( j ) m B ( j +1) m = ( − j mA ( j ) m ± ∓ = m B ( j ) m ± ± (cid:88) p =1 ,...,jp =odd (cid:18) jp (cid:19) B ( j − p ) m ± A ( p )1 = m B ( j ) m ± ∓ (cid:88) p =1 ,...,jp =odd (cid:18) jp (cid:19) p − (cid:89) s =0 (1 − (2 s ) ) B ( j − p ) m ± (D.7)which is derived from the expansion A ( j ) m − n = ( − j j (cid:88) p =0 (cid:18) jp (cid:19) B ( p ) m A ( j − p ) n ( m > n ) . (D.8)taking into account that A ( p ) − = − A ( p )1 and A (2 p )1 = δ p, . From (D.7) we obtain a recursiveformula for the corresponding matrices, B ( j +1) m (( MC ) j +1 ) m,n = (cid:88) ± ± ( B ( j ) m ± [( MC ) j ] m ± ,n − (cid:88) p =1 ,...,jp =odd (cid:18) jp (cid:19) p − (cid:89) s =0 (1 − (2 s ) ) (cid:88) ± ( B ( j − p ) m ± [( MC ) j − p ( MC ) p ] m ± ,n (D.9)If we assume the relation (D.5), then (D.9) implies that the coefficients X ( j ) satisfy therecurrence relation (D.9) which takes the form( j + 1)! X ( j +1) = j ! CX ( j ) − j ( j − X ( j − ( SMC ) + A , (D.10)– 26 –here we wrote explicitly only the first two terms. If we can prove independently that(D.10) is satisfied, this would imply (D.5). To do that, let us first notice that the lhs of(D.10) equals the sum of the first two terms in the rhs. This follows from the explicit formof X ( j ) , eq. (D.3). Therefore to prove (D.5) it is sufficient to show that A = 0. One cancheck, after tedious algebra, that this is indeed the case. Let us only write down a basiccommutator used: [ S , ( SMC ) k ] = − k − (cid:88) r =0 a kr ( SMC ) k − r ( MC ) r +1 ,a kr = r (cid:89) s =1 (2 s − (cid:18) kr + 1 (cid:19) , a k = k. (D.11)One of the nice features of the exponential form (D.1) is that it renders the symplecticproperty C = UCU T almost obvious, UCU T = (cid:16) P e e − σ CMS + P o e − σ CSM (cid:17) PCP (cid:16) e σ SMC P e + e σ MCS P o (cid:17) = P e e − σ CMS P e σ CMS C + P o e − σ CSM P e σ CSM C = ( P e + P o ) PC = C . (D.12) References [1] B. 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