Form-factors of the finite quantum XY-chain

F orm-factors of the finite quan tum XY-c hain Nik olai Iorgo v Bogolyub ov Institute for Theoretical Ph ysics , Kiev 036 80, Ukraine E-mail: iorgov @bitp.kie v .ua Abstract. Explicit fac torized formulas for the matrix element s (form-facto rs) of the spin op erators σ x and σ y betw een the eige n vectors of the Hamiltonian of the finite quantum perio dic XY-chain in a tra nsv er se field w ere derived. The deriv atio n is based on the rela tio ns b et ween three mo dels: the mo del o f quantum XY-chain, Ising mo del on 2D lattice and N = 2 Baxter–B azhano v–Stro g ano v τ (2) -mo del. Due to these relations we transfer the formulas for the form-factors of the latter mo del recently obtained by the use o f separation o f v aria ble s method to the mo de l of quan tum XY- chain. Ho pefully , the for m ulas for the form-factors will help in ana lysis o f m ultip oint dynamic correlation functions a t a finite tempera ture. As an example, we re - deriv e the a symptotics o f tw o-p oint corre la tion function in the diso rdered pha se without the use of the T oeplitz determinants a nd the Wiener–Hopf factorization metho d. P A CS num b ers: 75.10 Jm, 75.1 0.Pq, 05 .50+q, 02.30 Ik Submitted to: J. Phys. A: Math. Gen. 1. In t roduction The quan tum XY-c hain is one of the simplest mo dels whic h is ric h enough from the p oin t of view of ph ysics and at the same time p ermits strict mathematical analysis. The study of this mo del w as started in [1] where it was rewritten in terms of f ermionic op erators b y means of the Jordan–Wigner transformation. Now this relation is a standa r d mean to study differen t pro perties (the sp ectrum of the Hamiltonian [1, 2 ], the correlatio n functions [3, 4, 5, 6, 7], the emptiness formation probabilit y [8], the en tanglemen t en trop y [9, 10, 11, 12, 13]) , quan tum quenc hes [1 4 ] in XY-chain. Although the Hamilto nian of the mo del is equiv alen t to the Hamiltonia n of a free fermionic system, the spin op erators σ x and σ y are expressed in terms of fermionic o perators in a non- local w ay . Th us the study of cor r elation functions of suc h o p erators is a non-trivial problem. F o r example, the correlation function h σ x 0 σ x d i can b e written through T o eplitz determinan t o f size d and the deriv ation of t he asymptotics d → ∞ requires [3 ] the use of Szeg¨ o theorem a nd the Wiener–Hopf factorization metho d. In this pap er w e pro pose an alternativ e w ay to study correlation functions of the XY-mo del: w e derive the formulas for the matrix elemen ts of spin op erators σ x and F orm-factors of the finite quantum XY-chain 2 σ y b et w een the eigenv ectors of the Hamiltonian of the finite quantum XY-chain in a transv erse field. These form ulas allow to obtain at least formal expression for m ultip oin t dynamic correlatio n functions at a finite temp erature. F or this aim, it is enough to insert t he resolution of iden tity o perator as a sum of pro jectors to the eigenspaces of Hamiltonian. Hop efully t he correlation f unctions in terms of these sums will b e more easily a na lyz ed. As an application of the form ulas f or form-fa cto r s, w e re-derive the asymptotics of correlation function h σ x 0 σ x d i at d → ∞ . The idea o f deriv ation of form- f actors of the quantum finite XY-chain is to use the relations b et w een three mo dels: the mo del of quantum XY-c hain, the Ising mo del o n 2D lattice and N = 2 Baxter–Bazhano v–Str o gano v (BBS) mo del [15, 16]. The relation b et w een the first and the second mo del w as o bse rv ed in [17] (the relation (16) in this pap er b et we en the energies of fermionic excitations of t hese tw o mo dels seems to b e new), the relation b et wee n the second and the third mo del w as found in [18, 19]. T he parameters of the mo dels are ( h, κ ), ( K x , K y ) and ( a, b ), resp ec tiv ely . Due to these relations we tra nsfer t he f o rm ulas for the f orm-factors of N = 2 BBS mo del, recen tly obtained [20, 21] by the use of separation of v ariables metho d, to the mo del of quan tum XY-c hain. The main form ula s are (31), (32) together with (20), (2 9 ), ( 1 6). The separation of v ariables metho d for the quantum in tegrable systems (with basic example b eing the T oda c ha in) was in tro duced by Skly anin [22] and further dev elop ed b y Kharc hev and Leb edev [23]. In [19], t his metho d w as adapted for BBS mo del (a Z N - symmetric quan tum spin system) to obtain the eigen v ectors of this mo del. A t N = 2 and sp ecial v alues of pa rameters the BBS mo del reduces [18, 19, 24] to the Ising mo del. The eigen ve ctors o f transfer-matrix of Ising mo del obtained b y the separation of v ariables metho d allo w ed [20, 21] to prov e the conjectural form ula [25, 2 6] for the matrix elemen ts of spin op erator for finite Ising mo del. This deriv ation ha d pro vided first pro of of the form ula. A summarizing o v erview of the results on separatio n o f v ariables for BBS mo del is given in [24 ]. It is in teresting tha t f a ctorized formulas fo r the matrix elemen ts of spin op erators exist also for sup erin tegrable Z N -symmetric c hiral P otts quan tum c ha in [27, 28]. In Sect. 2 w e remind the definition of the finite quantum XY-c hain in a transv erse field, its phase diag ram, eigen v alues of the Hamiltonian and giv e general commen ts on the mat r ix elemen ts of spin op erators b et w een the eigen vectors of the Hamiltonian. Sect. 3 is dev oted to the description of relations b et w een three mo dels: the mo del of quan tum XY-c hain, the Ising mo del on 2D lattice and N = 2 Baxter–Bazhanov – Stroganov mo del. Using these relat io ns, in Sect. 4 we deriv e formulas for the matrix elemen ts (form-f a ctors) of the spin op erators σ x and σ y b et w een the eigen v ectors of the Hamiltonian of the finite quantum XY-chain. In Sect. 5, t hese formulas are rewritten to the case of chain of infinite length. In Sect. 6, as an application of the form ulas for form-factors, w e r e- deriv e the asymptotics [3] of correlation function h σ x 0 σ x d i a t d → ∞ without the use of the T o eplitz determinan ts and Wiener–Hopf fa cto r izat io n metho d. F orm-factors of the finite quantum XY-chain 3 2. Definition of the finite quan tum XY-c hain in a tr an sv erse field 2.1. The Ham iltonian an d phase diagr am The Hamiltonian of the XY-chain of length n in a tra nsve rse field h is [1, 2] H = − 1 2 n X k =1  1 + κ 2 σ x k σ x k +1 + 1 − κ 2 σ y k σ y k +1 + h σ z k  , (1) where σ i k are Pauli matrices, κ is the anisotropy . In the case κ = 0 w e get XX-chain (isotropic case). The v alue κ = 1 cor r es p onds to t he quantum Ising chain in a transv erse field. In what follows we restrict ourselv es to the case κ > 0, h ≥ 0. O ther signs of κ and h can b e obtained using a utomorphisms of the algebra of Pauli matrices. Also w e will supp ose the p erio dic b oundary condition σ i k = σ i k + n . In [29] it is sho wn that the form ulas for the matrix elemen t s of spin op erators obtained in Sect. 4 are also a pplicable for the an tip erio dic b oundary condition σ x k + n = − σ x k , σ y k + n = − σ y k and σ z k + n = σ z k . No w ab out the v alues of h . Due to the relatio n of XY c hain with 2D Ising mo del, whic h will b e discussed in the next section, the coupling constan t h plays the role of a temp erature-lik e v ariable. The v alue h > 1 corresp onds to the paramag netic (disordered) phase. The v alue 0 ≤ h < 1 corresp onds to the f erromagnetic (ordered) phase. If 0 ≤ h < (1 − κ 2 ) 1 / 2 , it is oscillatory regio n (b ecause of oscillatory b eha vior of tw o-p oin t correlation function). Another p ec uliarity related to this regio n is t he follo wing. A t fixed κ , 0 < κ ≤ 1, in the regio n where h > (1 − κ 2 ) 1 / 2 the NS- v acuum energy is low er t han R-v acuum energy (a sy mptotically , at n → ∞ , they b ecome coinciding). In the r egio n 0 ≤ h ≤ (1 − κ 2 ) 1 / 2 there are in t ers ections at sp ecial v alues of h of these v acuum lev els ev en at finite n . The num b er of these in t ersections grow s with n . F or a detailed analysis o f t he oscillatory region see [3, 30]. In this pap er we deriv e the form ulas for the matrix elemen ts in paramagnetic phase ( h > 1, 0 < κ < 1) and add commen ts on the mo dification of t he form ulas for o ther v alues of parameters. 2.2. Eigenvalues and e i g enve ctors of the Hamiltonian of XY-chain Using Jordan– Wig ner and Bogoliub o v transformations the Hamilto nia n H of the XY- c hain can b e rewritten a s the Hamiltonian o f the system of f ree fermions and diagonalized [1, 2]. The relation b et wee n energies ε ( q ) and momen ta q of the fermionic excitations is ε ( q ) =  ( h − cos q ) 2 + κ 2 sin 2 q  1 / 2 , q 6 = 0 , π , (2) ε (0) = h − 1 , ε ( π ) = h + 1 . The Ha milto nian H comm utes with the op erator V = σ z 1 σ z 2 · · · σ z n . Since V 2 = 1 , the eigen v ectors ar e separated to tw o sectors with resp ect to the eigen v alue of V . Belo w the sign + / − in fro nt o f ε ( q ) in the expression for energies E cor r es p onds to the absence/presence of the fermionic excitation with the momentum q . Eac h suc h excitation carries the energy ε ( q ). F orm-factors of the finite quantum XY-chain 4 • NS–sector: V → +1, the fermionic excitations ha v e “half-inte ger” quasimomen ta q ∈ NS =  2 π n ( j + 1 / 2) , j ∈ Z n  ⇒ E = − 1 2 X q ∈ NS ± ε ( q ) . This sector includes the states only with an ev en n um b er of excitations. • R–sector: V → − 1, the fermionic excitations hav e “integer” quasimomen t a q ∈ R =  2 π n j , j ∈ Z n  ⇒ E = − 1 2 X q ∈ R ± ε ( q ) . (3) In the paramagnetic phase this sector includes the states only with an o dd n um b er of excitations. In the ferromagen tic phase (0 ≤ h < 1) it is natural to r e-define the energy of zero-momen tum excitation as ε (0) = 1 − h to b e p ositiv e. F ro m the form ula (3) for energy E , this c ha nge of the sign of ε (0) in the ferromagnetic phase leads to a formal c hange b et w een absence/presence of zero- momen t um excitation in the lab elling of eigenstates. Th us although t he analytical expressions for energies E in terms of h a nd κ are the same in b oth phases, b ecause of the redefinition of ε (0) in the case of 0 ≤ h < 1 the n umber of excitations in the ferromagnetic phase is ev en. W e will denote the eigenstates | Φ i α model b y the set of v alues o f the excited quasi- momen ta Φ = { q 1 , q 2 , . . . , q L } , the lab el o f the sector α = NS or R and t he lab el of the mo del. F or example, the state in R -sector of the quan tum XY-c hain with n = 3 sites in the paramagnetic phase with all p ossible quasi-momen ta Φ = { 0 , 2 π / 3 , − 2 π / 3 } exc ited is | 0 , 2 π / 3 , − 2 π / 3 i R X Y . It has the energy E = − 1 2 (1 − h − ε (2 π / 3) − ε ( − 2 π/ 3)) . The same form ula for the energy in the ferromag netic phase corresp onds to the state | 2 π / 3 , − 2 π / 3 i R X Y . 2.3. Matrix elements o f s p in op er ators F or ma lly in order to calculate an y cor r elation function for XY-c hain it is sufficien t to find the matrix elemen ts of spin op erators σ x k , σ y k and σ z k b et w een the eigenstates of Hamiltonian H . • Matrix elements of σ z k . The op erator σ z k comm utes with V = σ z 1 σ z 2 · · · σ z n . Therefore the action of σ z k do es not c hange the sector. In f act the op erator σ z k can b e presen ted a s a bilinear com binatio n of o p erators o f creation and a nnihilat io n of the fermionic excitations. Hence the matrix elemen ts of σ z k b et w een eigen vec tors o f H can b e calculated easily (most of them are 0). W e will not consider suc h matrix elemen ts in this pap er. • Matrix elements of σ x k and σ y k . The op erators σ x k and σ y k an t icommute with V = σ z 1 σ z 2 · · · σ z n . Therefore their action c hanges the sector. The o perator s σ x k and σ y k can not b e presen ted in terms of F orm-factors of the finite quantum XY-chain 5 fermionic op erators in a lo cal w a y . All the matr ix elemen ts of them b et w een the eigen ve ctors of H fro m different sectors are non-zero! The aim of this pap er is to derive explicit factorized formula for the mat r ix elemen ts of σ x k and σ y k . The idea is to relate three models: the quantum XY-c hain in a transv erse field, the Ising mo del on 2D lattice and N = 2 BBS mo del. The relation b etw een the first and the second mo del is based on the observ ation by M. Suzuki [17]. The relation b et w een the sec ond a nd the third is based on [18]. The latter r elat io n tog ethe r with the results on separation of v ariables fo r BBS mo del allo w ed [21] to prov e the form ulas for the matrix elemen ts of spin op erator o f Ising mo del found by A. Bugrij and O. Liso vyy [25, 26]. In this pap er w e transfer these results on the matrix elemen ts to the case of XY-c hain. The parameters of these three mo dels are ( h, κ ), ( K x , K y ) and ( a, b ), resp ec tiv ely . 3. Relation b et ween three mo dels 3.1. R el a tion b etwe en quantum XY-chain and the Ising mo de l on a lattic e The ro w-to -ro w transfer-matrix of the tw o-dimensional Ising mo del with parameters K x and K y can b e c hosen as t XY := T 1 / 2 1 T 2 T 1 / 2 1 = exp n X k =1 K ∗ y 2 σ z k ! exp n X k =1 K x σ x k σ x k +1 ! exp n X k =1 K ∗ y 2 σ z k ! , (4) where the spin configurat io ns of the ro ws are c hosen to b e lab eled b y the eigen vec tors of the op erators σ x k , the parameter K ∗ y is dual to K y , that is tanh K y = exp( − 2 K ∗ y ), and T 1 = exp n X k =1 K ∗ y σ z k ! , T 2 = exp n X k =1 K x σ x k σ x k +1 ! . (5) In [17], M. Suzuki observ ed that if w e c ho ose K x and K ∗ y suc h that tanh 2 K x = √ 1 − κ 2 h , cosh 2 K ∗ y = 1 κ (6) then t he Hamiltonian (1) of XY-c hain will commute with the transfer-ma t rix of the 2D Ising mo del (4) and these tw o op erators ha v e a common set of eigen vec tors. 3.2. N = 2 BBS mo d e l and its r elation to the I sing mo del T o define N = 2 BBS mo del w e use the f ollo wing L -op erator ‡ [16, 3 1 ] L k ( λ ) = 1 + λ σ z k λ σ x k ( a − b σ z k ) σ x k ( a − b σ z k ) λa 2 + σ z k b 2 ! , (7) dep ending on parameters a , b and spectral parameter λ . I t satisfies the Y ang–Ba xter equation with (tw isted) quantum trigonometric R -matrix. In particular it means that ‡ In comparis on with [2 1] we interc hanged σ x k and σ z k in the L -op erator (7). It is just a nother representation of the W eyl algebra en tering the definition of L -op erator. F orm-factors of the finite quantum XY-chain 6 the eigen ve ctors of the transfer ma t r ix t ( λ ) = tr L 1 ( λ ) L 2 ( λ ) · · · L n ( λ ) built from suc h L -op erators are indep ende n t of λ . Fixing the spectral parameter to the v alue λ = b/a , the L -op erator (7) degenerates L k ( b/a ) = (1 + σ z k b/a ) 1 a σ x k !  1 , b σ x k  and the transfer matr ix t ( λ ) can b e put in to a nonsymme tric Ising form t ( b/a ) = n Y k =1 (1 + σ z k · b/a ) · n Y k =1 (1 + σ x k σ x k +1 · a b ) = (cosh K x cosh K ∗ y ) − n T 1 T 2 , t ( b/a ) ∼ T 1 T 2 = exp  P n k =1 K ∗ y σ z k  exp  P n k =1 K x σ x k σ x k +1  , (8) if w e use p erio dic bo undary condition σ i n + k = σ i k and iden t if y e − 2 K y = tanh K ∗ y = b/a , tanh K x = ab . (9) Th us at λ = b/a w e get the transfer-matrix of the Ising mo del. If we do not fix t he sp ectral parameter to this sp ecial v alue, w e shall talk o f the “generalized Ising mo del”. Ho wev er, tra nsfer matrix eigenstates are independen t of the c hoice of λ . In [1 9 , 20, 21] the eigen ve ctors for the no nsymmetric transfer mat r ix (8) and the matrix elemen t s of σ x k b et w een these eigenv ectors were deriv ed using metho d of separation of v ar iba les . Comparing (6 ) and ( 9) w e get the following simple relations f or the par ameters of XY-mo del and special BBS-mo del with L -op erator (7): κ = a 2 − b 2 a 2 + b 2 , h = 1 + a 2 b 2 a 2 + b 2 . (10) 3.3. R el a tion b etwe en the ener gies of e x citations for the Ising mo del and XY-chain In the previous subsections w e ha ve sho wn ho w the quan tum XY-c ha in in a transv erse field, the Is ing mo del on 2D lattice and N = 2 BBS mo del are related. The parameters of the mo dels are ( h, κ ), ( K x , K y ) and ( a, b ), resp ec tiv ely . The relations b et w een these pairs of parameters a r e give n b y (6), (9) and (10). Since the t r a nsfe r-matrices t XY = T 1 / 2 1 T 2 T 1 / 2 1 , t Is = T 1 / 2 2 T 1 T 1 / 2 2 , T 1 T 2 of 2D Ising mo del are related by similarity transformations, the en umeration of the eigenstates of all these transfer-matrices is the same as describ ed in Sect. 2.2 for the quantum XY-chain. They will b e denoted, respective ly , b y | Φ i α XY , | Φ i α Is , | Φ i α , where Φ = { p 1 , p 2 , . . . , p L } is the set o f v alues of the excited quasi-momen ta Φ = { p 1 , p 2 , . . . , p L } and α = NS or R is the lab el of the sector. Their eigen v alues e − γ Φ are the same: t XY | Φ i α XY = e − γ Φ | Φ i α XY , t Is | Φ i α Is = e − γ Φ | Φ i α Is , T 1 T 2 | Φ i α = e − γ Φ | Φ i α , (11) γ Φ = L X l =1 γ ( p l ) − 1 2 X p ∈ α γ ( p ) , (12) cosh γ ( p ) = ( t x + t − 1 x )( t y + t − 1 y ) 2( t − 1 x − t x ) − t − 1 y − t y t − 1 x − t x cos p , (13) F orm-factors of the finite quantum XY-chain 7 t x = tanh K x , t y = tanh K y . The eigen v alues of t he transfer matrix t ( λ ) of the BBS mo del with L -op erator (7) are prop ortional to Q p ( λ ± s p ) (see form ula (68) of [2 0 ]), s p =  b 4 − 2 b 2 cos p + 1 a 4 − 2 a 2 cos p + 1  1 / 2 , (14) where the sign + / − in the front of s p in the expression for the eigen v alues of t ( λ ) corresp onds to the absence/prese nce of the fermionic excitation with the momen tum p . The momen tum p runs ov er the same sets as in the case of the quan tum XY-c hain. Due to (8), (11) and (12 ) we ha ve the relation b et w een γ ( p ) and s p (see [21]): e γ ( p ) = as p + b as p − b (15) and a relation b et w een ε ( p ) and γ ( p ): using (1 0 ), (14) a nd (15 ) we get sinh γ ( p ) = 2 abs p a 2 s 2 p − b 2 = 2 ab ( a 2 − b 2 )(1 − a 2 b 2 ) p ( b 4 − 2 b 2 cos p + 1 )( a 4 − 2 a 2 cos p + 1) = 2 ab ( a 2 + b 2 ) ( a 2 − b 2 )(1 − a 2 b 2 ) ε ( p ) = √ 1 − κ 2 κ √ κ 2 + h 2 − 1 ε ( p ) . (16) The existence of the r elat io n b et w een γ ( p ) and ε ( p ) is surprising b ecause the commu- tativit y of the Hamiltonian (1) of the XY-c ha in and the t r ansfer matr ix (4) of the 2D Ising mo del do es not imply a priori an y relation b et w een their eigen v alues. 3.4. Uniformization of the disp ersion r elation (1 3 ) W e us e a parametrization of the dispersion relation (13) of the 2D Ising mo del in terms of elliptic function at h > 1, 0 < κ < 1 whic h corr esp onds to the parama g netic phase of Ising mo del. This parametrization is a mo dification o f parametrization from [32] giv en for the ferromagnetic phase of Ising mo del and corresp onding to 0 < h < 1, (1 − h 2 ) 1 / 2 < κ < 1 for XY-c ha in. W e introduce the mo dulus of elliptic curv e k − 1 b y k − 1 = sinh 2 K x sinh 2 K y = κ / √ κ 2 + h 2 − 1 = ( a 2 − b 2 ) / (1 − a 2 b 2 ) . In the paramag netic phase w e ha ve 0 ≤ k − 1 < 1. Complete elliptic in tegra ls f or k − 1 and for the supplemen tary modulus are K = K ( k − 1 ) and K ′ = K ((1 − k − 2 ) 1 / 2 ), respectiv ely . W e define real parameter a , 0 < a < K ′ / 2, b y one of the equiv alen t relations 1 / sinh 2 K x = i k / sn(2 i a , k − 1 ) , 1 / sinh 2 K y = − i sn(2i a , k − 1 ) . The following tw o elliptic functions λ ( u ) = sn( u − i a , k − 1 ) / sn( u + i a , k − 1 ) , z ( u ) = k − 1 sn( u − i a , k − 1 ) sn( u + i a , k − 1 ) satisfy the relation sinh 2 K x ( z + z − 1 ) + sinh 2 K y ( λ + λ − 1 ) = 2 cosh 2 K x cosh 2 K y . (17) F orm-factors of the finite quantum XY-chain 8 T o pro v e it w e note that t he left-hand side of (17) is a n elliptic function without p oles (the p oles at ± a and ± a + i K ′ are canceled) and therefore it is a constan t. Thus it is sufficien t to establish v alidit y of (1 7 ) at u = 0. F or this end w e use λ (0) = − 1, z (0) = − k − 1 sn 2 (i a , k − 1 ) = − k 1 − dn(2i a , k − 1 ) 1 + cn(2i a , k − 1 ) , z − 1 (0) = − k 1 + dn(2i a , k − 1 ) 1 − cn (2i a , k − 1 ) whic h follo w from the formulas of Example 6, Sect. 22.21 of [33], and cosh 2 K x = dn(2i a , k − 1 ) , cosh 2 K y = i cn(2i a , k − 1 ) / sn(2i a , k − 1 ) . The relation (17) coincides with the disp ersion relation (13) if one iden tifies z ( u ) = e − i p and λ ( u ) = e − γ ( p ) . The par a mete r u on the elliptic curv e is an analogue of rapidit y . No w if p runs from − π to π then u runs along the segmen t from i K ′ / 2 to 2 K + i K ′ / 2. There is a nother disp ersion relatio n corresp onding to the ev o lution in the transv erse direction on the Ising lat t ice: cosh ¯ γ ( ¯ p ) = ( t x + t − 1 x )( t y + t − 1 y ) 2( t − 1 y − t y ) − t − 1 x − t x t − 1 y − t y cos ¯ p . (18) It is uniformized by λ ( u ) = e i ¯ p and z ( u ) = e − ¯ γ ( ¯ p ) . Now if ¯ p runs from − π to π t hen u runs along the segmen t fro m 0 to 2 K . F ro m (10 ) , we ha ve a 2 = h − √ h 2 + κ 2 − 1 1 − κ , b 2 = h − √ h 2 + κ 2 − 1 1 + κ . (19) The p oin ts with z = a ± 2 and z = b ± 2 are the branchin g p oin ts of the sp ectral curv e (17) considered as λ ( z ). The parameters a 2 , b 2 corresp ond respectiv ely to λ 2 , λ − 1 1 of [3] and to α − 1 1 , α − 1 2 of [32]. W e hav e also a 2 = e − ¯ γ (0) , b 2 = e − ¯ γ ( π ) . 4. F orm ulas for the matrix elemen ts of spin op erators In this section w e will deriv e the formulas for the matrix elemen ts of spin op erators for quan tum XY-c ha in of finite length. The deriv ation for the basic region of parameters h > 1, 0 < κ < 1 is giv en in Sect. 4.1. The form ula s for other v alues of parameters can b e obtained b y analytic contin uation. The details of the con t inuation are give n in the follo wing subsections. 4.1. Par a m agnetic phase: h > 1 , 0 < κ < 1 W e use the Bugrij–Lisovyy formula ((4 0 ) of [26 ]) for the mat r ix elemen t of spin op erator b et w een the eigenv ectors | Φ 0 i Is = | q 1 , q 2 , . . . , q K i NS Is and | Φ 1 i Is = | p 1 , p 2 , . . . , p L i R Is of the symmetric tra nsfe r ma t r ix t Is = T 1 / 2 2 T 1 T 1 / 2 2 for t he finite 2D Ising mo del (the states are lab eled b y the momen ta of excited fermions as it is explained in Sects. 2.2 and 3.3): Ξ Φ 0 , Φ 1 = | NS Is h q 1 , q 2 , . . . , q K | σ x m | p 1 , p 2 , . . . , p L i R Is | 2 F orm-factors of the finite quantum XY-chain 9 = ξ ξ T K Y k =1 Q NS q 6 = q k sinh γ ( q k )+ γ ( q ) 2 n Q R p sinh γ ( q k )+ γ ( p ) 2 L Y l =1 Q R p 6 = p l sinh γ ( p l )+ γ ( p ) 2 n Q NS q sinh γ ( p l )+ γ ( q ) 2 ·  t y − t − 1 y t x − t − 1 x  ( K − L ) 2 / 2 × K Y k 0, then the co efficien t α L Φ is determined from the requiremen t XY h Φ | = | Φ i † XY . W e hav e also α L Φ α R Φ = h Φ | Φ i . Since h Φ | Φ i is real at real a and b , b oth α L Φ and α R Φ are real to o. The matrix elemen ts of BBS mo del are related to t he matrix elemen ts of XY-c ha in b y h Φ 0 | σ x m | Φ 1 i = α L Φ 0 α R Φ 1 · XY h Φ 0 | T − 1 / 2 1 σ x m T 1 / 2 1 | Φ 1 i XY (25) F orm-factors of the finite quantum XY-chain 10 = α L Φ 0 α R Φ 1 · e γ Φ 0 − γ Φ 1 XY h Φ 0 | T 1 / 2 1 σ x m T − 1 / 2 1 | Φ 1 i XY , (26) where the last relation follow s from the facts that | Φ i XY is eigen v ector of T 1 / 2 1 T 2 T 1 / 2 1 with the eigen v alue e − γ Φ (see (11)) and T 2 comm utes with σ x m . Complex conjugation of (26) together with (25) with in terc hanged Φ 0 and Φ 1 giv e h Φ 0 | σ x m | Φ 1 i = α L Φ 0 α R Φ 1 α L Φ 1 α R Φ 0 · e γ Φ 0 − γ Φ 1 XY h Φ 0 | T 1 / 2 1 σ x m T − 1 / 2 1 | Φ 1 i XY . (27) F ro m the other side, the formu las (5 6 ) and (57) of [21] giv e the f ollo wing f actorized presen tations for the matrix elemen t b et we en eigenstates of the tra nsfe r matrix T 1 T 2 (see fo otnote on p. 5): h Φ 0 | σ x m | Φ 1 i = f 1 ( b ) f 2 ( b 2 ) , h Φ 1 | σ x m | Φ 0 i = f 1 ( − b ) f 2 ( b 2 ) , where f 1 ( b ) is a r eal (we supp ose that a and b a r e real) and f 2 ( b 2 ) is a complex but in v ariant with respect to b → − b . Therefore h Φ 0 | σ x m | Φ 1 i h Φ 1 | σ x m | Φ 0 i = f 1 ( b ) f 1 ( − b ) =: C Φ 0 , Φ 1 . (28) It is easy to prov e using explicit form ula s for f 1 ( b ) from [21] that C Φ 0 , Φ 1 = Q p ∈ R e γ ( p ) / 2 Q q ∈ NS e γ ( q ) / 2 Q K k =1 e γ ( q k ) Q L l =1 e γ ( p l ) = e γ Φ 0 − γ Φ 1 (29) for | Φ 0 i = | q 1 , q 2 , . . . , q K i NS and | Φ 1 i = | p 1 , p 2 , . . . , p L i R . L et us consider, fo r example, the case of o dd n (the length of the c hain) and σ 0 = σ π ( σ q = 0/ σ q = 1 corresp onds to absence/presence of fermion excitation with momen tum q ). F rom Eq. ( 5 6) and the discussion in Section 6.1 o f [2 1 ] w e can choose f 1 ( b ) = (( − 1) σ 0 − a b ) Y k ∈ ˇ D (( − 1) k b + as π k/n ) Y k ∈ ˆ D (( − 1) k b − as π k/n ) , where s p is giv en b y (14) and the set ˆ D (resp. ˇ D ) consists of suc h k from { 1 , 2 , . . . , n − 1 } for whic h the fermions with b oth momen ta ± πk /n are excited (resp. not excited) in the states | Φ 0 i and | Φ 1 i . Using (15) we get C Φ 0 , Φ 1 = f 1 ( b ) f 1 ( − b ) = ( − 1) σ 0 − a b ( − 1) σ 0 + a b · Q p ∈ R ,p 6 =0 e γ ( p ) / 2 Q q ∈ NS ,q 6 = π e γ ( q ) / 2 · Q K k =1 ,q k 6 = π e γ ( q k ) Q L l =1 ,p l 6 =0 e γ ( p l ) . Th us it remains to v erify that the first fraction also fits (29) as contribution o f the momen ta 0 and π . F or this end we need just to tak e in to accoun t (14), (15) and s 0 = b 2 − 1 a 2 − 1 , s π = b 2 + 1 a 2 + 1 , e ( γ (0) − γ ( π )) / 2 = 1 − a b 1 + a b . The correct sign of the lat ter f orm ula can b e fixed from the limit b = 0 (the quantum Ising c hain limit) and then taking limit a → 0 (it corresp onds to the limit o f strong external field h → ∞ ). All the other three cases of o dd (ev en) n and σ 0 = σ π ( σ 0 6 = σ π ) can b e analyzed similarly . It prov es (29). F orm-factors of the finite quantum XY-chain 11 T aking into accoun t (2 4) and (28) w e get h Φ 0 | σ x m | Φ 1 i ( h Φ 0 | Φ 0 ih Φ 1 | Φ 1 i ) 1 / 2 = e i δ Φ 0 , Φ 1 ( C Φ 0 , Φ 1 Ξ Φ 0 , Φ 1 ) 1 / 2 , (30) where δ Φ 0 , Φ 1 is a phase related to a pa rticular normalizatio n o f eigen v ectors. T o relate these matrix elemen ts to the mat rix elemen ts of XY-c hain we o bse rv e that ( 2 7) and (28) imply α L Φ 0 α R Φ 1 = α L Φ 1 α R Φ 0 . Since α L Φ 0 α R Φ 0 α L Φ 1 α R Φ 1 = h Φ 0 | Φ 0 ih Φ 1 | Φ 1 i w e obtain α L Φ 0 α R Φ 1 = ( h Φ 0 | Φ 0 ih Φ 1 | Φ 1 i ) 1 / 2 . Thus from (25), (26), (29) and (30 ) we deriv e XY h Φ 0 | T − 1 / 2 1 σ x m T 1 / 2 1 | Φ 1 i XY = XY h Φ 0 |  σ x m cosh K ∗ y − i σ y m sinh K ∗ y  | Φ 1 i XY = e i δ Φ 0 , Φ 1 ( C Φ 0 , Φ 1 Ξ Φ 0 , Φ 1 ) 1 / 2 , XY h Φ 0 | T 1 / 2 1 σ x m T − 1 / 2 1 | Φ 1 i XY = XY h Φ 0 |  σ x m cosh K ∗ y + i σ y m sinh K ∗ y  | Φ 1 i XY = e i δ Φ 0 , Φ 1  C − 1 Φ 0 , Φ 1 Ξ Φ 0 , Φ 1  1 / 2 . Finally taking appropriate linear com binations of these tw o form ulas w e get the main result o f the pap er: t he matrix elemen ts of spin op erators b et we en the eigen v ectors | Φ 0 i XY = | q 1 , q 2 , . . . , q K i NS XY from the NS-sector and | Φ 1 i XY = | p 1 , p 2 , . . . , p L i R XY from the R-sector of the Hamiltonian (1) of XY-c hain are | XY h Φ 0 | σ x m | Φ 1 i XY | 2 = κ 2(1 + κ )  C 1 / 2 Φ 0 , Φ 1 + C − 1 / 2 Φ 0 , Φ 1  2 Ξ Φ 0 , Φ 1 = 2 κ 1 + κ cosh 2 γ Φ 0 − γ Φ 1 2 Ξ Φ 0 , Φ 1 , (31) | XY h Φ 0 | σ y m | Φ 1 i XY | 2 = κ 2(1 − κ )  C 1 / 2 Φ 0 , Φ 1 − C − 1 / 2 Φ 0 , Φ 1  2 Ξ Φ 0 , Φ 1 = 2 κ 1 − κ sinh 2 γ Φ 0 − γ Φ 1 2 Ξ Φ 0 , Φ 1 , (32) where Ξ Φ 0 , Φ 1 , C Φ 0 , Φ 1 and γ Φ are giv en by (20), (29), (12) and (16). W e also used sinh K ∗ y = b √ a 2 − b 2 = r 1 − κ 2 κ , cosh K ∗ y = a √ a 2 − b 2 = r 1 + κ 2 κ . (33) 4.2. F err omagnetic p h ase: 0 < κ < 1 , √ 1 − κ 2 < h < 1 Let us give a g eneral commen t o n t he contin uation o f the formulas from the region h > 1 to region 0 ≤ h < 1 . F ormally , all the formulas fo r matr ix elemen ts of spin op erators are correct for the paramagnetic phase where h > 1 and for the f erro magnetic phase whe re 0 ≤ h < 1. But for the case 0 ≤ h < 1 it is natural to c hange the sign of ε (0 ) (and also of γ (0 ) ) o f zero-momen tum excitation to b e p ositiv e: ε (0 ) = 1 − h . F rom (3), this ch ange of t he sign of ε (0) (a nd of γ (0 )) in the ferromagnetic phase leads to a formal c hang e b et w een absence/presen ce of zero-momen tum excitation in the lab elling of eigenstates. Therefore the nu m b er of the excitations in each sector ( NS and R) b ecomes ev en. Direct calculation shows that the c hange o f the sign of γ (0) in (31) and (32) can b e absorb ed F orm-factors of the finite quantum XY-chain 12 to obtain formally the same form ula s (31) and (32) but with new γ (0), ev en L (the n umber of the excitations in R - sec tor) and new ξ = (1 − k 2 ) 1 / 4 = ((1 − h 2 ) / κ 2 ) 1 / 4 . The explicit form ulas for the matrix elemen ts for the r e gion of p ar ameters 0 < κ < 1, √ 1 − κ 2 < h < 1 are giv en [37]. 4.3. R e gi o n κ > 1 F ro m the relation (16) b etw een the energies of XY-chain and Ising mo del excitations it follo ws that the energies γ ( p ) of the Ising mo del excitations b ecome complex and it is useful to in tro duce γ ( p ) = i ˜ γ ( p ) suc h that sin ˜ γ ( p ) = √ κ 2 − 1 κ √ κ 2 + h 2 − 1 ε ( p ) . Here ˜ γ ( p ) should b e c hosen to b e monotonically increasing function of p when p runs from 0 to π . If h ≥ κ 2 − 1, the energy ε ( p ) is mo no t onically increasing function of p with minim um at p = 0 and maxim um at p = π . In this case 0 < ˜ γ ( p ) ≤ π / 2. If 0 ≤ h < κ 2 − 1, the energy ε ( p ) is non-monotonical function having additional extrem um (maxim um) a t p = p c , cos p c = h/ ( 1 − κ 2 ), ˜ γ ( p c ) = π / 2. In this case 0 < ˜ γ ( p ) < π / 2 for 0 ≤ p < p c and ˜ γ ( p ) > π / 2 for p c < p ≤ π . W e contin ue analytically t he formulas (3 1) and (32) a nd write them in terms of ˜ γ ( p ). All the c hanges in t he final form ulas are the following: | XY h Φ 0 | σ x m | Φ 1 i XY | 2 = 2 κ κ + 1 cos 2 ˜ γ Φ 0 − ˜ γ Φ 1 2 ˜ Ξ Φ 0 , Φ 1 , | XY h Φ 0 | σ y m | Φ 1 i XY | 2 = 2 κ κ − 1 sin 2 ˜ γ Φ 0 − ˜ γ Φ 1 2 ˜ Ξ Φ 0 , Φ 1 , where ˜ Ξ Φ 0 , Φ 1 is giv en b y (20) with substitutions sinh γ ( p ) + γ ( q ) 2 → sin ˜ γ ( p ) + ˜ γ ( q ) 2 , t y − t − 1 y t x − t − 1 x → κ 2 − 1 κ √ κ 2 + h 2 − 1 and ˜ γ Φ is give n by (12) with γ ( p ) → ˜ γ ( p ). Also, for 0 ≤ h < 1, one has to tak e in to accoun t the mo difications related to the zero mo de describ ed in Sect. 4 .2 . 4.4. Oscil la tory r e gion 0 < κ < 1 , 0 ≤ h < √ 1 − κ 2 Similarly to the region κ > 1, the energies γ ( p ) of the Ising mo del excitations b ec ome complex and it is useful to rewrite the matrix elemen ts of spin op erators in terms of ˜ γ ( p ) = γ ( p ) + i π / 2: cosh ˜ γ ( p ) = √ 1 − κ 2 κ √ 1 − κ 2 − h 2 ε ( p ) . Here ˜ γ ( p ) should b e c hosen to b e monotonically increasing function of p when p runs from 0 to π . If 1 − κ 2 ≤ h < √ 1 − κ 2 , the energy ε ( p ) is monotonically increasing function of p with minimum at p = 0 and maxim um at p = π . In this case ˜ γ ( p ) ≥ 0. If 0 ≤ h < 1 − κ 2 , the energy ε ( p ) is non-mono tonical function having additional extrem um F orm-factors of the finite quantum XY-chain 13 (minim um) at p = p c , cos p c = h/ (1 − κ 2 ), ˜ γ ( p c ) = 0. In this case ˜ γ ( p ) < 0 f o r 0 ≤ p < p c and ˜ γ ( p ) > 0 for p c < p ≤ π . W e contin ue analytically the form ula s (31 ) and (32) and write them in terms of ˜ γ ( p ). All the c hanges in the final formu las are the following: | XY h Φ 0 | σ x m | Φ 1 i XY | 2 = κ 2(1 + κ )  e ˜ γ Φ 0 − ˜ γ Φ 1 2 + ( − 1) K − L 2 e − ˜ γ Φ 0 − ˜ γ Φ 1 2  2 ˜ Ξ Φ 0 , Φ 1 , | XY h Φ 0 | σ y m | Φ 1 i XY | 2 = κ 2(1 − κ )  e ˜ γ Φ 0 − ˜ γ Φ 1 2 − ( − 1) K − L 2 e − ˜ γ Φ 0 − ˜ γ Φ 1 2  2 ˜ Ξ Φ 0 , Φ 1 , where ˜ Ξ Φ 0 , Φ 1 is giv en b y (20) with ξ = (1 − k 2 ) 1 / 4 = ((1 − h 2 ) / κ 2 ) 1 / 4 (since it is the ferromagnetic phase) a nd with the repacemen ts sinh γ ( p ) + γ ( q ) 2 → cosh ˜ γ ( p ) + ˜ γ ( q ) 2 , t y − t − 1 y t x − t − 1 x → 1 − κ 2 κ √ 1 − κ 2 − h 2 , ˜ γ Φ is giv en by (1 2) with γ ( p ) → ˜ γ ( p ). Also o ne has to tak e in to accoun t the mo difications related to the zero mo de describ ed in Sect. 4.2. 4.5. Other values o f p a r ameters 4.5.1. Quantum Isin g chain: κ = 1 . In the case of the quan tum Ising chain ( κ = 1) the form ula for the matrix elemen ts of spin op erator σ x m can b e deriv ed by a limiting pro cedure. The final for m ula w as derive d in [21, 34] and it is expressed in terms of the energies of excitations ε ( q ). In the case of general XY-c hain we w ere not able to find an analogous form ula for the ma t r ix elem en ts in t erms o f ε ( q ). 4.5.2. Boundary of o scil lator r e gion: κ 2 + h 2 = 1 . One of the p ec uliarities of XY-c ha in when the parameters b elong to the curv e κ 2 + h 2 = 1 is tha t the gro und states |i NS XY and |i R XY are degenerate and eac h of them can b e presen ted as a sum of tw o pure tensors [35]. In [3], for suc h v alues of parameters, the t wo-po in t correlation function w as found explicitly . Here w e give some commen ts on the matrix elemen ts o f spin op erators. They can b e derived from the general formulas for the region 0 < κ < 1, √ 1 − κ 2 < h < 1 b y a limiting pro cedure. F rom (16), denoting ζ = √ κ 2 + h 2 − 1 we get in the limit ζ → 0 e γ ( p ) = 2 h ζ √ 1 − h 2 ε ( p ) , ε ( p ) = 1 − h cos p , sinh γ ( p ) + γ ( q ) 2 = e ( γ ( p )+ γ ( q )) / 2 2 , ξ = 1 , ξ T = 1 . Using these formulas it is easy to tak e the limit ζ → 0 in the g ene ral fo rm ulas fo r the matrix elemen ts. W e g et, in particular, that the matrix elemen ts are non-zero if and only if K = L or K − L = ± 2. F orm-factors of the finite quantum XY-chain 14 5. Asymptotics of form-factors in the limit of infinite cha in In t his section w e ana lyz e the asymptotics of differen t parts of form-factors in the limit of infinite length ( n → ∞ ) of XY-c hain. F or this end it is helpful to use t he follow ing in tegra l represe n tations for differen t parts of fo r m- factors at finite n [26]. F or Λ − 1 = 1 2 X q ∈ NS γ ( q ) − X p ∈ R γ ( p ) ! , e η ( q ) = Q p ∈ NS  1 − e − γ ( q ) − γ ( p )  Q p ∈ R (1 − e − γ ( q ) − γ ( p ) ) and ξ T (see (22)) w e ha v e Λ − 1 = 1 π Z π 0 dp lo g coth n ¯ γ ( p ) 2 , η ( q ) = 1 π Z π 0 dp cos p − e − γ ( q ) cosh γ ( q ) − cos p log coth n ¯ γ ( p ) 2 , ξ T = n 2 2 π 2 Z π 0 Z π 0 dp dq ¯ γ ′ ( p ) ¯ γ ′ ( q ) sinh n ¯ γ ( p ) sinh n ¯ γ ( p ) log     sin(( p + q ) / 2) sin(( p − q ) / 2)     . Let us sho w that Λ − 1 → 0 if n → ∞ . In fact w e hav e the follo wing asymptotics Λ − 1 → 2 π Z π 0 dp e − n ¯ γ ( p ) = 2 π Z π 0 dp e − n (¯ γ (0)+¯ γ ′′ (0) p 2 / 2+ ··· ) ≃ e − n ¯ γ (0) π Z ∞ −∞ dp e − n ¯ γ ′′ (0) p 2 / 2 = e − n ¯ γ (0) s 2 nπ ¯ γ ′′ (0) → 0 , where w e used t he f a ct that ¯ γ ( p ) > 0 (there is a gap in the sp ectrum for the non-critical parameters). Similarly w e get η ( q ) → 0, ξ T → 1 at n → ∞ . F or the deriv ation of these form ulas tog ethe r with the more precise asymptotics at n → ∞ , see [36]. Another wa y to get the asymptotics in the limit of infinite length of the c hain is giv en in [37]. In the limit of infinite XY-c hain the formulas (3 1), (32) for the matrix elemen t s of spin op erators b et w een the eigenstates | Φ 0 i XY = | q 1 , q 2 , . . . , q K i NS XY from the NS-sector and | Φ 1 i XY = | p 1 , p 2 , . . . , p L i R XY from the R-sector, and (20) b ecome | XY h Φ 0 | σ x m | Φ 1 i XY | 2 = Ξ Φ 0 , Φ 1 2 κ 1 + κ cosh 2 P K k =1 γ ( q k ) − P L l =1 γ ( p l ) 2 , (34) | XY h Φ 0 | σ y m | Φ 1 i XY | 2 = Ξ Φ 0 , Φ 1 2 κ 1 − κ sinh 2 P K k =1 γ ( q k ) − P L l =1 γ ( p l ) 2 , (35) Ξ Φ 0 , Φ 1 = ξ K Y k =1 1 n sinh γ ( q k ) L Y l =1 1 n sinh γ ( p l ) ·  t y − t − 1 y t x − t − 1 x  ( K − L ) 2 / 2 × K Y k