Matrix KP hierarchy and spin generalization of trigonometric Calogero-Moser hierarchy

Matrix KP hierarc h y and spin generalization of trigonometric Calogero-Moser hierarc h y V. Prokofev ∗ 1,2 and A. Zabro din † 3 ,4 1 Mosco w Institute of P h ysics and T ec hnology , Dolgo prudn y , Institutsky p er., 9, Mosco w region, 14 1700, Russia 2 Sk olk o v o Institute of Science and T echnology , 1430 2 6 Mosco w, Russian F ederation 3 National Researc h Univ ersit y Higher School of Economics, 20 My asnitsk ay a Ulitsa, Mosco w 101000, Russian F ederation 4 Steklo v Mathematical Institute of Russian Academ y of Sciences, Gubkina str. 8, Mosco w, 1199 9 1, Russian F ederation September 2019 ITEP-TH-29/19 Abstract W e consid er solutions of the matrix KP hierarc h y that are tr igonometric func- tions of th e firs t hierarc hical time t 1 = x and establish the corresp ondence w ith the spin generalizat ion of the trigonometric Calogero-Moser system on th e lev el of hierarc hies. Namely , the ev olution of p oles x i and matrix residues at the p oles a α i b β i of the solutions with resp ect to th e k -th hierarc hical time of the matrix KP hierarc h y is sho wn to b e g iv en by t he Hamiltonian flo w with the Ha miltonian which is a linear com bination of the fi rst k higher Hamiltonians of the sp in trigonomet- ric Calogero- Moser system with co ordinates x i and with spin degrees of freedom a α i , b β i . By considering ev olution of p oles according to the discrete time m atrix KP hierarc h y w e al so in tro duce the in teg rable discrete time v er s ion of the trigonometric spin Calogero- Moser system. 1 In tro du ction The matrix generalization of the Kado mtsev -P etviash vili (KP) hierarch y is an infinite set of compatible nonlinear differen tial equations with infinitely man y indep enden t (time) v ariables t = { t 1 , t 2 , t 3 , . . . } and matrix dep enden t v aria bles. It is a subhierarc hy of the ∗ v a dim.prokofev@ph ystech.edu † zabro din@itep.ru 1 m ulti-comp onen t KP hierarc h y [1, 2, 3 , 4]. Among all solutions to these equations, of sp ecial in terest are solutions whic h ha v e a finite num b er of p oles in the v ariable x = t 1 in a fundamen tal domain of the complex plane. In particular, one can consider solutions whic h are trigonometric or hyperb olic functions o f x with p oles dep ending on the times t 2 , t 3 , . . . . The dynamics of p oles of singular solutions to nonlinear integrable equations is a w ell kno wn sub ject in mathematical ph ysics [5, 6 , 7 , 8]. It w as sho wn that the p oles of solutions to the KP equation a s functions of the t ime t 2 mo v e as particles o f the integrable Calogero-Moser many-bo dy system [9, 10, 11, 12]. Rational, trigo no metric and elliptic solutions corresp ond resp ectiv ely to rationa l, trigonometric or elliptic Calogero-Moser systems . The further progress w as ac hiev ed in [13], where it w as sho wn that the corresp ondence b et w een rational solutions to the KP equation and the Calogero-Moser system with rational p otential can be extended to the lev el of hierarc hies. Namely , the ev olution of p oles with resp ect to the higher time t m of the KP hierarch y w as show n to b e giv en b y the higher Ha miltonian flow of the in tegrable Calo gero-Moser system with the Hamiltonian H m = tr L m , where L is the Lax matrix. Later this corresp ondence was generalized to trigono metric solutions of the KP hierarc h y (see [14, 15]). It was sho wn tha t the dynamics of p oles with resp ect to the higher t ime t m is give n b y the Hamiltonian flo w with t he Hamilto nia n H m = 1 2( m + 1) γ tr  ( L + γ I ) m +1 − ( L − γ I ) m +1  , (1.1) where I is the unity matrix and γ is a parameter suc h t ha t π i/γ is the p erio d of the trigonometric or h yp erb olic f unctions. Clearly , the Hamiltonian H m is a linear com bina- tion of the Hamiltonians H k = tr L k . In this pap er w e generalize t his result to trigonometric solutions of the matrix KP hierarc h y . The singular (in general, elliptic) solutio ns to the matrix KP equation w ere in v estigated in [16]. It was sho wn that the ev olution of da t a of suc h solutions ( p ositions of p oles and some in ternal degrees of freedom) with resp ect to the time t 2 is isomorphic to the dynamics of a spin generalization of the Calogero-Moser system (the Gibb ons- Hermsen system [17]). It is a system of N particles with co or dinat es x i with in ternal degrees of freedom give n b y N -dimensional column v ectors a i , b i whic h pairwise interact with each o ther. The Hamiltonia n is H = N X i =1 p 2 i − γ 2 X i 6 = k ( b T i a k )( b T k a i ) sinh 2 ( γ ( x i − x k )) (1.2) (here b T i is the transp osed row -v ector) with the non-v anishing Pois son bra c k ets { x i , p k } = δ ik , { a α i , b β k } = δ αβ δ ik . The mo del is know n t o b e integrable, with the higher Hamiltonians (in tegrals of motion in in v olution) H k = tr L k , where L is the Lax matrix o f the mo del giv en b y L j k = − p j δ j k − (1 − δ j k ) γ b T j a k sinh( γ ( x j − x k )) . (1.3) Our main result in this pap er is that the dynamics of p oles x i and v ectors a i , b i (whic h parametrize matrix residues at the p oles) with resp ect to the higher time t m is g iv en b y 2 the Hamiltonian flo w with the Hamiltonian (1 .1) and with the Lax matrix (1.3). The corresp onding result for rational solutions ( γ = 0 ) w as established in [18]. W e use the metho d suggested by Kric hev er [7 ] for elliptic solutions of the KP equation. It consists in substituting the solutio n no t in the KP equation itself but in the auxiliary linear problem for it (this implies a suitable p ole ansatz fo r the w a v e fuction). This metho d allo ws one to obtain the equations of motion to gether with the Lax represen ta tion for t hem. Another result of this pap er is t he time discretization of the tr ig onometric spin Calogero-Moser (Gibb ons-Hermsen) mo del. ( The time discretization of the rational spin Calogero-Moser system within the same a pproac h w as suggested in [19].) Because of the precise correspondence b etw een the trigonometric solutions of the matrix KP hierarc h y and the trig o nometric spin Calogero- Moser hierarc h y , the in tegrable time discre tization of the Calog ero-Moser system and its spin generalization can b e o bta ined from dynamics of p oles of trigonometric solutions to semi-discrete solito n equations. (“Semi” means that the time b ecomes discrete while the space v ariable x , with respect to whic h one considers p ole solutions, remains con tin uous.) At the same time, it is known that in tegrable dis- cretizations of soliton equations can b e regarded as b elonging to the same hierarc h y as their con tin uous coun terparts. Namely , the discrete t ime step is equ iv a len t to a special shift of infinitely man y con tin uous hierarchical times. This fact lies in the basis of the metho d of g enerating discrete soliton equations dev eloped in [20]. F or in tegrable time discretization of many-bo dy systems see [21, 22, 23, 24 , 2 5]. In this pap er, w e deriv e equations of motion in discrete t ime p for the spin generalization of the trigo nometric Calogero-Moser mo del: X j coth( γ ( x i ( p ) − x j ( p + 1))( b T i ( p ) a j ( p + 1))( b T j ( p + 1) a i ( p )) + X j coth( γ ( x i ( p ) − x j ( p − 1))( b T i ( p ) a j ( p − 1))( b T j ( p − 1) a i ( p )) = 2 X j 6 = i coth( γ ( x i ( p ) − x j ( p ))( b T i ( p ) a j ( p ))( b T j ( p ) a i ( p )) , (1.4) where a i ( p ), b i ( p ) ar e spin v aria bles. In the limit γ → 0 the result of [1 9] is repro duced. 2 The matrix KP hi erarc h y Here w e briefly review the main facts ab out the m ulti-comp onen t a nd matrix KP hier- arc hies following [3, 4]. W e start from the more g eneral m ulti-comp onen t KP hierarch y . The indep enden t v aria bles are N infinite sets of contin uous “times” t = { t 1 , t 2 , . . . , t N } , t α = { t α, 1 , t α, 2 , t α, 3 , . . . } , α = 1 , . . . , N and N discrete in teger v ariables s = { s 1 , s 2 , . . . , s N } (“c harges”) constrained b y the condition N X α =1 s α = 0. In what f o llo ws, w e will mostly put s α = 0 since w e are interes ted in the dynamics in the con tin uous times. 3 In the bilinear formalism, the dep enden t v ariable is the tau- function τ ( s ; t ). W e also in tro duce the tau-functions τ αβ ( t ) = τ ( e α − e β ; t ) , (2.1) where e α is the v ector whose α th comp onen t is 1 and all other entries are equal to zero. The N -comp onen t KP hierarc h y is the infinite set of bilinear equations for the tau-functions whic h are enco ded in the basic bilinear relation N X ν =1 ǫ αν ǫ β ν I C ∞ dz z δ αν + δ β ν − 2 e ξ ( t ν − t ′ ν , z ) τ αν  t − [ z − 1 ] ν  τ ν β  t ′ + [ z − 1 ] ν  = 0 (2.2) v alid for any t , t ′ . Here ǫ αβ is a sign factor: ǫ αβ = 1 if α ≤ β , ǫ αβ = − 1 if α > β . In (2.2) w e use the fo llowing standard not a tion: ξ ( t γ , z ) = X k ≥ 1 t γ ,k z k ,  t ± [ z − 1 ] γ  αk = t α,k ± δ αγ z − k k . The in tegration con tour C ∞ is a big circle aro und ∞ . Hereafter, w e omit the v ariables s in the notation for t he ta u- functions. An imp ortan t role in the theory of in tegrable hierarc hies is pla y ed b y the w a v e func- tion. In the m ulti-comp onen t KP hierarc h y , the w a v e function Ψ( t ; z ) and its adjoint Ψ † ( t ; z ) are N × N matrices with the comp o nents Ψ αβ ( t ; z ) = ǫ αβ τ αβ ( t − [ z − 1 ] β ) τ ( t ) z δ αβ − 1 e ξ ( t β ,z ) , Ψ † αβ ( t ; z ) = ǫ β α τ αβ ( t + [ z − 1 ] α ) τ ( t ) z δ αβ − 1 e − ξ ( t α ,z ) (2.3) (here and b elo w † do es not mean the Hermitian conjugatio n) . The complex v ariable z is called the sp ectral parameter. Around z = ∞ , the w av e function Ψ can b e represen ted in the form of the series Ψ αβ ( t ; z ) =   δ αβ + X k ≥ 1 w ( k ) αβ ( t ) z k   e ξ ( t β ,z ) , (2.4) where w ( k ) ( t ) a r e some matrix functions. In terms of t he wa ve functions, the bilinear relation (2.2) can b e written as I C ∞ dz Ψ( t ; z )Ψ † ( t ′ ; z ) = 0 . (2.5) Another (equiv alen t) approac h to the m ulti-comp onen t KP hierarc h y is based o n matrix pseudo-differen tial op erators. The hierarc h y can b e understo o d as an infinite set of ev olution equations in t he times t for matrix functions of a v ar ia ble x . F or example, the co efficien ts w ( k ) of t he w a v e function can b e tak en a s suc h matrix functions, the ev o lution b eing w ( k ) ( x ) → w ( k ) ( x, t ). In what follows w e denote τ ( x, t ), w ( k ) ( x, t ) simply a s τ ( t ), 4 w ( k ) ( t ), suppres sing the dep endence on x . Let us in tro duce the matrix pseudo-differen tial “w a v e op erator” W with matrix elemen ts W αβ = δ αβ + X k ≥ 1 w ( k ) αβ ( t ) ∂ − k x , (2.6) where w ( k ) αβ ( t ) are the same matrix functions as in (2 .4). The w a v e function is a result of action of the w a v e o p erator to the exp onential f unction: Ψ( t ; z ) = W exp  xz I + N X α =1 E α ξ ( t α , z )  , (2.7) where E α is the N × N matrix with the comp o nen ts ( E α ) β γ = δ αβ δ αγ . The adjoint w a v e function can b e written as Ψ † ( t ; z ) = exp  − xz I − N X α =1 E α ξ ( t α , z )  W − 1 . (2.8) Here the o p erators ∂ x whic h ente r W − 1 act to the left (the left action is defined as f ∂ x ≡ − ∂ x f ). It is pro v ed in [4] t ha t the w a v e f unction a nd it s adj o in t satisfy the linear equations ∂ t α,m Ψ( t ; z ) = B αm Ψ( t ; z ) , − ∂ t α,m Ψ † ( t ; z ) = Ψ † ( t ; z ) B αm , (2.9) where B αm is the differen tia l op erato r B αm =  W E α ∂ m x W − 1  + . The no tation ( . . . ) + means the differen tial part of a pseudo-differen tial op erator, i.e. the sum of a ll terms with ∂ k x , where k ≥ 0. Ag ain, the opera t o r B αm in the sec ond equation in (2.9) acts to the left. In particular, it follow s from (2.9) at m = 1 that N X α =1 ∂ t α, 1 Ψ( t ; z ) = ∂ x Ψ( t ; z ) , N X α =1 ∂ t α, 1 Ψ † ( t ; z ) = ∂ x Ψ † ( t ; z ) , (2.10) so the v ector field ∂ x can b e iden tified with the v ector field P α ∂ t α, 1 . The matrix KP hierarc h y is a subhierarc h y of the m ulti-comp onen t KP one whic h is obtained by a restriction of the time v ariables in the following manner: t α,m = t m for eac h α and m . The corresp o nding vec tor fields are related as ∂ t m = P N α =1 ∂ t α,m . The w a v e function for the matrix KP hierar c h y has the expansion Ψ αβ ( t ; z ) =  δ αβ + w (1) αβ ( t ) z − 1 + O ( z − 2 )  e xz + ξ ( t ,z ) , (2.11) where ξ ( t , z ) = X k ≥ 1 t k z k . The co efficien t w (1) αβ ( t ) pla ys an imp or tan t role in what follows. Equations (2.9) imply that the wa ve function o f the matrix KP hierarc h y and its adjoin t satisfy the linear equations ∂ t m Ψ( t ; z ) = B m Ψ( t ; z ) , − ∂ t m Ψ † ( t ; z ) = Ψ † ( t ; z ) B m , m ≥ 1 , (2.12) where B m is the differen tial op erator B m =  W ∂ m x W − 1  + . At m = 1 w e ha v e ∂ t 1 Ψ = ∂ x Ψ, so we can iden tify ∂ x = ∂ t 1 = N X α =1 ∂ t α, 1 and the ev olution in t 1 is simply a shift of the 5 v ariable x : w ( k ) ( x, t 1 , t 2 , . . . ) = w ( k ) ( x + t 1 , t 2 , . . . ) . A t m = 2 equations (2.12) turn in to the linear problems ∂ t 2 Ψ = ∂ 2 x Ψ + V ( t )Ψ , (2.13) − ∂ t 2 Ψ † = ∂ 2 x Ψ † + Ψ † V ( t ) (2.14) whic h ha v e the form of the matrix non- stationary Sc hr¨ odinger equations with V ( t ) = − 2 ∂ x w (1) ( t ) . (2.15) Let us deriv e a useful corollary of the bilinear r elat io n (2.2). Differen tiating it with resp ect to t m and putting t ′ = t after t his, w e obtain: 1 2 π i N X ν =1 I C ∞ dz z m Ψ αν ( t ; z )Ψ † ν β ( t ; z ) = − ∂ t m w (1) αβ ( t ) (2.16) or, equiv alen tly , res ∞  z m Ψ αν Ψ † ν β  = − ∂ t m w (1) αβ . (2.17) Here and b elow the summation from 1 to N o v er rep eated Greek indices is implied. The residue at infinity is defined according to res ∞ ( z − n ) = δ n 1 . A t the end of this section let us mak e some remarks on the disc rete time v ersion of the matrix KP hierar ch y . The discrete t ime ev o lution is defined as a sp ecial shift of the infinite n um b er of con tin uous time v ariables according to the rule [20] τ p = τ t − p N X α =1 [ µ − 1 ] α ! , Ψ p = Ψ t − p N X α =1 [ µ − 1 ] α ; z ! . Here p is the discrete time v ariable and µ is a contin uous parameter. Eac h µ corr esponds to its o wn discrete time flo w. The limit µ → ∞ is the con tin uous limit. One can sho w, using the explicit expressions of the w a v e functions through the tau-f unction and some corollaries of the bilinear relation (see [1 9]) that the corresp onding linear problems hav e the form µ Ψ p αβ − µ Ψ p +1 αβ = ∂ x Ψ p αβ +  w (1) αν ( p + 1) − w (1) αν ( p )  Ψ p ν β , (2.18) µ Ψ † p αβ − µ Ψ † p − 1 αβ = − ∂ x Ψ † p αβ + Ψ † p αν  w (1) ν β ( p ) − w (1) ν β ( p − 1)  . (2.19) 3 T rigonomet ric solutio ns of the matrix KP hierar- c h y: dynamics of p oles in t 2 Our aim is to study solutions to the matrix KP hierarc h y whic h a re trigonometric func- tions of the v ariable x (and, therefore, t 1 ). F or the trig onometric solutions, the tau- function has the form τ = C N Y i =1 ( e 2 γ x − e 2 γ x i ) , (3.1) where γ is a parameter. The p erio d of the function is π i/γ . R eal (resp ectiv ely , imaginary) γ corresp onds to h yp erb olic ( r esp ectiv ely , trigonometric) functions. In the limit γ → 0 6 one obtains rational solutions. The N ro ots x i (assumed to b e distinct) depend on the times t . It is conv enien t to pass to the exp onen tiated v ariables w = e 2 γ x , w i = e 2 γ x i , (3.2) then the t a u-function b ecomes a p olynomial with the ro ots w i : τ = C Q i ( w − w i ). Clearly , w e ha v e ∂ x = 2 γ ∂ w , ∂ 2 x = 4 γ 2 ( w 2 ∂ 2 w + w ∂ w ). It is clear from (2.3) that the wa ve functions Ψ, Ψ † (and th us the co efficien t w (1) ), as functions o f x , hav e simple p oles a t x = x i . It is sho wn in [18] that the residues at these p oles are matrices of rank 1 . W e para metrize them through the column ve ctors a i = ( a 1 i , a 2 i , . . . , a N i ) T , b i = ( b 1 i , b 2 i , . . . , b N i ) T , c i = ( c 1 i , c 2 i , . . . , c N i ) T ( T means transp osition) and the ro w v ector c ∗ i = ( c ∗ 1 i , c ∗ 2 i , . . . , c ∗ N i ): Ψ αβ = e xz + ξ ( t ,z )   C αβ + X i 2 γ w 1 / 2 i a α i c β i w − w i   , (3.3) Ψ † αβ = e − xz − ξ ( t ,z )   C − 1 αβ + X i 2 γ w 1 / 2 i c ∗ α i b β i w − w i   , ( 3 .4) where the matrix C αβ do es not dep end o n x . Note t ha t the constan t term in t he a djoin t w a v e function is the inv erse matrix C − 1 αβ . This follo ws fro m (2.8). F or the matrices w (1) and V = − 2 ∂ x w (1) w e ha v e w (1) αβ = S αβ − X i 2 γ w i a α i b β i w − w i , V αβ = − 8 γ 2 X i w w i a α i b β i ( w − w i ) 2 , (3.5) where the matrix S αβ do es not depend on x . T ending w → ∞ in (2.17), one concludes that ∂ t m S αβ = 0 for all m ≥ 1, so t he matr ix S αβ do es not dep end on all the times. The comp onen ts of the vec tors a i , b i are going to b e spin v ariables o f the Gibb ons-Hermsen mo del. W e first consider the dynamics of p oles with resp ect to the time t 2 . The pro cedure is similar to the rational case [18]. F ollowing Kriche v er’s approac h, w e consid er the linear problems (2.13), (2.14), ∂ t 2 Ψ αβ = ∂ 2 x Ψ αβ − 8 γ 2 N X i =1 w w i a α i b ν i ( w − w i ) 2 Ψ ν β , − ∂ t 2 Ψ † αβ = ∂ 2 x Ψ † αβ − 8 γ 2 Ψ † αν N X i =1 w w i a ν i b β i ( w − w i ) 2 and substitute here the p ole ansatz (3.3), (3 .4) for the wa ve functions. Consider first the equation fo r Ψ. First o f all, comparing the b eha vior of b oth sides as w → ∞ , w e conclude tha t ∂ t 2 C αβ = 0 , so C αβ do es not dep end o n t 2 (in a similar wa y , from the higher linear problems one can see tha t C αβ do es not dep end o n all the times t m ). Aft er the substitution, we see that the expression has p oles at w = w i up to the third order. Equating co efficien ts at the p oles of differen t orders at w = w i , w e get the conditions: • A t 1 ( w − w i ) 3 : b ν i a ν i = 1; 7 • A t 1 ( w − w i ) 2 : − 1 2 ˙ x i c β i − 2 γ X k 6 = i w 1 / 2 i w 1 / 2 k b ν i a ν k c β k w i − w k − ( z − γ ) c β i = w 1 / 2 i ˜ b β i ; • A t 1 w − w i : ∂ t 2 ( w 1 / 2 i a α i c β i ) = 2 γ w 1 / 2 i ˙ x i a α i c β i + 8 γ 2 X k 6 = i w 2 i w 1 / 2 k a α i b ν i a ν k c β k ( w i − w k ) 2 − 8 γ 2 X k 6 = i w 3 / 2 i w k a α k b ν k a ν i c β i ( w i − w k ) 2 , where ˜ b β i = b ν i C ν β , and ˙ x i = ∂ t 2 x i . Similar calculations for the linear problem for Ψ † lead to the conditions • A t 1 ( w − w i ) 3 : b ν i a ν i = 1; • A t 1 ( w − w i ) 2 : − 1 2 ˙ x i c ∗ α i − 2 γ X k 6 = i w 1 / 2 i w 1 / 2 k c ∗ α k b ν k a ν i w k − w i − ( z + γ ) c ∗ α i = − w 1 / 2 i ˜ a α i ; • A t 1 w − w i : ∂ t 2 ( w 1 / 2 i c ∗ α i b β i ) = − 2 γ w 1 / 2 i ˙ x i c ∗ α i b β i + 8 γ 2 X k 6 = i w 2 i w 1 / 2 k c ∗ α k a ν i b ν k b β i ( w i − w k ) 2 − 8 γ 2 X k 6 = i w 3 / 2 i w k c ∗ α i a ν k b ν i b β k ( w i − w k ) 2 , where ˜ a α i = C − 1 αν a ν i . The conditions coming from the third order p oles are constrain ts on the vec tors a i , b i . The other conditions can b e written in the matrix form      ( z I − ( L + γ I )) c β = − W 1 / 2 ˜ b β ˙ c β = M c β , (3.6)      c ∗ α ( z I − ( L − γ I )) = ˜ a αT W 1 / 2 ˙ c ∗ α = c ∗ α ˜ M , (3.7) where c β = ( c β 1 , . . . c β N ) T , c ∗ α = ( c ∗ α 1 , . . . c ∗ α N ), ˜ b β = ( ˜ b β 1 , . . . ˜ b β N ) T , ˜ a α = ( ˜ a α 1 , . . . ˜ a α N ) are N -dimens ional v ectors, I is the unity matrix, W = diag ( w 1 , w 2 , . . . w N ) and L , M , ˜ M are N × N matrices of the form L ik = − 1 2 ˙ x i δ ik − 2 γ (1 − δ ik ) w 1 / 2 i w 1 / 2 k b ν i a ν k w i − w k , (3.8) M ik = ( γ ˙ x i − Λ i ) δ ik + 8 γ 2 (1 − δ ik ) w 3 / 2 i w 1 / 2 k b ν i a ν k ( w i − w k ) 2 , (3.9) ˜ M ik = ( γ ˙ x i + Λ ∗ i ) δ ik − 8 γ 2 (1 − δ ik ) w 1 / 2 i w 3 / 2 k b ν i a ν k ( w i − w k ) 2 . (3.10) Here Λ i = ˙ a α i a α i + 8 γ 2 X k 6 = i w i w k a α k b ν k a ν i a α i ( w i − w k ) 2 , − Λ ∗ i = ˙ b α i b α i − 8 γ 2 X k 6 = i w i w k b ν i a ν k b α k b α i ( w i − w k ) 2 (3.11) 8 do not dep end on the index α . In fact one can see that Λ i = Λ ∗ i . Indeed, m ultiplying equations (3.11) b y a α i b α i (no summation here!), summing o v er α and summ ing the tw o equations, w e get Λ i − Λ ∗ i = ∂ t 2 ( a α i b α i ) = 0 b y virtue of the constrain t a α i b α i = 1. Differen tiating the first equation in (3.6) by t 2 , we get, after some calculations, t he compatibilit y condition of equations (3.6): ( ˙ L + [ L, M ]) c β = 0 . (3.12) One can see, taking into account equations (3.11) , whic h w e write here in the form ˙ a α i = Λ i a α i − 2 γ 2 X k 6 = i a α k b ν k a ν i sinh 2 ( γ ( x i − x k )) , (3.13) ˙ b α i = − Λ i b α i + 2 γ 2 X k 6 = i b ν i a ν k b α k sinh 2 ( γ ( x i − x k )) (3.14) (in this f o rm they are equations of motion for the spin degrees of freedom) that the off- diagonal elemen ts of the ma t r ix ˙ L + [ L, M ] are equal to zero. V anishing o f the diagonal elemen ts yields equations of motion for the p oles x i : ¨ x i = − 8 γ 3 X k 6 = i cosh( γ ( x i − x k )) sinh 3 ( γ ( x i − x k )) b µ i a µ k b ν k a ν i . (3.15) The gauge transformation a α i → a α i q i , b α i → b α i q − 1 i with q i = exp  Z t 2 Λ i dt  eliminates the terms with Λ i in (3.13 ), (3.14), so w e can put Λ i = 0. This give s the equations of motion ˙ a α i = − 2 γ 2 X k 6 = i a α k b ν k a ν i sinh 2 ( γ ( x i − x k )) , ˙ b α i = 2 γ 2 X k 6 = i b ν i a ν k b α k sinh 2 ( γ ( x i − x k )) . (3.16) T ogether with (3.15) they are equations of motio n of the trigonometric G ibb ons-Hermsen mo del. Their Lax represen tation is giv en b y the matrix equation ˙ L = [ M , L ]. It states that the time ev olution of the Lax matrix is an isosp ectral transformat ion. It follows that the quantities H k = tr L k are in tegrals of motion. In particular, H 2 = N X i =1 p 2 i − γ 2 X i 6 = k b µ i a µ k b ν k a ν i sinh 2 ( γ ( x i − x k )) = tr L 2 (3.17) is the Hamiltonian of the Gibb o ns-Hermsen mo del. Equations of motion (3.16), (3.15) are equiv alen t to the Hamiltonian equations ˙ x i = ∂ H 2 ∂ p i , ˙ p i = − ∂ H 2 ∂ x i , ˙ a α i = ∂ H 2 ∂ b α i , ˙ b α i = − ∂ H 2 ∂ a α i . (3.18) 9 4 Dynamics of p o les in t he h igher times The main to ol for the analysis of the dynamics in the higher times is the relation (2.17) whic h, after substitution of (3.3), (3.4) and (3.5) take s the form res ∞   z m  C αν + X i 2 γ w 1 / 2 i a α i c ν i w − w i  C − 1 ν β + X k 2 γ w 1 / 2 k c ∗ ν k b β k w − w k    = 2 γ X i ∂ t m ( w i a α i b β i ) w − w i + 4 γ 2 X i ∂ t m x i w 2 i a α i b β i ( w − w i ) 2 . (4.1) The b oth sides are rational functions of w with p oles at w = w i v anishing at infinit y . Iden tifying the co efficien ts in front of the second order p oles, w e obta in ∂ t m x i = res ∞  z m c ν i w − 1 i c ∗ ν i  . (4.2) Solving the linear equations (3.6), (3.7), w e get c ν i = − X k ( z I − ( L + γ I )) − 1 ik w 1 / 2 k ˜ b ν k , c ∗ ν i = X k ˜ a ν k w 1 / 2 k ( z I − ( L − γ I )) − 1 k i , (4.3) and, therefore, (4.2) reads ∂ t m x i = − res ∞ X k ,k ′ z m ˜ a ν k ˜ b ν k ′ w 1 / 2 k  1 z I − ( L − γ I )  k i w − 1 i  1 z I − ( L + γ I )  ik ′ w 1 / 2 k ′ ! = − res ∞ tr z m W 1 / 2 RW 1 / 2 1 z I − ( L − γ I ) W − 1 E i 1 z I − ( L + γ I ) ! , where E i is the dia g onal matrix with matrix elemen ts ( E i ) j k = δ ij δ ik and R is the N ×N matrix R ik = ˜ b ν i ˜ a ν k = b ν i a ν k . (4.4) The follow ing comm utation relation can b e c hec k ed directly: [ L, W ] = 2 γ ( W 1 / 2 RW 1 / 2 − W ) . (4.5) Note that E i = − ∂ L/∂ p i . The rest of the calculation is similar to the one done in [15]. W e hav e, using (4 .5): ∂ t m x i = 1 2 γ res ∞ tr z m ( LW − W L + 2 γ W ) 1 z I − ( L − γ I ) W − 1 ∂ L ∂ p i 1 z I − ( L + γ I ) ! = 1 2 γ res ∞ tr z m ∂ L ∂ p i 1 z I − ( L + γ I ) − ∂ L ∂ p i 1 z I − ( L − γ I ) !! = 1 2 γ tr ∂ L ∂ p i ( L + γ I ) m − ∂ L ∂ p i ( L − γ I ) m ! = 1 2( m + 1) γ ∂ ∂ p i tr  ( L + γ I ) m +1 − ( L − γ I ) m +1  = ∂ H m ∂ p i , 10 where H m is giv en b y (1.1). Note t ha t H 2 = H 2 + const. W e ha v e obtained one part of the Hamiltonian equations fo r the higher time flow s. In the case γ → 0 (rational solutions) the r esult of the pap er [18] is repro duced. In order to obtain another part o f the Hamiltonian equations, let us differen tiate (4.2) with r esp ect to t 2 : ∂ t m ˙ x i = − 2 γ res ∞  z m c ∗ ν i ˙ x i w − 1 i c ν i  + res ∞  z m ( c ν i w − 1 i ∂ t 2 c ν i + ∂ t 2 c ∗ ν i w − 1 i c ν i )  = res ∞ X k  z m ( c ∗ ν i w − 1 i B ik c ν k − c ∗ ν k w − 1 k B k i c ν i )  , where B j k = 8 γ 2 (1 − δ j k ) w 3 / 2 j w 1 / 2 k b ν j a ν k ( w i − w k ) 2 . Therefore, w e hav e, using (4.3): ∂ t m p i = 1 2 ∂ t m ˙ x i = − res ∞ " z m tr W 1 / 2 RW 1 / 2 1 z I − ( L − γ I ) G ( i ) 1 z I − ( L + γ I ) !# , where the matrix G ( i ) is g iv en b y G ( i ) j k = 4 γ 2 ( δ ij − δ ik ) w 1 / 2 j w 1 / 2 k b ν j a ν k ( w i − w k ) 2 . It is straigh tforw ard to c hec k the iden tities ( W G ( i ) − G ( i ) W ) j k = − 2 γ L j k ( δ ij − δ ik ) , W G ( i ) + G ( i ) W = 2 ∂ L ∂ x i . A direct calculatio n whic h literally rep eats t he one done in [15] show s that ∂ t m p i == − 1 2 γ res ∞ " z m tr ( LW − W L + 2 γ W ) 1 z I − ( L − γ I ) G ( i ) 1 z I − ( L + γ I ) !# = − 1 2 γ res ∞ " z m tr W G ( i ) 1 z I − ( L + γ I ) − G ( i ) W 1 z I − ( L − γ I ) !# = − 1 2 γ res ∞ " z m tr ∂ L ∂ x i 1 z I − ( L + γ I ) − 1 z I − ( L − γ I ) !!# = − 1 2 γ tr ∂ L ∂ x i ( L + γ I ) m − ∂ L ∂ x i ( L − γ I ) m ! = − ∂ H m ∂ x i . W e hav e established the remaining part of the Hamiltonian equations for the higher t ime dynamics of the x i ’s. 11 5 Dynamics of spin v ariables in the high er times Comparison of the first order p oles in (4.1 ) give s the following relation: ∂ t m ( w i a α i b β i ) = res ∞   z m  w 1 / 2 i C αν c ∗ ν i b β i + w 1 / 2 i a α i c ν i C − 1 ν β +2 γ X k 6 = i w 1 / 2 i w 1 / 2 k w i − w k ( a α i b β k c ν i c ∗ ν k + a α k b β i c ν k c ∗ ν i )    . Using ( 4 .3), we can rewrite it in the form b β i " − ∂ t m a α i + res ∞  z m  X k a α k w − 1 / 2 i w 1 / 2 k  1 z I − ( L − γ I )  k i − 2 γ X k 6 = i X l,n w − 1 / 2 i w 1 / 2 k w i − w k a α k a ν l w 1 / 2 l  1 z I − ( L − γ I )  li  1 z I − ( L + γ I )  k n w 1 / 2 n b ν n    − a α i " ∂ t m b β i + res ∞  z m  X k b β k w − 1 / 2 i w 1 / 2 k  1 z I − ( L + γ I )  ik +2 γ X k 6 = i X l,n w − 1 / 2 i w 1 / 2 k w i − w k b β k a ν l w 1 / 2 l  1 z I − ( L − γ I )  lk  1 z I − ( L + γ I )  in w 1 / 2 n b ν n    = 2 γ ∂ t m x i a α i b β i . Separating the terms a t k = i in the sums ov er k in the first and the third lines, and taking in to accoun t that 2 γ ∂ t m x i = res ∞ tr " z m E i  1 z I − ( L − γ I ) − 1 z I − ( L + γ I )  # = res ∞ " z m  1 z I − ( L − γ I )  ii − z m  1 z I − ( L + γ I )  ii # , w e represen t this equation as follows : b β i P α i − a α i Q β i = 0 , (5.1) where P α i = − ∂ t m a α i + res ∞   z m   X k 6 = i a α k w − 1 / 2 i w 1 / 2 k  1 z I − ( L − γ I )  k i +tr W 1 / 2 RW 1 / 2 1 z I − ( L − γ I ) W − 1 ∂ L ∂ b α i 1 z I − ( L + γ I ) !!# , Q β i = ∂ t m b β i + res ∞   z m   X k 6 = i b β k w − 1 / 2 i w 1 / 2 k  1 z I − ( L + γ I )  ik +tr W 1 / 2 RW 1 / 2 1 z I − ( L − γ I ) ∂ L ∂ a β i W − 1 1 z I − ( L + γ I ) !!# , 12 Here we to ok in to account that ∂ L j k ∂ b α i = − 2 γ δ ij (1 − δ j k ) w 1 / 2 i w 1 / 2 k a α k w i − w k , ∂ L j k ∂ a α i = − 2 γ δ ik (1 − δ j k ) w 1 / 2 j w 1 / 2 i b α j w j − w i . It then follo ws f rom (5.1 ) that P α i a α i = Q β i b β i = − Λ ( m ) i (5.2) do es not dep end on the indices α , β . Let us tra nsform the expressions for P α i , Q β i using the commutation relation (4.5 ), i.e., substituting W 1 / 2 RW 1 / 2 = 1 2 γ ( LW − W L + 2 γ W ) . W e hav e: P α i = − ∂ t m a α i + 1 2 γ res ∞ " z m tr ∂ L ∂ b α i 1 z I − ( L + γ I ) − ∂ L ∂ b α i 1 z I − ( L − γ I ) ! +2 γ X k 6 = i a α k w − 1 / 2 i w 1 / 2 k  1 z I − ( L − γ I )  k i + tr  ∂ L ∂ b α i − W − 1 ∂ L ∂ b α i W  1 z I − ( L − γ I )     . But ∂ L ∂ b α i − W − 1 ∂ L ∂ b α i W ! j k = − 2 γ δ ij (1 − δ j k ) w − 1 / 2 i w 1 / 2 k a α k , ans so the second line is equal to zero. W e are left with P α i = − ∂ t m a α i + ∂ H m ∂ b α i . (5.3) A similar calculation for Q α i yields Q α i = ∂ t m b α i + ∂ H m ∂ a α i . (5.4) Therefore, from (5.2) w e ha v e the equations of motion ∂ t m a α i = ∂ H m ∂ b α i + Λ ( m ) i a α i , ∂ t m b α i = − ∂ H m ∂ a α i − Λ ( m ) i b α i . The gauge transformation a α i → a α i q ( m ) i , b α i → b α i ( q ( m ) i ) − 1 with q ( m ) i = exp  Z t m Λ ( m ) i dt  eliminates the terms with Λ ( m ) i and so we can put Λ ( m ) i = 0. In this w a y w e obta in the Hamiltonian equations of motion for spin v ar iables in the higher times: ∂ t m a α i = ∂ H m ∂ b α i , ∂ t m b α i = − ∂ H m ∂ a α i . (5.5) with H m giv en b y (1.1). 13 6 Time discretizatio n o f t he t rigonometri c Gibb ons- Hermsen mo d el Our strategy is to substitute the p ole a nsatz fo r the discrete time w a v e functions Ψ p αβ =  1 − z µ  p e xz   C αβ + X i 2 γ w 1 / 2 i ( p ) a α i ( p ) c β i ( p ) w − w i ( p )   , (6.1) Ψ † p αβ =  1 − z µ  − p e − xz   C − 1 αβ + X i 2 γ w 1 / 2 i ( p ) c ∗ α i ( p ) b β i ( p ) w − w i ( p )   (6.2) and w (1) αβ (see (3.5)) into the linear problems (2.18), (2.19) and identify the co efficien ts in fron t of the p oles ( w − w i ( p )) − 2 , ( w − w i ( p ± 1 ) ) − 1 and ( w − w i ( p )) − 1 . (Note that the constan t t erm S αβ in w (1) αβ ( p ) cancels in t he com bination w (1) αβ ( p + 1) − w (1) αβ ( p ) b ecause the shift p → p + 1 is equiv alent to a shift of times a nd S αβ do es not dep end on the times.) W e b egin with the linear problem (2.18) for Ψ. F ro m cancellation of differen t p oles w e ha v e the follow ing conditions: • A t 1 ( w − w i ( p )) 2 : b ν i ( p ) a ν i ( p ) = 1; • A t 1 w − w i ( p +1) : ( z − µ ) c β i ( p + 1) = − w 1 / 2 i ( p ) ˜ b β i ( p + 1) − 2 γ X j w 1 / 2 i ( p ) w 1 / 2 j ( p ) b ν i ( p + 1) a ν j ( p ) c β j ( p ) w i ( p + 1) − w j ( p ) ; • A t 1 w − w i ( p ) : ( z − µ − 2 γ ) a α i ( p ) c β i ( p ) + w 1 / 2 i ( p ) a α i ( p ) ˜ b β i ( p ) − 2 γ X j w j ( p + 1) a α j ( p + 1) b ν j ( p + 1) a ν i ( p ) c β i ( p ) w i ( p ) − w j ( p + 1) +2 γ X j 6 = i w 1 / 2 i ( p ) w 1 / 2 j ( p ) a α i ( p ) b ν i ( p ) a ν j ( p ) c β j ( p ) w i ( p ) − w j ( p ) + 2 γ X j 6 = i w j ( p ) a α j ( p ) b ν j ( p ) a ν i ( p ) c β i ( p ) w i ( p ) − w j ( p ) = 0 . In tro duce the matrices L ij ( p ) = − δ ij ˙ x i ( p ) 2 − 2 γ (1 − δ ij ) w 1 / 2 i ( p ) w 1 / 2 j ( p ) b ν i ( p ) a ν j ( p ) w i ( p ) − w j ( p ) (6.3) (the same La x matr ix a s in (3 .8)) a nd M ij ( p ) = 2 γ w 1 / 2 i ( p + 1) w 1 / 2 j ( p ) b ν i ( p + 1) a ν j ( p ) w i ( p + 1) − w j ( p ) , (6.4) 14 then t he ab o v e conditions can b e written as                                                        ( z − µ ) c β i ( p + 1) = − w 1 / 2 i ( p + 1) ˜ b β i ( p + 1) − X j M ij ( p ) c β j ( p ) a α i ( p )   X j  ( z − γ ) δ ij − L ij ( p )  c β j ( p ) + w 1 / 2 i ( p ) ˜ b β i ( p )   | {z } =0 + c β i ( p )   X j a α j ( p + 1)( W 1 / 2 ( p + 1) M ( p ) W − 1 / 2 ( p )) j i + X j a α j ( p )( W 1 / 2 ( p ) L ( p ) W − 1 / 2 ( p )) j i − ( µ + γ ) a α i ( p )   = 0 . (6.5) The first line in t he second equation v anishes by virtue of (3.6). Therefore, w e hav e the follo wing equations: ( z − µ ) c β ( p + 1) = − W 1 / 2 ( p + 1) ˜ b β ( p + 1) − M ( p ) c β ( p ) , (6.6) a αT ( p + 1) W 1 / 2 ( p + 1) M ( p ) W − 1 / 2 ( p ) + a αT ( p ) W 1 / 2 ( p ) L ( p ) W − 1 / 2 ( p ) = ( µ + γ ) a αT ( p ) . (6.7) A similar solution of the linear problem (2.19) for Ψ † giv es the equations ( z − µ ) c ∗ α ( p − 1) = ˜ a αT ( p − 1) W 1 / 2 ( p − 1) − c ∗ α ( p ) M ( p − 1) , (6.8) W − 1 / 2 ( p ) M ( p − 1) W 1 / 2 ( p − 1) b β ( p − 1) + W − 1 / 2 ( p ) L ( p ) W 1 / 2 ( p ) b β ( p ) = ( µ − γ ) b β ( p ) . (6.9) A simple calculation, similar to the one done in [19], shows that the compat ibility condition of equations (3.6), (6.6) is the discrete Lax equation L ( p + 1) M ( p ) = M ( p ) L ( p ) (6.10) whic h holds true pro vided equations (6.7), (6.9) are satisfied. Equations (6.7), (6.9) are equations of mot ion of the discrete time trigono metric Gibb ons-Hermsen mo del. Let us consider equation (6.7 ) and represen t it in a somewhat b etter form. In order to do this, write it in the form 2 γ X k w i ( p ) a α k ( p + 1) b ν k ( p + 1) a ν i ( p ) w k ( p + 1) − w i ( p ) + 2 γ X k 6 = i w i ( p ) a α k ( p ) b ν k ( p ) a ν i ( p ) w i ( p ) − w k ( p ) +2 γ X k a α k ( p + 1) b ν k ( p + 1) a ν i ( p ) − 2 γ X k 6 = i a α k ( p ) b ν k ( p ) a ν i ( p ) − ( µ + γ ) a α i ( p ) − ˙ x i ( p ) 2 a α i ( p ) = 0 and add it t o the origina l equation taking into accoun t that X k a α k ( p + 1) b ν k ( p + 1) = X k a α k ( p ) b ν k ( p ) . 15 This fo llows from the fact that X i a α i b β i is an integral of motion, i.e., ∂ t m  X i a α i b β i  = 0 for a ll m . Indeed, we hav e ∂ t m  X i a α i b β i  = X i b β i ∂ H m ∂ b α i − a α i ∂ H m ∂ a β i ! and this is zero b ecause H m is a linear combination of H k = tr L k and X i b β i tr  ∂ L ∂ b α i L m − 1  − a α i tr  ∂ L ∂ a β i L m − 1  ! = X i X j,k b β i ∂ L j k ∂ b α i L m − 1 k j − a α i ∂ L j k ∂ a β i L m − 1 k j ! = 2 γ X i X j 6 = k ( δ ik − δ ij ) w 1 / 2 j w 1 / 2 k b β j a α k w j − w k L m − 1 k j = 0 . As a result, we obtain the equation γ X k coth( γ ( x k ( p + 1) − x i ( p )) a α k ( p + 1) b ν k ( p + 1) a ν i ( p ) = γ X k 6 = i coth( γ ( x k ( p ) − x i ( p )) a α k ( p ) b ν k ( p ) a ν i ( p ) + ˙ x i ( p ) 2 a α i ( p ) + µa α i ( p ) . (6.11) A similar transformation of equation (6.9) leads to the equation γ X k coth( γ ( x i ( p ) − x k ( p − 1)) b α k ( p − 1) b ν i ( p ) a ν k ( p − 1) = γ X k 6 = i coth( γ ( x i ( p ) − x k ( p )) b α k ( p ) b ν i ( p ) a ν k ( p ) + ˙ x i ( p ) 2 b α i ( p ) + µb α i ( p ) . (6.12) Multiply the first eq uation b y b α i ( p ) and sum o v er α , then multiply the second equation b y a α i ( p ), sum ov er α and tak e into accoun t the constraint b ν i a ν i = 1. Subtracting the resulting equations, w e eliminate ˙ x i ( p ) and o btain the equations of motion ( 1 .4): X j coth( γ ( x i ( p ) − x j ( p + 1)) b ν i ( p ) a ν j ( p + 1)) b β j ( p + 1) a β i ( p ) + X j coth( γ ( x i ( p ) − x j ( p − 1)) b ν i ( p ) a ν j ( p − 1)) b β j ( p − 1) a β i ( p ) = 2 X j 6 = i coth( γ ( x i ( p ) − x j ( p )) b ν i ( p ) a ν j ( p )) b β j ( p ) a β i ( p ) . (6.13) These equations of motion generalize the o nes f o r the rational G ibb ons-Hermsen mo del obtained in [19]. They lo ok lik e the Bethe ansatz equations for the quan tum trigonometric Gaudin mo del “dressed” b y the spin v ariables. 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