KP and Toda tau functions in Bethe ansatz

March 29, 2 022 7:52 WSPC - Proceedings T rim Size: 9in x 6in infan09 1 KP and T o da tau functions in Bethe ansatz Kanehisa T ak asaki Gr aduate Scho ol of Human and Enviro nment al Studies Kyoto U niversity Y oshida, Sakyo, Ky oto, 606-8501, Jap an takasaki@math.h.kyoto-u.ac.jp Recen t work of F oda and his group on a connection betw een classical i n tegrable hierarchies (the KP and 2D T oda hierarc hies) and some quan tum integrable systems (the 6-ve rtex mo del wi th DWBC, the finite XXZ chain of spin 1 / 2, the phase m o del on a finite cha in, etc.) is reviewed. Some additional inform ation on this iss ue is also presen ted. Keywor ds : six-vertex model; XXZ mo del; domain wall b oundary condition; Bethe ansatz; KP hierarch y; T oda hierarch y; tau function 1. In tro d uctio n Searching for a connection b etw een class ic al and qua nt um in tegrable sys- tems is an o ld a nd new sub ject, o ccas ionally leading to a breakthro ugh to- wards a new a rea of rese a rch. One of the landmarks in this sense is the quan- tum inv erse sca ttering metho d, also known as the a lgebraic Bethe ansa tz. Stemming from the classical inverse scattering metho d, the a lgebraic Bethe ansatz cov e r s a wide class o f integrable systems including so lv able mo dels of statistical mec ha nics on the basis o f the Y ang -Baxter equations. 1 More- ov er, rema rk ably , it was recognized later that a kind of cla ssical integrable systems (discrete Hirota equations) show up in the so called nested Bethe ansatz. 2 Recently a new connection was found b y F oda and his group. 3–7 They observed that specia l solutio ns of the classical integrable hier archies (the KP and 2D T oda hier archies) a re hidden in quantum (or s tatistical) inte- grable sy stems such as the 6-vertex mo del under the domain wall b oundary condition (D WBC), 3 the finite XXZ chain of spin 1 / 2 , 4,5 and some other quantum int egrable systems. 6,7 Their r esults are bas e d on a determinant formula of physical quantities, namely , the Izergin- Korepin formula for the March 29, 2 022 7:52 WSPC - Proceedings T rim Size: 9in x 6in infan09 2 partition function o f the 6-vertex mo del 8–10 and the Slavnov for mula for the scalar pro duct o f Bethe states in the XXZ spin chain. 11,12 Those for mulae contain a set of free v ariables, and the determinant in the formula is divided by the V a ndermonde determinant of these v ariables. F oda et a l. interpreted the quotient of the determinant by the V ander mo nde determinants as a tau function of the KP (or 2 D T oda ) hierarch y expressed in the s o called “Miwa v ariables”. In this paper , we r eview these results along with some additiona l infor - mation on this issue. W e are particular ly int erested in the rele v a nce o f the 2-comp onent K P (2-KP) and 2D T oda hier archies. Unfortunately , this re- search is s till in a n early stage, and we cannot definitely s ay which dir ection this research leads us to. A mo des t go al will b e to unders tand the algebraic Bethe a nsatz b etter in the p er sp ective of class ical integrable hierarchies. This pap er is orga nized a s follows. In Section 2, we start with a brief account of the notion of Sch ur functions that play a fundamen tal role in the theory of integrable hierarchies, and in tro duce the tau function o f the KP , 2-KP and 2D T oda hierarchies as a function of b oth the usual time v ariables and the Miwa v aria bles. Sectio n 3 deals with the par tition function of the 6-v ertex model with D WBC. F ollo wing the pro cedure o f F o da et al., we rewrite the Izergin-Kor epin for mula into an almost rational form and sho w that a main par t o f the par titio n function can be in terpreted as a KP ta u function. Actually , the pa r tition function allows tw o differ ent int erpretations tha t corresp ond to t wo choices of the Miw a v ar iables. W e examine a unified interpretation of the partition function as a tau function of the 2-KP (o r 2D T o da ) hierarch y . In Section 4, we turn to the finite XXZ chain of s pin 1 / 2, and present a similar interpretation to the scalar pr o duct of Bethe states (one of which dep ends on free v ariables ) on the basis of the Slavnov formula. Section 5 is devoted to some other mo dels including the phase mo del, 13 which is a ls o studied by the gro up of F o da. 6 F or those mo dels, a determinant formula is known to hold for the sc a lar pr o duct of Bethe s ta tes b o th of whic h depend on free parameters . 14 W e consider a sp ecial cas e related to enumeration of b oxed plane pa rtitions. 2. T au functions 2.1. Schu r functions Let us review the notion o f Sch ur functions. W e mostly follow the notatio ns of Ma c donald’s b o ok. 15 F or N v ariable s x = ( x 1 , . . . , x N ) and a partition λ = ( λ 1 , λ 2 , . . . , λ N ) March 29, 2 022 7:52 WSPC - Proceedings T rim Size: 9in x 6in infan09 3 ( λ 1 ≥ λ 2 ≥ · · · ≥ λ N ≥ 0) of length ( λ ) ≤ N , the Sch ur function s λ ( x ) can be defined b y W eyl’s character formula s λ ( x ) = det( x λ i − i + N j ) N i,j =1 ∆( x ) , (1) where ∆( x ) is the V ander monde determinant ∆( x ) = det( x − i + N j ) N i,j =1 = Y 1 ≤ i l ( λ ) and j ≤ l ( λ ) is zero a nd the low er right blo ck for i, j > l ( λ ) is an upper triang ular matrix w ith 1 on the diagona l line. This is a pla c e where a connectio n with the KP hierarchy 17 shows up. Namely , the v a r iables t = ( t 1 , t 2 , . . . ) a r e nothing but the “time v ar iables” of the K P hierarchy , and the Sch ur functions s λ [ t ] ar e sp ecial ta u functions. As first p ointed o ut by Miwa, 18 viewing the tau function as a function of the x v a riables leads to a discr ete (or difference) a nalogue of the KP hiera rch y . F or this reason, the x v a riables are sometimes referre d to as “ Miwa v aria bles” in the literature o f integrable systems. 2.2. T au functions of KP hier ar chy Let us use the notation τ [ t ] for the tau function in the usual sense (namely , a function of t ), and let τ ( x ) denote the function o btained from τ [ t ] b y the c ha nge of v aria bles (4). It is the latter that plays a central role in this pap er. A general tau function o f the KP hierar ch y is a linear co mbination of the Sch ur functions τ [ t ] = X λ c λ s λ [ t ] , (7) where the coefficie nt s c λ are Pl ¨ uck er co ordina tes of a po int of an infinite dimensional Grass ma nn manifold (Sato Gra ssmannian). 19 Roughly spea k- ing, the Sato Gra s smannian cons is ts of linear subspaces W ≃ C N of a fixed a These conv enient notations are b orrow ed from Zi nn-Justin’s pap er. 16 March 29, 2 022 7:52 WSPC - Proceedings T rim Size: 9in x 6in infan09 5 linear s pa ce V ≃ C Z . W e s hall not pursue those fully gener al tau functions in the following. W e are int erested in a smaller (but yet infinite dimensional) clas s of tau functions such that c λ = 0 for all partitions with l ( λ ) > N . This co rresp o nds to a submanifold Gr( N , ∞ ) of the full Sato Grass mannian. The Pl¨ uck er co ordinates c λ are la b elled by partitions of the for m λ = ( λ 1 , . . . , λ N ), and given by finite determina nts as c λ = det( f i,l j ) N i,j =1 , l i := λ i − i + N . (8) Note that the sequence s λ i ’s and l i ’s of non-neg a tive in tegers are in one-to- one cor resp ondence: ∞ > λ 1 ≥ · · · λ N ≥ 0 ← → ∞ > l 1 > · · · > l N ≥ 0 . The N × ∞ matr ix F = ( f ij ) i =1 ,...,N , j =0 , 1 ,... of para meter s represent a p oint of the Grassmann manifold Gr( N , ∞ ). By the Ca uch y-Binet form ula, the tau function τ ( x ) in the x -picture can b e expressed as τ ( x ) = X ∞ >l 1 > ··· >l N ≥ 0 det( f i,l j ) N i,j =1 det( x l j i ) N i,j =1 ∆( x ) = det( f i ( x j )) N i,j =1 ∆( x ) , (9 ) where f i ( x )’s a re the p ower se ries of the form f i ( x ) = ∞ X l =0 f il x l . In par ticular, if there is a p ositive integer M suc h that f ij = 0 for i ≥ M + N (in other words, f i ( x )’s ar e poly nomials of degree less than M + N ), the Pl ¨ uc ker co o rdinate c λ v a nishes for a ll Y oung dia grams not co nt ained in the N × M recta ngular Y oung dia gram, namely , c λ = 0 for λ 6⊆ ( M N ) := ( M , . . . , M | {z } N ) The tau function τ [ t ] thereby b ecomes a linea r combination o f a finite num- ber o f Sch ur function, hence a po lynomial in t . Geometrically , these so lu- tions o f the KP hierarch y sit on the finite dimensional Grass mann manifold Gr ( N , N + M ) of the Sato Gra s smannian. March 29, 2 022 7:52 WSPC - Proceedings T rim Size: 9in x 6in infan09 6 2.3. T au functions of 2-KP hier ar chy The tau function τ [ t , ¯ t ] of the 2-comp o nent KP (2- KP) hierar ch y is a func- tion of tw o sequences t = ( t 1 , t 2 , . . . ) and ¯ t = ( ¯ t 1 , ¯ t 2 , . . . ) of time v ariables, and ca n b e expressed as τ [ t , ¯ t ] = X λ,µ c λµ s λ [ t ] s µ [ ¯ t ] , (10) where c λµ ’s are Pl ¨ uc ker coo rdinates of a p oint of a 2-comp onent ana logue of the Sato Grassmannnia n (which is, actually , isomorphic to the one- comp onent version). 19 The a forementioned class of tau functions of the KP hier arch y can be generalized to the 2-comp onent case . Such tau functions corresp ond to po ints of the submanifold Gr( M + N , 2 ∞ ) of the 2-co mp o ne nt Sa to Grass- mannian. F or those tau functions, the P l ¨ uc ker co ordina tes c λµ v a nish if l ( λ ) > M or l ( µ ) > N ; the rema ining Pl ¨ uck er co ordina tes are given by finite determinants o f a matrix with tw o rectangular blo cks of size ( M + N ) × M and ( M + N ) × N a s c λµ = det( f i,l j | g i,m k ) , (11) where i is the row index ranging ov er i = 1 , . . . , M + N and j, k are column indices in the tw o blo cks r anging o ver j = 1 , . . . , M and k = 1 , . . . , N , resp ectively . l j ’s and m k ’s ar e rela ted to the par ts of λ = ( λ j ) M j =1 and µ = ( µ i ) N j =1 as l j = λ j − j + M , m k = µ k − k + N . By the c ha nge of v aria bles from x and y to t n = 1 n M X j =1 x n j , ¯ t n = 1 n N X k =1 y n k , (12) the tau function τ [ t , ¯ t ] is conv er ted to the ( x , y )-picture τ ( x , y ). Aga in by the Cauch y-Binet formula, τ ( x , y ) turns out to b e a quotient of tw o determinants as τ ( x , y ) = det( f i ( x j ) | g i ( y k )) ∆( x )∆( y ) , (13) where the deno minator is the determinan t with the same blo ck structure as (11), and f i ( x ) a nd g j ( y ) are power s eries of the form f i ( x ) = ∞ X l =0 f il x l , g i ( y ) = ∞ X l =0 g il y l . March 29, 2 022 7:52 WSPC - Proceedings T rim Size: 9in x 6in infan09 7 2.4. T au function of 2D T o da hier ar chy The 2 -KP hierarch y is closely r e lated to the 2D T o da hiera rch y. 20 The tau function τ s [ t , ¯ t ] of the 2D T o da hierarch y depends on a discre te v ar iable (lattice co ordinate) s alongside the t wo series of time v ariables t and ¯ t . F or eac h v alue of s , τ s [ t , ¯ t ] is a tau function of the 2-KP hierar chy , and these 2-KP tau functions are m utually connected by a kind o f B¨ acklund transformatio ns. Consequently , τ s [ t , ¯ t ] can b e expressed as sho w n in (10) with the co efficients c sλµ depe nding on s . Actually , it is more natural to use s λ [ t ] s µ [ − ¯ t ] ra ther tha n s λ [ t ] s µ [ ¯ t ] for the Schur function ex pa nsion of the T o da tau function. 21 (Note that s µ [ − ¯ t ] can b e rewritten as s µ [ − ¯ t ] = ( − 1) | µ | s t µ [ t ] , where t µ denotes the transp ose of µ .) Expanded in these product of tau functions a s τ s [ t , ¯ t ] = X λ,µ c sλµ s λ [ t ] s µ [ − ¯ t ] , (14) the co efficie nts c sλµ bec ome Pl ¨ uck er co or dinates of a n infinite dimensional flag manifold. In tuitively , they a re minor determinants c sλµ = det( g λ i − i + s,µ j − j + s ) ∞ i,j =1 (15) of a n infinite matrix g = ( g ij ) i,j ∈ Z , thoug h this definition r equires justifi- cation. 21 In pa rticular, if g is a diagona l matrix, the co efficients c sλµ are also diago nal (na mely , c sλµ ∝ δ λµ ) and the Sch ur function expansio n (14) simplifies to the “diag onal” form τ s [ t , ¯ t ] = X λ c sλ s λ [ t ] s λ [ − ¯ t ] , c sλ = ∞ Y i =1 g λ i − i + s . (16) If we reformulate the 2D T o da hierarch y on the s e mi- infinite lattice s ≥ 0 , the infinite determinants defining c λ ’s ar e r eplaced by finite determi- nants, and τ s [ t , ¯ t ] itself b ecomes a finite deter minant. W e shall e ncounter an example of such ta u functions in the next section. 3. 6-v ertex mo del with DWBC 3.1. Setup of mo del W e consider the 6-vertex mo del on an N × N square lattice with inho- mogeneity parameter s u = ( u 1 , . . . , u N ) and v = ( v 1 , . . . , v N ) assigned to March 29, 2 022 7:52 WSPC - Proceedings T rim Size: 9in x 6in infan09 8 ✲ ✲ ✲ ✲ ✲ ✲ ✛ ✛ ✛ ✛ ✛ ✛ ❄ ❄ ❄ ❄ ❄ ❄ ✻ ✻ ✻ ✻ ✻ ✻ u 1 u 2 · · · u N v 1 v 2 · · · v N Fig. 1. Square lattice with DWBC the rows a nd columns. The b o undary of the lattice is supplemen ted with extra edg es p ointing outw a rds, and the domain-wall bounda r y conditio n (D WBC) is impo sed on these extra edges. Namely , the a rrows on the extra edges on top and b ottom of the b oundary are po inting outw ards, a nd those on the other extra edges are po int ing inw ar ds (see fig ure 1). The v ertex at the in ter section of the i -th row and the j -th column is given the following weigh t w ij determined by the co nfiguration of a rrows on the adjacent edg es: ✲ ✲ ✻ ✻ ✛ ✛ ❄ ❄ w ij = a ( u i − v j ) ✲ ✲ ❄ ❄ ✛ ✛ ✻ ✻ w ij = b ( u i − v j ) ✲✛ ❄ ✻ ✛ ✲ ✻ ❄ w ij = c ( u i − v j ) The weight functions a ( u ) , b ( u ) , c ( u ) are defined as a ( u ) = sinh( u + γ ) , b ( u ) = sinh u, c ( u ) = sinh γ , (17) where γ is a parameter . Thus the pa r tition function of this mo del is defined March 29, 2 022 7:52 WSPC - Proceedings T rim Size: 9in x 6in infan09 9 as a function of the inhomoge ne ity pa rameters u and v : Z N = Z N ( u , v ) = X configuratio n N Y i,j =1 w ij . 3.2. Izer gin-Kor epin formula for Z N According to the r e sult of Korepin 8 and Izergin, 9 the partition function Z N has the determinant formula Z N = Q N i,j =1 sinh( u i − v j + γ ) sinh( u i − v j ) Q 1 ≤ i