T-system and thermodynamic Bethe ansatz equations for solvable lattice models associated with superalgebras

T -system and thermo dynamic Bethe ansatz equations for solv able lattice mo dels asso ciated with sup eralgebras ∗ Zengo Tsub oi Graduat e Sc ho ol of Mathemat ical Sciences, Universit y of T okyo † Abstract An analytic Bethe ansatz is carried out related to the Lie sup eral- gebra osp (1 | 2 s ). W e present an eigenv alue form ula of a transfer matrix in dressed v acuum form (D VF) l ab eled by a Y oun g (sup er) diagram. Remark able dualit y a mong D VFs is found . A complete set of transfer matrix functional r elations ( T -system) is p rop osed as a reduction of a Hirota-Miw a equation. W e also derive a thermo dy namic Bethe ans atz (TBA) equation from this T -system and the quan tum trans fer matrix metho d. This TBA equation is iden tical to the one from the string h yp othesis. Journal-ref: RIMS Kokyuroku 1280 (200 2 ) 19-34 URL: h ttp://rep o sitor y .kulib.ky o to-u.ac.jp/dspace/bitstream/2433/423 56/1/1280 03.p df 1 In tro du ction Solv able lattice mo dels related to Lie superalg ebras [1] hav e r eceiv ed m uc h atten tions [2, 3, 4, 5, 6, 7, 8, 9]. T o construct eigen v alue form ulae of tra nsfer matrices for suc h mo dels is an imp ortan t problem in mathematical physic s. T o ac hiev e this prog ram, the Bethe ansatz has b een often used. ∗ This is a review pap er submitted to the pro ceedings of the workshop: ‘Bilinear Metho d in the Study of In tegr able Sys tems and Related T opics’, RIMS, K yoto, July , 2001 (URL: ht tp://rep os itory .kulib.kyoto-u.ac.jp/dspa ce/handle/24 33/40854 ). F or more details , see the original pa per s [3 1, 40]. † present address (on January 2010): Ok ay ama Institute for Quantum Physics, 1-9-1 Kyo yama, Ok ayama 700 -0015 , Japan 1 No w adays, there is muc h literature (see for example, [4 , 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25] and referenc es therein.) on Bethe ansatz analysis fo r solv able lattice mo dels related to Lie sup eralgebras. Ho w eve r, most of it deals only with models related to simple represen tations lik e f undamen tal ones. Only a few p eople (see f or example, [16, 18]) tried to deal with more complicated mo dels suc h as fusion mo dels [26] b y the Bethe ansatz; while there w as no systematic study on this sub j ect. T o address suc h situatio ns, w e ha v e recen tly executed [27, 28, 29 , 30, 31] an analytic Bethe ansatz [32, 33, 34, 35] systematically related to the Lie sup eralgebras sl ( r + 1 | s + 1) , B ( r | s ) , C ( s ) , D ( r | s ) cases. Namely , w e hav e prop osed a set of dressed v acuum forms ( DVFs) and a class of functional rela- tions ( T -system) for it. Moreov er w e ha v e also studied thermo dynamic Bethe ansatz (TBA) equations [36] related to osp (1 | 2) [37, 38, 39] and o sp (1 | 2 s ) [40] from the p o int of view of the string h yp othesis [41, 42] and the quan tum transfer matrix (QTM) metho d [43, 44, 45, 46 , 47, 22]. In this pap er, w e briefly review the T - system a nd the TBA equation related to the Lie sup eralgebra osp (1 | 2 s ) = B (0 | s ) based o n [3 1, 40]. A f- ter a brief review on the Lie sup eralgebra osp (1 | 2 s ), w e introduce a QTM for osp (1 | 2 s ) mo del[17] in section 3 . In section 4, w e carry out an analytic Bethe ansatz based on the Bethe ansatz equation (BAE) (1 3) and obtain the eigenv alue f o rm ula for the QTM. W e define the dressed v acuum f orm (D VF) T λ ⊂ µ ( v ) lab eled b y a sk ew-Y o ung (sup er) diagram λ ⊂ µ as a sum- mation ov er semi-standard tableaux. This D VF has a determinant expression (quan tum sup ersymmetric Jacobi-T rudi formula). In particular, for a rect- angular Y oung (sup er) diagram, this D VF satisfies a kind o f Hirota- Miw a equation[48, 4 9]. By considering a reduction to this equation, we deriv e t he osp (1 | 2 s ) version of the T -system. Based on this T - system, w e deriv e the TBA equation from the QTM metho d in section 5. Namely , w e consider a dep endan t v ariable transformation, and deriv e the Y -system from the T - system. Then w e t r a nsform the Y - system with certain analytical conditions in t o the TBA equation. Moreo ver w e find that this TBA equation coincides with the one from the string hy p othesis. This indicates the v alidit y o f the string h yp othesis for the osp (1 | 2 s ) mo del. 2 The Lie su p eralg ebra osp (1 | 2 s ) In this section, w e briefly men tion the Lie sup eralgebra B (0 | s ) = osp (1 | 2 s ) for s ∈ Z ≥ 1 (see for example [1, 50, 51]). In con trast to o ther Lie sup eralgebras, the simple ro ot system of osp (1 | 2 s ) is unique and given as follo ws (see Figure 1): 2 ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ α 1 α 2 α s − 2 ✍✌ ✎☞ ⑦ α s − 1 α s ❅ ❅   Figure 1 : Dynkin diagram for the Lie sup eralgebra B (0 | s ) = osp ( 1 | 2 s ) ( s ≥ 1): white circles denote ev en ro ots; a blac k circle denotes a n o dd ro ot. α i = δ i − δ i +1 for i = 1 , 2 , . . . , s − 1 , α s = δ s (1) where δ 1 , . . . , δ s are the bases of the dual space of the Cartan subalgebra with the bilinear f o rm ( | ) suc h that ( δ i | δ j ) = − δ i j (2) { α i } i 6 = s are ev en ro ots and α s is an o dd ro o t with ( α s | α s ) 6 = 0. Let λ ⊂ µ b e a sk ew-Y oung (sup er) diagram lab eled b y the sequence s of non-nega t ive in t egers λ = ( λ 1 , λ 2 , . . . ) and µ = ( µ 1 , µ 2 , . . . ) suc h that µ i ≥ λ i : i = 1 , 2 , . . . ; λ 1 ≥ λ 2 ≥ · · · ≥ 0; µ 1 ≥ µ 2 ≥ · · · ≥ 0 and µ ′ = ( µ ′ 1 , µ ′ 2 , . . . ) be the conjugate of µ . In particular, f or λ = φ , µ 1 ≤ s case, the Kac-Dynkin lab el [ b 1 , b 2 , . . . , b s ] is related t o the Y oung (sup er) diag ram with shap e µ = ( µ 1 , µ 2 , . . . ) as follo ws: b i = µ ′ i − µ ′ i +1 for i ∈ { 1 , 2 , . . . , s − 1 } , b s = 2 µ ′ s . (3) An irreducible represen tation with the Kac-D ynkin lab el [ b 1 , b 2 , . . . , b s ] is fi- nite dimensional if and only if b j ∈ Z ≥ 0 for j ∈ { 1 , 2 , . . . , s − 1 } , b s ∈ 2 Z ≥ 0 . (4) 3 osp (1 | 2 s ) mo d el and QTM met ho d In this section, w e introduce an in tegrable spin c hain[17 , 19 ] asso ciated with the fundamen ta l represen tat io n of osp (1 | 2 s ), and define a QTM. The ˇ R - matrix[5, 8, 9, 19 ] of the mo del is giv en as ˇ R ( v ) = I + v P − 2 v 2 v − g E , (5) 3 where g = 2 s + 1; P cd ab = ( − 1) p ( a ) p ( b ) δ ad δ bc ; E cd ab = α ab ( α − 1 ) cd ; a, b, c, d ∈ J = { 1 , 2 , . . . , s, 0 , s, . . . , 2 , 1 } (1 ≺ 2 ≺ · · · ≺ s ≺ 0 ≺ s ≺ · · · ≺ 2 ≺ 1); α is (2 s + 1) × (2 s + 1 ) anti-diagonal matrix whose non-zero elemen ts are α a, a = 1 for a ∈ { 1 , 2 , . . . , s, 0 } and α a,a = − 1 for a ∈ { s, s − 1 , . . . , 1 } ; a = a ; p ( a ) = 0 for a = 0; p ( a ) = 1 for a ∈ { 1 , 2 , . . . , s } ⊔ { s, . . . , 2 , 1 } . The Hamiltonian of the presen t mo del for the p erio dic b oundary condition is g iven b y H = J L X k =1  P k ,k +1 + 2 g E k ,k +1  , (6) where L is the n um b er of the lattice sites ; P k ,k +1 and E k ,k +1 act non tr ivially on the k th site and k + 1 th site. There ar e sev eral form ulations of QTM for graded v ertex mo dels. W e consider the case where the transfer m atrix is defined as the o r dinary trace of a mono drom y mat r ix. The QTM is defined as T (1) 1 ( u, v ) = T r j N 2 Y k =1 R a 2 k ,j ( u + iv ) e R a 2 k − 1 ,j ( u − iv ) , (7) where R cd ab ( v ) = ˇ R cd ba ( v ); e R j k ( v ) = t k R k j ( v ) ( t k is the tra nsp osition in the k -th space); N is t he T rotter num b er and assumed to ev en. By using the la r gest eigen v alue T (1) 1 ( u N , 0) of the QTM (7), the fr ee energy densit y is expressed as F = − 1 β lim N →∞ log T (1) 1 ( u N , 0) , (8) where u N = − J β N ( β = 1 / ( k B T ); k B : the Boltzmann constan t; T : the temp erature). F rom no w on, w e abbreviate t he parameter u in T (1) 1 ( u, v ). 4 Analytic Bethe ansatz and T -s ystem for QTM One can obtain the eigen v alue formu lae of the QTM (7) b y replacing the v acuum part of the DVF for the r ow-to-ro w transfer mat r ix [17, 19] with that of the QTM. Explicitly w e ha v e T (1) 1 ( v ) = X a ∈ J a v , (9) 4 where the functions { a v } a ∈ J are defined as a v = ψ a ( v ) Q a − 1 ( v + i 2 ( a + 1) ) Q a ( v + i 2 ( a − 2)) Q a − 1 ( v + i 2 ( a − 1)) Q a ( v + i 2 a ) for a ∈ { 1 , 2 , . . . , s } , 0 v = ψ 0 ( v ) Q s ( v + i 2 ( s − 1)) Q s ( v + i 2 ( s + 2)) Q s ( v + i 2 ( s + 1)) Q s ( v + i 2 s ) , (10) a v = ψ a ( v ) Q a − 1 ( v − i 2 ( a − 2 s )) Q a ( v − i 2 ( a − 2 s − 3)) Q a − 1 ( v − i 2 ( a − 2 s − 2)) Q a ( v − i 2 ( a − 2 s − 1)) for a ∈ { 1 , 2 , . . . , s } , where Q 0 ( v ) := 1; ψ a ( v ) is the v acuum part ψ a ( v ) =        ζ 1 φ + ( v ) φ − ( v + i ) φ + ( v − 2 s − 1 2 i ) φ + ( v − 2 s +1 2 i ) for a = 1 , ζ a φ + ( v ) φ − ( v ) for 2  a  2 , ζ 1 φ − ( v ) φ + ( v − i ) φ − ( v + 2 s − 1 2 i ) φ − ( v + 2 s +1 2 i ) for a = 1 , (11) where φ ± ( v ) = ( v ± iu ) N 2 ; ζ a is a phase factor: ζ a =            ( − 1) N − M 1 if a = 1 ( − 1) M a − 1 − M a if a ∈ { 2 , 3 , . . . , s } 1 if a = 0 ( − 1) M a − 1 − M a if a ∈ { s, . . . , 3 , 2 } ( − 1) N − M 1 if a = 1 , (12) where a = a . The complex v a riables { v ( a ) k } a r e ro ots of the following Bethe ansatz equation N Y j =1 v ( a ) k − w ( a ) j + i 2 δ a 1 v ( a ) k − w ( a ) j − i 2 δ a 1 ! = − ( − 1) M a − 1 − M σ ( a +1) s +1 Y d =1 Q σ ( d ) ( v ( a ) k + i 2 B ad ) Q σ ( d ) ( v ( a ) k − i 2 B ad ) , (1 3) where k ∈ { 1 , 2 , . . . , M a } ; a ∈ { 1 , 2 , . . . , s } ; σ ( d ) = d fo r 1 ≤ d ≤ s ; σ ( s + 1) = s ; B ad = 2 δ ad − δ a,d +1 − δ a,d − 1 ; Q a ( v ) = Q M a k =1 ( v − v ( a ) k ); M a ∈ Z ≥ 0 ; M 0 = N . The parameter σ expresses an effect of ‘a p eculiar tw o-b o dy self- in t era ctio n for the r o ot { v ( s ) k } ’ [1 9], whic h originates from the o dd simple ro ot α s with ( α s | α s ) 6 = 0. One may interpret the QTM a s a transfer matrix of an inhomogeneous v ertex mo del. In our case, the inhomogeneity para meters w ( a ) j ∈ C tak e the v alues: w ( a ) j = iuδ a 1 for j ∈ 2 Z ≥ 1 ; w ( a ) j = ( − iu + ig 2 ) δ a 1 for j ∈ 2 Z ≥ 0 + 1. The dress part of the D VF (9) is free of po les under the BAE (13). This is a requiremen t from the analytic Bethe ansatz [32]. 5 No w we will presen t a DVF T λ ⊂ µ ( v ) fo r a ‘fusion QTM’. W e can deriv e the explicit expression o f T λ ⊂ µ ( v ) b y mo difying the v acuum part of t he D VF in Ref. [31] so that the v a cuum part is c ompatible with the left hand side of the BAE (13 ) . W e assign co ordinates ( i, j ) ∈ Z 2 on the sk ew-Y o ung (sup er) diagram λ ⊂ µ suc h tha t the ro w index i increases as w e go down w ards and the column index j increases as w e go f r o m the left to the right and that (1 , 1) is on the top left corner of µ . W e define an admissible tableau b on the sk ew-Y oung (sup er) diagram λ ⊂ µ as a set of elemen ts b ( i, j ) ∈ J lab eled b y the co or dina t es ( i, j ) men tioned a b o v e, w ith the follo wing rule (admis sibilit y conditions). b ( i, j ) ≺ b ( i, j + 1 ) , b ( i, j )  b ( i + 1 , j ) . (14) Let B ( λ ⊂ µ ) b e the set of admissible tableaux o n λ ⊂ µ . F or any ske w- Y o ung (sup er) diagram λ ⊂ µ , define T λ ⊂ µ ( v ) as fo llo ws T λ ⊂ µ ( v ) = X b ∈ B ( λ ⊂ µ ) Y ( j,k ) ∈ ( λ ⊂ µ ) b ( j, k ) v − i 2 ( − µ 1 + µ ′ 1 − 2 j +2 k ) , (15) where the pro duct is tak en o v er the co ordinates ( j, k ) on λ ⊂ µ . Let T ( a ) m ( v ) := T ( a m ) ( v ). The follow ing determinant form ula (quan tum sup er- symmetric Jacobi-T rudi formula) should b e v alid (cf. [3 5 ]). T λ ⊂ µ ( v ) = det 1 ≤ j,k ≤ µ ′ 1 ( T ( µ k − λ j + j − k ) 1 ( v − i 2 ( − µ 1 + µ ′ 1 + µ ′ k + λ ′ j − j − k + 1))) . (16) W e ma y think o f (15) as an osp (1 | 2 s ) ve rsion of the Bazhanov and Reshetikhin’s eigen v alue formu la [33 ]. In particular, f o r λ = φ , µ 1 ≤ s case, the ‘top term’ of T µ ( v ) will b e t he term corresp onding t o the tableau b ( i, j ) = j (1 ≤ i ≤ µ ′ j , 1 ≤ j ≤ s ). This term carries the osp (1 | 2 s ) we igh t with the Kac-Dynkin lab el (3) (in the sense in R ef. [3 4]). DVFs hav e so called Bethe-s tr ap structures [34] and w e confirmed, for sev eral examples, t ha t T λ ⊂ µ ( v ) coincides with the Bethe-strap of the minimal connected comp onen t whic h inc ludes the to p term as the examples in Figure 2, Figure 3 and Fig ure 4. T λ ⊂ µ ( v ) ma y b e view ed as a prototype o f a ‘ q - sup ercharacter’ (cf. [52]). No w w e intro duce the functional relations among DVFs. The followin g relation follows from the determinan t formula (16). T ( a ) m ( v + i 2 ) T ( a ) m ( v − i 2 ) = T ( a ) m +1 ( v ) T ( a ) m − 1 ( v ) + T ( a − 1) m ( v ) T ( a +1) m ( v ) , ( 1 7) where a, m ∈ Z ≥ 1 . This functiona l relation is a kind of Hirot a -Miw a equation [48, 4 9] and can b e prov ed b y the Jacobi iden tit y . The followin g theorem follo ws from the a dmissible condition (14). 6 1 ✲ (1 , 1) 2 ✲ (2 , 2) 0 ✲ (2 , 3) 2 ✲ (1 , 4) 1 Figure 2: The Bethe-strap structure of T (1) 1 ( v ) for osp (1 | 4): The pair ( a, b ) denotes the common p ole v ( a ) k − i 2 b of the pair of the ta bleaux connected by the ar r ow. This common p ole v anishes under the BAE ( 1 3). The leftmost tableau corresp onds to the ‘highest w eigh t ’, whic h is called the top term . This term carries the osp (1 | 4) w eight δ 1 . Theorem 1 T λ ⊂ µ ( v ) = 0 if λ ⊂ µ c ontains m × a r e ctangular sub diagr am ( m : the numb er of r ow, a : the numb er of c olumn) w ith a ∈ Z ≥ 2 s +2 and m ∈ Z ≥ 1 . In p articular, we have T ( a ) m ( v ) = 0 if a ∈ Z ≥ 2 s +2 and m ∈ Z ≥ 1 . (18) There is a remark able duality for T ( a ) m ( v ). Theorem 2 F or any a ∈ { 1 , . . . , s } and m ∈ Z ≥ 0 , we have T ( a ) m ( v ) = M ( a ) m ( v ) T (2 s − a +1) m ( v ) , (19) wher e M ( a ) m ( v ) is give n as M ( a ) m ( v ) = m Y j =1  ψ 1 ( v − i 2 ( m − a − 2 j + 2)) ψ 1 ( v − i 2 ( m − 2 s + a − 2 j + 1)) × Q a k =2 ψ 2 ( v − i 2 ( m − a − 2 j + 2 k )) Q 2 s − a +1 k =2 ψ 2 ( v − i 2 ( m − 2 s + a − 2 j + 2 k − 1)) ) . (20) F or a ∈ { 1 , 2 , . . . , s } and m ∈ Z ≥ 1 , w e define a no rmalization function N ( a ) m ( v ) = Q m j =1 Q a k =1 φ − ( v − m − a − 2 j +2 k 2 i ) φ + ( v − m − a − 2 j +2 k 2 i ) φ − ( v − m − a 2 i ) φ + ( v + m − a 2 i ) . (21) W e reset T ( a ) m ( v ) / N ( a ) m ( v ) to T ( a ) m ( v ), where T ( a ) m ( v ) is defined by (15) . By using t he Theorem 1,2, w e can obtain the T - system as a reduction of the Hirota-Miw a equation (1 7 ). T ( a ) m ( v + i 2 ) T ( a ) m ( v − i 2 ) = T ( a ) m +1 ( v ) T ( a ) m − 1 ( v ) + T ( a − 1) m ( v ) T ( a +1) m ( v ) for a ∈ 1 , 2 , . . . , s − 1 , (22) T ( s ) m ( v + i 2 ) T ( s ) m ( v − i 2 ) = T ( s ) m +1 ( v ) T ( s ) m − 1 ( v ) + g ( s ) m ( v ) T ( s − 1) m ( v ) T ( s ) m ( v ) , 7 1 1 2 1 2 2 0 1 0 2 2 1 0 0 2 2 1 1 2 0 1 2 2 2 1 0 1 2 1 1 ◗ ◗ ◗ ◗ s (1 , 5) ◗ ◗ ◗ ◗ s (2 , 4) ◗ ◗ ◗ ◗ s (1 , 5) ◗ ◗ ◗ ◗ s (2 , 4) ◗ ◗ ◗ ◗ s (1 , 5) ◗ ◗ ◗ ◗ s (2 , 3) ✑ ✑ ✑ ✑ ✰ (2 , 2) ✑ ✑ ✑ ✑ ✰ (2 , 1) ✑ ✑ ✑ ✑ ✰ (1 , 0) ✑ ✑ ✑ ✑ ✰ (2 , 1) ✑ ✑ ✑ ✑ ✰ (1 , 0) ✑ ✑ ✑ ✑ ✰ (1 , 0) ❄ (1 , 3) ❄ (2 , 2) ❄ (2 , 1) ❄ (2 , 3) ❄ (2 , 4) ❄ (1 , 2) P P P P P P P P P P q (1 , 5) ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✮ (1 , 0) Figure 3: The Bethe-strap structure o f T (1) 2 ( v ) for os p (1 | 4): The topmost tableau corresp onds to the ‘highest w eigh t ’, whic h is called the top term . This term carries the osp (1 | 4) w eight 2 δ 1 8 2 1 0 1 0 2 2 1 2 2 1 1 2 0 1 2 1 0 1 2 ❄ (2 , 3) ❄ (1 , 2) ❄ (2 , 2) ❄ (1 , 3) ❄ (2 , 4) ❄ (2 , 1) ❅ ❅ ❅ ❘ (1 , 3)    ✠ (2 , 3) ❅ ❅ ❅ ❘ (2 , 2)    ✠ (1 , 2) P P P P P P P P P q (1 , 3) ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✮ (1 , 2) Figure 4: The Bethe-strap structure o f T (2) 1 ( v ) for os p (1 | 4): The topmost tableau corresp onds to the ‘highest w eigh t ’, whic h is called the top term . This term carries the osp (1 | 4) w eight δ 1 + δ 2 9 where T ( a ) 0 ( v ) = φ − ( v + a 2 i ) φ + ( v − a 2 i ) for a ∈ Z ≥ 1 , T (0) m ( v ) = φ − ( v − m 2 i ) φ + ( v + m 2 i ) for m ∈ Z ≥ 1 , (23) g ( s ) m ( v ) = φ − ( v + m + s +1 2 i ) φ + ( v − m + s +1 2 i ) φ − ( v + m + s 2 i ) φ + ( v − m + s 2 i ) for m ∈ Z ≥ 1 . F or s = 1, g (1) m ( v ) T (0) m ( v ) coincides with the function T (0) m ( v ) in Ref. [38 ]. Since the dress part of the D VF T ( a ) m ( v ) is same as the row-to-ro w case, this functional equation (22) has essen tially the same for m as the osp (1 | 2 s ) T -system in Ref.[31]. 5 TBA e quation F or m ∈ Z ≥ 1 , w e define the Y -functions: Y ( a ) m ( v ) = T ( a ) m +1 ( v ) T ( a ) m − 1 ( v ) T ( a − 1) m ( v ) T ( a +1) m ( v ) for a ∈ { 1 , 2 , . . . , s − 1 } , Y ( s ) m ( v ) = T ( s ) m +1 ( v ) T ( s ) m − 1 ( v ) g ( s ) m ( v ) T ( s − 1) m ( v ) T ( s ) m ( v ) . (24) By using the T -system (22), one can show that the Y -functions satisfy t he follo wing Y -system: Y ( a ) m ( v + i 2 ) Y ( a ) m ( v − i 2 ) = (1 + Y ( a ) m +1 ( v ))(1 + Y ( a ) m − 1 ( v )) Q s d =1 (1 + ( Y ( d ) m ( v )) − 1 ) I ad , (25) where Y ( a ) 0 ( v ) = 0, a ∈ { 1 , 2 , . . . , s } and m ∈ Z ≥ 1 ; I ad = δ a,d − 1 + δ a,d +1 + δ ad δ as . A nume rical analysis for finite N , u, s indicates that a tw o-string solution (f o r ev ery color) in the sector N = M 1 = M 2 = · · · = M s of the BAE (13) pro vides the largest eigen v alue of the QTM (7) at v = 0. Moreov er, w e exp ect the follo wing conjecture is v alid for this tw o-string solution. Conjecture 1 F or smal l u ( | u | ≪ 1 ) and a ∈ { 1 , 2 , . . . , s } , every z er o of T ( a ) m ( v ) is lo c ate d o utside of the physic al strip Im v ∈ [ − 1 2 , 1 2 ] . Based on this conjecture, w e shall establish the ANZC prop ert y in some domain for the Y -functions (24) to transform the Y - system (25) to nonlin- ear integral equations. Here ANZC means Ana lytic NonZero and Constant 10 asymptotics in the limit | v | → ∞ . One can show that the Y -function has the follow ing asymptotic v alue lim | v | →∞ Y ( a ) m ( v ) = m ( g + m ) a ( g − a ) , (26) whic h is identifie d to the solution of the constant Y -system ( Y ( a ) m ) 2 = (1 + Y ( a ) m − 1 )(1 + Y ( a ) m +1 ) Q s d =1 (1 + ( Y ( d ) m ) − 1 ) I ad , (27) where Y ( a ) 0 := 0, a ∈ { 1 , 2 , . . . , s } and m ∈ Z ≥ 1 . F rom the Conjecture 1 and (26), w e find that t he functions 1 + Y ( a ) m ( v ), 1 + ( Y ( a ) m ( v )) − 1 in the domain Im v ∈ [ − δ , δ ] (0 < δ ≪ 1) and Y ( a ) m ( v ) for ( a, m ) 6 = (1 , 1) in the domain Im v ∈ [ − 1 2 , 1 2 ] (ph ysical strip) hav e the ANZC prop ert y . On the other hand, Y (1) 1 ( v ) has zeros of order N / 2 at ± i ( 1 2 − u ) if u > 0 ( J < 0), p oles of order N/ 2 at ± i ( 1 2 + u ) if u < 0 ( J > 0 ) in the ph ysical strip. Then we m ust mo dify Y (1) 1 ( v ) as e Y ( a ) m ( v ) = Y ( a ) m ( v )  tanh π 2 ( v + i ( 1 2 ± u )) tanh π 2 ( v − i ( 1 2 ± u ) )  ± N δ a 1 δ m 1 2 , (28) where the sign ± is identical to that of − u . T aking note on the relation tanh π 4 ( v + i ) tanh π 4 ( v − i ) = 1 , (29) one can mo dify the lhs of the Y -system (25) as e Y ( a ) m ( v − i 2 ) e Y ( a ) m ( v + i 2 ) = (1 + Y ( a ) m +1 ( v ))(1 + Y ( a ) m − 1 ( v )) Q s d =1 (1 + ( Y ( d ) m ( v )) − 1 ) I ad , (30) for m ∈ Z ≥ 1 and a ∈ { 1 , 2 , . . . , s } . No w that the ANZC pro p ert y has b een establis hed for t he Y -system, we can transform (30 ) in to a system of nonlinear integral equations by a standard pro cedure. log Y ( a ) m ( v ) = ∓ N δ a 1 δ m 1 2 log  tanh π 2 ( v + i ( 1 2 ± u )) tanh π 2 ( v − i ( 1 2 ± u ))  + K ∗ log ( (1 + Y ( a ) m − 1 )(1 + Y ( a ) m +1 ) Q s d =1 (1 + ( Y ( d ) m ) − 1 ) I ad ) ( v ) , (31) 11 where Y ( a ) 0 ( v ) = 0, a ∈ { 1 , 2 , . . . , s } and m ∈ Z ≥ 1 ; ∗ is a con v o lution ( f ∗ h )( v ) = Z ∞ −∞ dw f ( v − w ) h ( w ) , (32) and the k ernel is K ( v ) = 1 2 cosh π v . (33) Substituting u = − β J N and taking the T rott er limit N → ∞ , w e obtain the TBA equation log Y ( a ) m ( v ) = π J β δ ap δ mb cosh π v + K ∗ log ( (1 + Y ( a ) m − 1 )(1 + Y ( a ) m +1 ) Q s d =1 (1 + ( Y ( d ) m ) − 1 ) I ad ) ( v ) , (34) where a ∈ { 1 , 2 , . . . , s } , m ∈ Z ≥ 1 , Y ( a ) 0 ( v ) := 0. This TBA equation ( 34) is iden tical to the one from the string h yp othesis. T a king note on the relations C ad ( v ) = min( a,d ) X l =1 G | a − d | +2 l − 1 ( v ) , G a ( v ) = 4 2 s + 1 cos (2 s +1 − 2 a ) π 4 s +2 cosh 2 π v 2 s +1 cos (2 s +1 − 2 a ) π 2 s +1 + cosh 4 π v 2 s +1 , b C ad ( k ) = Z ∞ −∞ d v C ad ( v ) e − ik v , s X c =1 b C ac ( k ) b D cd ( k ) = δ ad , (35) b D cd ( k ) = 2 δ cd cosh k 2 − I cd , one can also rewrite this TBA equation as log Y ( a ) m ( v ) = 2 π β J δ m 1 G a ( v ) + s X b =1 C ab ∗ log ( (1 + Y ( b ) m − 1 )(1 + Y ( b ) m +1 ) Q s d =1 (1 + Y ( d ) m ) I bd ) ( v ) , (36) where Y ( a ) 0 ( v ) = 0, a ∈ { 1 , 2 , . . . , s } and m ∈ Z ≥ 1 . In con trast to (34), (36) do es not con tain 1 + ( Y ( a ) m ( v )) − 1 whic h is not relev an t t o ev aluate the central 12 c harg e for the case J < 0. One can also deriv e the f ollo wing relation fro m (22) for m = 1, (24) and (35). log T (1) 1 ( v ) = log φ − ( v + i ) φ + ( v − i ) + s X a =1 G a ∗ log (1 + Y ( a ) 1 ) + N Z ∞ 0 d k 2 e − k 2 sinh( k u ) cos( kv ) cosh( 2 s − 1 4 k ) k cosh( 2 s +1 4 k ) . (37) T aking the T rotter limit N → ∞ with u = − J β N , we obtain the free energy densit y F = − 1 β log T (1) 1 (0) without infinite sum. F = J  2 2 s + 1  2 log 2 − ψ ( 1 2 s + 1 ) + ψ ( 3 + 2 s 2 + 4 s )  − 1  − k B T s X a =1 Z ∞ −∞ d v G a ( v ) lo g(1 + Y ( a ) 1 ( v )) , (38) where ψ ( z ) is the digamma function ψ ( z ) = d dz log Γ( z ) . (39) The first term in the rhs of (38) for J = − 1 coincides with the grand state energy of the osp (1 | 2 s ) mo del in [17]. Using the result of this section, w e can sho w that the cen tral c ha rge of the corresp onding system is s . 6 Discuss ion In this paper, w e ha v e derived the TBA equation from the osp (1 | 2 s ) version of the T -system. The osp ( r | 2 s ) in tegrable spin c hain is related t o in teresting ph ysical problems, suc h as the lo op mo del whic h is related to statistical prop erties of p o lymers[23], and the fractiona l quan tum Hall effect [53], etc. So it is desirable to study the os p ( r | 2 s ) in tegrable spin c hain b ey ond the osp (1 | 2 s ) case. F or r > 0 case, we ha v e only the T - system for tensor-lik e represen tations [31]. T o construct a complete set of the T -system whic h is relev ant for the QTM metho d, we ha v e to treat spinorial represen tations. In closing this pap er, w e shall men tion the s l ( r + 1 | s + 1) ve rsion of the T -system [27, 28, 29] whic h is omitted in this pap er. The osp (1 | 2 s ) T -system is obta ined as a reduction of a kind of Hirota-Miw a equation. This is also the case with sl ( r + 1 | s + 1). F or m, a ∈ Z ≥ 1 , sl ( r + 1 | s + 1) T - system leads 13 as follow s. T ( a ) m ( v − 1 ) T ( a ) m ( v + 1) = T ( a ) m +1 ( v ) T ( a ) m − 1 ( v ) + T ( a − 1) m ( v ) T ( a +1) m ( v ) for 1 ≤ a ≤ r or 1 ≤ m ≤ s or ( a, m ) = ( r + 1 , s + 1) , T ( r +1) m ( v − 1) T ( r +1) m ( v + 1) = T ( r +1) m +1 ( v ) T ( r +1) m − 1 ( v ) for m ≥ s + 2 , T ( a ) s +1 ( v − 1) T ( a ) s +1 ( v + 1) = T ( a +1) s +1 ( v ) T ( a − 1) s +1 ( v ) for a ≥ r + 2 . where, T ( a ) s +1 ( v ) = ǫ a T ( r +1) a + s − r ( v ) for a ≥ r + 1 , T (0) m ( v ) = T ( a ) 0 ( v ) = 1 . Here w e omit the v acuum part whic h can b e easily reco vered so as to b e compatible with the lhs (v acuum part) of the BAE. The phase factor ǫ a de- p ends on the definition of the transfer matrix. F o r example, if the transfer matrix is defined as a supertra ce of a mono drom y matrix, w e hav e ǫ a = ( − 1) ( s +1)( a + r +1) . Note that ab o v e functional equation reduces to the T - system for sl r +1 [33] (see also [54, 55]) if w e set s = − 1. References [1] V. Kac, L e ctur e Notes in Mathematics 676 , 597 (1978 ). [2] J. H. H. P erk and C. L. Sc h ultz, Phys. L ett. 84A , 407 (1 9 81). [3] P . P . Kulish and E. K. Skly anin, J. Sov. Math 19 , 1596 (1982) . [4] P . P . Kulish, J. Sov. Math. 35 , 2648 (1986). [5] V. V. Bazhanov and A. G . Shadrik ov, T he or. Math. Phys. 73 , 130 2 (1988). [6] T. D eguc hi, A. F ujii and K. Ito, Phys. L ett. B 238 , 242 (1990). [7] H. Saleur, Nucl. Phys. B336 , 363 (1 990). [8] R. B. Zhang, A. J. Brac ken a nd M. D . Go uld, Phys. L ett. B257 , 133 (1991). [9] M. J. Martins a nd P . B. Ramos, J. Phys. A: Math. Gen. 27 , L703 (1994). 14 [10] P . Schlottmann, Phys. R ev. B36 , 5177 (1 987). [11] 1 H. J. de V ega and E. Lop es, Phys. R ev. L ett. 67 , 489 (19 91). [12] F . H. L. Essler and V. E. Korepin, Ph ys. R ev. B46 , 914 7 (1992). [13] F . H. L. Essler, V. E. Korepin and K . Sch outens, Phys. R ev. L ett. 68 , 2960 (1992) . [14] A. F o erster and M. Karows ki, Nucl. Phys. B396 , 611 (1993 ). [15] F . H. L. Essler and V. E. Korepin, Int. J. Mo d. Phys. B8 , 3243 (1994). [16] Z . Maassarani, J. Phys. A: Math. Gen. 28 ,1 305 (1995). [17] M. J. Martins, Phys L ett. B359 , 334 (1995). [18] M. P . Pfannm ¨ uller and H. F rahm, Nucl. Phys. B479 , 575 (1 9 96). [19] M. J. Martins and P . B. Ramos, Nucl. Ph ys. B500 , 579 (1997). [20] Y. K. Zhou and M. T. Batchelor, Nucl. Phys. B490 , 576 (1997). [21] F . G¨ ohmann and S. Murak ami, J. Phys. A: Math. Gen. 31 7729, (199 8). [22] G . J ¨ uttner, A. Kl ¨ ump er and J. Suzuki, Nucl. Phys. B512 , 581 (1998). [23] M. J. Martins, B. Nienhuis a nd R. Rietman, Phys. R ev. L ett. 81 , 5 0 4 (1998). [24] H. F rahm, Nucl. Phys. B559 , 613 (199 9). [25] H. Saleur, Nucl. Phys. B578 , 55 2 ( 2 000). [26] P . P . Kulish, N. Y u. Reshetikhin and E. K. Skly anin, L ett. Math. Phys. 5 393 (1981). [27] Z . Tsuboi, J. Phys. A: Math. Gen. 30 , 79 7 5 (1997). [28] Z . Tsuboi, Physic a A 252 , 565 (1998). [29] Z . Tsuboi, J. Phys. A: Math. Gen. 31 , 54 8 5 (1998). [30] Z . Tsuboi, Physic a A 267 , 173 (1999). [31] Z . Tsuboi, J. Phys. A: Math. Gen. 32 , 7175 (1999). 1 added o n 5 Ja n uary 20 10 15 [32] N. Y u. Reshetikhin , Sov. Phys. JETP 57 , 691 (1983). [33] V. V. Bazhano v and N. Y u. Reshetikhin, J. Phys. A Math. Gen. 23 , 1477 (1990) . [34] A. Kuniba and J. Suzuki, C ommun. Math. Phys. 173 , 225 (1995). [35] A. Kuniba, Y. Oh t a and J. Suzuki, J. Phys. A: Math. Gen. 28 , 621 1 (1995). [36] C. N. Y ang and C. P . Y ang, J. Math. Phys. 10 , 111 5 (1969). [37] K . Sak ai and Z. Tsub oi, Mo d. Phys. L ett. A14 , 2427 (1999 ). [38] K . Sak ai and Z. Tsub oi, Int. J. Mo d. Ph ys. A15 , 2329 (2000 ). [39] K . Sak ai and Z. Tsub oi, J. Phys. S o c. Jpn . 70 , 367 (2001). [40] Z . Tsuboi, cond-mat/01 08358; Int.J.Mo d. Phys. A17 , 2351 (2002). [41] M. T ak ahashi, Pr o g. The or. Phys. 46 , 40 1 (1 971). [42] M. Gaudin, Phys. R ev. L ett. 26 , 1301 (1971). [43] M. Suzuki, Ph ys. Rev. B 31 , 2957 (1985) . [44] M. Suzuki and M. Inoue, Prog . Theor. Ph ys. 78 , 787 ( 1 987). [45] T. Koma, Pr o g. The or. Phys. 78 , 12 13 (1987). [46] J. Suzuki, Y. Akutsu a nd M. W adati, J. Phys. So c. Jpn. 59 , 2667 (1990) . [47] A. Kl ¨ ump er, A nn. Physik 1 , 540 (19 92). [48] R . Hirota, J. Phys. So c. Jpn. 50 , 378 5 ( 1 981). [49] T. Miw a, Pro c. Japan. Acad. 58 , 9 (1982). [50] R . J. F armer a nd P . D . Jarvis, J. Phys. A: Math. Gen. 17 , 2365 (198 4). [51] B. Morel, A. Sciarrino and P . Sorba, J. Phys. A: Math. Gen. 18 , 1 5 97 (1985). [52] E. F renk el and E. Mukhin, Commun. Math. Phys. 216 , 23 (2001 ). [53] F . D. M. Haldane and E. H. R ezayi, Phys. R ev. L ett. 60 , 956 (19 8 8). [54] A. Kl ¨ ump er and P . P earce, Physic a A183 , 30 4 (1992). 16 [55] A. Kuniba, T. Nak anishi and J. Suzuki, Int. J. Mo d. Phys. , A9 , 5215 (1994). 17