Incoherent bound states in an infinite $XXZ$ chain at $Delta=-1/2$

Incoheren t b ound states in an in fin i te X X Z c hain at ∆ = − 1 / 2 P . N. Bibiko v Sankt-Petersbur g State Un iversity Octob er 24, 2018 Abstract F or an infinite X X Z c hain with ∆ = − 1 / 2 we ha ve obtained a family of trans- lationary inv arian t th r ee -magnon states which do not satisfy the s tr i ng conjecture. All of them hav e the same energy . 1 In t ro duction W e shall study an infinite X X Z spin ch ain [1] related to the Ha m iltonian H = ∞ X n = −∞ H n,n +1 , (1) where H n,n +1 = S x n S x n +1 + S y n S y n +1 + ∆  S z n S z n +1 − 1 4  . (2) The corresp onding Hilb ert space is a n infinite tensor pro duct of C 2 spaces a s so ciated with the lattice sites . In eve ry suc h space w e shall use the fo llo wing basis S z n |±i n = ± 1 2 |±i n . (3) 1 Here a nd in (2) S n denotes a triple of S = 1 / 2 spin o p erators asso ciat ed with n -th site. ∆ is a real parameter. The follo wing transformation H → − U H U − 1 , (4) where U = Y n σ z 2 n , (5) ( σ j n = 2 S j n for j = x, y , z ar e the Pauli matrices) is equiv alen t to the substitution ∆ → − ∆. The result o f our pap er corresp onds to the sp ecial case ∆ = − 1 / 2. T raditiona lly the mo del (1) is treated on a finite c hain related to the Hilb ert space ( C 2 ) ⊗ N ( N is the n umber of sites). Usually there supp osed perio dic b oundary conditio ns [2]-[4] H ( per iod ) = N X n =1 H n,n +1 , N + 1 ≡ 1 , (6) (see ho w ev er [5] where the ch ain with op en b oundaries w as studied). Since b oth the Hamiltonians (1) and (6) commute with S z the z comp onen t of the total spin S = X n S n (7) their sp ectrums split on subsectors corr esp onding to differen t v alues of S z . Bethe Ansatze is used as a n effectiv e metho d for treating the Hamiltonian (1) [1] or (6) [2]-[4] separately in all subsectors. Within this approach first of all is considered the highest S z state | Ω i = Y n | + i n , (8) whic h is an eigen vector of b oth (1) and (6). The next sector is generated b y quasiparticles (magnons). Since b oth the Hamiltonians (1) a nd (6) comm ute with la ttice translations one can readily obtain an explicit f o rm of the one-magnon state with quasi momen tum k | 1 , k i = X n e ik n  n − 1 Y m = n min ⊗| + i m  ⊗ |−i n ⊗  n max Y m = n +1 ⊗| + i m  , (9) where for the infinite ch ain n min = −∞ , n max = ∞ while f o r the finite one n min = 1, n max = N . The corresp onding dispersion E magn ( k ) = cos k − ∆ , (10) 2 readily follow s from the lo cal action form ulas H n,n +1 . . . |∓i n |±i n +1 . . . = − ∆ 2 . . . |∓i n |±i n +1 . . . + 1 2 . . . |±i n |∓i n +1 . . . , H n,n +1 . . . |±i n |±i n +1 . . . = 0 , (11) whic h are conse quences of (2). The exp onen t e ik n in (9 ) is a one-magnon w av e function. Within Bethe Ansatze wa v e functions of all eigenstates ar e represen ted as sums of Bethe exponents. F or example for a general tw o-mag non state | 2 , k 1 , k 2 i = X m 0 ) . Since in (1 3 ) m < n the second term is ”bad” (un b ounded). So there should b e C 21 ( k 1 , k 2 ) = 0. This condition pro duce a relation b et w een u and v . All magnons within the same complex ha v e a similar sp a t ial dep endenc e of phase . That is wh y a complex ma y b e considered as a c oher ent b ound state of the corresp onding magnons Usually it is assumed that for the X X Z mo del the string conjecture is r ig h t. Within this assumption thermo dyn amics of the infinite X X Z c hain w as studied in [6] for | ∆ | ≥ 1 and in [7] for | ∆ | < 1. Ho w ev er in the presen t pap er w e sho w that the string conjecture fails in the sp ec ial p oin t ∆ = − 1 / 2. Namely w e shall presen t a family of three-magnon infinite-c hain inc oh e r ent b ound stat es with total zer o q uasi momentum . The ∆ = − 1 / 2 X X Z c hain is now intens iv ely studied in v arious asp ects (see the recen t articles [8]- [10] and references therein). W e b eliev e that our result shed an additional lig ht on this mo del. 2 Three-magnon incohe ren t b ou nd states First of all let us utilize the translation in v ariance and represen t a three magnon state with total quasimomen tum k in the following g e neral form | 3 , k i = X m 0 but ma y b e contin ued to m = 0 , n > 0 a nd m > 0 , n = 0 according to Bethe conditions 2∆ a ( k , 1 , n ) = e ik / 3 a ( k , 0 , n ) + e − ik / 3 a ( k , 0 , n + 1) , 2∆ a ( k , m, 1) = e − ik / 3 a ( k , m, 0) + e ik / 3 a ( k , m + 1 , 0) . (16) 4 Under (1 6) the Sc hr¨ odinger equation in the whole region m, n > 0 ta k es the following form − 3∆ a ( k , m, n ) + 1 2 h e − ik / 3 a ( k , m + 1 , n ) + e ik / 3 a ( k , m − 1 , n ) +e − ik / 3 a ( k , m − 1 , n + 1) + e ik / 3 a ( k , m + 1 , n − 1) + e − ik / 3 a ( k , m, n − 1) +e ik / 3 a ( k , m, n + 1) i = E a ( k , m, n ) . (17) The follo wing trial b ounde d wa v e function a ( m, n ) = e ( iu 1 − v 1 ) m +( iu 2 − v 2 ) n , (18) satisfy (17) for E ( k , u 1 , u 2 , v 1 , v 2 ) = cosh v 1 cos ( k / 3 − u 1 ) + cosh v 2 cos ( k / 3 + u 2 ) + cosh ( v 1 − v 2 ) cos ( k / 3 + u 1 − u 2 ) − 3∆ . (19) Normalization condition X | a ( m, n ) | 2 < ∞ , (20) results in v 1 , 2 > 0 . (21) F o rm the other side t he system (16) giv es x 1 = e i ( k/ 3+ u 1 ) − v 1 + e − ik / 3 − 2∆ 1 e iu 2 − v 2 = 0 , x 2 = e ik / 3 + e i ( u 2 − k / 3 ) − v 2 − 2∆ 1 e iu 1 − v 1 = 0 . (22) T reating x 1 − ¯ x 2 one ma y readily obtain 2∆ F = − ¯ F e ik / 3 , (23) where F = e iu 2 − v 2 − e − iu 1 − v 1 . A t 4 ∆ 2 6 = 1 Eq. (23) giv es F = 0 or equiv alen tly u 1 = − u 2 and v 1 = v 2 . In this case the string conjecture is satisfied. How ev er in tw o sp ecial p oin ts ∆ = ± 1 / 2 connected b y the symmetry (4) there should b e additional solutions. T aking ∆ = − 1 / 2 and treat ing x 1 − e ik / 3 x 2 one gets k = 0 . (24) No w the sys tem (22) results in e − v 1 cos u 1 + e − v 2 cos u 2 = − 1 , e − v 1 sin u 1 = − e − v 2 sin u 2 , (25) 5 or e v 1 = sin ( u 1 − u 2 ) sin u 2 , e v 2 = sin ( u 2 − u 1 ) sin u 1 . (26) According to (26) sin u 1 and sin u 2 ha v e opp osite signs. Without loss of generalit y o ne ma y put 0 < u 1 < π , − π < u 2 < 0 . (27) Under this assumption b oth cos u 1 , 2 / 2 > 0 and the system (2 1 ) is reducible to sin  u 1 − u 2 2  < 0 , sin  u 1 2 − u 2  < 0 , (28) or equiv alen tly π < u 1 − u 2 2 < 2 π , π < u 1 2 − u 2 < 2 π . (29) It ma y b e readily prov ed f r om (19 ) and (26) that all these states hav e zero energy . According to (15) and ( 1 8) they describ e mag no n triples with corresp onding quasi mo- men tums k 1 = − u 1 − iv 1 , k 2 = u 1 − u 2 + i ( v 1 − v 2 ) , k 3 = u 2 + iv 2 . (30) The string conjecture obviously is fa ile d. The author thanks P . P . Kulish for careful reading o f the man uscript. References [1] D. Babbitt, E. G utk in, The Plancher el formula for the infin i te XXZ Heisenb er g chain Lett. Math. Phy s. 20 (1990 ), 91-99 [2] M. Gaudin L a F onction D’onde d e Bethe , Masson, 1983 [3] F addeev L D 19 9 8 How algebr aic Bethe Ansatz works for in t e gr able mo dels , Quantum symmetries /Symmetries quantique, Pr o c e e dings of the L es Houches s umm er scho ol Session LXIV, eds. A. Connes, K. Ga w edzki and J. 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