Quasi-Exactly Solvable Models Derived from the Quasi-Gaudin Algebra

Quasi-Exactly Solv able Mo dels Deriv ed from the Quasi-Gaudin Algebra Y uan-Harng Lee, Jon Links, and Y ao-Zhong Zhang Scho ol o f Ma thematics a nd Physics, The University of Que ensland, Brisb ane, Q l d 4072, Austr alia Abstract The quasi-Gaudin algebra w as introduced t o construct integra ble systems which are only quasi- exactly solv able. Using a suitable represen tation of the q uasi -Gaudin algebra, w e obtain a class of b o sonic models whic h exhibit this cu r ious prop ert y . These models hav e the notable feature that they do not p re serve U (1) symmetry , which is t ypically associated to a non-conserv ation of particle num b er. An exact solution for the eigenv alues within the q uasi -exactly solv ab le sector is obtained via the algebraic Beth e ansatz formalism. P ACS Numbers: 02.30. Ik, 03.65.Fd, 05.30.J p. 1 In tro duction In [1, 2] Ushveridze prop osed a metho d for studying quasi-exactly solv a ble (QE S) sy s tems [3 –5] fro m the presp ective of integrable systems a nd the Qua ntum Inv ers e Scattering Metho d (QISM) [6]. The a pproach, which is called the p artial alg ebraic Bethe a ns atz (ABA), r elies on deforming the Y ang-Ba xter algebra in such a wa y that it retains most of the features required for the QISM but leads to g enerating functions of int egrable s ystems which are only Q E S. This defor mation of the Y ang- B axter algebr as led to new classes of hitherto unknown algebra s. A limiting case is the (ra tional) quasi- Gaudin algebra which will b e the fo cus of this study . Exactly solv able models hav e found many successe s in v a rious branc hes o f ph ysics and mathematics. Over recent years they hav e co ntin ued to find new applications in diverse fields such as Bose- E instein condensates a nd degenerate F ermi g ases, quantum optics, s uper conductivity , a nd nuclear pa ir ing among other thing s e.g. [7 –14]. Ther e has also b een significant in ter est in QES mo dels , with new applications of these b eing found in problems relating to matr ix pro duct states [15], and in dissipative systems [16]. How ever by co mpa rison the partial ABA approa ch see ms to hav e received little a tten tion and remains essentially undeveloped. An app ea ling pr o p erty of the par tial ABA is that it provides us with a con- structive alg ebraic approa ch for obtaining Q ES mo dels which hav e multiple degrees o f freedom. One pa rticular asp ect of the ABA which b ears some r elev ance to our present exp osition is the study of quantum integrable models w hich do not preserve U (1) symmetry . Such mo dels are interesting for a nu m ber of reasons. In the c ontext of spin- bo son Hamiltonia ns of the T a vis-Cummings for m, these mo dels corres p o nd to physical systems without the rotating wa ve appr oximation. Diagonalisatio n o f such mo dels is a s o mewhat co mplicated a ffair within the ABA metho d due to the lack of reference states, often re- quiring the us e o f functional Bethe ansatz or Sklyanin’s se paration of v aria ble technique [9, 11]. Non U (1) preserving mo dels are also r elev ant to the study o f o p en quan tum sy stems whereby the U (1) symmetry is bro ken due to co upling to an environmen t. An example of this is fo und in the spin-b oson Hamilto- nian of Leg gett et al [1 7] which has found applicatio ns ra nging from q uantum-state engineering [1 8] to 1 biomolecular systems [19]. In the present pap er, we w ill study QES b osonic models descending from suitable realisations o f the quasi-Gaudin algebra. It will b e shown tha t suc h models cor resp ond to a n ex tension of the su (1 , 1) Dick e Hamiltonian [2 0] by the addition of U(1) symmetry-break ing terms. The Hamiltonian can be written a s H = H 0 + H 1 (1.1) with H 0 and H 1 refering to the Dick e Hamiltonian a nd the U(1) symmetry-br e a king comp onent resp ec- tively . Explicitly , they hav e the form H 0 = wN b + m X i =1 2 ǫ i S z i + g m X i =1 bS + i + b † S − i ! , H 1 = g ( b + b † ) n + f z − m X i =1 S z i ! − b † b 2 − ( b † ) 2 b ! . (1.2) Here N b , b, b † are standard b osonic op erato rs, f z is a r epresentation depe ndent pa rameter, w 0 , ǫ i , g are free parameters , n is an integer a nd S z , ± i are e ither single- mo de or double-mo de represe ntations of su (1 , 1) generator s (r efer to equations (3.2 2) and (3.2 3) b elow). The Hamiltonian H 1 may b e int erpreted a s a coupling of the su (1 , 1) Dick e mo del to an e x ternal sys tem. Our pap er is structured as follows. In Section 2 we will briefly review the partial ABA metho d of obtaining quas i- exact solutions for models asso cia ted to the quasi-Ga udin alge br a. In Section 3 we will use a suitable representation of the quasi-Ga udin algebra to obtain the integrable b osonic mo del (1.1 ). W e then der ive the P artial ABA so lutio n of the Hamiltonian and dis c uss a sp ects relating to the qua si-exact solv ability . Finally in Section 4 w e summarise o ur r esults and discuss p ossible future lines of work. 2 Quasi-Gaudin Algebra and B ethe Ansatz Solution Let us first introduce the r ational (rank 1) Gaudin algebra and the a sso ciated abstract, in teg rable mo de ls befo re defining its quasi counterpart. The rational Gaudin model is a parameter- dep e ndent infinit e- dimensional Lie algebra satisfying the following commutation r elations: S z ( λ ) S z ( µ ) − S z ( µ ) S z ( λ ) = 0 , S ± ( λ ) S ± ( µ ) − S ± ( µ ) S ± ( λ ) = 0 , S z ( λ ) S ± ( µ ) − S ± ( µ ) S z ( λ ) = ± S ± ( λ ) − S ± ( µ ) µ − λ , S − ( λ ) S + ( µ ) − S + ( µ ) S − ( λ ) = 2 S z ( λ ) − S z ( µ ) µ − λ , whereby λ and µ are complex sp ectral par ameters. F r om these relations, it can be shown that H ( λ ) = S z ( λ ) S z ( λ ) − 1 2 S + ( λ ) S − ( λ ) − 1 2 S − ( λ ) S + ( λ ) (2.3) satisfies the following commutation relations [ H ( λ ) , H ( µ )] = 0 (2.4) and therefore a cts a s a generato r of co mmu ting o pe rators in an abstract integrable s ystem. Assuming the existence o f a suitable reference state, the spe ctrum of H ( λ ) can b e obtained via the s tandard ABA [2]. 2 Analogous to the Ga udin alg ebra is the so - called qua s i-Gaudin alg ebra. It is defined by the following parameter de p endent set of relations [1, 2] S z n ( λ ) S z n ( µ ) − S z n ( µ ) S z n ( λ ) = 0 , S ± n ± 1 ( λ ) S ± n ( µ ) − S ± n ± 1 ( µ ) S ± n ( λ ) = 0 , S z n ± 1 ( λ ) S ± n ( µ ) − S ± n ( µ ) S z n ( λ ) = ± S ± n ( λ ) − S ± n ( µ ) µ − λ , S − n +1 ( λ ) S + n ( µ ) − S + n − 1 ( µ ) S − n ( λ ) = 2 S z n ( λ ) − S z n ( µ ) µ − λ (2.5) whereby n is an in teger and λ , µ are co mplex parameters. While (2.5) app ea rs to b e similar to the Gaudin alg ebra, we stress that there ar e imp ortant qualitative differ e nce b etw e e n the tw o . Importantly , note that (2.5) do not define comm utation r elations and a re therefore not L ie algebr aic rela tions. Despite lo oking somewha t ar bitary , the quasi-Gaudin alg ebra ca n be understo o d as a grading deformation on the original Gaudin algebra. W e refer the reader to [2] for a more de ta iled discuss ion. Similar to the Gaudin algebra , there exists a generating function of co mmuting operato rs for the quasi-Gaudin a lg ebra. It has the for m H n ( λ ) = S z n ( λ ) S z n ( λ ) − 1 2 S − n +1 ( λ ) S + n ( λ ) − 1 2 S + n − 1 ( λ ) S − n ( λ ) (2.6) and ca n be shown to form a commutativ e family with resp ect to the sp ectral par ameters, i.e. [ H n ( λ ) , H n ( µ )] = 0 . (2.7) Note that the commutation rela tion (2.7) do es not extend to the genera l cas e where H n ( λ ) and H m ( µ ) hav e different integer v alues of n and m . This is due to the lack of a defining r elations b etw een elements of the a lgebra with arbitrar y integer indexes. The ABA solution for the genera ting function H n ( λ ) of the qua si-Gaudin algebra has b e en obtained in [1, 2]. As wtih the standard Gaudin algebra , the ABA diagonalis a tion o f H n ( λ ) works if the representation o f (2.5) s uppo rts a r eference state | 0 i , viz. S z 0 ( λ ) | 0 i = f ( λ ) | 0 i , S − 0 ( λ ) | 0 i = 0 (2.8) The Bethe v ector is g iven b y ψ ( µ 1 , · · · µ n ) = S + n − 1 ( µ n ) S + n − 2 ( µ n − 1 ) · · · S + 0 ( µ 1 ) | 0 i . (2.9) By succ e ssively applying the following relation H n ( λ ) S + n − 1 ( µ n ) = S + n − 1 ( µ n ) H n − 1 ( λ ) + 2 S + n − 1 ( µ n ) S z n − 1 ( λ ) − S + n − 1 ( λ ) S z n − 1 ( µ n ) λ − µ n (2.10) we can shift the op er ator H n ( λ ) to wards the right of the pro duct of S + i ( µ i +1 ) op era tors on the right- hand side of (2.9), s o that we finally hav e H n ( λ ) acting on the referenc e state. After having completed this pro c e dure, we p erfo r m the same op era tion for the v ario us S z i ( λ ) , S z i ( µ i +1 ) that were ge ne r ated as a byproduct o f shifting the H n ( λ ) thro ug h the pro duct of the S + i ( µ i +1 ). The fina l form is given by H ( λ ) ψ ( µ 1 , · · · µ n ) = A ( λ ) ψ ( µ 1 , · · · µ n ) + 2 X i B ( µ i ) ψ ( µ 1 , · · · , µ i − 1 , λ, µ i +1 , · · · , µ n ) (2.11) whereby A ( λ ) = f ( λ ) 2 + f ′ ( λ ) + 2 n X i =1 f ( λ ) λ − µ i + 2 n X i =1 1 λ − µ i X j 6 = i 1 µ i − µ j , B ( µ i ) = f ( µ i ) + X j 6 = i 1 µ i − µ j . (2.12) 3 By r equiring that the un w anted terms v anish we obtain the following Bethe ans atz equations: n X k =1 ,k 6 = i 1 µ i − µ k + f ( µ i ) = 0 , i = 1 , 2 , ..., n (2.13) with the e ig env alue for H n ( λ ) given by E n ( λ ) = f 2 ( λ ) + f ′ ( λ ) + 2 n X i =1 f ( λ ) − f ( µ i ) λ − µ i . (2.14) As a pr o of of existence, an explict representation for (2.5 ) is pr ovided in [1, 2]: S − n ( λ ) = S − ( λ ) + f z − S z + n λ − c , S 0 n ( λ ) = S 0 ( λ ) + f z − S z + n + d λ − c , S + n ( λ ) = S + ( λ ) + f z − S z + n + 2 d λ − c . ( 2.15) with c a nd d a s free parameters, S ± ,z ( λ ) are generators o f the Gaudin alg ebra, and S z and f z are defined as S z = lim λ →∞ λS z ( λ ) , S z | 0 i = f z | 0 i . (2.16) In terms o f this realisation, the g e nerating function H n ( λ ) takes the for m H n ( λ ) = S z ( λ ) S z ( λ ) − 1 2 S − ( λ ) S + ( λ ) − 1 2 S + ( λ ) S − ( λ ) + 2 S z ( λ )( n + d + f z − S z ) − S − ( λ )( n + 2 d + f z − S z ) − S + ( λ )( n + f z − S z ) λ − c − 1 4( λ − c ) 2 . (2.17) It can b e seen tha t the condition o f hermiticit y for (2.17) is satisfied when d = 1 / 2 and the representation for the Ga udin alg e bra is unitary , i.e. satisfying the condition S + ( λ ) † = S − ( λ ) , S z ( λ ) † = S z ( λ ) . (2.18) 3 Bosonic Represen tations of the Quasi-Gaudin A lgebra The quasi-Gaudin a lg ebra of the form (2.15) admits mixed representations, consis ting o f su (1 , 1) alg ebras and the Heisen ber g alge br a, with the following fo r m: S − n ( λ ) = 2 b g + m X i =1 S − i λ − ǫ j + f z − N b − P i S z i + n λ − c , S z n ( λ ) = w − 2 λ g 2 + m X i =1 S z i λ − ǫ j + f z − N b − P i S z i + n + 1 2 λ − c , S + n ( λ ) = 2 b † g + m X i =1 S + i λ − ǫ j + f z − N b − P i S z i + n + 1 λ − c . (3.19) The S ± ,z i and { N b , b, b † } are resp ectively the su (1 , 1) and Heisenberg algebr as, which ob ey the commu- tation r elations  S z i , S ± j  = ± S ± i δ ij ,  S − i , S + j  = 2 S z i δ ij  N b , b †  = b † , [ N b , b ] = − b ,  b, b †  = 1 (3.20) 4 and S z and f z are defined as S z = m X i =1 S z i + N b , S z | 0 i = f z | 0 i . (3.21) W e note her e that our definition for S z differs fro m that of (2.16) as the prior definition is div ergent for this par ticula r rea lis ation. The su (1 , 1) algebra s ha s tw o bosonic operator r ealisatio ns . The first is giv en by the single- mo de representation, wher eby S z i = a † i a i 2 + 1 4 = N a i 2 + 1 4 , S + i = ( a † i ) 2 2 , S − i = a 2 i 2 . (3.22) The seco nd one is given b y the t wo-mo de representation, S z i = 1 2  a † i a i + c † i c i  + 1 2 = ( N a i + N c i ) 2 + 1 2 , S + i = a † i c † i , S − i = a i c i . (3.23) There are m ultiple referenc e states for b o th b osonic realisatio ns. F or the single-mo de realisa tion, there are finitely man y of them. W e can ex pr ess them as | 0 , { l }i = m Y i =1 ( a † i ) l i | 0 i , l i = 0 or 1 (3.24) where { l } is a shorthand notatio n for the set { l 1 , · · · l m } and S z | 0 , { l }i = f z | 0 , { l }i = m X i =1 l i 2 + 1 4 ! | 0 , { l }i . (3.25) F or the tw o-mo de r ealisation, ther e are infinitely many reference states. Without loss of g enerality w e can wr ite them as | 0 , { l }i = m Y i =1 ( a † i ) l i | 0 i , l i = 0 , 1 , 2 , · · · (3.26) with S z | 0 , { l }i = f z | 0 , { l }i = m X i =1 l i 2 + 1 2 ! | 0 , { l }i . (3.27) It can be seen that ea ch reference state cor resp onds to a distinct eigenfunction of the Casimir op er ators for the su (1 , 1) generators S ± ,z i . As the su (1 , 1) Casimir oper ators acts as central elements with r esp ect to (3.22) and (3.23), we can use Sc h ur’s lemma to deduce that ea ch refere nc e sta te gives rise to a distinct irreducible re pr esentation. 4 Quasi-Exactly Solv able Hamiltonians W e now consider the genera ting function H n ( λ ) of the quasi-Gaudin alg ebra obtained from the represen- tation (3.1 9). Ass uming ǫ i 6 = ǫ j , it can be seen that H n ( λ ) = − 4 g 2  n + f z + 1 2  + 1 g 4 ( w − 2 λ ) 2 − 2 g 2   H c λ − c + m X j =1 H j λ − ǫ j   + m X i =1 K i ( λ − ǫ i ) 2 − 1 4( λ − c ) 2 (4.28) 5 with H j = (2 ǫ i − w ) S z j + g  b † S − j + bS − j  + m X i 6 = j 1 ǫ j − ǫ i  2 S z i S z j − S + i S − j − S − i S + j  + g 2  S z j  n + 1 2 + f 0 − S 0  − 1 2 S − j ( n + 1 + f z − S z ) − 1 2 S + j ( n + f z − S z )  ǫ j − c , H c = g 2 m X i =1 S z i  n + 1 2 + f z − P m i =1 S z i − N b  − 1 2 S + i ( n + f z − S z ) − 1 2 S − i ( n + 1 + f z − S z ) ( c − ǫ i ) +(2 c − w ) n + 1 2 + f 0 − m X i =1 S z i − N b ! + g b † n − m X i =1 S z i − N b ! + g b n + 1 − m X i =1 S z i − N b ! , K i = S z i S z i − 1 2  S − i S + i + S + i S − i  . (4.29) F rom (4.28) a nd the commutation re lation (2 .7), it fo llows that H i,c , K i,c form a set of mutually com- m uting op erator s. By considering the following linea r combination H = Υ + P i H i + H c and s e tting the co efficient c = 0, we obtained the desired b osonic hamiltonian. F or the single-mo de repr esentations, we hav e H = wN b + m X i =1 ǫ i N a i + g m X i =1  b ( a † i ) 2 + b † a 2 i  + g ( n + f z )( b † + b ) − ( b + b † ) m X i =1 N a i 2 − b † b 2 − ( b † ) 2 b ! (4.30) where Υ = w  n + 1 2 + f z  − m X i =1 ǫ i 2 . F or the t wo-mo de repr esentations, we obta in H = wN b + m X i =1 ǫ i ( N a i + N c i ) + g m X i =1  ba † i c † i + b † a i c i  + g ( n + f z )( b † + b ) − ( b + b † ) m X i =1 N a i 2 − b † b 2 − ( b † ) 2 b ! (4.31) with Υ = w  n + 1 2 + f z  − m X i =1 ǫ i . W e note that for the ca se when m = 1 , the mo dels co rresp ond to quasi-exactly solv able extensions for atom-molecule BE C mo dels co ntained [21]. The eigenv alues for the Ha milto nia ns can b e extracted from the Bethe ansa tz solution of (4.28): E n ( λ ) = f 2 ( λ ) + f ′ ( λ ) + 2 n X i =1 f ( λ ) − f ( µ i ) λ − µ i . (4.32) This is do ne by ev aluating the residues of the p o les ǫ i and c . Doing so yields E = Υ − w m X i =1 s z i + 1 2 ! + m X i =1 2 ǫ i s z i + g 2 2   m X j =1 n X i =1 s z j µ i − ǫ j + n X i =1 1 2( µ i − c )   (4.33) whereby s z i = (2 l i + 1) / 4 for the sing le -mo de representations and s z i = ( l i + 1) / 2 for the tw o -mo de representations. 6 W e now examine the quasi-e x actly s olv able nature of the Hamiltonians in more detail. F or the sake of clar ity , we sha ll only consider the Ha miltonian with the single-mo de boso nic repres entation (4.30), as results for the tw o -mo de r epresentation will follow analo gously . It is str aightfo ward to see that (4.30) acts on an infinite-dimensional Hilbert space V span by the follo wing ba sis states V ≡ span { ( b † ) l 0 ( a † 1 ) l 1 · · · ( a † m ) l m | 0 i} ≡ span {| l 0 , · · · , l m i} , l i ∈ Z + . (4.34) In or de r to identify the in v ariant subspa ce which characteris es the qua si-exact solv ability of the Hamil- tonian, let us write the Hamiltonian as H g = H 0 + H − + H + (4.35) whereby we hav e intro duced a gra ding structure on the Hamiltonian thro ugh setting H 0 = wN b + m X i =1 ǫ i N a i + g m X i =1 b ( a † i ) 2 + b † a 2 i + ( n + f z )( b † + b ) ! , H + = ( n + f z ) b † − b † m X i =1 N a i 2 − ( b † ) 2 b, H − = ( n + f z ) b − b m X i =1 N a i 2 − b † b 2 . (4.36) The assigned grading of ± , 0 is determined by the commutation relations of H ± , 0 with the U (1) charge S z = N b + P m i =1 (2 N a i + 1) / 4:  S z , H 0  = 0 ,  S z , H +  = H + ,  S z , H −  = − H − . (4.37) In ligh t of these relations, we may decompo se V into a direct sum o f eigenspa ce V i,p of the U (1) c harge S z and the Ca s imir op era tors of the su (1 , 1) alge br a K i = S z i ( S z i − 1) − S + i S − i , i.e. V = M i, { p } V i, { p } . (4.38) Explicitly , the subspace V i, { p } can b e written as V i, { p } ≡ s pan { ( b † ) l 0 ( a † 1 ) 2 l 1 + p 1 · · · ( a † m ) 2 l m + p m | 0 i} , m X j =0 l j = i , p i = 0 or 1 . (4.39) It can also be verified that S z V i, { p } =   i + m X j =1  p i 2 + 1 4    V i, { p } , K i V i, { p } =  p i 2 + 1 4   p i 2 − 3 4  V i, { p } . (4.40) F rom the commutation rela tions (4.37), we ther efore have H + V i, { p } ⊆ V i +1 , { p } , H 0 V i, { p } ⊆ V i, { p } , H − V i, { p } ⊆ V i − 1 , { p } . (4.41) The Q ES prop er ty of the Hamiltonian ar ises from the fact that for g iven integer v a lue of n and f z = P i ( l i + 1) / 4, we hav e H + V n, { l } = { 0 } . As a r esult, the Hamiltonian leav e s the following subspace inv a riant: V QES ≡ n M i =0 V i, { l } . (4.42) W e can indeed verify that the Bethe v ectors lie within this inv a riant subspace, by expanding the eigen- vectors (2.9 ) explicitly . It would b e int eresting to examine the p ossibility o f obtaining exact s olutions outside o f this sector . 7 5 Conclusion W e’ve inv estiga ted a class of QES, int egrable multi-mode b o sonic mo dels using the qua si-Gaudin alg e- bra. W e see that such mo dels are obtained via a mixed r e presentation consisting of commuting copies of su (1 , 1), and the Heisenber g algebra. Integrable Hamiltonia ns were extra cted fro m the gener a ting function o f commuting op era tors. A notable feature was that the Q ES Ha miltonians we obtain do not preserve U (1) symmetry . W e iden tified the QES sector o f the Ha miltonian as the dir e ct sum o f the eigensubspaces of the U (1) charge with eig env alues no greater than n . The ABA metho d leads to partial solutions of the Hamiltonians we’ve considered. Given the inte- grability o f the Hamiltonian, in the sense that H n ( λ ) acts as a gener ator of conserved op erator s, it w o uld be in ter esting to ex plo re the p ossibility of obtaining the entire sp ectrum via some other techniques. The dominating exp erience is that in tegrability and exact solv abilty go hand-in-hand. It is not apparent for these Hamiltonia ns whether the full sp ectrum is po tentially accessible. Finally w e note that due to the constraint ar ising from impos ing hermiticit y on the g enerating function H n ( λ ), the quas i-Gaudin for malism is at present limited to cas e s bas ed on underlying unitary represen- tations of su (1 , 1), or the Heis enberg algebra. 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