Quasi-Exactly Solvable Models Derived from the Quasi-Gaudin Algebra
aa r X i v : . [ n li n . S I] A ug Quasi-Exactly Solvable Models Derived from the Quasi-GaudinAlgebra
Yuan-Harng Lee, Jon Links, and Yao-Zhong Zhang
School of Mathematics and Physics, The University of Queensland, Brisbane, Qld 4072, Australia
Abstract
The quasi-Gaudin algebra was introduced to construct integrable systems which are only quasi-exactly solvable. Using a suitable representation of the quasi-Gaudin algebra, we obtain a class ofbosonic models which exhibit this curious property. These models have the notable feature thatthey do not preserve U (1) symmetry, which is typically associated to a non-conservation of particlenumber. An exact solution for the eigenvalues within the quasi-exactly solvable sector is obtainedvia the algebraic Bethe ansatz formalism.PACS Numbers: 02.30.Ik, 03.65.Fd, 05.30.Jp. In [1, 2] Ushveridze proposed a method for studying quasi-exactly solvable (QES) systems [3–5] from theprespective of integrable systems and the Quantum Inverse Scattering Method (QISM) [6]. The approach,which is called the partial algebraic Bethe ansatz (ABA), relies on deforming the Yang-Baxter algebra insuch a way that it retains most of the features required for the QISM but leads to generating functions ofintegrable systems which are only QES. This deformation of the Yang-Baxter algebras led to new classesof hitherto unknown algebras. A limiting case is the (rational) quasi-Gaudin algebra which will be thefocus of this study.Exactly solvable models have found many successes in various branches of physics and mathematics.Over recent years they have continued to find new applications in diverse fields such as Bose-Einsteincondensates and degenerate Fermi gases, quantum optics, superconductivity, and nuclear pairing amongother things e.g. [7–14]. There has also been significant interest in QES models, with new applicationsof these being found in problems relating to matrix product states [15], and in dissipative systems [16].However by comparison the partial ABA approach seems to have received little attention and remainsessentially undeveloped. An appealing property of the partial ABA is that it provides us with a con-structive algebraic approach for obtaining QES models which have multiple degrees of freedom.One particular aspect of the ABA which bears some relevance to our present exposition is the studyof quantum integrable models which do not preserve U (1) symmetry. Such models are interesting for anumber of reasons. In the context of spin-boson Hamiltonians of the Tavis-Cummings form, these modelscorrespond to physical systems without the rotating wave approximation. Diagonalisation of such modelsis a somewhat complicated affair within the ABA method due to the lack of reference states, often re-quiring the use of functional Bethe ansatz or Sklyanin’s separation of variable technique [9,11]. Non U (1)preserving models are also relevant to the study of open quantum systems whereby the U (1) symmetryis broken due to coupling to an environment. An example of this is found in the spin-boson Hamilto-nian of Leggett et al [17] which has found applications ranging from quantum-state engineering [18] to1iomolecular systems [19].In the present paper, we will study QES bosonic models descending from suitable realisations of thequasi-Gaudin algebra. It will be shown that such models correspond to an extension of the su (1 ,
1) DickeHamiltonian [20] by the addition of U(1) symmetry-breaking terms. The Hamiltonian can be written as H = H + H (1.1)with H and H refering to the Dicke Hamiltonian and the U(1) symmetry-breaking component respec-tively. Explicitly, they have the form H = wN b + m X i =1 ǫ i S zi + g m X i =1 bS + i + b † S − i ! ,H = g ( b + b † ) n + f z − m X i =1 S zi ! − b † b − ( b † ) b ! . (1.2)Here N b , b, b † are standard bosonic operators, f z is a representation dependent parameter, w , ǫ i , g arefree parameters, n is an integer and S z, ± i are either single-mode or double-mode representations of su (1 , H may be interpreted as acoupling of the su (1 ,
1) Dicke model to an external system.Our paper is structured as follows. In Section 2 we will briefly review the partial ABA method ofobtaining quasi-exact solutions for models associated to the quasi-Gaudin algebra. In Section 3 we willuse a suitable representation of the quasi-Gaudin algebra to obtain the integrable bosonic model (1.1). Wethen derive the Partial ABA solution of the Hamiltonian and discuss aspects relating to the quasi-exactsolvability. Finally in Section 4 we summarise our results and discuss possible future lines of work.
Let us first introduce the rational (rank 1) Gaudin algebra and the associated abstract, integrable modelsbefore defining its quasi counterpart. The rational Gaudin model is a parameter-dependent infinite-dimensional Lie algebra satisfying the following commutation relations: 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 spectral parameters. From these relations, it can be shown that H ( λ ) = S z ( λ ) S z ( λ ) − S + ( λ ) S − ( λ ) − S − ( λ ) S + ( λ ) (2.3)satisfies the following commutation relations[ H ( λ ) , H ( µ )] = 0 (2.4)and therefore acts as a generator of commuting operators in an abstract integrable system. Assuming theexistence of a suitable reference state, the spectrum of H ( λ ) can be obtained via the standard ABA [2].2nalogous to the Gaudin algebra is the so-called quasi-Gaudin algebra. It is defined by the followingparameter dependent set of relations [1, 2] S zn ( λ ) S zn ( µ ) − S zn ( µ ) S zn ( λ ) = 0 ,S ± n ± ( λ ) S ± n ( µ ) − S ± n ± ( µ ) S ± n ( λ ) = 0 ,S zn ± ( λ ) S ± n ( µ ) − S ± n ( µ ) S zn ( λ ) = ± S ± n ( λ ) − S ± n ( µ ) µ − λ ,S − n +1 ( λ ) S + n ( µ ) − S + n − ( µ ) S − n ( λ ) = 2 S zn ( λ ) − S zn ( µ ) µ − λ (2.5)whereby n is an integer and λ , µ are complex parameters. While (2.5) appears to be similar to theGaudin algebra, we stress that there are important qualitative difference between the two. Importantly,note that (2.5) do not define commutation relations and are therefore not Lie algebraic relations. Despitelooking somewhat arbitary, the quasi-Gaudin algebra can be understood as a grading deformation on theoriginal Gaudin algebra. We refer the reader to [2] for a more detailed discussion.Similar to the Gaudin algebra, there exists a generating function of commuting operators for thequasi-Gaudin algebra. It has the form H n ( λ ) = S zn ( λ ) S zn ( λ ) − S − n +1 ( λ ) S + n ( λ ) − S + n − ( λ ) S − n ( λ ) (2.6)and can be shown to form a commutative family with respect to the spectral parameters, i.e.[ H n ( λ ) , H n ( µ )] = 0 . (2.7)Note that the commutation relation (2.7) does not extend to the general case where H n ( λ ) and H m ( µ )have different integer values of n and m . This is due to the lack of a defining relations between elementsof the algebra with arbitrary integer indexes. The ABA solution for the generating function H n ( λ ) ofthe quasi-Gaudin algebra has been obtained in [1, 2]. As wtih the standard Gaudin algebra, the ABAdiagonalisation of H n ( λ ) works if the representation of (2.5) supports a reference state | i , viz. S z ( λ ) | i = f ( λ ) | i , S − ( λ ) | i = 0 (2.8)The Bethe vector is given by ψ ( µ , · · · µ n ) = S + n − ( µ n ) S + n − ( µ n − ) · · · S +0 ( µ ) | i . (2.9)By successively applying the following relation H n ( λ ) S + n − ( µ n ) = S + n − ( µ n ) H n − ( λ ) + 2 S + n − ( µ n ) S zn − ( λ ) − S + n − ( λ ) S zn − ( µ n ) λ − µ n (2.10)we can shift the operator H n ( λ ) towards the right of the product of S + i ( µ i +1 ) operators on the right-hand side of (2.9), so that we finally have H n ( λ ) acting on the reference state. After having completedthis procedure, we perform the same operation for the various S zi ( λ ) , S zi ( µ i +1 ) that were generated as abyproduct of shifting the H n ( λ ) through the product of the S + i ( µ i +1 ). The final form is given by H ( λ ) ψ ( µ , · · · µ n ) = A ( λ ) ψ ( µ , · · · µ n ) + 2 X i B ( µ i ) ψ ( µ , · · · , µ i − , λ, µ i +1 , · · · , µ n ) (2.11)whereby A ( λ ) = f ( λ ) + f ′ ( λ ) + 2 n X i =1 f ( λ ) λ − µ i + 2 n X i =1 λ − µ i X j = i µ i − µ j ,B ( µ i ) = f ( µ i ) + X j = i µ i − µ j . (2.12)3y requiring that the unwanted terms vanish we obtain the following Bethe ansatz equations: n X k =1 ,k = i µ i − µ k + f ( µ i ) = 0 , i = 1 , , ..., n (2.13)with the eigenvalue for H n ( λ ) given by E n ( λ ) = f ( λ ) + f ′ ( λ ) + 2 n X i =1 f ( λ ) − f ( µ i ) λ − µ i . (2.14)As a proof of existence, an explict representation for (2.5) is provided in [1, 2]: S − n ( λ ) = S − ( λ ) + f z − S z + nλ − c ,S n ( λ ) = S ( λ ) + f z − S z + n + dλ − c ,S + n ( λ ) = S + ( λ ) + f z − S z + n + 2 dλ − c . (2.15)with c and d as free parameters, S ± ,z ( λ ) are generators of the Gaudin algebra, and S z and f z are definedas S z = lim λ →∞ λS z ( λ ) , S z | i = f z | i . (2.16)In terms of this realisation, the generating function H n ( λ ) takes the form H n ( λ ) = S z ( λ ) S z ( λ ) − S − ( λ ) S + ( λ ) − 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 − λ − c ) . (2.17)It can be seen that the condition of hermiticity for (2.17) is satisfied when d = 1 / S + ( λ ) † = S − ( λ ) , S z ( λ ) † = S z ( λ ) . (2.18) The quasi-Gaudin algebra of the form (2.15) admits mixed representations, consisting of su (1 ,
1) algebrasand the Heisenberg algebra, with the following form: S − n ( λ ) = 2 bg + m X i =1 S − i λ − ǫ j + f z − N b − P i S zi + nλ − c ,S zn ( λ ) = w − λg + m X i =1 S zi λ − ǫ j + f z − N b − P i S zi + n + λ − c ,S + n ( λ ) = 2 b † g + m X i =1 S + i λ − ǫ j + f z − N b − P i S zi + n + 1 λ − c . (3.19)The S ± ,zi and { N b , b, b † } are respectively the su (1 ,
1) and Heisenberg algebras, which obey the commu-tation relations (cid:2) S zi , S ± j (cid:3) = ± S ± i δ ij , (cid:2) S − i , S + j (cid:3) = 2 S zi δ ij (cid:2) N b , b † (cid:3) = b † , [ N b , b ] = − b , (cid:2) b, b † (cid:3) = 1 (3.20)4nd S z and f z are defined as S z = m X i =1 S zi + N b , S z | i = f z | i . (3.21)We note here that our definition for S z differs from that of (2.16) as the prior definition is divergent forthis particular realisation.The su (1 ,
1) algebras has two bosonic operator realisations. The first is given by the single-moderepresentation, whereby S zi = a † i a i N a i , S + i = ( a † i ) , S − i = a i . (3.22)The second one is given by the two-mode representation, S zi = 12 (cid:16) a † i a i + c † i c i (cid:17) + 12 = ( N a i + N c i )2 + 12 , S + i = a † i c † i , S − i = a i c i . (3.23)There are multiple reference states for both bosonic realisations. For the single-mode realisation, thereare finitely many of them. We can express them as | , { l }i = m Y i =1 ( a † i ) l i | i , l i = 0 or 1 (3.24)where { l } is a shorthand notation for the set { l , · · · l m } and S z | , { l }i = f z | , { l }i = m X i =1 l i ! | , { l }i . (3.25)For the two-mode realisation, there are infinitely many reference states. Without loss of generality wecan write them as | , { l }i = m Y i =1 ( a † i ) l i | i , l i = 0 , , , · · · (3.26)with S z | , { l }i = f z | , { l }i = m X i =1 l i ! | , { l }i . (3.27)It can be seen that each reference state corresponds to a distinct eigenfunction of the Casimir operatorsfor the su (1 ,
1) generators S ± ,zi . As the su (1 ,
1) Casimir operators acts as central elements with respectto (3.22) and (3.23), we can use Schur’s lemma to deduce that each reference state gives rise to a distinctirreducible representation.
We now consider the generating function H n ( λ ) of the quasi-Gaudin algebra obtained from the represen-tation (3.19). Assuming ǫ i = ǫ j , it can be seen that H n ( λ ) = − g (cid:18) n + f z + 12 (cid:19) + 1 g ( w − λ ) − g H c λ − c + m X j =1 H j λ − ǫ j + m X i =1 K i ( λ − ǫ i ) − λ − c ) (4.28)5ith H j = (2 ǫ i − w ) S zj + g (cid:0) b † S − j + bS − j (cid:1) + m X i = j ǫ j − ǫ i (cid:0) S zi S zj − S + i S − j − S − i S + j (cid:1) + g (cid:0) S zj (cid:0) n + + f − S (cid:1) − S − j ( n + 1 + f z − S z ) − S + j ( n + f z − S z ) (cid:1) ǫ j − c ,H c = g m X i =1 S zi (cid:0) n + + f z − P mi =1 S zi − N b (cid:1) − S + i ( n + f z − S z ) − S − i ( n + 1 + f z − S z )( c − ǫ i )+(2 c − w ) n + 12 + f − m X i =1 S zi − N b ! + gb † n − m X i =1 S zi − N b ! + gb n + 1 − m X i =1 S zi − N b ! ,K i = S zi S zi − (cid:0) S − i S + i + S + i S − i (cid:1) . (4.29)From (4.28) and the commutation relation (2.7), it follows that H i,c , K i,c form a set of mutually com-muting operators. By considering the following linear combination H = Υ + P i H i + H c and setting thecoefficient c = 0, we obtained the desired bosonic hamiltonian. For the single-mode representations, wehave H = wN b + m X i =1 ǫ i N a i + g m X i =1 (cid:16) b ( a † i ) + b † a i (cid:17) + g ( n + f z )( b † + b ) − ( b + b † ) m X i =1 N a i − b † b − ( b † ) b ! (4.30)where Υ = w (cid:18) n + 12 + f z (cid:19) − m X i =1 ǫ i . For the two-mode representations, we obtain H = wN b + m X i =1 ǫ i ( N a i + N c i ) + g m X i =1 (cid:16) ba † i c † i + b † a i c i (cid:17) + g ( n + f z )( b † + b ) − ( b + b † ) m X i =1 N a i − b † b − ( b † ) b ! (4.31)with Υ = w (cid:18) n + 12 + f z (cid:19) − m X i =1 ǫ i . We note that for the case when m = 1, the models correspond to quasi-exactly solvable extensions foratom-molecule BEC models contained [21].The eigenvalues for the Hamiltonians can be extracted from the Bethe ansatz solution of (4.28): E n ( λ ) = f ( λ ) + f ′ ( λ ) + 2 n X i =1 f ( λ ) − f ( µ i ) λ − µ i . (4.32)This is done by evaluating the residues of the poles ǫ i and c . Doing so yields E = Υ − w m X i =1 s zi + 12 ! + m X i =1 ǫ i s zi + g m X j =1 n X i =1 s zj µ i − ǫ j + n X i =1 µ i − c ) (4.33)whereby s zi = (2 l i + 1) / s zi = ( l i + 1) / V span by the following basis states V ≡ span { ( b † ) l ( a † ) l · · · ( a † m ) l m | i} ≡ span {| l , · · · , l m i} , l i ∈ Z + . (4.34)In order to identify the invariant subspace which characterises the quasi-exact solvability of the Hamil-tonian, let us write the Hamiltonian as H g = H + H − + H + (4.35)whereby we have introduced a grading structure on the Hamiltonian through setting H = wN b + m X i =1 ǫ i N a i + g m X i =1 b ( a † i ) + b † a i + ( n + f z )( b † + b ) ! ,H + = ( n + f z ) b † − b † m X i =1 N a i − ( b † ) b,H − = ( n + f z ) b − b m X i =1 N a i − b † b . (4.36)The assigned grading of ± , H ± , with the U (1) charge S z = N b + P mi =1 (2 N a i + 1) / (cid:2) S z , H (cid:3) = 0 , (cid:2) S z , H + (cid:3) = H + , (cid:2) S z , H − (cid:3) = − H − . (4.37)In light of these relations, we may decompose V into a direct sum of eigenspace V i,p of the U (1) charge S z and the Casimir operators of the su (1 ,
1) algebra K i = S zi ( S zi − − S + i S − i , i.e. V = M i, { p } V i, { p } . (4.38)Explicitly, the subspace V i, { p } can be written as V i, { p } ≡ span { ( b † ) l ( a † ) l + p · · · ( a † m ) l m + p m | 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 (cid:18) p i (cid:19) V i, { p } , K i V i, { p } = (cid:18) p i (cid:19) (cid:18) p i − (cid:19) V i, { p } . (4.40)From the commutation relations (4.37), we therefore have H + V i, { p } ⊆ V i +1 , { p } , H V i, { p } ⊆ V i, { p } , H − V i, { p } ⊆ V i − , { p } . (4.41)The QES property of the Hamiltonian arises from the fact that for given integer value of n and f z = P i ( l i + 1) /
4, we have H + V n, { l } = { } . As a result, the Hamiltonian leaves the following subspace invariant: V QES ≡ n M i =0 V i, { l } . (4.42)We can indeed verify that the Bethe vectors lie within this invariant subspace, by expanding the eigen-vectors (2.9) explicitly. It would be interesting to examine the possibility of obtaining exact solutionsoutside of this sector. 7 Conclusion
We’ve investigated a class of QES, integrable multi-mode bosonic models using the quasi-Gaudin alge-bra. We see that such models are obtained via a mixed representation consisting of commuting copiesof su (1 , U (1) symmetry. We identified the QES sector of the Hamiltonian as the direct sum of theeigensubspaces of the U (1) charge with eigenvalues no greater than n .The ABA method leads to partial solutions of the Hamiltonians we’ve considered. Given the inte-grability of the Hamiltonian, in the sense that H n ( λ ) acts as a generator of conserved operators, it wouldbe interesting to explore the possibility of obtaining the entire spectrum via some other techniques. Thedominating experience is that integrability and exact solvabilty go hand-in-hand. It is not apparent forthese Hamiltonians whether the full spectrum is potentially accessible.Finally we note that due to the constraint arising from imposing hermiticity on the generating function H n ( λ ), the quasi-Gaudin formalism is at present limited to cases based on underlying unitary represen-tations of su (1 , su (1 , Aknowledgements
This work was supported by the Australian Research Council through Discovery Projects DP11013434and DP110101414.
References [1] A. Ushveridze, Mod. Phys. Lett.
A13 (1998) 281-292[2] A. Ushveridze, Ann. Phys. (1998) 81-134[3] A. Turbiner, Commun. Maths. Phys. (1988) 467-474[4] V.V. Ulyanov and O.B. Zaslavskii, Phys. Rep. (1992) 179-251[5] A.G. Ushveridze,
Quasi-Exactly Solvable Models in Quantum Mechanics , Institute of Physics Pub-lishing, London, 1994.[6] V.E. Korepin, N.M. Bogoliubov, and A.G. Izergin,
Quantum Inverse Scattering Method and Corre-lation Functions , Cambridge University Press, 1993.[7] G. Ortiz, R. Somma, J. Dukelsky, and S. Rombouts, Nucl. Phys. B (2005) 421-457[8] A. Foerster and E. Ragoucy, Nucl. Phys. B (2007) 373-403[9] L. Amico, H. Frahm, A. Osterloh, and G.A.P. Ribeiro, Nucl. Phys. B (2007) 283-300[10] F. Pan, M.-X. Xie, X. Guan, L.-R. Dai, and J.P. Draayer, Phys. Rev. C (2009) 044306[11] L. Amico, H. Frahm, A. Osterloh, and T. Wirth, Nucl. Phys. B (2010) 604-626812] M.T. Batchelor, A. Foerster, X.-W. Guan, and C.C.N. Kuhn, J. Stat. Mech.: Theor. Exp. (2010)P12014[13] C. Dunning, M. Iba˜nez, J. Links, G. Sierra, and S.-Y. Zhao, J. Stat. Mech.: Theor. Exp. (2010)P08025[14] Y.-H. Lee, J. Links, and Y.-Z. Zhang, Nonlinearity (2011) 1975-1986[15] M. Sanz, M.M. Wolf, D. P´erez-Garca, and J.I. Cirac, Phys. Rev. A (2009) 042308[16] S.H. Jacobsen and P.D. Jarvis, J. Phys. A: Math. Theor. (2010) 255305[17] A.J. Leggett, S. Chakravaty, A.T. Dorsey, M.P.A. Fisher, A. Garg, and W. Zwerger, Rev. Mod.Phys. (1987) 1-85[18] Y. Makhlin, G. Sch¨on, and A. Shnirman, Rev. Mod. Phys. (2001) 357-400[19] J. Gilmore and R.H. McKenzie, J. Phys.: Condens. Matter (2005) 17351746[20] I. Tikhonenkov, E. Pazy, Y. B. Band, and A. Vardi, Phys. Rev. A (2008) 063624[21] J. Links, H.-Q. Zhou, R.H. McKenzie, and M.D. Gould, J. Phys. A: Math. Gen.36