Quasi-solvability of Calogero-Sutherland model with Anti-periodic Boundary Condition Arindam Chakraborty Dept. of Physics, Jadavpur University, Calcutta 700 032, India Subhankar Ray ∗ Dept. of Physics, Jadavpur University, Calcutta 700 032, India and C N Yang Institute for Theoretical Physics, Stony Brook, NY 11794, USA 7 0 J. Shamanna † 0 2 Physics Department, University of Calcutta, Calcutta 700 009, India n a (Dated: 30 December 2006) J 5 The U(1) Calogero-Sutherland Model with anti-periodic boundary condition is 1 studied. This model is obtained by applying a vertical magnetic field perpendicu- 1 v 7 lar to the plane of one dimensional ring of particles. The trigonometric form of the 3 1 Hamiltonian isrecastbyusingasuitablesimilarity transformation. Thetransformed 1 0 Hamiltonian is shown to be integrable by constructing a set of momentum operators 7 0 which commutes with the Hamiltonian and amongst themselves. The function space / h t of monomials of several variables remains invariant under the action of these oper- - p e ators. The above properties imply the quasi-solvability of the Hamiltonian under h : consideration. v i X r a I. INTRODUCTION The study of Calogero-Sutherland system has inspired significant research activity since the pioneering work of Calogero and Sutherland1,2. The integrability of the model has been studied for different root systems over the past few decades3. A few of the classical and spin varieties of the model are found to be exactly solvable and the solutions in terms of their eigenvalues and eigenfunctions have been used extensively to describe physical properties of several condensed matter systems. The study of Calogero systems is also related to various other research areas inphysics andmathematics, e.g., Yang-Millstheories4,5, solitontheory6, 2 random matrix model7, multivariable orthogonal polynomials8, Selberg integral formula9, W algebra10 etc. ∞ Thisarticleinvestigates theCalogero-SutherlandModel(CSM) withanti-periodicbound- ary condition. The anti-periodic boundary condition is a special case of the general twisted boundary condition which arises when a one dimensional chain of particles is placed in a transverse magnetic field. A one dimensional chain of particles with a periodic boundary condition is topologically equivalent to a one dimensional ring. A particle transported adia- batically around this ring an integral number of times returns to the same point. In absence of a magnetic field this implies that the particle returns to the same quantum state. How- ever, in the presence of a transverse magnetic field, one adiabatic transportation around the ring introduces a phase factor exp(iφ). This is called a twisted boundary condition. When the phase factor is exp(iφ) = 1, it is called an anti-periodic boundary condition. Though − the introduction of a magnetic field is physically important in this context, the model be- comes mathematically more involved; and the CSM with anti-periodic boundary condition remains less extensively investigated. The original version of the Calogero system incorporates long-range interaction by con- sidering a two-body inverse square potential. The integrability of such systems was initially studied by Calogero and Perelomov11,12 by means of Lax pair formulation. The integrability of CSM has since been investigated in a variety of ways3,13,14,15. The general form of CSM Hamiltonian is often represented by the following equation: N H = ∂ 2 λ(λ 1) U(x ) (1) N j − − −jk Xj=1 Xj,k j=k 6 The two-body potential, represented by U(x ) is a long-range interaction in a chain of −jk spinless nonrelativistic particles in one dimension. Here, λ is a dimensionless interaction parameter, x and x denote the coordinates of the j-th and k-th particle respectively and j k x = x x . While studying the solvability ofA -type Calogero model, the Hamiltonian −jk j− k N−1 is operated on a partially ordered state space of all symmetric polynomials of several vari- ables. This results in an upper triangular representation of the Hamiltonian. The diagonal terms of this matrix are the eigenvalues of the Hamiltonian. The orthonormaleigenfunctions are expressed in terms of Jack symmetric polynomials8 which are very useful in determining the various physical properties of many particle systems with long-range interactions16. 3 The search for an exact form of eigenfunction sometimes leads to partial diagonaliza- tion of the Hamiltonian17,18. Among the one dimensional systems with periodic boundary condition, several such quasi-solvable models exist. The eigenvalues and eigenfunctions for many of them have been obtained19,20. The model with sl(2) structure was first discovered by Turbiner and Ushveridze21. It was also observed that the well known N body Calogero- Sutherland models2,22,23 have similar Lie algebraic structure of sl(N +1). It may be noted that these models are in fact different generalizations of the classically integrable Inozemtsev model24,25. The common feature of these models with some under- lying Lie algebraic structure is the existence of an invariant finite dimensional module of the associated Lie algebra. Post and Turbiner26 studied a classification of linear differential operators of a single variable which have a finite dimensional invariant subspace spanned by monomials. One of the basic advantages of quasi-solvability is that, one can restrict the study to a finite dimensional submanifold of the full Hamiltonian. The finite dimensional matrix elements can be calculated by allowing the Hamiltonian to act on finite-dimensional subspaces of a Hilbert space on which it is originally defined. When the Hamiltonian oper- ator preserves an infinite number of subsequences of such finite dimensional subspaces17 it becomes solvable. The exact solvability of a model is ensured when the closure property is imposed on the space on which the Hamiltonian is allowed to act. In this article we study the integrability and solvability of a spinless non-relativistic Calogero-Sutherland model (CSM) with anti-periodic boundary condition. The two-body long-range interaction incorporating the anti-periodic boundary condition is derived. The Hamiltonian so obtained is reduced to a more apparent integrable form, using a similarity transformation. The integrability is then verified by constructing a set of mutually com- muting momentum-like differential operators which further commute with the Hamiltonian. Finally, the concept of quasi-solvability is discussed for a model of many particle system. For CSM with anti-periodic boundary condition the quasi-solvability is studied by operating the Hamiltonian on a multivariable polynomial space18. The momentum operators in the anti-periodic model remind us of the well known Dunkl operator27,28 which resembles the Laplace-Beltrami-type operator acting on a symmetric Riemannian space. These operators areextensively usedinthestudyofintegrability andsolvabilityofCalogero-Sutherlandmod- els. It is shown that these commuting momentum operators preserve the space spanned by all monomials of degree n, i.e., zℓi , where ℓ 0 and ℓ = n, n being a non-negative { i i } i ≥ i Q P 4 integer. This property ensures the quasi-solvability of the Hamiltonian under study. II. TRIGONOMETRIC VERSION OF CSM HAMILTONIAN Let us first consider the periodic CSM with inverse square long-range interaction in the absence of a magnetic field. The topological representation of a one dimensional chain of particles with periodic boundary condition is simply a circular ring. A particle when trans- ported adiabatically around the ring an integral number of times, does not take up any phase factor, and so the eigenfunctions retain their initial form. Then the pairwise interac- tion summed around a unitarily equivalent circle of circumference L, an infinite number of times is given as, + ∞ 1 1 = (2) (x+nL)2 d(x)2 n= X−∞ where, as shown in the figure, x is the interparticle distance along the ring and d(x) is the chord length. It is easy to verify that d(x) = L/πsin(πx/L). Therefore, the potential x k θ d(x − x ) d(x) j+1 j x x j j+1 x FIG. 1: Inter-particle distances d(x) and d(x x ) for particles on a circular chain j+1 j − U(x) = (π2/L2)sin 2x is an inverse trigonometric function of the inter-particle distance x. − The Hamiltonian with the above potential is given by, N π2 1 H = ∂ 2 λ(λ 1) (3) N j − − L2 sin2(πx /L) Xj=1 Xj,k −jk j=k 6 where x = x x . Using standard trigonometric identity, making a change of variable −jk j − k (π/2L)x x and rescaling the Hamiltonian (4L2/π2)H H , Eq. (3) may be written j j N N → → 5 as, N 1 1 H = ∂ 2 λ(λ 1) + . (4) N j − − sin2(x ) cos2(x ) Xj=1 Xj,k −jk −jk ! j=k 6 Letusnowconsider theanti-periodiccase. Whenamagneticfieldisintroducedtransverse to theonedimensionalring,ageneraltwistedboundaryconditionarises. Aparticletransported adiabatically around the entire system n number of times picks up a net phase exp(inφ). The pairwise interaction summed around a unitarily equivalent circle of circumference L, an infinite number of times, is now given as, + ∞ exp(iφn) . (5) (x+nL)2 n= X−∞ The above summation can be performed by making the choice, φ = 2πp/q, with p, q relative primes, and n = jq + k, with j, k integers ( < j < + and 0 k q 1). The −∞ ∞ ≤ ≤ − interaction term becomes, + q 1 + q 1 ∞ exp(iφn) − ∞ exp(i2πpj)exp(i2πpk/q) − exp(i2πpk/q) = = . (6) n= (x+nL)2 k=0j= [(x+kL)+(qL)j]2 k=0 (qL/π)sin π(x+kL) 2 X−∞ X X−∞ X qL h i The last expression represents an interaction with a general twisted boundary condition. The model can be viewed as a system of interacting particles residing on a circle with circumference qL. For p/q = 1/2 the sum becomes, q−1 exp(i2πpk/q) 1 exp(iπk) 1 ( 1)k = = − . (7) 2 ⇒ 2 2 k=0 (qL/π)sin π(x+kL) k=0 (qL/π)sin π(x+kL) k=0 (2L/π)sin π(x+kL) X qL X qL X 2L h i h i h i This corresponds to an anti-periodic boundary condition16. The potential then takes the following form cos(πx/L) U(x) = . (8) (L2/π2)sin2(πx/L) Thus the Hamiltonian with anti-periodic boundary condition, using standard trigonometric identity, and scale changes (π/2L)x x and (4L2/π2)H H , becomes j j N N → → N 1 1 Hap = ∂ 2 λ(λ 1) . (9) N j − − sin2(x ) − cos2(x ) Xj=1 Xj,k −jk −jk ! j=k 6 6 III. INTEGRABILITY OF THE MODEL HAMILTONIAN The integrability of such types of Hamiltonian is established by constructing a complete set of commuting momentum operators that also commute with the model Hamiltonian. These operators were initially introduced in the study of Calogero-Sutherland model with periodic boundary conditions (both spin and classical cases) and are known as Dunkl oper- ators. Similar operators have been used to study Calogero-Sutherland-type models derived from different root systems and are called Dunkl-type operators18,29. To construct commutative Dunkl-type operators we introduce variables, z = exp(2ix ). j j Using this substitution, the anti-periodic Hamiltonian becomes, N z z z z Hap = (z ∂ )2 λ(λ 1) j k λ(λ 1) j k , where z = z z . (10) N j j − − (z )2 − − (z+)2 j±k j ± k Xj=1 Xj,k j−k Xj,k jk j=k j=k 6 6 Let us apply the following similarity transformation, H = ∆ 1Hap∆ N − N e where, ∆ = (zj−k)µ1(λ)(zj+k)µ2(λ) , and µ (λ) = λ,1 λ , µ (λ) = 1[1 √1+4λ 4λ2]. (z z )( µ1(λ)+µ2(λ) /2) 1 − 2 2 ± − j k { } Yj,k j<k Thus, the anti-periodic Hamiltonian becomes H = N (z ∂ )2 + µ1(λ) zj+k(z ∂ z ∂ )+ µ2(λ) zj−k(z ∂ z ∂ ) (11) N j j 2 z j j − k k 2 z+ j j − k k Xj=1 Xj,k j−k Xj,k jk j=k j=k e 6 6 The term µ (λ) is real for all λ, however, µ (λ) is real only for 1 + 4λ 4λ2 0, i.e., 1 2 − ≥ λ 1 1 . Under this restriction, the Hamiltonian H becomes hermitian. In the | − 2| ≤ √2 N following, the integrability of the Hamiltonian is studied for different allowed values of λ. e Let usintroduce thecoordinateexchange operators Λ j,k = 1,..N;j = k andthesign jk { | 6 } reversing operators Λ j,k = 1,..N . The coordinateexchange operator acting onthe coor- j { | } dinatesofj-thandk-thparticlemaybedefinedbytheoperationΛ f(z ,..,z ,..,z ,..,z ) = jk 1 j k N f(z ,..,z ,..,z ,..,z ). This operator is (i) self-adjoint, (ii) unitary, and satisfies (iii) 1 k j N Λ Λ = Λ Λ = Λ Λ , (iv) Λ Λ = Λ Λ , (v) Λ z ∂ = z ∂ . ij jk ik ij jk ik ij kl kl ij jk k k j j 7 The sign reversing operator Λ may be defined by its action on the coordinates of the j j-th particle as Λ f(z ,..,z ,..,z ) = f(z ,.., z ,..,z ). This operator is (i) self-inverse j 1 j N 1 j N − Λ 1 = Λ , (ii) mutually commuting [Λ ,Λ ] = 0, and satisfies (iii) [Λ ,Λ ] = 0 , i = j = k, −j j j k ij k 6 6 (iv) Λ Λ = Λ Λ . ij j i ij In terms of Λ and Λ , we introduce operator Λ = Λ Λ Λ , which is (i) self-adjoint jk j jk j k jk and (ii) unitary. In addition it satisfies (iii) Λ Λ = Λ Λ = Λ Λ , (iv) Λ Λ = Λ Λ , ij jk ik ij jk ik ij kl kl ij e (v) Λ z ∂ = z ∂ . In terms of the above mentioned operators, the Dunkl-type momentum jk k k j j e e e e e e e e e e operators D j = 1,...,N may be represented by the following equation, j e { | } µ1(λ) zj+k µ2(λ) zj−k D = z ∂ + (1 Λ ).+ (1 Λ ) (12) j j j 2 z − jk 2 z+ − jk k(=j) j−k k(=j) jk X6 X6 e H = D2. TheseDunkl-typeoperatorscommutewiththecoordinateexchangeoperators N j j and thPe operators Λjk, i.e; [Dj,Λjk] = 0, [Dj,Λjk] = 0. They also commute among them- e selves and because of the very nature of their construction, commute with the Hamiltonian; e e [D ,D ] = 0, [D ,H ] = 0. The existence of such an operator establishes the integrability j k j N of the system. e IV. QUASI-SOLVABILITY OF THE MODEL HAMILTONIAN The integrability does not necessarily imply the existence of a function space involv- ing the variables z j = 1,...,N such that H can be represented in a diagonal form. j N { | } However, sometimes it may so happen that operators like H , acting on a suitably chosen N e function subspace can preserve the space partially. In such cases we introduce the term e quasi-solvability. A linear differential operator H of several variables z j = 1,...,N , is N j { | } said to be quasi-solvable if it preserves a finite dimensional function space V whose basis ν admits an analytic expression in a closed form i.e., H V V , dimV = n(ν) < , where V = v (z),...,v (z) . (13) N ν ν ν ν 1 n(ν) ⊆ ∞ h i One of the advantages of quasi-solvability is that one can explicitly evaluate finite dimen- sional matrix elements A defined by kl n(ν) H v = A v , (k = 1,...,n(ν)). N k kl l l=1 X 8 The finite dimensional submatrices A may be diagonalizable even when the entire H is kl N not. If the space V is the subspace of a Hilbert space on which the operator H is defined, ν N the spectrum of H can be computed algebraically, so as to obtain the exact eigenvectors N of H that belong entirely to V . This is the typical nature of quasi-solvability. N ν A quasi-solvable operator is said to be solvable if the quasi-solvability condition holds for an infinite number of sequences of finite dimensional proper subspaces each containing its previous descendant. V V V ... 1 2 ν ⊂ ⊂ ··· ⊂ ⊂ Moreover, if the closure of V , as ν , is the Hilbert space on which H acts, we call ν N → ∞ H to be exactly solvable. N Now, we shall show that the Dunkl-type momentum operators obtained in Eq.(12) pre- serve the space i.e., the space spanned by all monomials of the form zℓi, where ℓ 0 Rn i i i ≥ and iℓi = n, n being a non negative integer. Q ItPis easy to verify that the operator (zj∂j) preserves the space n. R (z ∂ ) zℓi = ℓ zℓi (14) j j i j i i i Y Y We shall show that the second and third operators in Eq. (12) preserve , i.e., (z + n j R z )/(z z )(1 Λ ) zℓi and (z z )/(z +z )(1 Λ) zℓi . They can k j − k − jk i i ∈ Rn j − k j k − jk i i ∈ Rn be rewritten as Q Q e z +z |ℓj−ℓk|−1 zj zk(1−Λjk) ziℓi =  ziℓi(zj+zk)(zjzk)min(ℓj,ℓk)sign(ℓj−ℓk) zj|ℓj−ℓk|−1−rzkr j k − i i(=j,k) r=0 Y Y6 X   (15) and z z |ℓj−ℓk|−1 zj −+zk(1−Λ)jk ziℓi =  ziℓi(zj−zk)(zjzk)min(ℓj,ℓk)κ(ℓj,ℓk) (−zj)|ℓj−ℓk|−1−rzkr j k i i(=j,k) r=0 Y Y6 X e   (16) where sign(0) = 0 and sign(α) = α/ α for α = 0. And | | 6 1 α < β κ(α,β) =  0 α = β     ( 1)α+β α > β − −     9 The right hand side of Eq.(15) can be expressed as a sum of the following two terms, |ℓj−ℓk|−1 ziℓi zj(zjzk)min(ℓj,ℓk)sign(ℓj −ℓk) zj|ℓj−ℓk|−1−rzkr (17) ! i=j,k r=0 Y6 X and |ℓj−ℓk|−1 ziℓi zk(zjzk)min(ℓj,ℓk)sign(ℓj −ℓk) zj|ℓj−ℓk|−1−rzkr. (18) ! i=j,k r=0 Y6 X Let pj and pk denote the powers of z and z in the r-th summand of Eq. (17). Then r r j k pj = max(ℓ ,ℓ ) r and pk = min(ℓ ,ℓ )+r . Thus, pj +pk = ℓ +ℓ . Therefore, the sum r j k − r j k r r j k of powers of z and z , in the r-th summand is ( ℓ )+ℓ +ℓ = ℓ . j k i=j,k i j k i i 6 Hence, the expression (17) is a member of Pn. Similar calculationPshows that the ex- R pression (18) also belongs to a space spanned by monomials of degree n. Thus, the second operator in Eq. (12) preserves . In a similar manner it can be verified that the third n R operator in Eq. (12) also preserves . As the operators D j = 1,...,N are linear n j R { | } and preserve the space , the Hamiltonian H (= D2) also preserves , and hence is Rn N j Rn quasi-solvable. P e V. CONCLUSION In this article we have studied the behaviour of one dimensional chain of particles in a magnetic field interacting through an inverse square potential. The anti-periodic boundary condition allows one to analyze a special form of the above situation. The extension of the model to anti-periodic case reduces the algebraic symmetry of the root systems. This makes this model mathematically more challenging. Here, we recast the Hamiltonian to a new form by using a suitable similarity transformation. The transformed Hamiltonian is shown to be integrable in the sense that there exists a complete set of commuting momentum operators which also commute with the Hamiltonian. It is observed that the momentum operators are hermitian for a certain range of the interaction parameter. The new form of the Hamiltonian and its constituent momentum operators indicate the existence of a multivariable polynomial space which is invariant under the action of the Hamiltonian. Indeed, it is observed that the momentum operatorsconstructed in this article keep the monomial space invariant. This invariance demonstrates the quasi-solvability n R of the model. 10 Acknowledgment AC wishes to acknowledge the Council of Scientific and Industrial Research, India (CSIR) for fellowship support. Electronic address: [email protected] ∗ Electronic address: [email protected] † 1 F. Calogero, J. Math. Phys. 10 (1969) 2191; 10 (1969) 2197. 2 B. Sutherland, J. Math. Phys. 12 (1971) 246; 12 (1971) 251. 3 M. A. Olshanetsky, A. M. Perelomov, Phys. Rep. 94 (1983) 313. 4 A. Gorsky and N. Nekrasov, Nucl. Phys. B 414 (1994) 213. 5 J. A. Minahan and A. P. Polychronakos, Phys. Lett. B 326 (1994) 288. 6 A. P. Polychronakos, Phys. Rev. Lett. 74 (1995) 5153. 7 F. J. Dyson, J. Math. Phys. 3 (1962) 140, 157, 166. 8 H. Jack, Proc. Roy. Soc. (Edinburgh) A 69 (1970) 1. 9 P. J. Forrester, Phys. Lett. A 179 (1993) 127. 10 K. Hikami, M. Wadati, J. Phys. Soc. Japan 62 (1993) 469. 11 F. Calogero, Lett. Nuovo Cim. 13 (1975) 411. 12 A. M. Perelomov,Lett. Math. Phys. 1 (1977) 531. 13 J. A. Minahan, A. P. 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