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Algebraic Models of Many-Body Systems and Their Dynamic Symmetries and Supersymmetries

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Algebraic Models of Many-Body Systems and Their Dynamic Symmetries and Supersymmetries Journal of Physics: Conference Series PAPER • OPEN ACCESS Algebraic models of many-body systems and their dynamic symmetries and supersymmetries To cite this article: Francesco Iachello 2019 J. Phys.: Conf. Ser. 1194 012048 View the article online for updates and enhancements. This content was downloaded from IP address 131.169.5.251 on 06/05/2019 at 21:02 Group32 IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 1194 (2019) 012048 doi:10.1088/1742-6596/1194/1/012048 Algebraic models of many-body systems and their dynamic symmetries and supersymmetries Francesco Iachello Center for Theoretical Physics, Sloane Laboratory, Yale University, New Haven, CT 06520-8120, USA [email protected] Abstract. An overview of the method of spectrum generating algebras (SGA) and dynamical symmetries (DS) is given and applied to a class of models (the so-called s-b boson models) with SGAu(n). Quantum phase transitions (QPT) in these systems are discussed by introducing the coset spaces U(n)/U(n-1)!U(1). Finally, spectrum generating superalgebras (SGSA) and dynamical supersymmetries (SUSY) of a class of Bose-Fermi systems with SGAu(n/m) are introduced and applied to s-b Bose-Fermi models. 1. Introduction Algebraic methods have been used extensively in physics since their introduction by Heisenberg (1932), Wigner (1936) and Racah (1942). Since 1974, a general formulation of algebraic methods has emerged. (See, for example, [1]). In this formulation, a quantum mechanical many-body system is mapped onto an algebraic structure. The logic of the method is: Many-body system Quantization in terms of boson and/or fermion operators Algebraic structure (Lie algebra, Graded Lie algebra, Kac-Moody algebra, q-deformed algebra,...) Computation of observables (Spectra, Transitions, ...) Comparison with experiments 2. Algebraic models An algebraic model [2] is an expansion of the Hamiltonian and other operators in terms of elements, G!, of an algebra (often a Lie algebra g, or a contraction of it). The algebra Gg + is called the spectrum generating algebra (SGA). In most applications, the expansion is a polynomial in the elements, Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 Group32 IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 1194 (2019) 012048 doi:10.1088/1742-6596/1194/1/012048 HE0 BB G uGG ... , (1) Tt0 B tG ... but more complicated forms have been considered HfG () . (2) An interesting situation occurs when H does not contain all elements of g, but only the invariant (Casimir) operators of a chain of algebras gg&&' g " & ... H E0 Cg() ' Cg (') " Cg (")... , (3) called a dynamical symmetry (DS). (For the concept of DS, see also [3]). In this case, the energy eigenvalues can be written explicitly in terms of quantum numbers labeling the representations of g EHE0 Cg( ) ' Cg ( ') " Cg ( ") ... (4) Also here more complicated functionals of invariant operators have been considered HfC()i . (5) When a dynamic symmetry occurs, one can also calculate matrix elements of all operators in explicit analytic form, in terms of Clebsch-Gordan coefficients (isoscalar factors) of the algebras gg&&' g " & ... Dynamic symmetries were introduced implicitly by Pauli in 1926 [4] and later by Fock [5] for the hydrogen atom. The Hamiltonian with Coulomb interaction is invariant under a set of transformations, G, larger than rotations (Runge-Lenz transformations, SO(4)). (In this article, algebras will be denoted by lowercase letters, g, and groups by capital letters, G). It can be written in terms of Casimir operators of G, pe22 A H , (6) 2((4))1mr CSO2 with eigenvalues given by the Bohr formula A En(,, m ) . (7) n2 States can be classified by representations of sososo(4)&& (3) (2) with quantum number nm,, respectively. The corresponding spectrum is shown in figure 1 where it is compared with experiment. Apart from small relativistic corrections non included in Eq.(7), the agreement is excellent. Figure 1. The spectrum of the hydrogen atom is shown as evidence of SO(4) symmetry. In this case, the expansion is 1/H fC (i ). Pauli’s construction has been generalized to any number of dimensions 2 with DS so(+1) and to scattering states with DS so(*,1). In this case, only the Casimir operator of g appears, and therefore the algebra is often called degeneracy algebra, gd. The 2 Group32 IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 1194 (2019) 012048 doi:10.1088/1742-6596/1194/1/012048 embedding of gd into a larger algebra (the so-called dynamical algebra) which contains all the states of the system, is discussed in several books [2, 6]. DS assumed an important role in physics with the introduction of flavor symmetry by Gell’mann and Ne’eman [7-8], SUf(3). States can be classified by representations of &"& " sufIY(3) su (2) u (1) spin IY (2) u (1) with mass formula FV 1 2 MabCU12((1)) cCSUGW ( (2)) CU 1 ((1)) HX4 , (8) FV 1 2 MYII(,,3 ) a bY cGW II ( 1) Y HX4 where Y, I, I3 are the hypercharge, isotopic spin and third component of the isotopic spin labeling the representations of ususpinYI(1), (2), I (2) respectively. The corresponding spectrum is shown in figure 2 where is compared with experiment. Figure 2. The spectrum of the baryon decuplet is shown as an example of dynamic SUf(3) symmetry in baryons. 2.1. Bosonic systems It is convenient to write the elements Gg + as bilinear products of creation and annihilation operators (Jordan-Schwinger realization). For bosonic systems † Gbb , 1,...,n . (9) FV† FV FV†† From HXbb, , HXbb,,0HX bb , one obtains the commutation relations FV HXGG, G G (10) which define the real form of g=u(n) [or gl(n)]. The basis upon which the elements act is the totally symmetric representation, with one-row Young tableaux, 1 ††nn' NN [ ,0,0,...,0] Nbb ( ) ( ) ... 0 (11) N ! ' characterized by the total number of bosons N. 2.2. Fermionic systems Fermionic systems can also be treated algebraically in terms of bilinear products of anti-commuting operators † Gaaij i j ij, 1,..., m . (12) † †† From aaij, ij, aaij,,0 a i a j , one obtains 3 Group32 IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 1194 (2019) 012048 doi:10.1088/1742-6596/1194/1/012048 FV HXGGij, kl jk G il il G jk , (13) spanning the Lie algebra u(m) as before. The basis upon which these elements act is, however, the totally anti-symmetric representation, with one-column Young tableaux 1 †† N F [1,1,...,1,0,...,0] NaaFii'... 0 (14) N F ! characterized by the total number of fermions NF. 2.3. Mixed Bose-Fermi systems Mixtures of bosonic and fermionic systems can be treated in terms of bilinear products † Gbb Gaa † ,1,..., n ik i k (15) †† ij,1,..., m Fiiba † Fiiab often placed in matrix form CSbb†† ba DT i DT†† . (16) EUabiij aa The bilinear products above generate the graded Lie algebra (also called superalgebra) U(n/m). (For a classification of superalgebras see [9-10]. The superalgebra su(n/m) is A(n-1,m-1) in Kac’s classification). In a single index notation, the commutation relations of the graded algebra are FV j HXXX, B cX ; XY, iijB cY ; YYik, B f ik X (17) j where X are bosonic operators and Y are fermionic operators, together with the Jacobi identities. The basis upon which the elements act is the totally supersymmetric representation, with supertableux NNBF N[ N ,0,...,0} (18) where N is the total number of particles bosons+fermions. A symbol (square-curly bracket) different from Eq.(11) has been used in Eq.(18) to indicate that these representations are totally symmetric under the interchange of bosons and totally anti-symmetric under the interchange of fermions [11]. 3. Geometry of algebraic models Geometry can be associated to algebraic models with algebra g by constructing appropriate coset spaces, obtained by splitting g (Cartan decomposition) into g "hp (19) where h, a subalgebra of g, is called the stability algebra and p the remainder (not closed with respect to commutation). For models with u(n) structure, the appropriate coset space (g/h) is un()/( un" 1) u (1) . (20) This space is a globally symmetric Riemannian space [2, chapter 5] with dimension 2(n-1). For bosonic models, the algebra h can be constructed by selecting one boson, b1, and choosing †† hbbbb11, , 2,...,n . (21) †† 2,...,n p bb11, bb Associated with the Cartan decomposition, there are geometric variables i defined by 4 Group32 IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 1194 (2019) 012048 doi:10.1088/1742-6596/1194/1/012048 FV ## $+ iexp GWB ip i ext , pi p . (22) HXi For bosonic systems 1 N ### FV%&†*†%& † Nbbbbb;expHX11 1 0 . (23) N ! For systems with fixed value of N it is convenient to introduce projective coherent states in terms of projective variables [12] '' 1 FV††N Nbb;0HX1 . (24) N ! For algebraic models written in terms of boson operators bs1 and bmm ( 2,..., n ) , it is convenient to rewrite the coherent state as CSN 1 †† Nsb;0DTB mm (25) N ! EUm with normalization CSN 2 NN;;DT 1B m . (26) EUm The semiclassical energy functional associated with the quantum algebraic Hamiltonian is NHN;; EN(;) . (27) NN;; This energy functional depends on the complex coordinates m which can be split into real coordinates, qm and momenta, pm.
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