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arXiv:hep-th/0203245 · Physical review. B, Condensed matter

Algebra of observables in the Calogero model and in the Chern-Simons matrix model

Larisa Jonke, Stjepan Meljanac

Published Nov 12, 2002• 6 citations
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Abstract

The algebra of observables of an N-body Calogero model is represented on the ${S}_{N}$-symmetric subspace of the positive-definite Fock space. We discuss some general properties of the algebra and construct four different realizations of the dynamical symmetry algebra of the Calogero model. Using the fact that the minimal algebra of observables is common to the Calogero model and the finite Chern-Simons (CS) matrix model, we extend our analysis to the CS matrix model. We point out the algebraic similarities and distinctions of these models.

DOI
10.1103/physrevb.66.205313

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We describe the implications of permutation symmetry for the state space and dynamics of quantum mechanical systems of matrices of general size NNN. We solve the general 11-parameter permutation invariant quantum matrix harmonic oscillator Hamiltonian and calculate the canonical partition function. The permutation invariant sector of the Hilbert space, for general Hamiltonians, can be described using partition algebra diagrams forming the bases of a tower of partition algebras Pk(N){P}_{k}(N)Pk​(N). The integer kkk is interpreted as the degree of matrix oscillator polynomials in the quantum mechanics. Families of interacting Hamiltonians are described which are diagonalized by a representation theoretic basis for the permutation invariant subspace which we construct for $N\ensuremath{\ge}2k$. These include Hamiltonians for which the low-energy states are permutation invariant and can give rise to large ground state degeneracies related to the dimensions of partition algebras. A symmetry-based mechanism for quantum many body scars discussed in the literature can be realized in these matrix systems with permutation symmetry. A mapping of the matrix index values to lattice sites allows a realization of the mechanism in the context of modified Bose-Hubbard models. Extremal correlators analogous to those studied in AdS/CFT\mathrm{AdS}/\mathrm{CFT}AdS/CFT are shown to obey selection rules based on Clebsch-Gordan multiplicites (Kronecker coefficients) of symmetric groups.
We study a multispecies one-dimensional Calogero model with two- and three-body interactions. Using an algebraic approach (Fock space analysis), we construct ladder operators and find infinitely many, but not all, exact eigenstates of the model Hamiltonian. Besides the ground state energy, we deduce energies of the excited states. It turns out that the spectrum is linear in quantum numbers and that the higher-energy levels are degenerate. The dynamical symmetry responsible for degeneracy is SU(2). We also find the universal critical point at which the model exhibits singular behaviour. Finally, we make contact with some special cases mentioned in the literature.
In Quantum Mechanics operators must be hermitian and, in a direct product space, symmetric. These properties are saved by Lie algebra operators but not by those of quantum algebras. A possible correspondence between observables and quantum algebra operators is suggested by extending the definition of matrix elements of a physical observable, including the eventual projection on the appropriate symmetric space. This allows to build in the Lie space of representations one-parameter families of operators belonging to the enveloping Lie algebra that satisfy an approximate symmetry and have the properties required by physics.
Spin generalization of the relativistic Calogero - Sutherland model is constructed by using the affine Hecke algebra and shown to possess the quantum affine symmetry . The spinless model is exactly diagonalized by means of the Macdonald symmetric polynomials. The dynamical density - density correlation function, as well as the one-particle Green function, are evaluated exactly. We also investigate the finite-size scaling of the model and show that the low-energy behaviour is described by the C = 1 Gaussian theory with a new selection rule. The results indicate that the excitations obey the fractional exclusion statistics and also exhibit the Tomonaga - Luttinger liquid behaviour.
An algebraic construction of the eigenstates for the quantum Calogero model is investigated. Extending the method of Lapointe and Vinet, we construct the eigenstates for the second conserved operator of the quantum Calogero model. All the eigenstates can be factorized into symmetric polynomials which we call “Hi-Jack symmetric polynomials” and the ground state wave function. The conjectured formula for the eigenvalue of the second conserved operator is confirmed. The Hi-Jack polynomials are strong candidates for the orthogonal basis of the quantum Calogero model.