Person: Rodríguez González, Miguel Ángel
Universidad Complutense de Madrid
Faculty / Institute
Now showing 1 - 10 of 37
PublicationQuasi-exactly solvable potentials on the line and orthogonal polynomials(American Institute of Physics, 1996-08) Finkel Morgenstern, Federico; González López, Artemio; Rodríguez González, Miguel ÁngelIn this paper we show that a quasi-exactly solvable (normalizable or periodic) one-dimensional Hamiltonian satisfying very mild conditions defines a family of weakly orthogonal polynomials which obey a three-term recursion relation. in particular, we prove that (normalizable) exactly solvable one-dimensional systems are characterized by the fact that their associated polynomials satisfy a two-term recursion relation. We study the properties of the family of weakly orthogonal polynomials defined by an arbitrary one-dimensional quasi-exactly solvable Hamiltonian, showing in particular that its associated Stieltjes measure is supported on a finite set. From this we deduce that the corresponding moment problem is determined, and that the kth moment grows Like the kth power of a constant as k tends to infinity. We also show that the moments satisfy a constant coefficient linear difference equation, and that this property actually characterizes weakly orthogonal polynomial systems. PublicationQuasi-exactly solvable n-body spin hamiltonians with short-ange interaction potentials(Institute of Mathematics of National Academy of Science of Ukraine, 2006-11-03) Enciso, Alberto; Finkel Morgenstern, Federico; González López, Artemio; Rodríguez González, Miguel ÁngelWe review some recent results on quasi-exactly solvable spin models presenting near-neighbors interactions. These systems can be understood as cyclic generalizations of the usual Calogero-Sutherland models. A nontrivial modification of the exchange operator formalism is used to obtain several infinite families of eigenfunctions of these models in closed form. PublicationGeneralized isotropic Lipkin-Meshkov-Glick models: ground state entanglement and quantum entropies(IOP Publishing, 2016-03) Carrasco, José A.; Finkel Morgenstern, Federico; González López, Artemio; Rodríguez González, Miguel Ángel; Tempesta, PiergiulioWe introduce a new class of generalized isotropic Lipkin–Meshkov–Glick models with su(m+1) spin and long-range non-constant interactions, whose non-degenerate ground state is a Dicke state of su(m+1) type. We evaluate in closed form the reduced density matrix of a block of Lspins when the whole system is in its ground state, and study the corresponding von Neumann and Rényi entanglement entropies in the thermodynamic limit. We show that both of these entropies scale as a log L when L tends to infinity, where the coefficient a is equal to (m − k)/2 in the ground state phase with k vanishing magnon densities. In particular, our results show that none of these generalized Lipkin–Meshkov–Glick models are critical, since when L-->∞ their Rényi entropy R_q becomes independent of the parameter q. We have also computed the Tsallis entanglement entropy of the ground state of these generalized su(m+1) Lipkin–Meshkov–Glick models, finding that it can be made extensive by an appropriate choice of its parameter only when m-k≥3. Finally, in the su(3) case we construct in detail the phase diagram of the ground state in parameter space, showing that it is determined in a simple way by the weights of the fundamental representation of su(3). This is also true in the su(m+1) case; for instance, we prove that the region for which all the magnon densities are non-vanishing is an (m + 1)-simplex in R^m whose vertices are the weights of the fundamental representation of su(m+1). PublicationOn higher-dimensional superintegrable systems: a new family of classical and quantum Hamiltonian models(IOP Publishing Ltd, 2022-12-16) Rodríguez González, Miguel Ángel; Tempesta, PiergiulioWe introduce a family of n-dimensional Hamiltonian systems which, contain, as special reductions, several superintegrable systems as the Tremblay-Turbiner-Winternitz system, a generalized Kepler potential and the anisotropic harmonic oscillator with Rosochatius terms. We conjecture that there exist special values in the space of parameters, apart from those leading to known cases, for which this new Hamiltonian family is superintegrable. PublicationNew spin Calogero-Sutherland models related to B-N-type Dunkl operators(Elsevier, 2001-10-15) Finkel Morgenstern, Federico; Gómez-Ullate Otaiza, David; González López, Artemio; Rodríguez González, Miguel ÁngelWe construct several new families of exactly and quasi-exactly solvable BCN-type Calogero-Sutherland models with internal degrees of freedom. Our approach is based on the introduction of a new family of Dunkl operators of B-N type which, together with the original B-N-type Dunkl operators, are shown to preserve certain polynomial subspaces of finite dimension. We prove that a wide class of quadratic combinations involving these three sets of Dunkl operators always yields a spin Calogero-Sutherland model, which is (quasi-)exactly solvable by construction. We show that all the spin Calogero-Sutherland models obtainable within this framework can be expressed in a unified way in terms of a Weierstrass rho function with suitable half-periods. This provides a natural spin counterpart of the well-known general formula for a scalar completely integrable potential of BCN type due to Olshanetsky and Perelomov. As an illustration of our method, we exactly compute several energy levels and their corresponding wavefunctions of an elliptic quasi-exactly solvable potential for two and three particles of spin 1/2. PublicationCritical behavior of su(1|1) supersymmetric spin chains with long-range interactions(American Physical Society, 2016-06-01) Carrasco, José A.; Finkel Morgenstern, Federico; González López, Artemio; Rodríguez González, Miguel Ángel; Tempesta, PiergiulioWe introduce a general class of su(1|1) supersymmetric spin chains with long-range interactions which includes as particular cases the su(1|1) Inozemtsev (elliptic) and Haldane-Shastry chains, as well as the XX model. We show that this class of models can be fermionized with the help of the algebraic properties of the su(1|1) permutation operator and take advantage of this fact to analyze their quantum criticality when a chemical potential term is present in the Hamiltonian. We first study the low-energy excitations and the low-temperature behavior of the free energy, which coincides with that of a (1+1)-dimensional conformal field theory (CFT) with central charge c=1 when the chemical potential lies in the critical interval (0,E(π)), E(p) being the dispersion relation. We also analyze the von Neumann and Rényi ground state entanglement entropies, showing that they exhibit the logarithmic scaling with the size of the block of spins characteristic of a one-boson (1+1)-dimensional CFT. Our results thus show that the models under study are quantum critical when the chemical potential belongs to the critical interval, with central charge c=1. From the analysis of the fermion density at zero temperature, we also conclude that there is a quantum phase transition at both ends of the critical interval. This is further confirmed by the behavior of the fermion density at finite temperature, which is studied analytically (at low temperature), as well as numerically for the su(1|1) elliptic chain. PublicationA_(N)-type Dunkl operators and new spin Calogero-Sutherland models(Springer, 2001-08) Finkel Morgenstern, Federico; Gómez-Ullate Otaiza, David; González López, Artemio; Rodríguez González, Miguel ÁngelA new family of AN-type Dunkl operators preserving a polynomial subspace of finite dimension is constructed. Using a general quadratic combination of these operators and the usual Dunkl operators, several new families of exactly and quasi-exactly solvable quantum spin Calogero-Sutherland models are obtained. These include, in particular, three families of quasi-exactly solvable elliptic spin Hamiltonians. PublicationThe Berry-Tabor conjecture for spin chains of Haldane-Shastry type(EPL Association, European Physical Society, 2008-07) Barba, J. C.; Finkel Morgenstern, Federico; González López, Artemio; Rodríguez González, Miguel ÁngelAccording to a long-standing conjecture of Berry and Tabor, the distribution of the spacings between consecutive levels of a "generic" integrable model should follow Poisson's law. In contrast, the spacings distribution of chaotic systems typically follows Wigner's law. An important exception to the Berry-Tabor conjecture is the integrable spin chain with long-range interactions introduced by Haldane and Shastry in 1988, whose spacings distribution is neither Poissonian nor of Wigner's type. In this letter we argue that the cumulative spacings distribution of this chain should follow the "square root of a logarithm" law recently proposed by us as a characteristic feature of all spin chains of Haldane-Shastry type. We also show in detail that the latter law is valid for the rational counterpart of the Haldane-Shastry chain introduced by Polychronakos. PublicationNew algebraic quantum many-body problems(IOP Publishing, 2000-10-20) Gómez-Ullate Otaiza, David; González López, Artemio; Rodríguez González, Miguel ÁngelWe develop a systematic procedure for constructing quantum many-body problems whose spectrum can be partially or totally computed by purely algebraic means. The exactly solvable models include rational and hyperbolic potentials related to root systems, in some cases with an additional external field. The quasi-exactly solvable models can be considered as deformations of the previous ones which share their algebraic character. PublicationOn the families of orthogonal polynomials associated to the Razavy potential(IOP Publishing LTD, 1999-10-01) Finkel Morgenstern, Federico; González López, Artemio; Rodríguez González, Miguel ÁngelWe show that there are two different families of (weakly) orthogonal polynomials associated to the quasi-exactly solvable Razavy potential V_ (x) = (ζ cosh 2x−M)^ 2 (ζ > 0, M ∈ N). One of these families encompasses the four sets of orthogonal polynomials recently found by Khare and Mandal, while the other one is new. These results are extended to the related periodic potential U_(x) = −(ζ cos 2x − M) ^2 , for which we also construct two different families of weakly orthogonal polynomials. We prove that either of these two families yields the ground state (when M is odd) and the lowest lying gaps in the energy spectrum of the latter periodic potential up to and including the (M − 1)_(th) gap and having the same parity as M − 1. Moreover, we show that the algebraic eigenfunctions obtained in this way are the well-known finite solutions of the Whittaker–Hill (or Hill’s three-term) periodic differential equation. Thus, the foregoing results provide a Lie-algebraic justification of the fact that the Whittaker– Hill equation (unlike, for instance, Mathieu’s equation) admits finite solutions.