Person: González López, Artemio
Universidad Complutense de Madrid
Faculty / Institute
Now showing 1 - 10 of 55
PublicationNew quasi-exactly solvable hamiltonians in 2 dimensions(Springer, 1994-01) González López, Artemio; Kamran, Niky; Olver, Peter J.Quasi-exactly solvable Schrodinger operators have the remarkable property that a part of their spectrum can be computed by algebraic methods. Such operators lie in the enveloping algebra of a finite-dimensional Lie algebra of first order differential operators-the" hidden symmetry algebra. "In this paper we develop some general techniques for constructing quasi-exactly solvable operators. Our methods are applied to provide a wide variety of new explicit two-dimensional examples (on both flat and curved spaces) of quasi-exactly solvable Hamiltonians, corresponding to both semisimple and more general classes of Lie algebras. PublicationA novel class of translationally invariant spin chains with long-range interactions(Springer, 2020-08-21) Basur Mallick, B.; Finkel Morgenstern, Federico; González López, ArtemioWe introduce a new class of open, translationally invariant spin chains with long-range interactions depending on both spin permutation and (polarized) spin reversal operators, which includes the Haldane-Shastry chain as a particular degenerate case. The new class is characterized by the fact that the Hamiltonian is invariant under “twisted” translations, combining an ordinary translation with a spin flip at one end of the chain. It includes a remarkable model with elliptic spin-spin interactions, smoothly interpolating between the XXX Heisenberg model with anti-periodic boundary conditions and a new open chain with sites uniformly spaced on a half-circle and interactions inversely proportional to the square of the distance between the spins. We are able to compute in closed form the partition function of the latter chain, thereby obtaining a complete description of its spectrum in terms of a pair of independent su(1|1) and su(m/2) motifs when the number m of internal degrees of freedom is even. This implies that the even m model is invariant under the direct sum of the Yangians Y (gl(1|1)) and Y (gl(0|m/2)). We also analyze several statistical properties of the new chain’s spectrum. In particular, we show that it is highly degenerate, which strongly suggests the existence of an underlying (twisted) Yangian symmetry also for odd m. 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). 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. PublicationA duality principle for the multi-block entanglement entropy of free fermion systems(Nature Publishing Group, 2017-09-11) Carrasco, J. A.; Finkel Morgenstern, Federico; González López, Artemio; Tempesta, PiergiulioThe analysis of the entanglement entropy of a subsystem of a one-dimensional quantum system is a powerful tool for unravelling its critical nature. For instance, the scaling behaviour of the entanglement entropy determines the central charge of the associated Virasoro algebra. For a free fermion system, the entanglement entropy depends essentially on two sets, namely the set A of sites of the subsystem considered and the set K of excited momentum modes. In this work we make use of a general duality principle establishing the invariance of the entanglement entropy under exchange of the sets A and K to tackle complex problems by studying their dual counterparts. The duality principle is also a key ingredient in the formulation of a novel conjecture for the asymptotic behavior of the entanglement entropy of a free fermion system in the general case in which both sets A and K consist of an arbitrary number of blocks. We have verified that this conjecture reproduces the numerical results with excellent precision for all the configurations analyzed. We have also applied the conjecture to deduce several asymptotic formulas for the mutual and r-partite information generalizing the known ones for the single block case. PublicationInvariant differential equations and the Adler-Gel'fand-Dikii bracket(American Institute of Physics, 1997-11) González López, Artemio; Hernández Heredero, Rafael; Beffa, Gloria MaríIn this paper we find an explicit formula for the most general vector evolution of curves on RPn−1 invariant under the projective action of SL(n, R). When this formula is applied to the projectivization of solution curves of scalar Lax operators with periodic coefficients, one obtains a corresponding evolution in the space of such operators. We conjecture that this evolution is identical to the second KdV Hamiltonian evolution under appropriate conditions. These conditions give a Hamiltonian interpretation of general vector differential invariants for the projective action of SL(n, R), namely, the SL(n, R) invariant evolution can be written so that a general vector differential invariant corresponds to the Hamiltonian pseudo-differential operator. We find common coordinates and simplify both evolutions so that one can attempt to prove the equivalence for arbitrary n . 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. PublicationThe Multidimensional Darboux transformation(Elsevier, 1998-07) González López, Artemio; Kamran, NikyA generalization of the classical one-dimensional Darboux transformation to arbitrary n- dimensional oriented Riemannian manifolds is constructed using an intrinsic formulation based on the properties of twisted Hodge Laplacians. The classical two-dimensional Moutard transformation is also generalized to non-compact oriented Riemannian manifolds of dimension n ≥ 2. New examples of quasi-exactly solvable multidimensional matrix Schrödinger operators on curved manifolds are obtained by applying the above results.