Federico Finkel

Global properties of the spectrum of the Haldane-Shastry spin chain

Departamento de F´isica Te´orica II, Universidad Complutense, 28040 Madrid, Spain(Dated: May 23, 2005; revised September 1, 2005)We derive an exact expression for the partition function of the su(m) Haldane–Shastry spin chain,which we use to study the density of levels and the distribution of the spacing between consecutivelevels. Our computations show that when the number of sites N is large enough the level density isGaussian to a very high degree of approximation. More surprisingly, we also find that the nearest-neighbor spacing distribution is not Poissonian, so that this model departs from the typical behaviorfor an integrable system. We show that the cumulative spacing distribution of the model can bewell approximated by a simple functional law involving only three parameters.

Quasi‐exactly solvable potentials on the line and orthogonal polynomials

In 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.

On form-preserving transformations for the time-dependent Schrodinger equation

In this paper we point out a close connection between the Darboux transformation and the group of point transformations which preserve the form of the time-dependent Schrodinger equation (TDSE). In our main result, we prove that any pair of time-dependent real potentials related by a Darboux transformation for the TDSE may be transformed by a suitable point transformation into a pair of time-independent potentials related by a usual Darboux transformation for the stationary Schrodinger equation. Thus, any (real) potential solvable via a time-dependent Darboux transformation can alternatively be solved by applying an appropriate form-preserving point transformation of the TDSE to a time-independent potential. The pre-eminent role of the latter type of transformations in the solution of the TDSE is illustrated with a family of quasi-exactly solvable time-dependent anharmonic potentials.

A new algebraization of the Laméequation

We develop a new way of writing the LameHamiltonian in Lie-algebraic form. This yields, in a natural way, an explicit formula for both the Lamepolynomials and the classical non-meromorphic Lame functions in terms of Chebyshev polynomials and of a certain family of weakly orthogonal polynomials.

ON THE FAMILIES OF ORTHOGONAL POLYNOMIALS ASSOCIATED TO THE RAZAVY POTENTIAL

We show that there are two different families of (weakly) orthogonal polynomials associated to the quasi-exactly solvable Razavy potential V(x) = (cos 2x-M)2 (>0, M). 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) = -(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 Hills 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, Mathieus equation) admits finite solutions.

Symmetries of the Fokker-Planck equation with a constant diffusion matrix in 2 + 1 dimensions

We completely classify the symmetries of the Fokker-Planck equation in two spatial dimensions with a constant positive-definite diffusion matrix. We apply these results to construct group-invariant solutions for a physically interesting family of Fokker-Planck equations.

Level density of spin chains of Haldane-Shastry type.

We provide a rigorous proof of the fact that the level density of all known su(m) spin chains of Haldane-Shastry type associated with the A(N-1) root system approaches a Gaussian distribution as the number of spins N tends to infinity. Our approach is based on the study of the large-N limit of the characteristic function of the level density, using the description of the spectrum in terms of motifs and the asymptotic behavior of the transfer matrix.

Quasi-exactly solvable spin 1/2 Schrödinger operators

The algebraic structures underlying quasi-exact solvability for spin 1/2 Hamiltonians in one dimension are studied in detail. Necessary and sufficient conditions for a matrix second-order differential operator preserving a space of wave functions with polynomial components to be equivalent to a Schrodinger operator are found. Systematic simplifications of these conditions are analyzed, and are then applied to the construction of new examples of multi-parameter QES spin 1/2 Hamiltonians in one dimension.

Supersymmetric spin chains with nonmonotonic dispersion relation: criticality and entanglement entropy

We study the critical behavior and the ground-state entanglement of a large class of su(1|1) supersymmetric spin chains with a general (not necessarily monotonic) dispersion relation. We show that this class includes several relevant models, with both short- and long-range interactions of a simple form. We determine the low temperature behavior of the free energy per spin, and deduce that the models considered have a critical phase in the same universality class as a (1+1)-dimensional conformal field theory (CFT) with central charge equal to the number of connected components of the Fermi sea. We also study the Rényi entanglement entropy of the ground state, deriving its asymptotic behavior as the block size tends to infinity. In particular, we show that this entropy exhibits the logarithmic growth characteristic of (1+1)-dimensional CFTs and one-dimensional (fermionic) critical lattice models, with a central charge consistent with the low-temperature behavior of the free energy. Our results confirm the widely believed conjecture that the critical behavior of fermionic lattice models is completely determined by the topology of their Fermi surface.

Yangian-invariant spin models and Fibonacci numbers

We study a wide class of finite-dimensional su(m|n)-supersymmetric models closely related to the representations of the Yangian Y(sl(m|n)) labeled by border strips. We quantitatively analyze the degree of degeneracy of these models arising from their Yangian invariance, measured by the average degeneracy of the spectrum. We compute in closed form the minimum average degeneracy of any such model, and show that in the non-supersymmetric case it can be expressed in terms of generalized Fibonacci numbers. Using several properties of these numbers, we show that (except in the simpler su(1|1) case) the minimum average degeneracy grows exponentially with the number of spins. We apply our results to several well-known spin chains of Haldane-Shastry type, quantitatively showing that their degree of degeneracy is much higher than expected for a generic Yangian-invariant spin model. Finally, we show that the set of distinct levels of a Yangian-invariant spin model is described by an effective model of quasi-particles. We study this effective model, discussing its connections to one-dimensional anyons and properties of generalized Fibonacci numbers.