29,869 research outputs found
A combinatorial Li-Yau inequality and rational points on curves
We present a method to control gonality of nonarchimedean curves based on graph theory. Let k denote a complete nonarchimedean valued field.We first prove a lower bound for the gonality of a curve over the algebraic closure of k in terms of the minimal degree of a class of graph maps, namely: one should minimize over all so-called finite harmonic graph morphisms to trees, that originate from any refinement of the dual graph of the stable model of the curve. Next comes our main result: we prove a lower bound for the degree of such a graph morphism in terms of the first eigenvalue of the Laplacian and some “volume” of the original graph; this can be seen as a substitute for graphs of the Li–Yau inequality from differential geometry, although we also prove that the strict analogue of the original inequality fails for general graphs. Finally,we apply the results to give a lower bound for the gonality of arbitraryDrinfeld modular curves over finite fields and for general congruence subgroups Γ of Γ (1) that is linear in the index [Γ (1) : Γ ], with a constant that only depends on the residue field degree and the degree of the chosen “infinite” place. This is a function field analogue of a theorem of Abramovich for classical modular curves. We present applications to uniform boundedness of torsion of rank two Drinfeld modules that improve upon existing results, and to lower bounds on the modular degree of certain elliptic curves over function fields that solve a problem of Papikian
Finite-Size Scaling for Quantum Criticality above the Upper Critical Dimension: Superfluid-Mott-Insulator Transition in Three Dimensions
Validity of modified finite-size scaling above the upper critical dimension
is demonstrated for the quantum phase transition whose dynamical critical
exponent is . We consider the -component Bose-Hubbard model, which is
exactly solvable and exhibits mean-field type critical phenomena in the
large- limit. The modified finite-size scaling holds exactly in that limit.
However, the usual procedure, taking the large system-size limit with fixed
temperature, does not lead to the expected (and correct) mean-field critical
behavior due to the limited range of applicability of the finite-size scaling
form. By quantum Monte Carlo simulation, it is shown that the same holds in the
case of N=1.Comment: 18 pages, 4 figure
Clustering of Conditional Mutual Information for Quantum Gibbs States above a Threshold Temperature
We prove that the quantum Gibbs states of spin systems above a certain threshold temperature are approximate quantum Markov networks, meaning that the conditional mutual information decays rapidly with distance. We demonstrate the exponential decay for short-ranged interacting systems and power-law decay for long-ranged interacting systems. Consequently, we establish the efficiency of quantum Gibbs sampling algorithms, a strong version of the area law, the quasilocality of effective Hamiltonians on subsystems, a clustering theorem for mutual information, and a polynomial-time algorithm for classical Gibbs state simulations
Optimal entanglement manipulation via coherent-state transmission
We derive an optimal bound for arbitrary entanglement manipulation based on
the transmission of a pulse in coherent states over a lossy channel followed by
local operations and unlimited classical communication (LOCC). This stands on a
theorem to reduce LOCC via a local unital qubit channel to local filtering. We
also present an optimal protocol based on beam splitters and a quantum
nondemolition (QND) measurement on photons. Even if we replace the QND
measurement with photon detectors, the protocol outperforms known entanglement
generation schemes.Comment: 5 pages, 1 figur
Particle Propagator of Spin Calogero-Sutherland Model
Explicit-exact expressions for the particle propagator of the spin 1/2
Calogero-Sutherland model are derived for the system of a finite number of
particles and for that in the thermodynamic limit. Derivation of the expression
in the thermodynamic limit is also presented in detail. Combining this result
with the hole propagator obtained in earlier studies, we calculate the spectral
function of the single particle Green's function in the full range of the
energy and momentum space. The resultant spectral function exhibits power-law
singularity characteristic to correlated particle systems in one dimension.Comment: 43 pages, 6 figure
Derivation of Green's Function of Spin Calogero-Sutherland Model by Uglov's Method
Hole propagator of spin 1/2 Calogero-Sutherland model is derived using
Uglov's method, which maps the exact eigenfunctions of the model, called
Yangian Gelfand-Zetlin basis, to a limit of Macdonald polynomials (gl_2-Jack
polynomials). To apply this mapping method to the calculation of 1-particle
Green's function, we confirm that the sum of the field annihilation operator on
Yangian Gelfand-Zetlin basis is transformed to the field annihilation operator
on gl_2-Jack polynomials by the mapping. The resultant expression for hole
propagator for finite-size system is written in terms of renormalized momenta
and spin of quasi-holes and the expression in the thermodynamic limit coincides
with the earlier result derived by another method. We also discuss the
singularity of the spectral function for a specific coupling parameter where
the hole propagator of spin Calogero-Sutherland model becomes equivalent to
dynamical colour correlation function of SU(3) Haldane-Shastry model.Comment: 36 pages, 8 figure
Norm kernels and the closeness relation for Pauli-allowed basis functions
The norm kernel of the generator-coordinate method is shown to be a symmetric
kernel of an integral equation with eigenfunctions defined in the
Fock--Bargmann space and forming a complete set of orthonormalized states
(classified with the use of SU(3) symmetry indices) satisfying the Pauli
exclusion principle. This interpretation allows to develop a method which, even
in the presence of the SU(3) degeneracy, provides for a consistent way to
introduce additional quantum numbers for the classification of the basis
states. In order to set the asymptotic boundary conditions for the expansion
coefficients of a wave function in the SU(3) basis, a complementary basis of
functions with partial angular momenta as good quantum numbers is needed. Norm
kernels of the binary systems 6He+p, 6He+n, 6He+4He, and 8He+4He are considered
in detail.Comment: 25 pages; submitted to Few-Body System
Existence of Density Functionals for Excited States and Resonances
We show how every bound state of a finite system of identical fermions,
whether a ground state or an excited one, defines a density functional.
Degeneracies created by a symmetry group can be trivially lifted by a
pseudo-Zeeman effect. When complex scaling can be used to regularize a
resonance into a square integrable state, a DF also exists.Comment: 4 pages, no figure
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