33,282 research outputs found

    Number-parity effect for confined fermions in one dimension

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    For NN spin-polarized fermions with harmonic pair interactions in a 11-dimensional trap an odd-even effect is found. The spectrum of the 11-particle reduced density matrix of the system's ground state differs qualitatively for NN odd and NN even. This effect does only occur for strong attractive and repulsive interactions. Since it does not exists for bosons, it must originate from the repulsive nature implied by the fermionic exchange statistics. In contrast to the spectrum, the 11-particle density and correlation function for strong attractive interactions do not show any sensitivity on the number parity. This also suggests that reduced-density-matrix-functional theory has a more subtle NN-dependency than density functional theory.Comment: published versio

    Duality of reduced density matrices and their eigenvalues

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    For states of quantum systems of NN particles with harmonic interactions we prove that each reduced density matrix ρ\rho obeys a duality condition. This condition implies duality relations for the eigenvalues λk\lambda_k of ρ\rho and relates a harmonic model with length scales l1,l2,,lNl_1,l_2, \ldots, l_N with another one with inverse lengths 1/l1,1/l2,,1/lN1/l_1, 1/l_2,\ldots, 1/l_N. Entanglement entropies and correlation functions inherit duality from ρ\rho. Self-duality can only occur for noninteracting particles in an isotropic harmonic trap

    New fermionic formula for unrestricted Kostka polynomials

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    A new fermionic formula for the unrestricted Kostka polynomials of type An1(1)A_{n-1}^{(1)} is presented. This formula is different from the one given by Hatayama et al. and is valid for all crystal paths based on Kirillov-Reshetihkin modules, not just for the symmetric and anti-symmetric case. The fermionic formula can be interpreted in terms of a new set of unrestricted rigged configurations. For the proof a statistics preserving bijection from this new set of unrestricted rigged configurations to the set of unrestricted crystal paths is given which generalizes a bijection of Kirillov and Reshetikhin.Comment: 35 pages; reference adde

    A bijection between type D_n^{(1)} crystals and rigged configurations

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    Hatayama et al. conjectured fermionic formulas associated with tensor products of U'_q(g)-crystals B^{r,s}. The crystals B^{r,s} correspond to the Kirillov--Reshetikhin modules which are certain finite dimensional U'_q(g)-modules. In this paper we present a combinatorial description of the affine crystals B^{r,1} of type D_n^{(1)}. A statistic preserving bijection between crystal paths for these crystals and rigged configurations is given, thereby proving the fermionic formula in this case. This bijection reflects two different methods to solve lattice models in statistical mechanics: the corner-transfer-matrix method and the Bethe Ansatz.Comment: 38 pages; version to appear in J. Algebr

    Hubbard model: Pinning of occupation numbers and role of symmetries

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    Fermionic natural occupation numbers do not only obey Pauli's exclusion principle, but are even further restricted by so-called generalized Pauli constraints. Such restrictions are particularly relevant whenever they are saturated by given natural occupation numbers λ=(λi)\vec{\lambda}=(\lambda_i). For few-site Hubbard models we explore the occurrence of this pinning effect. By varying the on-site interaction UU for the fermions we find sharp transitions from pinning of λ\vec{\lambda} to the boundary of the allowed region to nonpinning. We analyze the origin of this phenomenon which turns out be either a crossing of natural occupation numbers λi(U),λi+1(U)\lambda_{i}(U), \lambda_{i+1}(U) or a crossing of NN-particle energies. Furthermore, we emphasize the relevance of symmetries for the occurrence of pinning. Based on recent progress in the field of ultracold atoms our findings suggest an experimental set-up for the realization of the pinning effect.Comment: published versio

    Crystal structure on rigged configurations

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    Rigged configurations are combinatorial objects originating from the Bethe Ansatz, that label highest weight crystal elements. In this paper a new unrestricted set of rigged configurations is introduced for types ADE by constructing a crystal structure on the set of rigged configurations. In type A an explicit characterization of unrestricted rigged configurations is provided which leads to a new fermionic formula for unrestricted Kostka polynomials or q-supernomial coefficients. The affine crystal structure for type A is obtained as well.Comment: 20 pages, 1 figure, axodraw and youngtab style file necessar

    Richard Stanley through a crystal lens and from a random angle

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    We review Stanley's seminal work on the number of reduced words of the longest element of the symmetric group and his Stanley symmetric functions. We shed new light on this by giving a crystal theoretic interpretation in terms of decreasing factorizations of permutations. Whereas crystal operators on tableaux are coplactic operators, the crystal operators on decreasing factorization intertwine with the Edelman-Greene insertion. We also view this from a random perspective and study a Markov chain on reduced words of the longest element in a finite Coxeter group, in particular the symmetric group, and mention a generalization to a poset setting.Comment: 11 pages; 3 figures; v2 updated references and added discussion on Coxeter-Knuth grap

    q-Supernomial coefficients: From riggings to ribbons

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    q-Supernomial coefficients are generalizations of the q-binomial coefficients. They can be defined as the coefficients of the Hall-Littlewood symmetric function in a product of the complete symmetric functions or the elementary symmetric functions. Hatayama et al. give explicit expressions for these q-supernomial coefficients. A combinatorial expression as the generating function of ribbon tableaux with (co)spin statistic follows from the work of Lascoux, Leclerc and Thibon. In this paper we interpret the formulas by Hatayama et al. in terms of rigged configurations and provide an explicit statistic preserving bijection between rigged configurations and ribbon tableaux thereby establishing a new direct link between these combinatorial objects.Comment: 19 pages, svcon2e.sty file require
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