1,111 research outputs found

    Hook formulas for skew shapes III. Multivariate and product formulas

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    We give new product formulas for the number of standard Young tableaux of certain skew shapes and for the principal evaluation of the certain Schubert polynomials. These are proved by utilizing symmetries for evaluations of factorial Schur functions, extensively studied in the first two papers in the series "Hook formulas for skew shapes" [arxiv:1512.08348, arxiv:1610.04744]. We also apply our technology to obtain determinantal and product formulas for the partition function of certain weighted lozenge tilings, and give various probabilistic and asymptotic applications.Comment: 40 pages, 17 figures. This is the third paper in the series "Hook formulas for skew shapes"; v2 added reference to [KO1] (arxiv:1409.1317) where the formula in Corollary 1.1 had previously appeared; v3 Corollary 5.10 added, resembles published versio

    Degenerate perturbation theory of quantum fluctuations in a pyrochlore antiferromagnet

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    We study the effect of quantum fluctuations on the half-polarized magnetization plateau of a pyrochlore antiferromagnet. We argue that an expansion around the easy axis limit is appropriate for discussing the ground state selection amongst the classically degenerate manifold of collinear states with a 3:1 ratio of spins parallel/anti-parallel to the magnetization axis. A general approach to the necessary degenerate perturbation theory is presented, and an effective quantum dimer model within this degenerate manifold is derived for arbitrary spin ss. We also generalize the existing semiclassical analysis of Hizi and Henley [Phys. Rev. B {\bf 73}, 054403 (2006)] to the easy axis limit, and show that both approaches agree at large ss. We show that under rather general conditions, the first non-constant terms in the effective Hamiltonian for s≥1s\geq 1 occur only at {\sl sixth} order in the transverse exchange coupling. For s≥3/2s\geq 3/2, the effective Hamiltonian predicts a magnetically ordered state. For s≤1s\leq 1 more exotic possibilities may be realized, though an analytical solution of the resulting quantum dimer model is not possible

    Generalized Hardcore Dimer Models approach to low-energy Heisenberg frustrated antiferromagnets: general properties and application to the kagome antiferromagnet

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    We propose a general non-perturbative scheme that quantitatively maps the low-energy sector of spin-1/2 frustrated Heisenberg antiferromagnets to effective Generalized Quantum Dimer Models. We develop the formal lattice independent frame and establish some important results on (i) the locality of the generated Hamiltonians (ii) how full resummations can be performed in this renormalization scheme. The method is then applied to the much debated kagome antiferromagnet for which a fully resummed effective Hamiltonian - shown to capture the essential properties and provide deep insights on the microscopic model [D. Poilblanc, M. Mambrini and D. Schwandt, arXiv:0912.0724] - is derived.Comment: 26 pages, 4 figures, EPAPS inlined, manuscript revised, corrected minor typos (notably figure 2)
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