3,929 research outputs found

    Galois groups of multivariate Tutte polynomials

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    The multivariate Tutte polynomial Z^M\hat Z_M of a matroid MM is a generalization of the standard two-variable version, obtained by assigning a separate variable vev_e to each element ee of the ground set EE. It encodes the full structure of MM. Let \bv = \{v_e\}_{e\in E}, let KK be an arbitrary field, and suppose MM is connected. We show that Z^M\hat Z_M is irreducible over K(\bv), and give three self-contained proofs that the Galois group of Z^M\hat Z_M over K(\bv) is the symmetric group of degree nn, where nn is the rank of MM. An immediate consequence of this result is that the Galois group of the multivariate Tutte polynomial of any matroid is a direct product of symmetric groups. Finally, we conjecture a similar result for the standard Tutte polynomial of a connected matroid.Comment: 8 pages, final version, to appear in J. Alg. Comb. Substantial revisions, including the addition of two alternative proofs of the main resul

    Doping evolution of spin and charge excitations in the Hubbard model

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    To shed light on how electronic correlations vary across the phase diagram of the cuprate superconductors, we examine the doping evolution of spin and charge excitations in the single-band Hubbard model using determinant quantum Monte Carlo (DQMC). In the single-particle response, we observe that the effects of correlations weaken rapidly with doping, such that one may expect the random phase approximation (RPA) to provide an adequate description of the two-particle response. In contrast, when compared to RPA, we find that significant residual correlations in the two-particle excitations persist up to 40%40\% hole and 15%15\% electron doping (the range of dopings achieved in the cuprates). These fundamental differences between the doping evolution of single- and multi-particle renormalizations show that conclusions drawn from single-particle processes cannot necessarily be applied to multi-particle excitations. Eventually, the system smoothly transitions via a momentum-dependent crossover into a weakly correlated metallic state where the spin and charge excitation spectra exhibit similar behavior and where RPA provides an adequate description.Comment: 5 pages, 4 figures, plus supplementary materia
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