Mayer expansions and the Hamilton-Jacobi equation. II. Fermions, dimensional reduction formulas

Brydges, David C.; Wright, J. D.

doi:10.1007/bf01028465

Cited by 18 publications

(17 citation statements)

References 6 publications

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“…The original functionals G and V are recovered in the limit Λ → 0. It is not hard to show that the effective interaction satisfies the following exact renormalization group equation [20,25] …”

Section: B Wick-ordered Continuous Rgmentioning

confidence: 99%

Renormalization-group analysis of the two-dimensional Hubbard model

2000

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Salmhofer [Commun. Math. Phys. 194, 249 (1998)] has recently developed a new renormalization group method for interacting Fermi systems, where the complete flow from the bare action of a microscopic model to the effective low-energy action, as a function of a continuously decreasing infrared cutoff, is given by a differential flow equation which is local in the flow parameter. We apply this approach to the repulsive two-dimensional Hubbard model with nearest and next-nearest neighbor hopping amplitudes. The flow equation for the effective interaction is evaluated numerically on 1-loop level. The effective interactions diverge at a finite energy scale which is exponentially small for small bare interactions. To analyze the nature of the instabilities signalled by the diverging interactions we extend Salmhofers renormalization group for the calculation of susceptibilities. We compute the singlet superconducting susceptibilities for various pairing symmetries and also charge and spin density susceptibilities. Depending on the choice of the model parameters (hopping amplitudes, interaction strength and band-filling) we find commensurate and incommensurate antiferromagnetic instabilities or d-wave superconductivity as leading instability. We present the resulting phase diagram in the vicinity of half-filling and also results for the density dependence of the critical energy scale.

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Section: B Wick-ordered Continuous Rgmentioning

confidence: 99%

Renormalization-group analysis of the two-dimensional Hubbard model

2000

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“…This construction, which is termed the "polymer expansion" or "cluster expansion", is an important tool in mathematical statistical mechanics [97,29,47,31,24], and much effort has been devoted to finding complex polydiscs in which the polymer expansion is convergent, i.e. in which the (polymer-)lattice-gas partition function is nonvanishing [88,33,97,29,72,30,32,100,37,38,89,90,31,82,25,101,102,104]. One goal of this paper is to make a modest contribution to this line of development.…”

Section: Introductionmentioning

confidence: 99%

“…Results of this type have traditionally been proven [88,33,97,29,30,32,100,31,104] by explicitly bounding the terms in the Mayer expansion (1.4); this requires some rather nontrivial combinatorics (for example, Proposition 2.4 below together with the counting of trees). Once this is done, an immediate consequence is that Z W is nonvanishing in any polydisc where the series for log Z W is convergent.…”

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confidence: 99%

The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lov�sz Local Lemma

2005

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We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {p x } if and only if the independent-set polynomial for G is nonvanishing in the polydisc of radii {p x }. Furthermore, we show that the usual proof of the Lovász local lemma -which provides a sufficient condition for this to occur -corresponds to a simple inductive argument for the nonvanishing of the independent-set polynomial in a polydisc, which was discovered implicitly by Shearer [98] and explicitly by Dobrushin [37,38]. We also present some refinements and extensions of both arguments, including a generalization of the Lovász local lemma that allows for "soft" dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternating-sign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.

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“…This result was first proved for two-body interactions in [BW88]. A simpler proof based on the Forest-Root formula will be given here for arbitrary J(X).…”

Section: A Fundamental Theorem Of Calculusmentioning

confidence: 79%

Branched polymers and dimensional reduction

2003

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We establish an exact relation between self-avoiding branched polymers in D + 2 continuum dimensions and the hard-core continuum gas at negative activity in D dimensions. We review conjectures and results on critical exponents for D + 2 = 2, 3, 4 and show that they are corollaries of our result. We explain the connection (first proposed by Parisi and Sourlas) between branched polymers in D + 2 dimensions and the Yang-Lee edge singularity in D dimensions.

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Mayer expansions and the Hamilton-Jacobi equation. II. Fermions, dimensional reduction formulas

Cited by 18 publications

References 6 publications

Renormalization-group analysis of the two-dimensional Hubbard model

Renormalization-group analysis of the two-dimensional Hubbard model

The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lov�sz Local Lemma

Branched polymers and dimensional reduction

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