In this work we put forward an effective Gaussian free field description of critical wavefunctions at the transition between plateaus of the integer quantum Hall effect. To this end, we expound our earlier proposal that powers of critical wave intensities prepared via point contacts behave as pure scaling fields obeying an Abelian operator product expansion. Our arguments employ the framework of conformal field theory and, in particular, lead to a multifractality spectrum which is parabolic. We also derive a number of old and new identities that hold exactly at the lattice level and hinge on the correspondence between the Chalker-Coddington network model and a supersymmetric vertex model.
We compute Haar ensemble averages of ratios of random characteristic polynomials for the classical Lie groups $$K = \mathrm {O}_N\,$$ , $$\mathrm {SO}_N\,$$ , and $$\mathrm {USp}_N\,$$ . To that end, we start from the Clifford–Weyl algebra in its canonical realization on the complex $$\fancyscript {A}_V$$ of holomorphic differential forms for a $$\mathbb {C} $$ -vector space $$V_0\,$$ . From it we construct the Fock representation of an orthosymplectic Lie superalgebra $$\mathfrak {osp}$$ associated to $$V_0\,$$ . Particular attention is paid to defining Howe’s oscillator semigroup and the representation that partially exponentiates the Lie algebra representation of $$\mathfrak {sp} \subset \mathfrak {osp}$$ . In the process, by pushing the semigroup representation to its boundary and arguing by continuity, we provide a construction of the Shale–Weil–Segal representation of the metaplectic group. To deal with a product of n ratios of characteristic polynomials, we let $$V_0 = \mathbb {C}^n \otimes \mathbb {C}^N$$ where $$\mathbb {C}^N$$ is equipped with the standard K-representation, and focus on the subspace $$\fancyscript {A}_V^K$$ of K-equivariant forms. By Howe duality, this is a highest-weight irreducible representation of the centralizer $$\mathfrak {g}$$ of $$\mathrm {Lie}(K)$$ in $$\mathfrak {osp}$$ . We identify the K-Haar expectation of n ratios with the character of this $$\mathfrak {g} $$ -representation, which we show to be uniquely determined by analyticity, Weyl-group invariance, certain weight constraints, and a system of differential equations coming from the Laplace-Casimir invariants of $$\mathfrak {g}\,$$ . We find an explicit solution to the problem posed by all these conditions. In this way, we prove that the said Haar expectations are expressed by a Weyl-type character formula for all integers $$N \ge 1$$ . This completes earlier work of Conrey, Farmer, and Zirnbauer for the case of $$\mathrm {U}_N\,$$ .
Abstract The Migdal‐Kadanoff approximation is applied to a nonlinear σ model with fields taking values in a perfectly graded noncompact superspace G/K of rank one. It is shown how to implement the spherical transform for this model space numerically in a controlled manner. The correlation length exponent ν, and the dimensions of a series of G ‐invariant scaling operators at the fixed point, are computed for 2 < d ⩽ 3. No perturbative instability of the type found by Kravtsov et al. and Wegner is observed.
Using the quasiparticle random phase approximation, we calculate the nuclear matrix elements governing two-neutrino and neutrinoless double-beta decay. We show that a consistent treatment, including both particle-hole and particle-particle interactions, helps to resolve the longstanding discrepancy between experimental and calculated two-neutrino decay rates. The particle-particle force, which allows us to bring calculated EC/${\ensuremath{\beta}}^{+}$ decay rates in semimagic nuclei into closer agreement with experiment, is in large part responsible for suppressing calculated two-neutrino decay rates that are otherwise too fast. We test the validity of our procedure by comparing quasiparticle random phase approximation results with exact solutions for a solvable model, in which the suppression of two-neutrino decay by the particle-particle interaction is confirmed. We then extend our approach to the neutrinoless decay associated with a finite Majorana neutrino mass and, conceivably, with majoron emission, and demonstrate that the nuclear matrix elements governing these processes are also suppressed. We present predicted half-lives for both two-neutrino and neutrinoless double-beta decay in several candidate nuclei, and discuss the difficulties associated with the calculation of such highly suppressed quantities.
The color-flavor transformation is a mathematical result that has applications to problems as diverse as lattice gauge theory, random network models, and dynamical systems. Several variants are described, and an outline of the proof is given. It is then shown how to use the transformation to set up a field theoretic formalism for quantum chaotic maps.
