Optimal quasi-free approximation:reconstructing the spectrum from ground state energies

The sequence of ground state energy density at finite size, e_{L}, provides much more information than usually believed. Having at disposal e_{L} for short lattice sizes, we show how to re-construct an approximate quasi-particle dispersion for any interacting model. The accuracy of this method relies on the best possible quasi-free approximation to the model, consistent with the observed values of the energy e_{L}. We also provide a simple criterion to assess whether such a quasi-free approximation is valid. As a side effect, our method is able to assess whether the nature of the quasi-particles is fermionic or bosonic together with the effective boundary conditions of the model. When applied to the spin-1/2 Heisenberg model, the method produces a band of Fermi quasi-particles very close to the exact one of des Cloizeaux and Pearson. The method is further tested on a spin-1/2 Heisenberg model with explicit dimerization and on a spin-1 chain with single ion anisotropy. A connection with the Riemann Hypothesis is also pointed out.Comment: 9 pages, 5 figures. One figure added showing convergence spee

Grassmann-Gaussian integrals and generalized star products

In quantum scattering on networks there is a non-linear composition rule for on-shell scattering matrices which serves as a replacement for the multiplicative rule of transfer matrices valid in other physical contexts. In this article, we show how this composition rule is obtained using Berezin integration theory with Grassmann variables.Comment: 14 pages, 2 figures. In memory of Al.B. Zamolodichiko

On the gap between the complex structured singular value and its convex upper bound

Published versio

Surface Properties of Aperiodic Ising Quantum Chains

We consider Ising quantum chains with quenched aperiodic disorder of the coupling constants given through general substitution rules. The critical scaling behaviour of several bulk and surface quantities is obtained by exact real space renormalization.Comment: 4 pages, RevTex, reference update

Communications and Related Projects

Contains reports on three research projects.Office of Scientific Research and Development (OSRD) OEMsr-26

Kirchhoff's Rule for Quantum Wires

In this article we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with $n$ open ends and where we define the Hamiltonian to be (minus) the Laplace operator with general boundary conditions at the vertices. This results in a scattering theory with $n$ channels. The corresponding on-shell S-matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy $E>0$ is explicitly given in terms of the boundary conditions and the lengths of the internal lines. It is shown to be unitary, which may be viewed as the quantum version of Kirchhoff's law. We exhibit covariance and symmetry properties. It is symmetric if the boundary conditions are real. Also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory. Finally we provide a composition rule by which the on-shell S-matrix of a graph is factorizable in terms of the S-matrices of its subgraphs. All proofs only use known facts from the theory of self-adjoint extensions, standard linear algebra, complex function theory and elementary arguments from the theory of Hermitean symplectic forms.Comment: 40 page

An optimal gap theorem

By solving the Cauchy problem for the Hodge-Laplace heat equation for $d$-closed, positive $(1, 1)$-forms, we prove an optimal gap theorem for K\"ahler manifolds with nonnegative bisectional curvature which asserts that the manifold is flat if the average of the scalar curvature over balls of radius $r$ centered at any fixed point $o$ is a function of $o(r^{-2})$. Furthermore via a relative monotonicity estimate we obtain a stronger statement, namely a `positive mass' type result, asserting that if $(M, g)$ is not flat, then $\liminf_{r\to \infty} \frac{r^2}{V_o(r)}\int_{B_o(r)}\mathcal{S}(y)\, d\mu(y)>0$ for any $o\in M$

Lie families: theory and applications

We analyze families of non-autonomous systems of first-order ordinary differential equations admitting a common time-dependent superposition rule, i.e., a time-dependent map expressing any solution of each of these systems in terms of a generic set of particular solutions of the system and some constants. We next study relations of these families, called Lie families, with the theory of Lie and quasi-Lie systems and apply our theory to provide common time-dependent superposition rules for certain Lie families.Comment: 23 pages, revised version to appear in J. Phys. A: Math. Theo

Systematic Procedure to Avoid Unintended Polarity Mismatch in the Cascade Connection of Multiport Devices with Symmetric Feeding Lines

This paper is a postprint of a paper submitted to and accepted for publication in [journal] and is subject to Institution of Engineering and Technology Copyright. The copy of record is available at IET Digital LibraryThe traditional cascading of generalised scattering matrices (GSMs) assumes that the modal sets at the connected ports of a cascaded network are strictly equal. This implies a careful selection of the modal polarities, or the reference systems, of every port. Usually, the connection scheme of every device is known a priori. Then, the individual GSMs are pre-processed, or auxiliary devices, which correct possible modal mismatches at the ports, are included in appropriate positions among the cascade, so that the traditional cascading-by-pairs approach can be directly applied. This scheme clearly complicates the reutilisation of previously calculated GSMs, and mixes the cascading with the solution of the individual building blocks. In this study, a systematic procedure is proposed to define the polarity of the modes at the ports of a device fed with transmission lines or waveguides showing a single or double symmetry. The modified expressions to calculate the scattering parameters of the cascade of two multiport devices, incorporating the regular modal corrections to apply when this criterion is used to define the modal polarity at the ports, is also presented in this study. This strategy is more convenient from the point of view of programming, less error-prone and easier to implement.This work was supported by the Ministry of Science and Innovation, Spanish Government, under Research Projects TEC2013-47037-C05-3-R and TEC2013-47037-C05-1-R.Belenguer Martínez, Á.; Borja, A.; Díaz Caballero, E.; Esteban González, H.; Boria Esbert, VE. (2015). Systematic Procedure to Avoid Unintended Polarity Mismatch in the Cascade Connection of Multiport Devices with Symmetric Feeding Lines. IET Microwaves Antennas and Propagation. 9(11):1128-1135. https://doi.org/10.1049/iet-map.2014.0167S1128113591