81 research outputs found
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
On the gap between the complex structured singular value and its convex upper bound
Published versio
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
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 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 channels. The
corresponding on-shell S-matrix formed by the reflection and transmission
amplitudes for incoming plane waves of energy 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
Aperiodic Ising Quantum Chains
Some years ago, Luck proposed a relevance criterion for the effect of
aperiodic disorder on the critical behaviour of ferromagnetic Ising systems. In
this article, we show how Luck's criterion can be derived within an exact
renormalisation scheme for Ising quantum chains with coupling constants
modulated according to substitution rules. Luck's conjectures for this case are
confirmed and refined. Among other outcomes, we give an exact formula for the
correlation length critical exponent for arbitrary two-letter substitution
sequences with marginal fluctuations of the coupling constants.Comment: 27 pages, LaTeX, 1 Postscript figure included, using epsf.sty and
amssymb.sty (one error corrected, some minor changes
Superposition rules for higher-order systems and their applications
Superposition rules form a class of functions that describe general solutions
of systems of first-order ordinary differential equations in terms of generic
families of particular solutions and certain constants. In this work we extend
this notion and other related ones to systems of higher-order differential
equations and analyse their properties. Several results concerning the
existence of various types of superposition rules for higher-order systems are
proved and illustrated with examples extracted from the physics and mathematics
literature. In particular, two new superposition rules for second- and
third-order Kummer--Schwarz equations are derived.Comment: (v2) 33 pages, some typos corrected, added some references and minor
commentarie
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
- …