22,449 research outputs found
Standing Waves in a Non-linear 1D Lattice : Floquet Multipliers, Krein Signatures, and Stability
We construct a class of exact commensurate and incommensurate standing wave
(SW) solutions in a piecewise smooth analogue of the discrete non-linear
Schr\"{o}dinger (DNLS) model and present their linear stability analysis. In
the case of the commensurate SW solutions the analysis reduces to the
eigenvalue problem of a transfer matrix depending parametrically on the
eigenfrequency. The spectrum of eigenfrequencies and the corresponding
eigenmodes can thereby be determined exactly. The spatial periodicity of a
commensurate SW implies that the eigenmodes are of the Bloch form,
characterised by an even number of Floquet multipliers. The spectrum is made up
of bands that, in general, include a number of transition points corresponding
to changes in the disposition of the Floquet multipliers. The latter
characterise the different band segments. An alternative characterisation of
the segments is in terms of the Krein signatures associated with the
eigenfrequencies. When one or more parameters characterising the SW solution is
made to vary, one occasionally encounters collisions between the band-edges or
the intra-band transition points and, depending on the the Krein signatures of
the colliding bands or segments, the spectrum may stretch out in the complex
plane, leading to the onset of instability. We elucidate the correlation
between the disposition of Floquet multipliers and the Krein signatures,
presenting two specific examples where the SW possesses a definite window of
stability, as distinct from the SW's obtained close to the anticontinuous and
linear limits of the DNLS model.Comment: 31 pages, 11 figure
On the construction of a digital transfer function from its real part on unit circle
It is shown in this correspondence that the system function H(z) of a linear time invariant (LTI) causal digital filter with real impulse response coefficients can be obtained from the real part of its frequency response HR(ejω) given in the form of a rational trigonomentric function, using algebraic methods rather than complex contour integration techniques
Transfer Matrices and Excitations with Matrix Product States
We investigate the relation between static correlation functions in the
ground state of local quantum many-body Hamiltonians and the dispersion
relations of the corresponding low energy excitations using the formalism of
tensor network states. In particular, we show that the Matrix Product State
Transfer Matrix (MPS-TM) - a central object in the computation of static
correlation functions - provides important information about the location and
magnitude of the minima of the low energy dispersion relation(s) and present
supporting numerical data for one-dimensional lattice and continuum models as
well as two-dimensional lattice models on a cylinder. We elaborate on the
peculiar structure of the MPS-TM's eigenspectrum and give several arguments for
the close relation between the structure of the low energy spectrum of the
system and the form of static correlation functions. Finally, we discuss how
the MPS-TM connects to the exact Quantum Transfer Matrix (QTM) of the model at
zero temperature. We present a renormalization group argument for obtaining
finite bond dimension approximations of MPS, which allows to reinterpret
variational MPS techniques (such as the Density Matrix Renormalization Group)
as an application of Wilson's Numerical Renormalization Group along the virtual
(imaginary time) dimension of the system.Comment: 39 pages (+8 pages appendix), 14 figure
The Asymmetric Pupil Fourier Wavefront Sensor
This paper introduces a novel wavefront sensing approach that relies on the
Fourier analysis of a single conventional direct image. In the high Strehl
ratio regime, the relation between the phase measured in the Fourier plane and
the wavefront errors in the pupil can be linearized, as was shown in a previous
work that introduced the notion of generalized closure-phase, or kernel-phase.
The technique, to be usable as presented requires two conditions to be met: (1)
the wavefront errors must be kept small (of the order of one radian or less)
and (2) the pupil must include some asymmetry, that can be introduced with a
mask, for the problem to become solvable. Simulations show that this asymmetric
pupil Fourier wavefront sensing or APF-WFS technique can improve the Strehl
ratio from 50 to over 90 % in just a few iterations, with excellent photon
noise sensitivity properties, suggesting that on-sky close loop APF-WFS is
possible with an extreme adaptive optics system.Comment: 5 figures, accepted for publication by PAS
Thin Fisher Zeroes
Biskup et al. [Phys. Rev. Lett. 84 (2000) 4794] have recently suggested that
the loci of partition function zeroes can profitably be regarded as phase
boundaries in the complex temperature or field planes.
We obtain the Fisher zeroes for Ising and Potts models on non-planar
(``thin'') regular random graphs using this approach, and note that the locus
of Fisher zeroes on a Bethe lattice is identical to the corresponding random
graph. Since the number of states appears as a parameter in the Potts solution
the limiting locus of chromatic zeroes is also accessible.Comment: 10 pages, 4 figure
Discrete Riemann Surfaces and the Ising model
We define a new theory of discrete Riemann surfaces and present its basic
results. The key idea is to consider not only a cellular decomposition of a
surface, but the union with its dual. Discrete holomorphy is defined by a
straightforward discretisation of the Cauchy-Riemann equation. A lot of
classical results in Riemann theory have a discrete counterpart, Hodge star,
harmonicity, Hodge theorem, Weyl's lemma, Cauchy integral formula, existence of
holomorphic forms with prescribed holonomies. Giving a geometrical meaning to
the construction on a Riemann surface, we define a notion of criticality on
which we prove a continuous limit theorem. We investigate its connection with
criticality in the Ising model. We set up a Dirac equation on a discrete
universal spin structure and we prove that the existence of a Dirac spinor is
equivalent to criticality
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