54 research outputs found
Eigenvalue Attraction
We prove that the complex conjugate (c.c.) eigenvalues of a smoothly varying
real matrix attract (Eq. 15). We offer a dynamical perspective on the motion
and interaction of the eigenvalues in the complex plane, derive their governing
equations and discuss applications. C.c. pairs closest to the real axis, or
those that are ill-conditioned, attract most strongly and can collide to become
exactly real. As an application we consider random perturbations of a fixed
matrix . If is Normal, the total expected force on any eigenvalue is
shown to be only the attraction of its c.c. (Eq. 24) and when is circulant
the strength of interaction can be related to the power spectrum of white
noise. We extend this by calculating the expected force (Eq. 41) for real
stochastic processes with zero-mean and independent intervals. To quantify the
dominance of the c.c. attraction, we calculate the variance of other forces. We
apply the results to the Hatano-Nelson model and provide other numerical
illustrations. It is our hope that the simple dynamical perspective herein
might help better understanding of the aggregation and low density of the
eigenvalues of real random matrices on and near the real line respectively. In
the appendix we provide a Matlab code for plotting the trajectories of the
eigenvalues.Comment: v1:15 pages, 12 figures, 1 Matlab code. v2: very minor changes, fixed
a reference. v3: 25 pages, 17 figures and one Matlab code. The results have
been extended and generalized in various ways v4: 26 pages, 10 figures and a
Matlab Code. Journal Reference Added.
http://link.springer.com/article/10.1007%2Fs10955-015-1424-
Stability of Periodically Driven Topological Phases against Disorder
In recent experiments, time-dependent periodic fields are used to create
exotic topological phases of matter with potential applications ranging from
quantum transport to quantum computing. These nonequilibrium states, at high
driving frequencies, exhibit the quintessential robustness against local
disorder similar to equilibrium topological phases. However, proving the
existence of such topological phases in a general setting is an open problem.
We propose a universal effective theory that leverages on modern free
probability theory and ideas in random matrices to analytically predict the
existence of the topological phase for finite driving frequencies and across a
range of disorder. We find that, depending on the strength of disorder, such
systems may be topological or trivial and that there is a transition between
the two. In particular, the theory predicts the critical point for the
transition between the two phases and provides the critical exponents. We
corroborate our results by comparing them to exact diagonalizations for
driven-disordered 1D Kitaev chain and 2D Bernevig-Hughes-Zhang models and find
excellent agreement. This Letter may guide the experimental efforts for
exploring topological phases.Comment: 5 pages + 9 pages supplementary material. 4 Figure
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