15,518 research outputs found
Chern-Simons theory, exactly solvable models and free fermions at finite temperature
We show that matrix models in Chern-Simons theory admit an interpretation as
1D exactly solvable models, paralleling the relationship between the Gaussian
matrix model and the Calogero model. We compute the corresponding Hamiltonians,
ground-state wavefunctions and ground-state energies and point out that the
models can be interpreted as quasi-1D Coulomb plasmas. We also study the
relationship between Chern-Simons theory on and a system of N
one-dimensional fermions at finite temperature with harmonic confinement. In
particular we show that the Chern-Simons partition function can be described by
the density matrix of the free fermions in a very particular, crystalline,
configuration. For this, we both use the Brownian motion and the matrix model
description of Chern-Simons theory and find several common features with c=1
theory at finite temperature. Finally, using the exactly solvable model result,
we show that the finite temperature effect can be described with a specific
two-body interaction term in the Hamiltonian, with 1D Coulombic behavior at
large separations.Comment: 19 pages, v2: references adde
Probability density of determinants of random matrices
In this brief paper the probability density of a random real, complex and
quaternion determinant is rederived using singular values. The behaviour of
suitably rescaled random determinants is studied in the limit of infinite order
of the matrices
Exact results for the star lattice chiral spin liquid
We examine the star lattice Kitaev model whose ground state is a a chiral
spin liquid. We fermionize the model such that the fermionic vacua are toric
code states on an effective Kagome lattice. This implies that the Abelian phase
of the system is inherited from the fermionic vacua and that time reversal
symmetry is spontaneously broken at the level of the vacuum. In terms of these
fermions we derive the Bloch-matrix Hamiltonians for the vortex free sector and
its time reversed counterpart and illuminate the relationships between the
sectors. The phase diagram for the model is shown to be a sphere in the space
of coupling parameters around the triangles of the lattices. The abelian phase
lies inside the sphere and the critical boundary between topologically distinct
Abelian and non-Abelian phases lies on the surface. Outside the sphere the
system is generically gapped except in the planes where the coupling parameters
are zero. These cases correspond to bipartite lattice structures and the
dispersion relations are similar to that of the original Kitaev honeycomb
model. In a further analysis we demonstrate the three-fold non-Abelian
groundstate degeneracy on a torus by explicit calculation.Comment: 7 pages, 8 figure
Ehrenfest-time dependence of counting statistics for chaotic ballistic systems
Transport properties of open chaotic ballistic systems and their statistics
can be expressed in terms of the scattering matrix connecting incoming and
outgoing wavefunctions. Here we calculate the dependence of correlation
functions of arbitrarily many pairs of scattering matrices at different
energies on the Ehrenfest time using trajectory based semiclassical methods.
This enables us to verify the prediction from effective random matrix theory
that one part of the correlation function obtains an exponential damping
depending on the Ehrenfest time, while also allowing us to obtain the
additional contribution which arises from bands of always correlated
trajectories. The resulting Ehrenfest-time dependence, responsible e.g. for
secondary gaps in the density of states of Andreev billiards, can also be seen
to have strong effects on other transport quantities like the distribution of
delay times.Comment: Refereed version. 15 pages, 14 figure
Density of critical points for a Gaussian random function
Critical points of a scalar quantitiy are either extremal points or saddle
points. The character of the critical points is determined by the sign
distribution of the eigenvalues of the Hessian matrix. For a two-dimensional
homogeneous and isotropic random function topological arguments are sufficient
to show that all possible sign combinations are equidistributed or with other
words, the density of the saddle points and extrema agree. This argument breaks
down in three dimensions. All ratios of the densities of saddle points and
extrema larger than one are possible. For a homogeneous Gaussian random field
one finds no longer an equidistribution of signs, saddle points are slightly
more frequent.Comment: 11 pages 1 figure, changes in list of references, corrected typo
Quantum ergodicity and entanglement in kicked coupled-tops
We study the dynamical generation of entanglement as a signature of chaos in
a system of periodically kicked coupled-tops, where chaos and entanglement
arise from the same physical mechanism. The long-time averaged entanglement as
a function of the position of an initially localized wave packet very closely
correlates with the classical phase space surface of section -- it is nearly
uniform in the chaotic sea, and reproduces the detailed structure of the
regular islands. The uniform value in the chaotic sea is explained by the
random state conjecture. As classically chaotic dynamics take localized
distributions in phase space to random distributions, quantized versions take
localized coherent states to pseudo-random states in Hilbert space. Such random
states are highly entangled, with an average value near that of the maximally
entangled state. For a map with global chaos, we derive that value based on new
analytic results for the typical entanglement in a subspace defined by the
symmetries of the system.
For a mixed phase space, we use the Percival conjecture to identify a
"chaotic subspace" of the Hilbert space. The typical entanglement, averaged
over the unitarily invariant Haar measure in this subspace, agrees with the
long-time averaged entanglement for initial states in the chaotic sea. In all
cases the dynamically generated entanglement is predicted by a unitary ensemble
of random states, even though the system is time-reversal invariant, and the
Floquet operator is a member of the circular orthogonal ensemble.Comment: 12 pages with 8 figure
Random Hierarchical Matrices: Spectral Properties and Relation to Polymers on Disordered Trees
We study the statistical and dynamic properties of the systems characterized
by an ultrametric space of states and translationary non-invariant symmetric
transition matrices of the Parisi type subjected to "locally constant"
randomization. Using the explicit expression for eigenvalues of such matrices,
we compute the spectral density for the Gaussian distribution of matrix
elements. We also compute the averaged "survival probability" (SP) having sense
of the probability to find a system in the initial state by time . Using the
similarity between the averaged SP for locally constant randomized Parisi
matrices and the partition function of directed polymers on disordered trees,
we show that for times (where is some critical
time) a "lacunary" structure of the ultrametric space occurs with the
probability . This means that the escape from some bounded
areas of the ultrametric space of states is locked and the kinetics is confined
in these areas for infinitely long time.Comment: 7 pages, 2 figures (the paper is essentially reworked
Fluctuations of the correlation dimension at metal-insulator transitions
We investigate numerically the inverse participation ratio, , of the 3D
Anderson model and of the power-law random banded matrix (PRBM) model at
criticality. We found that the variance of scales with system size
as , being the
correlation dimension and the system dimension. Therefore the concept of a
correlation dimension is well defined in the two models considered. The 3D
Anderson transition and the PRBM transition for (see the text for the
definition of ) are fairly similar with respect to all critical magnitudes
studied.Comment: RevTex, 5 pages, 4 eps figures, to be published in Phys. Rev. Let
Estimation of Mean Time between Failures in Two Unit Parallel Repairable System
Mean time between failures is a method for estimating the reliability parameters of any repairable system. MTBF is also helpful in performing decision analysis in parallel and series systems and subsystems. The MTBF is the reciprocal of the failure rate when each component which fails is replaced immediately with another having the identical failure rate. There are situations when the assumption of a constant failure rate is not realistic and in many of these situations one assumes instead that the failure rate function increases or decreases smoothly with time i.e. there are no discontinuity or turning points. In this paper, we have tried to estimate MTBF taking real failure rate in case of two unit parallel repairable system and study its consistency with either the initial or the last stage of the failure rate curve
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