16,846 research outputs found
Projective Hilbert space structures at exceptional points
A non-Hermitian complex symmetric 2x2 matrix toy model is used to study
projective Hilbert space structures in the vicinity of exceptional points
(EPs). The bi-orthogonal eigenvectors of a diagonalizable matrix are
Puiseux-expanded in terms of the root vectors at the EP. It is shown that the
apparent contradiction between the two incompatible normalization conditions
with finite and singular behavior in the EP-limit can be resolved by
projectively extending the original Hilbert space. The complementary
normalization conditions correspond then to two different affine charts of this
enlarged projective Hilbert space. Geometric phase and phase jump behavior are
analyzed and the usefulness of the phase rigidity as measure for the distance
to EP configurations is demonstrated. Finally, EP-related aspects of
PT-symmetrically extended Quantum Mechanics are discussed and a conjecture
concerning the quantum brachistochrone problem is formulated.Comment: 20 pages; discussion extended, refs added; bug correcte
Replica theory of the rigidity of structural glasses
We present a first principle scheme to compute the rigidity, i. e. the
shear-modulus of structural glasses at finite temperatures using the cloned
liquid theory, which combines the replica theory and the liquid theory. With
the aid of the replica method which enables disentanglement of thermal
fluctuations in liquids into intra-state and inter-state fluctuations, we
extract the rigidity of metastable amorphous solid states in the supercooled
liquid and glass phases. The result can be understood intuitively without
replicas. As a test case, we apply the scheme to the supercooled and glassy
state of a binary mixture of soft-spheres. The result compares well with the
shear-modulus obtained by a previous molecular dynamic simulation. The rigidity
of metastable states is significantly reduced with respect to the instantaneous
rigidity, namely the Born term, due to non-affine responses caused by
displacements of particles inside cages at all temperatures down to T=0. It
becomes nearly independent of temperature below the Kauzmann temperature T_K.
At higher temperatures in the supercooled liquid state, the non-affine
correction to the rigidity becomes stronger suggesting melting of the
metastable solid state. Inter-state part of the static response implies jerky,
intermittent stress-strain curves with static analogue of yielding at
mesoscopic scales.Comment: 52 pages, 10 figure
Special Lagrangian fibrations, mirror symmetry and Calabi-Yau double covers
The first part of this paper is a review of the Strominger-Yau-Zaslow
conjecture in various settings. In particular, we summarize how, given a pair
(X,D) consisting of a Kahler manifold and an anticanonical divisor, families of
special Lagrangian tori in X-D and weighted counts of holomorphic discs in X
can be used to build a Landau-Ginzburg model mirror to X. In the second part we
turn to more speculative considerations about Calabi-Yau manifolds with
holomorphic involutions and their quotients. Namely, given a hypersurface H
representing twice the anticanonical class in a Kahler manifold X, we attempt
to relate special Lagrangian fibrations on X-H and on the (Calabi-Yau) double
cover of X branched along H; unfortunately, the implications for mirror
symmetry are far from clear.Comment: 27 pages, 1 figur
Black hole singularities: a new critical phenomenon
The singularitiy inside a spherical charged black hole, coupled to a
spherical, massless scalar field is studied numerically. The profile of the
characteristic scalar field was taken to be a power of advanced time with an
exponent . A critical exponent exists. For
exponents below the critical one () the singularity
is a union of spacelike and null sectors, as is also the case for data with
compact support. For exponents greater than the critical one
() an all-encompassing, spacelike singularity
evolves, which completely blocks the ``tunnel'' inside the black hole,
preventing the use of the black hole as a portal for hyperspace travel.Comment: 4 pages, 5 eps figures; An Erratum is added. The main conclusions of
the original Letter are unchange
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