21 research outputs found
Aspects of holographic entanglement entropy: shape dependence and hyperscaling violating backgrounds
This thesis collects three works about holographic computations of entanglement entropy. In the first one it is shown how to compute numerically, following the Ryu-Takayanagi prescription, the entanglement entropy for arbitrarily shaped entangling regions in three dimensional conformal field theories. The other two focus on holographic theories with hyperscaling violating exponents: the time dependence after a holographic quench is analyzed and the arising of new universal terms due to the presence of non smooth boundaries is shown
On the Shape of Things: From holography to elastica
We explore the question of which shape a manifold is compelled to take when
immersed in another one, provided it must be the extremum of some functional.
We consider a family of functionals which depend quadratically on the extrinsic
curvatures and on projections of the ambient curvatures. These functionals
capture a number of physical setups ranging from holography to the study of
membranes and elastica. We present a detailed derivation of the equations of
motion, known as the shape equations, placing particular emphasis on the issue
of gauge freedom in the choice of normal frame. We apply these equations to the
particular case of holographic entanglement entropy for higher curvature three
dimensional gravity and find new classes of entangling curves. In particular,
we discuss the case of New Massive Gravity where we show that non-geodesic
entangling curves have always a smaller on-shell value of the entropy
functional. Then we apply this formalism to the computation of the entanglement
entropy for dual logarithmic CFTs. Nevertheless, the correct value for the
entanglement entropy is provided by geodesics. Then, we discuss the importance
of these equations in the context of classical elastica and comment on terms
that break gauge invariance.Comment: 54 pages, 8 figures. Significantly improved version, accepted for
publication in Annals of Physics. New section on logarithmic CFTs. Detailed
derivation of the shape equations added in appendix B. Typos corrected,
clarifications adde
Spinning probes and helices in AdS
We study extremal curves associated with a functional which is linear in the
curve's torsion. The functional in question is known to capture the properties
of entanglement entropy for two-dimensional conformal field theories with
chiral anomalies and has potential applications in elucidating the equilibrium
shape of elastic linear structures. We derive the equations that determine the
shape of its extremal curves in general ambient spaces in terms of geometric
quantities. We show that the solutions to these shape equations correspond to a
three-dimensional version of Mathisson's helical motions for the centers of
mass of spinning probes. Thereafter, we focus on the case of maximally
symmetric spaces, where solutions correspond to cylindrical helices and find
that the Lancret ratio of these equals the relative speed between the
Mathisson-Pirani and the Tulczyjew-Dixon observers. Finally, we construct all
possible helical motions in three-dimensional manifolds with constant negative
curvature. In particular, we discover a rich space of helices in AdS which
we explore in detail.Comment: 28 pages, 5 figure
Entanglement Entropy for Singular Surfaces in Hyperscaling violating Theories
We study the holographic entanglement entropy for singular surfaces in
theories described holographically by hyperscaling violating backgrounds. We
consider singular surfaces consisting of cones or creases in diverse
dimensions. The structure of UV divergences of entanglement entropy exhibits
new logarithmic terms whose coefficients, being cut-off independent, could be
used to define new central charges in the nearly smooth limit. We also show
that there is a relation between these central charges and the one appearing in
the two-point function of the energy-momentum tensor. Finally we examine how
this relation is affected by considering higher-curvature terms in the
gravitational action.Comment: 27 pages, 4 figures, v2: typos corrected, references adde
Dislocation screening in crystals with spherical topology
Whereas disclination defects are energetically prohibitive in two-dimensional
flat crystals, their existence is necessary in crystals with spherical
topology, such as viral capsids, colloidosomes or fullerenes. Such a
geometrical frustration gives rise to large elastic stresses, which render the
crystal unstable when its size is significantly larger than the typical lattice
spacing. Depending on the compliance of the crystal with respect to stretching
and bending deformations, these stresses are alleviated by either a local
increase of the intrinsic curvature in proximity of the disclinations or by the
proliferation of excess dislocations, often organized in the form of
one-dimensional chains known as "scars". The associated strain field of the
scars is such to counterbalance the one resulting from the isolated
disclinations. Here, we develop a continuum theory of dislocation screening in
two-dimensional closed crystals with genus one. Upon modeling the flux of scars
emanating from a given disclination as an independent scalar field, we
demonstrate that the elastic energy of closed two-dimensional crystals with
various degrees of asphericity can be expressed as a simple quadratic function
of the screened topological charge of the disclinations, both at zero and
finite temperature. This allows us to predict the optimal density of the excess
dislocations as well as the minimal stretching energy attained by the crystal
Measuring Gaussian rigidity using curved substrates
The Gaussian (saddle splay) rigidity of fluid membranes controls their
equilibrium topology but is notoriously difficult to measure. In lipid
mixtures, typical of living cells, linear interfaces separate liquid ordered
(LO) from liquid disordered (LD) bilayer phases at subcritical temperatures.
Here we consider such membranes supported by curved supports that thereby
control the membrane curvatures. We show how spectral analysis of the
fluctuations of the LO-LD interface provides a novel way of measuring the
difference in Gaussian rigidity between the two phases. We provide a number of
conditions for such interface fluctuations to be both experimentally measurable
and sufficiently sensitive to the value of the Gaussian rigidity, whilst
remaining in the perturbative regime of our analysis.Comment: 5 pages, 3 figures. v2: version accepted for publicatio
Interface geometry of binary mixtures on curved substrates
Motivated by recent experimental work on multicomponent lipid membranes
supported by colloidal scaffolds, we report an exhaustive theoretical
investigation of the equilibrium configurations of binary mixtures on curved
substrates. Starting from the J\"ulicher-Lipowsky generalization of the
Canham-Helfrich free energy to multicomponent membranes, we derive a number of
exact relations governing the structure of an interface separating two lipid
phases on arbitrarily shaped substrates and its stability. We then restrict our
analysis to four classes of surfaces of both applied and conceptual interest:
the sphere, axisymmetric surfaces, minimal surfaces and developable surfaces.
For each class we investigate how the structure of the geometry and topology of
the interface is affected by the shape of the substrate and we make various
testable predictions. Our work sheds light on the subtle interaction mechanism
between membrane shape and its chemical composition and provides a solid
framework for interpreting results from experiments on supported lipid
bilayers.Comment: 26 pages, 10 figure
Thermodynamic equilibrium of binary mixtures on curved surfaces
We study the global influence of curvature on the free energy landscape of
two-dimensional binary mixtures confined on closed surfaces. Starting from a
generic effective free energy, constructed on the basis of symmetry
considerations and conservation laws, we identify several model-independent
phenomena, such as a curvature-dependent line tension and local shifts in the
binodal concentrations. To shed light on the origin of the phenomenological
parameters appearing in the effective free energy, we further construct a
lattice-gas model of binary mixtures on non-trivial substrates, based on the
curved-space generalization of the two-dimensional Ising model. This allows us
to decompose the interaction between the local concentration of the mixture and
the substrate curvature into four distinct contributions, as a result of which
the phase diagram splits into critical sub-diagrams. The resulting free energy
landscape can admit, as stable equilibria, strongly inhomogeneous mixed phases,
which we refer to as antimixed states below the critical temperature. We
corroborate our semi-analytical findings with phase-field numerical simulations
on realistic curved lattices. Despite this work being primarily motivated by
recent experimental observations of multi-component lipid vesicles supported by
colloidal scaffolds, our results are applicable to any binary mixture confined
on closed surfaces of arbitrary geometry.Comment: 20 Pages, 7 Figures; comments and references added, typos correcte