473 research outputs found
AdS holography and the SYK model
These are lecture notes based on a series of lectures presented at the XIII
Modave Summer School in Mathematical physics aimed at PhD students and young
postdocs. The goal is to give an introduction to some of the recent
developments in understanding holography in two bulk dimensions, and its
connection to microscopics of near extremal black holes. The first part reviews
the motivation to study, and the problems (and their interpretations) with
holography for AdS spaces. The second part is about the Jackiw-Teitelboim
theory and nearly-AdS spaces. The third part introduces the
Sachdev-Ye-Kitaev model, reviews some of the basic calculations and discusses
what features make the model exciting.Comment: Lecture notes, 58 pages. v6: further small correction
Modular Hamiltonians of excited states, OPE blocks and emergent bulk fields
We study the entanglement entropy and the modular Hamiltonian of slightly
excited states reduced to a ball shaped region in generic conformal field
theories. We set up a formal expansion in the one point functions of the state
in which all orders are explicitly given in terms of integrals of multi-point
functions along the vacuum modular flow, without a need for replica index
analytic continuation. We show that the quadratic order contributions in this
expansion can be calculated in a way expected from holography, namely via the
bulk canonical energy for the entanglement entropy, and its variation for the
modular Hamiltonian. The bulk fields contributing to the canonical energy are
defined via the HKLL procedure. In terms of CFT variables, the contribution of
each such bulk field to the modular Hamiltonian is given by the OPE block
corresponding to the dual operator integrated along the vacuum modular flow.
These results do not rely on assuming large or other special properties of
the CFT and therefore they are purely kinematic.Comment: 40 pages, 2 figures. v3: some typos corrected, references added,
extended discussion on convergence and holographic interpretatio
Warped Weyl fermion partition functions
Warped conformal field theories (WCFTs) are a novel class of non-relativistic
theories. A simple, yet non-trivial, example of such theory is a massive Weyl
fermion in -dimensions, which we study in detail. We derive general
properties of the spectrum and modular properties of partition functions of
WCFTs. The periodic (Ramond) sector of this fermionic system is non-trivial,
and we build two novel partition functions for this sector which have no
counterpart in a CFT. The thermodynamical properties of WCFTs are revisited
in the canonical and micro-canonical ensemble.Comment: 41 page
Chaos and relative entropy
One characterization of a chaotic system is the quick delocalization of
quantum information (fast scrambling). One therefore expects that in such a
system a state quickly becomes locally indistinguishable from its
perturbations. In this paper we study the time dependence of the relative
entropy between the reduced density matrices of the thermofield double state
and its perturbations in two dimensional conformal field theories. We show that
in a CFT with a gravity dual, this relative entropy exponentially decays until
the scrambling time. This decay is not uniform. We argue that the early time
exponent is universal while the late time exponent is sensitive to the
butterfly effect. This large answer breaks down at the scrambling time,
therefore we also study the relative entropy in a class of spin chain models
numerically. We find a similar universal exponential decay at early times,
while at later times we observe that the relative entropy has large revivals in
integrable models, whereas there are no revivals in non-integrable models.Comment: 34+11 pages, 8 figure
Echoes of chaos from string theory black holes
The strongly coupled D1-D5 conformal field theory is a microscopic model of
black holes which is expected to have chaotic dynamics. Here, we study the weak
coupling limit of the theory where it is integrable rather than chaotic. In
this limit, the operators creating microstates of the lowest mass black hole
are known exactly. We consider the time-ordered two-point function of light
probes in these microstates, normalized by the same two-point function in
vacuum. These correlators display a universal early-time decay followed by
late-time sporadic behavior. To find a prescription for temporal
coarse-graining of these late fluctuations we appeal to random matrix theory,
where we show that a progressive time-average smooths the spectral form factor
(a proxy for the 2-point function) in a typical draw of a random matrix. This
coarse-grained quantity reproduces the matrix ensemble average to a good
approximation. Employing this coarse-graining in the D1-D5 system, we find that
the early-time decay is followed by a dip, a ramp and a plateau, in remarkable
qualitative agreement with recent studies of the Sachdev-Ye-Kitaev (SYK) model.
We study the timescales involved, comment on similarities and differences
between our integrable model and the chaotic SYK model, and suggest ways to
extend our results away from the integrable limit.Comment: 26 pages, 9 figures, v3: discussion of dip time adde
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