22 research outputs found
Entanglement Scrambling in 2d Conformal Field Theory
We investigate how entanglement spreads in time-dependent states of a 1+1
dimensional conformal field theory (CFT). The results depend qualitatively on
the value of the central charge. In rational CFTs, which have central charge
below a critical value, entanglement entropy behaves as if correlations were
carried by free quasiparticles. This leads to long-term memory effects, such as
spikes in the mutual information of widely separated regions at late times.
When the central charge is above the critical value, the quasiparticle picture
fails. Assuming no extended symmetry algebra, any theory with has
diminished memory effects compared to the rational models. In holographic CFTs,
with , these memory effects are eliminated altogether at strong
coupling, but reappear after the scrambling time at
weak coupling.Comment: 52 pages, 7 figure; v2: references adde
Heavy-Heavy-Light-Light correlators in Liouville theory
We compute four-point functions of two heavy and two "perturbatively heavy"
operators in the semiclassical limit of Liouville theory on the sphere. We
obtain these "Heavy-Heavy-Light-Light" (HHLL) correlators to leading order in
the conformal weights of the light insertions in two ways: (a) via a path
integral approach, combining different methods to evaluate correlation
functions from complex solutions for the Liouville field, and (b) via the
conformal block expansion. This latter approach identifies an integral over the
continuum of normalizable states and a sum over an infinite tower of lighter
discrete states, whose contribution we extract by analytically continuing
standard results to our HHLL setting. The sum over this tower reproduces the
sum over those complex saddlepoints of the path integral that contribute to the
correlator. Our path integral computations reveal that when the two light
operators are inserted at equal time in radial quantization, the leading-order
HHLL correlator is independent of their separation, and more generally that at
this order there is no short-distance singularity as the two light operators
approach each other. The conformal block expansion likewise shows that in the
discrete sum short-distance singularities are indeed absent for all
intermediate states that contribute. In particular, the Virasoro vacuum block,
which would have been singular at short distances, is not exchanged. The
separation-independence of equal-time correlators is due to cancelations
between the discrete contributions. These features lead to a Lorentzian
singularity that, in conformal theories with anti-de Sitter (AdS) duals, would
be associated to locality below the AdS scale.Comment: 40 pages, 1 figure; v2: clarifications added, minor typos corrected,
published versio
Holographic Entanglement Entropy from 2d CFT: Heavy States and Local Quenches
We consider the entanglement entropy in 2d conformal field theory in a class
of excited states produced by the insertion of a heavy local operator. These
include both high-energy eigenstates of the Hamiltonian and time-dependent
local quenches. We compute the universal contribution from the stress tensor to
the single interval Renyi entropies and entanglement entropy, and conjecture
that this dominates the answer in theories with a large central charge and a
sparse spectrum of low-dimension operators. The resulting entanglement
entropies agree precisely with holographic calculations in three-dimensional
gravity. High-energy eigenstates are dual to microstates of the BTZ black hole,
so the corresponding holographic calculation is a geodesic length in the black
hole geometry; agreement between these two answers demonstrates that
entanglement entropy thermalizes in individual microstates of holographic CFTs.
For local quenches, the dual geometry is a highly boosted black hole or conical
defect. On the CFT side, the rise in entanglement entropy after a quench is
directly related to the monodromy of a Virasoro conformal block.Comment: 30 pages, 5 figures; v2: minor clarifications and references adde
Holographic second laws of black hole thermodynamics
Recently, it has been shown that for out-of-equilibrium systems, there are
additional constraints on thermodynamical evolution besides the ordinary second
law. These form a new family of second laws of thermodynamics, which are
equivalent to the monotonicity of quantum R\'enyi divergences. In black hole
thermodynamics, the usual second law is manifest as the area increase theorem.
Hence one may ask if these additional laws imply new restrictions for
gravitational dynamics, such as for out-of-equilibrium black holes? Inspired by
this question, we study these constraints within the AdS/CFT correspondence.
First, we show that the R\'enyi divergence can be computed via a Euclidean path
integral for a certain class of excited CFT states. Applying this construction
to the boundary CFT, the R\'enyi divergence is evaluated as the renormalized
action for a particular bulk solution of a minimally coupled gravity-scalar
system. Further, within this framework, we show that there exist transitions
which are allowed by the traditional second law, but forbidden by the
additional thermodynamical constraints. We speculate on the implications of our
findings.Comment: 81 pages, 19 figures; v2: clarifications and reference added, minor
typos corrected, published versio
First law of holographic complexity
We investigate the variation of holographic complexity for two nearby target
states. Based on Nielsen's geometric approach, we find the variation only
depends on the end point of the optimal trajectory, a result which we designate
the first law of complexity. As an example, we examine the complexity=action
conjecture when the AdS vacuum is perturbed by a scalar field excitation, which
corresponds to a coherent state. Remarkably, the gravitational contributions
completely cancel and the final variation reduces to a boundary term coming
entirely from the scalar field action. Hence the null boundary of
Wheeler-DeWitt patch appears to act like the "end of the quantum circuit".Comment: 7 pages + supplemental material, 2 figures; v2: clarifications and
reference added, published versio
Aspects of The First Law of Complexity
We investigate the first law of complexity proposed in arXiv:1903.04511,
i.e., the variation of complexity when the target state is perturbed, in more
detail. Based on Nielsen's geometric approach to quantum circuit complexity, we
find the variation only depends on the end of the optimal circuit. We apply the
first law to gain new insights into the quantum circuits and complexity models
underlying holographic complexity. In particular, we examine the variation of
the holographic complexity for both the complexity=action and complexity=volume
conjectures in perturbing the AdS vacuum with coherent state excitations of a
free scalar field. We also examine the variations of circuit complexity
produced by the same excitations for the free scalar field theory in a fixed
AdS background. In this case, our work extends the existing treatment of
Gaussian coherent states to properly include the time dependence of the
complexity variation. We comment on the similarities and differences of the
holographic and QFT results.Comment: 108 pages, 15 figures; v2: references adde
Thermalization of Strongly Coupled Field Theories
Using the holographic mapping to a gravity dual, we calculate 2-point functions, Wilson loops, and entanglement entropy in strongly coupled field theories in d = 2, 3, and 4 to probe the scale dependence of thermalization following a sudden injection of energy. For homogeneous initial conditions, the entanglement entropy thermalizes slowest and sets a time scale for equilibration that saturates a causality bound. The growth rate of entanglement entropy density is nearly volume-independent for small volumes but slows for larger volumes. In this setting, the UV thermalizes first