11 research outputs found
Emergent spacetime and the ergodic hierarchy
Various diagnostics of the emergence of an arrow of time in the bulk
description of a holographic theory have been proposed, including the decay of
some real time correlation functions and the appearance of type III von
Neumann algebras carrying half-sided modular inclusions. This note puts forward
a close parallel between these diagnostics and a quantum formulation of the
ergodic hierarchy of dynamical systems. Theories with an emergent spacetime
appear to sit near the top of this hierarchy.Comment: 8 pages, 2 tables, 1 figur
Thermal states are vital: Entanglement Wedge Reconstruction from Operator-Pushing
We give a general construction of a setup that verifies bulk reconstruction,
conservation of relative entropies, and equality of modular flows between the
bulk and the boundary, for infinite-dimensional systems with operator-pushing.
In our setup, a bulk-to-boundary map is defined at the level of the
-algebras of state-independent observables. We then show that if the
boundary dynamics allow for the existence of a KMS state, physically relevant
Hilbert spaces and von Neumann algebras can be constructed directly from our
framework. Our construction should be seen as a state-dependent construction of
the other side of a wormhole and clarifies the meaning of black hole
reconstruction claims such as the Papadodimas-Raju proposal. As an
illustration, we apply our result to construct a wormhole based on the HaPPY
code, which satisfies all properties of entanglement wedge reconstruction.Comment: 38 pages, 4 figures, 1 tabl
Large von Neumann algebras and the renormalization of Newton's constant
I derive a family of Ryu--Takayanagi formulae that are valid in the large
limit of holographic quantum error-correcting codes, and parameterized by a
choice of UV cutoff in the bulk. The bulk entropy terms are matched with a
family of von Neumann factors nested inside the large von Neumann algebra
describing the bulk effective field theory. These factors are mapped onto one
another by a family of conditional expectations, which are interpreted as a
renormalization group flow for the code subspace. Under this flow, I show that
the renormalizations of the area term and the bulk entropy term exactly
compensate each other. This result provides a concrete realization of the
ER=EPR paradigm, as well as an explicit proof of a conjecture due to Susskind
and Uglum.Comment: 33 pages + appendix; minor clarifications and figures added in v
Thermal states are vital: Entanglement Wedge Reconstruction from Operator-Pushing
We give a general construction of a setup that verifies bulk reconstruction, conservation of relative entropies, and equality of modular flows between the bulk and the boundary, for infinite-dimensional systems with operator-pushing. In our setup, a bulk-to-boundary map is defined at the level of the Cβ-algebras of state-independent observables. We then show that if the boundary dynamics allow for the existence of a KMS state, physically relevant Hilbert spaces and von Neumann algebras can be constructed directly from our framework. Our construction should be seen as a state-dependent construction of the other side of a wormhole and clarifies the meaning of black hole reconstruction claims such as the Papadodimas-Raju proposal. As an illustration, we apply our result to construct a wormhole based on the HaPPY code, which satisfies all properties of entanglement wedge reconstruction
The infinite-dimensional HaPPY code: entanglement wedge reconstruction and dynamics
We construct an infinite-dimensional analog of the HaPPY code as a growing
series of stabilizer codes defined respective to their Hilbert spaces. The
Hilbert spaces are related by isometric maps, which we define explicitly. We
construct a Hamiltonian that is compatible with the infinite-dimensional HaPPY
code and further study the stabilizer of our code, which has an inherent
fractal structure. We use this result to study the dynamics of the code and map
a nontrivial bulk Hamiltonian to the boundary. We find that the image of the
mapping is scale invariant, but does not create any long-range entanglement in
the boundary, therefore failing to reproduce the features of a CFT. This result
shows the limits of the HaPPY code as a model of the AdS/CFT correspondence,
but also hints that the relevance of quantum error correction in quantum
gravity may not be limited to the CFT context.Comment: 49 pages+references+appendix, 24 figures, 5 table
Holographic tensor networks from hyperbolic buildings
We introduce a unifying framework for the construction of holographic tensor
networks, based on the theory of hyperbolic buildings. The underlying dualities
relate a bulk space to a boundary which can be homeomorphic to a sphere, but
also to more general spaces like a Menger sponge type fractal. In this general
setting, we give a precise construction of a large family of bulk regions that
satisfy complementary recovery. For these regions, our networks obey a
Ryu--Takayanagi formula. The areas of Ryu--Takayanagi surfaces are controlled
by the Hausdorff dimension of the boundary, and consistently generalize the
behavior of holographic entanglement entropy in integer dimensions to the
non-integer case. Our construction recovers HaPPY--like codes in all
dimensions, and generalizes the geometry of Bruhat--Tits trees. It also
provides examples of infinite-dimensional nets of holographic conditional
expectations, and opens a path towards the study of conformal field theory and
holography on fractal spaces.Comment: 29 pages + appendices and references, 4 figures, a few
clarifications, matches published versio
Bounds on spectral gaps of Hyperbolic spin surfaces
We describe a method for constraining Laplacian and Dirac spectra of two
dimensional compact orientable hyperbolic spin manifolds and orbifolds. The key
ingredient is an infinite family of identities satisfied by the spectra. These
spectral identities follow from the consistency between 1) the spectral
decomposition of functions on the spin bundle into irreducible representations
of and 2) associativity of pointwise multiplication
of functions. Applying semidefinite programming methods to our identities
produces rigorous upper bounds on the Laplacian spectral gap as well as on the
Dirac spectral gap conditioned on the former. In several examples, our bounds
are nearly sharp; a numerical algorithm based on the Selberg trace formula
shows that the orbifold, a particular surface with signature
, and the Bolza surface nearly saturate the bounds at genus , and
respectively. Under additional assumptions on the number of harmonic
spinors carried by the spin-surface, we obtain more restrictive bounds on the
Laplacian spectral gap. In particular, these bounds apply to hyperelliptic
surfaces. We also determine the set of Laplacian spectral gaps attained by all
compact orientable two-dimensional hyperbolic spin orbifolds. We show that this
set is upper bounded by ; this bound is nearly saturated by the
orbifold, whose first non-zero Laplacian eigenvalue is
.Comment: 60 Pages, 3 Tables, 11 Figure