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arXiv:2211.09398

More on Half-Wormholes and Ensemble Average

Tian, Jia, Yingyu Yang

Published Nov 17, 2022• 0 citations
arXivHTMLPDF

Abstract

We continue our study about the half-wormhole proposal. By generalizing the original proposal of half-wormhole we propose a new way to detect half-wormholes. The crucial idea is to decompose the observables into self-averaged sector and non-self-averaged sectors. We find the contributions from different sectors have interesting statistics in the semi-classical limit. In particular, dominant sectors tend to condense and the condensation explains the emergence of half-wormholes and we expect that the appearance of condensation is a signal of possible bulk description. We also initiate the study of multi-linked-half-wormholes using our approach.

DOI
10.48550/arxiv.2211.09398

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We continue our study about the half-wormhole proposal. By generalizing the original proposal of half-wormhole we propose a new way to detect half-wormholes. The crucial idea is to decompose the observables into self-averaged sector and non-self-averaged sectors. We find the contributions from different sectors have interesting statistics in the semi-classical limit. In particular, dominant sectors tend to condense and the condensation explains the emergence of half-wormholes and we expect that the appearance of condensation is a signal of possible bulk description. We also initiate the study of multi-linked-half-wormholes using our approach.
Abstract We continue our study Half-Wormholes and Ensemble Averages about the half-wormhole proposal. By generalizing the original proposal of the half-wormhole, we propose a new way to detect half-wormholes. The crucial idea is to decompose the observables into self-averaged sectors and non-self-averaged sectors. We find the contributions from different sectors have interesting statistics in the semi-classical limit. In particular, dominant sectors tend to condense and the condensation explains the emergence of half-wormholes and we expect that the appearance of condensation is a signal of possible bulk description. We also initiate the study of multi-linked half-wormholes using our approach.
A bstract After averaging over fermion couplings, SYK has a collective field description that sometimes has “wormhole” solutions. We study the fate of these wormholes when the couplings are fixed. Working mainly in a simple model, we find that the wormhole saddles persist, but that new saddles also appear elsewhere in the integration space — “half-wormholes.” The wormhole contributions depend only weakly on the specific choice of couplings, while the half-wormhole contributions are strongly sensitive. The half-wormholes are crucial for factorization of decoupled systems with fixed couplings, but they vanish after averaging, leaving the non-factorizing wormhole behind.
Abstract In this paper, we introduce a “partially disorder-averaged” SYK model. This model has a real parameter that smoothly interpolates between the ordinary totally disorder-averaged SYK model and the totally fixed-coupling model. For the large N effective description, in addition to the usual bi-local collective fields, we also introduce a new additional set of local collective fields. These local fields can be understood as “half” of the bi-local collective fields, and in the totally fixed-coupling limit, they represent the “half-wormholes” which were found in recent studies. We find that the large N saddles of these local fields vanish in the total-disorder-averaged limit, while they develop nontrivial profiles as we gradually fix the coupling constants. We argue that the bulk picture of these local collective fields represents a correlation between a spacetime brane and the asymptotic AdS boundary. This illuminates how the half-wormhole saddles emerge in the SYK model with fixed couplings.
An important question about black holes is to what extent a typical pure state differs from the ensemble average. We show that this question can be answered within semi-classical gravity. We focus on the quantum deviation, which measures the fluctuations in the expectation value of an operator in an ensemble of pure states. For a large class of ensembles and observables, these fluctuations are calculated by a correlation function in the eternal black hole background, which can be reliably calculated within semi-classical gravity. This implements the idea of [arXiv:2002.02971] that wormholes can arise from averages over states rather than theories. As an application, we calculate the size of the long-time correlation function \langle A(t) A(0)\rangle ⟨ A ( t ) A ( 0 ) ⟩ .
Recently it has been shown by Almheiri and Lin [1] that the reconstruction of a black-hole interior is sensitive to knowing the exact coupling of the boundary theory even if the coupling is irrelevant. This motivates us to enlarge the set of the one-time and two-time toy models inspired from the Sachdev-Ye-Kitaev by deforming the same with ``irrelevant'' coupling. We find that half-wormholes, as well as the wormholes, persist in the presence of the deformation, leading to a similar mechanism for curing the factorization problem. While for the one-time case, the deformed partition function and its moments change by an overall factor, which can completely be absorbed into a renormalization of coupling; for the two-time (or coupled one-time) Sachdev-Ye-Kitaev we find nontrivial dynamics of the saddles as the couplings are varied. Curiously, the irrelevant deformations that we consider can also be thought of as an ensemble average over an overall scaling of the original undeformed Hamiltonian with an appropriate probability distribution; this allows for the possibility that half-wormholes may also be present in suitably defined ensemble of theories.
The Sachdev--Ye--Kitaev (SYK) model has a wormholelike solution after averaging over the fermionic couplings in the nearly AdS2{\mathrm{AdS}}_{2}AdS2​ space. Even when the couplings are fixed the contribution of these wormholes continues to exist and new saddle points appear which are interpreted as ``half-wormholes.'' In this paper, we will study the fate of these wormholes in a model without quenched disorder, namely, a tensor model with $O(N{)}^{q\ensuremath{-}1}$ gauge symmetry whose correlation function and thermodynamics in the large NNN limit are the same as that of the SYK model. We will restate the factorization problem linked with the wormhole threaded Wilson operator, in terms of global charges or nontrivial cobordism classes associated with disconnected wormholes. Therefore for the partition function to factorize especially at short distances, there must exist certain topological defects which break the global symmetry associated with wormholes and make the theory devoid of global symmetries. We will interpret these wormholes with added topological defects as our ``half-wormholes.'' We will also comment on the late time behavior of the spectral form factor, particularly its leading and subleading order contributions coming from higher genus wormholes in the gravitational sector. Finally we will show how the other nontrivial saddles from half-wormhole dominate and give rise to unusual thermodynamics in the bulk sector due to nonperturbative effects.