Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2105.08207 · SciPost Physics

Half-wormholes in SYK with one time point

Baur Mukhametzhanov

Published Jan 19, 2022• 21 citations
arXivHTMLPDF

Abstract

In this note we study the SYK model with one time point, recently considered by Saad, Shenker, Stanford, and Yao. Working in a collective field description, they derived a remarkable identity: the square of the partition function with fixed couplings is well approximated by a ``wormhole'' saddle plus a ``pair of linked half-wormholes'' saddle. It explains factorization of decoupled systems. Here, we derive an explicit formula for the half-wormhole contribution. It is expressed through a hyperpfaffian of the tensor of SYK couplings. We then develop a perturbative expansion around the half-wormhole saddle. This expansion truncates at a finite order and gives the exact answer. The last term in the perturbative expansion turns out to coincide with the wormhole contribution. In this sense the wormhole saddle in this model does not need to be added separately, but instead can be viewed as a large fluctuation around the linked half-wormholes.

DOI
10.21468/scipostphys.12.1.029

Related papers

Wormhole calculus without averaging from O(N)q−1 tensor model

Sayantan Choudhury, K. Shirish

The Sachdev--Ye--Kitaev (SYK) model has a wormholelike solution after averaging over the fermionic couplings in the nearly ${\mathrm{AdS}}_{2}$ 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 $N$ 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.

arXiv:2106.14886•Physical review. D/Physical review. D.•Feb 2, 2022•10 citations

Factorizing wormholes in a partially disorder-averaged SYK model

Kanato Goto, Kenta Suzuki, Tomonori Ugajin

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.

arXiv:2111.11705•Journal of High Energy Physics•Sep 8, 2022•4 citations

Wormholes without averaging

Phil Saad, Stephen H. Shenker, Douglas Stanford +1 more

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.

arXiv:2103.16754•Journal of High Energy Physics•Sep 19, 2024•82 citations

Half-wormholes in a complex SYK model

Yingyu Yang

Abstract We compute the half-wormhole contribution in a complex SYK model with one time point by exact computations and saddle point analysis. The chemical potential significantly influences the computation and the relative magnitude of the half-wormhole. When the chemical potential is large compared to the random coupling, the disk dominates so that there is no wormhole and half-wormhole. When the chemical potential is relatively small we find out the wormhole and the half-wormhole contributions.

Europhysics Letters (EPL)•Aug 20, 2025•1 citation

Revealing Nonperturbative Effects in the SYK Model

I. Ya. Aref’eva, I. V. Volovich, M. A. Khramtsov

In the large-N limit, we study saddle points of two SYK chains coupled by an interaction that is nonlocal in Euclidean time. We study the free model with the order of the fermionic interaction q = 2 analytically and also investigate the model with interaction in the case q = 4 numerically. We show that in both cases, there is a nontrivial phase structure with an infinite number of phases. Each phase corresponds to a saddle point in the noninteracting two-replica SYK. The nontrivial saddle points have a nonzero value of the replica-nondiagonal correlator in the sense of quasiaveraging if the coupling between replicas is turned off. The nonlocal interaction between replicas thus provides a protocol for turning the nonperturbatively subleading effects in SYK into nonequilibrium configurations that dominate at large N. For comparison, we also study two SYK chains with local interaction for q = 2 and q = 4 . We show that the q= 2 model has a similar phase structure, while the phase structure differs in the q = 4 model, dual to the traversable wormhole.

arXiv:1905.04203•Theoretical and Mathematical Physics•Nov 1, 2019•6 citations

Wormholes and half wormholes under irrelevant deformation

Diptarka Das, Sridip Pal, Anurag Sarkar

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.

arXiv:2203.14988•Physical review. D/Physical review. D.•Sep 15, 2022•7 citations
In this note we study the SYK model with one time point, recently considered by Saad, Shenker, Stanford, and Yao. Working in a collective field description, they derived a remarkable identity: the square of the partition function with fixed couplings is well approximated by a wormhole'' saddle plus a pair of linked half-wormholes'' saddle. It explains factorization of decoupled systems. Here, we derive an explicit formula for the half-wormhole contribution. It is expressed through a hyperpfaffian of the tensor of SYK couplings. We then develop a perturbative expansion around the half-wormhole saddle. This expansion truncates at a finite order and gives the exact answer. The last term in the perturbative expansion turns out to coincide with the wormhole contribution. In this sense the wormhole saddle in this model does not need to be added separately, but instead can be viewed as a large fluctuation around the linked half-wormholes.
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.
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.
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 We compute the half-wormhole contribution in a complex SYK model with one time point by exact computations and saddle point analysis. The chemical potential significantly influences the computation and the relative magnitude of the half-wormhole. When the chemical potential is large compared to the random coupling, the disk dominates so that there is no wormhole and half-wormhole. When the chemical potential is relatively small we find out the wormhole and the half-wormhole contributions.
In the large-N limit, we study saddle points of two SYK chains coupled by an interaction that is nonlocal in Euclidean time. We study the free model with the order of the fermionic interaction q = 2 analytically and also investigate the model with interaction in the case q = 4 numerically. We show that in both cases, there is a nontrivial phase structure with an infinite number of phases. Each phase corresponds to a saddle point in the noninteracting two-replica SYK. The nontrivial saddle points have a nonzero value of the replica-nondiagonal correlator in the sense of quasiaveraging if the coupling between replicas is turned off. The nonlocal interaction between replicas thus provides a protocol for turning the nonperturbatively subleading effects in SYK into nonequilibrium configurations that dominate at large N. For comparison, we also study two SYK chains with local interaction for q = 2 and q = 4 . We show that the q= 2 model has a similar phase structure, while the phase structure differs in the q = 4 model, dual to the traversable wormhole.
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.