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:2412.14238 · Journal of High Energy Physics

Twisted times, the Schwarzian and its deformations in DSSYK

Micha Berkooz A, Ronny Frumkin, Ohad Mamroud, Josef Seitz

Published May 8, 2025• 7 citations
arXivHTMLPDF

Abstract

A bstract The IR dynamics of SYK is that of the Schwarzian theory, the effective theory of broken reparametrization invariance. In the double scaling limit, SYK is completely solvable by chord diagrams, whose generating functional is a bilocal Liouville theory. At low temperatures a set of modes in this description becomes soft. We interpret them as reparametrization of some twisted time coordinates, and show explicitly that they lead to the nonlinear Schwarzian theory. We further consider deformations of the theory in the double scaling limit, giving rise to diagrams with multiple species of chords, and show that the generating functional is now a Liouville theory with multiple fields. These deformations can be tracked to the IR and we discuss how they affect the Schwarzian.

DOI
10.1007/jhep05(2025)080

Related papers

Near conformal perturbation theory in SYK type models

Sumit R. Das, Animik Ghosh, Antal Jevicki +1 more

Abstract We present a systematic procedure to extract the dynamics of the low energy soft mode in SYK type models with a single energy scale J and emergent reparametrization symmetry in the IR. This is given in the framework of the perturbative scheme of arXiv:1608.07567 based on a specific (off-shell) breaking of conformal invariance in the UV, adjusted to yield the exact large-N saddle point. While this breaking term formally vanishes on-shell, it has a non-trivial effect on correlation functions and the effective action. In particular, it leads to the Schwarzian action with a specific coupling to bi-local matter. The method is applied to the evaluation of O(1) corrections to the correlation function of bi-locals. As a byproduct we confirm precise agreement with the explicit, symmetry breaking procedure. We provide a verification in the large q limit (Liouville theory), where the correlators can be calculated exactly at all length scales. In this case, our scheme illuminates how the enhanced O(J) and the subleading O(1) contributions originate from the Schwarzian dynamics of the soft mode and its interaction with h = 2 (bi-local) matter.

arXiv:2006.13149•Journal of High Energy Physics•Dec 1, 2020•9 citations

Nonlinear soft mode action for the large-p SYK model

Marta Bucca, Márk Mezei

A bstract The physics of the SYK model at low temperatures is dominated by a soft mode governed by the Schwarzian action. In [1] the linearised action was derived from the soft mode contribution to the four-point function, and physical arguments were presented for its nonlinear completion to the Schwarzian. In this paper, we give two derivations of the full nonlinear effective action in the large p limit, where p is the number of fermions in the interaction terms of the Hamiltonian. The first derivation uses that the collective field action of the large- p SYK model is Liouville theory with a non-conformal boundary condition that we study in conformal perturbation theory. This derivation can be viewed as an explicit version of the renormalisation group argument for the nonlinear soft mode action in [2]. The second derivation uses an Ansatz for how the soft mode embeds into the microscopic configuration space of the collective fields. We generalise our results for the large- p SYK chain and obtain a “Schwarzian chain” effective action for it. These derivations showcase that the large- p SYK model is a rare system, in which there is sufficient control over the microscopic dynamics, so that an effective description can be derived for it without the need for extra assumptions or matching (in the effective field theory sense).

arXiv:2412.14799•Journal of High Energy Physics•Mar 12, 2025•2 citations

Path integral for chord diagrams and chaotic-integrable transitions in double scaled SYK

Micha Berkooz A, Nadav Brukner, Yiyang Jia +1 more

We study transitions from chaotic to integrable Hamiltonians in the double scaled Sachdev-Ye-Kitaev (SYK) and p -spin systems. The dynamics of our models is described by chord diagrams with two species. We begin by developing a path integral formalism of coarse graining chord diagrams with a single species of chords, which has the same equations of motion as the bilocal ( G Σ ) Liouville action, yet appears otherwise to be different and in particular well defined. We then develop a similar formalism for two types of chords, allowing us to study different types of deformations of double scaled SYK and in particular a deformation by an integrable Hamiltonian. The system has two distinct thermodynamic phases: one is continuously connected to the chaotic SYK Hamiltonian, the other is continuously connected to the integrable Hamiltonian, separated at low temperature by a first order phase transition. We also analyze the phase diagram for generic deformations, which in some cases includes a zero-temperature phase transition. Published by the American Physical Society 2024

arXiv:2403.05980•Physical review. D/Physical review. D.•Nov 26, 2024•14 citations

Large N Fields and Holography

Animik Ghosh

We study (nearly) AdS/CFT holography within the context of the Sachdev-Ye- Kitaev (SYK) model. We present a systematic procedure to extract the dynamics of the low energy Schwarzian mode in SYK type models with a single energy scale J and emergent reparametrization symmetry in the infrared within the framework of perturbation theory. We develop a systematic approach using Feynman diagrams in bilocal theory to obtain a formal expression for the enhanced and O(1) corrections to the bilocal propagator and apply this general technique to large q SYK. We show that the Schwarzian theory describes a sector of a general class of two dimensional CFTs using conformal bootstrap techniques. We also provide a gravitational interpretation of this fact by showing that the dynamics in the near horizon throat of a specific class of BTZ black holes is described very well by Jackiw-Teitelboim (JT) gravity which gives rise to the Schwarzian action. We study the highly nontrivial conformal matter spectrum of the SYK model at arbitrary q. We provide a three dimensional bulk interpretation and carry out Kaluza-Klein (KK) reduction to reproduce the SYK spectrum and more significantly, the conformal bilocal propagator. We provide an interpretation of the two dimensional dual spacetime of the conformal sector of the SYK model using techniques from bilocal holography. We present a resolution of a conundrum about the signature of the dual spacetime using nonlocal integral transformations.

•Aug 4, 2020•0 citations

Towards a full solution of the large N double-scaled SYK model

Micha Berkooz A, Mikhail Isachenkov, Vladimir Narovlansky +1 more

Abstract We compute the exact, all energy scale, 4-point function of the large N doublescaled SYK model, by using only combinatorial tools and relating the correlation functions to sums over chord diagrams. We apply the result to obtain corrections to the maximal Lyapunov exponent at low temperatures. We present the rules for the non-perturbative diagrammatic description of correlation functions of the entire model. The latter indicate that the model can be solved by a reduction of a quantum deformation of SL(2), that generalizes the Schwarzian to the complete range of energies.

arXiv:1811.02584•Journal of High Energy Physics•Mar 1, 2019•188 citations

q-Askey Deformations of Double-Scaled SYK

Sergio E. Aguilar-Gutierrez, Trivko Kukolj, Josef Seitz

We construct families of deformations of the double-scaled SYK (DSSYK) model and investigate their bulk interpretation. We introduce microscopic deformations of the SYK model which, after ensemble averaging and in the double-scaling limit, are described by a transfer matrix encoding the recurrence relations of basic orthogonal polynomials in the q-Askey scheme. For certain families of deformations in the semiclassical limit at finite temperature, the chord number (encoding Krylov complexity) corresponds to the length of an Einstein-Rosen bridge connecting an End-Of-The-World brane to an anti-de Sitter asymptotic boundary. By increasing one of the deformation parameters, the models eventually exhibit discrete energy levels, signaling a new geometric transition in sine dilaton gravity. Via the SYK-Schur duality, Krylov complexity also admits a representation-theoretic interpretation as the spread of the SU(2) spin in the index of an $\mathcal{N}=2$ SU(2) gauge theory. We study the operator algebras of the deformed theories. The algebras can be type II$_1$ or type I$_\infty$ factors, depending on the operators that are included. The entanglement entropy between the type II$_1$ algebras for a pure state manifests as an extremal surface through the Ryu-Takayanagi formula. We discuss connections between our results and the emergence of baby universes in the bulk.

hep-thCross-listedarXiv:2605.13956•May 15, 2026
A bstract The IR dynamics of SYK is that of the Schwarzian theory, the effective theory of broken reparametrization invariance. In the double scaling limit, SYK is completely solvable by chord diagrams, whose generating functional is a bilocal Liouville theory. At low temperatures a set of modes in this description becomes soft. We interpret them as reparametrization of some twisted time coordinates, and show explicitly that they lead to the nonlinear Schwarzian theory. We further consider deformations of the theory in the double scaling limit, giving rise to diagrams with multiple species of chords, and show that the generating functional is now a Liouville theory with multiple fields. These deformations can be tracked to the IR and we discuss how they affect the Schwarzian.
Abstract We present a systematic procedure to extract the dynamics of the low energy soft mode in SYK type models with a single energy scale J and emergent reparametrization symmetry in the IR. This is given in the framework of the perturbative scheme of arXiv:1608.07567 based on a specific (off-shell) breaking of conformal invariance in the UV, adjusted to yield the exact large-N saddle point. While this breaking term formally vanishes on-shell, it has a non-trivial effect on correlation functions and the effective action. In particular, it leads to the Schwarzian action with a specific coupling to bi-local matter. The method is applied to the evaluation of O(1) corrections to the correlation function of bi-locals. As a byproduct we confirm precise agreement with the explicit, symmetry breaking procedure. We provide a verification in the large q limit (Liouville theory), where the correlators can be calculated exactly at all length scales. In this case, our scheme illuminates how the enhanced O(J) and the subleading O(1) contributions originate from the Schwarzian dynamics of the soft mode and its interaction with h = 2 (bi-local) matter.
A bstract The physics of the SYK model at low temperatures is dominated by a soft mode governed by the Schwarzian action. In [1] the linearised action was derived from the soft mode contribution to the four-point function, and physical arguments were presented for its nonlinear completion to the Schwarzian. In this paper, we give two derivations of the full nonlinear effective action in the large p limit, where p is the number of fermions in the interaction terms of the Hamiltonian. The first derivation uses that the collective field action of the large- p SYK model is Liouville theory with a non-conformal boundary condition that we study in conformal perturbation theory. This derivation can be viewed as an explicit version of the renormalisation group argument for the nonlinear soft mode action in [2]. The second derivation uses an Ansatz for how the soft mode embeds into the microscopic configuration space of the collective fields. We generalise our results for the large- p SYK chain and obtain a “Schwarzian chain” effective action for it. These derivations showcase that the large- p SYK model is a rare system, in which there is sufficient control over the microscopic dynamics, so that an effective description can be derived for it without the need for extra assumptions or matching (in the effective field theory sense).
We study transitions from chaotic to integrable Hamiltonians in the double scaled Sachdev-Ye-Kitaev (SYK) and p -spin systems. The dynamics of our models is described by chord diagrams with two species. We begin by developing a path integral formalism of coarse graining chord diagrams with a single species of chords, which has the same equations of motion as the bilocal ( G Σ ) Liouville action, yet appears otherwise to be different and in particular well defined. We then develop a similar formalism for two types of chords, allowing us to study different types of deformations of double scaled SYK and in particular a deformation by an integrable Hamiltonian. The system has two distinct thermodynamic phases: one is continuously connected to the chaotic SYK Hamiltonian, the other is continuously connected to the integrable Hamiltonian, separated at low temperature by a first order phase transition. We also analyze the phase diagram for generic deformations, which in some cases includes a zero-temperature phase transition. Published by the American Physical Society 2024
We study (nearly) AdS/CFT holography within the context of the Sachdev-Ye- Kitaev (SYK) model. We present a systematic procedure to extract the dynamics of the low energy Schwarzian mode in SYK type models with a single energy scale J and emergent reparametrization symmetry in the infrared within the framework of perturbation theory. We develop a systematic approach using Feynman diagrams in bilocal theory to obtain a formal expression for the enhanced and O(1) corrections to the bilocal propagator and apply this general technique to large q SYK. We show that the Schwarzian theory describes a sector of a general class of two dimensional CFTs using conformal bootstrap techniques. We also provide a gravitational interpretation of this fact by showing that the dynamics in the near horizon throat of a specific class of BTZ black holes is described very well by Jackiw-Teitelboim (JT) gravity which gives rise to the Schwarzian action. We study the highly nontrivial conformal matter spectrum of the SYK model at arbitrary q. We provide a three dimensional bulk interpretation and carry out Kaluza-Klein (KK) reduction to reproduce the SYK spectrum and more significantly, the conformal bilocal propagator. We provide an interpretation of the two dimensional dual spacetime of the conformal sector of the SYK model using techniques from bilocal holography. We present a resolution of a conundrum about the signature of the dual spacetime using nonlocal integral transformations.
Abstract We compute the exact, all energy scale, 4-point function of the large N doublescaled SYK model, by using only combinatorial tools and relating the correlation functions to sums over chord diagrams. We apply the result to obtain corrections to the maximal Lyapunov exponent at low temperatures. We present the rules for the non-perturbative diagrammatic description of correlation functions of the entire model. The latter indicate that the model can be solved by a reduction of a quantum deformation of SL(2), that generalizes the Schwarzian to the complete range of energies.
We construct families of deformations of the double-scaled SYK (DSSYK) model and investigate their bulk interpretation. We introduce microscopic deformations of the SYK model which, after ensemble averaging and in the double-scaling limit, are described by a transfer matrix encoding the recurrence relations of basic orthogonal polynomials in the q-Askey scheme. For certain families of deformations in the semiclassical limit at finite temperature, the chord number (encoding Krylov complexity) corresponds to the length of an Einstein-Rosen bridge connecting an End-Of-The-World brane to an anti-de Sitter asymptotic boundary. By increasing one of the deformation parameters, the models eventually exhibit discrete energy levels, signaling a new geometric transition in sine dilaton gravity. Via the SYK-Schur duality, Krylov complexity also admits a representation-theoretic interpretation as the spread of the SU(2) spin in the index of an N=2\mathcal{N}=2N=2 SU(2) gauge theory. We study the operator algebras of the deformed theories. The algebras can be type II1_11​ or type I∞_\infty∞​ factors, depending on the operators that are included. The entanglement entropy between the type II1_11​ algebras for a pure state manifests as an extremal surface through the Ryu-Takayanagi formula. We discuss connections between our results and the emergence of baby universes in the bulk.