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

Physical review. D/Physical review. D.

Supersymmetric Sachdev-Ye-Kitaev models

Wenbo Fu, Davide Gaiotto A, JUAN M. MALDACENA, Subir Sachdev A

Published Jan 13, 2017• 334 citations
Publisher

Abstract

We discuss a supersymmetric generalization of the Sachdev-Ye-Kitaev (SYK) model. These are quantum mechanical models involving $N$ Majorana fermions. The supercharge is given by a polynomial expression in terms of the Majorana fermions with random coefficients. The Hamiltonian is the square of the supercharge. The $\mathcal{N}=1$ model with a single supercharge has unbroken supersymmetry at large $N$, but nonperturbatively spontaneously broken supersymmetry in the exact theory. We analyze the model by looking at the large $N$ equation, and also by performing numerical computations for small values of $N$. We also compute the large $N$ spectrum of ``singlet'' operators, where we find a structure qualitatively similar to the ordinary SYK model. We also discuss an $\mathcal{N}=2$ version. In this case, the model preserves supersymmetry in the exact theory and we can compute a suitably weighted Witten index to count the number of ground states, which agrees with the large $N$ computation of the entropy. In both cases, we discuss the supersymmetric generalizations of the Schwarzian action which give the dominant effects at low energies.

DOI
10.1103/physrevd.95.026009

Related papers

Publisher’s Note: Supersymmetric Sachdev-Ye-Kitaev models [Phys. Rev. D 95 , 026009 (2017)]

Wenbo Fu, Davide Gaiotto A, JUAN M. MALDACENA +1 more

We discuss a supersymmetric generalization of the Sachdev-Ye-Kitaev model. These are quantum mechanical models involving $N$ Majorana fermions. The supercharge is given by a polynomial expression in terms of the Majorana fermions with random coefficients. The Hamiltonian is the square of the supercharge. The ${\cal N}=1$ model with a single supercharge has unbroken supersymmetry at large $N$, but non-perturbatively spontaneously broken supersymmetry in the exact theory. We analyze the model by looking at the large $N$ equation, and also by performing numerical computations for small values of $N$. We also compute the large $N$ spectrum of "singlet" operators, where we find a structure qualitatively similar to the ordinary SYK model. We also discuss an ${\cal N}=2$ version. In this case, the model preserves supersymmetry in the exact theory and we can compute a suitably weighted Witten index to count the number of ground states, which agrees with the large $N$ computation of the entropy. In both cases, we discuss the supersymmetric generalizations of the Schwarzian action which give the dominant effects at low energies.

arXiv:1610.08917•Physical review. D/Physical review. D.•Mar 23, 2017•44 citations

Supersymmetry in the Standard Sachdev-Ye-Kitaev Model

Jan Behrends, Benjámin Béri

Supersymmetry is a powerful concept in quantum many-body physics. It helps to illuminate ground-state properties of complex quantum systems and gives relations between correlation functions. In this Letter, we show that the Sachdev-Ye-Kitaev model, in its simplest form of Majorana fermions with random four-body interactions, is supersymmetric. In contrast to existing explicitly supersymmetric extensions of the model, the supersymmetry we find requires no relations between couplings. The type of supersymmetry and the structure of the supercharges are entirely set by the number of interacting Majorana modes and are thus fundamentally linked to the model's Altland-Zirnbauer classification. The supersymmetry we uncover has a natural interpretation in terms of a one-dimensional topological phase supporting Sachdev-Ye-Kitaev boundary physics and has consequences away from the ground state, including in q-body dynamical correlation functions.

arXiv:1908.00995•Physical Review Letters•Jun 12, 2020•23 citations

A note on Sachdev–Ye–Kitaev like model without random coupling

Takahiro Nishinaka, Seiji Terashima

We study a description of the large N limit of the Sachdev–Ye–Kitaev (SYK) model in terms of quantum mechanics without quenched disorder. Instead of random couplings, we introduce massive scalar fields coupled to fermions, and study a small mass limit of the theory. We show that, under a certain condition, the correlation functions of fermions reproduce those of the SYK model with a temperature dependent coupling constant in the large N limit. We also discuss a supersymmetric generalization of our quantum mechanical model. As a byproduct, we develop an efficient way of estimating the large N behavior of correlators in the SYK model.

arXiv:1611.10290•Nuclear Physics B•Nov 20, 2017•47 citations

N = 2 SYK models with dynamical bosons and fermions

Francesco Benini, Tomás Reis, Saman Soltani +1 more

We study a class of Sachdev, Ye, and Kitaev (SYK) models with N = 2 supersymmetry, described by N fermions in chiral Fermi multiplets, as well as α N first-order bosons in chiral multiplets. The interactions are characterized by two integers ( p , q ) . We focus on the large N and low energy limit of these models. Despite the presence of dynamical bosons, we find conformal behavior akin to the standard SYK model. We use I -extremization of a Witten index to study the supersymmetric solutions. In particular, we find an exact expression for the entropy, which matches the numerical solutions to the Schwinger-Dyson equations. We further solve the model both in the large p and large p , q limits. Numerically, we verify our analytical results and obtain estimates for the Schwarzian coupling in the near zero-temperature limit. We also study the low-lying spectrum of operators to determine the parameter ranges where the Schwarzian mode dominates the IR dynamics. Lastly, we study out-of-time-ordered correlators to show that the model is maximally chaotic.

arXiv:2402.08414•Physical review. D/Physical review. D.•Jul 29, 2025•3 citations

The chiral SYK model

Biao Lian, Shivaji L. Sondhi, Zhenbin Yang

Abstract We study the generalization of the Sachdev-Ye-Kitaev (SYK) model to a 1 + 1 dimensional chiral SYK model of N flavors of right-moving chiral Majorana fermions with all-to-all random 4-fermion interactions. The interactions in this model are exactly marginal, leading to an exact scaling symmetry. We show the Schwinger-Dyson equation of this model in the large N limit is exactly solvable. In addition, we show this model is integrable for small N ≤ 6 by bosonization. Surprisingly, the two point function in the large N limit has exactly the same form as that for N = 4, although the four point functions of the two cases are quite different. The ground state entropy in the large N limit is the same as that of N free chiral Majorana fermions, leading to a zero ground state entropy density. The OTOC of the model in the large N limit exhibits a non-trivial spacetime structure reminscent of that found by Gu and Kitaev [1] for generic SYK-like models. Specifically we find a Lyapunov regime inside an asymmetric butterfly cone, which are signatures of quantum chaos, and that the maximal velocity dependent Lyapunov exponent approaches the chaos bound 2π/β as the interaction strength approaches its physical upper bound. Finally, the model is integrable for (at least) N ≤ 6 but chaotic in the large N limit, leading us to conjecture that there is a transition from integrability to chaos as N increases past a critical value.

arXiv:1906.03308•Journal of High Energy Physics•Sep 1, 2019•24 citations

Universality and Thouless energy in the supersymmetric Sachdev-Ye-Kitaev model

Antonio M. Garcı́a-Garcı́a, Yiyang Jia, J. J. M. Verbaarschot

We investigate the supersymmetric Sachdev-Ye-Kitaev (SYK) model, $N$ Majorana fermions with infinite range interactions in $0+1$ dimensions. We have found that, close to the ground state $E\ensuremath{\approx}0$, discrete symmetries alter qualitatively the spectral properties with respect to the non-supersymmetric SYK model. The average spectral density at finite $N$, which we compute analytically and numerically, grows exponentially with $N$ for $E\ensuremath{\approx}0$. However the chiral condensate, which is normalized with respect the total number of eigenvalues, vanishes in the thermodynamic limit. Slightly above $E\ensuremath{\approx}0$, the spectral density grows exponentially with the energy. Deep in the quantum regime, corresponding to the first $O(N)$ eigenvalues, the average spectral density is universal and well described by random matrix ensembles with chiral and superconducting discrete symmetries. The dynamics for $E\ensuremath{\approx}0$ is investigated by level fluctuations. Also in this case we find excellent agreement with the prediction of chiral and superconducting random matrix ensembles for eigenvalue separations smaller than the Thouless energy, which seems to scale linearly with $N$. Deviations beyond the Thouless energy, which describes how ergodicity is approached, are universally characterized by a quadratic growth of the number variance. In the time domain, we have found analytically that the spectral form factor $g(t)$, obtained from the connected two-level correlation function of the unfolded spectrum, decays as $1/{t}^{2}$ for times shorter but comparable to the Thouless time with $g(0)$ related to the coefficient of the quadratic growth of the number variance. Our results provide further support that quantum black holes are ergodic and therefore can be classified by random matrix theory.

arXiv:1801.01071•Physical review. D/Physical review. D.•May 4, 2018•62 citations
We discuss a supersymmetric generalization of the Sachdev-Ye-Kitaev (SYK) model. These are quantum mechanical models involving NNN Majorana fermions. The supercharge is given by a polynomial expression in terms of the Majorana fermions with random coefficients. The Hamiltonian is the square of the supercharge. The N=1\mathcal{N}=1N=1 model with a single supercharge has unbroken supersymmetry at large NNN, but nonperturbatively spontaneously broken supersymmetry in the exact theory. We analyze the model by looking at the large NNN equation, and also by performing numerical computations for small values of NNN. We also compute the large NNN spectrum of ``singlet'' operators, where we find a structure qualitatively similar to the ordinary SYK model. We also discuss an N=2\mathcal{N}=2N=2 version. In this case, the model preserves supersymmetry in the exact theory and we can compute a suitably weighted Witten index to count the number of ground states, which agrees with the large NNN computation of the entropy. In both cases, we discuss the supersymmetric generalizations of the Schwarzian action which give the dominant effects at low energies.
We discuss a supersymmetric generalization of the Sachdev-Ye-Kitaev model. These are quantum mechanical models involving NNN Majorana fermions. The supercharge is given by a polynomial expression in terms of the Majorana fermions with random coefficients. The Hamiltonian is the square of the supercharge. The N=1{\cal N}=1N=1 model with a single supercharge has unbroken supersymmetry at large NNN, but non-perturbatively spontaneously broken supersymmetry in the exact theory. We analyze the model by looking at the large NNN equation, and also by performing numerical computations for small values of NNN. We also compute the large NNN spectrum of "singlet" operators, where we find a structure qualitatively similar to the ordinary SYK model. We also discuss an N=2{\cal N}=2N=2 version. In this case, the model preserves supersymmetry in the exact theory and we can compute a suitably weighted Witten index to count the number of ground states, which agrees with the large NNN computation of the entropy. In both cases, we discuss the supersymmetric generalizations of the Schwarzian action which give the dominant effects at low energies.
Supersymmetry is a powerful concept in quantum many-body physics. It helps to illuminate ground-state properties of complex quantum systems and gives relations between correlation functions. In this Letter, we show that the Sachdev-Ye-Kitaev model, in its simplest form of Majorana fermions with random four-body interactions, is supersymmetric. In contrast to existing explicitly supersymmetric extensions of the model, the supersymmetry we find requires no relations between couplings. The type of supersymmetry and the structure of the supercharges are entirely set by the number of interacting Majorana modes and are thus fundamentally linked to the model's Altland-Zirnbauer classification. The supersymmetry we uncover has a natural interpretation in terms of a one-dimensional topological phase supporting Sachdev-Ye-Kitaev boundary physics and has consequences away from the ground state, including in q-body dynamical correlation functions.
We study a description of the large N limit of the Sachdev–Ye–Kitaev (SYK) model in terms of quantum mechanics without quenched disorder. Instead of random couplings, we introduce massive scalar fields coupled to fermions, and study a small mass limit of the theory. We show that, under a certain condition, the correlation functions of fermions reproduce those of the SYK model with a temperature dependent coupling constant in the large N limit. We also discuss a supersymmetric generalization of our quantum mechanical model. As a byproduct, we develop an efficient way of estimating the large N behavior of correlators in the SYK model.
We study a class of Sachdev, Ye, and Kitaev (SYK) models with N = 2 supersymmetry, described by N fermions in chiral Fermi multiplets, as well as α N first-order bosons in chiral multiplets. The interactions are characterized by two integers ( p , q ) . We focus on the large N and low energy limit of these models. Despite the presence of dynamical bosons, we find conformal behavior akin to the standard SYK model. We use I -extremization of a Witten index to study the supersymmetric solutions. In particular, we find an exact expression for the entropy, which matches the numerical solutions to the Schwinger-Dyson equations. We further solve the model both in the large p and large p , q limits. Numerically, we verify our analytical results and obtain estimates for the Schwarzian coupling in the near zero-temperature limit. We also study the low-lying spectrum of operators to determine the parameter ranges where the Schwarzian mode dominates the IR dynamics. Lastly, we study out-of-time-ordered correlators to show that the model is maximally chaotic.
Abstract We study the generalization of the Sachdev-Ye-Kitaev (SYK) model to a 1 + 1 dimensional chiral SYK model of N flavors of right-moving chiral Majorana fermions with all-to-all random 4-fermion interactions. The interactions in this model are exactly marginal, leading to an exact scaling symmetry. We show the Schwinger-Dyson equation of this model in the large N limit is exactly solvable. In addition, we show this model is integrable for small N ≤ 6 by bosonization. Surprisingly, the two point function in the large N limit has exactly the same form as that for N = 4, although the four point functions of the two cases are quite different. The ground state entropy in the large N limit is the same as that of N free chiral Majorana fermions, leading to a zero ground state entropy density. The OTOC of the model in the large N limit exhibits a non-trivial spacetime structure reminscent of that found by Gu and Kitaev [1] for generic SYK-like models. Specifically we find a Lyapunov regime inside an asymmetric butterfly cone, which are signatures of quantum chaos, and that the maximal velocity dependent Lyapunov exponent approaches the chaos bound 2π/β as the interaction strength approaches its physical upper bound. Finally, the model is integrable for (at least) N ≤ 6 but chaotic in the large N limit, leading us to conjecture that there is a transition from integrability to chaos as N increases past a critical value.
We investigate the supersymmetric Sachdev-Ye-Kitaev (SYK) model, NNN Majorana fermions with infinite range interactions in 0+10+10+1 dimensions. We have found that, close to the ground state $E\ensuremath{\approx}0$, discrete symmetries alter qualitatively the spectral properties with respect to the non-supersymmetric SYK model. The average spectral density at finite NNN, which we compute analytically and numerically, grows exponentially with NNN for $E\ensuremath{\approx}0$. However the chiral condensate, which is normalized with respect the total number of eigenvalues, vanishes in the thermodynamic limit. Slightly above $E\ensuremath{\approx}0$, the spectral density grows exponentially with the energy. Deep in the quantum regime, corresponding to the first O(N)O(N)O(N) eigenvalues, the average spectral density is universal and well described by random matrix ensembles with chiral and superconducting discrete symmetries. The dynamics for $E\ensuremath{\approx}0$ is investigated by level fluctuations. Also in this case we find excellent agreement with the prediction of chiral and superconducting random matrix ensembles for eigenvalue separations smaller than the Thouless energy, which seems to scale linearly with NNN. Deviations beyond the Thouless energy, which describes how ergodicity is approached, are universally characterized by a quadratic growth of the number variance. In the time domain, we have found analytically that the spectral form factor g(t)g(t)g(t), obtained from the connected two-level correlation function of the unfolded spectrum, decays as 1/t21/{t}^{2}1/t2 for times shorter but comparable to the Thouless time with g(0)g(0)g(0) related to the coefficient of the quadratic growth of the number variance. Our results provide further support that quantum black holes are ergodic and therefore can be classified by random matrix theory.