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:2206.13547 · Physical review. B./Physical review. B

Precise low-temperature expansions for the Sachdev-Ye-Kitaev model

Erick Arguello Cruz, Grigory Tarnopolsky

Published Jul 5, 2023• 3 citations
arXivHTMLPDF

Abstract

We solve numerically the large-$N$ Dyson-Schwinger equations for the Sachdev-Ye-Kitaev (SYK) model utilizing the Legendre polynomial decomposition and reaching ${10}^{\ensuremath{-}36}$ accuracy. Using this we compute the energy of the SYK model at low temperatures $T\ensuremath{\ll}J$ and obtain its series expansion up to ${T}^{7.54}$. While it was suggested that the expansion contains terms ${T}^{3.77}$ and ${T}^{5.68}$, we find that the first noninteger power of temperature is ${T}^{6.54}$, which comes from the two-point function of the fermion bilinear operator ${O}_{{h}_{1}}=\ensuremath{\chi}{\ensuremath{\partial}}_{\ensuremath{\tau}}^{3}\ensuremath{\chi}$ with scaling dimension ${h}_{1}\ensuremath{\approx}3.77$. The coefficient in front of ${T}^{6.54}$ term agrees well with the prediction of the conformal perturbation theory. We conclude that the conformal perturbation theory appears to work even though the SYK model is not strictly conformal.

DOI
10.1103/physrevb.108.035103

Related papers

Solving Lévy Sachdev-Ye-Kitaev Model

Budhaditya Bhattacharjee, William. E. Salazar, Alexei Andreanov +1 more

We present an exact solution in the large-$N$ limit of the Lévy Sachdev-Ye-Kitaev (LSYK) model introduced in Ref. [1], wherein the couplings are drawn from a Lévy Stable distribution parameterized by a tail exponent $μ\in [0, 2]$. Starting from the Hamiltonian and its associated partition function, we highlight the key differences from the standard Gaussian SYK model and derive the large-$N$ Schwinger-Dyson equations via a bosonic oscillator representation of the action. These equations are solved both numerically and analytically in the large-$q$ and infrared limits. We subsequently analyze the chaotic properties of the model by computing the Krylov exponent from the large-$q$ Green's function and extracting the Lyapunov exponent from the $4$-point function. The parameter $μ$ continuously interpolates between a free theory at $μ= 0$ and the conventional, maximally chaotic Gaussian SYK model at $μ= 2$, with non-maximal chaos persisting throughout the intermediate regime $0 < μ< 2$. Thermodynamic quantities, including the entropy, free energy, average energy, and specific heat capacity, are computed and compared with their Gaussian SYK counterparts. The interpretations of the thermodynamics are discussed with respect to the holographic dual and non-Fermi liquid theory. Finally, we discuss an alternative representation of the LSYK model based on a distinct decomposition of the Lévy Stable distribution, which establishes a non-trivial connection to Gaussian SYK, and provide supporting analytical and numerical results in the appendices.

hep-thCross-listedarXiv:2604.01320•Apr 3, 2026•0 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 ${\imath }\epsilon $ prescription in the SYK model

Razvan Gurău

We introduce an prescription for the Sachdev–Ye–Kitaev model both at finite and at zero temperature. This prescription regularizes all the naive ultraviolet divergences of the model. As expected the prescription breaks the conformal invariance, but the latter is restored in the limit. We prove rigorously that the Schwinger Dyson equation of the resummed two point function at large N and low momentum is recovered in this limit. Based on this prescription we introduce an effective field theory Lagrangian for the infrared SYK model.

arXiv:1705.08581•Journal of Physics Communications•Nov 17, 2017•20 citations

Supersymmetric Sachdev-Ye-Kitaev models

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

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.

Physical review. D/Physical review. D.•Jan 13, 2017•334 citations

Extension of the SYK model to 1+1 dimensions in the strong coupling limit

Andy A. V. Yunge, Alessandro De Martino, A. Tagliacozzo

We study a $1+1$ (space-time)-dimensional extension of the $0+1$-dimensional Sachdev-Ye-Kitaev model for $N$ Majorana fermions, with random all-to-all quartic interactions, averaged over disorder. At large interaction couplings and large $N$, the conformal symmetry of the effective action emerges, which is not broken spontaneously as in the original $0+1$ SYK model. Two-point correlators are obtained from a coupling expansion of the Schwinger-Dyson equations. For $N=4$, the model can be mapped onto complex fermions and solved exactly via the bosonization technique, featuring two branches of excitations: a gapped ``pseudocharge'' mode and a gapless ``pseudospin'' mode. We give an approximate analytic form of the two-point correlators at large distances and zero temperature for $N=4$, which is adopted heuristically to evaluate an approximation to the large-$N$ free energy in the zero temperature limit, numerically. The fact that this energy displays an absolute minimum at a finite value of the gap, though in a restricted range of parameters, suggests that a gapped branch of excitations is also present in our extension of the model at least in that range of parameters.

Physical review. B./Physical review. B•Aug 23, 2023•1 citation

Single-Instance Observables in the Sachdev-Ye-Kitaev Model

Brian Swingle

We present a diagrammatic expansion for thermal expectation values of low-weight Majorana strings in a fixed realization of the $N$-fermion Sachdev--Ye--Kitaev (SYK) model. At each order in the expansion, the observable is expressed as a finite sum of polynomials in the couplings multiplied by temperature-dependent kernels built from the large-$N$ melonic propagator. At fixed $βJ$, the expansion is organized in powers of $N^{-1/2}$, and each order can be efficiently evaluated with classical resources polynomial in $N$. The expansion also extends to correlations at arbitrary imaginary times and sufficiently short real times. For the four-body SYK model, we show that the diagrammatic predictions give an excellent account of exact diagonalization results at sizes up to $N=24$. We also apply the method to the Maldacena--Qi (MQ) model formed from two identical copies of a fixed SYK realization. This enables the identification of a nearby SYK-like Hamiltonian whose thermofield double state has higher fidelity with the Maldacena--Qi ground state than the thermofield double of the original SYK Hamiltonian.

hep-thCross-listedarXiv:2609.16130•Sep 16, 2026•0 citations
We solve numerically the large-NNN Dyson-Schwinger equations for the Sachdev-Ye-Kitaev (SYK) model utilizing the Legendre polynomial decomposition and reaching 10\ensuremath−36{10}^{\ensuremath{-}36}10\ensuremath−36 accuracy. Using this we compute the energy of the SYK model at low temperatures T\ensuremath≪JT\ensuremath{\ll}JT\ensuremath≪J and obtain its series expansion up to T7.54{T}^{7.54}T7.54. While it was suggested that the expansion contains terms T3.77{T}^{3.77}T3.77 and T5.68{T}^{5.68}T5.68, we find that the first noninteger power of temperature is T6.54{T}^{6.54}T6.54, which comes from the two-point function of the fermion bilinear operator Oh1=\ensuremathχ\ensuremath∂\ensuremathτ3\ensuremathχ{O}_{{h}_{1}}=\ensuremath{\chi}{\ensuremath{\partial}}_{\ensuremath{\tau}}^{3}\ensuremath{\chi}Oh1​​=\ensuremathχ\ensuremath∂\ensuremathτ3​\ensuremathχ with scaling dimension h1\ensuremath≈3.77{h}_{1}\ensuremath{\approx}3.77h1​\ensuremath≈3.77. The coefficient in front of T6.54{T}^{6.54}T6.54 term agrees well with the prediction of the conformal perturbation theory. We conclude that the conformal perturbation theory appears to work even though the SYK model is not strictly conformal.
We present an exact solution in the large-NNN limit of the Lévy Sachdev-Ye-Kitaev (LSYK) model introduced in Ref. [1], wherein the couplings are drawn from a Lévy Stable distribution parameterized by a tail exponent μ∈[0,2]μ\in [0, 2]μ∈[0,2]. Starting from the Hamiltonian and its associated partition function, we highlight the key differences from the standard Gaussian SYK model and derive the large-NNN Schwinger-Dyson equations via a bosonic oscillator representation of the action. These equations are solved both numerically and analytically in the large-qqq and infrared limits. We subsequently analyze the chaotic properties of the model by computing the Krylov exponent from the large-qqq Green's function and extracting the Lyapunov exponent from the 444-point function. The parameter μμμ continuously interpolates between a free theory at μ=0μ= 0μ=0 and the conventional, maximally chaotic Gaussian SYK model at μ=2μ= 2μ=2, with non-maximal chaos persisting throughout the intermediate regime 0<μ<20 < μ< 20<μ<2. Thermodynamic quantities, including the entropy, free energy, average energy, and specific heat capacity, are computed and compared with their Gaussian SYK counterparts. The interpretations of the thermodynamics are discussed with respect to the holographic dual and non-Fermi liquid theory. Finally, we discuss an alternative representation of the LSYK model based on a distinct decomposition of the Lévy Stable distribution, which establishes a non-trivial connection to Gaussian SYK, and provide supporting analytical and numerical results in the appendices.
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.
We introduce an prescription for the Sachdev–Ye–Kitaev model both at finite and at zero temperature. This prescription regularizes all the naive ultraviolet divergences of the model. As expected the prescription breaks the conformal invariance, but the latter is restored in the limit. We prove rigorously that the Schwinger Dyson equation of the resummed two point function at large N and low momentum is recovered in this limit. Based on this prescription we introduce an effective field theory Lagrangian for the infrared SYK model.
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 study a 1+11+11+1 (space-time)-dimensional extension of the 0+10+10+1-dimensional Sachdev-Ye-Kitaev model for NNN Majorana fermions, with random all-to-all quartic interactions, averaged over disorder. At large interaction couplings and large NNN, the conformal symmetry of the effective action emerges, which is not broken spontaneously as in the original 0+10+10+1 SYK model. Two-point correlators are obtained from a coupling expansion of the Schwinger-Dyson equations. For N=4N=4N=4, the model can be mapped onto complex fermions and solved exactly via the bosonization technique, featuring two branches of excitations: a gapped ``pseudocharge'' mode and a gapless ``pseudospin'' mode. We give an approximate analytic form of the two-point correlators at large distances and zero temperature for N=4N=4N=4, which is adopted heuristically to evaluate an approximation to the large-NNN free energy in the zero temperature limit, numerically. The fact that this energy displays an absolute minimum at a finite value of the gap, though in a restricted range of parameters, suggests that a gapped branch of excitations is also present in our extension of the model at least in that range of parameters.
We present a diagrammatic expansion for thermal expectation values of low-weight Majorana strings in a fixed realization of the NNN-fermion Sachdev--Ye--Kitaev (SYK) model. At each order in the expansion, the observable is expressed as a finite sum of polynomials in the couplings multiplied by temperature-dependent kernels built from the large-NNN melonic propagator. At fixed βJβJβJ, the expansion is organized in powers of N−1/2N^{-1/2}N−1/2, and each order can be efficiently evaluated with classical resources polynomial in NNN. The expansion also extends to correlations at arbitrary imaginary times and sufficiently short real times. For the four-body SYK model, we show that the diagrammatic predictions give an excellent account of exact diagonalization results at sizes up to N=24N=24N=24. We also apply the method to the Maldacena--Qi (MQ) model formed from two identical copies of a fixed SYK realization. This enables the identification of a nearby SYK-like Hamiltonian whose thermofield double state has higher fidelity with the Maldacena--Qi ground state than the thermofield double of the original SYK Hamiltonian.