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arXiv:2212.04944 · Journal of High Energy Physics

Renormalisation group flows of deformed SYK models

Dionysios Anninos, Damián A. Galante, Sameer U. Sheorey

Published Nov 27, 2023• 17 citations
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Abstract

Abstract We explore computationally tractable deformations of the SYK model. The deformed theories are described by the sum of two SYK Hamiltonians with differing numbers, q and $$ \overset{\sim }{q} $$ q ~ , of interacting fermions. In the large N limit, employing analytic and numerical tools, we compute finite temperature correlation functions and thermodynamic quantities. We identify a novel analytically solvable model in the large q limit. We find that, under certain circumstances, the thermal RG flow in the strongly coupled infrared phase exhibits two regions of linear-in-temperature entropy, which we interpret in terms of Schwarzian actions. Using conformal perturbation theory we compute the leading relevant correction away from the intermediate near-conformal fixed point. Holographic spacetimes in two spacetime dimensions that reproduce the thermodynamics of the microphysical theory are discussed. These are flow geometries that interpolate between two Euclidean near-AdS2 spacetimes with different radii. The Schwarzian soft mode corresponding to the AdS2 region in the deep interior resides entirely within the geometric regime.

DOI
10.1007/jhep11(2023)197

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Abstract We explore computationally tractable deformations of the SYK model. The deformed theories are described by the sum of two SYK Hamiltonians with differing numbers, q and q∼\overset{\sim }{q}q∼​ q ~ , of interacting fermions. In the large N limit, employing analytic and numerical tools, we compute finite temperature correlation functions and thermodynamic quantities. We identify a novel analytically solvable model in the large q limit. We find that, under certain circumstances, the thermal RG flow in the strongly coupled infrared phase exhibits two regions of linear-in-temperature entropy, which we interpret in terms of Schwarzian actions. Using conformal perturbation theory we compute the leading relevant correction away from the intermediate near-conformal fixed point. Holographic spacetimes in two spacetime dimensions that reproduce the thermodynamics of the microphysical theory are discussed. These are flow geometries that interpolate between two Euclidean near-AdS2 spacetimes with different radii. The Schwarzian soft mode corresponding to the AdS2 region in the deep interior resides entirely within the geometric regime.
We analyse a class of SYK models whose Hamiltonian is the sum of two SYK Hamiltonians with different numbers of fermions q,q~q, \tilde qq,q~​ in each interaction. We consider both Euclidean and Lorentzian probes of the quantum system in the large NNN limit. In the strong coupling phase, the entropy provides a diagnostic of the thermal renormalisation group flow. Under certain conditions, two parametrically separated regimes of near-conformal behaviour emerge. The first reproduces the standard linear-in-temperature scaling characteristic of the single SYK model. The system then flows to another near-fixed point whose entropy scaling depends on the ratio n=q/q~n = q/\tilde qn=q/q~​. For n<3/2n<3/2n<3/2, the entropy exhibits anomalous, stronger-than-linear scaling in temperature. At n=3/2n=3/2n=3/2, there is an additional logarithmic enhancement. Using conformal perturbation theory, we argue that in the infrared regime of the SYK model, there may exist disordered conformal operators with dimensions 1<Δ≤3/21 < Δ\leq 3/21<Δ≤3/2. In Lorentzian signature, we study the out-of-time-ordered correlator and show that these deformed theories exhibit near-maximal chaos in both regimes (when they exist). We comment on the relation between the anomalous scalings found here and those observed in certain near-extremal black holes in two and higher dimensions.
A bstract We analyse a class of SYK models whose Hamiltonian is the sum of two SYK Hamiltonians with different numbers of fermions q, q∼ \overset{\sim }{q} q∼​ q ~ in each interaction. We consider both Euclidean and Lorentzian probes of the quantum system in the large N limit. In the strong coupling phase, the entropy provides a diagnostic of the thermal renormalisation group flow. Under certain conditions, two parametrically separated regimes of near-conformal behaviour emerge. The first reproduces the standard linear-in-temperature scaling characteristic of the single SYK model. The system then flows to another near-fixed point whose entropy scaling depends on the ratio n=q/q∼ n=q/\overset{\sim }{q} n=q/q∼​ n = q / q ~ . For n < 3 / 2, the entropy exhibits anomalous, stronger-than-linear scaling in temperature. At n = 3 / 2, there is an additional logarithmic enhancement. Using conformal perturbation theory, we argue that in the infrared regime of the SYK model, there may exist disordered conformal operators with dimensions 1 < ∆ ≤ 3 / 2. In Lorentzian signature, we study the out-of-time-ordered correlator and show that these deformed theories exhibit near-maximal chaos in both regimes (when they exist). We comment on the relation between the anomalous scalings found here and those observed in certain near-extremal black holes in two and higher dimensions.
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.
Notes on complex q=2q=2q=2 SYK
This note clarifies and extends results on the complex SYK model to the solvable q = 2 case. We calculate the four point function OPE of fermions in the low energy CFT, implying the existence of a tower of integer-weight operators in the IR. We comment on the lack of a mode breaking conformal symmetry in this special case of SYK and the consequences for deformations of the theory near the conformal fixed point. We use the nearly-free structure of the model to provide a closed form expression for OPE coefficients of the integer-weight operators. We also discuss analytic and numerical results relevant to the thermodynamics of q = 2 SYK in both the complex and real case. The tower of operators transform in the discrete series of representations of SL(2,R), the representations shared by dS2 and AdS2. In this work we continue discussion of holographic models including these representations.
Abstract It has recently been demonstrated that the large N limit of a model of fermions charged under the global/gauge symmetry group O(N) q−1 agrees with the large N limit of the SYK model. In these notes we investigate aspects of the dynamics of the O(N) q−1 theories that differ from their SYK counterparts. We argue that the spectrum of fluctuations about the finite temperature saddle point in these theories has (q−1)N22 \left(q-1\right)\frac{N^2}{2} (q−1)2N2​ q − 1 N 2 2 new light modes in addition to the light Schwarzian mode that exists even in the SYK model, suggesting that the bulk dual description of theories differ significantly if they both exist. We also study the thermal partition function of a mass deformed version of the SYK model. At large mass we show that the effective entropy of this theory grows with energy like E ln E (i.e. faster than Hagedorn) up to energies of order N 2. The canonical partition function of the model displays a deconfinement or Hawking Page type phase transition at temperatures of order 1/ln N. We derive these results in the large mass limit but argue that they are qualitatively robust to small corrections in J/m.
Abstract Sachdev-Ye-Kitaev (SYK) model, which describes N randomly interacting Majorana fermions in 0+1 dimension, is found to be an solvable UV-complete toy model for holographic duality in nearly AdS2 dilaton gravity. Ref. [1] proposed a modified model by coupling two identical SYK models, which at low-energy limit is dual to a global AdS2 geometry. This geometry is an “eternal wormhole” because the two boundaries are causally connected. Increasing the temperature drives a Hawking-Page like transition from the eternal wormhole geometry to two disconnected black holes with coupled matter field. To gain more understanding of the coupled SYK model, in this work, we study the finite temperature spectral function of this system by numerical solving the Schwinger-Dyson equation in real-time. We find in the low-temperature phase the system is well described by weakly interacting fermions with renormalized single-particle gap, while in the high temperature phase the system is strongly interacting and the single-particle peaks merge. We also study the q dependence of the spectral function.