Schwarzian correction to quantum correlation in SYK model

TL;DR

This work analyzes Schwarzian corrections to quantum correlations in a class of SYK-type models in the large-$N$ limit, using a gravity dual to derive the Schwarzian effective action. By evaluating soft-mode propagators and loop corrections, the authors show that Schwarzian dynamics transfer spectral weight from a quasiparticle peak to a Hubbard band in the DOS, and they compute higher-point OTOCs and local spin susceptibilities within a Schwinger-Keldysh framework. The results span NFL, quantum-liquid, and spin-glass-like regimes (characterized by conformal dimension $\Delta$) and extend to higher-point functions (4-, 6-, 8-point), revealing chaotic growth with Lyapunov exponent $\lambda_L=2\pi/\beta$ in the appropriate regimes. The findings connect IR Schwarzian physics to observable spectral features and dynamical susceptibility in strongly correlated quantum liquids, offering a holographic perspective consistent with DMFT insights and potentially relevant to disordered metals and spin-glass materials.

Abstract

We study a class of SYK-type models in large N limit from the gravity dual side in terms of Schwarzian action analytically. The quantum correction to two point correlation function due to the Schwarzian action produces transfer of degree of freedom from the quasiparticle peak to Hubbard band in density of states (DOS), a signature strong correlation. In Schwinger-Keldysh (SK) formalism, we calculate higher point thermal out-of-time order correlation (OTOC) functions, which indicate quantum chaos by having Lyapunov exponent. Higher order local spin-spin correlations are also calculated, which can be related to the dynamical local susceptibility of quantum liquids such as spin glasses, disordered metals.

Figures (13)

• Figure 1: Feynman diagrams (a) Two point functions of scalars (dashed line) $G(t_1,t_2)$ and the corrections from gravitational soft mode (double wave lines) for ${\mathcal{C}}_1(t_{12})$ and ${\mathcal{C}}_2(t_{12})$ as shown in Eq.(\ref{['Eq:G12_C12']}). (b) Feynman diagrams for two point correlation functions $G_2(t_1,t_2)$ of scalar fields with loop corrections from soft modes as shown in Eq.(\ref{['Eq:2pt-4pt-6pt-8pt-T!=0']}).
• Figure 2: Dynamical susceptibility or retarded Green's functions of Schwarzian NFL with $\Delta=1/4$: $\bar{\chi}(\omega) = -G^R(\omega)/\pi$ given in Eq.(\ref{['Eq:GR_loop']}): (a) Evolution with coupling strength $(2\pi C)^{-1}$ : $C=1/3\pi$ (blue/purple solid line), $C=1/2\pi$ (cyan/magenta dashed line); $C=1/\pi$ (green/orange dashed line) and $C=+\infty$ (black/red dotted line). (b) Evolution with temperature $T=\beta^{-1}$: In front of $\bar{\chi}(\omega)$, we have multiplying a temperature depending factor $\pi/\beta$. For different $\beta$ : $\beta=2\pi$ (blue/purple solid line), $\beta=20\pi/3$ (cyan/magenta dashed line), $\beta=20\pi$ (green/orange dashed line); $\beta=200\pi$ (black/red dotted line). We have chosen input parameters as $\beta=2\pi$.
• Figure 3: Dynamical susceptibility or thermal retarded Green's functions $\chi(\omega) = G^R(\omega)$ given in Eq.(\ref{['Eq:GR_loop']}) for quantum liquid with (solid line) or without (dashed line) Schwarzian correction. For $\Delta=1/4$ cases: (a) -Re $G^R(\omega)$ (blue line); (b) -Im $G^R(\omega)$ (orange line). We have chosen $\beta=2\pi$ and $C=1/(2\pi)$. Local dynamical spin-spin correlation functions of quantum liquid $\chi_{\text{loc}}^{(2)}(\omega)$ as given in Eq.(\ref{['Eq:S_Delta-omega']}) in high temperature case with $\beta=2\pi$ (solid green/pink line) and low temperature case with $\beta=20\pi$ (dashed cyan/magenta line): (c) Re $\chi_{\text{loc}}^{(2)}(\omega)$, (d) Im $\chi_{\text{loc}}^{(2)}(\omega) \sim \tanh(\omega\beta/2)$. To avoid singularity of $\chi_{\text{loc}}^{(2)}(\omega)$ at $\Delta=1/4$, we have chosen $\Delta=1/4-\epsilon$ with $\epsilon = 10^{-3}$.
• Figure 4: Susceptibility or thermal retarded Green's functions $\chi(\omega) = G^R(\omega)$ of quantum liquid with (solid line) or without (dashed line) Schwarzian correction as given in Eq.(\ref{['Eq:GR_loop']}). For $\Delta=1/3$ case: (a) -Re $G^R(\omega)$ (black line); (b) -Im $G^R(\omega)$ (red line). We have chosen $\beta=20\pi$ and $C=1/(2\pi)$. Local dynamical spin-spin correlation functions of quantum liquid $\chi_{\text{loc}}^{(2)}(\omega)$ as given in Eq.(\ref{['Eq:S_Delta-omega']}): (c) Re $\chi_{\text{loc}}^{(2)}(\omega)$, (d) Im $\chi_{\text{loc}}^{(2)}(\omega)$ at high temperature with $\beta=2\pi$ (solid green/pink line) or at low temperature with $\beta=20\pi$ (dashed cyan/magenta line)
• Figure 5: Dynamical susceptibility or retarded Green's functions of Schwarzian spin glass with $\Delta=1/2$: $\chi(\omega) = -\bar{G}^R(\omega)/\pi$ is given in Eq.(\ref{['Eq:GR_loop']}). (a) Evolution with different coupling strength $(2\pi C)^{-1}$: $C=1/3\pi$ (blue/purple solid lines), $C=1/2\pi$ (cyan/magenta dashed line); $C=1/\pi$ (green/orange dashed line) and $C=+\infty$ (black/red dotted line). (b) Evolution with different temperature $T$: In front of Eq.(\ref{['Eq:GR_loop']}), we have multiplying a temperature depending factor $\pi/\beta$. For different $T=\beta^{-1}$ : $\beta=2\pi$ (blue/purple solid lines), $\beta=20\pi/3$ (cyan/magenta dashed line); $\beta=20\pi$ (green/orange dashed line); $\beta=200\pi$ (black/red dotted line). We have chosen input parameters as $\beta=2\pi$.