Out-of-time-ordered correlators of the Hubbard model: Sachdev-Ye-Kitaev strange metal in the spin-freezing crossover region

The Sachdev-Ye-Kitaev (SYK) model describes a strange metal that shows peculiar non-Fermi-liquid properties without quasiparticles. It exhibits a maximally chaotic behavior characterized by out-of-time-ordered correlators (OTOCs), and is expected to be a holographic dual to black holes. While a faithful realization of the SYK model in condensed-matter systems may be involved, a striking similarity between the SYK model and the Hund-coupling-induced spin-freezing crossover in multiorbital Hubbard models has recently been pointed out. To further explore this connection, we study OTOCs for fermionic single-orbital and multiorbital Hubbard models, which are prototypical models for strongly correlated electrons in solids. We introduce an imaginary-time four-point correlation function with an appropriate time ordering, which by means of the spectral representation and the out-of-time-order fluctuation-dissipation theorem can be analytically continued to real-time OTOCs. Based on this approach, we numerically evaluate real-time OTOCs for Hubbard models in the thermodynamic limit, using the dynamical mean-field theory in combination with a numerically exact continuous-time Monte Carlo impurity solver. The results for the single-orbital model show that a certain spin-related OTOC captures local moment formation in the vicinity of the metal-insulator transition, while the self-energy does not show SYK-type non-Fermi-liquid behavior. On the other hand, for the two- and three-orbital models with nonzero Hund coupling we find that the OTOC exhibits a rapid damping at short times and an approximate power-law decay at longer times in the spin-freezing crossover regime characterized by fluctuating local moments and a non-Fermi-liquid self-energy $\mathrm{\Sigma }\left(\omega \right)\sim \sqrt{\omega }$. These results are in a good agreement with the behavior of the SYK model, providing firm evidence for the close relation between the spin-freezing crossover physics of multiorbital Hubbard models and the SYK strange metal.

Figure 1

Schematic illustration of the spin-freezing crossover from (a) a Fermi-liquid state with a Kondo singlet via (b) a fluctuating moment state to (c) a frozen moment state in multiorbital systems with nonzero Hund coupling. Here, we focus on a single lattice site (orange) and represent the rest of the lattice by an electron bath (light blue).

Figure 2

Sketch of the phase diagram of the two-orbital Hubbard model in the space of the interaction $U$ and filling per orbital and spin $n_{\sigma }$ for a fixed ratio $J/U>0$ and temperature of the order of $1/100$ of the bandwidth (left panel), and in the space of the temperature $T$ and $n_{\sigma }$ at fixed $U$ (right panel). The bold black lines indicate Mott insulating solutions appearing at integer total fillings. Doping of the half-filled Mott insulator ($n_{\sigma }=0.5$) results in an incoherent metal state with frozen magnetic moments (blue shaded region). At lower fillings, there is a crossover to a Fermi-liquid metal. The crossover region (pink) is characterized by the non-Fermi-liquid self-energy $\mathrm{\Sigma }\left(\omega \right)\sim \sqrt{\omega }$. The dashed lines show the parameter region that we explore in Sec. 4.

Figure 3

The procedure to compute the out-of-time-ordered correlation function $\langle A\left(t\right)B\left(0\right),A\left(t\right)B\left(0\right)\rangle \pm \langle B\left(0\right)A\left(t\right),B\left(0\right)A\left(t\right)\rangle$ from the imaginary-time data obtained by QMC calculations.

Figure 4

Left panel: comparison between the exact solution and the result obtained by analytical continuation for the real and imaginary parts of the OTOC $\langle c_{\sigma }^{†}\left(t\right)c_{\sigma }\left(0\right),c_{\sigma }^{†}\left(t\right)c_{\sigma }\left(0\right)\rangle$ in the noninteracting case ($U=0,\beta \hslash =50$) of the single-orbital Hubbard model. Right panel: comparison of the long-time behavior for the modulus $|\langle c_{\sigma }^{†}\left(t\right)c_{\sigma }\left(0\right),c_{\sigma }^{†}\left(t\right)c_{\sigma }\left(0\right)\rangle |$. The exact result shows a power-law decay ($\sim 1/t^3$).

Figure 5

Top panels: Real and imaginary parts of the OTOC $\langle c_{\sigma }^{†}\left(t\right)c_{\sigma }\left(0\right),c_{\sigma }^{†}\left(t\right)c_{\sigma }\left(0\right)\rangle$ for the single-orbital Hubbard model with $\beta =50$ at half filling. The thin lines correspond to metallic solutions, while the thick line ($U=6$) corresponds to a Mott insulating solution. Bottom left panel: Spectral function ${\mathcal{A}}_{{\left(c_{\sigma }^{†}{c}_{\sigma }\right)}^2}\left(\omega \right)=-\frac{1}{\pi }\mathrm{Im}{C}_{{\left(c_{\sigma }^{†}{c}_{\sigma }\right)}^2}^R\left(\omega \right)$ multiplied by $coth\left(\frac{\beta \omega }{4}\right)$. Bottom right panel: Modulus of the OTOC $|\langle c_{\sigma }^{†}\left(t\right)c_{\sigma }\left(0\right),c_{\sigma }^{†}\left(t\right)c_{\sigma }\left(0\right)\rangle |$ (normalized at $t=0$) on a log-log scale. The dashed line is the result for $U=0$.

Figure 6

Top panels: Real and imaginary parts of the OTOC $\langle \widehat{n}\left(t\right)\widehat{n}\left(0\right),\widehat{n}\left(t\right)\widehat{n}\left(0\right)\rangle -1$ for the single-orbital Hubbard model with $\beta =50$ at half filling. The thin lines correspond to metallic solutions, while the thick line ($U=6$) corresponds to a Mott insulating solution. Bottom left panel: Spectral function ${\mathcal{A}}_{{\left(nn\right)}^2}\left(\omega \right)=-\frac{1}{\pi }\mathrm{Im}{C}_{{\left(nn\right)}^2}^R\left(\omega \right)$ multiplied by $coth\left(\frac{\beta \omega }{4}\right)$. Bottom right panel: Modulus of the OTOC $|\langle \widehat{n}\left(t\right)\widehat{n}\left(0\right),\widehat{n}\left(t\right)\widehat{n}\left(0\right)\rangle -1|$ (normalized at $t=0$) on a log-log scale.

Figure 7

Top panels: real and imaginary parts of the OTOC $\langle {\widehat{n}}_{\sigma }\left(t\right){\widehat{n}}_{\sigma }\left(0\right),{\widehat{n}}_{\sigma }\left(t\right){\widehat{n}}_{\sigma }\left(0\right)\rangle -{\langle {\widehat{n}}_{\sigma }\rangle }^4$ for the single-orbital Hubbard model with $\beta =50$ at half-filling. All the colored lines correspond to metallic solutions. Bottom left panel: spectral function ${\mathcal{A}}_{{\left(n_{\sigma }{n}_{\sigma }\right)}^2}\left(\omega \right)=-\frac{1}{\pi }\mathrm{Im}{C}_{{\left(n_{\sigma }{n}_{\sigma }\right)}^2}^R\left(\omega \right)$ multiplied by $coth\left(\frac{\beta \omega }{4}\right)$. In the inset, we plot the imaginary-time four-point function $\langle {\widehat{n}}_{\sigma }\left(\tau \right){\widehat{n}}_{\sigma }\left(0\right),{\widehat{n}}_{\sigma }\left(\tau \right){\widehat{n}}_{\sigma }\left(0\right)\rangle -{\langle {\widehat{n}}_{\sigma }\rangle }^4$, with the black line corresponding to the Mott insulating solution at $U=6$. Bottom right panel: modulus of the OTOC $|\langle {\widehat{n}}_{\sigma }\left(t\right){\widehat{n}}_{\sigma }\left(0\right),{\widehat{n}}_{\sigma }\left(t\right){\widehat{n}}_{\sigma }\left(0\right)\rangle -{\langle {\widehat{n}}_{\sigma }\rangle }^4|$ (normalized at $t=0$) on a log-log scale.

Figure 8

Left panel: the imaginary-time four-point function $\langle {\widehat{n}}_{\sigma }\left(\tau \right){\widehat{n}}_{\sigma }\left(0\right),{\widehat{n}}_{\sigma }\left(\tau \right){\widehat{n}}_{\sigma }\left(0\right)\rangle$, the squared density-density correlation function ${\langle {\widehat{n}}_{\sigma }\left(\tau \right){\widehat{n}}_{\sigma }\left(0\right)\rangle }^2$, and their difference for the single-orbital Hubbard model with $U=5.5$ and $\beta =50$. The dashed curve shows a fit of the difference with the function $f_{\text{fit}}\left(\tau \right)=a+b\tau +c{\tau }^2$ in the interval $\tau /\beta \in [0:0.2]$. Right panel: fitting coefficient $c$ as a function of $U$ for indicated values of $\beta$.

Figure 9

Left panel: imaginary part of the self-energy as a function of Matsubara frequency for the two-orbital Hubbard model with $U=8, J=U/4$, and $\beta =50$ on a log-log scale, with the dashed line indicating $\sqrt{{\omega }_n}$ behavior. Right panel: the coefficient $c$ extracted from fitting $\langle {\widehat{n}}_{\sigma }\left(\tau \right){\widehat{n}}_{\sigma }\left(0\right),{\widehat{n}}_{\sigma }\left(\tau \right){\widehat{n}}_{\sigma }\left(0\right)\rangle -{\langle {\widehat{n}}_{\sigma }\left(\tau \right){\widehat{n}}_{\sigma }\left(0\right)\rangle }^2$ with a function $f_{\text{fit}}\left(\tau \right)=a+b\tau +c{\tau }^2$ in the interval $\tau /\beta \in [0:0.05]$ for the two-orbital Hubbard model. The spin-freezing crossover occurs roughly at the filling $n_{\sigma }\approx 0.28$ (blue curve in the left panel and blue arrow in the right panel).

Figure 10

Top panels: real and imaginary parts of the OTOC $\langle {\widehat{n}}_{\sigma }\left(t\right){\widehat{n}}_{\sigma }\left(0\right),{\widehat{n}}_{\sigma }\left(t\right){\widehat{n}}_{\sigma }\left(0\right)\rangle -{\langle {\widehat{n}}_{\sigma }\rangle }^4$ for the two-orbital Hubbard model with $U=8, J=U/4, \beta =50$ and indicated fillings. Bottom left panel: spectral function ${\mathcal{A}}_{{\left(n_{\sigma }{n}_{\sigma }\right)}^2}\left(\omega \right)=-\frac{1}{\pi }\mathrm{Im}{C}_{{\left(n_{\sigma }{n}_{\sigma }\right)}^2}^R\left(\omega \right)$ multiplied by $coth\left(\frac{\beta \omega }{4}\right)$. In the inset, we plot the imaginary-time four-point function $\langle {\widehat{n}}_{\sigma }\left(\tau \right){\widehat{n}}_{\sigma }\left(0\right),{\widehat{n}}_{\sigma }\left(\tau \right){\widehat{n}}_{\sigma }\left(0\right)\rangle -{\langle {\widehat{n}}_{\sigma }\rangle }^4$. Bottom right panel: modulus of the OTOC $|\langle {\widehat{n}}_{\sigma }\left(t\right){\widehat{n}}_{\sigma }\left(0\right),{\widehat{n}}_{\sigma }\left(t\right){\widehat{n}}_{\sigma }\left(0\right)\rangle -{\langle {\widehat{n}}_{\sigma }\rangle }^4|$ (normalized at $t=0$) on a log-log scale. In the inset of the bottom right panel, we illustrate the approximate data collapse obtained by considering the spin-freezing crossover regimes for different values of $J$. In all the panels, the thick blue curves represent the results for the spin-freezing crossover region.

Figure 11

Left panel: the imaginary-time four-point function $\overline{\langle {\widehat{n}}_i\left(\tau \right){\widehat{n}}_i\left(0\right),{\widehat{n}}_i\left(\tau \right){\widehat{n}}_i\left(0\right)\rangle }$, the density-density correlation function $\overline{{\langle {\widehat{n}}_i\left(\tau \right){\widehat{n}}_i\left(0\right)\rangle }^2}$, and their difference for the SYK model with $N=12$ and $\beta J_{\mathrm{SYK}}=10$. The dashed curve shows a fit of the difference with a function $f_{\text{fit}}\left(\tau \right)=a+b\tau +c{\tau }^2$ in the interval $\tau /\beta \in [0:0.2]$. Right panel: fitting coefficient $c$ as a function of $T/J_{\mathrm{SYK}}$.

Figure 12

The density-density OTOC $\overline{\langle {\widehat{n}}_i\left(t\right){\widehat{n}}_i\left(0\right),{\widehat{n}}_i\left(t\right){\widehat{n}}_i\left(0\right)\rangle -{\langle {\widehat{n}}_i\rangle }^4}$ for the particle-hole symmetric SYK model averaged over ${10}^2$ samples with $\beta J_{\mathrm{SYK}}=100$ and various $N$. The top left and right panels show the real and imaginary parts of the OTOC, while the bottom left and right panels show the log and log-log plots of the modulus of the OTOC (normalized at $t=0$). The dashed line shows a power-law decay $\propto t^{-1.6}$ for comparison.

Figure 13

Temperature dependence of the density-density OTOC $\overline{\langle {\widehat{n}}_i\left(t\right){\widehat{n}}_i\left(0\right),{\widehat{n}}_i\left(t\right){\widehat{n}}_i\left(0\right)\rangle -{\langle {\widehat{n}}_i\rangle }^4}$ for the particle-hole symmetric SYK model averaged over ${10}^2$ samples with $N=12$. The top left and right panels show the real and imaginary parts of the OTOC, while the bottom left and right panels show the log and log-log plots of the modulus of the OTOC. The dashed line shows a power-law decay $\propto t^{-1.6}$ for comparison.

Figure 14

Results for the three-orbital Hubbard model with $U=8, J=U/4$, and $\beta =50$. Left panel: imaginary part of the self-energy as a function of Matsubara frequency ${\omega }_n$. The dashed line corresponds to $\sqrt{{\omega }_n}$. The spin-freezing crossover is located near $n_{\sigma }=0.220$. Right panel: the modulus of the OTOC $|\langle {\widehat{n}}_{\sigma }\left(t\right){\widehat{n}}_{\sigma }\left(0\right),{\widehat{n}}_{\sigma }\left(t\right){\widehat{n}}_{\sigma }\left(0\right)\rangle -{\langle {\widehat{n}}_{\sigma }\rangle }^4|$ (normalized at $t=0$) on a log-log scale. In the spin-freezing crossover region, there is an approximate power-law behavior $\sim 1/t^{1.75}$ indicated by the dashed line.