Transport and chaos in lattice Sachdev-Ye-Kitaev models

TL;DR

The paper investigates transport and chaos in a lattice of Sachdev-Ye-Kitaev islands connected by one-body hopping and coupled to many local phonons, focusing on linear-$T$ resistivity and the spread of chaos. It combines SYK-based many-body dynamics with Keldysh/Bethe-Salpeter approaches to extract OTOCs, Lyapunov exponents, and butterfly velocities, uncovering two regimes: a weak electron-phonon coupling incoherent metal with near-maximal chaos and chaos diffusion closely tied to the energy diffusion $D_E$, and a strong-coupling regime where electron–phonon scattering is nearly elastic and chaos propagates diffusively, far from maximal. The study shows that long-distance chaos diffusion $D_{\rm chaos}$ tracks $D_E$ across regimes, while short-distance scrambling $D_*$ behaves differently, and it reveals how phonons modify the linear-$T$ resistivity, induce phonon drag in thermal transport, and cause substantial violations of Wiedemann–Franz law. Using Keldysh formalism and ladder resummations, the work identifies scaling relations controlled by the dimensionless parameters $gt_0/U$ and $T/E_c$, providing a unified picture of transport and chaos in strongly interacting electron–phonon systems with SYK-like physics. The results offer insights into how chaos imposes hydrodynamic-like constraints on transport in non-Fermi liquids and suggest experimental signatures in resistivity, optical conductivity, and thermal transport in strongly correlated materials.

Abstract

We compute the transport and chaos properties of lattices of quantum Sachdev-Ye-Kitaev islands coupled by single fermion hopping, and with the islands coupled to a large number of local, low energy phonons. We find two distinct regimes of linear-in-temperature ($T$) resistivity, and describe the crossover between them. When the electron-phonon coupling is weak, we obtain the `incoherent metal' regime, where there is near-maximal chaos with front propagation at a butterfly velocity $v_B$, and the associated diffusivity $D_{\rm chaos} = v_B^2/(2 πT)$ closely tracks the energy diffusivity. On the other hand, when the electron-phonon coupling is strong, and the linear resistivity is largely due to near-elastic scattering of electrons off nearly free phonons, we find that the chaos is far from maximal and spreads diffusively. We also describe the crossovers to low $T$ regimes where the electronic quasiparticles are well defined.

Figures (26)

• Figure 1: Schematic of the lattice of SYK islands, each with $N$ orbitals with two-body interaction $U$. The islands are coupled with one-body hopping $t_0$.
• Figure 2: Crossovers as a function $T$ for $gt_0/U \ll 1$ and $gt_0/U \gg 1$, where $g$ is a dimensionless measure of the electron-phonon coupling. The two chaos velocities $v_\ast$ and $v_B$ are defined as in Ref. Gu2. The resistivity is $\rho$ (in units of $h/e^2$), the thermal conductivity is $\kappa$ (in units of $k_B^2 T/\hbar$), and the thermal diffusivity is $D_E$. The chaos exponent $\lambda_L$, and the diffusivities $D_{\rm chaos}$ and $D_\ast$ are defined in Section \ref{['sec:scrambling']}. There is near-maximal chaos and front propagation with velocity $v_B$ only in the "incoherent metal" regime which has $v_\ast < v_B$. The other regimes have $v_\ast > v_B$ and diffusive propagation of far-from-maximal chaos. Here $\kappa_{\rm phonon}$ refers to the phonon drag correction, discussed in Section \ref{['sec:phonondrag']}. The values above do not include the saturation effects discussed in Section \ref{['sec:saturation']}.
• Figure 3: Chaos crossovers in spacetime for $v_\ast < v_B$, adapted from Ref. Gu2. In region A, the OTOC is diffusive, see \ref{['eq:OTOC_diffusive']}. In region B, the OTOC shows a wave-front propagation as well as maximal chaos, see \ref{['eq:OTOCfront']}. In region C, the OTOC does not grow. In regiond D, the OTOC has saturated.
• Figure 4: Schematic behavior of Lyapunov exponent $\lambda_L(q)$ on the imaginary $q$-axis at strong and weak interaction respectively (from Ref. Gu2). The pole in the prefactor of the OTOC sits at $q_1$, where $\lambda_L(q_1)=2\pi T$. The butterfly velocity $v_B$\ref{['OTOC3']} is the slope of blue lines. The threshold velocity $v_*$ is the tangent slope (red lines) of $\lambda_L(q)$ at $q_1$. At strong interaction (a) $v_*<v_B$ and at weak interaction (b) $v_*>v_B$.
• Figure 5: The time contour $C$ for calculating OTOC. The contour is drawn such that the real time goes to the left, which is convenient when acting by operators on left.