Periodically Driven Sachdev-Ye-Kitaev Models

TL;DR

This paper investigates the heating and thermalization dynamics of a periodically driven generalized Sachdev-Ye-Kitaev model, revealing distinct behaviors in Fermi liquid and non-Fermi liquid regimes and discussing potential experimental implications.

Contribution

The study introduces an exact field theoretic approach to analyze heating in a driven SYK model, uncovering how quasi-particles influence thermalization timescales.

Findings

Non-Fermi liquids heat exponentially and thermalize rapidly.

Heavy Fermi liquids resist heating and thermalize slowly.

Distinct regimes of heating dynamics are identified.

Abstract

Periodically driven quantum matter can realize exotic dynamical phases. In order to understand how ubiquitous and robust these phases are, it is pertinent to investigate the heating dynamics of generic interacting quantum systems. Here we study the thermalization in a periodically-driven generalized Sachdev-Ye-Kitaev (SYK)-model, which realizes a crossover from a heavy Fermi liquid (FL) to a non-Fermi liquid (NFL) at a tunable energy scale. Developing an exact field theoretic approach, we determine two distinct regimes in the heating dynamics. While the NFL heats exponentially and thermalizes rapidly, we report that the presence of quasi-particles in the heavy FL obstructs heating and thermalization over comparatively long time scales. Prethermal high-frequency dynamics and possible experimental realizations of non-equilibrium SYK physics are discussed as well.

Equations

$$\hat{H}$$

$$\begin{array}{c}\text{fmfgraph}\ast (30,20)\text{fmfleft}i\text{fmfright}o\text{fmf}fermioni,o\end{array}=\begin{array}{c}\text{fmfgraph}\ast (30,20)\text{fmfleft}i\text{fmfright}o\text{fmf}fermion,fore=(0.7,,0.7,,0.7)i,o\end{array}+\begin{array}{c}\text{fmfgraph}(70,10)\text{fmfleft}i\text{fmfright}o\text{fmf}phantomi,v1,v2,v3,v4,o\text{fmf}dots,left=1.5,tension=0.01v2,v4\text{fmf}fermion,fore=(0.7,,0.7,,0.7),tension=3.5i,v2\text{fmf}fermion,tension=3.5v4,o\text{fmf}fermion,left,tension=1v2,v4,v2\text{fmf}fermionv2,v4\text{fmfdot}v2,v4\end{array}+\begin{array}{c}\text{fmfgraph}(70,20)\text{fmfleft}i\text{fmfright}o\text{fmf}phantomi,v1,v4,o\text{fmf}dots,left=1.,tension=0.2v1,v4\text{fmf}fermion,fore=(0.7,,0.7,,0.7),tension=2.i,v1\text{fmf}fermion,tension=2.v4,o\text{fmf}fermionv1,v4\text{fmfdot}v1,v4\end{array},$$

$$G(t,t^{\prime })$$

$$\Sigma (1,2)$$

$$ϵ(t)$$

$$=-i{\int }_{\mathcal{C}}d{t}^{\prime }\frac{J^2}{2}{G}^{\mathcal{C}}(t,t^{\prime }{)}^2{G}^{\mathcal{C}}(t^{\prime },t{)}^2+K^2{G}^{\mathcal{C}}(t,t^{\prime })G^{\mathcal{C}}(t^{\prime },t),$$

$$|{ϵ}_{\mathrm{SYK}}(t)|\sim e^{-\Gamma (a,\Omega )t}$$

$$G^K(t,\omega )=\tanh \left(\frac{{\beta }_{eff.}(t)\omega }{2}\right)iA(t,\omega ).$$

$$ϵ(t)=⟨\hat{H}{⟩}_{T_{\text{inst.}}(t)}.$$

$$H_{\mathrm{F}}$$

$${\hat{H}}_{\mathrm{F}}=$$

$$+\frac{-i}{\Omega }\frac{1}{\sqrt{2^3{N}^4}}\sum _{ijkl}{\tilde{J}}_{ijkl}{c}_i^{†}{c}_j^{†}{c}_k{c}_l+\mathcal{O}(1/{\Omega }^2),$$

$$G^{>}(t,t^{\prime })=-i⟨c(t)c^{†}(t^{\prime })⟩,G^{<}(t,t^{\prime })=i⟨c(t^{\prime }{)}^{†}c(t)⟩.$$

$$i\frac{d}{dt}{G}^{\lessgtr }(t,t^{\prime })={\int }_{t_0}^{max(t,t^{\prime })}d\tau {\Sigma }^R(t,\tau )G^{\lessgtr }(\tau ,t^{\prime })+{\Sigma }^{\lessgtr }(t,\tau )G^A(\tau ,t^{\prime }).$$

$$G^{\lessgtr }(T+\Delta t,t^{\prime })=G^{\lessgtr }(T,t^{\prime })-i\frac{\Delta t}{2}\left[{\int }_{-\infty }^{max(T,t^{\prime })}d{t}^{\prime \prime }{\Sigma }^R{G}^{\lessgtr }+{\Sigma }^{\lessgtr }{G}^A+{\int }_{-\infty }^{max(T+\Delta t,t^{\prime })}d{t}^{\prime \prime }{\Sigma }^R{G}^{\lessgtr }+{\Sigma }^{\lessgtr }{G}^A\right].$$

$${\hat{H}}_{\text{lab}}=-J\sum _{⟨ij⟩}{e}^{i\vec{A}(t)({\vec{r}}_i-{\vec{r}}_j)}{c}_i^{†}{c}_j+\frac{U}{2}\sum {\hat{n}}_{i}V(i-j){\hat{n}}_j,$$

$$V(t)=e^{-i{\vec{A}}_{light}(t)\sum {\vec{r}}_i{\hat{n}}_i}.$$

$$H_{\text{rot}}={\partial }_t\vec{A}(t)\sum {\vec{r}}_i\widehat{n_i}-J\sum _{⟨ij⟩}{e}^{i{\vec{A}}^{\prime }({\vec{r}}_i-{\vec{r}}_j)}{c}_i^{†}{c}_j+\frac{1}{2}\sum {\hat{n}}_{i}V(|i-j|){\hat{n}}_j,$$

$$H^{\text{LLL}}(t)=\frac{1}{2!}\sum _{ijkl=1}^N{J}_{ij;kl}{c}_i^{†}{c}_j^{†}{c}_k{c}_l+\sum _{\alpha \beta =1}^N{F}_{\alpha \beta }{c}_{\alpha }^{†}{c}_{\beta }+f(t)\sum _{\alpha \beta =1}^N{A}_{\alpha \beta }{c}_{\alpha }^{†}{c}_{\beta },$$

Full text

In this work, we propose that the generalized Sachde-Ye-Kitaev (SYK) model Sachdev and Ye (1993); Kitaev (2015) is a prototypical model for fast heating in periodically driven quantum systems. The SYK model was originally introduced as a solvable model for a non-Fermi liquid Sachdev and Ye (1993); Parcollet and Georges (1999). Recently, work initiated by Kitaev Kitaev (2015); Polchinski and Rosenhaus (2016); Maldacena and Stanford (2016); Bagrets Dmitry (2016) studied quantum dynamics, chaos, and information scrambling in the SYK model and showed that it saturates bounds for the operator growth in out-of-time ordered correlation functions Maldacena and Stanford (2016). Remarkably, generalizations of SYK models feature a crossover from a heavy Fermi liquid (FL) to a non-Fermi liquid (NFL) and can be solved exactly Song et al. (2017); Chen et al. (2017); Banerjee and Altman (2017). By periodically driving these models we study the heating dynamics of a heavy FL and a NFL in the thermodynamic limit; see Fig. 1 for an illustration. While we find the NFL to rapidly thermalize, the existence of well-defined quasiparticles dramatically slows down the full thermalization of the heavy FL. Furthermore, heating can be suppressed in both cases by driving with sufficiently high frequencies. The system then enters a prethermal regime characterized by an effective temperature which is stable over long times. We also explore a possible experimental realization of our system by measuring the absorption of a THz laser that impinges on an irregularly shaped and perforated graphene flake Chen et al. (2018).

**The Model.—**We focus on a generalization of the SYK model. It consists of $N$ sites with corresponding complex fermions $c_{i}$, $i\in\{0,\dots,N\}$ and is described by the following Hamiltonian:

[TABLE]

where $K_{ij}$ and $J_{ij;kl}$ are chosen to be random, uncorrelated all-to-all couplings which are drawn from Gaussian distributions with means $\overline{K_{ij}},\overline{J_{ij;kl}}=0$ and variances of $\overline{K_{ij}K_{ij}^{*}}=|K|^{2}$ and $\overline{J_{ij;kl}J_{ij;kl}^{*}}=|J|^{2}$, respectively. The line denotes a disorder average over realizations of $K_{ij}$ and $J_{ij;kl}$. We drive the system out of equilibrium by modulating the hopping term with a periodic drive $f(t)$ of period $\bar{T}=2\pi/\Omega$, see Fig. 1. In equilibrium and at zero hopping, $K_{ij}=0$, the single-particle correlation function exhibits a scaling solution $G(t)\sim 1/\sqrt{|{t}|}$ and the fermions acquire an anomalous dimension of $1/4$. In this regime quasiparticles do not exist and the density of states (DOS) diverges as $\sim 1/\sqrt{\omega}$, which is a strong NFL feature. However, in the thermodynamic limit any finite hopping $K_{ij}\neq 0$ constitutes a relevant perturbation, such that the low-energy physics is described by long-lived fermions. Their lifetime is of the usual Fermi liquid form and diverges for low temperatures as $\tau_{\mathrm{LT}}\sim 1/T^{2}$. Although the system is a FL at the lowest energies, it crosses over to the NFL scaling solutions above the crossover scale $E_{c}\sim K^{2}/J$ Song et al. (2017).

**Non-equilibrium formalism.—**Using an exact field-theoretic treatment, we solve for the heating dynamics of the SYK model (1) as it is periodically modulated in time with a sinusoidal drive of frequency $\Omega$ and amplitude $a$. In the thermodynamic limit $N\rightarrow\infty$ only a small subset of Feynman diagrams contributes, which can be resummed. The Kadanoff Baym equations (KBEs) that govern the evolution of the disorder-averaged Green’s function are then given by the following diagrams:

{fmffile}

sde_feynman

[TABLE]

where convolutions in time are performed over the Kadanoff-Baym round-trip contour $\mathcal{C}$ going to times infinity and back. Gray lines denote the free Green’s function, $G_{0}$; black lines the disorder averaged full Green’s function, $G(t,t^{\prime})=-i\overline{\langle\mathrm{T}_{\mathcal{C}}\{c_{i}(t)c_{j}^{\dagger}(t^{\prime})\}\rangle}$ (where $\mathrm{T}_{\mathcal{C}}$ denotes contour time ordering); and dotted lines represent disorder averages of $K_{ij}$ and $J_{ij;kl}$, respectively. The precise form of our KBEs is read off from the diagrams to be

[TABLE]

As initial conditions we fix the Green’s function in the lower quadrant of the two-time plane $(t,t^{\prime})$ with low-temperature initial states, which are determined by numerically iterating the equilibrium Schwinger-Dyson equations Maldacena and Stanford (2016); Babadi (2013); Eberlein et al. (2017).

We then switch on the drive at time $t=0$ and follow the evolution of the system. The resulting equations are integrated numerically employing a predictor-corrector scheme; see Supplemental Material for details sup . Following this formalism we obtain self-consistent and conserving solutions for the Green’s functions arbitrarily far from equilibrium.

Heating dynamics of a heavy Fermi liquid.—

We periodically drive both the NFL (also referred to as strange metal) which arises at low temperatures when the hopping is negligible, and the heavy FL, which is the relevant low-energy description for finite hopping $K_{ij}\neq 0$. Both cases are expected to be discernible from their short time dynamics, which will be dominated by the low-energy excitations that are vastly different in the two cases. We initially prepare the system in a low-temperature state $T\ll\mathrm{min}(J,K)$, and then compute the absorbed energy as a function of time from a diagrammatic expansion to leading order in $N$, {fmffile}energy_feynman

[TABLE]

where the first diagram accounts for interactions and the second for the kinetic energy.

We first study the heating of the strange metal phase, by setting $f(t)=a\sin(\Omega t)$ in Eq. 1, which corresponds to a drive with zero mean. The system is prepared at low temperatures $\beta J=50$, which realizes an approximately conformal state. Because of the strong scattering and the absence of quasiparticles the absorbed energy increases exponentially comparatively quickly after switching on the drive and remains so as it crosses over to the free ultraviolet theory, see Fig. 2 (a). This exponential heating is of the form

[TABLE]

and arises with a frequency and amplitude dependent rate $\Gamma(a,\Omega)$. The parameter dependence of $\Gamma(a,\Omega)$ is studied in the Supplemental Material for close to resonant driving. We find a universal scaling with the driving amplitude $\Gamma(a,\Omega)\sim a^{2}f(\Omega)$.

The heating of the heavy Fermi liquid can be richer. We again prepare the system in a low-temperature state with $\beta J=\beta K=50$ which lies deep within the FL regime. We drive the system by modulating the hopping term $K_{ij}$ periodically around a mean value, $f(t)=1+a\sin(\Omega t)$. In this case the drive commutes with the hopping, but not with the interaction. Initially, quasiparticles which are excited are long-lived, since $\tau_{\mathrm{FL}}\sim 1/T^{2}$. For low drive frequency $\Omega$, the system therefore exhibits two heating regimes: (I) subexponential heating at early times and (II) late time exponential heating. The crossover scale for the transition from slow (I) to fast (II) heating is tuned by $\Omega$, Fig. 2(b) and is set by $E_{c}\sim K^{2}/J$. While the absorption at short times depends on the nature of the drive, we find an exponential heating at late times as a universal property.

**Thermalization.—**To understand the heating of the system beyond the dynamics of the energy density, we also study observables that more directly probe the many-body state. In particular, we will evaluate generalized fluctuation dissipation relations (FDR) to extract effective temperatures. To this end, we transform the Green’s function to center of mass coordinates, $t=(t_{1}+t_{2})/2$, and relative coordinates, $\Delta t=t_{1}-t_{2}$, and Fourier transform the latter. Subsequently we compare the statistical or Keldysh component $G^{K}=-i\langle[c(t),c^{\dagger}(t^{\prime})]\rangle$ and the spectral components $A=\langle\{c(t),c^{\dagger}(t^{\prime})\}\rangle$ of the Green’s function as follows

[TABLE]

Provided that the FDRs are fulfilled at low energies, we can extract the time-dependent effective temperature $T_{\mathrm{eff.}}=1/\beta_{\mathrm{eff.}}$ to characterize the state.

We find the FDRs for the strange metal to be rapidly fulfilled for comparably weak driving, yielding an effective time dependent temperature $T_{\text{eff.}}(t)$ shortly after the drive is switched on. We can compare this temperature, with a thermal state evaluated at the instantaneous energy density via the self-consistent relation

[TABLE]

Comparing, the two temperatures $T_{\text{eff.}}(t)$ and $T_{\text{inst.}}(t)$ tells us how thermal the system is at a certain time $t$.

The strange metal shows almost instantaneous thermalization, since the two temperatures agree already at short times, see of Fig. 3(a) and the FDRs are fulfilled over a reasonably large energy regime (b). Thus the system quickly falls back to its equilibrium albeit at a temperature that increases with time. At late times the temperature grows exponentially as $T_{\mathrm{eff.}}^{\mathrm{SYK}}(t)\sim e^{\Gamma(a,\Omega)t}$. This is connected to the exponential energy absorption by the diverging specific heat $C_{V}^{\mathrm{SYK}}\sim 1/T^{2}+\mathcal{O}(1/T^{4})$.

By contrast, the presence of quasiparticles in the heavy Fermi liquid obstructs thermalization on short timescales and FDRs are fulfilled only for small energies, see Fig. 3(d), where the FDRs exhibit kinks at low frequencies which are reminiscent of the low-temperature initial state. Extracting an effective temperature scale from the low-energy behavior shows that for long time spans the two different notions of temperature strongly disagree, Fig. 3 (c). Only at late times the system thermalizes and well-defined FDRs emerge, indicating that the system is in thermal equilibrium.

For weak driving we can estimate the emergent temperature scale $T^{*}$ at which the system thermalizes by comparing the dissipative response $\Gamma_{E}\sim a^{2}\Omega\chi^{\prime\prime}(\Omega)$, where $\chi^{\prime\prime}(\Omega)$ is the imaginary part of the kinetic energy correlator, with the decay rate of excitations in the heavy FL $1/\tau_{\text{LT}}\sim T^{2}$, which gives $T^{*}\sim a\sqrt{\Omega\chi^{\prime\prime}(\Omega)}$. The effective thermalization timescale is then set by $t^{*}\sim 1/T^{*}$ and is thus inversely proportional to the driving strength. The linear scaling of $T^{*}$ is confirmed by our exact numerical simulations in the Supplemental Material. By contrast in the strange metal the inverse life-time scales as $1/\tau^{\text{NFL}}_{\text{LT}}\sim T$, leading to a $T^{*}\sim a^{2}\Omega\chi^{\prime\prime}(\Omega)$ that is strongly suppressed at small drive amplitudes. The temperature of our initial state is therefore already of the same order of magnitude as $T^{*}$ supporting the rapid thermalization of the NFL. Since our model of Eq. 1 is rather generic, we expect that slow thermalization can also be present in more complex physical systems with weakly interacting quasiparticles, providing an alternative route of controlling heating dynamics in Floquet systems.

High frequency dynamics.—

So far, we have considered drive frequencies that are slow compared to the bandwidth of the couplings. Heating can be slowed down by increasing the driving frequency $\Omega$. The intermediate-time, stroboscopic, evolution of the system is governed by the Floquet Hamiltonian, $H_{\mathrm{F}}:=i\ln[U(\bar{T})]/\bar{T}$, where $U(T)$ is the unitary time evolution operator over one driving period. $H_{\mathrm{F}}$ can be determined perturbatively in the inverse frequency, $1/\Omega$, of the drive using the Magnus expansion (ME)

[TABLE]

with $H_{\mathrm{F}}^{0}=\frac{1}{\bar{T}}\int_{0}^{\bar{T}}H(t)$ and $H_{\mathrm{F}}^{1}=\frac{1}{2!\bar{T}i}\int_{0}^{\bar{T}}dt_{1}\int_{t_{0}}^{t_{1}}dt_{2}[H(t_{1}),H(t_{2})]$. This effective high-frequency Hamiltonian is conserved up to exponentially long times $t^{*}\sim e^{\mathcal{O}(\Omega/J)}$ in $\Omega$. For times $t\ll t^{*}$ the system is therefore expected to be approximately static and well described by the ME Hamiltonian.

We find that the fast drive simply renormalizes the coupling $J_{ij;kl}$ of the SYK model. To first order, the ME yields the new effective Hamiltonian:

[TABLE]

where $\tilde{J}_{ijkl}$ denote the interactions induced by the drive, which are evaluated by explicitly using commutation relations. We find the following couplings $\tilde{J}_{ijkl}=2\sum_{\mu}K_{i\mu}J_{\mu jkl}-2\sum_{\mu}J_{ijk\mu}K_{\mu l}$. To this order these couplings can be treated as independent from the original $J_{ijkl}$ due to disorder averages over $K_{ij}$. As the $\tilde{J}_{ijkl}$ are sums over products of Gaussian distributed variables, $\tilde{J}_{ijkl}$ are again Gaussian distributed, by the central limit theorem with mean $\overline{\tilde{J}_{ij;kl}}=0$ and variance $\overline{\tilde{J}_{ij;kl}\tilde{J}_{ij;kl}^{*}}=4|J|^{2}|K|^{2}$.

Starting from a low-temperature state $\beta J=50$ and switching on a rapidly oscillating drive, we find that the dynamics is divided into two regimes (Fig. 4): (I) At short times, the rapid drive acts as a perturbation that rearranges the many-body state and leads to some finite energy absorption. The stroboscopic dynamics is approximately described by a quantum quench to the Floquet Hamiltonian, $\hat{H}_{\mathrm{F}}$. (II) After that the energy absorption is strongly suppressed and the system enters a prethermal regime. By solving for the exact dynamics we find that the system respects effective FDRs, see Fig. 4 (a) inset. In the prethermal phase the system realizes a NFL with a rapid relaxation rate given by $1/\tau_{\text{NFL}}\sim T$. These fast scattering processes have to be compared to the weak absorption, which we find to be superexponentially suppressed in the driving frequency (see Supplemental Material). This separation of scales guarantees the emergence of long-lived prethermal dynamics and approximate FDRs. We have confirmed that the effective temperature increases steadily, albeit at a strongly suppressed rate. Although the system slowly heats to infinite temperature, it remains trapped in a quasistatic strongly correlated state for long times and evolves according to the effective Floquet Hamiltonian.

**Graphene flakes.—**Graphene flakes with sufficiently disordered boundaries, in strong background magnetic fields have recently been proposed to realize exotic SYK physics Chen et al. (2018). Here, we generalize these ideas to include driving by classical laser light that impinges onto the graphene flake; see Supplemental Material sup . This leads to a periodically driven SYK Hamiltonian as in Eq. (1), where $N$ is proportional to the number of zero modes in the lowest Landau level which scales with the flux through the sample. As the strange metal behavior appears for $N\gg 1$, high magnetic fields are required. The flux can, however, also be increased by considering larger flakes that contain irregularly shaped holes in their interior ata , allowing one to obtain the same number of modes $N$ in the lowest Landau level albeit at smaller applied magnetic fields. The light-induced driving term then dissipates energy in the system and leads to heating, which can, for example, be measured by transmission spectroscopy. To qualitatively predict the behavior of the dynamics, one needs to study the disorder parameters for realized flake geometries.

**Conclusions and Outlook.—**We studied the heating dynamics in periodically driven heavy Fermi liquids and strange metals. We focus on generalized SYK models, which remain exactly solvable using non-equilibrium field theory. These systems absorb energy exponentially quickly at late times, as they approach their infinite temperature states. While the strange metal thermalizes almost instantaneously, the heavy Fermi liquid requires a comparatively long time to equilibrate. We argue that this behavior is a result of relatively weakly interacting quasiparticle excitations whose lifetime scales as $1/T^{2}$ at low temperatures. This effect provides an alternative route to suppress heating in more complex periodically driven systems in the future. We also report a prethermal regime for high-frequency drives and propose experimental opportunities to probe non-equilibrium SYK physics irradiating irregularly shaped graphene flakes.

Our approach can be readily generalized to higher-dimensional SYK systems with potentially topologically nontrivial band structures. Such models would offer new routes to explore the interplay of transport topology, periodic driving and strong correlations.

S0.1 Schwinger Dyson equations on the Kelydsh contour

We obtain the non-equilibrium correlation functions of our system by solving the exact Schwinger Dyson equations on a Keldysh contour. Here we sketch the iterative numerical solution to the equations. The Schwinger Dyson equations in real time are a set of integro-differential equations, which are derived by acting with the free inverse Green’s function $G_{0}^{\mathcal{C}}(t,t^{\prime})^{-1}=i\partial_{t}i\delta^{\mathcal{C}}(t-t^{\prime})$ in Eq. (2). Out of equilibrium one defines the greater and lesser component of the Green’s function to be

[TABLE]

Contour functions and their convolutions can be decomposed into these components by applying the Langreth rules Rammer and Smith (1986). This decomposition leads to the following evolution equations for the correlation functions

[TABLE]

We integrate Eq. S2 using the trapezoid rule and obtain the prescription

[TABLE]

As both sides explicitly depend on $G^{\lessgtr}(T+\Delta t)$, we employ a predictor corrector scheme following Ref. Babadi et al., 2015. Step by step the procedure is implemented as follows:

1. (I)

Start with some initial Green’s functions $G^{\lessgtr}(t_{1},t_{2})$ defined on a subset of the $t_{1}-t_{2}$ plane $\{(t_{1},t_{2})|-\infty<t_{1,2}\leq T\}$ which characterizes the (thermal) initial state 2. (II)

Predict the entries of the Green’s functions $G^{\lessgtr}_{p}(T+\Delta t)$ along the line $t^{\prime}\in(-\infty,T]$ for some arbitrary quench protocol by setting $G^{\lessgtr}(T+\Delta t)\approx G^{\lessgtr}(T,t^{\prime})$ on the right hand side of Eq. S3 3. (III)

Calculate the predicted self energies $\Sigma^{\lessgtr}_{p}=\Sigma^{\lessgtr}[G_{p}(T+\Delta t,t^{\prime})]$ 4. (IV)

Determine the corrected entries of the Green’s functions $G^{\lessgtr}_{c}(T+\Delta t)$ by setting $G^{\lessgtr}(T+\Delta t)=G^{\lessgtr}_{p}(T+\Delta t)$ on the right hand side of the evolution equation.

This procedure is iterated until $|G^{\lessgtr}_{c}-G^{\lessgtr}_{p}|<\delta$ converged below some tolerance. $\delta$, $G^{\lessgtr}(T+\delta t)$ is then set equal to $G^{\lessgtr}_{c}$ and we continue by predicting the next time slice. For our purposes we chose a typical grid-size of the $t_{1}-t_{2}$ plane of $8000\times 8000$ and set $\delta=10^{-8}$. The initial conditions are found by iterating the SDE in frequency space Maldacena and Stanford (2016); Eberlein et al. (2017).

S0.2 Graphene flakes

Here we study the effect of laser light impinging on an irregularly shaped graphene flake by following the construction of Ref. Chen et al., 2018. As discussed in the main text, such a setup may provide an experimental realization of the model in Eq. (1).

The classical light impinging on the flake (see Fig. S1) is modelled by a Peierls substitution in the underlying tight-binding model. Introducing a screened Coulomb interaction between the electrons, the Hamiltonian of the graphene flake takes the form

[TABLE]

where $\vec{A}=(\cos(\Omega t)-1/2By,\cos(\Omega t)+1/2Bx,0)^{T}$ is the vector potential for the light field with frequency $\Omega$ and $B$ is the background magnetic field. $V(i-j)$ is the potential for density-density interactions. It is advantageous to remove the light-induced hopping amplitude in favour of a time-dependent modulation of the local density. As in Ref. Bukov et al. (2015) this is achieved by transforming to a rotating frame by means of a unitary

[TABLE]

In the rotating frame all expectation values remain unchanged and the time-evolution is generated by the transformed Hamiltonian

[TABLE]

now with a constant vector potential $\vec{A}^{\prime}=(-1/2By,1/2Bx,0)^{T}$. By projecting out all but the lowest Landau level (LLL) and by switching to the eigenbasis of the non-interacting problem, the flake is mapped onto a charged SYK Hamiltonian

[TABLE]

where $J_{ij;kl}$, $A_{\alpha,\beta}$ and $F_{\alpha\beta}$ are random all-to-all couplings. They are approximately gaussian distributed, with zero mean and variances $\overline{|J_{ij;kl}|^{2}}=\frac{J^{2}}{N^{3}}$, $\overline{|A_{\alpha\beta}|^{2}}=\frac{A^{2}}{N}$ and $\overline{|F_{\alpha\beta}|^{2}}=\frac{F^{2}}{N}$ Chen et al. (2018). $N$ is proportional to the number of zero modes in the LLL and scales with the flux through the sample. The strange metal behavior appears as $N\gg 1$ such that large magnetic fields are necessary. As discussed in the main text, the flux may also be increased by considering larger flakes which contain disordered interiors ata . The light-induced driving term dissipates energy in the system and leads to heating. This may realize an experimental probe of non-equilibrium dynamics in the SYK model. The energy density is accessible by transmission spectroscopy. The heating dynamics could indicate whether the system realizes a clean SYK model, or whether quadratic perturbations dominate the dynamics. While the phases are always distinguishable by studying the thermal properties of the system, FL and NFL cannot be discerned by studying the energy absorption alone if drive and hopping terms would turn out to be completely uncorrelated. In this case both phases would heat exponentially. As the correlations between $F_{ij}$ and $A_{ij}$ determine the functional form of the heating, they need to be analyzed on a concrete flake geometry.

S0.3 Thermalization timescale

Floquet engineering exotic phases of matter often relies on the equilibration of the modulated system, as the dynamics of periodically driven many-body systems can be complex. This thermalization is well understood in the asymptotic limit of fast driving, where the system is effectively static and heating is strongly suppressed. Alternatively, when a system relaxes sufficiently quickly it remains close to an equilibrium state at all times. Such approximations can be justified above a crossover temperature $T^{*}$, which emerges dynamically and determines the onset of thermalization in Floquet systems Genske and Rosch (2015). Here we analyze this scale in detail for a heavy Fermi liquid. Our numerics show a linear dependence of $T^{*}$ on the driving amplitude $a$ as long as $a$ is small, Fig. S2. The scale $T^{*}$ is determined as the lowest temperature of the driven system at which well-defined FDRs are found. As discussed in the main text, this result is explained as a competition of relaxation and energy dissipation. Generically energy is dissipated at a rate $1/\tau_{\text{diss.}}\sim\Omega f(\Omega)a^{2}$, while scattering of quasiparticles scales as $1/\tau_{\text{scatter}}\sim T^{2}$. In Fermi liquids one therefore finds $T^{*}\sim a$. As scattering in NFL’s is a much faster process which scales $\sim T$, one finds $T^{*}_{\text{NFL}}\sim a^{2}$, which is consistent with the rapid thermalization of the NFL phase found in the main text. Corrections to the linear behavior arise due to non-linear absorption when $a$ becomes large, and once the relaxation rate depends significantly on the frequency of the excitations induced by the drive.

S0.4 Heating rates in linear response theory

The the NFL phase absorbs energy exponentially $\epsilon(t)\sim\exp(-\Gamma t)$ for a wide regime of drives, where $\Gamma(\Omega,a)$ is a drive-dependent rate. In these cases the heating is almost independent on the initial state, despite being a dynamical property of a strongly interacting system. In the following we study $\Gamma(\Omega,a)$ as a function of drive parameters and demonstrate a universal scaling of $\Gamma\sim a^{2}$. Fig. S3 shows the collapse of rates after rescaling $\Gamma(\Omega,a)$ by $a^{-2}$. The frequency dependence is fixed by a non-universal function $f(\Omega)$. We estimate the rates within a linear response analysis around a high-temperature state and compare to the numerically exact results. To leading order in the driving strength, the absorbed energy per driving cycle is then given by the dissipative part of the response function: $\overline{\mathrm{d}E/\mathrm{d}t}=a^{2}\Omega\,\chi^{\prime\prime}(\Omega)$, which is determined by the commutator $D(t,t^{\prime})=-i\Theta(t-t^{\prime})\langle\psi|[H(t),H_{\text{drive}}(t^{\prime})]|\psi\rangle$. In the SYK model the susceptibility factorizes into equilibrium two-point functions $D(t)\sim G(t)G(-t)$ to leading order in $N$, which we determine by numerically iterating the SDEs. We find that linear response theory accurately describes the heating of the NFL phase. For fast drives $\Omega\gg J$, the system is quenched and rests in a long-lived prethermal NFL state. This regime emerges due to strongly suppressed heating and will not be captured by an expansion around a high temperature state. Accordingly, our linear response analysis shows a super-exponential suppression of $\Gamma$ in this regime. This result falls well within exponential bounds on the energy absorption $\dot{E}/N\leq\,Ce^{-|\kappa|\Omega}$, where $C$ and $\kappa$ are system dependent constants Abanin et al. (2015); Mori et al. (2016).