Reaching Sachdev-Ye-Kitaev physics by shaking the Hubbard model

Abstract

The Sachdev-Ye-Kitaev (SYK) model has attracted widespread attention due to its relevance to diverse areas of physics, such as high temperature superconductivity, black holes, and quantum chaos. The model is, however, extremely challenging to realize experimentally. In this work, we show how a particular form of Floquet engineering, termed ``kinetic driving'', effectively eliminates single-particle processes and creates quasi-random all-to-all interactions when applied to models of Hubbard type. For the specific case of the Bose-Hubbard model, we explicitly verify that the driven system indeed reproduces SYK physics by direct comparison of the spectral form factor and out-of-time ordered correlation functions (OTOCs). Our findings indicate that a cold-atom realization of kinetic driving -- achieved through modulation of hopping amplitudes in an optical lattice -- offers a practical and accurate platform for quantum simulation of the SYK model.

Figures (5)

• Figure 1: (a) Particles held in an optical lattice can be well described by the Hubbard model (Eq. \ref{['hubbard']}), with nearest neighbor hopping $J$ and onsite repulsion $U$. Shaking the lattice allows us to make $J$ a time periodic function At high driving frequencies this produces a static effective model with long-ranged interaction terms of the form $Q_{i j k l} a^\dagger_i a^\dagger_j a_k a_l$ describing processes like doublon hopping and assisted tunneling. (b) Schematic representation of the interaction amplitudes of the KDBH model (Eq. \ref{['effective_H']}) shaded according to their magnitude, $\left| Q_{i j k l} \right|$. The strongest links are between nearest neighbors, and their magnitude falls as the distance between the sites increases. Inset: The cumulative distribution of $Q_{i j k l}$ for $0.6 \leq \kappa \leq 0.9$, where $\kappa$ is sampled in steps of 0.01. The histogram is plotted on a logarithmic scale. The distribution is strongly peaked at small $Q$, and unlike the distribution of the SYK model, has gaps. (c) In the Bose-Hubbard model a given site $i$ is only connected to its nearest-neighbors by single-particle tunneling processes. (d) As in (b) but for the SYK model (Eq. \ref{['H_SYK']}). Every site is connected to every other site by a random tunneling amplitude. Inset: In contrast the the effective model, the distribution of the amplitudes $J_{i j; k l}$ is dense and is of Gaussian form, so that when plotted logarithmically the histogram is parabolic.
• Figure 2: Spectral form function (SFF) of the KDBH model for four different lattice sizes and bosons numbers; the notation $(N,L)$ denotes $N$ bosons on an $L$-site lattice. Multiplying the SFF by the dimension of the Hilbert space $D$, and scaling the time by $1/D$ collapses the curves to a universal form -- an initial drop, then a linear ramp, followed by a constant plateau. Each curve is labeled by the corresponding values of $D$, and we see the ramp behavior occurring earlier as $D$ increases. Inset - We define the ramp time as the time at which the scaled SFF falls to a value of 0.1. The red-dashed line drops as $D^{-1}$, and is included as a guide to the eye.
• Figure 3: (a) OTOCs for the BH model for 6 particles on a 6-site lattice with periodic boundary conditions. When the operators are separated by a single lattice spacing ($d=1$) the OTOC rapidly decays to its asymptotic value. For larger spacings, $d=2$ and $d=3$ there is a delay until the OTOC begins its decay, corresponding to information spreading through the lattice at a finite speed. (b) As in (a) but for the KDBH model, with $\kappa = 0.8$. The OTOCs show no sign of a time delay, beginning their decay immediately. This indicates the lack of locality in this model. (c) Gray curves indicate the OTOCs obtained for the KDBH model for $\kappa=0.6, 0.7, 0.8, 0.9$ for the three different separations. Their average, shown in black has the same form as the SYK OTOC, averaged over 25 random samples of $J_{i j;k l}$. To obtain quantitative agreement, the time coordinate of SYK data has been scaled by a factor of 9 (see text).
• Figure EM1: Distributions of the $Q$ amplitudes in an 8-site KDBH model for $\kappa = 0.6$. (a) Clean system: as seen previously, the distribution is non-Gaussian and contains several gaps. (b) Introducing a weak link in the parent BH Hamiltonian with $J = \left( J - \epsilon \right)$ produces a distribution much closer to the target Gaussian form.
• Figure EM2: (a) The SFF for the KDBH model ($\kappa = 0.8$ and $(N,L)=(11,11)$) follows the universal form: a sharp decay at the Thouless time, followed by a linear ramp before finally reaching a plateau. The gray curve shows the raw data, the black line is obtained by a 50-point running average to smoothen the oscillations. (b) In contrast, the SFF for the BH model ($U=0.2 J$ and $(12,12)$) does not have the universal form. Following the initial decay, the SFF first oscillates about a constant value, before going through the ramp-plateau behavior at a much later time. Red dashed lines indicate the value of $D^{-1}$ (where $D$ is the dimension of the Hilbert space); the blue dot-dashed line is a guide to the eye indicating the first plateau of the BH model.