Sachdev-Ye-Kitaev physics from the Hubbard model: A Floquet engineering approach

This paper demonstrates that applying a "kinetic driving" Floquet engineering technique to the Hubbard model, specifically the Bose-Hubbard model, effectively suppresses single-particle processes to generate quasi-random all-to-all interactions, thereby enabling a practical cold-atom quantum simulation of Sachdev-Ye-Kitaev (SYK) physics.

The Gist

The Big Picture: Building a "Quantum Black Hole" in a Lab

Imagine you want to study the physics of a black hole or a strange metal (a material that conducts electricity in weird ways). Physicists have a mathematical recipe for this called the SYK model. It's famous because it describes a world where particles don't just bump into their neighbors; they interact with everyone in the room at once, in a completely random and chaotic way.

The problem? Building this in a real lab is incredibly hard. It's like trying to build a house where every single brick is glued to every other brick in the building, not just the ones next to it. Real materials usually only interact with their immediate neighbors.

The Solution: The authors of this paper found a clever trick using light and shaking to force a simple system of atoms to act like this complex, chaotic black hole model.

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The Ingredients: The "Hubbard Model" (The Starting Point)

Think of the starting point as a crowded dance floor (an optical lattice) where particles (dancers) are trapped.

• The Rules: Normally, a dancer can only move to the spot immediately next to them (hopping). They can also bump into the person standing on the same spot (repulsion).
• The Goal: We want to stop the "hopping" to the next spot and instead make every dancer interact with every other dancer on the floor, randomly and all at once.

The Magic Trick: "Kinetic Driving" (Shaking the Floor)

The authors propose a method called "Kinetic Driving." Imagine you are on that dance floor, but instead of just standing there, you start shaking the entire floor back and forth very, very fast.

1. The "Cancel Out" Effect: You shake the floor so fast and in such a specific rhythm that, on average, the dancers can't actually move to the next spot. It's like trying to walk forward on a treadmill that is moving backward at the exact same speed; you stay in place. This effectively erases the "hopping" between neighbors.
2. The "Ghost" Interactions: Even though the dancers can't move, the shaking creates a strange side effect. Because the floor is vibrating, the dancers start "feeling" each other across the room. The shaking creates invisible, random bridges connecting every single dancer to every other dancer.

The paper calls this new, shaken system the KDBH model (Kinetically Driven Bose-Hubbard).

The Proof: Does it Actually Work?

The authors didn't just guess; they did the math and ran computer simulations to see if their "shaken dance floor" actually behaved like the theoretical "black hole" model (SYK). They looked at three specific things:

1. The Chaos Test (Spectral Form Factor):

   • Analogy: Imagine listening to the music of the dance floor. In a normal room, the notes are predictable. In a chaotic black hole room, the notes are a messy, random jumble that follows a very specific statistical pattern.
   • Result: The shaken system produced that exact same "messy but patterned" sound. It was chaotic in the right way.

2. The Speed of Information (OTOCs):

   • Analogy: If you whisper a secret to one dancer, how fast does the whole room know?
   • Normal Room: The whisper travels slowly, person to person, like a wave. It takes time to reach the back of the room.
   • Black Hole Room: The whisper is instantly heard by everyone. There is no "travel time" because everyone is connected.
   • Result: In their shaken system, the "whisper" spread instantly. There was no delay, proving the system had lost its "local" boundaries and became fully connected, just like the SYK model.

3. The "Sparse" Connection:

   • Analogy: In a perfect SYK model, everyone is connected to everyone. In the shaken system, the connections are a bit "sparse" (some links are weaker or missing), like a social network where you have many friends, but not every friend of a friend is your friend.
   • Result: The authors found that even with these missing links, the system still behaved exactly like the perfect black hole model. It was robust enough to handle the gaps.

The Conclusion

The paper concludes that by simply shaking an optical lattice (a grid of light holding atoms), scientists can turn a simple, local system into a complex, chaotic system that mimics the physics of black holes and strange metals.

• For Bosons (particles that like to clump together): They proved this works perfectly.
• For Fermions (particles that avoid each other): They showed the math works the same way, so it should work for them too.

In short: You don't need to build a black hole to study one. You just need a box of atoms, a laser, and a very fast, very precise shake. The shaking creates a "virtual" world where the rules of the universe are rewritten to be chaotic and all-connected.

Technical Summary

Technical Summary: Sachdev-Ye-Kitaev physics from the Hubbard model: A Floquet engineering approach

Problem Statement
The Sachdev-Ye-Kitaev (SYK) model is a paradigmatic system for studying non-Fermi liquid physics, quantum chaos, and holographic duality (black holes). It is characterized by random, all-to-all interactions between particles with no single-particle hopping. Despite its theoretical importance, experimental realization is extremely challenging due to the requirement for long-range, random interactions. While solid-state proposals (e.g., graphene flakes, Majorana modes) and digital quantum simulations exist, they face significant implementation hurdles. There is a need for a practical analog platform, particularly within ultracold atom systems, that can effectively engineer the specific interaction structure of the SYK model without requiring complex long-range coupling hardware.

Methodology
The authors propose a "kinetic driving" approach, a specific form of Floquet engineering applied to Hubbard-type lattice models. The core mechanism involves periodically modulating the kinetic energy term (hopping amplitude $J$) of a Bose-Hubbard (BH) model such that its time-average over a driving period vanishes. Specifically, the hopping is modulated as $J(t) = J_0 \cos(\omega t)$ in the high-frequency regime ($\omega \gg U$, where $U$ is the onsite repulsion).

Using the Magnus expansion (lowest-order term), the authors derive an effective static Hamiltonian ($H_{KDBH}$) for the driven system. In this limit, the single-particle hopping processes are eliminated, and the effective Hamiltonian reduces to a sum of all-to-all quartic interaction terms:
$$H_{KDBH} = U \sum_{i,j,k,l} Q_{ijkl} b^\dagger_i b^\dagger_j b_k b_l$$
where the coupling amplitudes $Q_{ijkl}$ are determined by the driving parameters (specifically the ratio $\kappa = J_0/\omega$) and Bessel functions. The authors explicitly verify that this method applies to both bosonic and spinless fermionic systems, though their numerical analysis focuses on the Bose-Hubbard case.

Key Contributions and Results
The paper validates that the kinetically driven Bose-Hubbard (KDBH) model reproduces the essential physics of the bosonic SYK model through three primary diagnostics:

1. Spectral Statistics and Random Matrix Theory (RMT):
   The authors confirm that for driving parameters $\kappa > 0.4$, the spectral statistics of the KDBH model follow the Gaussian Orthogonal Ensemble (GOE), consistent with the bosonic SYK model. This indicates the presence of quantum chaos.

2. Spectral Form Factor (SFF):
   A critical test for holographic behavior is the universal "drop-ramp-plateau" structure of the SFF. The authors demonstrate that the KDBH model exhibits this structure, unlike the standard undriven Bose-Hubbard model (which shows chaos in spectral statistics but lacks the universal SFF form). The ramp time in the KDBH model scales as $1/D$ (where $D$ is the Hilbert space dimension), matching SYK expectations. This confirms that the system possesses the necessary spectral rigidity and long-time correlations associated with SYK physics.

3. Out-of-Time Ordered Correlation Functions (OTOCs):
   The OTOC measures information scrambling and locality.

   • Standard BH Model: Exhibits spatial locality; information spreads with a finite "butterfly velocity," resulting in a time delay before decay for distant operators.
   • KDBH Model: Shows no time delay for operators at different separations; the decay is immediate and independent of distance. This indicates a lack of spatial locality and rapid scrambling, characteristic of the all-to-all connectivity in the SYK model.
   • Comparison: The averaged OTOC of the KDBH model (over different $\kappa$ values) aligns quantitatively with the SYK model, requiring only a rescaling of the time axis to account for the difference in energy scales between the Gaussian distribution of SYK couplings and the non-Gaussian distribution of KDBH couplings.

Addressing Model Differences
The authors acknowledge that the KDBH model is not a perfect replica of the ideal SYK model. The coupling amplitudes $Q_{ijkl}$ are not random Gaussian variables but depend deterministically on driving parameters, and the distribution is sparse (with gaps) rather than dense. However, the paper argues that the sparsity (approximately 50% non-zero elements due to symmetry) and the non-Gaussian nature of the couplings do not prevent the system from emulating SYK physics with high fidelity, consistent with previous studies on sparse SYK models.

Significance and Claims
The paper claims to offer a practical and accurate platform for the quantum simulation of the SYK model using existing cold-atom technology. The proposed scheme relies on "shaking" an optical lattice, a technique already demonstrated in experiments to control tunneling and induce synthetic gauge fields.

• Experimental Feasibility: The authors propose that kinetic driving can be implemented by modulating the lattice shaking strength $A(t)$ at a low frequency $\omega_2$ near the root of a Bessel function (where the effective hopping $J_{eff} \approx 0$), while maintaining a high-frequency drive $\omega_1$.
• Robustness: The study shows that heating effects (typically a concern in Floquet systems) do not manifest on the time scales required to probe SYK physics in the high-frequency regime. Furthermore, the introduction of a weak link to break translational symmetry refines the distribution of couplings closer to the Gaussian form, suggesting a pathway to improve experimental fidelity.
• Scope: The work establishes that the bosonic SYK model, which has been less explored than its fermionic counterpart, can be effectively realized and studied using this method. The approach is generalizable to fermionic systems as well.

In summary, the paper demonstrates that a simple periodic modulation of kinetic energy in a Hubbard model can effectively eliminate single-particle dynamics and generate the complex, all-to-all interactions necessary to simulate SYK physics, providing a viable route for experimental realization in optical lattices.