Floquet dynamical chiral spin liquid at finite frequency

Chiral Spin Liquids (CSL) are quantum spin analogs of electronic Fractional Chern Insulators. Their realizations on ultracold-atom or Rydberg-atom platforms remain very challenging. Recently, a setup of time-periodic modulations of nearest-neighbor Heisenberg couplings applied on an initial genuine spin liquid state on the square lattice has been proposed to stabilize a (Abelian) $\mathbb{Z}_2$ CSL phase. In the high-frequency limit, it was shown that time evolution can be described in terms of a static effective chiral Hamiltonian. Here we revisit this proposal and consider drives at lower frequency in a regime where the high-frequency Magnus expansion fails. We show that a Dynamical CSL (DCSL) is nevertheless stabilized in a finite range of frequency. The topological nature of this dynamical phase, as well as its instability below a critical frequency, is connected to specific features of the Floquet pseudo-energy spectrum. We also show that the DCSL can be represented faithfully by a two-dimensional time-periodic tensor network and, as in the static case, topological order is associated to a tensor gauge symmetry ($\mathbb{Z}_2$ in that case).

💡 Research Summary

The authors revisit a Floquet‑engineering scheme originally proposed to create an Abelian ℤ₂ chiral spin liquid (CSL) on a square lattice by periodically modulating nearest‑neighbor Heisenberg couplings. In the high‑frequency limit (ω ≫ J) the dynamics can be captured by a static effective Hamiltonian obtained from a Magnus expansion; this Hamiltonian contains a four‑site cyclic permutation term that generates a chiral interaction and stabilizes a CSL with ℤ₂ topological order.

The central question addressed here is whether the CSL survives when the drive frequency is lowered to a regime where the Magnus expansion no longer converges. The authors consider a sinusoidal drive H_drive(t)=cos(ωt) Hₓ + sin(ωt) Hᵧ acting on one sublattice, together with a smooth ramp λ(t) that turns the drive on adiabatically. They perform exact (up to negligible Trotter error) time‑evolution simulations on a 4 × 4 torus (N = 16 spins) in the total‑Sᶻ = 0 sector (dimension ≈ 1.3 × 10⁴).

A simple bandwidth estimate shows that the Floquet quasi‑energy bandwidth scales as W_N(ω) ≈ a N J²/(4ω) with a ≈ 2.4. When this bandwidth reaches the Floquet Brillouin‑zone width ω, the quasi‑energy gap closes, leading to heating and quantum chaotic dynamics. For the 4 × 4 system the critical frequency is found numerically to be ω_c ≈ 2.5–3 J.

For ω ≥ ω_c the system reaches a stationary regime after the ramp. The evolved state can be written as a superposition of two momentum sectors, \