Detuning the Floquet anomalous chiral spin liquid

TL;DR

The work analyzes how frequency detuning affects Swap-Model realizations of Floquet anomalous chiral spin liquids on a square lattice. By examining the quasi-energy and average-energy spectra, edge spectroscopy, and a geometrical winding number, the authors identify finite-size, intermediate, and heating regimes as detuning grows, and demonstrate that the anomalous CSL is not smoothly connected to the high-frequency CSL, suggesting possible prethermal windows and a chaotic crossover due to resonances. They further connect the high-frequency limit to known CSL realizations via an effective Hamiltonian and provide an interpolation framework showing how the system transitions across frequency scales. The study advances understanding of dynamical topological phases in interacting many-body systems and highlights the intricate role of resonances and boundaries in Floquet engineering.

Abstract

At high-frequency a periodically-driven quantum spin-1/2 system can emulate a chiral spin liquid (CSL) described by an effective static local chiral hamiltonian. In contrast, at low-frequency {\it anomalous} CSL can be realized in Swap Models, in which one-way spin transport occurs at the edge although the bulk time-evolution operator over one period is trivial. In this work we explicitly construct a family of Floquet quantum spin-1/2 models on the square lattice implementing Swap Models to investigate the stability of the anomalous CSL under frequency detuning and the transition to the high-frequency regime. We have used the average-energy spectrum on finite-size torus and cylinders to unfold the Floquet quasi-energy spectrum over the whole frequency range and obtain the geometrical Berry phases. This enabled us to identify three regimes upon increasing detuning: i) a finite-size regime (with no folding of the Floquet spectrum), ii) an intermediate (narrow) regime with folding and very few resonances and iii) a regime with an increased density of resonances suggesting heating. At small detuning, edge modes are revealed by spectroscopic tools and from the diamagnetic response of the system giving access to the anomalous winding number. The analysis of all the data suggests that the anomalous CSL is not continuously connected to the high-frequency CSL. We also discuss the possible occurrence of a long-lived prethermal anomalous CSL.

Figures (20)

• Figure 1: Schematic representation of the drive Hamiltonian $H_{\rm drive}(t)$ involving four XXZ nearest-neighbor couplings $H_a$, $H_b$, $H_c$ and $H_d$ acting successively during a time $T/4$ on the 4 sets of bonds labelled by $a$, $b$, $c$ and $d$.
• Figure 2: Fined-tuned gates map in the $(\omega,\Delta)$ plane. Trivial gate instances are represented as blue dots and swap gates as red dots. These two cases are denoted respectively as trivial points and swap points in the text.
• Figure 3: $4\times4$ square lattice cluster with (a) periodic boundary conditions along $x$ and $y$ axes (torus), (b) along $x$ axis (cylinder), and (c) open boundary conditions.
• Figure 4: Periodic micromotions in fine-tuned models on a $L\times L$ ($L=4$) system -- The fidelity $|\langle\phi(0)|\phi(t)\rangle|$ at $n=1/2$ boson density (the initial state being the RVB state, see text) is plotted vs reduced time. Top: torus geometry. Heisenberg, XY and Ising drives are considered. Bottom: cylinder and open geometries showing emerging $T_{\rm cyl}=(L/2) T$ and $T_{\rm open}=(2L-1)T$ time-periodicities (shown between arrows), respectively.
• Figure 5: Energy spectra on a $4\times 4$ torus around the tuned frequency $\omega_s=0.5$ (for $p=q=0$) in the finite-size unfolded regime $|\delta\omega|\lesssim \delta\omega_{\rm fold}(N)$ and slightly beyond, i.e. $\omega\in [0.43,0.6]$. Top row: quasi-energy spectra in units of $\omega/2=\pi/T$ (the Floquet energies being defined modulo 2 in $\omega/2$ units, ${\cal E}=\pm 1$ must be identified). Middle row: average-energy spectra in units of $J$. The eight lowest average-energy states (some of them being degenerate belonging to related momentum sectors) and the ferromagnetic state are represented by red lines and green lines, respectively, in both the Floquet and average-energy Æ spectra. Note that for $\omega=0.5$ the average-energy spectrum obtained by taking the limit $\delta \omega\rightarrow 0$ (in practice $\delta \omega=10^{-6}$) does not see the massive degeneracy of the Floquet spectrum. Bottom row: geometric Berry phases $\Phi_n= T({\cal E}_n - {\text{\ae}}_n)\;\;{\text{mod}} [2 \pi]$ of the nine states mentioned above for the corresponding values of $\omega$.