Floquet dynamical chiral spin liquid at finite frequency

This paper demonstrates that a Dynamical Chiral Spin Liquid (DCSL) with Z2 topological order can be stabilized at finite driving frequencies on a square lattice, where the high-frequency Magnus expansion fails, by showing that the system remains in a stationary regime characterized by specific Floquet quasi-energy features and a tensor network representation with Z2 gauge symmetry, until a critical frequency is reached where heating and chaotic behavior ensue.

The Gist

Imagine you have a giant, complex dance floor made of tiny magnets (spins). Normally, these magnets just want to point in opposite directions to their neighbors, like a calm, orderly crowd. But physicists are interested in a very special, chaotic state called a Chiral Spin Liquid (CSL).

Think of a CSL not as a calm crowd, but as a perpetual, swirling vortex of magnetic spins. In this state, the spins are constantly "dancing" in a specific, twisted direction (chirality), creating a hidden, robust order that is incredibly hard to break. It's like a whirlpool in a river that keeps spinning even if you throw a rock in it. This state is fascinating because it could be the key to building super-powerful, unbreakable quantum computers.

The problem? Making this "whirlpool" in a real lab is incredibly difficult. It's like trying to keep a perfect tornado spinning in a wind tunnel without it collapsing.

The Solution: The "Shaking" Technique (Floquet Engineering)

In this paper, the authors propose a clever trick to create this state. Instead of trying to build a static machine that holds the whirlpool, they suggest shaking the dance floor rhythmically.

They take a standard grid of magnets and apply a periodic "kick" or drive to the connections between them. Imagine you are shaking a tray of Jell-O. If you shake it at just the right speed, the Jell-O might settle into a new, stable shape that wouldn't exist if the tray were still.

• The High-Frequency Limit (The Old Way): Previously, scientists thought this only worked if you shook the tray extremely fast. If you shake it fast enough, the Jell-O doesn't have time to react to the individual shakes; it just sees an "average" smooth surface that naturally forms the whirlpool. This is called the "high-frequency limit."
• The New Discovery (The "Goldilocks" Zone): The authors asked: What if we shake it slower? They found that even if you slow down the shaking (lower the frequency), the whirlpool (the CSL) still forms. It's a "Dynamical Chiral Spin Liquid" (DCSL).

The Magic of the "Slow Shake"

Here is where the story gets interesting. When you shake the tray too slowly, the Jell-O usually gets messy and chaotic (heating up). But the authors found a "Goldilocks zone" of frequency:

1. Too Fast: The whirlpool forms, but it's a boring, static version.
2. Too Slow: The system breaks down, gets hot, and the order is lost (chaos).
3. Just Right (The Middle Ground): The system enters a dynamic steady state. It's not static; it's constantly moving and adjusting to the shake, but it maintains the perfect, swirling order of the whirlpool.

The "Rabi Oscillation" Dance

In this middle ground, the system doesn't just sit still. It performs a specific dance called Rabi oscillations.

Imagine two dancers, Alice and Bob, holding hands and spinning.

• In the "fast shake" world, only Alice spins perfectly.
• In the "just right" world, Alice and Bob take turns leading the spin. The system constantly swaps energy between two different patterns of movement. It's like a pendulum that never stops swinging between two states, creating a stable, rhythmic motion that preserves the special "whirlpool" order.

The Secret Map: Tensor Networks

How do we know this isn't just random noise? The authors used a mathematical tool called a Tensor Network (specifically PEPS).

Think of a Tensor Network as a highly efficient blueprint or a compression algorithm for the quantum state. Just as a JPEG file compresses a complex photo into a smaller file without losing the essence of the image, this blueprint compresses the complex, swirling dance of the magnets into a simple set of rules.

They found that even in this "slow shake" regime, the blueprint reveals a hidden Z2 symmetry. This is like a secret code in the dance floor's design that guarantees the whirlpool is topologically protected. It means the state is robust; you can't easily destroy it without tearing the whole dance floor apart.

Why Does This Matter?

1. It's More Realistic: Real quantum computers and simulators (like those using cold atoms) can't always shake things at "infinite" speeds. They have limits. This paper shows that we don't need impossible speeds to create these exotic states; we just need to find the right "Goldilocks" frequency.
2. It's Robust: The state survives even when the shaking isn't perfect. It has a "safety margin" before it turns into chaos.
3. It's a New Phase of Matter: This proves that you can have a topological state of matter that is inherently dynamic. It doesn't just exist in a static crystal; it exists because of the rhythm of time itself.

The Bottom Line

The authors discovered that you don't need a super-fast, perfect machine to create a quantum "whirlpool" of spins. By shaking the system at a moderate, rhythmic pace, you can stabilize a complex, swirling state of matter that is protected by deep mathematical laws. It's like finding that you can keep a perfect tornado spinning not by using a super-fan, but by gently, rhythmically tapping the ground beneath it.

Technical Summary

Here is a detailed technical summary of the paper "Floquet dynamical chiral spin liquid at finite frequency" by Poilblanc, Mambrini, and Goldman.

1. Problem Statement

Chiral Spin Liquids (CSL) are topologically ordered quantum phases of matter that serve as spin analogs to Fractional Chern Insulators. While static CSLs have been theoretically proposed in specific lattice models (e.g., Kagome or square lattices with symmetry-breaking terms), their experimental realization is challenging. Recent proposals utilize Floquet engineering (periodic driving) to stabilize CSL phases in ultracold atom or Rydberg atom platforms.

Previous work (Ref. [25]) demonstrated that applying a time-periodic modulation to nearest-neighbor Heisenberg couplings on a square lattice could stabilize an Abelian $\mathbb{Z}_2$ CSL phase, but this analysis relied on the high-frequency limit. In this limit, the system is described by a static effective Hamiltonian derived via the Magnus expansion.

The Core Question: Does this topological phase survive when the driving frequency is lowered to a regime where the high-frequency Magnus expansion fails (i.e., where the expansion converges slowly or not at all)? Specifically, does the system heat up into a chaotic state, or does a stable "Dynamical CSL" (DCSL) exist at intermediate frequencies?

2. Methodology

The authors investigate a spin-1/2 Heisenberg model on a square lattice subjected to a time-periodic drive.

• Model Hamiltonian:
  The system starts with an antiferromagnetic Heisenberg Hamiltonian $H_0$. A periodic drive $H_{drive}(t)$ is added, modulating the nearest-neighbor couplings with a phase shift of $2\pi/4$across four bond types. The drive breaks parity ($P$) and time-reversal ($T$) individually but preserves the product$PT$.
  $$H(t) = H_0 + \lambda(t)H_{drive}(t)$$
  where $\lambda(t)$ is a slow ramp function (adiabatic protocol) increasing from 0 to 1.

• Numerical Approach:

  • Exact Diagonalization (ED): Simulations are performed on a finite $4 \times 4$periodic torus ($N=16$spins) in the$S^z=0$sector (dimension$\approx 12,870$).
  • Time Evolution: The time evolution operator is computed using a high-order Trotter-Suzuki decomposition (64 steps per drive period) to minimize discretization errors.
  • Floquet Analysis: The Floquet operator $U(T_{drive})$ is diagonalized to obtain the quasi-energy spectrum and Floquet eigenstates.
  • Tensor Networks: The authors construct a Projected Entangled Pair State (PEPS) ansatz with bond dimension $D=3$ to represent the stationary state and verify its topological properties.

3. Key Contributions and Results

A. Existence of a Dynamical CSL (DCSL) at Finite Frequency

The authors identify a regime of intermediate frequencies ($\omega > \omega_c$) where a stable DCSL phase exists, distinct from the static CSL found in the infinite-frequency limit.

• Critical Frequency: They estimate a critical frequency $\omega_c \approx 2.7J$ (for the specific parameters used).
  • Above $\omega_c$: The system reaches a stationary regime with a non-zero plaquette chirality.
  • Below $\omega_c$: The Floquet bandwidth exceeds the drive frequency, the quasi-energy gap closes, and the system exhibits chaotic heating behavior (no plateau in chirality).

B. Structure of the Stationary State

Unlike the high-frequency limit where the ground state is a single component with specific symmetries, the finite-frequency DCSL is a complex superposition of four components.

• Rabi Oscillations: The time-evolved state exhibits Rabi oscillations between two primary sectors: a momentum $Q=(0,0)$ sector and a $Q=(\pi, \pi)$ sector.
• Symmetry Decomposition: The stationary state $|\Psi(t)\rangle$ is decomposed into four time-periodic components belonging to different irreducible representations (IRREPs) of the chiral space group:
  $$|\Psi(t)\rangle \approx |\phi^A_0\rangle + e^{2i\omega t}|\phi^B_0\rangle + e^{i\omega t}|\phi^E_\pi\rangle + e^{-i\omega t}|\phi^{E'}_\pi\rangle$$
  Here, $A$ and $B$ correspond to $Q=(0,0)$, while $E$ and $E'$ correspond to $Q=(\pi, \pi)$. As $\omega \to \infty$, the weights of $B, E,$ and $E'$ vanish, recovering the static CSL. At finite $\omega$, all four components coexist with significant weights.

C. Topological Order and PEPS Representation

The authors demonstrate that the DCSL retains $\mathbb{Z}_2$ topological order despite the dynamical nature of the drive.

• PEPS Construction: They successfully represent the time-evolved state using a chiral, time-periodic PEPS with bond dimension $D=3$. The ansatz captures the correct symmetry structure and the weights of the momentum components.
• Topological Degeneracy: By inserting $\mathbb{Z}_2$ fluxes through the torus holes, they construct four PEPS states. Diagonalization of the overlap matrix reveals that these states span a two-dimensional Hilbert space, confirming a two-fold topological degeneracy characteristic of $\mathbb{Z}_2$ topological order.
• Floquet Partners: The adiabatically prepared state overlaps significantly with the lowest-energy Floquet eigenstate ($\alpha=0$), while the orthogonal topological partner overlaps with the first excited Floquet state ($\alpha=1$).

D. Heating Mechanism

The transition to chaos is explained by the closing of the "Floquet gap."

• The bandwidth of the Floquet quasi-energy spectrum scales as $W_N \propto N/\omega$.
• When $W_N$ exceeds the drive frequency $\omega$ (i.e., the spectrum covers the full Floquet-Brillouin zone), the system can no longer be described by a prethermal state, leading to rapid heating and loss of topological order.

4. Significance

• Beyond High-Frequency Approximation: This work proves that Floquet-engineered topological phases are not restricted to the high-frequency limit. It establishes the existence of a robust "Dynamical CSL" at experimentally accessible intermediate frequencies.
• Experimental Feasibility: By identifying a stable frequency window ($\omega > \omega_c$) and providing a specific adiabatic ramping protocol, the paper offers a concrete roadmap for realizing CSLs in ultracold atom and Rydberg atom simulators, which often operate at finite frequencies.
• Theoretical Framework: The paper provides a rigorous characterization of the micromotion in driven topological systems, showing how time-periodic symmetries lead to complex superpositions of momentum sectors while preserving topological order.
• Tensor Network Validation: The successful mapping of a complex, time-dependent many-body state to a compact PEPS ansatz validates the use of tensor networks for analyzing non-equilibrium topological phases.

In conclusion, the authors demonstrate that a Dynamical Chiral Spin Liquid is a stable phase of matter at finite driving frequencies, characterized by a specific four-component wavefunction structure and robust $\mathbb{Z}_2$ topological order, provided the drive frequency remains above a critical threshold to avoid heating.