Floquet Spin Chain Dynamics

• Floquet spin chains are quantum systems with a time-periodic Hamiltonian that generate nonequilibrium phases and transitions unseen in static models.
• The discrete Lyapunov equation and covariance matrix formulation provide a powerful method to analyze stationary Floquet states and spectral bifurcations.
• Experimental implementations using platforms like trapped ions and superconducting circuits enable controlled studies of dynamic quantum phase transitions.

A Floquet spin chain is a quantum spin chain whose Hamiltonian is periodic in time. The system is engineered and analyzed using the framework of Floquet theory, enabling the investigation of rich nonequilibrium phases and transitions that arise solely due to the time-periodic driving. Many of the physical phenomena in Floquet spin chains, including long-range order, topological phase transitions, information propagation, and dynamical stabilization, have no static analogs. These systems can be realized as periodically kicked or modulated spin-1/2 XY or Ising chains and are central to current research in nonequilibrium quantum many-body physics.

1. Formulation of Floquet States via Discrete Lyapunov Equation

In a periodically driven open many-body quantum spin system described by a Markovian master equation with time-periodic Liouvillian $\mathcal{L}(t+τ) = \mathcal{L}(t)$,

$\frac{d\rho}{dt} = \mathcal{L}(t)\rho$

the stationary Floquet state $\rho_F$ is defined via invariance under the one-period propagator: $(τ,0)\rho_F = \rho_F$. For quasi-free systems, such as the XY spin chain mapped by Jordan-Wigner transformation to quadratic Majorana fermions, the state is encoded in a covariance matrix $$C_{jk}(t) = \mathrm{tr}[w_j w_k \rho(t)] - \delta_{jk}$$, which evolves according to a matrix Riccati equation:

$\dot C(t) = -X(t)C(t) - C(t)X^T(t) + Y(t)$

where $X(t)$ and $Y(t)$ are $2n \times 2n$ matrices defined by the system’s Hamiltonian and dissipative processes.

In the case of a kicked system ($X(t) = X^0 + \delta_\tau(t)X^1$), the covariance after one period can be expressed as

$C(\tau) = Q\,C(0)\,Q^T - P\,Q^T$

with $Q,\,P$ constructed from $X^0$,X^1$$and solutions to the steady-state Lyapunov equation$$X^0 Z + Z(X^0)^T = Y$. Imposing periodicity ($C_F = C(0) = C(\tau)$$) yields a discrete Lyapunov equation for the stationary state:</p> <p>$$Q(\tau) C_F - C_F Q^{-T}(\tau) = P(\tau)$</p> <p>The solution$C_F$$gives the steady Floquet state. This form is crucial for both numerical and analytical investigation of steady states in open, periodically driven quadratic systems (<a href="/papers/1103.4710" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Prosen et al., 2011</a>).</p> <h2 class='paper-heading' id='phase-diagram-exponential-versus-long-range-spin-correlations'>2. Phase Diagram: Exponential versus Long-Range Spin Correlations</h2> <p>Applying this formulation to the periodically kicked XY spin-1/2 chain (with alternating exchange and delta-function transverse field), the stationary state exhibits two distinct regimes in its spin–spin correlation functions:</p> <ul> <li><strong>Exponential decay:</strong> The residual correlator$$C_\text{res} = \frac{\sum_{j,k:|j-k|\geq n/2} |C_{j,k}|}{\# \text{ pairs}}$$falls off rapidly with system size, signifying short-range order.</li> <li><strong>Long-range magnetic correlations (LRMC):</strong>$$C_\text{res}$scales as$1/n$(for chain size$n$$), indicating correlations persist at long distances.</li> </ul> <p>The phase diagram, as a function of anisotropy$$\gamma$, kick period$\tau$, and magnetic field strength$h$$, is highly nontrivial: rather than a single transition, there are complex, re-entrant “ribs” or bands of LRMC phases interspersed with short-range phases. That is, by varying$$\tau$or$h$(with fixed$\gamma$$), the system can repeatedly transition between these phases, a phenomenon unique to Floquet driving.</p> <h2 class='paper-heading' id='quasiparticle-dispersion-and-stationary-points-origin-of-phases'>3. Quasiparticle Dispersion and Stationary Points: Origin of Phases</h2> <p>The re-entrant structure of the phase diagram is explained via the properties of the Floquet quasiparticle dispersion relation. In the absence of boundary dissipation, the problem reduces (via translational invariance) to the following Floquet eigenvalue equation:</p> <p>$$\theta_{1,2}(\kappa) = \pm \arccos \left[\cos(2\tau h) \cos(2\tau \epsilon(\kappa)) + \sin(2\tau h)\sin(2\tau \epsilon(\kappa)) \frac{\cos\kappa}{\epsilon(\kappa)}\right]$</p> <p>with$\epsilon(\kappa) = \sqrt{\cos^2\kappa + \gamma^2\sin^2\kappa}$$.</p> <p>A central result is that the transition from exponential to long-range order is tightly linked to the appearance of nontrivial stationary points$$\kappa^*$(with$\kappa^* \neq 0, \pi$) where$d\theta/d\kappa=0$$. The LRMC phase arises when at least one such stationary point exists.</p> <p>Near a phase boundary, expanding$$\theta(\kappa)$in powers of$\kappa$:</p> <p>$\theta(\kappa) = \theta_0 + a(\gamma,\tau,h)\kappa^2 + b(\gamma,\tau,h)\kappa^4 + \cdots$</p> <p>A change in sign of$a(\gamma,\tau,h)$$signals the emergence of nontrivial stationary points, and the corresponding critical field$$h_c$satisfies$a(\gamma,\tau,h_c)=0$. The total number$n_\sharp + 2$$of stationary points (including the trivial ones at$$0$and$\pi$$) matches precisely the structure of the LRMC and exponential phases as well as the scaling of operator block entropy in Floquet operator space.</p> <p>This correspondence provides a microscopic criterion for long-range order in periodically driven chains: <strong>Re-entrant LRMC phases are a direct consequence of “Floquet bifurcations” in the quasienergy spectrum</strong>, not present in undriven systems.</p> <h2 class='paper-heading' id='mathematical-and-physical-implications'>4. Mathematical and Physical Implications</h2> <p>The discrete Lyapunov equation encapsulates the time-periodic steady state and is tractable due to the quasi-free nature of the model. The key mathematical structures and physical consequences can be summarized as follows:</p> <div class='overflow-x-auto max-w-full my-4'><table class='table border-collapse w-full' style='table-layout: fixed'><thead><tr> <th>Aspect</th> <th>Mathematical Expression</th> <th>Physical Impact</th> </tr> </thead><tbody><tr> <td>Stationary Floquet state</td> <td>$$Q(\tau) C_F - C_F Q^{-T}(\tau) = P(\tau)$</td> <td>Determines unique steady-state covariance</td> </tr> <tr> <td>Dispersion &amp; phase</td> <td>$\theta(\kappa) = \pm \arccos( \ldots )$$</td> <td>Identifies stationary points, phase boundaries</td> </tr> <tr> <td>Residual correlator</td> <td>$$C_\text{res} \sim 1/n$(LRMC) or decays$\sim$exp($-n/\xi$) Characterizes long-range vs. exponential decay

The phase diagram, including LRMC and exponential regions, and their re-entrant sequence, is thus controlled by spectral properties rather than integrability or static frustration.

5. Broader Significance and Context

The periodically kicked XY chain discussed in (Prosen et al., 2011) exemplifies how nonequilibrium driving can generate far-from-trivial phase diagrams and physical phenomena. The approach generalizes to open quantum systems with Markovian baths and is directly applicable to engineered quantum systems (e.g., trapped ions, Rydberg chains, superconducting circuits) where dissipation and periodic driving are naturally present.

Beyond foundational interest, the link between stationary point counting of the Floquet spectrum and observable long-range order provides a rigorous, predictive framework for classifying nonequilibrium quantum phases. Floquet engineering of spin chains opens avenues for dynamically stabilized phases and transitions inaccessible in equilibrium, with direct implications for quantum simulation and control.

6. Limitations and Open Directions

The general framework based on the discrete Lyapunov equation is strictly valid for open, quadratic (quasi-free) systems. While many phenomena reminiscent of those in interacting chains may emerge, the description may not fully capture interaction-driven effects such as many-body localization or anomalous heating regimes found in more generic Floquet spin chains. The precise role of dissipation strength, boundary effects, and non-Markovian environments remains an important direction for further investigation.

This systematic structure of Floquet-driven XY chains—combining analytic tractability, spectral–correlator correspondence, and experimental relevance—illustrates the subtlety and richness of nonequilibrium quantum matter under periodic driving.