The use of Peres lattices in periodically driven systems

Abstract: We demonstrate the strength of the method of Peres lattices in periodically driven quantum systems. The method, which has previously been used mostly in stationary systems, enables us to efficiently detect resonances in the driven system, to monitor the onset of chaos, and to recognize critical properties of the Floquet modes. It also allows quick comparisons of the spectra of Floquet modes for various driving Hamiltonians and transparent tests of the iterative approximation techniques based on effective stationary Hamiltonians.

• The paper demonstrates that Floquet Peres lattices efficiently reveal resonances, chaos onset, surviving ESQPT signatures, parity effects, and the accuracy of effective Hamiltonian approximations in driven LMG systems.
• The authors show that weak driving preserves recognizable stationary lattice structures, while increasing drive strength spreads resonance-induced distortions until chaotic dynamics compress the lattice through near-ergodic phase-space coverage.
• The paper finds that truncated BCH Floquet Hamiltonians can miss resonances and diverge strongly for kicked systems, highlighting the need to validate effective-Hamiltonian methods against exact Floquet dynamics.

The paper under review extends the method of Peres lattices, originally formulated by Asher Peres in 1984 as a diagnostic of quantum chaos in stationary systems, to periodically driven quantum systems. The authors, working within the Lipkin–Meshkov–Glick (LMG) quasispin model subject to both delta-kicked and continuous periodic driving, demonstrate that Peres lattices constructed from Floquet modes serve as a compact visual instrument for detecting resonances, monitoring the onset of chaos, identifying surviving signatures of excited-state quantum phase transitions (ESQPTs), and benchmarking truncated Floquet Hamiltonian approximations. The central methodological observation is that a driven system with $f$ degrees of freedom must be treated as an effective $f{+}1$ degree-of-freedom system, so that even an $f=1$ kicked spin admits chaotic dynamics and requires a three-observable lattice construction.

Model and formalism

The unperturbed Hamiltonian is the LMG model $$\widehat{H}_0 = \widehat{J}_z - (\kappa/2j)\widehat{J}_x^2$$, acting in a fixed $(2j+1)$-dimensional eigenspace of $\widehat{J}^2$; it describes a fully connected set of $2j$ qubits. The driving takes the form $$\widehat{H}(t) = \widehat{H}_0 + \eta g(t)\widehat{H}'$$ with zero-mean periodic functions $g(t)$ (delta kick, single- and double-harmonic cosine, sawtooth, tent) and driving operators $$\widehat{H}' \in \{\widehat{J}_z,\ \widehat{J}_z^2/j,\ \widehat{J}_x\}$$. The classical limit is taken at $f{+}1$0 on the Bloch sphere, and the time-periodic Hamiltonian is embedded into a stationary two-degree-of-freedom extended Hamiltonian, making the stroboscopic return map formally equivalent to a Poincaré section.

The stationary system exhibits a second-order ground-state quantum phase transition at $f{+}1$1 and, for $f{+}1$2, an ESQPT at energy $f{+}1$3, both interpretable via spontaneous parity breaking ($f{+}1$4). Two parameter regimes are studied throughout: $f{+}1$5 (parity-conserving phase) and $f{+}1$6 (parity-breaking phase), the latter featuring a separatrix trajectory of infinite period through the hyperbolic point $f{+}1$7.

Floquet Peres lattices and resonances

Peres lattices for the driven system are defined as scatter plots of expectation values of pairs of observables in the Floquet modes $f{+}1$8, eigenvectors of the one-period evolution operator with quasienergies restricted to the first Brillouin zone. For weak driving, the lattices closely resemble those of the stationary system—one-dimensional chains of points—but decompose locally near energies satisfying the resonance condition $f{+}1$9 with small integers $f=1$0. The identified resonances include ratios $f=1$1, $f=1$2, and $f=1$3, whose locations coincide with chains of elliptic and hyperbolic periodic points in the classical Poincaré maps. Notably, resonances associated with trajectories already lying in the chaotic layer of the perturbed system do not stand out in the lattice.

Husimi distributions on the Bloch sphere confirm the correspondence: Floquet modes located at lattice distortions are localized near elliptic or hyperbolic periodic points, while the mode localized at the separatrix fixed point $f=1$4 preserves the cusp structure of the ESQPT in the perturbed lattice. The authors conclude that the ESQPT survives periodic driving, consistent with earlier findings [Bandyopadhyay2015, Bastidas2014a], though they explicitly note this holds only for not too large driving strengths.

Quantifying similarity and the route to chaos

A mean-squared-error-like measure $f=1$5 is introduced to quantify deviation of the driven lattice from the interpolated stationary lattice. Its dependence on driving parameters reveals several concrete results: similarity degrades non-monotonically with both $f=1$6 and $f=1$7; there is a threshold $f=1$8 below which $f=1$9, explained by the absence of accessible resonances given minimum classical periods $$\widehat{H}_0 = \widehat{J}_z - (\kappa/2j)\widehat{J}_x^2$$0 ($$\widehat{H}_0 = \widehat{J}_z - (\kappa/2j)\widehat{J}_x^2$$1) and $$\widehat{H}_0 = \widehat{J}_z - (\kappa/2j)\widehat{J}_x^2$$2 ($$\widehat{H}_0 = \widehat{J}_z - (\kappa/2j)\widehat{J}_x^2$$3); and $$\widehat{H}_0 = \widehat{J}_z - (\kappa/2j)\widehat{J}_x^2$$4 is systematically larger for $$\widehat{H}_0 = \widehat{J}_z - (\kappa/2j)\widehat{J}_x^2$$5, attributed to the infinite-period hyperbolic point inducing pairwise resonance arrangements that enhance vulnerability to driving.

At large driving strength, lattice distortions spread from resonances until ergodicity contracts the entire lattice into a narrow interval of expectation values, reflecting uniform phase-space coverage by chaotic Husimi distributions. A structurally important result is the exact parameter symmetry $$\widehat{H}_0 = \widehat{J}_z - (\kappa/2j)\widehat{J}_x^2$$6, which leaves the kicked evolution operator unchanged up to a phase. Consequently, full regularity reappears periodically at $$\widehat{H}_0 = \widehat{J}_z - (\kappa/2j)\widehat{J}_x^2$$7, where the system becomes equivalent to a regular undriven system with renormalized interaction strength—a nontrivial recurrence of order at large driving amplitudes.

Increasing $$\widehat{H}_0 = \widehat{J}_z - (\kappa/2j)\widehat{J}_x^2$$8 densifies the lattices: subresonances invisible at moderate size become resolvable (e.g., a resonance near $$\widehat{H}_0 = \widehat{J}_z - (\kappa/2j)\widehat{J}_x^2$$9 at $(2j+1)$0), and chaotic regions contract more tightly along the $(2j+1)$1 axis, consistent with ergodic averaging.

Changing the driving operator exposes symmetry effects: parity-conserving drivings ($(2j+1)$2, $(2j+1)$3) produce nearly degenerate parity doublets below $(2j+1)$4, whereas the parity-violating $(2j+1)$5 driving splits each string of points into two branches corresponding to Floquet modes localized in the left and right confined phase-space lobes. This provides a direct lattice-level signature of whether the driving conserves or violates the underlying discrete symmetry.

Failure of truncated Floquet Hamiltonians

For the kicked system, the Floquet Hamiltonian is obtained via the Baker–Campbell–Hausdorff (BCH) expansion of the logarithm of a product of exponentials. The comparison of Peres lattices built from truncated Floquet modes against exact ones yields a strongly negative result: the second-order truncation reproduces the overall lattice shape but misses all resonances, while the eighth-order truncation fails to reproduce even the gross shape, with significant departures appearing already at $(2j+1)$6 and persisting through $(2j+1)$7 (computed in octuple precision). The authors state plainly that this indicates strong divergence of the BCH series for the chosen parameters, in line with known convergence analyses. For continuous driving, the Rahav–Garary–Fishman scheme gives second-order corrections to $(2j+1)$8 with operator norms of only 0.14–0.18 (first parameter set) and 0.001 (second set); correspondingly, the truncated lattices are nearly indistinguishable from the stationary one and show no trace of the exact resonances. Higher-order terms would be computationally prohibitive, and divergent behavior analogous to the BCH case is anticipated but not demonstrated.

Limitations and open questions

Several caveats qualify the results. The survival of ESQPT cusps under driving is established only for weak driving; the critical strength beyond which dynamical criticality is destroyed is not determined. The reappearance of regularity at $(2j+1)$9 is specific to the delta-kicked form of the evolution operator and does not occur for the continuous drivings studied. The similarity measure $\widehat{J}^2$0 ignores horizontal deviations of lattice points, which matter at small $\widehat{J}^2$1. Resonance strengths and distortion shapes depend on the particular internal dynamics and driving type, so no universal statement about resonance vulnerability follows from this study. Finally, convergence of the Floquet Hamiltonian expansions is neither guaranteed nor observed here; establishing the parameter domain in which such expansions remain useful remains open.

Conclusion

The paper establishes Peres lattices as a computationally inexpensive, information-dense diagnostic for periodically driven finite quantum systems, capable of simultaneously exposing resonances, chaos onset, surviving criticality, symmetry breaking by the drive, and the quality of effective-Hamiltonian approximations—all within a single scatter plot over all Floquet modes. The demonstration that high-order BCH expansions diverge badly for realistic kicked-top parameters is a practically significant caution for Floquet engineering schemes relying on truncated effective Hamiltonians.