Floquet engineering of topological phases protected by emergent symmetries under resonant drives

Kaoru Mizuta; Kazuaki Takasan; Norio Kawakami

REVIEW 2 major objections 5 minor 46 references

Floquet engineering of topological phases protected by emergent symmetries under resonant drives

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A resonant drive on alternating sites gives a Z2-symmetric spin chain an emergent Z2×Z2 symmetry, and the sign of a static field switches the resulting symmetry-protected topological phase.

desk verdict A clear, well-derived Floquet protocol that uses an emergent Z2 symmetry to realize Z2 x Z2 SPT phases; worth serious refereeing, with requests for larger-scale numerics and a prethermal-lifetime estimate. read the letter →

arxiv 1908.04100 v3 pith:XMTSPV7Z submitted 2019-08-12 cond-mat.mes-hall cond-mat.quant-gas

Kaoru Mizuta , Kazuaki Takasan , Norio Kawakami This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.quant-gas

keywords Floquetengineeringemergentsymmetrysymmetry-protectedtopologicalorderresonantdrivevanVleckexpansionKitaevchainperioddoublingprethermalregime

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes a Floquet engineering scheme that adds a symmetry rather than assuming one: a resonant drive on alternating sites turns an Ising chain with only a global Z2 symmetry into an effective static system with an emergent Z2×Z2 symmetry. In that effective system, the third-order van Vleck Hamiltonian hosts symmetry-protected topological phases, and the nontrivial phase is controlled simply by the sign of a static transverse field. A small error in the resonant pulse becomes a second control knob, producing four distinct SPT phases that can be distinguished by a period-doubled oscillation of a nonlocal order parameter. If correct, this gives a route to realizing and switching topological phases in driven platforms where the protecting symmetry is absent in the undriven system.

What carries the argument

The central object is the emergent $\mathbb{Z}_2$ symmetry $X_o = \prod_{j:\text{odd}} \sigma^x_j$, generated by the resonant $\pi$-pulse that acts only on odd sites. Because the resonant drive satisfies $X_o^2 = 1$, the van Vleck effective Hamiltonian at every truncation order commutes with $X_o$, and the original global $\mathbb{Z}_2$ symmetry supplies the complementary factor $X_e = X_{\text{all}}X_o^{-1}$ on even sites, giving the $\mathbb{Z}_2\times\mathbb{Z}_2$ group. The load-bearing mechanism is that the third-order van Vleck term converts the ordinary Ising interaction into the $\mathbb{Z}_2\times\mathbb{Z}_2$-symmetric cluster interaction $\sigma^z_{j-1}\sigma^x_j\sigma^z_{j+1}$; a Jordan-Wigner transformation then maps the effective spin model to two decoupled Kitaev chains, one on odd and one on even sites, whose individual topological indices determine the phase.

What would settle it

Extend the exact diagonalization of $H_{\rm eff}=\frac{i}{2T}\log U(2T)$ to system sizes beyond $L=8$ and check whether the four-fold ground-state degeneracy persists within the predicted window $-2\gamma<g<0$; if higher-order terms shift the transitions so far that no $g$ in that interval is degenerate, the third-order truncation is not predictive.

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Extended reading notes

Core claim

The paper claims that a one-dimensional Ising spin chain with only a global $\mathbb{Z}_2$ symmetry can be made to host nontrivial topological phases protected by an emergent $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry. The driving protocol combines a resonant $\pi$-pulse on odd sites, which implants a new $\mathbb{Z}_2$ symmetry at every order of the high-frequency van Vleck expansion, with high-frequency drives whose third-order commutators generate the cluster-type term $\sigma^z_{j-1}\sigma^x_j\sigma^z_{j+1}$. The resulting effective Hamiltonian is $D_3 = (g+\gamma)\sum_j \sigma^x_j + \gamma \sum_j \sigma^z_{j-1}\sigma^x_j\sigma^z_{j+1}$ plus edge terms, with $\gamma = \frac{4J^2h}{3\pi\omega^2}\{2\sin\omega\tau - \omega\tau(1+\cos\omega\tau)\}$. For $\gamma>0$, the system is in a nontrivial $\mathbb{Z}_2\times\mathbb{Z}_2$ SPT phase for $-2\gamma < g < 0$ and trivial otherwise, so reversing the static field direction switches the phase; a perturbation of the resonant pulse yields four distinct phases with independently nontrivial odd- and even-site Kitaev chains.

Load-bearing premise

The whole construction rests on the theorem, taken from the authors' prior work, that a resonant pulse whose one-period unitary squares to the identity implants an emergent extra Z2 symmetry at every order of the high-frequency expansion and that the system stays in the effective static description for an exponentially long time before heating; if that theorem fails or the heating time is too short, the predicted Z2xZ2 SPT phases would not exist.

Editorial extensions

If this is right

A $\mathbb{Z}_2$-symmetric driven spin chain can realize $\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry-protected topological order without ever adding the second symmetry as a static Hamiltonian term.
The nontrivial SPT phase is switchable by reversing a static field: for $\gamma>0$ the system is nontrivial for $-2\gamma<g<0$ and trivial outside that window.
A slight deviation of the resonant pulse from a perfect $\pi$-rotation provides a second control parameter, yielding four distinct SPT phases characterized by the topological indices $(Z_o,Z_e)$ of the two Kitaev chains.
All four phases can be identified in real-time dynamics from the mean value and amplitude of a nonlocal order parameter that oscillates with period $2T$, in analogy with prethermal discrete time crystals.
The same symmetry-adding mechanism should extend to resonant drives with $\mathbb{Z}_N$ symmetry and to higher-dimensional systems, enabling SPT phases that are difficult to obtain in equilibrium.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Extension: because the emergent symmetry holds at every order of the van Vleck expansion, a natural next test is to replace the $\pi$-pulse on odd sites with a $2\pi/N$ pulse and search for $\mathbb{Z}_N$-protected or larger SPT phases; the paper sketches this generality but does not demonstrate it.
Extension: the period-doubled order parameter is prethermal, so an experimental identification of the phases should measure the lifetime of the $2T$ oscillation rather than only its presence, since heating will eventually destroy it.
Extension: the proposed cold-atom ladder maps the nonlocal order parameter to a two-site boundary observable, so the sign-switch prediction could be tested directly by watching $A(nT)$ while sweeping $g$ through the predicted window.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

You should know this paper is a clearly argued proposal for using a resonant drive to create an emergent Z2 symmetry, and thereby realize Z2 x Z2 SPT phases in a model that only has a global Z2 symmetry. The core idea is sound, and the paper earns its keep with an explicit van Vleck derivation and small-system numerics that show the expected four-fold degeneracies.

The genuinely new pieces are: the specific driving protocol (a periodic pi-pulse on odd sites plus high-frequency transverse field and Ising coupling), the effective Hamiltonian D3 and its phase diagram, the flip-error perturbation analysis that yields four distinct phases, and the detection scheme based on period-doubling of a nonlocal order parameter. The static model and the two-Kitaev-chain mapping come from Iadecola et al. [28], and the emergent-symmetry theorem is from the authors' own PRB [32], but the application is new.

What is done well: the appendix contains a complete calculation of the third-order van Vleck Hamiltonian, including the constant C. The mapping to two decoupled Kitaev chains is explained clearly. Exact diagonalization for L=8 shows the predicted four-fold degeneracy in the nontrivial regions, and the dynamics simulations match the correspondence between phases and the mean/amplitude of the oscillation.

Soft spots: the paper leans on the theorem from Ref. [32] that the resonant drive gives an emergent Z_N symmetry at all orders of the expansion, without re-deriving it. If you are not already persuaded by that theorem, this proposal inherits all of its assumptions. Note also that the numerical runs use J=h=0.9, T=1, N=2, meaning lambda N T = 1.8, which is outside the formal small-parameter regime of the van Vleck expansion. I don't see this as fatal, since the exact-diagonalization results are a non-perturbative check that actually finds the degeneracies, which is evidence of robustness. But the paper should say so and add finite-size scaling, because the degeneracy criterion (relative gap < 0.01) is ad hoc and the phase boundaries shift noticeably. The detection protocol uses a symmetry-broken initial state and a local proxy for a nonlocal order parameter; the paper is honest about this, but it never estimates the prethermal lifetime, which is important for any experimental claim.

Overall, the central argument holds up. This is a solid, useful theoretical proposal with real new ingredients. I would send it to peer review. A good referee will ask for larger-scale numerics, a clearer statement about the parameter regime, and an estimate of the prethermal time, but the core idea is believable.

Referee Report

2 major / 5 minor

Summary. The manuscript proposes a Floquet engineering scheme in which a one-dimensional Ising chain with only a global Z2 symmetry (plus discrete time-translation symmetry) is driven by a resonant π-pulse on odd sites together with high-frequency transverse fields, so that the stroboscopic 2T dynamics acquire an emergent Z2×Z2 symmetry. Using third-order van Vleck perturbation theory, the authors derive an effective Hamiltonian D3 (Eq. (22)) with a tunable ratio between the σx and σzσxσz terms, which realizes Z2×Z2-protected SPT phases switchable by the sign of a static field g. They verify fourfold ground-state degeneracy by exact diagonalization of the approximation-free effective Hamiltonian for L=8, extend the construction to a flip-error perturbation that yields four distinct phases from two inequivalent Kitaev chains, and propose a dynamical detection via period-doubled nonlocal order parameters. An ultracold-atom implementation is sketched.

Significance. If the claims hold, this is a conceptually interesting route to SPT phases whose protecting symmetry is absent from the original Hamiltonian and is instead generated by the resonant drive. The van Vleck calculation in the Appendix is explicit and internally consistent, and the effective parameters are obtained from microscopic drive amplitudes rather than fitted. The exact-diagonalization checks use the raw Floquet evolution rather than the truncated expansion, and the mapping to two decoupled Kitaev chains gives concrete, falsifiable predictions: fourfold degeneracy when both chains are nontrivial and period-doubled dynamics of a nonlocal order parameter. These are genuine strengths of the paper.

major comments (2)

[Section VI, Eqs. (42)-(46)] The derivation of the period-doubling signature asserts that '|GS⟩ is an eigenvector of U(2T)', but |GS⟩ was defined as the ground state of \tilde H_eff constructed from \tilde U(2T), which includes the symmetry-breaking 2T-periodic perturbation Hper(t). Unless the perturbation is infinitesimal and the state is confined to the (quasi)degenerate manifold of the unperturbed U(2T), or unless a different preparation protocol is intended, the equality in Eq. (46) does not follow. This is load-bearing for the central dynamical-detection claim, because the correspondence between the four phases and (φo, φe) rests on that equality. Please supply the missing argument, or modify the quench protocol so that the initial state is an eigenstate of the unperturbed U(2T) to the required accuracy.
[Section II, Eq. (10); Section IV, Fig. 2; Section V, Eq. (33)] The existence of the emergent Z2×Z2 symmetry and the exponentially long prethermal regime is taken as a theorem from Ref. [32] without a self-contained derivation or a quantitative statement of its domain of validity. The exact-diagonalization parameters J=h=0.9, T=1, N=2 used in Figs. 2-4 give λNT=1.8, which is not in the small-parameter regime of the van Vleck/prethermal expansion. The same concern applies to the flip-error analysis, where Eq. (33) requires ε/λT=O((λ/ω)^2) but the ε ranges shown in Figs. 3-4 appear to violate this condition. Since the paper explicitly acknowledges that the time-crystalline regime is prethermal, the physical proposal would be substantially strengthened by a clear statement of the theorem's hypotheses, a quantitative prethermal-lifetime estimate for the parameters used, or numerical/analytical evidence that the required emergent symmetry and phase structure persist for λNT>1 on accessible timescales.

minor comments (5)

[Section IV, after Eq. (23)] The text says 'sgn(g) = sgn(π−ωτ)'; the symbol g should presumably be γ, since it is the sign of γ that is being discussed.
[Footnote [33]] The word 'numebr' is a typo and should read 'number'.
[Section II, text after Eq. (10)] The word 'necessarrily' is a typo and should read 'necessarily'.
[Fig. 3(b) caption] The degeneracy criterion |E_n−E_0|/|E_0|<0.01 is scale-dependent and may misclassify cases where E_0 is accidentally small; an absolute criterion tied to the finite-size level spacing would be more robust.
[Section VI, Eq. (49)] The numerically employed observable (σz_1+σx_1σz_2)/2 is a boundary proxy for the nonlocal order parameter of Eq. (45); the text asserts the boundary effect is small, but for L=8 it would be useful to show a test of this assumption.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the effective Hamiltonian is derived from microscopic drive parameters and benchmarked against raw Floquet evolution.

full rationale

Walking the derivation chain, I find no step in which a prediction reduces to an input by construction. The effective Hamiltonian D3 (Eq. (22) and Eq. (A20)) is obtained by a third-order van Vleck calculation from the microscopic Hamiltonian H(t) of Eq. (13), with γ = 4J^2h/(3πω^2){2 sin ωτ − ωτ(1+cos ωτ)} fixed by the drive parameters; no parameter is fitted to the target phase diagram. The topological classification of Eq. (22) is then obtained by Jordan-Wigner mapping to two decoupled Kitaev chains, an exact solution of the model of Eq. (25), so the phase boundaries |g|=|2γ| are mathematical consequences rather than imported predictions. The numerical tests in Figs. 2–4 compute Heff = (i/2T) log U(2T) from exact diagonalization of the original time-ordered evolution, so they benchmark the effective-model claim against the raw Floquet dynamics. The only self-citation is Ref. [32], which supplies the general theorem (Eq. (10)) that a resonant drive obeying X^N=1 produces an emergent Z_N symmetry in D_n at every order and an exponentially long prethermal regime; this theorem is parameter-free, has stated assumptions that do not include the target Z2×Z2 SPT phase, and is not fitted to the present data, so it counts as independent support rather than circularity. Concerns about the numerical parameters lying outside the strict small-λNT regime are validation and correctness concerns, not evidence of circularity.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The paper introduces no free parameters that are fit to data; J, h, g, ω, τ, and ε are physical drive parameters, and γ is computed analytically from them. The central derivation rests on the authors' prior emergent-symmetry theorem [32], on the high-frequency expansion assumptions, and on the known two-Kitaev-chain classification of the static Z2 x Z2 model [28]. No new particles, forces, or other entities are postulated.

free parameters (2)

symmetry-breaking field strength h_z = not reported
In Section VI, the initial state is the ground state of the effective Hamiltonian including H_per(t) proportional to h_z (Eq. (41)). The paper says h_z is infinitesimal but never gives the value used for the numerical data in Fig. 5, leaving the exact protocol under-specified.
degeneracy threshold for numerical phase identification = 0.01 relative energy
In Section IV and Fig. 3(b), ground-state degeneracy is defined by |E_n - E_0|/|E_0| < 0.01. This hand-chosen threshold affects the apparent phase boundaries in the numerical phase diagram.

assumptions (7)

domain assumption van Vleck high-frequency expansion is valid and the prethermal regime is exponentially long (Ref. [32])
Invoked in Section II (Eq. (3)) to replace the full stroboscopic dynamics by the effective static Hamiltonian D_n at t = mNT; the paper relies on this for the existence of the effective description.
domain assumption Emergent symmetry theorem: D_n commutes with the resonant-drive operator X for every truncation order n (Eq. (10))
This is the load-bearing premise taken from the authors' prior work [32]; it guarantees the generated Z2 x Z2 symmetry in the effective model.
domain assumption Energy-scale separations g/λ = O((λ/ω)^2) (Eq. (21)) and ε/λT = O((λ/ω)^2) (Eq. (33)) allow neglecting certain third-order van Vleck terms
Used to obtain the simplified effective Hamiltonian D3 (Eq. (22) and Appendix Eq. (A5)); if violated, extra terms appear that could shift or destroy the predicted phases.
domain assumption The static Z2 x Z2 model (Eq. (25)) maps to two decoupled Kitaev chains and is nontrivial iff |a| < |b| (from Ref. [28])
The topological classification of the effective model is inherited from this known result; the paper invokes it to read off the phase boundaries.
standard math Jordan-Wigner transformation and ground-state degeneracy reliably diagnose 1D SPT order
Used in Sections IV and VI to map spin chains to Majorana fermions and to identify four-fold degeneracy as the signature of nontrivial Z2 x Z2 SPT phases.
domain assumption The one-period evolution U(T) is approximately the resonant π-pulse X_odd (Eq. (47))
Used in Section VI to derive the period-doubling correspondence between the stroboscopic order parameter and the topological indices; the approximation is checked numerically.
domain assumption The system starts in a symmetry-broken ground state of the perturbed effective Hamiltonian
The detection scheme in Section VI requires an initial state with spontaneous symmetry breaking; the paper states this can be prepared by an infinitesimal perturbation or measurement.

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Cite this review

Pith. "Pith review of Floquet engineering of topological phases protected by emergent symmetries under resonant drives." pith.science (2026). https://pith.science/paper/XMTSPV7Z

@misc{pith2026190804100,
  author       = {Pith},
  title        = {Pith review of: Floquet engineering of topological phases protected by emergent symmetries under resonant drives},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XMTSPV7Z}},
  note         = {Machine review of arXiv:1908.04100}
}

abstract

Floquet engineering is one of the most vigorous fields in periodically driven (Floquet) systems, with which we can control phases of matter usually by high-frequency drives. In this paper, with Floquet engineering by a combination of high-frequency drives and resonant drives, we propose a way to realize nontrivial topological phases protected by a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry only in the presence of a $\mathbb{Z}_2$ symmetry, using a robust emergent $\mathbb{Z}_2$ symmetry induced by the resonant drives. Moreover, the symmetry protected topological (SPT) phases are switchable between nontrivial and trivial phases only by the direction of a static transverse field, and even perturbations on the resonant drive can be utilized to realize richer SPT phases. We also discuss the real-time dynamics of the model, and find that which topological phases the system lies in can be distinguished by a period doubling of a nonlocal order parameter, as with discrete time crystals. A realization or a control of nontrivial SPT phases without the required symmetries by resonant drives, proposed in this paper, would shed a new light on the observation of topological phenomena in nonequilibrium setups.

Figures

Figures reproduced from arXiv: 1908.04100 by the authors.

**Figure 2.** FIG. 2. Excitation energies from the ground state to low energy excited states of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) Phase diagram of the effective static model [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. Excitation energies from the ground state to low energy states for (a) [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. Numerical results of the stroboscopic dynamics of the order parameter [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6. Possible experimental realization in ultracold atoms. [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

Discussion (0). Continue with ORCID to comment.

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(A12) results in the real constant C given by C = 1 4π{ωτ (1 + cosωτ )− 2 sinωτ}

(A18) Substituting this into Eq. (A12) results in the real constant C given by C = 1 4π{ωτ (1 + cosωτ )− 2 sinωτ}. (A19) Finally, we arrive at the third order van Vleck eﬀective Hamiltonian D3: D3 = (g +γ) L∑ j=1 σx j +γ L−1∑ j=2 σz j−1σx jσz j+1− γ 2 (σx 1 +σx L), (A20) γ = 4...

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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T. Mikami, S. Kitamura, K. Yasuda, N. Tsuji, T. Oka, and H. Aoki, Phys. Rev. B 93, 144307 (2016)

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Topological Materi- als Science

As a result, by taking only the lowest order term brought by Hper(t) into account, we arrive at the third- order van Vleck eﬀective Hamiltonian in the perturbed 8 FIG. 5. Numerical results of the stroboscopic dynamics of the order parameter A(nT ) for diﬀerent parameters which lie in the four distinct phases. In each graph, we choose J = 0.5,h =−0.5,τ = 1...

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