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Dynamic Realization of Majorana Zero Modes in a Particle-Conserving Ladder

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Pith's one-line read A time-periodic pulse sequence on a two-leg fermionic ladder produces an effective parity-conserving Hamiltonian whose attractive ground state is a topological superconductor with Majorana zero modes and a bulk gap of order 0.1τ.

desk verdict A clean Floquet scheme for a parity-preserving Majorana ladder, with the main caveat that the many-body phase diagram rests on a Trotter approximation that is only validated in a much weaker few-body regime. read the letter →

arxiv 2412.14886 v2 pith:BESEAPGD submitted 2024-12-19 quant-ph cond-mat.mes-hallcond-mat.quant-gascond-mat.str-el

Anais Defossez , Laurens Vanderstraeten , Lucila Peralta Gavensky , Nathan Goldman This is my paper · ORCID
classification quant-phcond-mat.mes-hallcond-mat.quant-gascond-mat.str-el
keywords MajoranazeromodesfermionicladderFloquetengineeringpairhoppingparitysymmetrybosonizationmatrixproductstatesopticallattices
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 route to Majorana zero modes that does not rely on a superconducting reservoir or on mean-field pairing: it works with two decoupled fermionic wires whose inter-leg hopping is switched on and off in a rapid, periodic sequence. At stroboscopic times the dynamics is captured by an effective Hamiltonian that contains inter-leg pair hopping and preserves a $Z_2$ leg-parity symmetry, exactly the ingredient known to protect Majorana zero modes in number-conserving ladder models. For attractive interactions $U_0<0$, bosonization and infinite-matrix-product-state simulations place the system in a topological superconducting phase with a bulk gap of about $0.1\tau$ and characteristic Majorana edge signatures. The value of the proposal is that the required ingredients—anisotropic nearest-neighbor interactions and pulsed tunneling control—are available in current optical-lattice experiments with ultracold atoms.

What carries the argument

The load-bearing object is the ladder written in spin operators $\hat{J}^j_\mu$, where $\hat{J}_x$ is the inter-leg single-particle hopping that the drive pulses turn on and off. The identity $e^{-i\eta \hat{J}_x} \hat{J}^j_z \hat{J}^{j+1}_z e^{i\eta \hat{J}_x}$ (Eq. (4)) separates the rotated interaction into parity-conserving pieces proportional to $\cos 2\eta$ and parity-breaking pieces proportional to $\sin 2\eta$, so the choice $\eta=\pi/2$ annihilates the $Z_2$-breaking terms and yields $H_{\mathrm{eff}}=\alpha H_0+(1-\alpha)H_1$ with pair hopping. The equilibrium analysis then uses bosonization in bonding/anti-bonding fields, reducing the anti-bonding sector to two competing sine-Gordon terms whose renormalization-group flow (Eq. (11)) opens the topological gap when $K_->1$. The numerical work uses infinite matrix-product states without imposing single-leg parity—so the topological order is signalled by spontaneous symmetry breaking in the thermodynamic limit—and a tangent-space excitation ansatz to extract the charge-sector gaps whose average defines $\Delta_{\mathrm{topo}}$.

What would settle it

Take the exact Floquet evolution operator of Eq. (3) on a finite ladder at $U_0=-1.5$, $\alpha=1/2$, filling $\nu=1/3$, choose a physically motivated $T$ such as $T=0.2$, and compute the stroboscopic many-body spectrum; if the lowest charged excitations do not show a gap near $0.1\tau$ or the stroboscopic parity-breaking probability grows on the preparation timescale, the phase diagram predicted from $H_{\mathrm{eff}}$ fails.

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Extended reading notes

Core claim

The central claim is that the pulsed time-evolution operator $U(T)=P^{\dagger} e^{-i(1-\alpha)T H_0} P e^{-i\alpha T H_0}$ with $P=e^{i\eta J_x}$, at $\eta=\pi/2$, is equivalent at stroboscopic times to the static parity-conserving Hamiltonian of Eq. (5), $H_{\mathrm{eff}}=\alpha H_0+(1-\alpha)H_1$, whose coupling constants are $U_1=U_0(1+\alpha)/2$ and $U_2=U_0(1-\alpha)/2$ and which contains inter-chain pair-hopping terms. The paper argues that in the attractive regime $U_0<0$ this effective model realizes a topological superconductor: the pair-tunneling coupling drives the anti-bonding sector toward the continuum limit of a Kitaev chain, producing a topological gap and Majorana zero modes, while the bonding sector remains gapless. The supported predictions include a topological gap $\Delta_{\mathrm{topo}}\approx0.1\tau$ at $\alpha=1/2$, $U_0=-1.5$, filling $\nu=1/3$; a transition at $U_0=0$ for all $0<\alpha<1$; and finite-chain signatures—twofold entanglement-spectrum degeneracies atop a free-boson CFT spectrum, and an end-of-chain revival in the correlation function $\langle \hat{a}^\dagger_1 \hat{a}_j\rangle$—which the authors read as direct evidence of Majorana edge modes.

Load-bearing premise

The paper assumes the high-frequency (Trotter) replacement of the pulsed evolution by the static effective Hamiltonian is accurate at the interaction strengths used in the many-body phase diagram, even though the approximation is explicitly checked only for two fermions on a four-site plaquette at $U_0=-0.7$ and the drive period $T$ used in the MPS simulations is not stated.

Editorial extensions

If this is right

  • At $\alpha=1/2$ and attractive interactions, the predicted topological gap reaches about $0.1\tau$, a scale well above the second-order corrections of the driving expansion, making the phase detectable in current cold-atom experiments.
  • Stroboscopic parity conservation survives small pulse-angle errors $\epsilon$, so the Majorana protection persists on time scales up to about $1/\epsilon$.
  • The topological sector and the gapless bosonic sector are decoupled, so Bragg or radio-frequency spectroscopy can probe the Majorana physics independently.
  • Starting from a staggered trivial insulator, a $\pi/2$ pulse followed by a slow ramp of the drive can prepare the topological ground state at filling $\nu=1/4$.

Reading between the lines

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

  • If the Trotter condition fails at the strong interactions used in the phase diagram, the actual Floquet state could differ from the predictions of $H_{\mathrm{eff}}$; an exact-Floquet calculation with an explicitly stated drive period would settle this and is not reported in the paper.
  • The same pulse logic could be transferred to a synthetic-dimension ladder, where the two legs are internal atomic states and inter-leg hopping is a Rabi coupling, relaxing the need for anisotropic real-space interactions.
  • The continuous-drive variant presented in Appendix B preserves the $Z_2$ symmetry for any parameters without fine tuning and is a natural alternative route worth testing numerically on the same quantities used in the main phase diagram.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The manuscript proposes a Floquet protocol for realizing a number-conserving fermionic ladder with Majorana zero modes. The pulse sequence in Eq. (3) is designed so that, at the special drive phase eta=pi/2 and in the high-frequency (Trotter) limit, the stroboscopic dynamics is governed by the static parity-conserving Hamiltonian Eq. (5), which contains the pair-hopping processes needed for topological superconductivity. The authors analyze this effective model with bosonization, obtaining a topological superconducting phase for attractive bare interactions U0<0, and support this with infinite-MPS calculations of the topological gap and with finite-MPS signatures of Majorana edge modes. The paper also discusses the role of micromotion, finite pulse duration, a continuous-drive variant, and an adiabatic preparation protocol.

Significance. The proposal is timely and, if the high-frequency reduction is controlled, offers an experimentally concrete route to number-conserving Majorana physics in optical lattices. The derivation of Eq. (5) is transparent, the Z2-preservation argument is elegant, and the bosonization and tensor-network results are mutually consistent with no fitted parameters entering the central claim. The appendices contain useful convergence checks and a treatment of finite pulse duration. The main barrier to accepting the quantitative phase diagram is the validity of the Floquet/Trotter description in the many-body parameter regime used for the central numerical results.

major comments (2)
  1. [Secs. III and V, Eqs. (3)-(5), Fig. 3] The central claim requires that the stroboscopic evolution of the pulse sequence in Eq. (3) is faithfully described by H_eff = alpha H0 + (1-alpha) H1. The only direct check of this Trotter/high-frequency approximation is the two-fermion plaquette in Fig. 2, at T=0.2, U0=-0.7, alpha=1/3. The many-body phase diagram in Fig. 3(c,d) is computed from the static H_eff at U0=-1.5, alpha=1/2, tau=1, and no drive period T is specified for these simulations; they are simulations of H_eff rather than of the driven ladder. For T=0.2, the leading Baker-Campbell-Hausdorff correction to H_eff has magnitude of order alpha(1-alpha) T |U0| tau, which is about 0.075 tau, comparable to the reported Delta_topo of about 0.1 tau, and the expansion parameter T(|U0|+tau) is about 0.5, not small. The authors should specify T, demonstrate convergence of the Floquet expansion at the parameters used for Fig. 3, or directly simulate the pulse sequence with infinite time-evolving block decimation/tDMRG to confirm that the topological gap survives in the stroboscopic dynamics.
  2. [Sec. VI and Fig. 3] The proposed adiabatic preparation protocol explicitly targets filling nu=1/4, whereas the bosonization phase diagram and all iMPS calculations are carried out at nu=1/3. At nu=1/4, one has ak_F=pi/4, so Eq. (10) gives gp = gbs = -4 U2/2; the two sine-Gordon couplings have equal magnitude, and the lowest-order RG no longer selects the pair-hopping term. The paper therefore does not establish that the state prepared at nu=1/4 lies in the MZM phase. The preparation protocol should either be analyzed at nu=1/4 or modified to reach nu=1/3, the filling at which the phase diagram was computed.
minor comments (4)
  1. [Eq. (5)] In the first line of Eq. (5), the b-operator hopping terms are missing hats; it should read \hat b^\dagger_j \hat b_{j+1} and its Hermitian conjugate.
  2. [Appendix B, Eqs. (B8)-(B10)] The statement that [O1,O2]=0 is not correct: for O1=sum_k(J^z_k J^z_{k+1}-J^y_k J^y_{k+1}) and O2=sum_k(J^y_k J^z_{k+1}+J^z_k J^y_{k+1}), a direct evaluation using [J^mu_j,J^nu_k]=i delta_{jk} epsilon^{munu rho} J^rho_k gives nonzero local and adjacent-bond commutators. The conclusion that [V_j,V_-j]=0 can nevertheless be rescued by noting that alpha_j beta_j=0 for every j due to J_{-j}(x)=(-1)^j J_j(x); the text should be corrected accordingly.
  3. [Sec. IV and Fig. 3] The strong-coupling threshold y_p(l*)=9 is arbitrary; the text should state explicitly that the bosonization phase diagram is qualitative and that this threshold affects the extracted length scale but not the location of the phase boundary.
  4. [Appendix C] The hopping amplitude is called t in Eq. (C4) and the truncation threshold is also called tau in Figs. 6-8, while the main text uses tau for the hopping; these conflicting notations should be distinguished.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: H_eff, bosonization, and iMPS results are derived from stated inputs; self-citations are method references only.

full rationale

The paper's derivation chain is self-contained. Eq. (3) defines the pulse sequence; H_eff=αH0+(1−α)H1 is obtained by a standard first-order Trotter expansion, and Eq. (5) follows from an explicit BCH rotation at η=π/2; no parameter in this step is fitted to the later gap predictions. The bosonization section uses the bare couplings U1, U2 and standard RG equations with a fixed strong-coupling threshold (yp(l*)=9); the threshold is a scale convention, not a fit to the MPS data, and only qualitative agreement is claimed. The infinite-MPS calculations solve Eq. (5) directly and report convergence with truncation threshold; they are not tuned to match bosonization. Self-citations (Refs. [32], [35], [39-42], [58]) provide the pulse ansatz or methodological tools, but the paper re-derives the effective Hamiltonian and numerically solves it, so the central topological-phase prediction does not reduce to a self-citation chain. The main caveat — that Trotter validity is checked only on a two-fermion plaquette and not at the many-body parameters of Sec. V — is a correctness/validity risk, not a circularity.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the high-frequency approximation for the Floquet sequence, the standard bosonization of the ladder in the K_- about 1 regime, and the tangent-space MPS excitation ansatz. There are no invented entities and only one hand-chosen numerical threshold (yp(l*)=9) in the analytic estimate of the gap scale.

free parameters (1)
  • RG strong-coupling threshold yp(l*) = 9
    In Sec. IV, the strong-coupling regime is assumed to be reached when yp,bs(l*) is of order 1, and the specific value yp(l*)=9 is chosen for the numerical integration to estimate the inverse correlation length. This choice affects the quantitative gap scale but not the qualitative phase boundary.
assumptions (5)
  • domain assumption Trotter (Suzuki) high-frequency expansion is valid at order O(T) for the pulse sequence in Eq. (3).
    Used in Sec. II to derive H_eff=alpha H_0+(1-alpha) H_1; the paper neglects O(T) corrections and benchmarks two-particle dynamics at T=0.2.
  • standard math The effective Hamiltonian H_eff preserves Z2 parity at all orders of the high-frequency expansion because H_0 and H_1 are parity-conserving and commutators of parity-conserving operators are parity-conserving.
    Stated in Sec. III; used to argue MZMs are protected. This is a theorem about commutators of parity-conserving operators.
  • standard math Bosonization of the two-leg ladder in the bonding/anti-bonding basis with Luttinger parameters K_+- and v_+- and the lowest-order RG equations (11) with K_- about 1.
    Sec. IV; standard 1D bosonization and RG within the marginal regime. The approximation K_- about 1 restricts the U0 range considered.
  • domain assumption The pair-tunneling term gp drives the anti-bonding sector into a Kitaev-chain-like topological phase, while backscattering gbs drives it trivial.
    Sec. IV, citing Refs. [14,17]; used to identify the topological phase from the RG flows.
  • domain assumption Infinite MPS with spontaneously broken single-leg parity correctly captures the topological order, and the MPS excitation ansatz gives the bulk gaps in the charge sectors.
    Sec. V and Appendix C; relies on the fermionic MPS formalism and tangent-space methods from Refs. [39-42,59,60].

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Cite this review

Pith. "Pith review of Dynamic Realization of Majorana Zero Modes in a Particle-Conserving Ladder." pith.science (2026). https://pith.science/paper/BESEAPGD

@misc{pith2026241214886,
  author       = {Pith},
  title        = {Pith review of: Dynamic Realization of Majorana Zero Modes in a Particle-Conserving Ladder},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BESEAPGD}},
  note         = {Machine review of arXiv:2412.14886}
}
read the original abstract

We present a scheme to realize a topological superconducting system supporting Majorana zero modes, within a number-conserving framework suitable for optical-lattice experiments. Our approach builds on the engineering of pair-hopping processes on a ladder geometry, using a sequence of pulses that activate single-particle hopping in a time-periodic manner. We demonstrate that this dynamic setting is well captured by an effective Hamiltonian that preserves the parity symmetry, a key requirement for the stabilization of Majorana zero modes. The phase diagram of our system is determined using a bosonization theory, which is then validated by a numerical study of the topological bulk gap and entanglement spectrum using matrix product states. Our results indicate that Majorana zero modes can be stabilized in a large parameter space, accessible in optical-lattice experiments.

Figures

Figures reproduced from arXiv: 2412.14886 by the authors.

Figure 1
Figure 1. (a) Sketch of the dynamic approach. The bare Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. (a) Rabi oscillations between the states [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. (a)-(b) Bosonization estimation of the inverse correlation [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Finite MPS simulations with the conservation of the [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: The dispersion ω(k) as a function of momentum k of the lowest-lying excitation energy in three different charge sectors, for the model in Eq. (C4) with W = −1.8 and filling ν = 1/3. We have shifted the chemical potential towards the middle of the gap between the Q = ±1…
Figure 7
Figure 7. Figure 7: Symmetry breaking (top) and topological gap (bottom) as [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 6
Figure 6. Figure 6: Symmetry breaking (top) and topological gap (bottom) as [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 9
Figure 9. Figure 9: (a)-(b) Time evolution of the mean probability of changing [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 8
Figure 8. Figure 8: Symmetry breaking (top) and topological gap (bottom) as [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Floquet-Engineered Hybrid Topological Orders with Majorana Edge Modes in Number-Conserving Fermionic Quantum Simulators

    cond-mat.quant-gas 2025-02 reject novelty 6.0 of 10

    A Floquet scheme is claimed to realize a new XTS topological phase with Majorana edge modes, but its parameter constraints contradict the studied phase diagram.

  2. Almost Strong Zero Modes at Finite Temperature

    cond-mat.str-el 2025-01 conditional novelty 6.0 of 10

    For the Kitaev-Hubbard chain, the finite-temperature decay rate of the Majorana edge mode follows an Arrhenius law with an effective gap systematically larger than the many-body gap.

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