Exact Strong Zero Mode Operators

• ESZM are quasilocal or boundary-localized operators that exactly commute with the full Hamiltonian or time-evolution operator, enforcing robust spectral degeneracies and infinite-temperature memory.
• They are constructed in various models including interacting Majorana chains, spin-1/2 XYZ/XXZ systems, Floquet-driven setups, and even certain stochastic classical processes.
• Their stability and exactness rely on integrability, suitable boundary conditions, and absence of resonances, which has significant implications for quantum memory and topological computation.

Exact strong zero mode operators (ESZM) are nontrivial, quasilocal or boundary-localized operators that commute exactly with the full Hamiltonian or Floquet/unitary time-evolution operator of an interacting quantum or stochastic many-body system, or with its classical Markov generator. Unlike topological ground-state edge modes, ESZMs act throughout the entire spectrum, enforcing robust spectral degeneracies and dynamical memory at infinite temperature. Their existence is closely tied to integrability, boundary conditions, and the absence of certain resonances, and they are realized in a wide variety of settings: from closed interacting Majorana chains and the spin-1/2 XYZ/XXZ models to locally driven Floquet systems, quantum dot arrays, and even certain stochastic classical processes.

1. Definition and Structural Properties

An exact strong zero mode operator $\Psi$ is characterized by the following properties:

• Commutation with the evolution operator: $[\Psi, H] = 0$ (Hamiltonian systems), $[\Psi, U] = 0$ (Floquet/unitary circuits), $[\Psi, W] = 0$ (classical Markov generators).
• (Anti)symmetry: $\{\Psi, \mathcal F\} = 0$ for $\mathbb{Z}_2$-symmetric models (e.g., spin flip or fermion parity), or $\Psi$ exchanges symmetry sectors.
• Normalization: $\Psi^2 = I$ (or $\Psi^n = I$ for generalized parafermion/strong zero modes).
• Localization: $\Psi$ is localized, typically at a boundary or interface, with its support decaying (often exponentially) into the bulk.
• Nontrivial action: $\Psi$ permutes or exchanges degenerate eigenstates in distinct symmetry sectors across the entire spectrum.

In closed chains of interacting Majorana fermions with an odd number of modes, ESZMs are constructed as explicit multinomials of Majorana operators; in spin chains, as infinite series of multi-body Pauli strings; in driven or open-boundary systems, as matrix-product operators exponentially localized at an edge. The existence of ESZMs is tied to exact commutation relations, yielding infinite-time coherence of appropriately chosen boundary observables.

2. Paradigmatic Constructions and Models

Closed Interacting Fermion Chains:

In systems of $2N+1$ interacting Majorana fermions, two anti-commuting ESZMs exist in the odd parity sector: the trivial total parity operator and a nontrivial operator continuously deformable from the noninteracting Majorana edge mode. The nontrivial ESZM is constructed by expanding in the full basis of fermionic monomials and selecting the null vector(s) of the commutator matrix $\mathcal H_{ab} = (\Upsilon_a, [H, \Upsilon_b])$ (Goldstein et al., 2011).

XYZ and XXZ Spin Chains:

For the open spin-1/2 XYZ chain (or the interacting Majorana chain), the ESZM $\Psi$ in the $z$-ordered regime $(|J_z| > |J_x|, |J_y|)$ is given as an infinite sum of strings of $\sigma^x, \sigma^y$, and a terminal $\sigma^z$. Explicitly,

$$\Psi = \sum_{S=0}^\infty \sum_{0 < a_1 < \cdots < a_{2S} < b} (XY)^{b-1}\,\sigma^z_b \prod_{s=1}^S [Y^{a_{2s-1}-a_{2s}}(1-X^{-2})\sigma^x_{a_{2s-1}}\sigma^x_{a_{2s}} + X^{a_{2s-1}-a_{2s}}(1-Y^{-2})\sigma^y_{a_{2s-1}}\sigma^y_{a_{2s}}],$$

with the property $[\Psi, H] = 0$, $\Psi^2 = \frac{1}{(1-X^2)(1-Y^2)} I$ for $X^2 < 1$, $Y^2 < 1$ (Fendley, 2015). In the XXZ chain with non-diagonal (U(1)-breaking) boundaries, the ESZM is uniquely specified by an MPO construction and remains exponentially localized at the edge, inducing infinite-temperature memory (Gehrmann et al., 7 Nov 2025).

Boundary and Interface ESZMs:

In the boundary Ising or Kitaev models, ESZMs arise at the interface between two ordered phases if and only if the single-particle bands remain nonoverlapping; otherwise, only approximate strong zero modes (almost SZMs) exist. Exact recursive and generating function methods yield closed forms for the boundary ESZM, which is exponentially localized unless a resonance proliferates (Olund et al., 2023).

Floquet-Engineered and Locally Driven Systems:

In nonintegrable staggered Heisenberg chains under local square-pulse Floquet driving, a partially resummed Floquet-Magnus expansion reveals an emergent approximate strong zero mode at the boundary. As the driven region grows, nonlocal multispin processes "shield" the boundary and the dephasing time of the boundary spin becomes exponentially large in the number of driven sites, approaching an ESZM in the thermodynamic limit (Mukherjee et al., 2023).

Short Interacting Chains and Parafermions:

In a two-site quantum dot chain with strong interaction and time-reversal symmetry, exact commuting Majorana Kramers-pair ESZMs and $\mathbb{Z}_3$ parafermion ESZMs emerge at a "sweet spot" in parameter space. The triply degenerate spectrum is protected by these operators, which generalize the notion of Majorana strong zero modes to parafermionic statistics (Bozkurt et al., 23 May 2024).

Stochastic Classical Chains:

In certain integrable classical Markov processes (e.g., exclusion and pair creation-annihilation chains), the generator $W$ admits an exact or almost strong zero mode $\Psi$ which is generally non-Hermitian and non-diagonal. Although such stochastic ESZMs do not protect long-lived physical observables, their conservation imposes exact dynamical constraints and off-diagonal correlation identities (Klobas et al., 2022).

3. Spectral Consequences and Degeneracies

The presence of an ESZM $\Psi$ in a system with a global symmetry $\mathcal F$ implies that the energy (or quasienergy) eigenstates are paired into sectors labeled by the eigenvalue of $\mathcal F$, with $\Psi$ exchanging the partners: $$H|\phi^\pm\rangle = \epsilon^\pm|\phi^\pm\rangle, \quad \mathcal F|\phi^\pm\rangle = \pm|\phi^\pm\rangle, \quad \Psi|\phi^\pm\rangle \propto |\phi^\mp\rangle.$$ Exact commutation $[H,\Psi]=0$ guarantees identical spectra in both symmetry sectors, i.e., strict spectral pairing. In the open Majorana chain, this yields $2^N$-fold degeneracy for $N$ vortices, with the ESZM algebra enforcing Ising anyon braid statistics under adiabatic exchange (Goldstein et al., 2011).

In Floquet systems with approximate SZMs, the residual splitting between partners is exponentially small in boundary length or number of driven sites, $\Delta_{\max} \sim e^{-c N}$, so while not strictly exact in finite systems, ESZMs emerge asymptotically in the combined thermodynamic/driven cluster limit (Mukherjee et al., 2023).

At quantum dot sweet spots (with $U\to\infty$), the entire spectrum is triply degenerate due to commuting parafermion ESZMs, enforcing robust $\mathbb{Z}_3$ structure throughout the spectrum (Bozkurt et al., 23 May 2024).

4. Conditions for Existence and Stability Criteria

The emergence of ESZMs is highly sensitive to model-specific details, boundary conditions, and the spectrum of bulk excitations:

• Integrability and nonresonance: In integrable models (XYZ, XXZ) and decoupled/ordered limits of free chains, the expansion for $\Psi$ converges and all commutators vanish in the thermodynamic limit, yielding exactness (Fendley, 2015, Gehrmann et al., 7 Nov 2025, Olund et al., 2023).
• Boundary/interface criteria: At a boundary between two ordered phases, an ESZM exists if and only if the single-particle bands do not overlap (precise analytic condition: see Eq. (S28) in (Olund et al., 2023)). The breakdown of this criterion signals the proliferation of resonant multi-body processes that destroy exact commutation.
• Driving and nonlocality: In periodically driven chains, nonlocal terms arising at higher order in the Floquet-Magnus expansion are essential to asymptotic exactness. The boundary spin memory is protected only when the driven region hosts a growing number of sites, eventually enforcing an exact discrete symmetry (Mukherjee et al., 2023).
• Boundary fields and circuit integrability: In the XXZ chain with U(1)-breaking but integrable boundary fields, rigorous transfer-matrix and MPO constructions show that the ESZM remains edge-localized and strictly commutes with the time-evolution or Hamiltonian (Gehrmann et al., 7 Nov 2025).

Loss of integrability, bulk U(1) symmetry breaking at generic boundaries, or band overlap at an interface typically only admits "almost" strong zero modes with exponentially or algebraically long, but finite, coherence times.

5. Physical Implications: Dynamics, Memory, and Quantum Information

Infinite-Temperature Memory:

The presence of an ESZM ensures that any local observable $\mathcal O$ with nonzero overlap with $\Psi$ retains nonzero infinite-time autocorrelation, even at infinite temperature: $$C_{\infty}^{\mathcal O}(t) = \frac{1}{2^N} \mathrm{Tr}[\mathcal O(t) \mathcal O(0)] \xrightarrow{t\to\infty} |\langle \Psi|\mathcal O \rangle|^2 > 0$$ (Gehrmann et al., 7 Nov 2025). This property enables infinite boundary coherence and renders ESZMs valuable for quantum memory.

Quantum Computation and Anyon Statistics:

ESZMs in Majorana or parafermion systems generate subspaces supporting non-Abelian statistics under vortex exchange:

• In Majorana networks, the ESZM algebra allows for the implementation of braid group representations, Clifford gates, and topological qubit encoding (Goldstein et al., 2011).
• In short quantum dot chains, parafermion ESZMs enable robust, degenerate ground states with nontrivial conduction properties and diagnostic spectroscopy signatures, providing minimal platforms for interaction-driven topological quantum computation (Bozkurt et al., 23 May 2024).

Floquet Engineering and Dynamical Decoupling:

In locally driven nonintegrable chains, cluster drives generate nonlocal virtual processes that "dynamically freeze" the boundary, suppressing relaxation and enabling long-lived quantum memories without disorder or topological protection. The emergence of an asymptotic ESZM is tied to the number of driven sites and the precise tuning of the driving frequency to "dynamical freezing" conditions (Mukherjee et al., 2023).

Stochastic Dynamics and Conservation Laws:

In classical Markov chains admitting a stochastic ESZM, hidden operator conservation imposes exact relations among nontrivial time-correlation functions, even though no long-lived observable emerges in the standard sense. This reflects the generalized role of zero mode operators in constraining nonequilibrium dynamics (Klobas et al., 2022).

6. Contrasts, Generalizations, and Limitations

• Exact SZMs vs. Almost SZMs: ESZMs are strictly defined by exact commutation with the full generator and are typically realizable only in integrable or finely-tuned models. In generic interacting or nonintegrable systems, resonances lead to "almost" SZMs, with $\|[H,\Psi]\|\sim e^{-cN}$ and only parametrically long coherence times (Olund et al., 2023, Mukherjee et al., 2023).
• Topological vs. Strong Zero Modes: While topological edge modes are ground-state-bound and protected by bulk spectral gaps, ESZMs act throughout the spectrum and are protected only by the absence of certain resonances or the presence of integrability. The former always survive at trivial/topological interfaces, while ESZMs can be killed by single or multi-body resonances at a boundary (Olund et al., 2023).
• Nonlocality under Mappings: In certain mappings (e.g., from XXZ to ASEP), ESZMs become highly nonlocal and lose physical significance for bulk observables, reflecting their fundamentally operator-localized (not physical observable-localized) nature (Gehrmann et al., 7 Nov 2025).

7. Outlook and Ongoing Directions

Recent advances in the explicit construction, stability analysis, and physical implementations of ESZMs point to several active directions:

• Realization and detection in quantum hardware (solid-state and cold atoms), exploiting Floquet engineering, quantum circuit design, or strongly interacting quantum dots.
• Systematic classification of systems and boundary/interface conditions supporting ESZMs, especially in higher-dimensional systems or with exotic symmetries.
• Extension to stochastic and classical systems, exploring operator conservation in nonequilibrium statistical mechanics.
• Utilization of parafermionic and other non-Abelian ESZMs for topologically protected quantum computation and error correction.

The existence of ESZMs provides a robust, integrability-based route to infinite-temperature boundary memory and operator-induced spectral degeneracies, bridging topological condensed matter, quantum information, and nonequilibrium statistical physics.