Almost Strong Zero Modes at Finite Temperature

Aditi Mitra; Matteo Rizzi; Niklas Tausendpfund

Almost Strong Zero Modes at Finite Temperature

T0 review · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read At finite temperature, the Majorana edge mode of the interacting Kitaev-Hubbard chain decays with lifetime $\exp(\Delta_{\rm eff}\beta)$, where $\Delta_{\rm eff}$ exceeds the many-body gap.

desk verdict Solid finite-temperature extension of the operator-Lanczos approach, with a credible but not yet airtight claim that Δ_eff > Δ; the quantitative exponent relies on fits in the large-β regime where the method is most fragile. read the letter →

arxiv 2501.11121 v3 pith:JDBXVAXL submitted 2025-01-19 cond-mat.str-el quant-ph

Niklas Tausendpfund , Aditi Mitra , Matteo Rizzi This is my paper · ORCID

classification cond-mat.str-elquant-ph

keywords AlmoststrongzeromodesMajoranafinitetemperatureKitaev-HubbardchainLanczosrecursiontensornetworkoperatorsedgedensityofstatestopologicalprotection

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

At zero temperature, Majorana zero modes at the ends of a gapped topological chain live forever; at infinite temperature, the same edge operators survive as long-lived "almost strong zero modes." This paper fills in the gap between those extremes for the interacting Kitaev-Hubbard chain. It argues that at finite temperature the edge mode decays exponentially slowly, with lifetime $\tau(\beta)=\exp(\Delta_{\rm eff}\beta)$, where the effective activation scale $\Delta_{\rm eff}$ is systematically larger than the thermodynamic many-body gap $\Delta$. The paper reads this as evidence that the degeneracy protecting the edge mode is not confined to the ground-state doublet but extends through a low-energy band of the spectrum. If true, this sharpens when cooling actually buys protection for topological qubits: the useful regime is set by $\Delta_{\rm eff}$, not by $\Delta$.

What carries the argument

The machinery is the Lanczos recursion for Heisenberg time evolution, lifted to finite temperature and implemented with matrix product operators. The commutation superoperator $[H,\cdot]$ turns the spreading of the seed operator $\gamma_{1,a}$ into a single-particle hopping problem on a semi-infinite chain; the Lanczos coefficients $b_n(\beta)$ are the hopping amplitudes, and the edge density of states $\nu^E_\beta(\omega)$ of this artificial chain is the Fourier dual of the autocorrelation function. The paper approximates the EDOS by a narrow Lorentzian of width $\gamma(\beta)$ plus a gapped incoherent background (Eq. (11)), and justifies this by mapping the Lanczos chain to a dimerized single-particle chain in its topological regime attached to a homogeneous lead (Eq. (12)), so that the width extracted from a four-parameter fit is the inverse lifetime. A tensor network ansatz with bond dimension up to $\chi=2000$ represents the operators and the thermal density matrix, making system sizes $L=22$ and arbitrarily long times accessible.

What would settle it

Compute the edge density of states at a fixed low temperature with a much higher bond dimension (or exact diagonalization for $L\approx 16$ to 20) and fit no assumed lineshape: if the zero-frequency feature is not Lorentzian, or if its width does not follow $\gamma=\gamma_0 e^{-\Delta_{\rm eff}\beta}$ over a range of $\beta$ spanning at least a decade in $\gamma$, the central claim fails. A direct check is to compare the autocorrelation function from the four-parameter fit with an independent long-time simulation: if the late-time decay deviates from the fitted exponential by more than the stated error, the effective gap is not a property of the model.

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Extended reading notes

Core claim

Framed as the authors state it: the damping rate $\gamma(\beta)$ of the edge-Majorana autocorrelation function obeys $\gamma(\beta)=\gamma_0 e^{-\Delta_{\rm eff}\beta}$ in the temperature range studied, and the effective gap extracted from the slope is always above the many-body gap. At the representative point $\mu/w=1.2$, $U/w=0.1$ they get $\Delta_{\rm eff}/w\approx 2.7$ against $\Delta/w\approx 1.09$. Along three cuts through the topological phase the ratio $\Delta_{\rm eff}/\Delta$ exceeds one, grows as $U\to 0$ where an exact strong zero mode exists, and turns upward again for larger $U$. Exact diagonalization supports the picture: an operator $\Gamma_\epsilon$ built from opposite-parity eigenstates below an energy cut $\epsilon$ has a commutator with $H$ that vanishes exponentially with system size for $\epsilon$ below about $\Delta_{\rm eff}$, and retains a size-independent overlap with the edge operator $\gamma_{1,a}$. The conclusion is that the low-energy sector below $\Delta_{\rm eff}$ hosts an approximate strong zero mode, so the topological protection extends over a finite energy window rather than only in the ground-state manifold.

Load-bearing premise

The load-bearing assumption is that the numerically computed edge density of states really is a narrow Lorentzian peak sitting on a gapped incoherent background, so that fitting its width $\gamma(\beta)$ to that shape yields the true decay rate; if the peak is non-Lorentzian or cannot be separated from the sidebands, the exponential scale $\Delta_{\rm eff}$ is an artifact of the fit.

Editorial extensions

If this is right

At any finite temperature the Majorana edge operator no longer has infinite lifetime, but the decay is exponentially slow with activation energy $\Delta_{\rm eff}$ rather than the thermodynamic gap $\Delta$.
The many-body spectrum below $\Delta_{\rm eff}$ is effectively doubly degenerate in opposite-parity pairs, so an edge operator projected to that window behaves like a strong zero mode.
Cooling a topological chain buys protection continuously: $\tau(\beta)=\exp(\Delta_{\rm eff}\beta)$ connects the zero-temperature infinite lifetime to the infinite-temperature almost strong zero mode.
As the interaction $U$ is tuned toward the integrable limit, $\Delta_{\rm eff}/\Delta$ grows, recovering the exactly protected strong zero mode; the same analysis applies to parafermion chains, Floquet circuits, and particle-conserving Majorana ladders.

Reading between the lines

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

Beyond the paper, the predicted $\Delta_{\rm eff}$ scale could be probed directly in edge-spin coherence experiments: the plateau height of the autocorrelation function and the subsequent decay time should show an activated dependence on temperature with slope $\Delta_{\rm eff}$.
The ratio $\Delta_{\rm eff}/\Delta$ turns upward at large $U$, which the authors flag as open; one testable extension is whether this tracks incipient localization or a second topological regime in the phase diagram.
A natural next calculation is to identify which low-energy states form the degenerate pairs below $\Delta_{\rm eff}$, for instance by parity-resolved spectroscopy or entanglement diagnostics, since the paper shows their existence but not their microscopic nature.
The method's efficiency suggests extending the same operator-Lanczos tensor network pipeline to two-dimensional ladders or dissipative settings, where the finite-temperature scalar product would need modification.

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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

Short version: this paper does something genuinely new—it runs the Lanczos algorithm on operators at finite temperature using MPOs, and uses it to extract the temperature-dependent decay rate of a Majorana edge mode in the Kitaev-Hubbard chain. The central observation is that the rate follows τ(β) ~ exp(Δ_eff β) with Δ_eff larger than the many-body gap. I think the qualitative claim is probably right, but the quantitative Δ_eff is softer than it looks.

What's good: The method is a real step beyond the infinite-temperature Lanczos studies of Yates et al., and the implementation appears careful. They check convergence with bond dimension, compare against TDVP at infinite temperature, and include a distinct exact-diagonalization probe: the low-energy projected operator Γ_ε has a commutator that decays exponentially for ε below ~Δ_eff and a crossover near Δ_eff. That ED check is the strongest evidence in the paper, and it makes the qualitative story credible. They also ship data and code.

The soft spot is the extraction of γ(β). The EDOS is fitted to a Lorentzian-plus-background ansatz, and then γ(β) is fitted to an exponential. The paper does not give error bars on either fit, and there are only a handful of temperature points. More importantly, the large-β regime—exactly the one that defines the slope—is where the method is least controlled. Figure 2 shows that for wβ = 2.0 and 2.35 the Lanczos coefficients are still growing at n = 200 and have not reached a plateau, while Appendix F1 states that the Lanczos instability sets in at smaller n_unstable for larger β. Truncating before the plateau means the homogeneous-lead continuation is unjustified; if the truncated chain still looks topological at the boundary, the Lorentzian width is underestimated, and since the bias grows with β, it steepens the log γ vs β slope. That would inflate Δ_eff. The authors' own K-window test (Fig. 12) is only shown at wβ = 0.4, not in the problematic regime, so it does not address the concern.

The ED check partially rescues them: it supports the existence of an ASZM below ~Δ_eff at one parameter point. But it doesn't validate the γ(β) thermometer across the phase diagram, and Fig. 6(b) shows Δ_eff/Δ for many parameter points without uncertainties.

Overall: worth publishing after revision, but a referee should push for error bars on the fits and a demonstration that the γ(β) extraction converges in the large-β regime, or at least an honest discussion of how the truncation bias affects the exponent. I'd take it to peer review.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Delta_eff is extracted from two successive fits and corroborated by an independent exact-diagonalization construction.

full rationale

The central result Delta_eff > Delta is not defined into the inputs. The paper computes Lanczos coefficients b_n(beta), forms the EDOS via a continued fraction with a homogeneous lead (Sec. IIB and Appendix D), fits the central peak to the Lorentzian of Eq. (11), and then fits the resulting width gamma(beta) to the Arrhenius form Eq. (16). Delta_eff is the slope of that second fit; the many-body gap Delta is extracted independently by DMRG (Appendix F4). Neither Eq. (11) nor Eq. (16) contains Delta, so the inequality Delta_eff > Delta is not a tautology. The EDOS ansatz (Eq. 11) is a modeling assumption, and the comparison of the fitted ACF (Eq. 15) with the Lanczos ACF is a self-consistency check rather than an independent prediction; this weakens the evidence but is not circular. The ED benchmark (Sec. IVC, Eq. 18) uses only spectral data and is external to the Lanczos pipeline; the crossover near Delta_eff is an observed consistency, not an imposed one. Self-citations to Refs. [16,17] supply the SSH/Lanczos structure, but the paper reproduces the continuum mapping in Appendix E and validates against TDVP and bond-dimension/system-size checks. Appendix F1 states a real robustness limitation: for larger beta the Lanczos instability sets in earlier while the staggering decay shifts to larger n, so truncation and continuation errors could bias the extracted gamma(beta) and hence Delta_eff; this is a numerical correctness risk, not circularity.

Assumptions & free parameters 8 free parameters · 7 assumptions · 0 invented entities

The paper's central claim rests on a chain of numerical approximations: the EDOS ansatz, the Arrhenius law, the Lanczos tail, and MPO truncation. The genuinely fitted physical quantities are γ(β), Δ_eff, and the DMRG gap; the rest are method choices with convergence checks. There are no new postulated particles or forces.

free parameters (8)

Lorentzian weight A(β) = per temperature, e.g., wβ = 1.4
One of four parameters in the EDOS fit Eq. (11); sets the spectral weight of the zero-energy peak.
Lorentzian width γ(β) = about 10^-4.5 to 10^-3.5 w in Fig. 6 range
Extracted from the central peak of the EDOS; the lifetime is defined as 1/γ(β). The central claim depends on this number.
Sideband center E0(β) = per temperature
Fitted in the semicircle approximation Eq. (12); affects short-time ACF, not the asymptotic decay.
Sideband width w*(β) = per temperature
Fitted in Eq. (12); width of the incoherent semicircle sidebands.
Arrhenius prefactor γ0 = from Fig. 6(a) fit
Prefactor in Eq. (16), fitted to the γ(β) data.
Effective gap Δ_eff = about 2.7w at μ/w = 1.2, U/w = 0.1; ratios 1.5 to 3.0 in Fig. 6(b)
Slope in the exponential fit Eq. (16); the central observable. Compared to the thermodynamic gap Δ.
Thermodynamic gap Δ∞ = e.g., 0.828w for μ/w = 1.2, U/w = 0.6
Extracted by fitting finite-size DMRG gaps m_p(L) = A/L^α + Δ∞ (Eq. F4); used as the baseline Δ to compare with Δ_eff.
Lanczos tail hopping b_N = plateau value, e.g., N around 200, w = i b_N
Choice for truncating the Lanczos series and attaching a semi-infinite homogeneous lead (Sec. IIB, App. D); shown to be insensitive in App. F1.

assumptions (7)

standard math The Lanczos recursion produces an orthonormal operator basis with respect to the finite-temperature scalar product Eq. (2), and the ACF is exactly the edge Green's function of the tridiagonal H_sp.
Used throughout Sec. IIB; no numerical approximation at this level.
domain assumption The symmetric finite-temperature scalar product <A|B> = (1/2) Tr[ρ{A†B + BA†}] is the physically relevant one for lifetimes.
Eq. (2); the paper notes other scalar products are possible (App. A) and checks the Wightman product gives qualitatively similar lifetimes (App. F2).
ad hoc to paper The EDOS has the form ν = A γ/(π(ω² + γ²)) + (1-A)ν̃ with ν̃ a gapped incoherent background (Eq. 11).
This ansatz defines what is meant by the Lorentzian width γ; the lifetime extraction is contingent on it. Proposed in Sec. IID and used for all fits.
domain assumption For large β the decay rate obeys γ(β) = γ0 exp(-Δ_eff β) (Eq. 16).
Taken from general finite-temperature perturbation arguments [31]; the accessible β range is limited, so the asymptotic regime is assumed rather than demonstrated.
domain assumption Unknown Lanczos coefficients for n > N are well approximated by a semi-infinite homogeneous chain with hopping b_N.
Sec. IIB and App. D; sensitivity to the averaging window is checked in Fig. 12.
ad hoc to paper A matrix product operator of fixed bond dimension χ with truncation based on the infinite-temperature scalar product faithfully represents the finite-temperature Lanczos basis operators.
Sec. IIC and App. G; the truncation is not variationally optimized for the finite-temperature scalar product, as the authors acknowledge.
domain assumption The seed operator γ1,a has a non-vanishing overlap with the true edge Majorana mode, so its ACF captures the edge-mode lifetime.
Sec. IV; the overlap is checked via the ED construction of Γϵ (Fig. 8), but the finite-size extrapolation is limited.

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Pith. "Pith review of Almost Strong Zero Modes at Finite Temperature." pith.science (2026). https://pith.science/paper/JDBXVAXL

@misc{pith2026250111121,
  author       = {Pith},
  title        = {Pith review of: Almost Strong Zero Modes at Finite Temperature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JDBXVAXL}},
  note         = {Machine review of arXiv:2501.11121}
}

abstract

Interacting fermionic chains exhibit extended regions of topological degeneracy of their ground states as a result of the presence of Majorana or parafermionic zero modes localized at the edges. In the opposite limit of infinite temperature, the corresponding non-integrable spin chains, obtained via generalized Jordan-Wigner mapping, are known to host so-called Almost Strong Zero Modes, which are long-lived with respect to any bulk excitations. Here, we study the fairly unexplored territory that bridges these two extreme cases of zero and infinite temperature. We blend two established techniques for states, the Lanczos series expansion and a tensor network ansatz, uplifting them to the level of operator algebra. This allows us to efficiently simulate large system sizes for arbitrarily long timescales and to extract the temperature-dependent decay rates. We observe that for the Kitaev-Hubbard model, the decay rate of the edge mode depends exponentially on the inverse temperature $\beta$, and on an effective energy scale $\Delta_{\rm eff}$ that is greater than the thermodynamic gap of the system $\Delta$.

Figures

Figures reproduced from arXiv: 2501.11121 by the authors.

**Figure 1.** (a) Sketch of the different phases of the Hamiltonian in Eq. (13) following [9, 11, 55]. For small µ/w and U/w, there exists an extended region (white) showing a topological ground-state degeneracy. Increasing either µ or U, one observes phase transitions into topologically trivial regions. For large µ, the state is described by a trivial band insulator, while for large U, one observes a transition into an incommen… view at source ↗

**Figure 2.** Lanczos coefficients for L = 22 at various inverse temperatures wβ = 1/T. The insert shows the staggered component h˜ n, Eq. (14), averaged over seven sites to reduce the noise in the data. the general behavior of the temperature dependent Lanczos series. The results are obtained by using the tensor network ansatz introduced in Sec. II C with a maximal bond dimension of χ = 2000 for the matrix product operator. Se… view at source ↗

**Figure 4.** EDOS obtained from the Lanczos coefficients at the finite temperatures shown in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

Figures from the paper (13 more)

**Figure 3.** Figure 3: (a) Comparison of the Lanczos series with increasing system sizes L = 16, 18, 20, 22, 30 at infinite temperature. The final plateau value increases with system size. (b) The Lanczos sequence for the same L and for wβ = 2.35. In contrast to the infinite temperature seq…

**Figure 5.** Figure 5: ACR for different temperatures with the same color code as in [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 7.** Figure 7: Commutator of the Hamiltonian H with the low energy projected ASZM Γϵ, see Eq. (18) for µ/w = 1.2 and U/w = 0.1. To check if the effective energy gap obtained in the previous section is reflected in the low energy part of the system, we study the model using exact diag…

**Figure 8.** Figure 8: Overlap of the projected ASZM Γϵ with the edge operator γ1,a for µ/w = 1.2 and U/w = 0.1. (a) Varying the energy cutoff ϵ over all scales. For large enough ϵ, we observe a decay of the overlap with respect to the system size. (b) A detailed plot for all energies ϵ/w <…

**Figure 9.** Figure 9: EDOS obtained for a short SSH chain with v = 0.5, w = 1.5 of length L = 20 attached to a semiinfinite homogeneous chain with hopping t = 1. The red line is the bulk contribution of the EDOS of a semiinfinite SSH chain with the same parameters, the orange dashed line…

**Figure 11.** Figure 11: EDOS obtained from the Lanczos series bn from [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]

**Figure 12.** Figure 12: EDOS obtained by varying the hopping strength w attaching the semi-infinite lead to the finite chain, with the latter described by the numerically obtained Lanczos coefficients bn. The different hopping parameters w are obtained by varying the averaging window K acco…

**Figure 14.** Figure 14: Comparison between the autocorrelation function obtained using the scalar product (red) and the Wightman scalar product (blue). In this appendix we demonstrate that the Wightman finite temperature scalar product defined in equation (A2) in appendix A gives qualitativ…

**Figure 13.** Figure 13: Energy resolved relative error ϵν of the EDOS, see Eq. (F2). Panel (a) [(b)] shows the error for wβ = 0 (wβ = 2.45). Finally, we demonstrate the convergence of our results with respect to the system size L. For this, we calculate the EDOS for the two system sizes L1 =…

**Figure 17.** Figure 17: Comparison between the autocorrelation function obtained from the Lanczos series (red dots) and the TDVP algorithm (blue crosses) at infinite temperature in the trivial insulating region (µ/w = 3, U/w = 0.1). Meaning of the panels are the same as in [PITH_FULL_IMAGE…

**Figure 16.** Figure 16: Comparison between the autocorrelation function obtained from the Lanczos series (red dots) and the TDVP algorithm (blue crosses) at infinite temperature in the Mott insulating region (µ/w = 0.2, U/w = 1.5). Meaning of the panels are the same as in [PITH_FULL_IMAGE:…

**Figure 18.** Figure 18: Energy gaps between different parity sectors for µ/w = 1.2, U/w = 0.6, and for various system sizes L. The energy difference between the two ground states of opposite parity, vanishes exponentially with the system size. Similarly, the energy difference between the t…

**Figure 20.** Figure 20: Finite state machine representation of the SZM of the XYZ model. Appendix I: Construction of Strong Zero Mode from spectral data In this appendix, we review the construction of an (almost) strong zero mode operator Γ from the full set of eigenstates and eigenenergies…

Discussion (0). Continue with ORCID to comment.

Reference graph

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In the following we discuss how these two approximations influence the results

Convergence Properties To calculate the EDOS and thus the lifetime of the edge modes at finite temperatures, we made two ap- proximations: The truncation of the bond dimension of the matrix product operators, and the modeling of the unknown Lanczos coefficients by a semi-infinite homoge- neous chain. In the following we discuss how these two approximation...

work page 2000
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Anti-commuting with the fermionic parity: {P, Γ} = 0,

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Commuting with the Hamiltonian:[Γ,H ]→ 0 for L→∞ . We also require theΓ to be localized at the edge of the system, so that the SZM has an exponentially decaying weight on operators with support away from the edge of the chain, similar to the MZM. It follows that a system possessing a SZM has an exact doubledegeneracyofthespectruminthethermodynamic limit: ...

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Wightman – Standard scalar product wt 100 102 104 C𝛽(t) 0.0 0.5 1.0 Standard Wightman Figure 14: Comparison between the autocorrelation function obtained using the scalar product (red) and the Wightman scalar product (blue). InthisappendixwedemonstratethattheWightmanfi- nite temperature scalar product defined in equation (A2) in appendix A gives qualitati...

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Panel (a) shows the autocorrelation func- tion

TDVP vs Lanczos wt 0 5 1 0 15 2 0 C𝛽(t) 0.4 0.5 0.6 0.7 0.8 0.9 1.0 wt 0 5 1 0 15 20 𝜖(t) 10−10 10−8 10−6 Lanczos TDVP Figure 15: Comparison between the autocorrelation function obtained from the Lanczos series (red dots) and the TDVP algorithm (blue crosses) at infinite temperature in the topological region (µ/w = 1.2, U/w = 0.1). Panel (a) shows the aut...

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With the DMRG we extracted the ground state of the Hamilto- nian for the even and odd parity sectors, together with the first excited states within each parity sector

Gap Extraction We extracted the gaps of the many-body spectrum of the Kitaev-Hubbard model (13) using the density ma- trix renormalization group (DMRG) [66, 67]. With the DMRG we extracted the ground state of the Hamilto- nian for the even and odd parity sectors, together with the first excited states within each parity sector. For the extraction of the s...

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

In the following we discuss how these two approximations influence the results

Convergence Properties To calculate the EDOS and thus the lifetime of the edge modes at finite temperatures, we made two ap- proximations: The truncation of the bond dimension of the matrix product operators, and the modeling of the unknown Lanczos coefficients by a semi-infinite homoge- neous chain. In the following we discuss how these two approximation...

work page 2000

[2] [2]

Anti-commuting with the fermionic parity: {P, Γ} = 0,

work page

[3] [3]

Commuting with the Hamiltonian:[Γ,H ]→ 0 for L→∞ . We also require theΓ to be localized at the edge of the system, so that the SZM has an exponentially decaying weight on operators with support away from the edge of the chain, similar to the MZM. It follows that a system possessing a SZM has an exact doubledegeneracyofthespectruminthethermodynamic limit: ...

work page 2000

[4] [4]

Wightman – Standard scalar product wt 100 102 104 C𝛽(t) 0.0 0.5 1.0 Standard Wightman Figure 14: Comparison between the autocorrelation function obtained using the scalar product (red) and the Wightman scalar product (blue). InthisappendixwedemonstratethattheWightmanfi- nite temperature scalar product defined in equation (A2) in appendix A gives qualitati...

work page

[5] [5]

Panel (a) shows the autocorrelation func- tion

TDVP vs Lanczos wt 0 5 1 0 15 2 0 C𝛽(t) 0.4 0.5 0.6 0.7 0.8 0.9 1.0 wt 0 5 1 0 15 20 𝜖(t) 10−10 10−8 10−6 Lanczos TDVP Figure 15: Comparison between the autocorrelation function obtained from the Lanczos series (red dots) and the TDVP algorithm (blue crosses) at infinite temperature in the topological region (µ/w = 1.2, U/w = 0.1). Panel (a) shows the aut...

work page 2000

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With the DMRG we extracted the ground state of the Hamilto- nian for the even and odd parity sectors, together with the first excited states within each parity sector

Gap Extraction We extracted the gaps of the many-body spectrum of the Kitaev-Hubbard model (13) using the density ma- trix renormalization group (DMRG) [66, 67]. With the DMRG we extracted the ground state of the Hamilto- nian for the even and odd parity sectors, together with the first excited states within each parity sector. For the extraction of the s...

work page

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