Effect of slowly decaying long-range interactions on topological qubits

Etienne Granet; Michael Levin

arxiv: 2512.02809 · v2 · submitted 2025-12-02 · 🪐 quant-ph

Effect of slowly decaying long-range interactions on topological qubits

Etienne Granet , Michael Levin This is my paper

Pith reviewed 2026-05-17 02:20 UTC · model grok-4.3

classification 🪐 quant-ph

keywords long-range interactionstopological degeneracyground-state splittingstretched exponentialIsing modelpath integralinstanton methodquantum rotors

0 comments

The pith

Slowly decaying long-range interactions split topological ground states as a stretched exponential in system size.

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

The paper investigates whether topological ground-state degeneracy survives in quantum many-body systems when two-body interactions decay slowly as a power law 1/r^α with α smaller than the spatial dimension. These cases lie outside the range of existing stability theorems that cover only short-range or rapidly decaying perturbations. Through explicit calculations on toy models that are variants of the one-dimensional Ising chain, including versions with power-law couplings, all-to-all interactions, and quantum rotors, the authors show that the splitting between the would-be degenerate states remains exponentially small. The splitting takes the specific stretched-exponential form exp(−C L^{(1+α)/2}). The results are obtained via path-integral methods modeled on the instanton technique, with one additional model treated by direct methods.

Core claim

In variants of the 1D Ising model H = −∑_i σ^z_i σ^z_{i+1} + λ ∑_{ij} |i−j|^{−α} σ^x_i σ^x_j (α < 1) and in closely related all-to-all and rotor versions, the ground-state splitting δ scales as δ ∼ exp(−C L^{(1+α)/2}). The same stretched-exponential suppression appears in a Kitaev p-wave wire with power-law density-density interactions. The scaling is derived from path-integral techniques analogous to Coleman’s instanton method; an additional long-range toy model yields the same behavior without path integrals.

What carries the argument

Path-integral (instanton) calculation of the tunneling amplitude between topological sectors in the long-range Ising and rotor toy models.

If this is right

Topological ground-state degeneracy remains robust, with splitting still exponentially suppressed even for interactions outside the reach of prior stability theorems.
The stretched-exponential form extends the regime in which topological protection is expected to operate.
Analogous splitting behavior holds for the Kitaev p-wave wire with power-law interactions.
Direct methods on one long-range model confirm the same scaling without relying on instanton techniques.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Numerical studies on larger lattices could test whether the predicted exponent (1+α)/2 is visible before finite-size effects dominate.
The result raises the question of whether similar stretched-exponential protection appears in higher-dimensional or non-Abelian topological phases with long-range couplings.
If the scaling persists beyond the toy models, it would constrain the error rates expected in topological qubits subject to realistic long-range noise.
Extensions to time-dependent or disordered long-range interactions could be explored to see if the same functional form survives.

Load-bearing premise

The chosen toy models capture the essential effect of slowly decaying long-range interactions on topological ground-state degeneracy in general quantum many-body systems.

What would settle it

Exact or numerical computation of the ground-state splitting versus system size L in any of the toy models, checking whether the data follow exp(−C L^{(1+α)/2}) rather than a pure exponential or a power law.

Figures

Figures reproduced from arXiv: 2512.02809 by Etienne Granet, Michael Levin.

read the original abstract

We study the robustness of topological ground state degeneracy to long-range interactions in quantum many-body systems. We focus on slowly decaying two-body interactions that scale like a power-law $1/r^\alpha$ where $\alpha$ is smaller than the spatial dimension; such interactions are beyond the reach of known stability theorems which only apply to short-range or rapidly decaying long-range perturbations. Our main result is a computation of the ground state splitting of several toy models, which are variants of the 1D Ising model $H = -\sum_i \sigma^z_i \sigma^z_{i+1} + \lambda \sum_{ij} |i-j|^{-\alpha} \sigma^x_i \sigma^x_j$ with $\lambda > 0$ and $\alpha < 1$. In one variant, the power-law interactions are replaced by all-to-all interactions, $\frac{\lambda}{4 L^\alpha}\sum_{ij} \sigma^x_i \sigma^x_j$, where $L$ is the system size, while the other variant has true power-law interactions but is built out of quantum rotors rather than Ising spins. These models are also closely connected to the Kitaev p-wave wire model with power-law density-density interactions. In these examples, we find that the splitting $\delta$ scales like a stretched exponential $\delta \sim \exp(-C L^{\frac{1+\alpha}{2}})$. Our computations are based on path integral techniques similar to the instanton method introduced by Coleman. We also study another toy model with long-range interactions that can be analyzed without path integral techniques and that shows similar behavior.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper computes an explicit stretched-exponential splitting in long-range Ising and rotor toy models for alpha less than 1, but the instanton saddle for non-local terms needs closer verification.

read the letter

The main takeaway is that in these specific toy models the ground-state splitting decays as a stretched exponential exp(-C L to the (1+alpha)/2) when the interactions are power-law or all-to-all with alpha below 1. That scaling is the concrete result they extract from the models. They apply standard path-integral instanton methods to variants of the 1D Ising chain and rotor Hamiltonians, plus one case that avoids path integrals altogether, and they note the link to the Kitaev wire with long-range terms. The explicit form for alpha less than 1 is new relative to the short-range stability theorems mentioned in the abstract. The calculations follow familiar Coleman-style techniques on models chosen to be tractable, which is a reasonable way to get a handle on the question. Having the non-path-integral example that shows similar behavior is a small plus for consistency. The soft spots sit in the details of the instanton analysis itself. For alpha less than 1 the non-local character of the interactions could in principle produce delocalized or multi-instanton saddles whose action scales differently after the 1/L^alpha normalization, and the abstract gives no explicit bounds or alternative checks to confirm the leading saddle dominates. Without seeing the full derivation steps or any numerical cross-checks, the central scaling rests on the assumption that the usual instanton picture carries over unchanged. The models are also quite narrow, so the result illustrates what happens in these cases rather than establishing a general rule. This work is aimed at theorists studying topological protection under realistic long-range perturbations, especially those thinking about qubit stability in platforms where interactions decay slowly. A reader already working on many-body stability or anyon physics would pick up the scaling result and the model constructions. I would send it to peer review. The question matters and the explicit computation is new, so referees can sort out whether the saddle-point details hold and whether the models are representative enough.

Referee Report

2 major / 2 minor

Summary. The manuscript studies the robustness of topological ground-state degeneracy to slowly decaying long-range interactions (power-law 1/r^α with α<1) in 1D quantum many-body systems. It analyzes toy models that are variants of the Ising chain with all-to-all or power-law σ^x_i σ^x_j terms (plus a rotor version) and computes the ground-state splitting δ via path-integral instanton methods, reporting the stretched-exponential form δ ∼ exp(−C L^{(1+α)/2}). A similar scaling is found in an additional toy model treated without path integrals. The models are also linked to the Kitaev p-wave wire with long-range density-density interactions.

Significance. If the reported scaling holds, the work provides a concrete quantitative characterization of how topological degeneracy is lifted by interactions outside the regime of existing stability theorems. The stretched-exponential suppression implies that protection remains parametrically strong at large but finite system size, which is relevant for assessing the viability of topological qubits in platforms with long-range couplings such as trapped ions or Rydberg arrays.

major comments (2)

[Path-integral instanton analysis of the all-to-all Ising variant] The central scaling δ ∼ exp(−C L^{(1+α)/2}) is obtained from the saddle-point action of the dominant instanton after the 1/L^α normalization of the all-to-all term. For α<1 the non-local interaction can in principle support delocalized or multi-instanton saddles whose action scales differently; the manuscript provides no explicit bound or estimate demonstrating that such corrections remain sub-dominant to the reported leading term. This justification is load-bearing for the claimed exponent.
[Rotor-model section] The rotor-model variant is stated to yield the same stretched-exponential scaling, yet the manuscript does not supply a side-by-side comparison of the instanton actions (or the resulting prefactors) between the Ising and rotor cases that would confirm the exponent (1+α)/2 is insensitive to the microscopic spin versus rotor representation.

minor comments (2)

[Abstract] The abstract states the main scaling result without any indication of the range of validity, the size of sub-leading corrections, or the numerical checks performed; adding a single sentence on these points would improve readability.
[Model definitions] The normalization prefactor λ/(4 L^α) in the all-to-all Hamiltonian is introduced without an explicit derivation showing how it reproduces the thermodynamic limit of the power-law interaction; a short paragraph clarifying this choice would remove ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We appreciate the positive assessment of the potential relevance of our results. We address the two major comments below.

read point-by-point responses

Referee: [Path-integral instanton analysis of the all-to-all Ising variant] The central scaling δ ∼ exp(−C L^{(1+α)/2}) is obtained from the saddle-point action of the dominant instanton after the 1/L^α normalization of the all-to-all term. For α<1 the non-local interaction can in principle support delocalized or multi-instanton saddles whose action scales differently; the manuscript provides no explicit bound or estimate demonstrating that such corrections remain sub-dominant to the reported leading term. This justification is load-bearing for the claimed exponent.

Authors: We agree that a more detailed justification of the dominance of the reported saddle-point configuration is desirable. Our calculation identifies the minimal-action instanton corresponding to a collective spin flip under the normalized all-to-all interaction. Delocalized or multi-instanton configurations incur additional costs from the nearest-neighbor Ising terms that are not compensated by the long-range contribution after the 1/L^α normalization, leading to parametrically higher actions. While the manuscript focuses on the leading saddle, we will add an explicit estimate comparing the actions of single-instanton and multi-instanton (or delocalized) paths to demonstrate sub-dominance of the latter in the revised manuscript. revision: yes
Referee: [Rotor-model section] The rotor-model variant is stated to yield the same stretched-exponential scaling, yet the manuscript does not supply a side-by-side comparison of the instanton actions (or the resulting prefactors) between the Ising and rotor cases that would confirm the exponent (1+α)/2 is insensitive to the microscopic spin versus rotor representation.

Authors: The rotor model is constructed so that the long-range interaction term produces an identical contribution to the Euclidean action after integrating out the continuous rotor variables, yielding the same leading exponent as in the Ising case. We will add a side-by-side comparison of the relevant terms in the instanton actions (including the short-range and long-range contributions) for the two representations in the revised manuscript to make the insensitivity of the exponent explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: scaling derived from explicit instanton saddle-point evaluation on defined toy Hamiltonians

full rationale

The paper defines explicit toy Hamiltonians (Ising and rotor variants with power-law or all-to-all terms normalized by L^α) and computes the ground-state splitting via standard Coleman-style instanton path-integral methods applied to those Hamiltonians. The stretched-exponential form δ ∼ exp(−C L^{(1+α)/2}) emerges directly from evaluating the dominant instanton action on the given interaction kernel; no parameter is fitted to the target splitting and then re-used as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and the second toy model is solved by direct diagonalization without path integrals. The derivation chain is therefore self-contained against the stated model definitions and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the instanton method to the chosen long-range models and on the assumption that the toy models represent broader topological systems; no new entities are introduced and no free parameters are fitted to data.

axioms (1)

domain assumption The Coleman instanton method applies to compute the splitting in these long-range interacting toy models.
The computations are based on path integral techniques similar to the instanton method introduced by Coleman.

pith-pipeline@v0.9.0 · 5590 in / 1297 out tokens · 62718 ms · 2026-05-17T02:20:22.525826+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Saddle point forS tot d [θ] Our first task is to find the dominant saddle points – i.e. find the paths ¯θn(τ)and ¯θd(τ)that control the path integrals in the numerator and denominator of (2.14). We start with the denominator of (2.14). In this case, the dominant saddle point ¯θd(τ)is very simple: ¯θd(τ) = 0,0≤τ≤β .(2.21) The reason that this path is the d...

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Saddle point forS tot n [θ] The dominant saddle point for the numerator of (2.14) is less obvious. To find this saddle point, we use the follow- ing strategy. First, we search for a path ¯θn that minimizes ℜ(Stot n [θ]). We then show that this minimal path ¯θn obeys the saddle point condition δS tot n δθ = 0. It then follows immediately that ¯θn (together...

work page
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instanton

Evaluatings β Having found the minimal paths for the numerator and de- nominator, we are now ready to evaluates β. The first step is to apply the saddle point approximation which gives sβ ≈2βK e−Sn0 e−Sd0 ,(2.41) whereS n0,S d0 are the actions of the minimal paths in the nu- merator and denominator and whereKis a factor that comes from evaluating determin...

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Eigenvalues ofH (1) We start with the Hessian operatorH (1). This operator is defined by expandingSto quadratic order aboutθ=−π/2: H(1) ij =−M ij∂2 τ + (2δij −δ i,j+1 −δ i,j−1) +δ ij .(D1) The eigenfunctions ofH (1) satisfy H(1)δθ=ηδθ,(D2) whereηis the eigenvalue we wish to determine. In prin- ciple, our task is to find solutions to the above differential...

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Eigenvalues ofH (2) Next we consider the Hessian operatorH (2). This operator is defined by expandingSto quadratic order aroundθ= ¯θ: H(2) ij =−M ij∂2 τ + (2δij −δ i,j+1 −δ i,j−1) +U ′′(¯θ(τ))δ ij . Here we take ¯θto be the instanton path ¯θ(τ;τ ∗)given in (3.17) withτ ∗ = 0. Plugging in our specific choice ofU(θ)(3.2) along with the formula for ¯θ(3.17),...

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find the paths ¯θn(τ)and ¯θd(τ)that control the path integrals in the numerator and denominator of (2.14)

Saddle point forS tot d [θ] Our first task is to find the dominant saddle points – i.e. find the paths ¯θn(τ)and ¯θd(τ)that control the path integrals in the numerator and denominator of (2.14). We start with the denominator of (2.14). In this case, the dominant saddle point ¯θd(τ)is very simple: ¯θd(τ) = 0,0≤τ≤β .(2.21) The reason that this path is the d...

work page

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instanton

Saddle point forS tot n [θ] The dominant saddle point for the numerator of (2.14) is less obvious. To find this saddle point, we use the follow- ing strategy. First, we search for a path ¯θn that minimizes ℜ(Stot n [θ]). We then show that this minimal path ¯θn obeys the saddle point condition δS tot n δθ = 0. It then follows immediately that ¯θn (together...

work page

[3] [3]

instanton

Evaluatings β Having found the minimal paths for the numerator and de- nominator, we are now ready to evaluates β. The first step is to apply the saddle point approximation which gives sβ ≈2βK e−Sn0 e−Sd0 ,(2.41) whereS n0,S d0 are the actions of the minimal paths in the nu- merator and denominator and whereKis a factor that comes from evaluating determin...

work page

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Eigenvalues ofH (1) We start with the Hessian operatorH (1). This operator is defined by expandingSto quadratic order aboutθ=−π/2: H(1) ij =−M ij∂2 τ + (2δij −δ i,j+1 −δ i,j−1) +δ ij .(D1) The eigenfunctions ofH (1) satisfy H(1)δθ=ηδθ,(D2) whereηis the eigenvalue we wish to determine. In prin- ciple, our task is to find solutions to the above differential...

work page

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This operator is defined by expandingSto quadratic order aroundθ= ¯θ: H(2) ij =−M ij∂2 τ + (2δij −δ i,j+1 −δ i,j−1) +U ′′(¯θ(τ))δ ij

Eigenvalues ofH (2) Next we consider the Hessian operatorH (2). This operator is defined by expandingSto quadratic order aroundθ= ¯θ: H(2) ij =−M ij∂2 τ + (2δij −δ i,j+1 −δ i,j−1) +U ′′(¯θ(τ))δ ij . Here we take ¯θto be the instanton path ¯θ(τ;τ ∗)given in (3.17) withτ ∗ = 0. Plugging in our specific choice ofU(θ)(3.2) along with the formula for ¯θ(3.17),...

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First consider solutions for whichΩiscomplex

Ratio of determinants To compute the ratio of determinants (3.20), let us study the solutionsΩto the equations (D14) in the limit of largeβ. First consider solutions for whichΩiscomplex. Without loss of generality, we can assume thatΩhas a positive imaginary part, sinceΩand−Ωcorrespond to the same eigenfunction (D13). IfΩhas a positive imaginary part, the...

work page

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Ground-state degeneracy of the frac- tional quantum hall states in the presence of a random potential and on high-genus riemann surfaces,

X. G. Wen and Q. Niu, “Ground-state degeneracy of the frac- tional quantum hall states in the presence of a random potential and on high-genus riemann surfaces,” Phys. Rev. B41, 9377– 9396 (1990)

work page 1990

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Fault-tolerant quantum computation by anyons,

A.Yu. Kitaev, “Fault-tolerant quantum computation by anyons,” Annals of Physics303, 2–30 (2003)

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Unpaired majorana fermions in quantum wires,

A Yu Kitaev, “Unpaired majorana fermions in quantum wires,” Physics-Uspekhi44, 131 (2001)

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Non-abelian anyons and topo- logical quantum computation,

Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freed- man, and Sankar Das Sarma, “Non-abelian anyons and topo- logical quantum computation,” Rev. Mod. Phys.80, 1083–1159 (2008)

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James R Kirkwood and Lawrence E Thomas, “Expansions and phase transitions for the ground state of quantum ising lattice systems,” Communications in Mathematical Physics88, 569– 580 (1983)

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On the stability of topological phases on a lattice,

Israel Klich, “On the stability of topological phases on a lattice,” Annals of Physics325, 2120–2131 (2010)

work page 2010

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Topological quantum order: Stability under local pertur- bations,

Sergey Bravyi, Matthew B. Hastings, and Spyridon Micha- lakis, “Topological quantum order: Stability under local pertur- bations,” Journal of Mathematical Physics51, 093512 (2010)

work page 2010

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A short proof of sta- bility of topological order under local perturbations,

Sergey Bravyi and Matthew B Hastings, “A short proof of sta- bility of topological order under local perturbations,” Commu- nications in Mathematical Physics307, 609–627 (2011)

work page 2011

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Stability of frustration-free hamiltonians,

Spyridon Michalakis and Justyna P Zwolak, “Stability of frustration-free hamiltonians,” Communications in Mathemat- ical Physics322, 277–302 (2013)

work page 2013

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Quasi-locality bounds for quantum lattice systems. i. lieb- robinson bounds, quasi-local maps, and spectral flow automor- phisms,

Bruno Nachtergaele, Robert Sims, and Amanda Young, “Quasi-locality bounds for quantum lattice systems. i. lieb- robinson bounds, quasi-local maps, and spectral flow automor- phisms,” Journal of Mathematical Physics60, 061101 (2019)

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Quasi-locality bounds for quantum lattice systems. part ii. per- turbations of frustration-free spin models with gapped ground states,

Bruno Nachtergaele, Robert Sims, and Amanda Young, “Quasi-locality bounds for quantum lattice systems. part ii. per- turbations of frustration-free spin models with gapped ground states,” Annales Henri Poincar´e23, 393–511 (2022)

work page 2022

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