Long-lived memory in sliding spin chains

Charles Stahl; Ethan Lake

REVIEW 3 major objections 4 minor 32 references

Sliding two Ising chains past one another can stretch the magnetic memory of a 1D magnet from e^{O(βJ)} to e^{O((βJ)^2 v ln v)}.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 02:14 UTC pith:2EB2G24H

load-bearing objection The shearing mechanism is plausible, but the central scaling law doesn't follow from the authors' own equations. the 3 major comments →

arxiv 2607.14383 v1 pith:2EB2G24H submitted 2026-07-15 cond-mat.stat-mech cond-mat.softnlin.CG

Long-lived memory in sliding spin chains

Charles Stahl , Ethan Lake This is my paper

classification cond-mat.stat-mech cond-mat.softnlin.CG MSC 82B2082C20

keywords magnetic frictionmemory timeIsing chainthermalizationnonequilibrium dynamicsmetastabilityGlauber dynamicssliding drive

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

At rest, a one-dimensional Ising ferromagnet can only remember its initial magnetization for a time that grows exponentially in the inverse temperature. This paper shows that when two such chains slide past one another at a modest relative speed, that memory time jumps to a much larger exponential, scaling as e^{(βJ)^2 v ln v}. The reason is that sliding shears apart the two-chain minority domains that would otherwise grow diffusively and erase the memory; once separated, the single-chain halves are quickly shrunk away by ballistic erosion. The paper supports this picture with analytic estimates of the erosion length and with forward-flux-sampling numerics, finding memory times around 21 orders of magnitude longer than the static case for representative parameters. If correct, this offers a simple nonequilibrium mechanism for substantially slowing thermalization in one dimension.

Core claim

The discovery is that sliding two Ising chains past one another converts the equilibrium diffusive growth of minority domains into a ballistic erosion mechanism. For v > v_c = 2/(1+e^{-2βJ}), a two-chain minority domain is sheared apart faster than interchain coupling can stitch it back together; the resulting single-chain halves shrink from their endpoints at velocity δ = (1+e^{-2βJ})^{-1}. The paper argues via a 'danger region' spacetime diagram that rare spin flips can only grow a domain by the multiplicative factor (1+δ/2v), which leads to a critical erosion length ξ_er ~ v e^{βJ}. Since nucleating a domain of this size requires about ln(ξ_er) rare events, the memory time becomes t_mem ~

What carries the argument

The central object is the danger region DR(l), a triangular spacetime region of area |DR(l)| ≈ l^2 δ/(4v(v−δ)) adjacent to a minority domain of size l. A single majority spin flip inside DR(l) can increase the domain size by at most the factor (1+δ/2v). These two geometric inputs turn the Arrhenius nucleation cost e^{2βJ} per flip into an exponent that grows as ln(ξ_er)/ln(1+δ/(2v)) ≈ 4(βJ)^2 v ln v, and they yield the erosion length ξ_er ~ v e^{βJ} as the balance point where a flip in DR becomes likely.

Load-bearing premise

The entire derivation leans on the geometric picture that the area of the danger region is l^2 δ/(4v(v−δ)) and that a spin flip inside it grows the domain by at most (1+δ/2v); these geometric estimates are asserted from the spacetime diagram rather than derived from the Glauber rates, so if the true stochastic geometry differs, the erosion length and the memory-time exponent would shift.

What would settle it

Perform a kinetic Monte Carlo simulation of a single minority domain in a sliding ladder and record the distribution of domain-size changes per spin flip; the paper's argument requires that a single flip inside the danger region grow the domain by no more than the factor (1+δ/2v), so any flip that grows the domain by a larger factor would falsify the predicted exponent 4(βJ)^2 v ln v.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

Sliding provides a purely mechanical knob to slow thermalization in a 1D magnet: increasing v to just above the threshold v_c takes the memory time from e^{O(βJ)} to e^{O(v(βJ)^2 ln v)}.
Below the erosion length ξ_er ~ v e^{βJ}, all minority domains are actively erased; only domains larger than this length behave like equilibrium diffusing droplets.
For speeds v = Ω(log L), the effective interchain coupling becomes all-to-all and the system develops genuine long-range order with infinite memory time in the thermodynamic limit.
Adding a symmetry-breaking magnetic field h simply replaces J by J − |h|, so the memory enhancement is independent of the Z2 symmetry.
At the representative parameters T = 0.6J and v = 5, the predicted memory time is ~10^{28}, about 21 orders of magnitude longer than at v = 0.

Where Pith is reading between the lines

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

The danger-region geometry is a specific case of a general principle: a drive that caps the per-event growth of a rare fluctuation to a factor (1+ε) converts a Boltzmann suppression e^{-βΔ} into an exponent proportional to ln(L_c)/ε, where L_c is the critical size. Designing other drives with small ε could give even longer memories.
The same shearing mechanism might be realized in two-dimensional sliding layers or in shaken granular systems, where the erosion length could be tuned by geometry; the paper only explores the 1D ladder.
The paper's phase diagram predicts that at fixed βJ the memory time grows exponentially with v; an experiment with cold atoms in optical lattices, where sliding can be engineered via a moving lattice, could test this directly.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

The shearing mechanism is plausible, but the central scaling law doesn't follow from the authors' own equations.

read the letter

Good idea, but the key formula is wrong. The advertised scaling t_mem ~ exp(4(βJ)^2 v ln v) does not come out of their own calculation. Eq. (8) gives an exponent ~4(βJ)^2 v/δ after dropping ln v; if you keep ln ξer = βJ + ln v, the leading term is still 4(βJ)^2 v/δ, and the ln v appears only in a subleading term of order βJ v ln v. So Eq. (4) is not derived; the paper's own math points to a different scaling.

What is new here: the shearing picture for sliding Ising chains is a natural and plausible mechanism. Past magnetic-friction work focused on the v→∞ mean-field limit; this paper starts to address finite v. The geometric argument for the erosion length ξer ~ v e^{βJ} is simple and appealing, and Fig. 3a supports it. They also ship code and data, and the FFS results are honest about the difficulties. The observation that steady-state energy can decrease with speed is worth noting.

The soft spots: the central scaling claim is load-bearing, and the numerics don't discriminate between v and v ln v. The temperature dependence is tested with a modified 'nucleationless' dynamics, which is a reasonable trick but not the original model. There are also arbitrary thresholds in the definitions of t_mem and ξer; those are probably secondary. If the corrected scaling is 4(βJ)^2 v/δ, the qualitative conclusion — sliding parametrically lengthens memory — still holds, so the paper could be fixed. But as written, the abstract and main text promise something the derivation and numerics don't deliver.

This is for people interested in nonequilibrium order, memory in 1D, magnetic friction, and metastability. It's a worthwhile read, but it needs a serious revision. I'd send it to peer review with clear instructions to fix the derivation and either correct the claimed scaling or redo the numerics to discriminate. If the authors can't reconcile the discrepancy, reject. Otherwise it could be a solid publication.

Takeaway: don't trust Eq. (4) as written; the mechanism is plausible, the quantitative claim is not.

Referee Report

3 major / 4 minor

Summary. The paper studies two ferromagnetic 1D Ising chains sliding past one another at relative velocity v and coupled to a thermal bath. It claims that for v above a small threshold the memory time of the magnetization increases from exp(O(βJ)) at v=0 to exp(O((βJ)^2 v ln v)) at finite v. The proposed mechanism is that sliding shears apart two-chain minority domains; the resulting single-chain domains are then ballistically eroded, unless the initial domain is larger than an erosion length ξer ∼ v exp(βJ). The paper derives Eq. (8), a lower bound for t_mem in terms of ξer, then states the main scaling Eq. (4), and reports forward-flux and modified-dynamics numerics in support.

Significance. If the central claim were correct, this would be a notable result: a simple, local, translation-invariant drive would parametrically enhance memory in a 1D spin system, going beyond the equilibrium Arrhenius bound. The paper is also commendable for making its code available and for using forward-flux sampling to probe extremely long time scales. However, the advertised quantitative scaling is not actually obtained from the paper's own derivation, and the numerical tests do not discriminate between the claimed v ln v dependence and a simpler v dependence. The qualitative shearing picture may survive, but the specific conclusion advertised in the abstract and Eq. (4) is unsupported as written.

major comments (3)

[§III, Eq. (8) versus Eq. (4)] Equation (4) does not follow from Eq. (8). With ξer = c v e^{βJ}, the number of multiplicative growth steps is n = ln(ξer)/ln(1+δ/2v), and Eq. (8) gives t_mem ≳ b exp(2βJ n). Expanding for δ/2v ≪ 1 yields exponent ≈ (4βJ v/δ) ln ξer = 4(βJ)^2 v/δ + 4βJ v ln v/δ. For large βJ, δ ≈ 1, so the leading term is 4(βJ)^2 v, not 4(βJ)^2 v ln v. The transition from Eq. (8) to Eq. (4) effectively drops the ln v contained in ln ξer and then reintroduces a v ln v factor outside the logarithm. The advertised e^{O((βJ)^2 v ln v)} scaling is therefore not derived, and the abstract, Fig. 1b, and the outlook overstate the result.
[Fig. 2 and §III, erosion-length argument] The central input to the argument is the danger-region geometry: the area |DR(l)| ≈ l^2 δ/(4v(v−δ)) and the bound that a successful spin flip enlarges the minority domain by at most a factor 1+δ/2v. These are asserted from the schematic in Fig. 2 rather than derived from the Glauber transition rates. Since both ξer ∼ v e^{βJ} and the t_mem exponent are consequences of this geometry, an independent derivation or a direct numerical measurement of |DR(l)| and of the maximal domain growth per flip is needed. Without it, the scaling prediction is not closed.
[Numerics, Fig. 3(b,c)] The numerical validation does not test the v ln v form. In Fig. 3b, linear fits of ln t_mem versus v at fixed βJ are equally consistent with exp(C v) and exp(C v ln v) because ln v is slowly varying over the fitted range. In Fig. 3c, the (βJ)^2 scaling is tested only after switching to a modified 'nucleationless' dynamics, and the text states that fits of ln tmem to (βJ)^a have lower R² for a=2 than for a=1. As written, that is evidence against the claimed quadratic law, not for it. Please report the actual R² values, justify the use of the modified dynamics, and provide a scaling collapse or direct comparison that can distinguish the candidate exponents.

minor comments (4)

[§III, text before Eq. (8)] The multiplicative growth factor is given as l → l(1+δ/2v), but the expression for n_{ξer} uses ln(1+2δ/v) and Eq. (8) later uses ln(1+δ/2v). These denominator conventions are inconsistent; the numerical prefactor in the exponent depends on this choice.
[Eq. (4)] The error term 'O(βJ)' is too weak: even in the corrected version of the argument, the subleading term is βJ v ln v / δ, which is not O(βJ) for fixed v > 1. The asymptotic statement should specify the dependence on v.
[Definitions in Eq. (1) and Eq. (9)] The thresholds 1/2 for t_mem and 3/4 for ξer are arbitrary, as is the factor 10 used to define the crossover in Fig. 1b. For asymptotic scaling exponents this is usually harmless, but the paper should state that the reported exponents are insensitive to these choices, or present a short sensitivity check.
[Threshold condition for shearing] The main result states v > 2/(1+e^{-2βJ}), while the numerics and some text use v > 2. These are close for large βJ but not identical; please make the condition used in each plot explicit.

Circularity Check

0 steps flagged

No circularity: central scaling is not forced by inputs, though Eq. (4) is not actually derived from Eq. (8) — a correctness gap, not a circular one.

full rationale

The paper's derivation of the memory-time scaling is not circular: the erosion length ξer is obtained from a stated geometric estimate of the danger-region area and the Glauber flip probability, then checked numerically, rather than fitted to the target tmem. The arbitrary thresholds (1/2, 3/4, 10) and the undetermined prefactor c affect only constant factors and do not re-inject the conclusion into the premises. Self-citations [8]–[11] are contextual, and [26] points to the authors' own supplementary material and code; the central geometry is described in the main text and is externally checkable, so this is not a load-bearing self-citation chain. The main internal issue is that the step from Eq. (8) to Eq. (4) is a non-sequitur: writing ln ξer ≈ βJ drops the ln v term, so the advertised e^{4(βJ)^2 v ln v} dependence does not follow from the paper's own equations. That is a correctness gap, not a circular reduction: the advertised scaling is not equivalent by construction to the inputs, nor is a fitted parameter renamed as a prediction. Hence no significant circularity is present.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The central claim rests on heuristic geometric arguments about deterministic erosion and danger-region growth, not on a rigorous derivation from the Glauber rates. Several constants and thresholds are chosen by hand, and the numerical validation uses a modified dynamics. No new physical entities are introduced.

free parameters (3)

prefactor c in ξ_er = c v e^{βJ}
The geometric estimate fixes the scaling form but not the numerical prefactor; numerical slope fits confirm the v and e^{βJ} dependence, but c is undetermined and does not enter the advertised exponent.
arbitrary thresholds 1/2 (t_mem), 3/4 (ξ_er), 10 (crossover)
Chosen by hand to define the memory time, erosion length, and crossover; the paper asserts insensitivity to these choices but provides no sensitivity analysis.
seeding cost c = 8 = 8
Energy cost to nucleate a size-1 minority domain (first flip ΔE=6J, second ΔE=2J); used to splice the e^{cβJ} nucleation factor onto the modified-dynamics result.

axioms (5)

domain assumption A single-chain minority domain is eroded ballistically from its endpoints at velocity δ=(1+e^{-2βJ})^{-1}.
Stated after Eq. (5); the endpoint velocity follows from Glauber rates, but treating erosion as deterministic is a simplification used throughout.
domain assumption If v/2 > δ, two-chain minority domains are sheared apart faster than the interchain coupling can stick them together.
Stated in 'The shearing mechanism'; no rigorous proof from the stochastic dynamics is given, yet it is the basis for the entire t_mem enhancement.
domain assumption The danger region has area |DR(l)| ≈ l²δ/(4v(v−δ)) and a successful flip in DR grows the domain by at most factor (1+δ/2v).
Assumed from the geometry of Fig. 2 in 'The erosion length'; this is the backbone of the ξ_er and t_mem derivation and is not derived from the Hamiltonian.
domain assumption Minority domains with size l > ξ_er are not ballistically eroded and their endpoints diffuse like equilibrium Ising domains.
Asserted after Eq. (7) and illustrated in Fig. 1d; no proof that erosion probability drops sharply at ξ_er.
ad hoc to paper The modified 'nucleationless' dynamics, which instantly inserts a size-2 minority domain at maximal/minimal magnetization, has the same growth/fission statistics as the original dynamics apart from the e^{cβJ} nucleation factor.
Introduced in the numerics section to remove the e^{cβJ} factor; the equivalence is assumed, not proven.

pith-pipeline@v1.3.0-alltime-deepseek · 7719 in / 18839 out tokens · 177419 ms · 2026-08-02T02:14:19.396253+00:00 · methodology

0 comments

read the original abstract

We study a system of two ferromagnetic one-dimensional Ising chains coupled to a thermal bath, which are driven out of equilibrium by being moved past one another at a constant speed. We show that even at modest speeds, magnetic friction between the two chains significantly increases the ability of the system to order. In particular, at inverse temperature $\beta$, Ising coupling $J$, and sliding speed $v$, the dynamics retains memory of its initial magnetization for a time that increases from $\exp(O(\beta J))$ at $v = 0$ to $\exp(O((\beta J)^2v\ln v))$ when $v>v_c$, where $v_c$ is a small constant. Magnetic friction thus provides a simple mechanism for parametrically slowing down thermalization in a one-dimensional magnet.

Figures

Figures reproduced from arXiv: 2607.14383 by Charles Stahl, Ethan Lake.

**Figure 2.** Figure 2: FIG. 2. How a minority domain of size [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: a shows the result of measuring (9) in numerics, and confirms the predicted scaling (7) at large enough βJ, v. We now confirm that tmem indeed scales as in (4). Doing so is infeasible with direct Monte Carlo updates, since the estimated values of tmem quickly become extremely large. Fortunately, forward flux sampling (FFS) [28]—a technique that selectively isolates updates that move the magnetization out … view at source ↗

discussion (0)

Reference graph

Works this paper leans on

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[19]

Thus in our units, if we use a lattice spacing of 10−10m, a velocity of v = 1 corresponds to a dimensionful speed of around 1 cm s−1

In typical magnetic systems, the time for a local moment to relax to its Weiss field is around 10 −8s [30]. Thus in our units, if we use a lattice spacing of 10−10m, a velocity of v = 1 corresponds to a dimensionful speed of around 1 cm s−1
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The present paper investigates the dynamics of a similar phenomenon in 1d

It was however observed in [20] that sliding two 3d Ising models along a 2d interface can increase the magnetiza- tion near the interface. The present paper investigates the dynamics of a similar phenomenon in 1d
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A. L. Toom, Nonergodic multidimensional system of au- tomata, Problemy Peredachi Informatsii 10, 70 (1974)

1974

[2] [2]

A. L. Toom, Stable and attractive trajectories in multi- component systems, Multicomponent random systems 6, 549 (1980)

1980

[3] [3]

P. Gacs, G. L. Kurdyumov, and L. A. Levin, One- dimensional uniform arrays that wash out finite islands, Problemy Peredachi Informatsii 14, 92 (1978)

1978

[4] [4]

Gacs, Reliable cellular automata with self- organization, Journal of Statistical Physics 103, 45 (2001)

P. Gacs, Reliable cellular automata with self- organization, Journal of Statistical Physics 103, 45 (2001)

2001

[5] [5]

B. Cirelson, Reliable storage of information in a system of unreliable components with local interactions, in Lo- cally Interacting Systems and Their Application in Biol- ogy (Springer, 2006) pp. 15–30

2006

[6] [6]

D. D. Tang and Y.-J. Lee, Magnetic memory: fundamen- tals and technology(Cambridge University Press, 2010)

2010

[7] [7]

Mordvintsev, E

A. Mordvintsev, E. Randazzo, E. Niklasson, and M. Levin, Growing neural cellular automata, Distill 5, e23 (2020)

2020

[8] [8]

Pajouheshgar, A

E. Pajouheshgar, A. Bhardwaj, N. Selub, and E. Lake, Exploring the landscape of nonequilibrium memories with neural cellular automata, Physical Review Letters 136, 037102 (2026)

2026

[9] [9]

Lake and S

E. Lake and S. Ro, Squeezing codes: robust fluctuation- stabilized memories, arXiv preprint arXiv:2509.20730 (2025)

Pith/arXiv arXiv 2025

[10] [10]

Stahl, B

C. Stahl, B. Placke, V. Khemani, and Y. Li, Slow mixing and emergent one-form symmetries in three-dimensional Z2 gauge theory, arXiv preprint arXiv:2601.06010 (2026)

arXiv 2026

[11] [11]

Stahl, E

C. Stahl, E. Lake, and V. Khemani, Brownian ratchets and pumps universally simulate many-body active dy- namics, arXiv preprint arXiv:2607.01231 (2026)

Pith/arXiv arXiv 2026

[12] [12]

Toom, Cellular automata with errors: problems for students of probability, Topics in contemporary proba- bility and its applications 117 (1995)

A. Toom, Cellular automata with errors: problems for students of probability, Topics in contemporary proba- bility and its applications 117 (1995)

1995

[13] [13]

Park, Ergodicity and mixing rate of one-dimensional cellular automata, Tech

K. Park, Ergodicity and mixing rate of one-dimensional cellular automata, Tech. Rep. (Boston University Com- puter Science Department, 1997)

1997

[14] [14]

Up to a factor of the system size, if one uses the usual convention for the mixing time in the mathematics liter- ature

[15] [15]

D. A. Levin and Y. Peres, Markov chains and mixing times (American Mathematical Society, 2026)

2026

[16] [16]

Bovier and F

A. Bovier and F. Den Hollander, Metastability: a potential-theoretic approach(Springer, 2016)

2016

[17] [17]

Temme, Thermalization time bounds for Pauli sta- bilizer Hamiltonians, Communications in Mathematical Physics 350, 603 (2017)

K. Temme, Thermalization time bounds for Pauli sta- bilizer Hamiltonians, Communications in Mathematical Physics 350, 603 (2017)

2017

[18] [18]

f (x) ∼ e−x2/2σ2 without qualitatively changing the physics

The floor and ceiling functions in this expression are not important, and can be replaced by a smoother kernel like P i P j f (i − j + vt)s(t) i s(b) j with e.g. f (x) ∼ e−x2/2σ2 without qualitatively changing the physics

[19] [19]

Thus in our units, if we use a lattice spacing of 10−10m, a velocity of v = 1 corresponds to a dimensionful speed of around 1 cm s−1

In typical magnetic systems, the time for a local moment to relax to its Weiss field is around 10 −8s [30]. Thus in our units, if we use a lattice spacing of 10−10m, a velocity of v = 1 corresponds to a dimensionful speed of around 1 cm s−1

[20] [20]

Kadau, A

D. Kadau, A. Hucht, and D. E. Wolf, Magnetic friction in Ising spin systems, Physical review letters 101, 137205 (2008)

2008

[21] [21]

Hucht, Nonequilibrium phase transition in an exactly solvable driven Ising model with friction, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics 80, 061138 (2009)

A. Hucht, Nonequilibrium phase transition in an exactly solvable driven Ising model with friction, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics 80, 061138 (2009)

2009

[22] [22]

Hilhorst, Two interacting Ising chains in relative mo- tion, Journal of Statistical Mechanics: Theory and Ex- 6 periment 2011, P04009 (2011)

H. Hilhorst, Two interacting Ising chains in relative mo- tion, Journal of Statistical Mechanics: Theory and Ex- 6 periment 2011, P04009 (2011)

2011

[23] [23]

The present paper investigates the dynamics of a similar phenomenon in 1d

It was however observed in [20] that sliding two 3d Ising models along a 2d interface can increase the magnetiza- tion near the interface. The present paper investigates the dynamics of a similar phenomenon in 1d

[24] [24]

In dimensionful units, typical parameters for magnetic solids would give a lifetime of ≈ 2.5×1011yr at T ∼ 150K and a sliding velocity of v ≈ 5cm s−1 [30]

[25] [25]

P. G. de S´ a and C. Maes, The Gacs-Kurdyumov-Levin automaton revisited, Journal of Statistical Physics 67, 507 (1992)

1992

[26] [26]

See the following github repository for the supplementary material and full codebase: github.com/ethanlake/sliding-layers (2026)

2026

[27] [27]

It is interesting to note that this line of reasoning, specif- ically the part where we argue that nξer ∼ ln(ξer), is sim- ilar to the approach used to show that the mixing time of the East model scales as eO(β2) [32]

[28] [28]

R. J. Allen, C. Valeriani, and P. Rein ten Wolde, For- ward flux sampling for rare event simulations, Journal of physics: Condensed matter 21, 463102 (2009)

2009

[29] [29]

Which, incidentally, is not the Gibbs state of any local Hamiltonian [22]

[30] [30]

Angst, A

S. Angst, A. Hucht, and D. E. Wolf, Strongly anisotropic nonequilibrium phase transition in Ising models with fric- tion, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics 85, 051120 (2012)

2012

[31] [31]

Igl´ oi, M

F. Igl´ oi, M. Pleimling, and L. Turban, Nonequilibrium phase transition in a driven Potts model with friction, Physical Review E—Statistical, Nonlinear, and Soft Mat- ter Physics 83, 041110 (2011)

2011

[32] [32]

J. P. Garrahan, P. Sollich, and C. Toninelli, Kinetically constrained models, Dynamical heterogeneities in glasses, colloids, and granular media 150, 111 (2011)

2011