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A thermally and perturbatively stable classical spin model in two dimensions

Shankar Balasubramanian, Margarita Davydova, Ting-Chun Lin, Ryan Thorngren

arXiv:2610.22548v1quant-phcond-mat.stat-mechcs.ITmath-phmath.ITmath.MP

TL;DR

Two-dimensional classical memories have been associated with thermal or perturbative instability, leaving open whether one can achieve both. This paper analyzes a recursively constructed spin model using alternating refinements and hierarchical gauge averaging. It shows that the model’s metastable states are thermally and perturbatively stable, while leaving quantum-perturbation questions for future work.

  • Problem

    In two dimensions, known classical memories face thermal or perturbative instability, leaving open whether a memory can be stable to both.

    Source: §1 · p. 2

  • Method

    The model alternates repetition and parity refinements, and the proof uses hierarchical gauge averaging to bound effective interactions.

    Source: §1.1.2 · p. 5, §3.5 · pp. 33–34

  • Results

    The constructed 2D model has exp(N^γ) metastable states, all shown to be thermally and perturbatively stable.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The paper establishes a two-dimensional classical spin model with many thermally and perturbatively stable metastable states.

    Source: §1 · p. 2

    The effect of quantum perturbations at nonzero temperature is asserted without proof here, and their zero-temperature behavior remains for future work.

    Source: §1.2 · p. 8

Abstract

from arXiv · show

We show that the 2D classical spin model constructed in arXiv:2605.10943 hosts many metastable states at non-zero temperature which are stable to arbitrary Hamiltonian perturbations, providing the first such example. Our proof relies on a recursive application of the cluster expansion.

1 Introduction

The paper answers whether a two-dimensional classical memory can be thermally stable and robust to Hamiltonian perturbations. It presents such a model, explains its alternating recursive construction, and outlines how hierarchical gauge averaging and a Peierls argument establish stability.

1 Introduction

  • The model has exp(N^γ) metastable states, all shown to be thermally and perturbatively stable, resolving the conjecture about the 2D code.The paper identifies the code with a classical spin model in two dimensions.Source: §1 · p. 2

1.1.1 Thermal dynamics and main theorem statement

  • The theorem gives a memory lifetime ≳exp(N^α) above an O(1) inverse-temperature threshold, with exponential lifetime preserved for sufficiently weak local classical perturbations.The perturbation strength is bounded using an exponentially weighted interaction norm.Source: §1.1.1 · p. 4

1.1.2 Overview of the construction

  • Alternating repetition refinements, which raise the energy barrier, with parity refinements, which add gauge codewords and local indistinguishability, targets both forms of stability.Each refinement preserves geometric locality in R2.Source: §1.1.2 · pp. 4–5

1.1.3 Proof outline

  • Hierarchical gauge averaging bounds the effective interaction from small perturbations, and a Peierls argument then bounds escape from the encoded logical states.The proof outline uses large witness syndromes to obtain an exponentially long lifetime in a power of N for a single encoded bit.Source: §1.1.3 · pp. 7–8
  • By including gauge transformations only from sufficiently high levels, the construction encodes O(N^γ) bits with thermal and perturbative stability.For k′=θn, the stated lifetime scales as exp(N^{α(1−θ)}).Source: §1.1.3 · p. 8

1.2 Outlook

  • The authors propose possible spin-glass-like properties, while noting that configurational-entropy scaling needs proof and quantum-perturbation questions remain future work.The paper also discusses a near self-duality and preliminary phase-diagram computations.Source: §1.2 · pp. 8–9

2 The 2D classical code

The code begins with a two-bit, two-check base code and is built by alternating repetition and parity refinements. These operations preserve 2D locality while respectively improving thermal and perturbative stability.

2 The 2D classical code

  • Alternating RRep and RPar refinements build the code from a base code of two bits and two checks arranged around a unit square.The first two iterations are C(1) = RRep(C(0)) and C(2) = RPar(C(1)).Source: §2 · p. 9, Figure 2 · p. 10
  • RRep introduces redundant bits that improve thermal stability, while RPar adds gauge generators that improve perturbative stability.Source: §2 · p. 9
  • Each replacement rule turns incidence edges into distinct ladder-like structures, with spins and parity checks represented by filled and open circles.Figure 1 shows the two rules separately for the basic square.Source: Figure 1 · p. 9

2.1 Classical gauge theory as a cochain complex

At each level, the code is represented as a cochain complex whose checks determine the syndrome and Hamiltonian energy. Its zero-energy configurations modulo symmetry form two logical classes encoding one bit.

2.1 Classical gauge theory as a cochain complex

  • The i-th code is described by a 3-term cochain complex over F2, with labels for symmetry generators, bits, and checks.Source: §2.1 · p. 10
  • The Hamiltonian is defined by the number of violated checks, and symmetry transformations preserve both syndrome and energy.The preservation follows from δ1δ0 = 0.Source: §2.1 · p. 11

2.1.1 The two logical states

  • At every level, the zero-energy configurations modulo symmetry form two non-trivial first-cohomology classes that encode one classical bit.A logical spin-flip exchanges the classes while preserving syndrome and energy under the unperturbed Hamiltonian.Source: §2.1.1 · p. 11

2.2 Tanner graph and embedding

Each code level has a bipartite Tanner graph recording spins and checks, but not symmetry generators. The graph is embedded in the plane with bounds controlling interaction range and spatial density.

2.2 Tanner graph and embedding

  • The Tanner graph records spins and checks but does not include symmetry generators.Source: §2.2 · p. 11
  • The embedding assigns positions to Tanner-graph vertices and extends linearly along edges, with locality controlled by bounds on interaction range and spatial density.Source: §2.2 · p. 11
  • For the paper’s embeddings, the stated locality constants are L = 1 and M = 2.These bounds are proved in Sec. 2.5.Source: §2.2 · p. 11

2.3 Construction: Base code C(0)

The base code C(0) has two spins and two identical, redundant checks, arranged on a unit square in the planar embedding.

2.3 Construction: Base code C(0)

  • The planar embedding places the two spins at (0, 0) and (1, 1), and the checks at (0, 1) and (1, 0).Source: §2.3 · p. 12
  • With no symmetry generators at this level, the zero-energy configurations (0, 0) and (1, 1) give H1(C(0)) ≅ F2, with α(0) = (1, 1).Source: §2.3 · p. 12
  • The base code C(0) contains two spins and two identical parity checks, which are redundant but retained for symmetry and further construction.Source: Figure 3 · p. 12

2.4 Construction: Replacement process: RRep and RPar

This section defines RRep and RPar as recursive code replacements, connecting embedded Tanner-graph rules to cochain constructions and comparison maps. It establishes that both replacements preserve the cochain-complex structure and that the comparison maps preserve H1, including nontrivial classes, with additional Par-iteration symmetry generators leaving this result unchanged.

2.4.1 Replacement in the graph framework

  • For odd ℓ≥7, each replacement turns an embedded bit–check edge into a shifted two-rail ladder with (ℓ−3)/2 type-specific rungs.RRep places checks at rungs between bits; RPar places bits at rungs between checks.Source: §2.4.1 · pp. 12–13

2.4.2 Review: spectral sequences and code embedding

  • The section reviews Ref. [1, Secs. 2.3]: connecting maps raise the order G < B < C, and successive cohomology with homological perturbation transfers the differential.The assembled differential is δ = δloc + κ, with δ² = 0.Source: §2.4.2 · pp. 13–14

2.4.3 Reduced complexes and comparison maps

  • The hat complex omits inherited symmetry generators, while the tilde complex contains only newly created generators; comparison maps extend to the full codes and induce H1 isomorphisms.The reduced map is eF_R,(i) = J_R,(i)ι_R,(i); restoring inherited generators gives F_R,(i).Source: §2.4.3 · pp. 14–15

2.4.4 RRep in code embedding formalism

  • RRep introduces no new symmetry generators, and its restored-code cochain map induces H1(C(i)) ≅ H1(C(i+1)) ≅ F2, transporting the nontrivial logical class.The reduced comparison map also induces an isomorphism on H1.Source: §2.4.4 · p. 16, §2.4.5 · p. 17

2.4.6 RPar in code embedding formalism

  • RPar adds bounded-size cycle and junction symmetry generators, while its restored-code cochain map induces an isomorphism on H1.Its reduced comparison map also induces an isomorphism on H1.Source: §2.4.6 · p. 18, §2.4.7 · pp. 19–20
  • For either RRep or RPar, the full code is a cochain complex and the comparison map induces an isomorphism on H1.The map is a cochain map.Source: §2.4.7 · p. 21
  • A nontrivial class α(i) in H1(C(i)) maps to a nontrivial class in H1(C(i+1)).The induced H1 isomorphism ensures that the class remains nontrivial after either replacement.Source: §2.4.7 · p. 21
  • Par iteration adds symmetry generators that are linear combinations of existing generators, leaving the image of δ0 and the comparison result unchanged.These additional generators are included for the memory time analysis.Source: §2.4.7 · p. 21

2.5 Main statements

Starting from a two-bit, two-check base code, the family alternates Rep and Par replacements for odd ℓ≥7. Every code in the family has locality bounds L=1 and M=2 and encodes one classical bit.

2.5 Main statements

  • The family alternates Rep and Par replacements from a two-bit, two-check base code, with odd ℓ≥7.R is RRep for even i and RPar for odd i.Source: §2.5 · p. 22
  • L=1 and M=2 give locality bounds independently of the iteration i.The code also encodes one classical bit, with H1(C(i)) ≅ F2 and [α(i)] ≠ 0.Source: §2.5 · p. 22
  • For odd ℓ≥7, either replacement preserves the stated geometric properties, including integral unit lattice edges and no repeated lattice segment.The geometric argument starts from degree-two-or-three vertices and uses the ladder structure of the replacement.Source: §2.5 · p. 22

2.6 Structure of the gauge group and Wilson operators

The model’s gauge group is organized as a hierarchy, and Wilson operators are represented by chains on the dual complex. The hierarchy bounds the support of partially gauge-invariant chains and implies a minimum weight for nontrivial global Wilson operators.

Hierarchical gauge group

  • A full hierarchy iteration is a Rep + Par step from C(i) to C(i+2), with C(2i) at level i.There are n levels in the hierarchy.Source: §2.6 · p. 23
  • The construction changes the basis of gauge generators by adding sums of generators whose embedded faces overlap, while leaving the gauge group unchanged.This basis is used to capture geometric locality in the embedding.Source: §2.6 · p. 23

Wilson operators

  • Chains represent supports of Pauli-Z operators, and chains in B1,(j) represent products of checks.A chain z specifies the Pauli-Z operator supported on its constituent bits.Source: §2.6 · p. 24
  • The lift through a Rep + Par step replaces Zb by Zbb0Zbb1 and commutes with the new Par gauge generators.Rep selects the bit in the unshifted copy, and Par takes both copies.Source: §2.6 · p. 24
  • The minimum weight of a nontrivial global Wilson operator is 2n.The proof proceeds by induction over Rep + Par iterations, starting from the fact that every weight-one Pauli-Z operator anticommutes with a level-n generator.Source: §2.6 · pp. 25–26
  • Every chain in cWi, the set of chains that are partially but not fully gauge invariant, has weight at least 2^(n−i+1).Ki is decomposed into fully gauge-invariant chains K1 and the remaining chains cWi.Source: §2.6 · p. 26

3 Effective perturbation strength

The section controls perturbation strength by averaging over gauge generators recursively and applying a strengthened cluster-expansion theorem at each hierarchy level. Block-spins and interaction graphs provide the bounded-degree, locality, and norm conditions needed for this iteration.

3.1 Hierarchical gauge averaging

  • Recursive gauge averaging splits the effective interaction into fully gauge-invariant terms and partially invariant terms, whose monomials have weight at least 2^(n−i+1).The fully invariant effective terms include products of unperturbed Hamiltonian terms and global Wilson operators; the same weight bound does not hold for them because the logarithmic expansion can generate smaller-weight invariant terms.Source: §3.1 · p. 27

3.2 Improved gauge-averaging theorem

  • The strengthened gauge-averaging theorem gives an absolutely convergent, exponentially local invariant interaction while allowing an arbitrarily small loss of decay.For 0 < m < η, |Veff|m ≤ 2|V|η when β|V|η is sufficiently small relative to η−m, without a fixed lower bound on η−m.Source: §3.2 · pp. 28–29

3.3.1 Block-spins Ei

  • Block-spins group bits connected by degree-two checks, and each block contains at most three bits.Gauge transformations flip an entire block-spin or leave it unchanged.Source: §3.3.1 · p. 29

3.3.2 Hierarchical interaction graphs

  • The hierarchical interaction graphs are connected and bounded-degree, with gauge generators of degree at most 16.The degree bounds are uniform across hierarchy levels.Source: §3.3.2 · p. 30

3.3.3 Hierarchical norms

  • Interactions at one level can be regrouped at the preceding level by summing terms with the same coarse shadow.The level-i norm is defined locally with respect to the gauge-interaction graph Qi.Source: §3.3.3 · p. 31

3.3.4 Geometric locality

  • The interaction graphs are designed to track geometric distance, and the section compares their local norms with geometrically local norms.The locality and norm-comparison lemmas give constants independent of hierarchy depth and ℓ≥7.Source: §3.3.4 · p. 31

3.4 Why gauge-averaging theorem can be applied in the induction step

  • At each induction step, block-spins Ei, the level-i gauge group, and the graph Qi instantiate the gauge-averaging theorem’s inputs.The theorem applies when the degree bound and smallness condition hold for the interaction regrouped at level i.Source: §3.4 · pp. 32–33

3.5 Main result

The main result bounds the effective interaction produced by hierarchical gauge averaging. Its induction controls the decay of partially invariant terms across levels, yielding bounds used to establish perturbative stability for sufficiently small local perturbations.

3.5 Main result

  • The hierarchy uses decay parameters mi that remain bounded above and below by positive constants independent of n.Each averaging step lowers the decay parameter from mi+1 to mi, and the sequence’s total decrease must remain bounded for locality.Source: §3.5 · pp. 33, 38, 39
  • The resulting bounds are used to show perturbative stability when β|V|η,loc is sufficiently small.The proof establishes an induction step for bounding effective-interaction strength across hierarchy levels.Source: §3.5 · pp. 33, 39
  • The strength of U(i) decreases exponentially with n−i under the recursive bound on |U(i)|mi,i.The induction treats the linear averaged contribution and the remainder, which begins at quadratic order in U(i+1), separately.Source: §3.5 · pp. 34–36
  • The linear contribution to U(i) is controlled by a norm decrease under coarse-graining, while the nonlinear remainder is bounded using smallness and holomorphicity of the cluster expansion.The proof combines the two bounds into a full induction step.Source: §3.5 · pp. 35, 36, 38

4 Peierls argument

This section develops a Peierls argument to bound memory lifetime by controlling unstable configurations, using hierarchical-decoder witnesses and localized fillings to analyze perturbation effects. At sufficiently low temperature, unstable configurations are exponentially rare in a power of system size, yielding a memory lifetime that is exponentially long in a power of system size.

4.1 Main idea

  • The memory lifetime is exponentially long in a power of the number of spins because paths between stable logical classes cross an unstable set of exponentially small probability.A Peierls estimate bounds the probability of large witness weight; together with lower bounds on the stable classes’ Gibbs masses, this makes their boundary-to-mass ratios exponentially small.Source: §4.1 · p. 41

4.5 Effect of the perturbation

  • The perturbation analysis bounds the energy change upon removing a witness in proportion to its size, with small coefficient ε, and Theorem 6 considers unstable configurations under Theorem 5’s assumptions.The hierarchy index assumes odd ℓ ≥ 7; Theorem 6 states that a constant α independent of n exists.Source: §4.1 · p. 42, §4.5 · pp. 46–47

4.2 Decoder

  • The decoder recursively applies local Rep and Par corrections, then uses a minimum-weight correction on a fixed-size repetition code and lifts the corrections to the physical code.Its decoded logical bit is the class of the corrected physical configuration in the logical-state space.Source: §4.2 · pp. 42–43

4.3 Decoding graph and induced subgraph

  • The decoding graph organizes the hierarchy’s local pieces, and its degree is independent of n for fixed ℓ, enabling bounded-size counting of witness subgraphs.The witness-syndrome count is at most Γ^m for fixed ℓ.Source: §4.3 · p. 44, §4.4 · p. 46

4.4 Unstable configurations and the witness

  • Configurations with zero level-1 syndrome are stable to any single spin flip, while every nonempty witness syndrome is an error coboundary and has weight at least 2n.For fixed ℓ, the number of distinct witness syndromes of weight m is at most Γ^m, with Γ independent of n.Source: §4.4 · pp. 44–46

4.5 Effect of the perturbation

  • A recursively coarse-grained construction gives a filling of z whose level-i diameter is O(li(Xi)), with a bound independent of levels i and n.The construction localizes the filling inside a refined region built from a star-like coarse region.Source: §4.5 · pp. 48–51

4.6 Peierls argument

  • Theorem 7 bounds the Gibbs probability of an unstable configuration by 2^-2n for β ≥ βc, assuming β|V|η,loc < ε for an n-independent ε > 0.The proof combines a bound on distinct witnesses with their minimum energy; βc = log(4Γ) + αε is independent of n.Source: §4.6 · pp. 53–54
  • The stable-sector probabilities yield tmem ≳ 2^(2n)/N, which is exponentially long in a power of N under the construction’s size bound.The argument compares sector weights under a logical flip and uses the unstable-probability bound to control the sector boundaries.Source: §4.6 · pp. 55–56

A Proof of the improved gauge-averaging theorem

The proof formulates gauge averaging as a polymer cluster expansion and verifies the Kotecky-Preiss criterion using pinned activities. An induction bounds the resulting quantity K, supporting a bound on the effective perturbation.

A Proof of the improved gauge-averaging theorem

  • Gauge averages factorize across independent polymers, allowing the effective interaction to be defined through a cluster expansion over connected incompatible tuples.The tuples may repeat polymers; disconnected tuples contribute zero.Source: A Proof of the improved… · pp. 56–57
  • The Kotecky-Preiss criterion supplies a sufficient condition for absolute convergence, and the proof checks it through a pinned-activity bound.The proof notes that the resulting bound K ≤ qη will also be useful below.Source: A Proof of the improved… · pp. 57–59
  • An induction bounds contributions from polymers by removing a label, bounding the resulting connected components, and correcting their overcounting with a 1/k! factor.The argument replaces equations (53)–(55) of Ref. [6] without counting involving ∆.Source: A Proof of the improved… · pp. 58–59
  • The proof uses the cluster-expansion estimate to bound the effective perturbation strength |Veff|m and concludes the lemma.The passage does not state the bound's value.Source: A Proof of the improved… · p. 59

B Geometry of the gauge-interaction graphs

The geometry proof establishes bounded generator and block-spin degrees, then relates fine and coarse hulls through the replacement map. These bounds support the geometric estimates used in the paper.

B Geometry of the gauge-interaction graphs

C Comparison of geometric locality statements

The proof compares graph distance in the gauge-interaction graph with lattice distance using bounded generator supports and paths in the embedded Tanner graph. It also relates generator-based and lattice hull lengths.

C Comparison of geometric locality statements

  • A universal constant c2 bounds gauge-graph distance above by lattice distance, using the bounded diameter of each gauge generator's support.The proof obtains a universal bound on diam∞Sg from locality of embedding and finite support size.Source: C Comparison of geometric locality… · p. 61
  • Lattice paths can be detoured along the embedded Tanner graph's boundary arcs, yielding the reverse distance bound with c1 = 1/(6A), where A = O(1).The construction uses the rectangular and corner regions left empty by the embedding.Source: C Comparison of geometric locality… · pp. 61–62
  • A spanning tree through shared generators bounds lattice-hull length using generator-hull length, while traversing the lattice hull bounds successive spin distances.The proof states ln(X) = |Hn(X)| and lloc(X) = |HZ2(X)|, and bounds the summed successive distances by 3lloc(X).Source: C Comparison of geometric locality… · p. 62
  • The proof also bounds interactions meeting a fixed block by choosing a generator acting on that block.Source: C Comparison of geometric locality… · p. 62
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