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LIT-889: Symmetries and phase diagrams with real-space mutual information neural estimation

4 min read

Properties5
StatusActive — read 2026-10-09 (NOTE-692); worth reading as the long companion of LIT-878 and the place where the real-space mutual-information method is shown on more than one model and its outputs are read as an ensemble: the maximal information counts the ordered sectors (exactly 1 bit in the 2D Ising ferromagnet, 2 bits in the columnar dimer phase), decays exponentially with the buffer in the Ising paramagnet and algebraically at criticality, and the optimal Ising filter goes uniform, boundary and random across T_c; 500 independent runs per temperature map an RSMI-degenerate subspace of filters whose projections show the broken C4 and translation symmetries of the dimer model and, at high temperature, its emergent U(1), and PCA of the ensemble recovers the pristine operators from a window where they never appear alone. The number of coarse variables is chosen where the information stops growing. A non-equilibrium chipping model is a one-figure validation, and its formal status is said to be missing.
Version2
Tagsnatural-sciences, information-theory, representation-learning, anthology-candidate
Cited as “paper” byNOTE-692 — Symmetries and phase diagrams with real-space mutual information neural estimation
Cited as “source” byTHEORY-194 — On Monte Carlo samples of the 2D Ising model and of the 2D dimer model with added noise spins, a block coarse-graining trained to maximize mutual information with the system beyond a buffer recovered the variables renormalization already uses (Kadanoff block spins; the dimers' low-momentum electric fields, with no weight on the noise), and iterated on 128 × 128 Ising samples gave the flow away from the critical point, locating it to about 1% and giving ν ≈ 1.0 ± 0.15, while a restricted Boltzmann machine trained by contrastive divergence on the noisy dimer data coupled to the noise and to columnar patterns instead

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Doruk Efe Gökmen, Zohar Ringel, Sebastian D. Huber and Maciej Koch-Janusz (2021), Physical Review E 104(6):064106 — ARXIV-2103.16887, DOI-10.1103/PhysRevE.104.064106

Key takeaways

  • The method, written out. The real-space mutual information I_Λ(H : E) between a block’s coarse variable H and its environment beyond a buffer is bounded below by InfoNCE with a separable critic f_Θ(h, e) = v(h)ᵀu(e) (two hidden layers of 32, embedding dimension 8); H = τ ∘ (Λ · v) with τ an annealed Gumbel-softmax; Λ and Θ are trained together by Adam at one learning rate. The paper derives the Barber–Agakov, NWJ and InfoNCE bounds, proves the Gumbel-max lemma, and states InfoNCE’s log K ceiling, which here is far above a few bits (mini-batches K = 100 for Ising, 400 for dimers). Runs take 8–35 s.
  • The information counts phases. In the 2D Ising model the maximal I_Λ is exactly 1 bit below T_c at every buffer, has a step at T_c that sharpens with the buffer, and decays exponentially with the buffer in the paramagnet; at T_c it decays algebraically (fitted exponent υ ≈ 0.09). In the dimer model it is 2 bits in the columnar phase and decays as L_B^(−υ) with υ ≈ 1.16 for free dimers. In ordered phases the optimal information counts the symmetry-broken sectors.
  • The Ising filters trace a flow without iteration. Optimal filters are uniform (the magnetization) at low T, staggered for the antiferromagnet, random in the paramagnet, and couple to the block’s boundary at T_c, where the shared information scales with the interface. A “boundaryness” ratio peaks at T_c and sharpens with L_B; just above and below T_c the filters separate toward the paramagnetic and ferromagnetic fixed points as L_B grows.
  • The ensemble of filters is the new object. Independent runs return different optimal filters. Projected onto the pristine filters, 500 runs per temperature show four plaquette peaks and a columnar peak forming representations of C4 (a Z4 cycle) and of lattice translations (Z2 × Z2) in the ordered phase; a second, diagonal representation just above T_BKT; and at high T a distribution invariant under continuous rotations of the electric-field pair, with RSMI constant in the angle: the emergent U(1) of the field theory, seen on a lattice and through a binary H. PCA of filters from 0.7 < T < 3.7, where no pristine filter appears alone, returns the pristine plaquette and staggered filters as the leading components.
  • The size of H is found, not assumed. Adding binary components until I_Λ stops growing gives one for Ising and two for dimers; one dimer component gives half the information, and extra components come out linearly dependent. Discretizing H leaves the high-T filters unchanged and acts as a regularizer in the ordered phase.
  • Out of equilibrium, a sketch. On the 1D chipping and aggregation model (L = 256, L_B = 8) the optimal filter averages the mass at every aggregation rate w, while the maximal information saturates at different values in the two phases and peaks slightly at the transition. The authors say the formal understanding of the filters out of equilibrium is missing.

Standing in the record

Filed on 2026-10-09 at the owner’s request, as one of the works cited by the batch on hierarchy and hyperbolic geometry (nucleation#113 and #114) that neither record held. The batch work that cites it is Gökmen et al.’s Statistical Physics through the Lens of Real-Space Mutual Information (LIT-878), which defers its Ising, symmetry and non-equilibrium examples here. Read on its own merits (NOTE-692); it files no THEORY.

It adds a second model and a second kind of evidence to the account LIT-873 grounds, THEORY-194, but every variable it finds (magnetization, staggered magnetization, the dimer order parameters and electric fields) was known beforehand, and the paper uses that knowledge to name them, so it does not meet that account’s promote_when. Like LIT-878 it neither iterates a flow nor fits exponents against an independent method; its one exponent pair (υ) is new and unchecked.

Its ensemble analysis bears on THEORY-019 and THEORY-017: it reads a symmetry off a degeneracy of the objective, which THEORY-019 says a degeneracy alone does not license; here the symmetry was known and the reading works because the projections are made onto a basis chosen from the known operators. NOTE-692 gives the detail. It does not bear on QUESTION-025 beyond analogy: its variables are lattice operators, not word attributes. Its estimator family is read in Poole et al. (LIT-246) and MINE (LIT-256), both held here and read, and InfoNCE’s source is held in the anthology (ANTH-LIT-589). Its formal antecedents are Lenggenhager et al. (LIT-881) and Gordon et al.’s optimality theorem (Phys. Rev. Lett. 126, 240601, not held).

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