NOTE-692: Symmetries and phase diagrams with real-space mutual information neural estimation

Contribution

LIT-878 announced the neural estimator for real-space mutual information (RSMI) and used it on one model. This paper is the full account of the algorithm and adds three things. It tests the information value itself as a phase diagnostic on a second model, the 2D Ising model, with a buffer-scaling analysis that separates ordered, critical and disordered phases. It introduces the ensemble of optimal filters from independent runs as an object of study, and shows that its geometry carries the system’s symmetries, broken and emergent, and that PCA of it recovers the pristine operators from data where they never appear alone. And it gives a procedure for the number of coarse variables, and a first run out of equilibrium.

Key insight

The RSMI objective does not pick out one filter; it picks out a set of equally good ones, and the shape of that set is information about the system. Where a symmetry acts on the filters and leaves the information unchanged, independent optimizations scatter over the symmetry’s orbit, so the scatter of many runs, projected onto the right basis, draws the representation of the symmetry: a discrete cycle for a broken lattice symmetry, a circle for an emergent continuous one.

Assumptions

• Everything in NOTE-681 carries over: a sampled distribution, a block–buffer–environment partition fixed in advance, translation invariance (one Λ for all blocks), a linear filter before discretization, and InfoNCE as a lower bound. Here the bound’s ceiling is stated: InfoNCE ≤ log K, so it cannot be tight when e^I > K (Eq. A10). With K = 100 (Ising) and 400 (dimers) the ceiling is about 4.6 and 6.0 nats against at most log 4. Tightness below the ceiling is still not measured.
• Geometry per model (Table I). Ising: L_V = 4 (ferromagnet) or 5 (antiferromagnet), L_B from 0 to 8, environment shell L_E = 10, one binary component, 64 × 64 lattice. Dimers: L_V = 8, L_B ∈ {2, 4, 6, 8}, L_E = 4, two binary components, 64 × 64 lattice, 32 400 samples.
• The coarse variable’s alphabet is chosen (binary). Its number of components is chosen by the saturation procedure of Sec. III D 2, not fixed blind.
• The optimality theorem is cited: that RSMI optima are the most relevant operators was “formally proven, at least for critical systems in equilibrium”, in Gordon et al. (2021). Off criticality and out of equilibrium the identification is by example.
• The symmetry readings need a basis. The projections of Fig. 9 are onto the pristine filters (P₁, P₂) and (E_x + E_y, E_x − E_y), which were identified using the known field theory.

Key results

• Ising phase diagram from the information (Figs. 3, 6). Below T_c the maximal I_Λ is exactly 1 bit at every buffer: it counts the two ferromagnetic sectors. At T_c it shows a step that sharpens as L_B grows. In the paramagnet it decays exponentially with L_B and is numerically zero for L_B ≥ 2 at T = 4. At the critical point it decays algebraically, fitted exponent υ ≈ 0.09.
• Dimer phase diagram (Figs. 5a, 6c). log 4 below T_BKT; algebraic decay above it, I_Λ(T → ∞) ~ L_B^(−υ) with υ ≈ 1.16. The authors read the continuity of the filters’ T-dependence above T_BKT as a further sign of a BKT-type transition.
• Ising filters (Figs. 3c, 7). Uniform at low T (the magnetization), staggered for the antiferromagnet, random at high T; at T_c they couple to the block’s boundary, because at criticality the shared information scales with the interface area (citing Wilms, Troyer and Verstraete, and Wolf et al.). The boundaryness ratio (Eq. 19) peaks at T_c and sharpens with L_B; for T slightly below and above T_c the filters separate with growing L_B toward the ferromagnetic and paramagnetic fixed points, “consistent with” the repulsive Ising fixed point. The transformations are not iterated.
• Dimer filters (Fig. 5b). As in LIT-878: columnar and plaquette at low T, staggered (electric fields) at high T, and in between a continuous interpolation between plaquette and staggered filters, attributed to the competition of the cosine and gradient terms of the sine-Gordon action in a finite system.
• PCA of the ensemble (Fig. 8). On filters from 0.7 < T < 3.7, where no pristine filter appears alone, the leading principal components are the pristine plaquette and staggered filters, so the operators are recoverable from a parameter window that never isolates them.
• Broken symmetries in the ensemble (Fig. 9a–c; 500 runs per temperature). Below T_BKT the projection onto (P₁, P₂) has four peaks (±P₁, ±P₂) and a central one (±C). C4 acts as a Z4 cycle on the peaks; translations T_x, T_y act on ±P₁,₂ as Z2 × Z2; ±C is invariant, a trivial representation. Columnar and plaquette pairs appear with equal frequency, reflecting their equal information. Just above T_BKT the central peak vanishes (C4 restored) and four diagonal peaks appear: the filters X = P₁ − P₂, Y = P₁ + P₂, which read site parity and give a second, one-dimensional representation of the broken translation. As T rises the ±P₁,₂ peaks broaden and go.
• Emergent U(1) (Fig. 9d–g). At T = 15 the projection onto the electric-field pair is approximately invariant under continuous rotations, and RSMI is constant in the rotation angle ϑ of (cos ϑ E_x − sin ϑ E_y, sin ϑ E_x + cos ϑ E_y). Below T_BKT the filters do not overlap the fields; just above it the U(1) has not yet emerged, because of plaquette correlations in the finite system.
• Why a continuous symmetry survives a discrete H (Sec. III D 1, Fig. 10, Eqs. 20–22). The filter acting on dimer bonds equals the lattice gradient of the block-averaged height, and block averaging turns the four-valued lattice height into a quasi-continuous field like the sine-Gordon one. The same filters are obtained without discretization; τ matters only in the ordered phase, where it shrinks the search space, an “a posteriori entropy cut-off”.
• How many components (Fig. 11). Information rises with the number of binary components until it saturates: two for dimers in both limits (log 4 below T_BKT, about ½ log 2 at T → ∞ for L_B = 4), one for Ising. One dimer component gives half the two-component value with a filter equal to one of the pair; extra components converge to filters linearly dependent on the first two.
• Chipping and aggregation (Sec. IV, Fig. 12). On the 1D model of Rajesh and Majumdar at density 1 (L = 256, L_B = 8), the optimal filter averages the block mass at every w; the maximal information saturates at different values in the two phases and peaks slightly at the transition.

Claims

id	claim	strength	support
C1	In ordered phases the maximal RSMI equals the log of the number of symmetry-broken sectors (1 bit Ising, 2 bits columnar dimers)	moderate	two models, Figs. 3a, 5a; the sector-counting reading is argued, not proved
C2	The scaling of maximal RSMI with the buffer distinguishes exponential (disordered) from algebraic (critical) correlations	moderate	Ising paramagnet vs Ising T_c and free dimers, Fig. 6; two fitted exponents, no error bars
C3	At the Ising critical point the optimal filter couples to the block’s boundary, and the boundaryness peak locates T_c more sharply as the buffer grows	moderate	Figs. 3c, 7; area-law explanation cited
C4	The ensemble of optimal filters from independent runs maps an RSMI-degenerate subspace whose structure carries the symmetries, broken and emergent	moderate for this model	Fig. 9, 500 runs per T; dimer model only; projections onto bases chosen from the known operators
C5	PCA of the ensemble over an intermediate window recovers the pristine operators	moderate	Fig. 8; one window, one model
C6	Discretizing H does not prevent a continuous symmetry from appearing in the ensemble and regularizes the ordered-phase search	moderate	Eqs. 21–22, Fig. 10; “the same filters are obtained” without discretization, stated without a figure
C7	The number of coarse components at which RSMI saturates is the number of relevant operators carrying the long-range information	moderate	Fig. 11, dimers in two limits; Ising asserted
C8	RSMI-NE detects a non-equilibrium phase transition from configurations alone	weak	one 1D model, one figure; the filter is mass averaging throughout; formal understanding said to be missing
C9	The physical interpretability comes from RSMI being a well-defined physical quantity, not from architectural choices	weak	asserted; architecture insensitivity stated, not shown

Method

As in NOTE-681, with the details this paper adds. Per mini-batch of K samples: coarse-grain each block, form the K × K score matrix F_ij = f_Θ(h_i, e_j), and ascend log Q + log K (Eq. 16) in Λ and Θ by Adam at learning rate 5 × 10⁻³. The Gumbel-softmax relaxation ε is annealed exponentially, ε_t = max(ε_min, ε_max e^(−rt)), with r = 5 × 10⁻³ and (ε_max, ε_min) = (0.5, 0.1) for Ising, (0.75, 0.1) for dimers. Training stops when successive epoch estimates differ by less than a threshold, with optional smoothing and an optional BFGS refinement of Λ alone. Dimer samples come from a directed-loop Monte Carlo started from Propp–Wilson random coverings, with bounce probabilities minimized by linear programming (Appendix C). The ensemble analyses repeat the optimization many times at each temperature and study the distribution of the returned filters: PCA over a temperature window, and projections onto pairs of pristine filters.

Concepts

• ensemble of filters: the distribution of RSMI-optimal Λ over independent runs at fixed physical parameters; introduced here as “a novel concept”.
• RSMI-degenerate subspace: the set of filters attaining the same maximal RSMI; the ensemble samples it.
• pristine filters: columnar C, plaquette P₁,₂ and staggered S₁,₂ (electric fields), found alone in the low- and high-T limits.
• boundaryness: |Σ_boundary Λ_i| / |Σ_bulk Λ_i| over the block (Eq. 19).
• compression rate: here set by the number and alphabet of H’s components; the relevant information is the system’s, not the user’s.
• site parity: which of two translation-related columnar states a region is in; read by the filters X = P₁ − P₂, Y = P₁ + P₂.

Connections

The algorithmic half reviews the variational bounds of Barber–Agakov, Nguyen–Wainwright–Jordan, MINE (LIT-256) and InfoNCE, following Poole et al. (LIT-246); the van den Oord et al. source of InfoNCE is held in the anthology (ANTH-LIT-589). The coarse-graining uses the Gumbel-softmax of Jang, Gu and Poole and Maddison et al. The physics rests on Koch-Janusz and Ringel (LIT-873), Lenggenhager et al. (LIT-881) and Gordon et al. (not held), and on Alet et al.’s interacting dimers. The PRL (LIT-878) is the short paper; this one shares its method and dimer results, as the arXiv listing’s “substantial text overlap” note says. For symmetries from learned objects it cites Bondesan and Lamacraft (not held).

Bearing on the record

• THEORY-194. Adds the 2D Ising model with a buffer-scaling analysis and a flow of filters toward the fixed points without iteration. All variables found were known, and the paper uses that knowledge to name them; no exponent is checked against an independent method. It does not meet promote_when. A candidate further source; I have not edited THEORY-194.
• THEORY-019. That account says symmetry forces degeneracy but degeneracy does not identify a symmetry. This paper reads symmetries off a degeneracy of the RSMI objective. It is consistent with THEORY-019 rather than against it: the symmetries it reports were known from the field theory, and they are drawn by projecting onto bases already identified with that theory’s operators; the paper claims to observe them, not to identify an unknown group from the ensemble alone. A case where the ensemble’s geometry identified a group nobody knew would test THEORY-019’s converse. My connection, not the paper’s.
• THEORY-017 and THEORY-002. NOTE-681 noted that the information objective fixes a subspace of filters and leaves the basis to symmetry or chance. Here the authors say so themselves (“an RSMI-degenerate subspace for the kernels”) and make the subspace the object of study. The basis remains an extra supplied from the known operators, as THEORY-017 says a basis is; and like THEORY-002’s kernels, the objective determines its optima only up to the symmetry group of what it sees.
• QUESTION-025 and THEORY-185. No direct bearing. The one transferable observation: the number of coarse variables is chosen by where the information stops growing, and extra components come out linearly dependent on the first ones, so “how many independent directions” is answered by saturation, not assumed. Nothing here involves attributes that imply each other.
• THEORY-036. As for LIT-873 and LIT-878: the buffer width is an explicit scale relative to which a description is real. My connection.
• LIT-881 and THEORY-198. Sector counting (C1) is a case of the full capture LIT-881 analyses: in the ordered phase two bits capture everything the block shares with the far environment.
• LIT-246, LIT-256. States the log K ceiling and the batch sizes NOTE-681 found missing, and shows the ceiling does not bind here.
• Anthology. A machine-learning method for physical data, built from estimators the anthology reads; its physical-sciences topic could hold it, hence anthology-candidate. No practice instruction.

Limitations

• Known answers throughout. Ising, antiferromagnet and dimer variables were all known; the symmetry projections use bases chosen from them.
• Two exponents, no error bars, and neither compared with a prediction: υ ≈ 0.09 for the Ising critical point and υ ≈ 1.16 for free dimers.
• One window, one model for the PCA and symmetry analyses; the Ising ensemble is not analysed.
• The non-equilibrium example is thin. One 1D model whose steady state is reached quickly; the optimal filter is mass averaging at every w; the transition is marked by “a slight peak”.
• Bound tightness unmeasured. The log K ceiling is cleared, but no comparison with an exactly known I_Λ is made.
• Small inconsistencies. The Fig. 2 caption says the RSMI of the antiferromagnet “at the critical point” converges to log 2, which sits oddly with the main text’s exactly 1 bit only below T_c and a step at T_c (it may be a small-buffer value); Sec. III B 1 says the Ising block is 4 × 4, Table I gives L_V = 5 for the antiferromagnet; the ensemble space for “8 × 8 two-component filters” is called 64-dimensional; one citation in Appendix B is printed “[30?]”.

Open questions

• Does the ensemble’s geometry identify a symmetry nobody knew, without projecting onto bases taken from the theory? That would test the converse THEORY-019 denies in general.
• How tight is InfoNCE here below its ceiling? An exactly computable I_Λ on a small Ising system would show it.
• What do the optimal filters mean out of equilibrium, and does a temporal buffer, which the authors propose, change the answer on a model far from equilibrium?
• Does the saturation count of components equal the number of relevant operators in a model where that number is not known?