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248 lines
9.1 KiB
Python
248 lines
9.1 KiB
Python
#!/usr/bin/env python3
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"""
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Pyrochlore–Sidon bridge: exact diagonalization for S=1 tetrahedron.
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Compares KMMC (S=5/2), NaCaNi₂F₇ (S=1), and the Sidon packing bound.
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Emits a receipt with the exact S=1 spectrum, closure fractions, and
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S-dependent trend.
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Usage:
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python3 pyrochlore_sidon_classify.py
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"""
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import hashlib
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import json
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import numpy as np
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from datetime import datetime, timezone
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Q16_SCALE = 65536
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def q16(f):
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return max(-2147483648, min(2147483647, int(f * Q16_SCALE)))
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# ─── Exact diagonalization: S=1 Heisenberg tetrahedron ────────────────────────
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dim = 3**4
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def decode(n):
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s = []
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for _ in range(4):
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s.append(n % 3 - 1)
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n //= 3
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return np.array(s)
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def encode(s):
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n = 0
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for k in range(3, -1, -1):
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n = n * 3 + int(s[k] + 1)
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return n
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def build_hamiltonian():
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H = np.zeros((dim, dim))
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for n in range(dim):
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s = decode(n)
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H[n, n] = sum(s[i] * s[j] for i in range(4) for j in range(i + 1, 4))
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for i in range(4):
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for j in range(i + 1, 4):
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if s[i] < 1 and s[j] > -1:
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fi = np.sqrt(max(0, 2 - s[i] * (s[i] + 1)))
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fj = np.sqrt(max(0, 2 - s[j] * (s[j] - 1)))
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s2 = s.copy()
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s2[i] += 1
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s2[j] -= 1
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H[n, encode(s2)] += 0.5 * fi * fj
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if s[i] > -1 and s[j] < 1:
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fi = np.sqrt(max(0, 2 - s[i] * (s[i] - 1)))
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fj = np.sqrt(max(0, 2 - s[j] * (s[j] + 1)))
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s2 = s.copy()
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s2[i] -= 1
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s2[j] += 1
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H[n, encode(s2)] += 0.5 * fi * fj
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return (H + H.T) / 2
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print("Building S=1 Heisenberg tetrahedron Hamiltonian (81×81)...")
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H = build_hamiltonian()
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e = np.linalg.eigh(H)[0]
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# Ground state analysis
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gs_degeneracy = np.sum(np.abs(e - e[0]) < 1e-10)
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gs_energy = e[0]
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print(f"Ground state energy: {gs_energy:.4f} J")
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print(f"Ground state degeneracy: {gs_degeneracy}")
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print(f"First excited: {(e[gs_degeneracy] - e[0]):.4f} J")
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# Thermodynamics
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betas = np.logspace(-3, 3, 500)
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Ts = 1 / betas
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entropy = np.array([
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np.log(np.sum(np.exp(-b * (e - e[0]))))
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+ b * np.sum((e - e[0]) * np.exp(-b * (e - e[0]))) / np.sum(np.exp(-b * (e - e[0])))
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for b in betas
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])
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spec_heat = np.array([
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(np.sum((e - e[0])**2 * np.exp(-b * (e - e[0]))) / np.sum(np.exp(-b * (e - e[0])))
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- (np.sum((e - e[0]) * np.exp(-b * (e - e[0]))) / np.sum(np.exp(-b * (e - e[0]))))**2)
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* b**2
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for b in betas
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])
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S0 = np.log(dim)
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closure = 1 - entropy / S0
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# Key temperatures for closure fractions
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results = {}
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for pct in [5, 10, 15, 20, 25]:
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target = pct / 100
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idx = np.argmin(np.abs(closure - target))
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results[f"closure_{pct}pct"] = {
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"T_J": round(Ts[idx], 4),
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"S_S0": round(entropy[idx] / S0, 4),
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}
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# Cv maximum location
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cv_max_idx = np.argmax(spec_heat)
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results["Cv_max"] = {"T_J": round(Ts[cv_max_idx], 4), "Cv": round(spec_heat[cv_max_idx], 4)}
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# Build receipt
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VERTICES = [1, 2, 4, 8]
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EDGES = [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]
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EDGE_SUMS = [VERTICES[i] + VERTICES[j] for i, j in EDGES]
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# Analytic P(L) comparison (classical S=∞ limit)
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def integrate_P(a, b):
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def F(x):
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return x**3 / 6 - 3 * x**4 / 64
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return F(b) - F(a)
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receipt = {
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"schema": "rrc_pyrochlore_sidon_v2",
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"claim_boundary": "exact_diagonalization_s1_tetrahedron;analytic_classical_limit;sidon_labels",
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"provenance": {
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"target_S1": "NaCaNi₂F₇ — Nature Physics 2018, arXiv:1711.07509",
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"target_S5_2": "K₃Mn(MoO₄)₂Cl — Adv. Mater. 2026, 2521218",
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"sidon_addresses": VERTICES,
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"edge_sums": EDGE_SUMS,
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"S1_hilbert_space": dim,
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},
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"exact_S1_spectrum": {
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"gs_energy_J": round(float(gs_energy), 4),
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"gs_degeneracy": int(gs_degeneracy),
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"first_excited_gap_J": round(float(e[gs_degeneracy] - e[0]), 4),
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"specific_heat_max": results["Cv_max"],
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},
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"thermodynamics": {
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"closure_fractions": results,
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"entropy_limit": round(float(S0), 4),
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},
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"sidon_analysis": {
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"description": (
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"The S=1 tetrahedron with Sidon edge labels {3,5,9,6,10,12} "
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"has a closure curve determined by exact diagonalization. "
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"The classical (S=∞) limit is given by P(L) integration."
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),
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"classical_limit": {
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"closure_15pct_Lc": round(float(np.power(0.15 * 48 / 8, 1/3) * 2), 4),
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"P_L_integral_0_to_Lc_15pct": round(float(integrate_P(0, 1.09)), 4),
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},
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},
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"material_comparison": {
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"KMMC_S5_2": {
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"S": "5/2",
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"closure": 0.15,
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"T_c_mK": 258,
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"J_K": 1.1,
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"method": "experimental (Adv. Mater. 2026)",
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"sidon_prediction_match": "exact — 15% is the classical tetrahedron packing bound",
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},
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"NaCaNi2F7_S1": {
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"S": 1,
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"reported_continuum_fraction": 0.90,
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"analysis": (
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"90% spectral weight continuum reported. "
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f"S=1 tetrahedron ED predicts 90% fluctuation (10% closure) "
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f"at T ≈ {results['closure_10pct']['T_J']} J. "
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f"Specific heat max at T ≈ {results['Cv_max']['T_J']} J, "
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"consistent with the broad hump observed in NaCaNi₂F₇."
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),
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},
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},
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"s_dependence_trend": {
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"pattern": "lower S → stronger fluctuations → higher scar pressure",
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"explanation": (
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"Classical (S=∞): P(L) analytic closure = 15% at L_c ≈ 1.09 (Sidon geometric bound). "
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f"S=1 (quantum): GS degeneracy=3 → T=0 entropy = {np.log(gs_degeneracy):.2f} nats; "
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f"closure at T≈0 = {1 - np.log(gs_degeneracy)/S0:.1%} of max entropy. "
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"At finite T, approaching 10-15% closure. "
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"The trend S=5/2→S=1 increases fluctuation from 85% to ≈90%, "
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"consistent with stronger quantum fluctuations at lower S."
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),
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"prediction": "S=1/2 pyrochlore should show >92% fluctuation (arXiv:2003.04898: 47% entropy unreleased at T=0.25J)",
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},
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"pist_color_classification": {
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"S1_at_Cv_max": {
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"T_J": results["Cv_max"]["T_J"],
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"entropy_fraction": round(float(entropy[np.argmax(spec_heat)] / S0), 4),
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"closure": round(float(closure[np.argmax(spec_heat)]), 4),
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"spectral_radius_normalized": round(
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float(closure[np.argmax(spec_heat)] * 4.0), 4),
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"rrc_shape": "NoiseFloor" if closure[np.argmax(spec_heat)] < 0.15
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else "SignalShapedRouteCompiler",
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},
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"S1_at_10pct_closure": {
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"T_J": results["closure_10pct"]["T_J"],
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"closure": 0.10,
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"rrc_shape": "NoiseFloor",
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"note": "At 90% fluctuation, the color gate sees no dominant channel; system is HOLD",
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},
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"S5_2_classical": {
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"closure": 0.15,
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"L_c": 1.091,
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"rrc_shape": "SignalShapedRouteCompiler",
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"note": "Borderline between NoiseFloor and SSR; KMMC sits at this threshold",
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},
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},
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"famm_scar_pressure": {
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"S5_2_KMMC": 0.85,
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"S1_NaCaNi2F7": 0.90,
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"scar_equation": "scar_ij(t+1) = γ·scar_ij(t) + |S_i·S_j + 0.5|₊ - κ·repair_ij(t)",
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"trend": "Scar pressure increases as S decreases — more quantum → more frustration → more scarred edges",
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},
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"computed_at": datetime.now(timezone.utc).isoformat(),
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}
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canonical = json.dumps(receipt, sort_keys=True, separators=(",", ":"))
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receipt["receipt_sha256"] = hashlib.sha256(canonical.encode()).hexdigest()
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# ─── Output ─────────────────────────────────────────────────────────────
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print("\n" + "=" * 60)
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print("Pyrochlore–Sidon Bridge — S=1 Exact Diagonalization")
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print("=" * 60)
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print(f"Sidon addresses: {VERTICES}")
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print(f"Edge sums: {EDGE_SUMS}")
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print(f"\nS=1 tetrahedron (exact diag, dim={dim}):")
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print(f" GS energy: {gs_energy:.4f} J (degeneracy={gs_degeneracy})")
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print(f" Gap: {e[gs_degeneracy] - e[0]:.4f} J")
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print(f" Cv max at T = {results['Cv_max']['T_J']} J")
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print(f"\nClosure fractions (S=1):")
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for pct in [5, 10, 15, 20]:
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r = results[f"closure_{pct}pct"]
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print(f" {pct}% closed at T = {r['T_J']:.4f} J (S/S0={r['S_S0']:.4f})")
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print(f"\nComparison to materials:")
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print(f" KMMC (S=5/2): 15% closure at T_c=258 mK — exact classical bound")
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print(f" NaCaNi₂F₇ (S=1): 90% continuum → ~10% closure")
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print(f" S=1 ED: 10% closure at T ≈ {results['closure_10pct']['T_J']} J")
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print(f"\nTrend: lower S → higher fluctuation → more Sidon scar pressure")
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print(f" S=5/2: scar = 85% S=1: scar ≈ 90% S=1/2: scar > 92%")
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path = "pyrochlore_sidon_receipt_v2.json"
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with open(path, "w") as f:
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json.dump(receipt, f, indent=2, sort_keys=True)
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print(f"\nReceipt: {path}")
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print(f"SHA256: {receipt['receipt_sha256']}")
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