README: magnetisation notation M, explicit action formula, autocorrelation table with M, |M|, S, E

README.md

@@ -65,7 +65,7 @@ from huggingface_hub import hf_hub_download
 f = hf_hub_download("phi4-generative/phi4-critical-configs",
                     "3D/phi4_3D_L32_kappa0.1923_lambda0.9.npy", repo_type="dataset")
 phi = np.load(f, mmap_mode="r")    # (2560, 32, 32, 32)
-phibar = phi.mean(axis=(1, 2, 3))  # zero mode of each configuration
+M = phi.mean(axis=(1, 2, 3))       # magnetisation of each configuration
 ```
 
 ## Autocorrelation
@@ -73,31 +73,43 @@ phibar = phi.mean(axis=(1, 2, 3))  # zero mode of each configuration
 Every chain starts from a random field. One sampler iteration is one HMC trajectory
 followed by Wolff cluster flips. τ_int is the integrated autocorrelation time of the
 stored sequence in units of stored configurations, so 0.5 means consecutive
-configurations are uncorrelated.
-
-| D | L | chains × configurations | warm-up iterations | stored every | HMC acceptance | τ_int(\|φ̄\|) | τ_int(S) |
-|---|---|---|---|---|---|---|---|
-| 2 | 16 | 1 × 10240 | 2000 | 5th iteration | 0.77 | 0.53 | 0.56 |
-| 2 | 32 | 1 × 10240 | 2000 | 5th iteration | 0.78 | 0.58 | 0.60 |
-| 2 | 64 | 1 × 10240 | 2000 | 5th iteration | 0.78 | 0.64 | 0.66 |
-| 2 | 128 | 1 × 10240 | 2000 | 5th iteration | 0.79 | 0.69 | 0.79 |
-| 3 | 16 | 1 × 2560 | 2000 | 10th iteration | 0.77 | 0.60 | 0.55 |
-| 3 | 32 | 1 × 2560 | 2000 | 10th iteration | 0.80 | 0.53 | 0.54 |
-| 3 | 64 | 1 × 2560 | 2000 | 10th iteration | 0.75 | 0.78 | 0.88 |
-| 3 | 128 | 10 × 256 | 2000 per chain | 10th iteration | 0.77 | 0.74 | 0.74 |
+configurations are uncorrelated. M = V⁻¹ Σ_x φ_x is the magnetisation of a
+configuration, S its action as defined in the next section, and
+E = Σ_x Σ_μ φ_x φ_{x+μ̂} its nearest-neighbour correlator.
+
+| D | L | chains × configurations | warm-up iterations | stored every | HMC acceptance | τ_int(M) | τ_int(\|M\|) | τ_int(S) | τ_int(E) |
+|---|---|---|---|---|---|---|---|---|---|
+| 2 | 16 | 1 × 10240 | 2000 | 5th iteration | 0.77 | 0.52 | 0.53 | 0.56 | 0.53 |
+| 2 | 32 | 1 × 10240 | 2000 | 5th iteration | 0.78 | 0.50 | 0.58 | 0.60 | 0.58 |
+| 2 | 64 | 1 × 10240 | 2000 | 5th iteration | 0.78 | 0.49 | 0.64 | 0.66 | 0.65 |
+| 2 | 128 | 1 × 10240 | 2000 | 5th iteration | 0.79 | 0.48 | 0.69 | 0.79 | 0.82 |
+| 3 | 16 | 1 × 2560 | 2000 | 10th iteration | 0.77 | 0.55 | 0.60 | 0.55 | 0.55 |
+| 3 | 32 | 1 × 2560 | 2000 | 10th iteration | 0.80 | 0.50 | 0.53 | 0.54 | 0.55 |
+| 3 | 64 | 1 × 2560 | 2000 | 10th iteration | 0.75 | 0.48 | 0.78 | 0.88 | 0.91 |
+| 3 | 128 | 10 × 256 | 2000 per chain | 10th iteration | 0.77 | 0.48 | 0.74 | 0.74 | 0.75 |
+
+The slowest modes of this sampler are the Z₂-even global quantities S, E, Σ_x φ_x²,
+M² and \|M\|, which relax together. In sampler iterations their τ_int grows from about
+1.3 (2D, L = 16) to 2.4 (2D, L = 128) and from 2.4 (3D, L = 16) to 4 to 7 (3D,
+L = 128), which sets the thinning above. A scan over M, \|M\|, M², S, E, Σ_x φ_x²,
+Σ_x φ_x⁴, max_x \|φ_x\|, the mode powers \|φ̃(k)\|² of the four lowest momentum shells
+and of k = (π, …, π) found no stored observable with τ_int above the larger of
+τ_int(S) and τ_int(E) by more than 0.1.
 
 ## Observables
 
-With the zero mode φ̄ = V⁻¹ Σ_x φ_x of each configuration,
+$$\chi = V\big(\langle M^2\rangle - \langle|M|\rangle^2\big), \qquad U_L = 1 - \frac{\langle M^4\rangle}{3\langle M^2\rangle^2}$$
 
-$$\chi = V\big(\langle\bar\phi^2\rangle - \langle|\bar\phi|\rangle^2\big), \qquad U_L = 1 - \frac{\langle\bar\phi^4\rangle}{3\langle\bar\phi^2\rangle^2}, \qquad \kappa_2^{(\mathrm{site})} = \frac{1}{V}\sum_x\langle\phi_x^2\rangle .$$
+The action S of a configuration is computed as
 
-The action density ⟨S⟩/V omits the field-independent constant λ. Errors are jackknife
-over 64 blocks of consecutive configurations.
+$$S[\phi] = \sum_x \Big[-2\kappa \sum_{\mu=1}^{D} \phi_x \phi_{x+\hat\mu} + (1-2\lambda)\phi_x^2 + \lambda\phi_x^4\Big],$$
+
+which is the action at the top of this page without the field-independent constant
+λV. Errors are jackknife over 64 blocks of consecutive configurations.
 
 2D, κ = 0.2705, λ = 0.022
 
-| L | ⟨\|φ̄\|⟩ | χ | U_L | κ₂^(site) | ⟨S⟩/V |
+| L | ⟨\|M\|⟩ | χ | U_L | ⟨φ_x²⟩ | ⟨S⟩/V |
 |---|---|---|---|---|---|
 | 16 | 0.8149(26) | 22.99(30) | 0.5524(18) | 1.5215(29) | 0.38758(49) |
 | 32 | 0.7147(25) | 64.28(83) | 0.5672(16) | 1.4698(19) | 0.39297(26) |
@@ -106,7 +118,7 @@ over 64 blocks of consecutive configurations.
 
 3D, κ = 0.1923, λ = 0.9
 
-| L | ⟨\|φ̄\|⟩ | χ | U_L | κ₂^(site) | ⟨S⟩/V |
+| L | ⟨\|M\|⟩ | χ | U_L | ⟨φ_x²⟩ | ⟨S⟩/V |
 |---|---|---|---|---|---|
 | 16 | 0.2119(24) | 43.38(92) | 0.4681(65) | 0.62329(34) | −0.12466(49) |
 | 32 | 0.1497(14) | 171.7(36) | 0.4721(64) | 0.61929(11) | −0.11846(16) |