README: observables as in arXiv:2607.08505, autocorrelation table, citation

README.md

@@ -15,14 +15,14 @@ size_categories:
 
 Monte Carlo configurations of the scalar φ⁴ theory on periodic lattices of volume V = L^D with action
 
-$$S[\phi] = \sum_x \Big[-2\kappa \sum_{\mu=1}^{D} \phi_x \phi_{x+\hat\mu} + (1-2\lambda)\phi_x^2 + \lambda\phi_x^4\Big]$$
+$$S[\phi] = \sum_x \Big[-2\kappa \sum_{\mu=1}^{D} \phi_x \phi_{x+\hat\mu} + \phi_x^2 + \lambda\big(\phi_x^2-1\big)^2\Big]$$
 
 | D | κ | λ | κ_c | L | configurations per L | total size |
 |---|---|---|---|---|---|---|
 | 2 | 0.2705 | 0.022 | 0.27088 | 16, 32, 64, 128 | 10240 | 1.8 GB |
 | 3 | 0.1923 | 0.9 | 0.19225 | 16, 32, 64, 128 | 2560 | 49 GB |
 
-κ_c is the critical point estimated from the Binder cumulant.
+κ_c is the critical point estimated from Binder-cumulant crossings in the reference below.
 
 ## Files
 
@@ -39,9 +39,9 @@ $$S[\phi] = \sum_x \Big[-2\kappa \sum_{\mu=1}^{D} \phi_x \phi_{x+\hat\mu} + (1-2
 
 Each array holds the float64 field values φ_x. Axis 0 runs over configurations in
 Monte Carlo order and the remaining axes are the periodic lattice directions. The
-configurations were generated with Wolff cluster updates combined with
-Fourier-accelerated HMC. The 3D L = 128 file consists of ten independent chains of
-256 configurations, concatenated along axis 0.
+configurations were generated with Fourier-accelerated HMC combined with Wolff cluster
+updates. The 3D L = 128 file consists of ten independent chains of 256 configurations,
+concatenated along axis 0.
 
 ## Download
 
@@ -65,39 +65,67 @@ 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)
-M = phi.mean(axis=(1, 2, 3))       # magnetisation per configuration
+phibar = phi.mean(axis=(1, 2, 3))  # zero mode of each configuration
 ```
 
-## Physical quantities
+## Autocorrelation
 
-$$M = \frac{1}{V}\sum_x \phi_x, \qquad \chi = V\langle M^2\rangle, \qquad U_4 = 1-\frac{\langle M^4\rangle}{3\langle M^2\rangle^2}, \qquad \xi_2 = \frac{\sqrt{\chi/F-1}}{2\sin(\pi/L)}$$
+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.
 
-$$F = \frac{1}{V}\Big\langle\Big|\sum_x e^{2\pi i x_\mu/L}\phi_x\Big|^2\Big\rangle$$
+| 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 |
 
-F is the propagator at the smallest nonzero momentum, averaged over the D directions
-μ, and ⟨S⟩/V is the action per site. Errors are jackknife over 64 blocks of
-consecutive configurations. The integrated autocorrelation time of |M| and S is at
-most 0.9 configurations in every file, so the configurations are effectively
-independent.
+## Observables
+
+With the zero mode φ̄ = V⁻¹ Σ_x φ_x of each configuration,
+
+$$\chi = V\big(\langle\bar\phi^2\rangle - \langle|\bar\phi|\rangle^2\big), \qquad U_L = 1 - \frac{\langle\bar\phi^4\rangle}{3\langle\bar\phi^2\rangle^2}, \qquad \kappa_2^{(\mathrm{site})} = \frac{1}{V}\sum_x\langle\phi_x^2\rangle .$$
+
+The action density ⟨S⟩/V omits the field-independent constant λ. Errors are jackknife
+over 64 blocks of consecutive configurations.
 
 2D, κ = 0.2705, λ = 0.022
 
-| L | ⟨φ²⟩ | ⟨S⟩/V | ⟨\|M\|⟩ | χ | U₄ | ξ₂/L |
-|---|---|---|---|---|---|---|
-| 16 | 1.5215(29) | 0.38758(49) | 0.8149(26) | 192.98(94) | 0.5524(18) | 0.7709(53) |
-| 32 | 1.4698(19) | 0.39297(26) | 0.7147(25) | 587.3(32) | 0.5672(16) | 0.7374(53) |
-| 64 | 1.4400(11) | 0.39619(18) | 0.6019(26) | 1696(11) | 0.5551(20) | 0.6512(55) |
-| 128 | 1.41936(61) | 0.398604(90) | 0.4446(23) | 4011(32) | 0.4815(27) | 0.4797(38) |
+| L | ⟨\|φ̄\|⟩ | χ | U_L | κ₂^(site) | ⟨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) |
+| 64 | 0.6019(26) | 211.8(30) | 0.5551(20) | 1.4400(11) | 0.39619(18) |
+| 128 | 0.4446(23) | 771.6(69) | 0.4815(27) | 1.41936(61) | 0.398604(90) |
 
 3D, κ = 0.1923, λ = 0.9
 
-| L | ⟨φ²⟩ | ⟨S⟩/V | ⟨\|M\|⟩ | χ | U₄ | ξ₂/L |
-|---|---|---|---|---|---|---|
-| 16 | 0.62329(34) | −0.12466(49) | 0.2119(24) | 227.3(42) | 0.4681(65) | 0.6478(99) |
-| 32 | 0.61929(11) | −0.11846(16) | 0.1497(14) | 906(12) | 0.4721(64) | 0.6576(75) |
-| 64 | 0.617963(58) | −0.116385(81) | 0.1125(12) | 3959(68) | 0.5032(48) | 0.7127(97) |
-| 128 | 0.617684(26) | −0.116003(38) | 0.09725(87) | 22205(327) | 0.5658(35) | 0.917(12) |
-
-At κ = κ_c and large L, U₄ and ξ₂/L approach the Ising values 0.6107 and 0.9050 in 2D
-(Salas and Sokal, J. Stat. Phys. 98, 551 (2000)) and 0.4655 and 0.6431 in 3D
-(Hasenbusch, Phys. Rev. B 82, 174433 (2010)).
+| L | ⟨\|φ̄\|⟩ | χ | U_L | κ₂^(site) | ⟨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) |
+| 64 | 0.1125(12) | 641(15) | 0.5032(48) | 0.617963(58) | −0.116385(81) |
+| 128 | 0.09725(87) | 2372(73) | 0.5658(35) | 0.617684(26) | −0.116003(38) |
+
+## Reference
+
+The ensembles follow the setup of
+
+```bibtex
+@article{Tan:2026ibs,
+    author = "Tan, Yang-yang and Aarts, Gert and Habibi, Diaa E. and Lucini, Biagio and Wang, Lingxiao",
+    title = "{Diffusion Models for Sampling Near Criticality in Lattice Field Theories}",
+    eprint = "2607.08505",
+    archivePrefix = "arXiv",
+    primaryClass = "hep-lat",
+    reportNumber = "RIKEN-iTHEMS-Report-26",
+    month = "7",
+    year = "2026"
+}
+```