README.md

metadata

license: cc-by-4.0
pretty_name: phi^4 lattice configurations near the critical point, 2D and 3D
tags:
  - physics
  - lattice-field-theory
  - monte-carlo
  - phase-transition
  - critical-point
size_categories:
  - 10K<n<100K

φ⁴ lattice configurations near the critical point

Monte Carlo configurations of the scalar φ⁴ theory on periodic lattices of volume V = L^D with action

$$S[\varphi ]=\sum _x[-2\kappa \sum _{\mu =1}^D{\varphi }_x{\varphi }_{x+\widehat{\mu }}+{\varphi }_x^2+\lambda ({\varphi }_x^2-1{)}^2]S[\backslash phi]\; =\; \backslash sum\_x\; \backslash Big[-2\backslash kappa\; \backslash sum\_\{\backslash mu=1\}^\{D\}\; \backslash phi\_x\; \backslash phi\_\{x+\backslash hat\backslash mu\}\; +\; \backslash phi\_x^2\; +\; \backslash lambda\backslash big(\backslash phi\_x^2-1\backslash big)^2\backslash Big]$$

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 Binder-cumulant crossings in the reference below. The folder DM/ holds the corresponding ensembles generated by the diffusion models of the same reference.

Files

2D/phi4_2D_L16_kappa0.2705_lambda0.022.npy     (10240, 16, 16)            21 MB
2D/phi4_2D_L32_kappa0.2705_lambda0.022.npy     (10240, 32, 32)            84 MB
2D/phi4_2D_L64_kappa0.2705_lambda0.022.npy     (10240, 64, 64)           336 MB
2D/phi4_2D_L128_kappa0.2705_lambda0.022.npy    (10240, 128, 128)        1342 MB
3D/phi4_3D_L16_kappa0.1923_lambda0.9.npy       (2560, 16, 16, 16)         84 MB
3D/phi4_3D_L32_kappa0.1923_lambda0.9.npy       (2560, 32, 32, 32)        671 MB
3D/phi4_3D_L64_kappa0.1923_lambda0.9.npy       (2560, 64, 64, 64)       5369 MB
3D/phi4_3D_L128_kappa0.1923_lambda0.9.npy      (2560, 128, 128, 128)   42950 MB

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 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

pip install -U huggingface_hub

# everything, 52 GB
hf download phi4-generative/phi4-critical-configs --repo-type dataset --local-dir phi4-critical-configs

# 2D Monte Carlo ensembles only, 1.8 GB
hf download phi4-generative/phi4-critical-configs --repo-type dataset --local-dir phi4-critical-configs --include "2D/*"

# diffusion-model ensembles only, 0.7 GB
hf download phi4-generative/phi4-critical-configs --repo-type dataset --local-dir phi4-critical-configs --include "DM/*"

# a single file
hf download phi4-generative/phi4-critical-configs --repo-type dataset --local-dir phi4-critical-configs --include "3D/*L64*"

import numpy as np
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 of each configuration

Autocorrelation

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. M is the magnetisation and S the action of a configuration, both defined in the next section.

D	L	chains × configurations	warm-up iterations	stored every	HMC acceptance	τ_int(M)	τ_int(|M|)	τ_int(S)
2	16	1 × 10240	2000	5th iteration	0.77	0.52	0.53	0.56
2	32	1 × 10240	2000	5th iteration	0.78	0.50	0.58	0.60
2	64	1 × 10240	2000	5th iteration	0.78	0.49	0.64	0.66
2	128	1 × 10240	2000	5th iteration	0.79	0.48	0.69	0.79
3	16	1 × 2560	2000	10th iteration	0.77	0.55	0.60	0.55
3	32	1 × 2560	2000	10th iteration	0.80	0.50	0.53	0.54
3	64	1 × 2560	2000	10th iteration	0.75	0.48	0.78	0.88
3	128	10 × 256	2000 per chain	10th iteration	0.77	0.48	0.74	0.74

The slowest modes of this sampler are the Z₂-even global quantities S, the nearest-neighbour correlator Σ_x Σ_μ φ_x φ_{x+μ̂}, 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 these quantities, M, 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 τ_int(S) by more than 0.1.

Observables

The magnetisation of a configuration and the quantities built from it are

$$M=\frac{1}{V}\sum _x{\varphi }_x,\chi =V(⟨M^2⟩-⟨\mathrm{\mid }M\mathrm{\mid }{⟩}^2),U_L=1-\frac{⟨M^4⟩}{3⟨M^2{⟩}^2}\mathrm{.}M\; =\; \backslash frac\{1\}\{V\}\backslash sum\_x\; \backslash phi\_x,\; \backslash qquad\; \backslash chi\; =\; V\backslash big(\backslash langle\; M^2\backslash rangle\; -\; \backslash langle|M|\backslash rangle^2\backslash big),\; \backslash qquad\; U\_L\; =\; 1\; -\; \backslash frac\{\backslash langle\; M^4\backslash rangle\}\{3\backslash langle\; M^2\backslash rangle^2\}\; .$$

The action S of a configuration is computed as

$$S[\varphi ]=\sum _x[-2\kappa \sum _{\mu =1}^D{\varphi }_x{\varphi }_{x+\widehat{\mu }}+(1-2\lambda ){\varphi }_x^2+\lambda {\varphi }_x^4],S[\backslash phi]\; =\; \backslash sum\_x\; \backslash Big[-2\backslash kappa\; \backslash sum\_\{\backslash mu=1\}^\{D\}\; \backslash phi\_x\; \backslash phi\_\{x+\backslash hat\backslash mu\}\; +\; (1-2\backslash lambda)\backslash phi\_x^2\; +\; \backslash lambda\backslash phi\_x^4\backslash Big],$$

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	⟨|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)
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	⟨|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)
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)

Diffusion-model ensembles

The folder DM/ holds the diffusion-model ensembles of the reference below at the same couplings. For each L a score network was trained on Monte Carlo configurations at that L. Each configuration is an independent reverse-SDE trajectory, integrated with 2000 Euler–Maruyama steps on a logarithmic time grid in 2D and a linear time grid in 3D. The arrays are float32 with the same axis order as the Monte Carlo files.

DM/2D/phi4_2D_L16_kappa0.2705_lambda0.022_DM.npy     (2048, 16, 16)          2 MB
DM/2D/phi4_2D_L32_kappa0.2705_lambda0.022_DM.npy     (2048, 32, 32)          8 MB
DM/2D/phi4_2D_L64_kappa0.2705_lambda0.022_DM.npy     (2048, 64, 64)         34 MB
DM/2D/phi4_2D_L128_kappa0.2705_lambda0.022_DM.npy    (2048, 128, 128)      134 MB
DM/3D/phi4_3D_L64_kappa0.1923_lambda0.9_DM.npy       (512, 64, 64, 64)     537 MB

The raw 3D output contains a few configurations outside the support of the Monte Carlo ensemble. As in the reference, the table below drops configurations with any site outside [−2.2576, 2.3138] or with |M| > 0.2297, the extremes of the L = 64 Monte Carlo ensemble of the reference. This removes 4 of the 512 configurations. The observables are defined as above, and the Monte Carlo values at the same L are in the tables above.

D	L	⟨|M|⟩	χ	U_L	⟨φ_x²⟩	⟨S⟩/V
2	16	0.7935(72)	24.69(74)	0.5357(51)	1.5010(79)	0.3882(11)
2	32	0.6961(52)	67.4(21)	0.5587(36)	1.4455(37)	0.38954(50)
2	64	0.5979(49)	207.5(66)	0.5535(42)	1.3698(29)	0.39970(37)
2	128	0.4387(46)	762(17)	0.4750(56)	1.4148(12)	0.39944(17)
3	64	0.1229(21)	674(38)	0.5169(91)	0.63820(39)	−0.09753(30)

Reference

The ensembles follow the setup of

@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"
}