---
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

$$S[\phi] = \sum_x \Big[-2\kappa \sum_{\mu=1}^{D} \phi_x \phi_{x+\hat\mu} + \phi_x^2 + \lambda\big(\phi_x^2-1\big)^2\Big]$$

| 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 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

```bash
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*"
```

```python
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

$$M = \frac{1}{V}\sum_x \phi_x, \qquad \chi = V\big(\langle M^2\rangle - \langle|M|\rangle^2\big), \qquad U_L = 1 - \frac{\langle M^4\rangle}{3\langle M^2\rangle^2} .$$

The action S of a configuration is computed as

$$S[\phi] = \sum_x \Big[-2\kappa \sum_{\mu=1}^{D} \phi_x \phi_{x+\hat\mu} + (1-2\lambda)\phi_x^2 + \lambda\phi_x^4\Big],$$

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 | ⟨\|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

```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"
}
```