Datasets:

phi4-generative
/

phi4-critical-configs

---
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"
}
```

	---
	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"
	}
	```