Ising phases and $T_c$¶

Machine learning the 2-D Ising transition, where the label noise is generated by the physics of finite-size crossover rather than injected by hand — so it can be measured, and so the bounded Tsallis loss has a physical knob.

What it shows¶

Classifying Monte Carlo spin configurations as ordered or disordered and reading the critical temperature off the classifier is the standard machine-learning approach to phases of matter (Carrasquilla & Melko, 2017). The q = 1 arm here is that method: an MLP trained with ordinary cross-entropy.

What makes it a Tsallis problem is where the labels come from. They are assigned by temperature, $y = \mathbb 1[T < T_c]$, but near the transition the correlation length $\xi \sim |T - T_c|^{-1}$ exceeds the lattice, so a configuration drawn just above $T_c$ is genuinely typical of the ordered phase. Its temperature label is wrong — and this is label noise generated by the physics, not injected as in label-noise robustness.

Being physical, it is measurable and it has a knob. The magnetization is the exact order parameter, so a threshold on $|m|$ is the best any function of the configuration can do; where it disagrees with the temperature label, the label is what is wrong. That disagreement rate $\eta_{\rm eff}(T, L)$ is the measured noise, and it shrinks as $L$ grows and the crossover narrows.

How it works¶

The loss is qjax.nn.tsallis_cross_entropy_loss with normalizer_q=1.0, which keeps an ordinary softmax under the deformed loss:

from qjax.nn import bounded_q, tsallis_cross_entropy_loss
import qjax.physics as qp

configurations = qp.sample_ising(key, size=16, temperatures=grid,
                                 num_samples=80, sweeps=700)

def loss_fn(params, x, y_onehot):
    q = bounded_q(params["q_raw"], 0.3, 1.3)   # or a fixed q
    return tsallis_cross_entropy_loss(
        logits(params, x), y_onehot, q=q, from_logits=True, normalizer_q=1.0
    )

Passing normalizer_q=1.0 is deliberate twice over: it avoids the 50-step tsallis_entmax bisection in the inner loop, and it rules out the genuine +inf that a sparse normalizer produces when it puts exactly zero mass on the true class.

Both halves of the data are augmented with their global spin flip. That is not data augmentation for its own sake: $-s$ has exactly the same energy as $s$, so without it the network can key on $\operatorname{sign}(m)$, which carries no phase information at all.

Result¶

Binder cumulant, the finite-size-scaling collapse, the confusion scan, and the scoreboard — (a) The Binder cumulant, a physics baseline with no machine learning in it. (b) Finite-size-scaling collapse at the *exact* exponents $\beta = 1/8$, $\nu = 1$ — a strong check that the sampler is equilibrated; the inset recovers $\nu$ from the classifier's own crossover width. (c) Learning by confusion: the balanced accuracy peaks at the true $T_c$ with no knowledge of it anywhere in the pipeline. (d) Distance from the exact $T_c$ for every estimator.

Validation¶

Laptop tier (L up to 16, 4 seeds), against exact values:

quantity	measured	exact	source of the exact value
$T_c$, best estimator (confusion peak, label-free)	2.2828	2.269185	Onsager (1944), closed form
$T_c$, best supervised arm ($q=1$, largest $L$)	2.2907 ± 0.0011	2.269185	"
$\nu$, from the crossover width $w(L)$	1.031	1	exact 2-D exponent
$\beta$, from $\langle	m	\rangle(T_c)$ vs $L$	0.1288
$u(T_c)/J$ at $L=16$	−1.459	−1.414214	Onsager; the residual is finite-size
sampled $\langle	m	\rangle$, $u$ at $L=4$	within 3σ
$\log Z$ at $L=4$	agree to $10^{-10}$	—	enumeration vs. transfer matrix

$T_c$ is recovered to 0.6 % by a label-free estimator and $\nu$ and $\beta$ to 3 % each, from raw spin configurations.

Two things about the method are worth stating outright rather than leaving in the source. First, the two physics baselines (the Binder crossing and the susceptibility peak) are read inside a fixed window $T \in [2.00, 2.62]$, where the temperature grid is dense. That window contains the exact $T_c$, so those two estimators are told roughly where to look — the restriction helps them, since a noisy Binder curve crosses many times far from $T_c$, which is why it is applied to them and not to the learned estimators. The confusion scan and the classifier crossings use the full grid. Second, equilibration was checked rather than assumed: following one set of chains at $T_c$ well past the shipped budget shows no residual drift in $u$ at any size up to $L = 64$, and test_physics_lattice.py pins the sampled $u(T_c)$ at $L = 8$ against the transfer matrix, which is exact for the finite periodic lattice.

What $q$ buys, and where it stops¶

Scored against the order parameter rather than against the temperature labels (the physical analogue of a clean test set), the result is a threshold rather than a smooth trend:

$L$	pooled noise $\bar\eta$	$q=1$ accuracy	$q=0.7$	$q=0.4$	learnable $q$
8	0.158	87.55 ± 0.43	+0.05 ± 0.10	+0.87 ± 0.28	+1.04 ± 0.28
10	0.130	87.78 ± 0.15	+0.81 ± 0.20	+1.38 ± 0.25	+1.29 ± 0.14
12	0.108	91.83 ± 0.03	−0.80 ± 0.23	−0.93 ± 0.35	−0.67 ± 0.32
16	0.080	93.20 ± 0.12	+0.11 ± 0.23	−0.75 ± 0.49	−0.29 ± 0.30

(gains in accuracy points over the $q=1$ arm, mean ± s.e. over 4 seeds)

Above roughly 13 % noise the bounded loss gains about a point, several standard errors clear. Below roughly 11 % it costs about the same. So nonextensivity pays here only above a measurable value of a physical parameter — and the sign change is reported rather than smoothed over.

The learnable $q$ is the arm that survives the regime change: at or near the top in the noisy regime, least penalized in the clean one. That is the practical argument for treating $q$ as a parameter to be learned rather than a constant to be chosen, in a setting where the right value depends on a physical scale the modeller does not control.

Takeaways¶

The pipeline recovers $T_c$, $\nu$ and $\beta$ from raw configurations, each against an exact value — so every claim below it is anchored.
Finite-size crossover is a physical source of label noise, measurable as $\eta_{\rm eff}(T, L)$ and tunable through $L$.
The bounded Tsallis loss helps in proportion to that noise, with a threshold near $\bar\eta \approx 0.12$; a learnable $q$ tracks the regime.
Two label-free estimators — the peak of qjax.tsallis_entropy on the network output, and learning by confusion — locate $T_c$ without ever being told it.

References¶

L. Onsager, Crystal statistics I, Phys. Rev. 65, 117 (1944).
J. Carrasquilla & R. G. Melko, Machine learning phases of matter, Nat. Phys. 13, 431 (2017).
E. P. L. van Nieuwenburg, Y.-H. Liu & S. D. Huber, Learning phase transitions by confusion, Nat. Phys. 13, 435 (2017).
Z. Zhang & M. R. Sabuncu, Generalized cross entropy loss for training deep neural networks with noisy labels, NeurIPS (2018).

Ising phases and \(T_c\)¶