[challenge]: Critical-exponent corrections near the long-range mean-field boundary σ=d/2

Released by

Youjin Deng, USTC

Contact email

yjdeng@ustc.edu.cn

Method

Analytical renormalization group / symbolic and ODE verification / long-range cluster Monte Carlo

Challenge issue

For an $O(n)$ model with ferromagnetic interaction $J(r)\sim r^{-(d+\sigma)}$, the line $\sigma=d/2$ separates the Gaussian mean-field regime from the interacting long-range regime. What are the corrections to $\eta$ and the other critical exponents at and near this boundary?

The main goals are: (1) derive a crossover RG description that is uniform on both sides of $\sigma=d/2$; (2) explain why the long-range anomalous dimension can remain $\eta=2-\sigma$ while thermodynamic exponents acquire nontrivial corrections; and (3) test the predicted power-law, logarithmic, and finite-size crossover behavior with controlled numerics.

The research question

Consider a classical long-range $O(n)$ model whose quadratic propagator is governed by $|k|^\sigma$:

$$
\mathcal S[\boldsymbol\phi]
=\frac12\int_k
\left(r_0+c_\sigma |k|^\sigma+c_2k^2\right)
|\boldsymbol\phi(k)|^2
+\frac{u_0}{4!}\int_x
\left(\boldsymbol\phi^2\right)^2+\cdots .
$$

Power counting gives

$$
[\phi]=\frac{d-\sigma}{2},
\qquad
[u_0]=2\sigma-d.
$$

Define the distance from the long-range upper-critical boundary by

$$
\epsilon=2\sigma-d.
$$

Then:

• $\epsilon<0$ is the Gaussian mean-field side;
• $\epsilon=0$ is the marginal line $\sigma=d/2$;
• $\epsilon>0$ is the interacting long-range side.

The papers arXiv:1611.06169 and arXiv:1705.08540 use the symbol $\alpha$ for the interaction exponent called $\sigma$ here. They rigorously establish, for small positive $\epsilon$ and weak coupling, the first-order correction to several thermodynamic exponents and the sticking of the critical two-point exponent at its long-range mean-field value.

The central puzzle is therefore not simply whether the exponents change. It is:

1. which exponents receive $O(\epsilon)$ corrections on the interacting side;
2. why $\eta$ behaves differently;
3. what replaces the $O(\epsilon)$ expansion exactly at $\epsilon=0$;
4. how power corrections turn into logarithmic corrections as $\epsilon\to0$;
5. which finite-size observables cleanly separate the bulk two-point exponent from dangerous-zero-mode effects.

This challenge concerns the mean-field/interacting-long-range boundary $\sigma=d/2$. It is distinct from the long-range/short-range crossover near $\sigma=2-\eta_{\rm SR}$ and does not assume a value for that separate boundary.

Analytical targets

1. Derive the crossover flow

Starting from the long-range $\phi^4$ theory, derive the beta function to the lowest order needed to resolve the boundary. After a documented normalization of the running coupling $g$, the flow should reduce to

$$
\frac{dg}{d\ell}
=\epsilon g-A_n g^2+O(g^3,\epsilon g^2),
\qquad
A_n>0.
$$

Derive rather than assume the coefficient $A_n$, and show how changes of coupling convention affect it.

The corresponding one-loop crossover solution is

$$
g(\ell)
=\frac{\epsilon g_0 e^{\epsilon\ell}}
{\epsilon+A_n g_0\left(e^{\epsilon\ell}-1\right)}.
$$

Its continuous marginal limit must be recovered:

$$
g(\ell)
=\frac{g_0}{1+A_n g_0\ell}
\quad\text{at}\quad \epsilon=0.
$$

Use this flow to identify:

• the Gaussian fixed point for $\epsilon\le0$;
• the interacting fixed point $g_*=\epsilon/A_n+O(\epsilon^2)$ for $\epsilon>0$;
• the correction-to-scaling exponent $\omega=\epsilon+O(\epsilon^2)$ on the interacting side;
• the marginal $1/\ln L$ corrections at $\epsilon=0$;
• the crossover variable $X=\epsilon\ln(L/L_0)$ and the associated exponentially large crossover scale as $|\epsilon|\to0$.

2. Derive the exponent corrections for $\epsilon>0$

For small positive $\epsilon$, reproduce the long-range $O(n)$ results

$$
\eta=2-\sigma,
$$

$$
\gamma
=1+\frac{n+2}{n+8}\frac{\epsilon}{\sigma}
+O(\epsilon^2),
$$

and, using scaling relations where their assumptions are valid,

$$
\nu
=\frac{\gamma}{\sigma}
=\frac{1}{\sigma}
+\frac{n+2}{n+8}\frac{\epsilon}{\sigma^2}
+O(\epsilon^2),
$$

$$
\alpha_H
=\frac{4-n}{n+8}\frac{\epsilon}{\sigma}
+O(\epsilon^2)
\qquad (n<4),
$$

$$
\beta
=\frac12
-\frac{3}{n+8}\frac{\epsilon}{\sigma}
+O(\epsilon^2).
$$

The derivation must state clearly which results are proved in the cited rigorous work, which follow from scaling or hyperscaling, and which rely only on perturbative RG.

3. Resolve the apparent absence of an $\eta$ correction

For the critical bulk two-point function, explain why the nonanalytic kinetic term is not renormalized into a new power and why

$$
G(r)\sim \frac{1}{r^{d-\sigma}}
$$

continues to imply

$$
2-\eta=\sigma
$$

slightly below the upper critical dimension.

The answer should distinguish the following statements:

• $\eta$ has no additional $O(\epsilon)$ anomalous correction beyond its explicit dependence on $\sigma$;
• if $\sigma=(d+\epsilon)/2$, then $\eta=2-d/2-\epsilon/2$ still varies kinematically with $\epsilon$;
• subleading corrections to $G(r)$ can remain even when the leading exponent sticks;
• a zero-momentum susceptibility can contain dangerous-variable factors that do not define a second bulk anomalous dimension.

Determine the leading correction to the effective two-point exponent

$$
\eta_{\rm eff}(r)
=2-d-\frac{d\ln G(r)}{d\ln r}
$$

for $\epsilon<0$, $\epsilon=0$, and $\epsilon>0$. In particular, decide whether the marginal theory has a multiplicative power of $\ln r$ or only corrections such as $1/\ln r$, and compute the first universal quantity that can be extracted from this approach.

4. Derive the marginal logarithms at $\epsilon=0$

At $\sigma=d/2$, derive the leading thermodynamic logarithmic corrections:

$$
\chi(t)
\sim |t|^{-1}
\left[\ln |t|^{-1}\right]^{\frac{n+2}{n+8}},
$$

$$
\xi(t)
\sim |t|^{-1/\sigma}
\left[\ln |t|^{-1}\right]^{\frac{n+2}{\sigma(n+8)}}.
$$

For $n<4$, also test

$$
C(t)
\sim
\left[\ln |t|^{-1}\right]^{\frac{4-n}{n+8}},
$$

and derive the appropriate $n=4$ and $n>4$ alternatives rather than extrapolating this expression blindly.

For periodic finite systems, derive separately the bulk nonzero-mode and dangerous-zero-mode predictions. A central test is

$$
S(k_{\min})\sim L^\sigma
\left[1+O\left(\frac{1}{\ln L}\right)\right],
$$

$$
S(0)\sim L^{d/2}(\ln L)^{1/2},
$$

so that at $d=2\sigma$

$$
\frac{S(0)}{S(k_{\min})}
\sim(\ln L)^{1/2}.
$$

Explain why a pure-power fit to $S(0)$ can produce a slowly drifting effective $\eta$ even though the bulk nonzero-momentum exponent remains $\eta=2-\sigma$.

Numerical task

Use the two-dimensional long-range Ising model as the primary test:

$$
H=-\sum_{i<j}
J_L(\mathbf r_i-\mathbf r_j;\sigma)s_i s_j,
\qquad
s_i=\pm1,
$$

with periodic image-summed interaction

$$
J_L(\mathbf r;\sigma)
=\sum_{\mathbf n\in\mathbb Z^2}
\frac{1}{|\mathbf r+L\mathbf n|^{2+\sigma}}.
$$

The main boundary is $d=2$, $\sigma=1$. Use a long-range cluster algorithm and test at least

$$
\sigma=0.90,\ 0.95,\ 1.00,\ 1.05,\ 1.10
$$

with

$$
L=32,\ 64,\ 128,\ 256,\ 512,
$$

and larger sizes when autocorrelation and wall-clock audits permit.

Measure:

• the zero-mode susceptibility $S(0)$;
• the first nonzero-momentum structure factor $S(k_{\min})$;
• their ratio $S(0)/S(k_{\min})$;
• Binder ratios and magnetization moments;
• thermal derivatives needed to estimate $\nu$;
• energy and specific heat;
• autocorrelation times and effective sample sizes.

The numerical analysis must compare, on equal footing:

1. a mean-field power law with corrections governed by $|\epsilon|$;
2. the marginal logarithmic form at $\epsilon=0$;
3. the interacting-long-range exponent expansion for $\epsilon>0$;
4. an unconstrained effective-power fit;
5. a joint crossover fit in $X=\epsilon\ln(L/L_0)$.

Do not infer a varying bulk $\eta$ from $S(0)$ alone. The primary momentum-resolved diagnostic is the comparison between $S(0)$ and $S(k_{\min})$ on the same configurations.

Validation

Before interpreting any exponent drift:

• reproduce the interaction matrix by an independent periodic-sum or Ewald implementation at small $L$;
• compare exact enumeration and Monte Carlo for $L=2,3$;
• verify the high-temperature magnetization moments;
• reproduce one published critical-temperature or Binder-ratio benchmark using exactly the same interaction normalization;
• test the fitting pipeline on synthetic datasets generated from power, logarithmic, and crossover forms;
• propagate uncertainty in $T_c$ through every exponent fit;
• repeat fits after removing the smallest size;
• report whether the available sizes can actually distinguish $L^{-\omega}$ from $1/\ln L$ when $\omega\simeq|\epsilon|$.

A drift that disappears under a change of $L_{\min}$ is not evidence for an anomalous $\eta$.

Fairness and five-day contract

1. Day-0 scaffold: provide the RG notebook, ODE solver, tested periodic coupling builder, long-range cluster sampler, and small-size exact-enumeration fixtures.
2. Frozen conventions: register the Hamiltonian normalization, definition of $t$, momentum convention, temperature windows, and size exclusions before production fits.
3. Shared data: release seeds, raw or blocked measurements, covariance estimates, autocorrelation diagnostics, and the exact code revision.
4. Competing hypotheses: fit power, logarithmic, and crossover forms with comparable flexibility; do not report only the preferred ansatz.
5. Allowed inconclusive result: if $|\epsilon|\ln L$ never becomes large enough to resolve the crossover, quantify the required next size or precision.

If a production Monte Carlo scaffold is unavailable, the five-day analytical floor is a complete one-loop crossover derivation, symbolic verification of the exponent relations, numerical integration of the RG flow, and synthetic-data identifiability tests.

Adjudication

Successful outcomes include:

1. Analytical resolution: a consistent derivation of the exponent corrections, marginal logarithms, and $\epsilon\ln L$ crossover.
2. Momentum-resolved numerical support: $S(k_{\min})$ and $S(0)$ exhibit the predicted distinct correction structure.
3. Controlled disagreement: a reproducible deviation survives normalization checks, $T_c$ propagation, size-window tests, and competing-fit comparison.
4. Quantified non-identifiability: the work establishes that accessible sizes cannot distinguish a small correction exponent from a logarithm.

Failure of the interaction normalization, exact-enumeration controls, or sample-quality gates is a validation failure rather than evidence against the RG prediction.

What is reproduction and what is new

Reproduction includes:

• recovering $d_c=2\sigma$ by power counting;
• reproducing the $O(\epsilon)$ formulas already established in the literature;
• confirming $\eta=2-\sigma$ slightly inside the interacting long-range regime;
• observing a visually plausible logarithmic collapse at $\sigma=d/2$.

Potentially new results include:

• a uniform and convention-explicit crossover function connecting both sides of $\sigma=d/2$;
• a controlled determination of the leading correction to $\eta_{\rm eff}$ at the marginal point;
• a momentum-resolved Monte Carlo test separating bulk and zero-mode corrections in one dataset;
• a quantitative resolution bound for distinguishing $L^{-|\epsilon|}$ from $1/\ln L$;
• a reproducible counterexample to the registered correction structure after all controls pass.

References

1. G. Slade, “Critical exponents for long-range $O(n)$ models below the upper critical dimension,” arXiv:1611.06169; Commun. Math. Phys. 358, 343–436 (2018).
2. M. Lohmann, G. Slade, and B. C. Wallace, “Critical two-point function for long-range $O(n)$ models below the upper critical dimension,” arXiv:1705.08540; J. Stat. Phys. 169, 1132–1161 (2017).
3. M. E. Fisher, S.-K. Ma, and B. G. Nickel, “Critical Exponents for Long-Range Interactions,” Phys. Rev. Lett. 29, 917 (1972).
4. E. Luijten and H. W. J. Blöte, “Classical critical behavior of spin models with long-range interactions,” Phys. Rev. B 56, 8945 (1997).
5. T. Horita, H. Suwa, and S. Todo, “Upper and Lower Critical Decay Exponents of Ising Ferromagnets with Long-range Interaction,” Phys. Rev. E 95, 012143 (2017).
6. K. Fukui and S. Todo, “Order-N Cluster Monte Carlo Method for Spin Systems with Long-range Interactions,” J. Comput. Phys. 228, 2629–2642 (2009).

No repository files or tests are named. Start with the requested RG notebook, ODE solver, periodic coupling builder, and small-size exact-enumeration fixtures; if Monte Carlo support exists, inspect the long-range cluster sampler next. Done means a documented crossover derivation and controlled analytical or numerical validation meeting the stated normalization, fitting, and reproducibility checks.