Critical φ⁴ Models and Logarithmic Corrections

For the weakly coupled $n$-component lattice $|\phi|^4$ model in four dimensions, the susceptibility has mean-field power $t^{-1}$ multiplied by the rigorously proved logarithm $(\log t^{-1})^{(n+2)/(n+8)}$. The result holds for $n\geq1$, small positive coupling, and approach to the model’s critical mass from the massive side. It is neither a three-dimensional formula nor evidence for an interacting four-dimensional continuum field.

Required background. Stable Manifolds and Relevant–Marginal Control supplies the tuned critical orbit and marginal flow. Finite-Range Decompositions and Multiscale Integration supplies the independent fluctuation scales.

Helpful background. Universality, Critical Manifolds, and Observable Control distinguishes exponent from amplitude. Thermodynamic Limits, Correlation Decay, and Phase Control supplies the infinite-volume setting.

The four-dimensional theorem

For $\phi_x\in\mathbb R^n$ on $\mathbb Z^4$, consider the Gibbs weight with Hamiltonian

$$H_{g,\nu}(\phi)= \sum_x\left[ \frac12\phi_x\cdot(-\Delta\phi)_x +\frac12\nu|\phi_x|^2 +\frac14g|\phi_x|^4 \right], \qquad g>0.$$

Let $\langle\cdot\rangle_{g,\nu}$ denote its infinite-volume state in the massive regime and define

$$\chi(g,\nu)=\sum_{x\in\mathbb Z^4} \langle\phi_0^1\phi_x^1\rangle_{g,\nu}.$$

For each $n\geq1$ and sufficiently small $g$, there is a critical $\nu_c(g)$ and a positive amplitude $A_g$ such that, with $t=\nu-\nu_c(g)\downarrow0$,

$$\chi(g,\nu_c+t) =A_g\,t^{-1} \bigl(\log t^{-1}\bigr)^{\frac{n+2}{n+8}} \bigl(1+o(1)\bigr).$$

Bauerschmidt, Brydges, and Slade prove this, together with pressure and specific-heat asymptotics and torus scaling limits, in Bauerschmidt, Brydges, and Slade 2014, Theorems 1.1–1.3, pp. 697–704. The theorem’s normalization of $g$ and $\nu$ fixes $A_g$ and $\nu_c$; changing the quartic convention changes those nonuniversal quantities, not the exponent.

How the logarithm is produced

Along the tuned orbit, the marginal quartic coupling obeys

$$g_{j+1}=g_j-\beta_jg_j^2+O(g_j^3), \qquad g_j\sim\frac{1}{\beta j}$$

below the mass scale. The mass derivative or susceptibility source receives a multiplicative correction

$$M_{j+1}=M_j \left(1+\gamma\beta_jg_j+O(g_j^2)\right), \qquad \gamma=\frac{n+2}{n+8}.$$

Taking logarithms and summing gives

$$\log\frac{M_j}{M_0} =\gamma\sum_{k<j}\frac1k+O(1) =\gamma\log j+O(1).$$

Thus $M_j\asymp j^\gamma$. The renormalized mass stops the flow at $j_m\asymp\log_Lm^{-1}$, so $M_{j_m}$ becomes a power of $\log m^{-1}$. The critical tuning relation between $m^2$ and $t$ converts this into the displayed logarithm in $t$.

This derivation identifies the exponent, but the theorem also needs uniform bounds on $K_j$, its mass derivative, the critical initial conditions, and the infinite-volume limit. Those estimates control the $O(1)$ accumulated error and prove the $1+o(1)$ remainder. The same mechanism yields different specific-heat behavior: fractional logarithms for $n=1,2,3$, a double logarithm for $n=4$, and boundedness for $n>4$ in the cited theorem.

Stopping the flow and recovering susceptibility

The connection between the scale recursion and the thermodynamic observable is itself a theorem step. Introduce a renormalized mass $m^2>0$ and define the mass scale $j_m$ by $L^{2j_m}m^2\asymp1$. Below $j_m$, the covariance behaves approximately masslessly and the harmonic sum of $g_j$ accumulates. Above $j_m$, massive covariance estimates improve by powers of $L^{-(j-j_m)}$, so the tail of the polymer expansion is summable. The susceptibility is obtained from the zero-momentum two-point function after the finite-volume limit, not by simply stopping a formal recursion.

Along the critical graph the bare displacement and the renormalized mass satisfy, at the precision needed here,

$$m^2\asymp t\bigl(\log t^{-1}\bigr)^{-\gamma}, \qquad \gamma=\frac{n+2}{n+8},$$

while the normalized two-point denominator is comparable to $m^2$. Hence $\chi\asymp m^{-2}$ yields the displayed $t^{-1}(\log t^{-1})^\gamma$ law. The proof controls the derivative of the critical graph and upgrades comparability to an asymptotic with $A_g(1+o(1))$. Omitting that tuning relation would leave the exponent in terms of an auxiliary mass, not the physical distance $t$ from criticality.

This is the theorem-level application of Landau–Ginzburg–Wilson Quantum Criticality: the upper-critical-dimension marginal flow modifies the susceptibility by the $n$-dependent logarithm while preserving its mean-field power.

Independent checks and failure test

At $n=0$, the formal exponent becomes $1/4$, matching the supersymmetric weakly self-avoiding-walk theorem on the next page; that is a useful cross-check, not an analytic continuation theorem for arbitrary observables. As $n\to\infty$, the exponent tends to one, consistent with the large-component saddle-point pattern. A change of logarithm base rescales $A_g$ only.

Move to $d=3$. The quartic coupling is relevant rather than marginal, so $g_j\sim1/j$ and the harmonic sum used above are absent. Substituting $d=3$ into the four-dimensional formula has no mathematical basis. Likewise, the small-coupling theorem cannot be extended to strong bare $g$ without an entrance estimate into its RG domain.

A second adversarial check is to change the sign of $g$. For $g<0$, the on-site quartic weight is not stable at large field, so the finite-volume Gibbs integral is not normalizable without a stabilizing higher interaction. The formal beta recursion still exists as a polynomial, but the probabilistic starting object and the large-field estimates do not. This cleanly separates algebraic manipulation of the flow from the constructive theorem.

Continuum status

The torus scaling limit proved in the cited weak-coupling analysis is Gaussian free field at critical scaling and white noise in the specified subcritical scaling. More broadly, Aizenman and Duminil-Copin prove Gaussianity of scaling limits for critical four-dimensional nearest-neighbor Ising-type and lattice-cutoff $\lambda\phi^4$ fields in their stated regimes Aizenman and Duminil-Copin 2021, Theorem 1.2 and § 1, pp. 163–177, with 2024 corrigendum. Logarithmic corrections therefore do not establish an interacting $\phi^4_4$ continuum QFT.

Exercise

Assume $g_j=(\beta j)^{-1}+O(j^{-2}\log j)$ and $M_{j+1}/M_j=1+\gamma\beta g_j+O(g_j^2)$. Show $M_j=Cj^\gamma(1+o(1))$ for some $C>0$.

Solution

Taking logarithms gives $\log(M_{j+1}/M_j)=\gamma/j+O(j^{-2}\log j)$. The error is summable, while $\sum_{k<j}k^{-1}=\log j+c+o(1)$. Hence $\log M_j=\gamma\log j+c'+o(1)$ and exponentiation gives the result.

References

• Aizenman, Michael, and Hugo Duminil-Copin. “Marginal Triviality of the Scaling Limits of Critical 4D Ising and $\phi_4^4$ Models.” Annals of Mathematics 194 (2021): 163–235; corrigendum 199 (2024): 479. DOI; Corrigendum.
• Bauerschmidt, Roland, David C. Brydges, and Gordon Slade. “Scaling Limits and Critical Behaviour of the 4-Dimensional $n$-Component $|\varphi|^4$ Spin Model.” Journal of Statistical Physics 157 (2014): 692–742. DOI; Open PDF.