Logarithmic finite-size corrections to Gu–Wen partition-function ratios at the c = 1 four-state Potts critical point

Gu–Wen partition-function ratios — such as $X_1 = Z_{L\times L}^2/Z_{2L\times L}$ and their cluster generalizations — are dimensionless observables that can be computed to very large system sizes by tensor renormalization group (TRG) methods and take universal values at criticality, predicted by the modular-invariant torus partition functions of conformal field theory (CFT). Morita and Kawashima (J. Phys. Soc. Jpn. 95, 044001 (2026)) confirmed this quantitatively for the Ising and three-state Potts models, but at the four-state Potts point — the canonical $c=1$ theory with a marginally irrelevant operator — the ratios drift from their universal values with a deviation consistent with a leading $(\ln L)^{-1}$ decay; no functional form, amplitude, or subleading structure was derived, and for $X_2$ finite-bond-dimension truncation errors mask the logarithmic regime entirely. The open problem is to determine the full asymptotic finite-size form of these ratios at $c=1$ (leading logarithmic power, nonuniversal logarithmic scale, and subleading terms) and to characterize the finite-bond-dimension ($\chi$) errors well enough that TRG data can be extrapolated jointly in $L$ and $\chi$ to the CFT universal values with controlled uncertainty. No published derivation or measurement of this structure is known.

Background

For a two-dimensional classical lattice model with periodic boundary conditions, let $Z_{L_1\times L_2}(T)$ be the partition function of an $L_1\times L_2$ torus at temperature $T$. Gu and Wen (2009) introduced the dimensionless ratios

$$X_1(T,L) \equiv \frac{Z_{L\times L}(T)^2}{Z_{2L\times L}(T)}, \qquad X_2(T,L) \equiv \frac{Z_{L\times L}(T)^2}{Z^{\mathrm{sk}}_{2L\times L}(T)},$$

where the denominator of $X_2$ is a skewed (45-degree tilted) system, a parallelogram with vertices $(0,0)$, $(2L,0)$, $(L,L)$, $(3L,L)$. Like the Binder parameter (the ratio of the fourth moment of the order parameter to the square of its second moment), these ratios converge to 1 in the disordered phase, to a symmetry-determined degeneracy in the ordered phase, and near criticality obey a one-parameter finite-size-scaling form $X(T,L)=f(L^{1/\nu}(T-T_c))$, from which the correlation-length exponent $\nu$ can be estimated. Their practical advantage is that they can be evaluated by tensor-network coarse graining: the partition function is written as a contraction of local tensors and computed by the tensor renormalization group (TRG) — here the bond-weighted variant (BWTRG), which redistributes discarded weight into bond weights and reaches larger bond dimension $\chi$ (the number of states kept per bond, which bounds the truncation error) at fixed cost — giving access to system sizes that are difficult to reach with Monte Carlo methods. A companion computation by the same authors, using the multi-impurity BWTRG method, showed that $X_1$ obeys the same finite-size-scaling law as the Binder parameter.

At criticality the ratios take universal values fixed by conformal field theory: on a torus with modular parameter $\tau$ (which encodes the aspect ratio), the critical partition function is the modular-invariant CFT partition function $Z^{\mathrm{CFT}}(\tau)$, so that, for example, $X_1 = Z^{\mathrm{CFT}}(i)^2/Z^{\mathrm{CFT}}(2i)$, and the universal value of $X_2$ reduces to $Z^{\mathrm{CFT}}(i)$ itself. Morita and Kawashima (2026) verified numerically, for the square-lattice Ising model ($c=1/2$) and three-state Potts model ($c=4/5$), where $c$ is the central charge, that BWTRG data for $X_1$, $X_2$, and the cluster generalizations $X_{m\times n}\equiv Z_{L\times L}^{mn}/Z_{mL\times nL}$ (plus skewed variants $X'$) flatten onto the predicted universal values over a wide range of $L$, with the plateau widening as $\chi$ increases.

The four-state Potts model is the hard case. Its critical point realizes a $c=1$ CFT — the $\mathbb{Z}_2$ orbifold of the compactified free boson (the four-state Potts point of the Ashkin–Teller critical line) — which, unlike the unitary minimal models, contains a marginally irrelevant operator: an operator of scaling dimension exactly equal to the space-time dimension 2, whose coupling is dimensionless and decreases only logarithmically under renormalization, $g(L)\sim 1/\ln(L/L_0)$ with a nonuniversal length scale $L_0$. As a consequence, finite-size quantities acquire logarithmic corrections. The scaling theory of Cardy, Nauenberg and Scalapino (1980), confirmed by the exact transfer-matrix and Monte Carlo analysis of Salas and Sokal (1997), established the structure of these corrections for the standard observables of the four-state Potts model: observables obtained as derivatives of the free energy carry multiplicative powers of $\ln L$ (for example, the susceptibility scales as $L^{7/4}(\ln L)^{-1/8}$), together with additive corrections of order $\ln\ln L/\ln L$ with universal coefficients and of order $1/\ln L$ with nonuniversal coefficients arising from the scale $L_0$; the correlation length carries no multiplicative logarithm. Analogous $1/\ln$ corrections have been measured for other four-state Potts observables such as cluster crossing probabilities (Fukushima and Sakai, 2019), and the renormalization-group theory of the vicinity of the four-state Potts fixed point has been verified in detail (Blöte, Guo and Nightingale, 2017). What this theory does not cover are amplitude-type quantities built directly from torus partition functions at different sizes and aspect ratios — precisely the Gu–Wen ratios.

Morita and Kawashima computed $X_1$ and $X_2$ (and larger-cluster ratios) for the four-state Potts model with BWTRG at $\chi=144$. Unlike the Ising and three-state cases, no plateau appears: $X_1(T_c,L)$ drifts monotonically toward the universal value $3.5995698\ldots$ with a deviation consistent with a leading $(\ln L)^{-1}$ decay near the estimated critical temperature; the paper posits a purely phenomenological form $X(T_c,L)\simeq g(1/\ln L)$ with the function $g$ unspecified, derives no amplitude or subleading structure, and exhibits only the leading trend over a limited range of $L$. For $X_2$ the situation is worse: finite-bond-dimension truncation errors, which grow with $L$ and are larger for $X_2$ than for $X_1$, for skewed than for untilted constructions, and for larger clusters, set in before the logarithmic regime is reached, so even the qualitative log-correction behavior of $X_2$ could not be confirmed (the estimated critical temperature itself drifts with $\chi$, by a few parts in $10^6$). The unresolved question is the precise asymptotic finite-size form of these ratios at $c=1$ and the controlled extrapolation of tensor-network data to the CFT universal values.

Problem Statement

Determine the asymptotic finite-size form of the Gu–Wen partition-function ratios at $c=1$ criticality, and thereby make their extrapolation to the conformal-field-theory universal values controlled. Concretely, for the two-dimensional four-state Potts model on the square lattice (the canonical realization of a $c=1$ CFT with a marginally irrelevant operator), establish the functional form of $X(T_c,L)$ for the whole family $X_1$, $X_2$, the cluster generalizations $X_{m\times n}$, and skewed variants $X'$, where it suffices to give the general form, worked out in full for $X_1$ and $X_2$ and specialized to the remaining constructions — including: (a) the leading logarithmic correction, i.e. whether the observed $(\ln L)^{-1}$ deviation is a pure logarithmic power $\propto(\ln L)^{p}$ (and if so with which power $p$) or rather $\propto(\ln(L/L_0))^{p}$ with a nonuniversal scale $L_0$; (b) the subleading structure, such as $1/(\ln L)^2$, $\ln\ln L/\ln L$, and power-law corrections to scaling, and which (if any) coefficient combinations are universal; and (c) the finite-bond-dimension behavior in TRG computations — how truncation error scales with $\chi$ relative to the $L$-dependence for the different ratio constructions — so that the $\chi\to\infty$ and $L\to\infty$ limits can be separated and the universal values (e.g. $X_1=3.5995698\ldots$, $X_2=4.0833494\ldots$ for the four-state Potts model) extracted with quantified uncertainty. The same question may optionally be addressed for other two-dimensional critical theories with marginally irrelevant operators, e.g. along the Ashkin–Teller critical line. An acceptable answer may be an analytic derivation (a renormalization-group/CFT treatment of the marginally irrelevant operator on the torus, matched to the lattice model), a systematic numerical characterization with a defensible extrapolation protocol, or a refutation of the posited leading $(\ln L)^{-1}$ form together with the correct structure. All three components (a)–(c) are required for a complete answer: any answer form that does not itself settle one of them — an analytic derivation does not supply the finite-$\chi$ behavior (c) — must establish that component by the complementary route. The components (a)–(c) jointly serve the single objective of controlled extraction of the universal values at $c=1$; the log-correction question cannot be settled numerically without the finite-$\chi$ characterization, since the paper shows truncation error contaminates exactly the large-$L$ data where the logarithmic structure would appear.

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