2026-10-15 · chapter Yang–Mills and the mass gapPart 13 The crossover bandvortex free energy, slab tension, the step function, the column bias · Kiichi, Shiori and Rin

The measured record

The lattice. The slab lattice is $(\mathbb Z/N_T)^2\times(\mathbb Z/M)^2$; the first two directions (size $N_T$) are the small $T^2$, and the remaining two (size $M$) stand in for the plane $\mathbb R^2$. The $T^2$ carries the 't Hooft twist $n_{12}=1$ (12-01). A symmetric torus is written $N^4$. The action is the Wilson action $-\frac\beta2\sum_pz_p\,\mathrm{tr}\,U_p$ ($SU(2)$) and its $U(1)$ version.

The vortex free energy. We write the free-energy difference with the sector in which the flux $n_{34}$ in the planar directions is changed as

and set the sector bias to $\theta=\tanh(\Delta F/2)$ (the same quantity as $F_v$ of 13-01 §04). The larger $\Delta F$, the costlier the vortex (the harder it is to enter), and $\theta$ approaches 1. In the confinement regime one expects $\Delta F\to0$.

How we measured. $\Delta F$ is obtained by replica exchange (exchanging configurations between replicas with interpolated couplings to produce samples that span the sectors) and by Bennett's acceptance-ratio method (BAR). The errors are statistical. At some points the difference between two runs with different seeds is 1–2 times the stated error, so the errors may come out on the small side. The string tension of the slab is read from the Creutz ratio

($W(R,T)$ is the $R\times T$ Wilson loop). The Creutz ratio at small $R$ includes the Coulomb term and overestimates the tension, so we read only the tendency, not the absolute value.

Notes on artefacts (in advance). Cautions common to all the tables are listed in §09. In particular, $N_T=2$ is the regime of lattice artefacts, $2^4$ is too small to see one-loop behaviour, and column-bias values of about 0.1 are at the statistical floor.

We hold the size $M$ in the planar directions fixed and widen only the size $N_T$ of the $T^2$. This is a test of the sign of the lateral-widening inequality (13-02 §02).

Table 1 $\Delta F(N_T)$, $M=8$ ($n_{12}=1$, replica exchange + BAR; number of seeds not recorded) [Reading] — the numbers are the same measurement given in 12-08 Table 3. We do not copy them here.

For $U(1)$, widening the $T^2$ makes $\Delta F$ increase sharply, and $\theta$ goes $0\to0.39\to0.78\to0.99$. $SU(2)$ goes $0\to0.63\to0.33\to1.51$, much more gently ($\theta$ goes $0\to0.31\to0.16\to0.64$). Under $N_T=4\to8$, $\ln(1/\theta)$ goes $0.93\to0.012$ for $U(1)$ (about 1/80), and $1.18\to0.45$ for $SU(2)$ (about 1/2.6). In this geometry, however, $\theta$ is of order 1 and the ceiling $\theta\le1$ weakens the power to discriminate; and $N_T=8=M$ for $SU(2)$ is the symmetric torus $8^4$ (physical size $L\sqrt\sigma\approx1.5$). $\beta=1.5$ for $U(1)$ and $\beta=2.5$ for $SU(2)$ are not matched to the same physical size.

Table 2 $\Delta F(N_T)$, $M=12$ (remeasured with a larger container) [Reading] — the numbers are the same measurement given in 12-08 Table 4. We do not copy them here.

For $U(1)$ at $N_T=8$, 24 replicas gave $0.54\pm0.53$ with poor mixing; we give the value with 40. The number of seeds is not recorded.

Table 3 $SU(2)$, $\beta=2.4$, $M=8$ [Reading] — the numbers are the same measurement given in 12-08 Table 5. We do not copy them here.

Reading [Reading]. In three settings ($M=8$ and $12$ at $\beta=2.5$, $M=8$ at $\beta=2.4$), the vortex free energy of $SU(2)$ does not increase when the transverse $T^2$ is widened (except for the symmetric point $N_T=M=8$ at $\beta=2.5$, $M=8$). For $U(1)$ it increases at both values of $M$. This is a numerical sign of the sign of the lateral-widening inequality. The errors are large, 0.1–0.5, and there are only two points in $\beta$. That $N_T=8$ at $\beta=2.4$ is smaller than the same point at $\beta=2.5$ fits with the same lattice being physically larger at $\beta=2.4$ ($L\sqrt\sigma\approx2.2$ against $1.5$).

One of the necessary conditions for the uniform slab tension (13-02) is that the string tension of the slab $\sigma_{\rm eff}(N_T)$ does not overshoot much in $N_T$. We measured the Creutz ratio (2) for $SU(2)$, $\beta=2.5$, $M=16$. The four-dimensional value is $\sigma a^2\approx0.036$.

Table 4 Creutz ratios of the slab, 400 measurements ($SU(2)$, $\beta=2.5$, $N_T\times N_T\times16\times16$; number of seeds not recorded) [Reading] — the numbers are the same measurement given in 12-08 Table 6. We do not copy them here.

Table 5 The same setting with five times the statistics (2000 measurements) and $N_T=10$ [Reading] — the numbers are the same measurement given in 12-08 Table 7. We do not copy them here.

Reading [Reading]. The peak in $\chi(4)$ at $N_T=6$ seen with 400 measurements ($0.089\pm0.013$, about 2.3σ from $N_T=8$) became $0.061\pm0.004$ with 2000 measurements; it was a statistical fluctuation. $\chi(4)$ increases over $N_T=4\to8$ and nearly saturates at $N_T=10$ with $0.071$ (1.7σ from $N_T=8$). No significant overshoot is visible. $\chi(4)$ includes the Coulomb term, so it comes out larger than the four-dimensional $\sigma a^2\approx0.036$. We read only the tendency in $N_T$, and $\beta$ is a single point.

Reading [Reading]. Within the errors, it increases monotonically in $\beta$. The reading is that this does not contradict β-monotonicity (13-06 §02) (β-monotonicity has not been shown for $SU(2)$). For $\beta\le2.3$ the vortex is nearly free ($\theta\approx0$), and at $\beta=2.6$–$2.7$ it becomes costly sharply. This can be read as the fixed lattice $4\times4\times8\times8$ becoming physically small and entering the dilute regime ($\theta\to1$). The points at 2.45 and 2.55 are interpolation points added for the fit in §05.

We test the renormalisation comparison (13-06 §05) directly. We measure $\Delta F$ on the fine lattice (all directions doubled) and read off, from the curve of the coarse lattice (Table 6 and others), the $\beta'$ of the coarse lattice that gives the same value. The prediction of asymptotic freedom is $\beta'\lt\beta$ for $SU(2)$, and $\beta'\ge\beta$ for $U(1)$, which does not run.

The coarse-lattice curve for $U(1)$ is $\Delta F(4,8,\beta')=0.42,0.55,0.83,0.92,1.46$ ($\beta'=1.3,1.4,1.5,1.6,1.7$). The first run for $U(1)$ (32 replicas) gave $-0.18\pm0.12$, but the minimum exchange acceptance rate was 0.03 and the mixing poor, so it is not used.

Reading [Reading]. For $SU(2)$, doubling lowers the vortex free energy ($2.97\to0.92$ at $\beta=2.7$), and the matching $\beta'$ is smaller than $\beta$. For $U(1)$ it rises ($0.83\to2.32$, about 3σ), and $\beta'\gt\beta$. The direction of the comparison separates between the two groups. The $+0.13$ at $\beta=2.7$ is somewhat smaller (about 1.5σ) than the one-loop 0.258 and than the roughly 0.23 expected from the known values of $a\sqrt\sigma$. The record notes that the reversed direction for $U(1)$ may also include artefacts of the small lattice.

Next, on the symmetric torus $N^4$, we look at the two steps $2^4\to4^4$ and $4^4\to8^4$.

The errors are 0.01–0.08 at $2^4$ and 0.03–0.26 at $4^4$. The $\beta'$ of the single step $2^4\to4^4$ (fitting the $4^4$ values to the $2^4$ curve) is as in Table 9.

At every readable $\beta$ (1.75–3.0), $\beta-\beta'\gt0$; that is, the step function of one step has no fixed point. The motion per step grows towards strong coupling; that it falls to 0.07 on the weak-coupling side is seen as an artefact of $2^4$ being too small for the one-loop regime.

Reading [Reading]. For $4^4\to8^4$ too, $\beta-\beta'\gt0$ at every $\beta$ measured, with a mean of about $+0.2$. This is close to the one-loop asymptotic value 0.258 and to the roughly 0.23 expected from the known values of $a\sqrt\sigma$. At some points the difference between seeds is 1–2 times the stated error (at $\beta=2.9$, $8^4$ gives 5.68 and 3.52), and we read the errors as quite likely on the small side. The reading with added seeds came closer to the asymptotic behaviour than the single-seed reading ($+0.08$–$+0.13$). It also fits the reading of the $8\times16$ slab at $\beta=2.7$ (Table 7, $+0.13$).

We carried out the same $2^4\to4^4$ scan for $U(1)$. Four-dimensional $U(1)$ has a phase transition at $\beta_c\approx1.01$ from the confining phase to the Coulomb phase.

Reading [Reading]. The sign of $\beta-\beta'$ for $U(1)$ flips between $\beta=1.0$ and $1.1$. This is the position of the known phase transition $\beta_c\approx1.01$. In the confining phase $\beta'\lt\beta$, and in the Coulomb phase $\beta'\gt\beta$. For $SU(2)$, the same scan gave $\beta-\beta'\gt0$ at every $\beta$ measured (Tables 9 and 10). The sign of the step function of one step works as a diagnostic that tells whether there is a phase transition. This is the numerical face of criterion (m) (13-06 §08), which an argument that closes the band must satisfy.

What this sign test can say is only that, on lattices up to $8^4$ and at the $\beta$ points of the table, the sign for $SU(2)$ did not change. It is not evidence that $SU(2)$ has no phase transition.

Relatedly, there are two readings [Reading] concerning the intermediate goal of seeing $\Delta F(L,L)\to0$ at any rate (13-06 §08). For $U(1)$ ($\beta=1.2$), $2^4\to4^4$ gives $3.10\to3.68$, flat, which does not contradict the picture of heading towards the classical half-flux value in the Coulomb phase. For $SU(2)$ ($\beta=2.75$), it falls, $5.95\to5.10$.

We measure the column face $f(\ell)$ (13-10 §02). The outer frame of the column $T^2_N\times\ell\times\ell$ is taken from an equilibrium configuration and held fixed; for each frame, the conditional free energy $\Delta F_B=\ln Z_B(+)/Z_B(-)$ when a sheet of $n_{34}$ is put inside the column, and the bias $\theta_B=\tanh(\Delta F_B/2)$, are obtained by replica exchange + BAR, and we take $\mathbb E|\theta_B|$ over the frames. It is this $\mathbb E|\theta_B|$ that enters the chessboard estimate (13-05). $SU(2)$ is at $\beta=2.3$, and $U(1)$ at $\beta=1.3$ in the Coulomb phase.

For $\ell\ge5$, the column bias of $SU(2)$ is 1/5 to 1/7 of that of $U(1)$. $U(1)$ decreases slowly (about $\ell^{-1.4}$ over $\ell=5\to8$), while $SU(2)$ drops to 1/3 over $\ell=4\to5$. The values of $SU(2)$ for $\ell\ge5$, however, are close to the floor made by the per-frame statistical errors (which bias $|\theta_B|$ positive), and at these sizes one cannot decide whether it is $e^{-\sigma\ell^2}$ or a power. The sign can change from frame to frame (for $SU(2)$ at $\ell=4$, the mean of $\Delta F_B$ is $-0.05$). In frames where the holonomy around the block is close to $-1$, the sector with the sheet is favoured.

With $\ell$ fixed, the bias approaches 1 with $N$ for both groups. While the bias is small, it grows roughly in proportion to $N^2$ ($\Delta F_B\propto N^2f(\ell)$), and the proportionality coefficient for $SU(2)$ is about 1/7 of that for $U(1)$. The difference between the groups should appear in how the $\ell$ needed to keep $\theta$ small grows, so we varied both $\ell$ and $N$.

Reading [Reading]. For $SU(2)$, at every one of $N=4,6,8$, the bias falls to the statistical floor (about 0.1) at $\ell\approx6$–$7$, and the needed $\ell$ hardly grows with $N$. For $U(1)$, even at $N=4$ it stays at 0.27 at $\ell=10$, and at $N=6$ it is 0.45 even at $\ell=10$; the needed $\ell$ grows with $N$, and the fall is slower. The 0.29 at $N=8$, $\ell=7$ comes from only 3 frames (cut off at the time limit); that it is larger than the 0.13 at $\ell=6$ we take as a fluctuation due to the small number of frames (all 3 frames have $\Delta F_B\gt0$, 0.43–0.85). This does not contradict the expectation ($\ell\sim\sqrt{\ln N}$ for $SU(2)$, $\ell\propto N$ for $U(1)$), but the lattices are small and the frames few, so the form of the growth cannot be determined.

Writing $U=\sigma V$ ($\sigma\in\mathbb Z_2$, $V$ a section of $SO(3)$), the twisted sector becomes "a $\mathbb Z_2$ random-plaquette gauge model on an $SO(3)$ background" (13-10 §03). The effective $\mathbb Z_2$ coupling per plaquette is $K_p=(\beta/2)|\mathrm{tr}\,U_p|$, and the frustration (a $\mathbb Z_2$ monopole) is a cube $c$ for which $\prod_{p\in\partial c}\mathrm{sign}\,\mathrm{tr}\,U_p=-1$.

$p_{\rm eff}$ is the frustration density $\rho$ of the cubes, converted to a probability of independent sign flips by $(1-2p_{\rm eff})^6=1-2\rho$. $K_c$ is the transition point of pure $\mathbb Z_2$ gauge theory, and the value to compare with is the Nishimori multicritical point $p_c\approx0.11$ of the four-dimensional random-plaquette gauge model (checking not yet done).

Reading [Reading]. For $\beta\gtrsim2.0$, the mean coupling $\langle K\rangle$ is 3–4 times $K_c$, and $p_{\rm eff}\lt0.11$. The $\mathbb Z_2$ glass is likely on the side of the ordered phase. For $\beta\lesssim1.5$, $p_{\rm eff}\ge0.16$, and soft plaquettes ($K\lt K_c$) are 28–49%: the disordered side. The conversion to independent signs is crude (soft plaquettes and frustration are correlated), so the boundary is a rough guide. From this diagnosis, the glass road is limited to the lower half of the band (C4 of 13-08 §05).

• $N_T=2$ (and $2^4$) is the regime of lattice artefacts. At $N_T=2$ one side of the $T^2$ is two lattice sites, the frustration of the twist is large, and it is far from the continuum picture (a gas of fractional instantons). That the $\beta-\beta'$ of the single step $2^4\to4^4$ falls to 0.07 on the weak-coupling side (Table 9) is also seen as due to $2^4$ being too small for the one-loop regime. Values at $N_T=2$ are kept as "a reference for the direction".
• The ceiling of $\theta$. In geometries where $\theta$ is of order 1 ($M=8$ of Table 1), $\theta\le1$ holds the ratio down and weakens the power to discriminate between the groups.
• The statistical floor. The column bias $\mathbb E|\theta_B|$ is biased positive by the per-frame statistical errors. Values of about 0.1 are close to the floor, and one cannot decide between an area exponent and a power (Tables 12 and 14).
• The Coulomb term. The Creutz ratio at small $R$ includes the Coulomb term and overestimates the tension (Tables 4 and 5).
• Poor mixing. Runs with a small minimum replica-exchange acceptance rate (replicas 24 and 32 for $U(1)$, the 0.10 of the second $SU(2)$ run) fluctuate strongly in value. Runs we did not use are noted under the tables.
• Number of seeds and errors. There are 1–3 seeds. At some points the difference between two runs with different seeds is 1–2 times the stated error, so the errors are quite likely on the small side.
• How $\beta$ is matched. The $\beta$ of $U(1)$ and of $SU(2)$ are not matched to the same physical size. What we compare is only the direction (sign) within each group.
• A withdrawn measurement. We have withdrawn reading the measurement that flips containers simultaneously as "the numerical face of the band". The flip probability of the one-step kernel levels off at about 1/2, and what comes out is only the weak form, which $U(1)$ also satisfies (C5 of 13-08 §05). The measurement itself is in 12-08.

All the rows of this page are numerical readings. The label is "Reading", with "(physics)" added for readings that include a physical estimate.

The dependency graph (the numbers of premises are smaller than the number of the conclusion):

[01] RDG WID     widening T²: U(1) rises, SU(2) does not              M = 8, 12
[02] RDG SLAB    slab tension increases and saturates                 β = 2.5, 2000 meas.
[03] RDG BETA    ΔF increases in β                                    4×4×8×8
[04] RDG DBL     doubling test: SU(2) β′ < β, U(1) the opposite ⟸ [03]  1–3 seeds
[05] RDG ASY     β − β′ ≈ +0.2 for 4⁴ → 8⁴ ⟸ [04]                     1–3 seeds
[06] RDG SIGN    sign of the step function: U(1) flips at β_c ⟸ [04]  2⁴ → 4⁴
[07] RDG COL     ℓ × N of the column bias                             N ≤ 8, ℓ ≤ 10
[08] RDG GLASS   diagnosis of the Z₂ glass                            8⁴
[09] RDG FIT     consistency of directions ⟸ [01][02][04][06][07]     tendency

The check is to see by eye that every number to the right of a "⟸" is smaller than the number on the left. There are no cycles. This page has no "Proved here", "Cited" or "Assumption" rows. Nor does it have nodes for the statements that cross the band (the uniform slab tension, the lateral-widening inequality, the renormalisation comparison); those are placed as assumptions in the ledger of Part 14 (14-02).

What this chapter can say

Sources and reproduction

Terms used in this chapter — the vortex free energy $\Delta F$, the sector bias $\theta$, the Creutz ratio, replica exchange, BAR, the step function, the column bias $\mathbb E|\theta_B|$, the $\mathbb Z_2$ glass, frustration. Their meanings are in A-02 Glossary.

Next chapter: 13-10 Candidate new viewpoints — into inequalities for which local quantities these tendencies might be grown.

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