2026-10-15 · chapter Yang–Mills and the mass gapPart 12 The second roadwork of driven insertion, vortex free energy, string tension of slabs · Kiichi, Shiori and Rin

The measured record

1. Driven insertion. On $SU(2)$ slabs, the cost of inserting a vortex in one go (0 steps) is large and proportional to the area of $T^2$, at 11.8, 24.2 and 44.9 ($N_T=2,3,4$, $\beta=2.3$), but driving it over 256 steps brought it down to 0.50–0.63 Reading (rows A, B and C of Table 1. In rows G and H, under different conditions, the work over 256 steps stays high, at 3.64 and 6.67). When the container (a disc) is smaller than the size of the vortex, increasing the number of steps does not bring it all the way down.
2. Fixing the container radius at 4 and widening to $N_T=4,6,8$ ($\beta=2.5$, 1024 steps), the work was 1.1–1.6, with no clear increase in $N_T$ Reading (10 configurations × 3 positions).
3. The vortex free energy. The free energy $\Delta F$ of the global flux sector increased as $T^2$ was widened for $U(1)$ ($\beta=1.5$), and did not increase, or decreased, for $SU(2)$ ($\beta=2.4,2.5$) Reading ($M=8$ and $12$; the errors are 0.04–0.78 for $U(1)$ and 0.1–0.36 for $SU(2)$).
4. String tension of slabs. For $SU(2)$, $\beta=2.5$, $M=16$, the Creutz ratio $\chi(4)$ increased for $N_T=4\to8$ and was nearly saturated at $N_T=10$. No significant overshoot is seen Reading (2000 measurements).
5. Simultaneous flips. The measurement driving containers simultaneously was found to be unusable as a numerical picture of the crossover band, and that positioning was withdrawn (12-06 §07). It remains only as a tool for lower bounds on the slab tension with the $T^2$ fixed.

• Lattice. The slab $(\mathbb Z/N_T)^2\times(\mathbb Z/M)^2$. Directions 1 and 2 form the twisted $T^2$ (twist $n_{12}=1$), and directions 3 and 4 the plane. $SU(2)$ uses the Wilson action, and $U(1)$, as a control, an action of the same form.
• Driven insertion. The work $W$ (equation (2) of 12-06) when only the inside of a container, a disc of the plane (radius $r$) × $T^2$, is updated by heat bath and the couplings of the vortex sheet are changed $1\to-1$ in $m$ steps. The acceptance rate is the mean of $\min(1,e^{-W})$.
• Vortex free energy. The $\Delta F=\ln(Z(1,0)/Z(1,1))$ of the two sectors of the flux $n_{34}\in\{0,1\}$ through the whole $(3,4)$ torus of the plane, estimated by replica exchange interpolating the coupling and Bennett's acceptance ratio (BAR). The bias is $\theta=\tanh(\Delta F/2)$. The errors are the record's estimates.
• String tension. The Creutz ratio $\chi(R)$ built from $R\times R$ Wilson loops in the plane. At small $R$ it contains the Coulomb term and overestimates the tension.
• Physical size. In the record, $N_T=8$ at $\beta=2.5$ is estimated as $L\sqrt\sigma\approx1.5$, $N_T=8$ at $\beta=2.4$ as $L\sqrt\sigma\approx2.2$, and $N_T=4$ at $\beta=2.7$ as $L\sqrt\sigma\approx0.4$ Reading (converted from known values of $a\sqrt\sigma$). Only the last is in the semiclassical region of a small $T^2$.
• $N_T=2$. The rows where the $T^2$ is 2 in lattice units are a region of large lattice artefacts. They are excluded when reading trends.

All are $SU(2)$, twist $n_{12}=1$. The number of samples is "number of configurations × number of insertion positions per configuration". The figures in brackets for $W$ are the standard deviations of the samples (from the records of the run outputs), not the errors of the means. Samples at different positions in the same configuration are not independent, so the error of the mean should be taken as larger than the standard deviation divided by the square root of the number of samples.

Row P is the measurement cited in 12-06 §04. Rows P, A and D are $N_T=2$ (the region of lattice artefacts).

Reading Reading: (i) The cost at 0 steps is roughly proportional to $N_T^2$ (11.8, 24.2, 44.9 for A, B, C). (ii) If the container is large enough, the work of driven insertion comes down to near 0. (iii) The container radius needed appears to roughly double each time the lattice spacing halves (2 at $\beta=2.3$, 4 at $2.5$; at $2.7$, 4 is not enough and even 8 does not reach near 0). This is consistent with the thickness of the vortex being fixed by a physical length. (iv) The work 0.51 at 1024 steps for H ($N_T=8$) is almost the same as 0.44 for E ($N_T=4$), but at 256 steps it is high, 6.67, depending strongly on the speed of driving.

Reading Reading: (i) I: fixing the container radius at 4 and widening to $N_T=4,6,8$, the work is 1.1–1.6, with no clear increase in $N_T$ (the error of the mean is the standard deviation $/\sqrt{30}$, about 0.2–0.3). The 0.44 and 0.51 of E and H in Table 1 were fluctuations from little statistics. (ii) J: at $\beta=2.7$, $r=8$, driving four times more slowly than G still gives $W\approx2.8$ (2.7 with 1024 steps for G), and the Jarzynski estimate is 2.4. Most of the work can be read not as dissipation due to the speed of driving but as a genuine free energy. $N_T=4$ at $\beta=2.7$ is a physically small $T^2$ ($L\sqrt\sigma\approx0.4$), on the side of the dilute region, so the reading is that the vortex really is expensive (physics).

On the Jarzynski estimate. $-\ln\langle e^{-W}\rangle$ is an estimate of the sector free energy by Corollary 3 of 12-06, but it is dominated by rare small $W$, so with few samples it fluctuates strongly. It is a column not shown in Table 1, but negative values have occurred (for example $-0.30$ for E at radius 4 with 1024 steps, $-0.085$ for A with 64 steps). This column is a guide for checking, and the values themselves are not reliable.

The limits of this series. What confinement on R⁴ needs is that the bias becomes small exponentially in the area as $L$ grows (13-01), and that is not visible in this measurement. $\beta$ takes a few values between 2.3 and 2.7, and the error estimates are still rough.

The container is the whole plane ($M\times M$), and the $T^2=N_T\times N_T$ is widened. It is a test of the sign of the lateral-widening inequality (13-02). All by replica exchange + BAR, twist $n_{12}=1$, 24 replicas (unless otherwise stated). The number of iterations is not in the record.

Reading Reading: (i) For $U(1)$, $\Delta F$ increases as the $T^2$ is widened (at $M=8$, $\theta$ goes $0\to0.39\to0.78\to0.99$; at $M=12$, $0.20\to1.35\to1.72$). (ii) For $SU(2)$, in three settings ($M=8$ and $12$ at $\beta=2.5$, $M=8$ at $\beta=2.4$), it does not increase or decreases when widened. The exception is 1.51 at $N_T=8=M$ (the symmetric torus $T^4_8$) for $\beta=2.5$, $M=8$. (iii) For $N_T=4\to8$, $\ln(1/\theta)$ goes 0.93 → 0.012 for $U(1)$ (about 1/80) and 1.18 → 0.45 for $SU(2)$ (about 1/2.6) ($M=8$). This can be read as a numerical sign of the sign of the lateral-widening inequality. (iv) That $N_T=8$ at $\beta=2.4$ is smaller than at $\beta=2.5$ is consistent with the same lattice being physically larger ($L\sqrt\sigma\approx2.2$ against 1.5).

Limits. This is a regime where $\theta$ is not close to 1 (it is $O(1)$), so the $U(1)$ ratio is also held down because $\theta\le1$, and in this geometry the lateral-widening inequality has little power to discriminate. The errors are large, 0.04–0.78 for $U(1)$ and 0.1–0.36 for $SU(2)$, and $\beta$ is one or two points. $\beta=1.5$ for $U(1)$ and $\beta=2.5$ for $SU(2)$ are not matched to "the same physics". The $N_T=2$ column is in the region of lattice artefacts.

$SU(2)$, $\beta=2.5$, plane $M=16$, twist $n_{12}=1$. We measured the Creutz ratio $\chi(R)$ of Wilson loops in the plane for each size $N_T$ of the $T^2$. It is a test of "monotonicity of the reduced two-dimensional string tension $\sigma_{\rm eff}(N_T)$" contained in the strong form of the lateral-widening inequality of 13-02, and of an "overshoot" in which the tension once becomes large at $L\sim1/\Lambda$. In the record, the four-dimensional value at $\beta=2.5$ is given as $\sigma a^2\approx0.036$ Reading.

Reading Reading: (i) With 400 measurements $\chi(4)$ was largest at $N_T=6$ (0.089), and its difference from $N_T=8$ was about 2.3 standard deviations. It looked like a sign of overshoot, but with 2000 measurements $N_T=6$ gives $0.061\pm0.004$, so it was a statistical fluctuation. (ii) With 2000 measurements $\chi(4)$ increases for $N_T=4\to8$, and at $N_T=10$ it is 0.071 (1.7 standard deviations from $N_T=8$), almost saturated. No significant overshoot is seen. This is consistent with monotonicity of the tension (a necessary condition of 13-02). (iii) $\chi(4)$ contains the Coulomb term, so it comes out larger than the four-dimensional $\sigma a^2\approx0.036$. We read only the trend in $N_T$, not the absolute value. (iv) The $N_T=2$ row (Table 6) is in the region of lattice artefacts, with only 400 measurements.

Limits. $\beta$ is the single value 2.5, and the plane the single size $M=16$. From $\chi(5)$ on the errors are large, and the value of the tension itself is not determined. $N_T=10$ is about 2 in $L\sqrt\sigma$, and the trend as $L\to\infty$ is an extrapolation.

A measurement that uses Theorem 10 of 12-06 ($\mathbb E\prod\theta\le\mathbb P(\text{no container flips})$) numerically. $SU(2)$, $\beta=2.5$, containers placed with spacing $B$, and driven insertions (512 steps) with acceptance $\min(1,e^{-W})$ performed simultaneously in 4 containers per configuration. We write $\bar a$ for the mean acceptance rate.

Reading Reading: (i) Table 8: widening the $T^2$ with the radius fixed, the work increases roughly in proportion to $N_T^2$, and the acceptance rate falls. The size of the vortex is fixed by the size of the $T^2$, so the reading is that a container of radius $r\lt L$ cannot hold the vortex, which then pays a cost proportional to the area as a sheet inside a thin tube. The lateral-widening inequality is a claim about containers of size at least $L$, so this is not a counterexample. The guaranteed tension ($-\ln\mathbb P(\text{no flip})$ divided by the number of containers and $B^2$) was about 0.005 at $N_T=4$. (ii) Table 9: if the containers are enlarged together with $L$, the acceptance rate is almost constant for $N_T=4\to6$ ($N_T=8$ cannot be judged from 3 configurations).

The withdrawn positioning. This measurement was originally designed as "a numerical picture of the uniform slab tension (the crossover band)", but that positioning has been withdrawn. The reason is as in 12-06 §07: the guarantee of the flow lemma is $\theta\le1-a$, and the $a$ of a one-step kernel levels off at about $\tfrac12$ (the acceptance rates of Tables 1, 2, 8 and 9 are all at most 0.46), so all that can be said is "for containers of size $L$, $\theta\le\theta_0$ with $\theta_0\sim\tfrac12$", and the resulting $\sigma\ge\ln(1/\theta_0)/L^2$ is the weak form that $U(1)$ also satisfies (13-01). Checking the form of the band $\theta\le e^{-cL^2}$ needs a kernel with $a\to1$. This measurement can be used only as a tool for measuring lower bounds on the slab tension with the $T^2$ fixed. The continuation of Table 9 at $N_T=6,8$ has been stopped.

(a) Proof. None of the numbers of this chapter is a proof, and the hypotheses of no inequality have been checked.

(b) The band. The trends of Table 2, Tables 3–5 and Table 7 do not show the crossover band ($\theta\le e^{-cL^2}$ uniform in $L$) numerically. The $T^2$ measured go only up to about $L\sqrt\sigma\approx2$, and $L\to\infty$ is an extrapolation.

(c) Errors. The error estimates of driven insertion are still rough (sample standard deviations only), the errors of replica exchange are the record's estimates, and the treatment of autocorrelation is not in the record.

(d) Lattice artefacts. The rows with $N_T=2$ are the region of lattice artefacts. Even at $N_T=4$, small-lattice corrections remain.

(e) Balance of the control. $\beta=1.5$ for $U(1)$ and $\beta=2.5$ for $SU(2)$ are not a comparison with matched physical sizes.

(f) Simultaneous flips. Reading Tables 8 and 9 as a numerical picture of the band has been withdrawn.

Only [13] is a row of logic rather than numerics: an application of [16] of 12-06, that only the weak form follows from $\theta\le1-a$; the reading [12] shows how far that takes effect.

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

[01] RDG COST0    cost at 0 steps ∝ N_T²                     β = 2.3, M = 8
[02] RDG DRIVE    driven work → near 0 in large containers   β = 2.3–2.5
[03] RDG RADIUS   radius needed doubles per halving of a ⟸ [02]
[04] RDG FIXR     radius 4, widening N_T: no increase        β = 2.5, 10 confs × 3
[05] RDG FREE     work at β = 2.7 is free energy (physics)   Jarzynski fluctuates
[06] RDG U1UP     ΔF of U(1) grows with T²                   β = 1.5
[07] RDG SU2FLAT  ΔF of SU(2) does not grow                  β = 2.4, 2.5
[08] RDG SIGN     sign of lateral widening (physics) ⟸ [06][07]
[09] RDG NOOVER   overshoot at N_T = 6 was a fluctuation     2000 measurements
[10] RDG SATUR    χ(4) grows and saturates ⟸ [09]           includes Coulomb term
[11] RDG AREA     fixed-radius simultaneous flips ∝ area
[12] RDG HALF     one-step acceptance ≤ 0.46 (range measured)
[13] PRV RETRACT  simultaneous flips cannot test band ⟸ [12]  flow lemma of 12-06

The way to check is to see by eye that every number to the right of "⟸" is smaller than the number on the left. There are no circles. There are no assumption rows. There is neither a node of confinement nor a node of the mass gap.

What this chapter can say

Sources and reproduction

Words used in this chapter — driven insertion, number of steps, work, acceptance rate, the free energy of the global flux, replica exchange, BAR, Creutz ratio, overshoot, simultaneous flips, guaranteed tension. Their meanings are in A-02 Glossary (the words of Part 12 are to be added). The numerics of Part 13 (the test doubling the whole lattice, step functions, column biases) are in 13-09.

Next chapter to read: 12-09 Where this road stops — confirms that widening the $T^2$ is exactly the crossover band, and summarises what the road from the direction of space has reached.

← 12-07 Why fixed maps failContents12-09 Where this road stops — widening the T² is the band →