[QMC] 🌠 Wander/漫步者: Issue #73 Ward Identities and Critical Scaling (Lately Completed)

[JunkaiWang-TheoPhy]

We returned to the same point, but the rain on the glass remembered the journey.

我们回到了同一点，玻璃上的雨却记得这段旅程。

第三幕 · 在地图尽头重新发问

幕间 · 玻璃上的雨记得旅程

我们回到同一点，却没有回到同一种理解。绕行一周的几何相位把临界涨落留在玻璃上，Ward 恒等式则替这道雨痕写下尺度。

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Team

Team name	Wander (漫步者)
Members	Chenxi Wan, Yedi Shen, Junkai Wang
Contact email	WangTheoPhys@outlook.com

Challenge

Row	Value
Challenge	Determine when the Berry curvature of the 2D TFIM is genuine second-moment quantum geometry and when it is constrained to an equilibrium response by a unitary-orbit Ward identity.
Catalog issue	Progress toward #73, proposed by @ChanceSiyuan; this PR does not close the issue.
Track	QMC, with ED validation and an iPEPS extension planned.

What this PR adds

• A public progress report deriving the exact finite-volume identity
  $F_{a\phi}=\partial_a\langle G\rangle$ for a unitary-conjugacy orbit.
• The infrared hierarchy
  $\omega^{-2}\rightarrow\omega^{-1}\rightarrow\omega^0$ for physical,
  mixed physical--orbit, and pure-orbit QGT components.
• ED validation of the spectral, Ward, Wilson-loop, and opposite-rate dynamic
  estimators.
• A sign-free SSE-QMC calculation comprising 272 completed production tasks
  at $\beta=4L$.
• Two numerical results: a magnetic exponent staircase compatible with
  adjacent spacing $z=1$, and a background-free thermal quotient that is
  $0.22\sigma$ from the Ward prediction and $4.42\sigma$ from the quoted
  finite-cusp reference.

Headline results

Test	Result	Interpretation
magnetic exponent gaps	$1.0491(319)$ and $1.0137(223)$	both compatible with $z=1$
thermal quotient $Q_F(6)$	$1.1485(900)$	Ward reference: $1.1288$
separation from cusp reference	$4.42\sigma$	passes the preregistered discriminator
ED Ward agreement	$8.9\times10^{-16}$ max absolute error	convention and implementation check

Claim boundary

The strong numerical claim is the fixed, additive-background-free quotient on
$L=6,12,24$, not a precision asymptotic exponent. Flexible corrected-power
fits are not independently decisive. Higher-statistics $\beta/L=6$, larger
$L$, iPEPS finite-correlation-length scaling, and thermodynamic finite-rate
scaling remain to be completed.

This work is complementary to #191: it focuses on the exact orbit reduction
and the equilibrium critical-scaling audit.

Checks

• Markdown whitespace and display-math delimiter checks pass.
• Public-content scan contains no credentials, account names, hostnames,
  scheduler identifiers, or private filesystem paths.
• Numerical values were cross-checked against the accepted jackknife analysis
  summary before publication.

Addresses #73 without closing it.

[JunkaiWang-TheoPhy]

@ChanceSiyuan, here is our current progress on #73.

We first derived an exact finite-volume Ward identity for any
unitary-conjugacy orbit,

$$ F_{a\phi}=\partial_a\langle G\rangle. $$

For the Kolodrubetz rotation this gives
$F_{s\phi}/V=(1/4)\partial_s m_X$ up to the orientation convention. It also
organizes the QGT into an infrared hierarchy:
$\omega^{-2}\rightarrow\omega^{-1}\rightarrow\omega^0$.

We then implemented and cross-checked four estimators with ED, including Ward,
spectral, Wilson-loop, and opposite-rate dynamic routes. The spectral and Ward
values agree to $8.9\times10^{-16}$ on the test sweep. A separate C++17
sign-free SSE-QMC implementation agrees with finite-temperature ED for seven
observables within $1.14$ combined standard errors.

The completed 272-task equilibrium production campaign at $\beta=4L$ gives
two headline results:

1. On $L=6,12,24$, the magnetic exponents are
   $3.0224(392)$, $1.9733(110)$, and $0.9597(213)$. Their adjacent gaps are
   $1.0491(319)$ and $1.0137(223)$, both compatible with the predicted $z=1$.
2. The thermal background-free quotient is
   $Q_F(6)=1.1485(900)$. It lies $0.22\sigma$ from the Ward weak-divergence
   reference $1.1288$ and $4.42\sigma$ from the earlier finite-cusp reference
   $0.7513$.

Our claim is deliberately limited: the fixed-triplet quotient passes its
preregistered discriminator, but flexible corrected-power fits are not yet
decisive. The next steps are larger $L$, higher-statistics aspect-ratio checks,
iPEPS finite-correlation-length scaling, and a thermodynamic finite-rate test.

The full public-safe derivation, numbers, audits, and limitations are in this
PR's progress report. We would especially welcome your feedback on the
unitary-orbit interpretation and on which iPEPS observable should be treated
as the primary Challenge #73 endpoint.