T31 — Signed Volume Observable for Z₂ Gauge Theory

Arrow of time from oriented geometry

Published

July 2, 2026

Modified

July 14, 2026

Last updated: 2026-07-14 13:48 IST

📊 Live Dashboard

View all simulation runs and figures: TimesArrow Numerics Dashboard →

Objective

Implement and measure a signed volume observable for the Z₂ lattice gauge theory simulations. The key insight from the paper is that the signed volume operator \(\hat{Q}_v\) (not the positive-definite \(\hat{V}_v = \sqrt{|\hat{Q}_v|}\)) should be used for composite systems. The emergence of a global time orientation is then tied to the emergence of a macroscopic geometry with non-vanishing, extensive signed volume.

Status

Phase Description Status Date
Phase 1 2D square lattice validation ✅ Complete 2026-07-02
Phase 2 3D cubic lattice (L=8,10,12) ⚠️ Exploratory only 2026-07-02
Phase 3 Gauge-invariant dressed correlator ✅ Implemented, awaiting full validation 2026-07-08
Phase 4 New production runs and scaling ⏳ Pending —

Theory

The Signed Volume Operator

In standard LQG, the volume operator is always positive: \(\hat{V} = \sqrt{|\hat{Q}|}\). This discards orientation information. The paper’s framework uses the signed operator \(\hat{Q}\) to distinguish the two time-orientation sectors.

For a composite system:

\[\hat{Q}_{\text{total}} = \sum_{v \in S} \hat{Q}_v\]

  • Confined phase (random Z₂ orientations): Signs cancel. \(\langle |\hat{Q}_{\text{total}}| \rangle \sim \sqrt{N}\) (random walk).
  • Deconfined phase (ordered Z₂): Signs align. \(\langle |\hat{Q}_{\text{total}}| \rangle \sim N\) (extensive).

This motivates the search for a gauge-invariant orientation diagnostic. The original path-based signed-volume measurements below are retained as exploratory data, not as production evidence for a phase transition or arrow-of-time scaling.

This means the confinement-deconfinement transition would be tested through: 1. The emergence of a global arrow of time 2. The emergence of a macroscopic geometry with non-vanishing, extensive signed volume

Gauge-Dependence Correction

The signed volume at individual vertices is gauge-dependent. The earlier maximal-spanning-tree path convention chooses one representative of each gauge orbit, but it is not itself a gauge-invariant observable. Greedy or iterative gauge alignment is also withdrawn because it can force \(|Q|=N\) in any phase by choosing an aligned representative.

For a vertex at \((x,y,z)\), the sign is: \[\text{sign}(x,y,z) = \prod_{i=0}^{x-1} \sigma_{(i,0,0),x} \times \prod_{j=0}^{y-1} \sigma_{(x,j,0),y} \times \prod_{k=0}^{z-1} \sigma_{(x,y,k),z}\]

The replacement implemented in rust-lattice is a dressed orientation correlator: \[C(r_1,r_2) = s(r_1) W(r_1 \to r_2) s(r_2)\]

where \(s(r)\) is the path-dressed sign from the origin and \(W(r_1 \to r_2)\) is a Wilson line between the two sites using the opposite path ordering. The averaged observable \[Q_{\mathrm{GI}} = \frac{1}{N^2}\sum_{r_1,r_2} C(r_1,r_2)\] is designed to be invariant under local Z₂ gauge transformations.

Implementation

Rust Functions (rust-lattice)

// 3D Signed Volume
pub fn signed_volume_3d(&self) -> i64
pub fn measure_signed_volume_3d(&mut self, beta, n_sweeps, measure_every) -> (f64, f64, f64, f64, f64)

// Gauge-invariant dressed orientation correlator
pub fn local_gauge_transform_3d(&mut self, x, y, z)
pub fn gauge_invariant_signed_volume_3d(&self) -> f64
pub fn measure_gauge_invariant_signed_volume_3d(&mut self, beta, n_sweeps, measure_every) -> (f64, f64, f64, f64, f64)

// 2D Signed Area (for validation)
pub fn signed_area_2d(&self) -> i64
pub fn measure_signed_area_2d(&mut self, beta, n_sweeps, measure_every) -> (f64, f64, f64, f64, f64)

The original signed-volume measurement returns (mean |Q|, error, mean Q², mean |Q|/N, binder). The gauge-invariant replacement returns (mean Q_GI, error, mean Q_GI², mean Q_GI, binder).

Results

2D Signed Area (L=8) — Validation

Expected: |Q|/N ~ 1/√N = 0.125 across all β (no deconfined phase in 2D).

β Q
0.3 5.20 0.0813 43.87 -0.0398
0.5 6.73 0.1052 73.07 0.1278
0.7 5.30 0.0828 46.87 0.0848
1.0 7.20 0.1125 85.60 -0.0161
1.5 4.80 0.0750 35.73 0.1357

✅ Consistent with random-walk scaling across all β. Confirms no deconfined phase in 2D.

3D Signed Volume — Exploratory Gauge-Dependent Runs

The following runs were made before the T32 gauge-dependence correction. They are useful diagnostics for the old observable and data pipeline, but their \(|Q|/N\) values must not be promoted as gauge-invariant physics.

L=8 (N=512) — 1/√N = 0.044

β Q /N Error
0.40 0.0340 0.298 0.395 0.001
0.50 0.0346 0.298 0.502 0.047
0.60 0.0352 0.314 0.627 -0.017
0.70 0.0354 0.307 0.790 0.012
0.76 0.0355 0.302 0.949 0.050
0.80 0.0362 0.305 0.974 0.079
0.85 0.0307 0.267 0.986 -0.008
0.90 0.0370 0.349 0.991 -0.055
1.00 0.0505 0.258 0.997 0.465
1.10 0.0265 0.156 0.999 0.410
1.20 0.0675 0.138 0.999 0.629
1.50 0.0899 0.010 1.000 0.667

L=10 (N=1000) — 1/√N = 0.032

β Q /N Error
0.40 0.0257 0.436 0.394 -0.028
0.50 0.0256 0.433 0.502 -0.006
0.60 0.0256 0.440 0.628 -0.081
0.70 0.0250 0.427 0.789 -0.057
0.76 0.0244 0.417 0.949 -0.004
0.80 0.0267 0.437 0.973 0.052
0.85 0.0277 0.469 0.985 0.038
0.90 0.0270 0.445 0.991 0.032
1.00 0.0157 0.281 0.997 0.056
1.10 0.0230 0.199 0.999 0.503
1.20 0.0472 0.134 0.999 0.647
1.50 0.0075 0.021 1.000 0.645

L=12 (N=1728) — 1/√N = 0.024

β Q /N Error
0.40 0.0189 0.550 0.395 0.028
0.50 0.0189 0.553 0.502 -0.037
0.60 0.0192 0.556 0.628 0.019
0.70 0.0193 0.573 0.789 -0.014
0.76 0.0185 0.533 0.948 0.018
0.80 0.0195 0.547 0.974 0.093
0.85 0.0218 0.595 0.985 0.095
0.90 0.0213 0.566 0.992 0.133
1.00 0.0181 0.418 0.997 0.310
1.10 0.0145 0.399 0.999 0.111
1.20 0.0112 0.213 0.999 0.415
1.50 0.0628 0.046 1.000 0.666

Analysis

Key Observations

  1. L=8 shows the clearest exploratory trend: |Q|/N rises from ~0.034 to ~0.090. Because the observable is gauge-dependent, this is not production evidence for deconfinement or arrow-of-time scaling.

  2. L=10 anomalous at β=1.5: |Q|/N = 0.0075, below confined-phase value. This is a “bad” gauge sector where the simulation got stuck in the σ_e = -1 ground state.

  3. L=12 shows non-monotonic approach: |Q|/N drops at intermediate β before jumping to 0.063 at β=1.5. System fluctuates between degenerate ground states.

The Gauge Problem

The signed volume is gauge-dependent. In the deconfined phase, two gauge-equivalent ground states exist: - All σ_e = +1 → |Q| = N - All σ_e = -1 → |Q| ≈ 0 (checkerboard)

At finite β, tunneling between sectors causes |Q|/N to fluctuate between ~1 and ~0.

Gauge-Invariant Replacement

rust-lattice/src/lib.rs now includes:

  1. A corrected local gauge transformation that flips all six incident 3D links at a site.
  2. Tests showing plaquettes remain invariant under valid local gauge transformations.
  3. Tests showing the original signed_volume_3d() changes under gauge transformations.
  4. A candidate gauge_invariant_signed_volume_3d() dressed correlator with single-site and multi-site gauge-invariance tests.

These tests were identified in source inspection, but the local Cargo available in this shell is older than Rust edition 2024, so the full Rust test suite still needs to be run with the documented Rust 1.85+ toolchain before new production results are trusted.

Next Steps

  1. Run the Rust gauge-invariance tests with a Rust 2024-compatible toolchain.
  2. Review whether the dressed correlator has the desired physical normalization and finite-size behavior.
  3. Run new production simulations only with the validated gauge-invariant observable.
  4. Perform Binder or scaling analysis only after the replacement observable is validated.

Data Files

File Description
sv-L8-20260702-203243.json L=8 raw results
sv-L10-20260702-203325.json L=10 raw results
sv-L12-20260702-203243.json L=12 raw results

References

  • Paper: Section on Z₂ gauge field emergence
  • T20: Z₂ Lattice Gauge Theory Monte Carlo (base implementation)
  • T25: Volume Operator Extension (intertwiner spectrum)