Quantitative Bulk–Boundary Correspondence for Anyon Condensation: Two Computable Indices with Explicit Arithmetic

In the paper's words

When a (2+1)-dimensional topologically ordered phase terminates at a boundary where a subset of anyons condenses, the boundary becomes gapped, and the bulk–boundary correspondence acquires sharp quantitative content: which boundary phenomena are forced by the bulk condensation data, and which remain contingent? Building on the two-index proposal of the QNFO framework [9], we develop a reconciled framework built on two computable quantities: the condensation strength I₁ = \|A\|/D, comparing the categorical size \|A\| = ∑_a ∈ A d_a² of the condensate algebra A to the bulk total quantum dimension D, and the Majorana count N_M = 2c_-, where c_- is the chiral central charge obtained from the Gauss–Milgram formula. We verify both indices by fully explicit arithmetic in three benchmark theories: the toric code condensing e, the Z₃ gauge theory condensing a charge, and the Ising topological order condensing its fermion ψ. [...]

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. Then $$\kappa_{xy} = \frac{(1/2)\,\pi^2 (1.381 \times 10^{-23})^2 (0.1)}{3 \times 6.626 \times 10^{-34}}.$$ Arithmetic: $(1.381 \times 10^{-23})^2 = 1.9072 \times 10^{-46}$; times $T = 0.1$ gives $1.9072 \times 10^{-47}$; times $c_- = 1/2$ gives $9.536 \times 10^{-48}$; times $\pi^2 = 9.8696$ gives $9.409 \times 10^{-47}$; divided by $3h = 1.9878 \times 10^{-33}$: $$\kappa_{xy} = \frac{9.409 \times 10^{-47}}{1.9878 \times 10^{-33}} \approx 4.73 \times 10^{-14}\;\text{W/K}.$$ This is a projection assuming a clean gapless chiral edge with full thermal equilibration; uncertainty is dominated by edge equilibration length, plausibly a factor of $2$ either way.

   from 4.3 Example 3: Ising topological order condensing $\psi$

2. Thermal Hall projection (Ising, gapless edge, $T = 0.1\;\text{K}$, clean-edge assumption, factor-of-$2$ uncertainty): $\kappa_{xy} \approx 4.73 \times 10^{-14}\;\text{W/K}$.

   from 5. Results

How it is done

1. The lesson is that "bulk–boundary correspondence" is not one theorem but a family, and one must state which boundary phenomena are claimed to be forced.

   from 2. Background and Related Work

2. Let $\mathcal{C}$ be a modular tensor category (MTC) describing a $(2+1)$D bulk topological order, with simple objects $a \in \mathcal{I}$, quantum dimensions $d_a$, topological spins $\theta_a = e^{2\pi i h_a}$, and modular $S$-matrix $S_{ab}$.

   from 3.1 Setup

3. For each benchmark we compute: (i) $\mathcal{D}$; (ii) $\|\mathcal{A}\|$ and $I_1$; (iii) the confined census (anyons with nontrivial monodromy against $\mathcal{A}$); (iv) $c_-$ via Gauss–Milgram and $N_M$; (v) for the Ising case, a labeled projection of the thermal Hall conductance $$\kappa_{xy} = \frac{c_- \,\pi^2 k_B^2 T}{3h},$$ with fundamental constants stated explicitly.

   from 3.4 Pipeline

4. The program converts the structural bulk–boundary correspondence of anyon condensation into arithmetic that an adjacent-field expert can audit line by line — and, more importantly, into predictions that experiment can fail.

   from 7. Conclusion

What would falsify it

1. We state precisely which boundary features remain undetermined by bulk data and which observations would falsify the framework.

   from Abstract

2. Our index framework is deliberately bidirectional in spirit: $I_1$ predicts boundary data from bulk data, and the inverse map (boundary $\to$ bulk) is where the framework can fail, as Section 6 discusses. [7] Conformal anomalies of CFTs with boundaries.

   from 2. Background and Related Work

3. This work provides a comprehensive framework for the failure of bulk Bloch invariants to predict boundary states in non-Hermitian systems, including boundary states appearing far from periodic-system gap closings.

   from 2. Background and Related Work

Where the claim stops

1. The condensation strength $I_1 = \|\mathcal{A}\|/\mathcal{D}$ and the Majorana count $N_M = 2c_-$ are computable from modular data alone, and explicit arithmetic in three closed examples shows that $I_1$ diagnoses the completeness of condensation while $N_M$ detects chiral Majorana anomalies that $I_1$ alone cannot see.

   from 7. Conclusion

2. The program converts the structural bulk–boundary correspondence of anyon condensation into arithmetic that an adjacent-field expert can audit line by line — and, more importantly, into predictions that experiment can fail.

   from 7. Conclusion

3. In non-Hermitian band theory, the correspondence famously fails in its naive form, and repairing it requires biorthogonal or point-gap refinements [2, 8].

   from 1. Introduction

Where to read

1. Abstract218 words · 1 min
2. 1. Introduction297 words · 1 min
3. 2. Background and Related Work606 words · 3 min
4. 3. Methods403 words · 2 min
5. 4. Analysis843 words · 4 min
6. 5. Results202 words · 1 min
7. 6. Discussion523 words · 2 min
8. 7. Conclusion142 words · 1 min
9. Appendix A. Divergence report106 words · 1 min