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Obstructions to Unitary Hamiltonians in Non-Unitary String-Net Models

Hanshi Yang

TL;DR

This work analyzes whether non-unitary spherical fusion categories can realize local Hamiltonians in a standard positive-definite Hilbert space. By combining a rigorous Levin-Wen string-net formalism with numerical optimization over $F$-symbols and an operator-algebraic analysis, it derives a sharp obstruction: a physical realization on a positive-definite space exists if and only if the category is pseudo-unitary. The Yang-Lee category, though satisfying pentagon equations, yields negative quantum dimensions that block unitary realizations and instead induce a Krein-space (indefinite metric) structure with η-unitary $F$-symbols and η-self-adjoint Hamiltonians. The results connect to $\mathcal{PT}$-symmetry, suggesting non-unitary TQFTs can describe open systems or non-Hermitian topological phases, while standard unitary TQFTs require pseudo-unitarity. Overall, the paper clarifies the precise algebraic and operator-theoretic obstructions to unitary Levin-Wen realizations and provides a framework for non-Hermitian topological models.

Abstract

The Levin-Wen string-net formalism provides a canonical mapping from spherical fusion categories to local Hamiltonians defining Topological Quantum Field Theories (TQFTs). While the topological invariance of the ground state is guaranteed by the pentagon identity, the realization of the model on a physical Hilbert space requires the category to be unitary. In this work, we investigate the obstructions arising when this construction is applied to non-unitary spherical categories, specifically the Yang-Lee model (the non-unitary minimal model $\mathcal{M}(2,5)$). We first validate our framework by explicitly constructing and verifying the Hamiltonians for rank-3 ($\text{Rep}(D_3)$), rank-5 ($\text{TY}(\mathbb{Z}_4)$), and Abelian ($\mathbb{Z}_7$) unitary categories. We then apply this machinery to the non-unitary Yang-Lee model. Using a custom gradient-descent optimization algorithm on the manifold of $F$-symbols, we demonstrate that the Yang-Lee fusion rules admit no unitary solution to the pentagon equations. We explain this failure analytically by proving that negative quantum dimensions impose an indefinite metric on the string-net space, realizing a Krein space rather than a Hilbert space. Finally, we invoke the theory of $\mathcal{PT}$-symmetric quantum mechanics to interpret the non-Hermitian Hamiltonian, establishing that the obstruction is intrinsic to the fusion ring and cannot be removed by unitary gauge transformations.

Obstructions to Unitary Hamiltonians in Non-Unitary String-Net Models

TL;DR

This work analyzes whether non-unitary spherical fusion categories can realize local Hamiltonians in a standard positive-definite Hilbert space. By combining a rigorous Levin-Wen string-net formalism with numerical optimization over -symbols and an operator-algebraic analysis, it derives a sharp obstruction: a physical realization on a positive-definite space exists if and only if the category is pseudo-unitary. The Yang-Lee category, though satisfying pentagon equations, yields negative quantum dimensions that block unitary realizations and instead induce a Krein-space (indefinite metric) structure with η-unitary -symbols and η-self-adjoint Hamiltonians. The results connect to -symmetry, suggesting non-unitary TQFTs can describe open systems or non-Hermitian topological phases, while standard unitary TQFTs require pseudo-unitarity. Overall, the paper clarifies the precise algebraic and operator-theoretic obstructions to unitary Levin-Wen realizations and provides a framework for non-Hermitian topological models.

Abstract

The Levin-Wen string-net formalism provides a canonical mapping from spherical fusion categories to local Hamiltonians defining Topological Quantum Field Theories (TQFTs). While the topological invariance of the ground state is guaranteed by the pentagon identity, the realization of the model on a physical Hilbert space requires the category to be unitary. In this work, we investigate the obstructions arising when this construction is applied to non-unitary spherical categories, specifically the Yang-Lee model (the non-unitary minimal model ). We first validate our framework by explicitly constructing and verifying the Hamiltonians for rank-3 (), rank-5 (), and Abelian () unitary categories. We then apply this machinery to the non-unitary Yang-Lee model. Using a custom gradient-descent optimization algorithm on the manifold of -symbols, we demonstrate that the Yang-Lee fusion rules admit no unitary solution to the pentagon equations. We explain this failure analytically by proving that negative quantum dimensions impose an indefinite metric on the string-net space, realizing a Krein space rather than a Hilbert space. Finally, we invoke the theory of -symmetric quantum mechanics to interpret the non-Hermitian Hamiltonian, establishing that the obstruction is intrinsic to the fusion ring and cannot be removed by unitary gauge transformations.
Paper Structure (26 sections, 6 theorems, 32 equations, 1 table)

This paper contains 26 sections, 6 theorems, 32 equations, 1 table.

Table of Contents

  1. Introduction
  2. Formalism
  3. Conventions and Normalization
  4. Verification on Unitary Categories
  5. Case I: Non-Abelian $\text{Rep}(D_3)$
  6. Case II: Tambara-Yamagami $\text{TY}(\mathbb{Z}_4)$
  7. Case III: Abelian Modular Category $\text{FR}^{7,6}_{1}$
  8. Numerical Investigation
  9. Methodology
  10. Stage 1: Combinatorial Possibilities.
  11. Stage 2: Manifold Optimization.
  12. Automated Discovery of Unitary Theories
  13. On the Yang-Lee Category
  14. Pseudo-Unitarity Analysis
  15. Indefinite Metric and Krein Space
  16. ...and 11 more sections

Key Result

Proposition 5.5

Let $\mathcal{C}_{YL}$ be the Yang-Lee fusion category. The $F$-matrices of $\mathcal{C}_{YL}$ are not unitary but are $\eta$-unitary with respect to the metric $\eta = \text{diag}(1, -1)$ in the fusion space $V_{\tau\tau\tau}^\tau$.

Theorems & Definitions (25)

  • Definition 2.1: String-Net Space
  • Definition 2.2: Vertex Operator
  • Definition 2.3: Plaquette Operator
  • Remark 2.4
  • Definition 4.1: The Yang-Lee Category $\mathcal{C}_{YL}$
  • Definition 5.1: Krein Space
  • Definition 5.2: The Categorical Metric
  • Remark 5.3
  • Definition 5.4: $\eta$-Unitarity
  • Proposition 5.5
  • ...and 15 more