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Statistical Mechanics and Categorical Entropy

Haiqi Wu, Kai Xu

TL;DR

The paper investigates the algebraicity conjecture for exp($h_0(F)$) of endofunctors on saturated $A_\infty$-categories and links it to von Neumann entropy in quantum lattice models via a gauged lattice framework. By deriving a lattice-based pathway from endofunctors, it shows that ground-state data obey integer-coefficient recurrences and that algebraicity of growth rates follows from spectral properties of local data, formalizing Condition B as a sufficient criterion. It then extends to gauged lattice models, defining physical space and entropy, and posits a main conjecture that the low-$T$, large-$N$ per-site entropy is the log of an algebraic integer, thereby tying categorical entropy to gauged von Neumann entropy. Together, these results suggest a structural bridge between $A_\infty$-categories and statistical mechanics, with potential impacts on symplectic geometry, algebraic geometry, and quantum information theory.

Abstract

This paper investigates the relationship between categorical entropy and von Neumann entropy of quantum lattices. We begin by studying the von Neumann entropy, proving that the average von Neumann entropy per site converges to the logarithm of an algebraic integer in the low-temperature and thermodynamic limits. Next, we turn to categorical entropy. Given an endofunctor of a saturated A-infinity-category, we construct a corresponding lattice model, through which the categorical entropy can be understood in terms of the information encoded in the model. Finally, by introducing a gauged lattice framework, we unify these two notions of entropy. This unification leads naturally to a sufficient condition for a conjectural algebraicity property of categorical entropy, suggesting a deeper structural connection between A-infinity-categories and statistical mechanics.

Statistical Mechanics and Categorical Entropy

TL;DR

The paper investigates the algebraicity conjecture for exp() of endofunctors on saturated -categories and links it to von Neumann entropy in quantum lattice models via a gauged lattice framework. By deriving a lattice-based pathway from endofunctors, it shows that ground-state data obey integer-coefficient recurrences and that algebraicity of growth rates follows from spectral properties of local data, formalizing Condition B as a sufficient criterion. It then extends to gauged lattice models, defining physical space and entropy, and posits a main conjecture that the low-, large- per-site entropy is the log of an algebraic integer, thereby tying categorical entropy to gauged von Neumann entropy. Together, these results suggest a structural bridge between -categories and statistical mechanics, with potential impacts on symplectic geometry, algebraic geometry, and quantum information theory.

Abstract

This paper investigates the relationship between categorical entropy and von Neumann entropy of quantum lattices. We begin by studying the von Neumann entropy, proving that the average von Neumann entropy per site converges to the logarithm of an algebraic integer in the low-temperature and thermodynamic limits. Next, we turn to categorical entropy. Given an endofunctor of a saturated A-infinity-category, we construct a corresponding lattice model, through which the categorical entropy can be understood in terms of the information encoded in the model. Finally, by introducing a gauged lattice framework, we unify these two notions of entropy. This unification leads naturally to a sufficient condition for a conjectural algebraicity property of categorical entropy, suggesting a deeper structural connection between A-infinity-categories and statistical mechanics.

Paper Structure

This paper contains 7 sections, 6 theorems, 50 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Lattice model and the von Neumann entropy
  3. The algebraicity in the case of a quantum lattice model
  4. Categorical entropy of $A_{\infty}$-categories
  5. Categorical entropy of saturated $A_{\infty}$-cateegories
  6. Categorical entropy from lattice model
  7. Conclusion

Key Result

Lemma 1

In the low temperature limit $T\longrightarrow 0$, the von Neumann entropy is given by the logarithm of the ground state degeneracy G: .

Figures (2)

  • Figure 1: A one-dimensional lattice with $V_{N+1}=V_1$
  • Figure 2: The action of $d_{loc}$ on the adjacent sites $V_N$ and $V_1$. This contributes the extra $R$ in the $LHS$ as well as the extra $V$ in the $RHS$ of (\ref{['dimH']}).

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Theorem 1
  • Lemma 2
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Theorem 2