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Modular theory and symmetry resolution in hyperfinite von Neumann algebras

Giuseppe Di Giulio, Moritz Dorband, Johanna Erdmenger, Henri Scheppach

TL;DR

This work extends symmetry-resolved modular theory from type I to hyperfinite von Neumann algebras (types II/III) by leveraging the Powers–Araki–Woods infinite tensor product construction and direct-integral decompositions. It shows that hyperfinite algebras can be resolved into subregion charge sectors, producing symmetry-resolved modular operators, modular flows, and modular correlation functions that obey a KMS condition sector-by-sector, provided the total modular structure satisfies the KMS property. The approach unifies fixed-charge sector analysis with continuous or discrete charge spectra via direct integrals, and demonstrates how a single effective charge sector emerges in large-N limits through a rescaled subregion charge. The results have potential applications in conformal field theory, condensed matter physics, and AdS/CFT bulk reconstruction, where boundary charge sectors can provide refined data for spacetime reconstruction and entanglement structure.

Abstract

We study modular theory in hyperfinite von Neumann algebras, i.e. in those of type II or type III, from the viewpoint of a subregion charge sector decomposition. We address this symmetry resolution by considering infinite tensor products of finite-dimensional algebras with fixed subregion charge values. An important ingredient is the combination of these algebras using direct integrals. This allows us to obtain the symmetry-resolved modular operator, modular flow, and modular correlation functions for hyperfinite algebras. Our approach establishes a mathematical foundation for recent results on symmetry resolution and modular theory in conformal field theory. Our analysis applies both to charges defined on a continuous range, or on a discrete set. The latter is of interest for condensed matter theory. Moreover, within the AdS/CFT correspondence we expect our findings to be relevant as a new ingredient for bulk spacetime reconstruction, including information from different boundary charge sectors.

Modular theory and symmetry resolution in hyperfinite von Neumann algebras

TL;DR

This work extends symmetry-resolved modular theory from type I to hyperfinite von Neumann algebras (types II/III) by leveraging the Powers–Araki–Woods infinite tensor product construction and direct-integral decompositions. It shows that hyperfinite algebras can be resolved into subregion charge sectors, producing symmetry-resolved modular operators, modular flows, and modular correlation functions that obey a KMS condition sector-by-sector, provided the total modular structure satisfies the KMS property. The approach unifies fixed-charge sector analysis with continuous or discrete charge spectra via direct integrals, and demonstrates how a single effective charge sector emerges in large-N limits through a rescaled subregion charge. The results have potential applications in conformal field theory, condensed matter physics, and AdS/CFT bulk reconstruction, where boundary charge sectors can provide refined data for spacetime reconstruction and entanglement structure.

Abstract

We study modular theory in hyperfinite von Neumann algebras, i.e. in those of type II or type III, from the viewpoint of a subregion charge sector decomposition. We address this symmetry resolution by considering infinite tensor products of finite-dimensional algebras with fixed subregion charge values. An important ingredient is the combination of these algebras using direct integrals. This allows us to obtain the symmetry-resolved modular operator, modular flow, and modular correlation functions for hyperfinite algebras. Our approach establishes a mathematical foundation for recent results on symmetry resolution and modular theory in conformal field theory. Our analysis applies both to charges defined on a continuous range, or on a discrete set. The latter is of interest for condensed matter theory. Moreover, within the AdS/CFT correspondence we expect our findings to be relevant as a new ingredient for bulk spacetime reconstruction, including information from different boundary charge sectors.

Paper Structure

This paper contains 28 sections, 150 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Setup and tools
  3. Review of modular theory
  4. Symmetry-resolved modular theory in type I algebras
  5. Review of the infinite tensor product construction
  6. Conserved charges in infinite tensor product algebras
  7. Local magnetization operators
  8. Rescaling of the charge
  9. Type III$_\lambda$:
  10. Type II$_1$ and III$_0$:
  11. Type III$_1$:
  12. Interpretation through symmetry-resolved entanglement
  13. Symmetry resolution of hyperfinite algebras
  14. General strategy
  15. Direct integrals of von Neumann algebras
  16. ...and 13 more sections

Figures (2)

  • Figure 1: Setup of the ITPFI construction. The Hilbert space is constructed from an infinite tower of pairwise entangled spins.
  • Figure 2: $C^{(N)}({\tilde{\lambda}_1})$ associated with the sequence \ref{['eq:seq_divergent']} plotted as a function of $N$.