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Threefold Way for Typical Entanglement

Haruki Yagi, Ken Mochizuki, Zongping Gong

TL;DR

The paper develops a symmetry-aware random-matrix framework for entanglement spectra, showing that typical symmetric states exhibit a direct-sum decomposition of the entanglement spectrum into Laguerre orthogonal (LOE), Laguerre unitary (LUE), and Laguerre symplectic (LSE) ensembles. By employing symmetry concentration, symmetry fractionalization via 2-cocycles, and irreducible representation theory, the authors derive a universal threefold way for entanglement spectra that generalizes Dyson's threefold way to the entanglement context. The construction yields explicit block-structure: WW^† decomposes into blocks labeled by irreps α of G0, with block types determined by the indicator ι_α and cohomology class, producing a rich degeneracy pattern dependent on the symmetry group and its projective representations. They extend the framework beyond the regular representation to arbitrary representations, showing the same LOE/LUE/LSE decomposition holds in general, with TRS and symmetry fractionalization producing distinct block counts and types. This work provides a foundational link between symmetry-enriched entanglement and random-matrix theory, with potential applications to symmetry-protected topological phases and anomaly structures in quantum many-body systems.

Abstract

A typical quantum state with no symmetry can be realized by letting a random unitary act on a fixed state, and the subsystem entanglement spectrum follows the Laguerre unitary ensemble (LUE). For integer-spin time reversal symmetry, we have an analogous scenario where we prepare a time-reversal symmetric state and let random orthogonal matrices act on it, leading to the Laguerre orthogonal ensemble (LOE). However, for half-integer-spin time reversal symmetry, a straightforward analogue leading to the Laguerre symplectic ensemble (LSE) is no longer valid due to that time reversal symmetric state is forbidden by the Kramers' theorem. We devise a system in which the global time reversal operator is fractionalized on the subsystems, and show that LSE arises in the system. Extending this idea, we incorporate general symmetry fractionalization into the system, and show that the statistics of the entanglement spectrum is decomposed into a direct sum of LOE, LUE, and/or LSE. Here, various degeneracies in the entanglement spectrum may appear, depending on the non-Abelian nature of the symmetry group and the cohomology class of the non-trivial projective representation on the subsystem. Our work establishes the entanglement counterpart of the Dyson's threefold way for Hamiltonians with symmetries.

Threefold Way for Typical Entanglement

TL;DR

The paper develops a symmetry-aware random-matrix framework for entanglement spectra, showing that typical symmetric states exhibit a direct-sum decomposition of the entanglement spectrum into Laguerre orthogonal (LOE), Laguerre unitary (LUE), and Laguerre symplectic (LSE) ensembles. By employing symmetry concentration, symmetry fractionalization via 2-cocycles, and irreducible representation theory, the authors derive a universal threefold way for entanglement spectra that generalizes Dyson's threefold way to the entanglement context. The construction yields explicit block-structure: WW^† decomposes into blocks labeled by irreps α of G0, with block types determined by the indicator ι_α and cohomology class, producing a rich degeneracy pattern dependent on the symmetry group and its projective representations. They extend the framework beyond the regular representation to arbitrary representations, showing the same LOE/LUE/LSE decomposition holds in general, with TRS and symmetry fractionalization producing distinct block counts and types. This work provides a foundational link between symmetry-enriched entanglement and random-matrix theory, with potential applications to symmetry-protected topological phases and anomaly structures in quantum many-body systems.

Abstract

A typical quantum state with no symmetry can be realized by letting a random unitary act on a fixed state, and the subsystem entanglement spectrum follows the Laguerre unitary ensemble (LUE). For integer-spin time reversal symmetry, we have an analogous scenario where we prepare a time-reversal symmetric state and let random orthogonal matrices act on it, leading to the Laguerre orthogonal ensemble (LOE). However, for half-integer-spin time reversal symmetry, a straightforward analogue leading to the Laguerre symplectic ensemble (LSE) is no longer valid due to that time reversal symmetric state is forbidden by the Kramers' theorem. We devise a system in which the global time reversal operator is fractionalized on the subsystems, and show that LSE arises in the system. Extending this idea, we incorporate general symmetry fractionalization into the system, and show that the statistics of the entanglement spectrum is decomposed into a direct sum of LOE, LUE, and/or LSE. Here, various degeneracies in the entanglement spectrum may appear, depending on the non-Abelian nature of the symmetry group and the cohomology class of the non-trivial projective representation on the subsystem. Our work establishes the entanglement counterpart of the Dyson's threefold way for Hamiltonians with symmetries.

Paper Structure

This paper contains 16 sections, 3 theorems, 100 equations, 3 figures, 1 table.

Table of Contents

  1. Definitions
  2. Projective representations of groups
  3. Theorems
  4. Symmetry concentration
  5. Setup and corollaries
  6. Classification of irreps
  7. Detailed proof of theorem \ref{['thm:espectrum']}
  8. Partial diagonalization
  9. Identification of the statistics
  10. $G=G_0$
  11. $G=G_0\rtimes\mathbb{Z}_2^\mathcal{T}$ in the absence of $\Upsilon$
  12. $G=G_0\rtimes\mathbb{Z}_2^\mathcal{T}$ in the presence of $\Upsilon$
  13. A theorem on ensemble types and their numbers
  14. Other representations
  15. $G=G_0$
  16. ...and 1 more sections

Key Result

Theorem 1

The matrix ensemble given in Eq. (wab) is completely decomposed into the direct sum of the threefold way. If $G=G_0$, the matrix ensemble of $WW^\dag$ is Otherwise $G=G_0\rtimes \mathbb{Z}_2^\mathcal{T}$, the ensemble of $WW^\dag$ is where $R_\pm$ is the set of (projective) irreps $\mathcal{D}^\alpha$ satisfying $\iota_\alpha=\pm\omega(t,t)$, while $R_0$ includes only one components ($\alpha$) o

Figures (3)

  • Figure 1: (a) Bipartitite system $a\cup b$ in an arbitrary (e.g., two) spatial dimension. The system is supposed to have an onsite symmetry $G$. (b) Local unitary conjugation of the onsite symmetry $D(g)^{\otimes N}$ gives $\mathbb{1}_L\otimes D(g)\otimes D(g)\otimes \mathbb{1}_R$, shown in the rearranged one-dimensional configuration. (c) Configuration of the system consisting of $L,l,r,R$. Here $P$ acting on $l\cup r$ is a projection from the $|G_0|^2$-dimensional space onto the $|G_0|$-dimensional $G_0$-symmetric space. Within this subspace, $\mathcal{U}$ is a unitary operator sampled from the Haar measure on some compact Lie groups. If $G=G_0$ ($G=G_0\rtimes\mathbb{Z}_2^\mathcal{T}$), that Lie group is the unitary group (isomorphic to the orthogonal group). Application of $\Omega$ (or/and $\Upsilon$) realizes the symmetry fractionalization of $G_0$ ($G_0\rtimes\mathbb{Z}_2^\mathcal{T}$).
  • Figure S1: Configuration of matrix block $W_{L,R}$. Here $G=G_0=C_{3v}$ was chosen for an example. $C_{3v}$ has three irreducible linear representations, two ($\alpha=1,2$) of which are 1D and the other $\alpha=3$ is 2D. $\alpha=1,2$ are 1D, thus they correspond to two independent $1\times1$ blocks $a$ and $b$. $\alpha=3$ is 2D, thus it corresponds to two identical $2\times2$ blocks $cdef$.
  • Figure S2: Let $G=G_0=C_{3v}$ again, and let $d_L=3,d_R=4$. Rearranging $W\mapsto W'$, we get block-diagonalized $d_L|G_0|\times d_R|G_0|$ rectangular matrix. The first (red) and the second (blue) blocks are mutually independent, while the third and the fourth (green) blocks are identical.

Theorems & Definitions (5)

  • Theorem 1
  • Theorem S1
  • proof
  • Theorem S2
  • proof