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Higher-order spectral form factors of circular unitary ensemble

Sohail, Youyi Huang, Lu Wei

TL;DR

The paper delivers exact finite-N expressions for the second- and third-order spectral form factors in the circular unitary ensemble, expressing them in terms of polygamma functions and exploiting the ensemble's translational invariance. It provides a closed-form for the second-order SFF valid for all real times, recovers the known integer-time result, and derives detailed asymptotics in regimes k ~ N and k = O(1). For the third-order SFF, it develops a semi-closed form (and closed forms in special cases) and introduces a generalized k,b framework to capture higher-order spectral correlations. Methodologically, it combines summation representations of the CUE kernel, telescopic polygamma identities, and finite-sum relations to produce exact results and rigorous asymptotics, offering deeper insight into quantum chaos and spectral universality beyond pairwise correlations.

Abstract

Spectral form factor (SFF), one of the key quantity from random matrix theory, serves as an important tool to probe universality in disordered quantum systems and quantum chaos. In this work, we present exact closed-form expressions for the second- and third-order SFFs in the circular unitary ensemble (CUE), valid for all real values of the time parameter, and analyze their asymptotic behavior in different regimes. In particular, for the second-order SFF, we derive an exact closed-form expression in terms of polygamma functions. In the limit of infinite matrix size, and when the time parameter is restricted to integer values, the second-order SFF reproduces the standard result established in earlier studies. When the time parameter is of order one relative to the matrix size, we demonstrate that the second-order SFF grows logarithmically with the ensemble dimension. For the third-order SFFs, a closed-form result in a special case is obtained by exploiting the translational invariance of CUE.

Higher-order spectral form factors of circular unitary ensemble

TL;DR

The paper delivers exact finite-N expressions for the second- and third-order spectral form factors in the circular unitary ensemble, expressing them in terms of polygamma functions and exploiting the ensemble's translational invariance. It provides a closed-form for the second-order SFF valid for all real times, recovers the known integer-time result, and derives detailed asymptotics in regimes k ~ N and k = O(1). For the third-order SFF, it develops a semi-closed form (and closed forms in special cases) and introduces a generalized k,b framework to capture higher-order spectral correlations. Methodologically, it combines summation representations of the CUE kernel, telescopic polygamma identities, and finite-sum relations to produce exact results and rigorous asymptotics, offering deeper insight into quantum chaos and spectral universality beyond pairwise correlations.

Abstract

Spectral form factor (SFF), one of the key quantity from random matrix theory, serves as an important tool to probe universality in disordered quantum systems and quantum chaos. In this work, we present exact closed-form expressions for the second- and third-order SFFs in the circular unitary ensemble (CUE), valid for all real values of the time parameter, and analyze their asymptotic behavior in different regimes. In particular, for the second-order SFF, we derive an exact closed-form expression in terms of polygamma functions. In the limit of infinite matrix size, and when the time parameter is restricted to integer values, the second-order SFF reproduces the standard result established in earlier studies. When the time parameter is of order one relative to the matrix size, we demonstrate that the second-order SFF grows logarithmically with the ensemble dimension. For the third-order SFFs, a closed-form result in a special case is obtained by exploiting the translational invariance of CUE.

Paper Structure

This paper contains 13 sections, 7 theorems, 76 equations, 2 figures.

Table of Contents

  1. Introduction and Main Results
  2. Problem formulation
  3. Main Results
  4. Second-order SFF
  5. Higher-order SFF
  6. Proofs of Main Results
  7. Proof of Proposition \ref{['thm:SFF']}
  8. Proof of Corollary \ref{['prop:k=Nt asymptotic']}
  9. Proof of Corollary \ref{['prop:k_O(1)']}
  10. Proof of Corollary \ref{['prop:Arbitrary_point_x_0']}
  11. Proof of Proposition \ref{['prop: 3rd_order_SFF_at_(k,-k)']}
  12. Proof of Proposition \ref{['3rd SFF k,floor k']}
  13. Conclusions

Key Result

Proposition 1

The closed-form expression for the second-order SFF in CUE, as defined in Eq. (eq:Second order SFF CUE), is given by

Figures (2)

  • Figure 1: Second-order SFF with $N=5$. The Blue curve represents the closed-form expression of the SFF given in Eq. (\ref{['exact_SFF_1']}), while the orange curve shows the discrete SFF at integer $k$ as given by Eq. (\ref{['exact_SFF_2']}).
  • Figure 2: The third order SFF as given by Eq. (\ref{['eq:SFF(k,b)']}) evaluated at the point $(k, \lfloor{k}\rfloor)$ with $N=5$.

Theorems & Definitions (17)

  • Definition 1: Forrester2021stat
  • Definition 2: Forrester2021stat
  • Proposition 1
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Definition 3
  • Proposition 2
  • Proposition 3
  • Lemma 1
  • ...and 7 more