[Paper Review] Trace formulas for Wiener--Hopf operators with applications to entropies of free fermionic equilibrium states

Hajo Leschke, Alexander V. Sobolev+1 more|arXiv (Cornell University)|May 14, 2016

Spectral Theory in Mathematical PhysicsMathematics29 references30 citations

TL;DR

This paper establishes uniform trace norm estimates and quasiclassical asymptotic formulas for non-smooth functions of one-dimensional Wiener–Hopf operators, extending Harold Widom's classical results to non-smooth functions and unbounded domains. The authors apply these trace formulas to derive sharp, uniform asymptotics for the entanglement entropy (EE) and local entropy of free fermionic systems at positive temperature, showing consistency with ground-state EE behavior in the zero-temperature limit. The key contribution is a unified asymptotic formula for thermal EE that remains sharp for both small T and large α (quasiclassical parameter), with a logarithmic divergence in T that matches known zero-temperature results upon identifying α ∼ 1/T.

ABSTRACT

We consider non-smooth functions of (truncated) Wiener--Hopf type operators on the Hilbert space $L^2(\mathbb R^d)$. Our main results are uniform estimates for trace norms ($d\ge 1$) and quasiclassical asymptotic formulas for traces of the resulting operators ($d=1$). Here, we follow Harold Widom's seminal ideas, who proved such formulas for smooth functions decades ago. The extension to non-smooth functions and the uniformity of the estimates in various (physical) parameters rest on recent advances by one of the authors (AVS). We use our results to obtain the large-scale behaviour of the local entropy and the spatially bipartite entanglement entropy (EE) of thermal equilibrium states of non-interacting fermions in position space $\mathbb R^d$ ($d\ge 1$) at positive temperature, $T>0$. In particular, our definition of the thermal EE leads to estimates that are simultaneously sharp for small $T$ and large scaling parameter $\alpha>0$ provided that the product $T\alpha$ remains bounded from below. Here $\alpha$ is the reciprocal quasiclassical parameter. For $d=1$ we obtain for the thermal EE an asymptotic formula which is consistent with the large-scale behaviour of the ground-state EE (at $T=0$), previously established by the authors for $d\ge 1$.

Motivation & Objective

• To extend quasiclassical trace asymptotics for Wiener–Hopf operators to non-smooth functions, overcoming limitations of prior smooth-function results.
• To establish uniform estimates for trace norms of operator differences involving non-smooth functions, valid across physical parameters such as temperature T and quasiclassical parameter α.
• To analyze the large-scale behavior of local and bipartite entanglement entropy (EE) in thermal equilibrium states of non-interacting fermions in position space Rd (d ≥1), particularly in the regime T > 0 and α → ∞.
• To derive asymptotic formulas for thermal EE that remain sharp in both the small-T and large-α limits, with a logarithmic dependence on T that matches zero-temperature results.

Proposed method

• Use of the Helffer–Sj¨ostrand formula to represent non-smooth functions f of self-adjoint operators via an integral over the complex plane, enabling analysis of operator differences.
• Introduction of multi-scale symbols a(ξ) to control the dependence of operator norms on physical parameters such as temperature T, ensuring uniform estimates in α and T.
• Application of recent advances in Schatten–von Neumann norm estimates for non-smooth functions of Wiener–Hopf operators to derive uniform bounds in the trace class (S1) and Schatten classes Sq.
• Derivation of quasiclassical asymptotic formulas for traces of the operator difference Dα(a, Λ; f) in the one-dimensional case, using a decomposition of f into smooth and singular parts via a smooth cutoff function.
• Use of the linearity of the trace and the operator difference map f ↦ Dα(a, Λ; f) to decompose the entanglement entropy into contributions from different parts of the symbol, enabling asymptotic analysis.
• Establishment of uniform bounds on the trace norm of the operator difference Dα(aT,µ, Λ; ηγ) for the γ-Rényi entropy function ηγ, with constants independent of α and T (for αT ≥ α0 > 0), using the multi-scale symbol framework.

Experimental results

Research questions

• RQ1 How can quasiclassical asymptotic formulas for traces of Wiener–Hopf operators be extended to non-smooth functions f, particularly those with a singularity at one point?
• RQ2 What uniform estimates can be derived for the trace norm of the operator difference Dα(a, Λ; f) that remain valid across varying physical parameters such as temperature T and quasiclassical parameter α?
• RQ3 What is the large-scale asymptotic behavior of the entanglement entropy (EE) and local entropy of free fermionic systems in thermal equilibrium at positive temperature T > 0?
• RQ4 How does the thermal EE behave in the joint limit T → 0 and α → ∞, and does it recover the known zero-temperature asymptotics?

Key findings

• For d = 1, the entanglement entropy Hγ(T, µ; αI) satisfies the asymptotic formula Hγ(T, µ; αI) = 2ωB(aT,µ, ηγ) + o(|log(T)| + 1) as α → ∞ and αT ≥ α0 > 0, with ω = ω(I) = ω(Ic).
• When I is bounded, the local γ-Rényi entropy satisfies Sγ(T, µ; αI) = αsγ(T, µ)|I| + 2KB(aT,µ; ηγ) + o(|log(T)| + 1), where sγ(T, µ) is the entropy density and K is a geometric factor.
• In the zero-temperature limit (T ↓ 0), the entanglement entropy asymptotically behaves as Hγ(T, µ; αI) = ωN(1 + γ/(6γ))|log(T)| + o(|log(T)| + 1), with N the number of connected components of the Fermi sea.
• The coefficient ωN(1 + γ/(6γ)) in the logarithmic term agrees exactly with the known zero-temperature result for ground-state EE, confirming consistency across T = 0 and T > 0 regimes.
• The paper establishes uniform trace norm bounds: ∥Dα(aT,µ, Λ; ηγ)∥S1 ≤ Cαd−1(|log(T)| + 1), with constants independent of α and T (for αT ≥ α0), which are sharp for the Fermi symbol aT,µ.
• The asymptotic formula for the trace of the operator difference Dα(a, Λ; f) is derived for non-smooth f via a decomposition f = fφ + f(1−φ), where φ is a smooth cutoff, and the resulting terms are estimated using the multi-scale symbol framework.