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An enhanced term in the Szegő-type asymptotics for the free massless Dirac operator

Leon Bollmann

TL;DR

This work develops Szegő-type asymptotics for the regularised Fermi projection of the massless Dirac operator in dimensions $d\ge 2$, focusing on cube domains and analytic test functions. By combining kernel-decay analysis, position-space localisation, and momentum-space commutation, the authors obtain a $d$-term expansion with a logarithmic remainder for analytic $g$, and, for polynomials of degree at most three, a $(d+1)$-term expansion with a logarithmic coefficient independent of the regularisation. The key novelty is handling the zero-dimensional symbol discontinuity at the origin, which necessitates fine-grained off-diagonal kernel decay estimates and local asymptotics on the unit sphere $S^{d-1}_+$. These results extend Widom–Sobolev-type expansions to a discontinuous matrix-valued symbol in higher dimensions and have potential implications for entanglement entropy scaling in relativistic free-fermion systems. A local asymptotic formula is established, providing an explicit boundary contribution to the log-term through a kernel evaluated on the positive quadrant boundary, and the log-coefficient is shown to be independent of the UV regularisation parameter $b$.

Abstract

We consider a regularised Fermi projection of the Hamiltonian of the massless Dirac equation at Fermi energy zero. The matrix-valued symbol of the resulting operator is discontinuous in the origin. For this operator, we prove Szegő-type asymptotics with the spatial cut-off domains given by $d$-dimensional cubes. For analytic test functions, we obtain a $d$-term asymptotic expansion and provide an upper bound of logarithmic order for the remaining terms. This bound does not depend on the regularisation. In the special case that the test function is given by a polynomial of degree less or equal than three, we prove a $(d+1)$-term asymptotic expansion with an error term of constant order. The additional term is of logarithmic order and its coefficient is independent of the regularisation.

An enhanced term in the Szegő-type asymptotics for the free massless Dirac operator

TL;DR

This work develops Szegő-type asymptotics for the regularised Fermi projection of the massless Dirac operator in dimensions , focusing on cube domains and analytic test functions. By combining kernel-decay analysis, position-space localisation, and momentum-space commutation, the authors obtain a -term expansion with a logarithmic remainder for analytic , and, for polynomials of degree at most three, a -term expansion with a logarithmic coefficient independent of the regularisation. The key novelty is handling the zero-dimensional symbol discontinuity at the origin, which necessitates fine-grained off-diagonal kernel decay estimates and local asymptotics on the unit sphere . These results extend Widom–Sobolev-type expansions to a discontinuous matrix-valued symbol in higher dimensions and have potential implications for entanglement entropy scaling in relativistic free-fermion systems. A local asymptotic formula is established, providing an explicit boundary contribution to the log-term through a kernel evaluated on the positive quadrant boundary, and the log-coefficient is shown to be independent of the UV regularisation parameter .

Abstract

We consider a regularised Fermi projection of the Hamiltonian of the massless Dirac equation at Fermi energy zero. The matrix-valued symbol of the resulting operator is discontinuous in the origin. For this operator, we prove Szegő-type asymptotics with the spatial cut-off domains given by -dimensional cubes. For analytic test functions, we obtain a -term asymptotic expansion and provide an upper bound of logarithmic order for the remaining terms. This bound does not depend on the regularisation. In the special case that the test function is given by a polynomial of degree less or equal than three, we prove a -term asymptotic expansion with an error term of constant order. The additional term is of logarithmic order and its coefficient is independent of the regularisation.

Paper Structure

This paper contains 19 sections, 21 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. The free massless Dirac Operator
  4. Main results
  5. Localisation and higher-order terms
  6. Estimates for the integral kernel decay
  7. Higher-order terms
  8. Localisation in position space
  9. Hilbert-Schmidt and trace norm estimates
  10. Proof of Theorem \ref{['theorem_higher_order_terms']}
  11. Estimates for smooth symbols and an upper bound
  12. Proof of Theorem \ref{['Theorem-Asymptotics-Analytic']}
  13. Commutation in momentum space
  14. Commutation for monomial test functions
  15. Proof of Theorem \ref{['Theorem-commutation']}
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $g:\mathbb{C}\rightarrow\mathbb{C}$ be an entire function satisfying $g(0)=0$. Let $b\in [0,\infty[\,$, then the following asymptotic formula holds as $L\rightarrow\infty$. The coefficients $A_{m,g,b}$ are defined in definition-A-coeff. The implied constant in the error term of order $\log L$ is independent of the ultraviolet cut-off parameter $b$.

Theorems & Definitions (24)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Lemma 3.1
  • Lemma 3.2
  • Remark 3.3
  • Theorem 3.4
  • Lemma 3.5
  • Lemma 3.6
  • ...and 14 more