[Paper Review] Entanglement entropies of an interval in the free Schr\"odinger field theory on the half line

Mihail Mintchev, Diego Pontello+1 more|arXiv (Cornell University)|Jun 13, 2022

Quantum many-body systems128 references11 citations

TL;DR

This paper investigates entanglement entropies of an interval adjacent to the boundary in a free spinless Schrödinger fermionic field theory on the half line at zero temperature and finite density. Using exact analytic expansions derived from Bessel kernel tau functions and Painlevé III transcendents, it shows that the entanglement entropy exhibits oscillatory behavior due to Friedel oscillations in the particle density, contrasting with the monotonic behavior on the full line. The key result is a precise analytic description of the entanglement entropy in both small and large dimensionless parameter regimes (η = RkF), validated numerically and extended to Lifshitz models with integer exponents.

ABSTRACT

We study the entanglement entropies of an interval adjacent to the boundary of the half line for the free fermionic spinless Schr\"odinger field theory at finite density and zero temperature, with either Neumann or Dirichlet boundary conditions. They are finite functions of the dimensionless parameter given by the product of the Fermi momentum and the length of the interval. The entanglement entropy displays an oscillatory behaviour, differently from the case of the interval on the whole line. This behaviour is related to the Friedel oscillations of the mean particle density on the half line at the entangling point. We find analytic expressions for the expansions of the entanglement entropies in the regimes of small and large values of the dimensionless parameter. They display a remarkable agreement with the curves obtained numerically. The analysis is extended to a family of free fermionic Lifshitz models labelled by their integer Lifshitz exponent, whose parity determines the properties of the entanglement entropies. The cumulants of the local charge operator and the Schatten norms of the underlying kernels are also explored.

Motivation & Objective

• To study entanglement entropies for an interval adjacent to the boundary in a free spinless Schrödinger fermionic field theory on the half line at finite density and zero temperature.
• To determine how boundary conditions (Neumann or Dirichlet) affect the entanglement entropy and its dependence on the dimensionless parameter η = RkF.
• To derive analytic expressions for the entanglement entropy in the small and large η regimes using spectral theory and special functions.
• To extend the analysis to a family of free fermionic Lifshitz models with integer Lifshitz exponents and explore their entanglement properties.

Proposed method

• The study employs the replica trick and Rényi entropies to compute the entanglement entropy via the trace of powers of the reduced density matrix.
• It uses the spectral theory of the Bessel kernel and Painlevé III transcendents to derive exact analytic expansions for the entanglement entropy in the small and large η regimes.
• The analysis is based on the solution of the sine kernel spectral problem using prolate spheroidal wave functions (PSWF) and their generalization to boundary-regularized systems.
• The authors apply the Kyiv formula method to Painlevé III equations to obtain complete asymptotic expansions of the Bessel kernel tau function.
• They derive analytic expressions for the entanglement entropy and its cumulants using the tau function approach and the PSWF method, validated through consistency checks.
• The results are extended to Lifshitz models with integer exponents, where the parity of the exponent determines the qualitative behavior of the entanglement entropy.

Experimental results

Research questions

• RQ1 How does the entanglement entropy of an interval adjacent to the boundary on the half line differ from that of an interval on the full line in a free fermionic Schrödinger field theory?
• RQ2 What is the origin of the oscillatory behavior in the entanglement entropy on the half line, and how is it related to Friedel oscillations in the particle density?
• RQ3 Can analytic expressions for the entanglement entropy be derived in both the small and large η (η = RkF) regimes using special functions and spectral theory?
• RQ4 How do the entanglement entropies and their cumulants behave in a family of free fermionic Lifshitz models with integer Lifshitz exponents?
• RQ5 To what extent do the results agree with lattice model continuum limits and consistency checks such as the replica limit and double scaling?

Key findings

• The entanglement entropy on the half line exhibits oscillatory behavior due to Friedel oscillations in the particle density, contrasting with the monotonic increase observed on the full line.
• The entanglement entropy is a finite function of the dimensionless parameter η = RkF, and its analytic expansion in the small η regime is derived using the PSWF and tau function approaches.
• In the large η regime, the entanglement entropy expansion is derived from the asymptotic behavior of the Bessel kernel tau function, with subleading terms computed via Painlevé III transcendents.
• The analytic expressions for the entanglement entropy in both small and large η regimes show remarkable agreement with numerical evaluations.
• The oscillatory structure is linked to the interference of fermionic modes near the boundary, and the amplitude of oscillations is determined by the boundary condition (Neumann or Dirichlet).
• For Lifshitz models with integer exponents, the parity of the exponent governs the qualitative behavior of the entanglement entropy, with distinct patterns emerging for even and odd exponents.