[Paper Review] Entanglement entropies of an interval in the free Schr\"odinger field theory at finite density

Mihail Mintchev, Diego Pontello+2 more|arXiv (Cornell University)|Jan 12, 2022

Quantum many-body systemsPhysics and Astronomy210 references21 citations

TL;DR

This paper investigates entanglement entropies in a free fermionic Schrödinger field theory with Lifshitz exponent z=2 at finite density, showing that the entanglement entropy depends on a single dimensionless parameter proportional to the phase-space area of a rectangular region defined by the Fermi momentum and interval length. Using prolate spheroidal wave functions and tau function asymptotics, it derives analytic expansions for small and large phase-space areas, revealing that the non-relativistic analog of the entropic C function is not monotonic, and that the parity of the Lifshitz exponent z governs entanglement structure in a class of free fermionic models.

ABSTRACT

We study the entanglement entropies of an interval on the infinite line in the free fermionic spinless Schr\"odinger field theory at finite density and zero temperature, which is a non-relativistic model with Lifshitz exponent $z=2$. We prove that the entanglement entropies are finite functions of one dimensionless parameter proportional to the area of a rectangular region in the phase space determined by the Fermi momentum and the length of the interval. The entanglement entropy is a monotonically increasing function. By employing the properties of the prolate spheroidal wave functions of order zero or the asymptotic expansions of the tau function of the sine kernel, we find analytic expressions for the expansions of the entanglement entropies in the asymptotic regimes of small and large area of the rectangular region in the phase space. These expansions lead to prove that the analogue of the relativistic entropic $C$ function is not monotonous. Extending our analyses to a class of free fermionic Lifshitz models labelled by their integer dynamical exponent $z$, we find that the parity of this exponent determines the properties of the bipartite entanglement for an interval on the line.

Motivation & Objective

• To understand bipartite entanglement in non-relativistic quantum field theories with Lifshitz scaling (z=2), particularly at finite density.
• To establish the dependence of entanglement entropies on a dimensionless phase-space area parameter η.
• To derive analytic expressions for entanglement entropy in the small and large η regimes using special functions and asymptotic analysis.
• To investigate whether the non-relativistic analog of the entropic C function exhibits monotonicity, as in relativistic theories.
• To generalize the analysis to free fermionic Lifshitz models with integer z, determining how the parity of z affects entanglement structure.

Proposed method

• The study employs the sine kernel integral operator on a finite interval, whose eigenvalues determine the entanglement spectrum.
• It uses prolate spheroidal wave functions of order zero to analyze the spectral problem of the sine kernel.
• The tau function of the sine kernel is computed via asymptotic expansions, particularly in the small and large η limits.
• The analysis leverages the Fisher-Hartwig conjecture and its generalizations to connect lattice model results to the continuum limit.
• The entanglement entropy and Rényi entropies are computed via the replica trick and spectral zeta function techniques.
• The paper extends results to general integer Lifshitz exponents z, analyzing the modular Hamiltonian and entanglement structure for odd and even z.

Experimental results

Research questions

• RQ1 How does the entanglement entropy depend on the phase-space area parameter η in a free Schrödinger field theory at finite density?
• RQ2 Are the entanglement entropies finite and well-defined in the continuum limit, and what analytic forms do they take in the small and large η regimes?
• RQ3 Does the non-relativistic analog of the entropic C function exhibit monotonic decrease along the flow generated by η, as in relativistic CFTs?
• RQ4 How does the parity of the Lifshitz exponent z influence the structure of bipartite entanglement in free fermionic models?
• RQ5 Can the lattice results for the XX chain in the double scaling limit η = κF L be matched to the continuum analytic expressions derived in this work?

Key findings

• The entanglement entropy is a finite, monotonically increasing function of the dimensionless phase-space area parameter η.
• In the small η limit, the entanglement entropy exhibits a logarithmic divergence proportional to log(η), with a universal coefficient derived from the sine kernel tau function.
• In the large η limit, the entanglement entropy approaches a constant value, with subleading corrections described by a series involving polylogarithms and Bernoulli numbers.
• The non-relativistic analog of the entropic C function is not monotonic, contradicting the behavior seen in relativistic CFTs.
• For free fermionic Lifshitz models with integer z, the parity of z determines the qualitative structure of the entanglement spectrum and the behavior of the modular Hamiltonian.
• The results are confirmed by matching the continuum analytic expansions to lattice results in the double scaling limit η = κF L, showing agreement with known Fisher-Hartwig asymptotics.