Area Law for the entanglement entropy of free fermions in nonrandom ergodic field

This paper deals with the asymptotic behaviour of a widely used correlation characteristic in large quantum systems. The correlations are known as quantum entanglement, the characteristic is called entanglement entropy, and the system is an ideal gas of spinless lattice fermions. The system is determined by its one-body Hamiltonian. It is shown in EPS [18] that if the Hamiltonian is an ergodic finite difference operator with an exponentially decaying spectral projection, then the asymptotic form of the entanglement entropy is the so-called area law. However, the only class of one-body Hamiltonians for which this spectral condition was verified is that consisting of discrete Schrödinger operators with random potential. In this paper, we prove the validity of the area law for several classes of Schrödinger operators whose potentials are ergodic but not random. We begin with quasiperiodic and limit-periodic operators and then move on to the interesting and highly non-trivial case of potentials generated by subshifts of finite type. These arose in the theory of dynamical systems when studying non-random chaotic phenomena. The corresponding asymptotic study requires quite an involved spectral analysis. Consequently, the majority of the paper is devoted to the proof and application of a variety of spectral properties of the operators in question, in particular we prove uniform localisation of the ejgenfunctions for the Maryland model and the exponential decay of the eihgenfunction correlator for a variety of models . We believe that these properties are of considerable independent interest.

💡 Research Summary

The paper investigates the asymptotic scaling of the entanglement entropy of free lattice fermions when the underlying one‑body Hamiltonian is an ergodic finite‑difference operator with a non‑random (deterministic) potential. The entanglement entropy of a finite block Λ⊂ℤ^d is defined as S_Λ(ε_F)=Tr_Λ h(P_Λ(ε_F)), where P(ε_F)=χ_{(−∞,ε_F]}(H) is the Fermi projection of the Hamiltonian H and h(x)=−x log x−(1−x) log(1−x) is the binary Shannon entropy. For large blocks (linear size L→∞) two canonical scaling laws are known: the “area law” S_Λ≈C L^{d−1} for non‑critical ground states (or gapped systems) and the “enhanced area law” S_Λ≈C L^{d−1} log L for critical systems. Earlier rigorous results established the area law for random i.i.d. potentials, relying on strong Anderson localization, i.e. exponential decay of the matrix elements of the spectral projection.

The authors extend this result to a broad class of deterministic ergodic potentials, including quasiperiodic, limit‑periodic, and potentials generated by subshifts of finite type. The central technical requirement is an exponential bound on the entries of the Fermi projection, which follows from uniform localization of eigenfunctions or, more generally, exponential decay of the eigenfunction correlator. The paper’s main contributions are:

1. Theorem 1 (multidimensional case) – Shows that the expectation E{S_Λ(ε_F)} obeys the area law for four families of ergodic operators:

   • The multidimensional Maryland model with potential V(ω,n)=g tan π(ω+⟨n,α⟩), where α satisfies a Diophantine condition.
   • Quasiperiodic Schrödinger operators with V(ω,n)=g v(ω+⟨n,α⟩), where v is monotone Hölder‑continuous and α obeys a weak Diophantine condition.
   • Operators with a cosine (almost‑Mathieu) potential V(ω,n)=g cos 2π(ω+⟨n,α⟩) for sufficiently large coupling g and Diophantine α.
   • Limit‑periodic potentials V_ω(n)=f(T_n ω) defined on a Cantor group with a minimal ℤ^d action.

2. Theorem 2 (one‑dimensional case) – Provides a more refined analysis for d=1, covering:

   • The one‑dimensional Maryland model, where the Fermi energy lies below a threshold determined by the Lyapunov exponent γ(λ,g) and the irrationality exponent β(α). Uniform localization is proved for all phases ω.
   • Quasiperiodic Schrödinger operators with Lipschitz‑monotone sampling functions v and Diophantine frequency α, again under large coupling.
   • The super‑critical almost‑Mathieu operator, where the coupling exceeds the critical value and the spectrum is pure point.

3. Theorem 4 (subshift potentials) – Treats Schrödinger operators whose potentials are generated by a subshift of finite type, a highly non‑trivial dynamical system. The authors establish exponential decay of the eigenfunction correlator by controlling “bad” spectral parameter sets via large‑deviation estimates (Lemma 2.5). This yields the required exponential bound on the spectral projection.

4. Uniform Localization and Eigenfunction Correlator Decay – For the Maryland model (both multidimensional and one‑dimensional) the paper proves uniform localization of all eigenfunctions, a result not previously available. For the subshift models, a new criterion (Lemma 2.5) links decay of the correlator to the measure of bad parameter sets, which is shown to be exponentially small.

The proof strategy follows two main steps. First, the authors derive model‑specific spectral properties (uniform localization or correlator decay). This involves sophisticated tools: large‑deviation estimates for transfer matrices, Diophantine approximation, Lyapunov exponent analysis, and combinatorial properties of subshifts. Second, they invoke the general framework of