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arXiv:2607.09467

Entanglement entropy of ground states of the Landau Hamiltonian on the half-plane

Paul E. Pfeiffer, Wolfgang Spitzer

Published Jul 10, 2026• 0 citations
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Abstract

We study the entanglement entropy of ground states of a Hamiltonian defined on a domain with a boundary. Surprisingly, boundary conditions can change the spectrum and the nature of the spectrum drastically but not the leading behaviour of the entanglement entropy. As is well-known, the Landau Hamiltonian on the full plane has pure point spectrum (the infinitely degenerate Landau levels) and ground states display a so-called strict area law. On the other hand, the Landau Hamiltonian on the half-plane has purely absolutely continuous spectrum and yet we prove a strict area law for its ground states. We raise the question of what extra or finer conditions on the absolutely continuous spectrum are necessary to guarantee a logarithmically enhanced area-law as we have for the Laplace operator.

DOI
10.48550/arxiv.2607.09467

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We study the entanglement entropy of ground states of a Hamiltonian defined on a domain with a boundary. Surprisingly, boundary conditions can change the spectrum and the nature of the spectrum drastically but not the leading behaviour of the entanglement entropy. As is well-known, the Landau Hamiltonian on the full plane has pure point spectrum (the infinitely degenerate Landau levels) and ground states display a so-called strict area law. On the other hand, the Landau Hamiltonian on the half-plane has purely absolutely continuous spectrum and yet we prove a strict area law for its ground states. We raise the question of what extra or finer conditions on the absolutely continuous spectrum are necessary to guarantee a logarithmically enhanced area-law as we have for the Laplace operator.
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The nature of entanglement in many-body systems is a focus of intense research with the observation that entanglement holds interesting information about quantum correlations in large systems and their relation to phase transitions. In particular, it is well known that although generic, many-body states have large, extensive entropy, ground states of reasonable local Hamiltonians carry much smaller entropy, often associated with the boundary length through the so-called area law. Here we introduce a continuous family of frustration-free Hamiltonians with exactly solvable ground states and uncover a remarkable quantum phase transition whereby the entanglement scaling changes from area law into extensively large entropy. This transition shows that entanglement in many-body systems may be enhanced under special circumstances with a potential for generating "useful" entanglement for the purpose of quantum computing and that the full implications of locality and its restrictions on possible ground states may hold further surprises.
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An area law is proved for the Renyi entanglement entropy of possibly degenerate ground states in one-dimensional gapped quantum systems. Suppose in a chain of nnn spins the ground states of a local Hamiltonian with energy gap εεε are constant-fold degenerate. Then, the Renyi entanglement entropy $R_α(0
We consider the two-dimensional ideal Fermi gas subject to a magnetic field which is perpendicular to the Euclidean plane R2\mathbb R^2R2 and whose strength B(x)B(x)B(x) at x∈R2x\in\mathbb R^2x∈R2 converges to some B0>0B_0>0B0​>0 as ∥x∥→∞\|x\|\to\infty∥x∥→∞. Furthermore, we allow for an electric potential VεV_\varepsilonVε​ which vanishes at infinity. They define the single-particle Landau Hamiltonian of our Fermi gas (up to gauge fixing). Starting from the ground state of this Fermi gas with chemical potential μ≥B0μ\ge B_0μ≥B0​ we study the asymptotic growth of its bipartite entanglement entropy associated to LΛLΛLΛ as L→∞L\to\inftyL→∞ for some fixed bounded region Λ⊂R2Λ\subset\mathbb R^2Λ⊂R2. We show that its leading order in LLL does not depend on the perturbations Bε:=B0−BB_\varepsilon := B_0 - BBε​:=B0​−B and VεV_\varepsilonVε​ if they satisfy some mild decay assumptions. Our result holds for all ααα-R\' enyi entropies α>1/3α>1/3α>1/3; for α≤1/3α\le 1/3α≤1/3, we have to assume in addition some differentiability of the perturbations BεB_\varepsilonBε​ and VεV_\varepsilonVε​. The case of a constant magnetic field Bε=0B_\varepsilon = 0Bε​=0 and with Vε=0V_\varepsilon= 0Vε​=0 was treated recently for general μμμ by Leschke, Sobolev and Spitzer. Our result thus proves the stability of that area law under the same regularity assumptions on the boundary ∂Λ\partial Λ∂Λ.