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arXiv:2403.14766 · Journal of Physics A Mathematical and Theoretical

Entanglement Hamiltonian for inhomogeneous free fermions

Riccarda Bonsignori, Viktor Eisler

Published Jun 6, 2024• 11 citations
arXivHTMLPDF

Abstract

Abstract We study the entanglement Hamiltonian for the ground state of one-dimensional free fermions in the presence of an inhomogeneous chemical potential. In particular, we consider a lattice with a linear, as well as a continuum system with a quadratic potential. It is shown that, for both models, conformal field theory predicts a Bisognano–Wichmann form for the entanglement Hamiltonian of a half-infinite system. Furthermore, despite being nonrelativistic, this result is inherited by our models in the form of operators that commute exactly with the entanglement Hamiltonian. After appropriate rescaling, they also yield an excellent approximation of the entanglement spectra, which becomes asymptotically exact in the bulk of the trapped Fermi gas. For the gradient chain, however, the conformal result is recovered only after taking a proper continuum limit.

DOI
10.1088/1751-8121/ad5501

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Abstract We study the entanglement Hamiltonian for the ground state of one-dimensional free fermions in the presence of an inhomogeneous chemical potential. In particular, we consider a lattice with a linear, as well as a continuum system with a quadratic potential. It is shown that, for both models, conformal field theory predicts a Bisognano–Wichmann form for the entanglement Hamiltonian of a half-infinite system. Furthermore, despite being nonrelativistic, this result is inherited by our models in the form of operators that commute exactly with the entanglement Hamiltonian. After appropriate rescaling, they also yield an excellent approximation of the entanglement spectra, which becomes asymptotically exact in the bulk of the trapped Fermi gas. For the gradient chain, however, the conformal result is recovered only after taking a proper continuum limit.
We study the entanglement Hamiltonian for a spherical domain in the ground state of a nonrelativistic free-fermion gas in arbitrary dimensions. Decomposed into a set of radial entanglement Hamiltonians, we show that the entanglement spectrum in each sector is identical to that of a hopping chain in a linear potential, with the angular momentum playing the role of the subsystem boundary. Furthermore, the eigenfunctions follow from a commuting differential operator that has exactly the form predicted by conformal field theory. Rescaled by the radial Fermi velocity, this operator gives a perfect approximation of the entanglement Hamiltonian, except for large angular momenta that belong to the edge regime in the analogous gradient chain. One thus finds that the conformal field theory result becomes asymptotically exact only in one dimension.
Abstract We study the ground-state entanglement Hamiltonian of free nonrelativistic fermions for semi-infinite domains in one dimension. This is encoded in the two-point correlations projected onto the subsystem, an operator that commutes with the linear deformation of the physical Hamiltonian. The corresponding eigenfunctions are shown to possess the exact same structure both in the continuum as well as on the lattice. Namely, they are superpositions of the occupied single-particle modes of the total Hamiltonian, weighted by the inverse of their energy as measured from the Fermi level, and multiplied by an extra phase proportional to the integrated weight. Using this ansatz, we prove that the Bisognano–Wichmann form of the entanglement Hamiltonian becomes exact, up to a nonuniversal prefactor that depends on the dispersion for gapped chains.
We study the entanglement Hamiltonian for free-fermion chains with a particular form of inhomogeneity. The hopping amplitudes and chemical potentials are chosen such that the single-particle eigenstates are related to discrete orthogonal polynomials of the Askey scheme. Due to the bispectral properties of these functions, one can construct an operator which commutes exactly with the entanglement Hamiltonian and corresponds to a linear or parabolic deformation of the physical one. We show that this deformation is interpreted as a local inverse temperature and can be obtained in the continuum limit via methods of conformal field theory. Using this prediction, the properly rescaled eigenvalues of the commuting operator are found to provide a very good approximation of the entanglement spectrum and entropy.
A bstract We investigate the dynamics of the entanglement Hamiltonian in a system of one-dimensional free fermions, following a local joining quench of two initially disconnected half-chains in their ground states. Applying techniques of conformal field theory, we obtain a local expression where the left- and right-moving components of the energy density are associated with different weight functions. The results are then compared to numerical calculations for the hopping chain, which requires to consider a proper continuum limit of the lattice entanglement Hamiltonian, obtaining a good agreement with the field-theory prediction.
Abstract We consider the entanglement Hamiltonian for an interval in a chain of free fermions in its ground state and show that the lattice expression goes over into the conformal one if one includes the hopping to distant neighbours in the continuum limit. For an infinite chain, this can be done analytically for arbitrary fillings and is shown to be the consequence of the particular structure of the entanglement Hamiltonian, while for finite rings or temperatures the result is based on numerical calculations.
We consider the entanglement Hamiltonian for an interval in a chain of free fermions in its ground state and show that the lattice expression goes over into the conformal one if one includes the hopping to distant neighbours in the continuum limit. For an infinite chain, this can be done analytically for arbitrary fillings and is shown to be the consequence of the particular structure of the entanglement Hamiltonian, while for finite rings or temperatures the result is based on numerical calculations.