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.
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.
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.
As a powerful theoretical construct, the entanglement Hamiltonian (EH) encapsulates the essential entanglement properties of a quantum many-body system. From the EH, one can extract a variety of entanglement quantities, such as entanglement entropies, negativity, and the entanglement spectrum. However, its general analytical form remains largely unknown. While the Bisognano-Wichmann theorem gives an exact EH form for Lorentz-invariant field theories, its validity on lattice systems is limited, especially when Lorentz invariance is absent. In this work, we propose a general scheme based on the lattice-Bisognano-Wichmann (LBW) ansatz and multi-replica-trick quantum Monte Carlo methods to numerically reconstruct the entanglement Hamiltonian in two-dimensional systems and systematically explore its applicability to systems without translational invariance, going beyond the original scope of the primordial Bisognano-Wichmann theorem. Various quantum phases--including gapped and gapless phases, critical points, and phases with either discrete or continuous symmetry breaking--are investigated, demonstrating the versatility of our method in reconstructing entanglement Hamiltonians. Furthermore, we find that when the entanglement boundary of a system is ordinary (i.e., free from surface anomalies), the LBW ansatz provides an accurate approximation well beyond Lorentz-invariant cases. Our work thus establishes a general framework for investigating the analytical structure of entanglement in the complex quantum many-body systems.
The modular (or entanglement) Hamiltonian correspondent to the half-space bipartition of a quantum state uniquely characterizes its entanglement properties. However, in the context of lattice models, its explicit form is analytically known only for the two spin chains and certain free theories in one dimension. In this work, we provide a thorough investigation of entanglement Hamiltonians in lattice models obtained via the Bisognano-Wichmann theorem, which provides an explicit functional form for the entanglement Hamiltonian itself in quantum field theory. Our study encompasses a variety of one- and two-dimensional models, supporting diverse quantum phases and critical points, and, most importantly, scanning several universality classes, including Ising, Potts, and Luttinger liquids. We carry out extensive numerical simulations based on the density matrix renormalization group method, exact diagonalization, and quantum Monte Carlo. In particular, we compare the exact entanglement properties and correlation functions to those obtained applying the Bisognano-Wichmann theorem on the lattice. We carry out this comparison on both the eigenvalues and eigenvectors of the entanglement Hamiltonian, and expectation values of correlation functions and order parameters. Our results evidence that as long as the low-energy description of the lattice model is well captured by a Lorentz-invariant quantum field theory, the Bisognano-Wichmann theorem provides a qualitatively and quantitatively accurate description of the lattice entanglement Hamiltonian. The resulting framework paves the way to direct studies of entanglement properties utilizing well-established statistical mechanics methods and experiments.
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.
Abstract We consider free-fermion chains in the ground state and the entanglement Hamiltonian for a subsystem consisting of two separated intervals. In this case, one has a peculiar long-range hopping between the intervals in addition to the well-known and dominant short-range hopping. We show how the continuum expressions can be recovered from the lattice results for general filling and arbitrary intervals. We also discuss the closely related case of a single interval located at a certain distance from the end of a semi-infinite chain and the continuum limit for this problem. Finally, we show that for the double interval in the continuum a commuting operator exists which can be used to find the eigenstates.