We derive a general relation between the ground state entanglement Hamiltonian and the physical stress tensor within the path integral formalism. For spherical entangling surfaces in a CFT, we reproduce the local ground state entanglement Hamiltonian derived by Casini, Huerta and Myers. The resulting reduced density matrix can be characterized by a spatially varying "entanglement temperature." Using the entanglement Hamiltonian, we calculate the first order change in the entanglement entropy due to changes in conserved charges of the ground state, and find a local first law-like relation for the entanglement entropy. Our approach provides a field theory derivation and generalization of recent results obtained by holographic techniques. However, we note a discrepancy between our field theoretically derived results for the entanglement entropy of excited states with a non-uniform energy density and current holographic results in the literature. Finally, we give a CFT derivation of a set of constraint equations obeyed by the entanglement entropy of excited states in any dimension. Previously, these equations were derived in the context of holography.

In this Letter, we study the effect of topological zero modes on Entanglement Hamiltonians and entropy of free chiral fermions in (1+1)d. We show how Riemann-Hilbert solutions combined with finite rank perturbation theory allow us to obtain exact expressions for Entanglement Hamiltonians. In the absence of the zero mode, the resulting Entanglement Hamiltonians consists of local and bilocal terms. In the periodic sector, the presence of a zero mode leads to an additional non-local contribution to the entanglement Hamiltonian. We derive an exact expression for this term and for the resulting change in the entanglement entropy.Entanglement Hamiltonians (EH) are the next object to explore in a series of advances in our understanding the quantum structure of many-body states in field theory and condensed matter [1][2][3][4][5][6][7][8]. Indeed, much work has been devoted to understanding entanglement entropy, in a variety of systems, and, subsequently, more detailed information about the spectrum of reduced density matrices is being investigated as well. Entanglement Hamiltonians go one step further in that they contain information about the entanglement spectrum, as well as about the associated eigenvectors, and the possibility to understand the reduced state as a thermal state. Perhaps the most striking of the recent results about Entanglement Hamiltonians is the realization that for a spherical entangling region in a conformal field theory (CFT), Entanglement Hamiltonians have a local form, that may be interpreted as the original hamiltonian energy density with a spatially dependent temperature [9][10][11]. This result is unusual in its simplicity, as, unfortunately, the computation of EHs is, in general, much more involved than entropy and spectrum and only a few results are available. Thus it is a grand challenge to find additional solvable cases.In this Letter, we present a new method for computing Entanglement Hamiltonians of free fermions in the presence of zero modes. Fermionic modes localized on topological defects often reflect the topological nature of the defect through unusual properties such as charge fractionalization [12] and non-abelian braiding [13] and appear in a variety of systems from polyacetylene [14] to defects in topological insulators and superconductors [15][16][17]. The modes are usually intimately tied to the appearance of ground state degeneracies of a topological nature. Here, we are able to present new results for the EHs for chiral fermions, and study in detail the effects of the boundary conditions (BC), periodic/anti-periodic for Majorana and generic for Dirac fermions, and of the zero modes (present in the case of periodic boundary conditions) on entanglement. The edge theory of the p + ip superconductor provides an explicit realization of such a scenario for Majorana fermions and serves as a concrete physical model for our calculation (see, e.g. [18]). Meanwhile, our calculation in the Dirac sector is relevant to the edge theory of the Quantum Hall state. Wherever available we m...

Abstract:What is the meaning of entanglement in a theory of extended objects such as strings? To address this question we consider the spatial entanglement between two intervals in the Gross-Taylor model, the string theory dual to two-dimensional Yang-Mills theory at large N . The string diagrams that contribute to the entanglement entropy describe open strings with endpoints anchored to the entangling surface, as first argued by Susskind. We develop a canonical theory of these open strings, and describe how closed strings are divided into open strings at the level of the Hilbert space. We derive the modular Hamiltonian for the Hartle-Hawking state and show that the corresponding reduced density matrix describes a thermal ensemble of open strings ending on an object at the entangling surface that we call an entanglement brane, or E-brane.

We elaborate on the extended Hilbert space factorization of Chern Simons theory and show how this arises naturally from a proper regularization of the entangling surface in the Euclidean path integral. The regularization amounts to stretching the entangling surface into a co-dimension one surface which hosts edge modes of the Chern Simons theory when quantized on a spatial subregion. The factorized state is a regularized Ishibashi state and reproduces the well known topological entanglement entropies. We illustrate how the same factorization arises from the gluing of two spatial subregions via the entangling product defined by Donnelly and Freidel [1].