Modular Hamiltonian of a chiral fermion on the torus

Abstract

We consider a chiral fermion at non-zero temperature on a circle (i.e., on a torus in the Euclidean formalism) and compute the modular Hamiltonian corresponding to a subregion of the circle. We do this by a very simple procedure based on the method of images, which is presumably generalizable to other situations. Our result is non-local even for a single interval, and even for Neveu-Schwarz boundary conditions. To the best of our knowledge, there are no previous examples of a modular Hamiltonian with this behavior.

Entanglement of a chiral fermion on the torus

“…The resolvent R 0V Λ is well-known in that case [19], so we can use it to obtain R V via the above equation. In [22] we showed that, in fact, R 0V Λ can be easily computed for generic temperatures, where V Λ is a collection of segments distributed all over the complex plane. In the next section we review the argument in a slightly simplified form.…”

“…In a recent paper [22] we computed a new modular Hamiltonian, namely that of a chiral fermion on a circle at non-zero temperature, i.e., on a torus in the Euclidean formalism (see [23] for a related result). The difficulty of the torus is that it cannot be mapped conformally to the plane.…”

Entanglement and relative entropy of a chiral fermion on the torus

“…The resolvent for the chiral fermion on the torus was recently obtained in [3] and also in [4]. As direct applications of the resolvent and modular Hamiltonian, we study entanglement entropy, mutual information, threepartite information and relative entropy.…”

Modular Hamiltonian for holographic excited states

“…Despite of this, its explicit form is known for the Rindler wedge in any QFT's vacuum state [8], as well as for a spherical entangling surface in CFTs [9] and in time dependent situations after a quantum quench [10]. Some new analytical results were also found recently for free theories and multiple intervals on Minkoski spacetime [11] and the torus [12]. Lastly, we mention that modular Hamiltonians have also been relevant in the context of the AdS=CFT correspondence [13][14][15], in studying the Bekenstein bound [16], the averaged null energy (ANEC) and quantum null energy (QNEC) conditions [17], and emergent gravity [18].…”

“…Typical examples of local flows are those produced by modular operators in the vacuum state and for the Rindler wedge [8], or the vacuum of a CFT on a sphere [9] and more recently the single interval case of fermions on a torus [12]. Typical examples of a nonlocal flow is when the region is on the real line and made of disjoint intervals [11] and as well when we study fermions in disjoint intervals on the torus [12]. The modular Hamiltonian K A is a self-adjoint operator that belongs to the algebra of operators, defined on the region A.…”