Modular Hamiltonians for the massless Dirac field in the presence of a defect

Published for SISSA by Springer Received: December 16, 2020 Accepted: January 31, 2021 Published: March 22, 2021 Mihail Mintcheva and Erik Tonnib a Dipartimento di Fisica, Università di Pisa and INFN Sezione di Pisa, largo Bruno Pontecorvo 3, 56127 Pisa, Italy b SISSA and INFN Sezione di Trieste, via Bonomea 265, 34136, Trieste, Italy E-mail: , Abstract: We study the massless Dirac field on the line in the presence of a point-like defect characterised by a unitary scattering matrix, that allows both reflection and transmission. Considering this system in its ground state, we derive the modular Hamiltonians of the subregion given by the union of two disjoint equal intervals at the same distance from the defect. The absence of energy dissipation at the defect implies the existence of two phases, where either the vector or the axial symmetry is preserved. Besides a local term, the densities of the modular Hamiltonians contain also a sum of scattering dependent bi-local terms, which involve two conjugate points generated by the reflection and the transmission. The modular flows of each component of the Dirac field mix the trajectory passing through a given initial point with the ones passing through its reflected and transmitted conjugate points. We derive the two-point correlation functions along the modular flows in both phases and show that they satisfy the Kubo-Martin-Schwinger condition. The entanglement entropies are also computed, finding that they do not depend on the scattering matrix. Keywords: Conformal Field Theory, Field Theories in Lower Dimensions ArXiv ePrint: 2012.01366 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP03(2021)205 JHEP03(2021)205 Modular Hamiltonians for the massless Dirac field in the presence of a defect Contents 1 2 Dirac fermions with a point-like defect on the line 2.1 General features 2.2 Auxiliary fields basis 3 3 7 3 Modular Hamiltonians 9 4 Entanglement entropies 11 5 Modular flows 13 6 Correlation functions along the modular flows 16 7 Special bipartitions 7.1 Two equal intervals at large separation distance 7.2 Interval with the defect in its center 7.3 Two semi-infinite lines 18 19 19 21 8 Modular evolution in the spacetime 22 9 Conclusions 24 1 Introduction The study of the geometric entanglement between complementary spatial regions has provided important insights in quantum field theory, quantum gravity, condensed matter and quantum information during the last few decades. Considering a quantum system in the state described by a density matrix ρ and assuming that its Hilbert space is factorised as H = HA ⊗ HB in correspondence with the spatial bipartition A ∪ B, the reduced density matrix ρA ∝ e−KA of the subregion A is a hermitean and positive semidefinite operator normalised by TrA ρA = 1. The hermitean operator KA is the modular Hamiltonian (also known as entanglement Hamiltonian) of the region A [1, 2] and its spectrum provides the entanglement entropy SA = − TrA (ρA log ρA ). The modular Hamiltonian KA leads to define the family of unitary operators U (τ ) = e−iτ KA , parameterised by the modular parameter τ ∈ R, that generates the modular flow O(τ ) ≡ U (τ ) O U (−τ ) of any operator O localised in A. This modular flow describes the intrinsic internal dynamics induced by the reduced density matrix. It is important analytic expressions for the modular Hamiltonians in terms of the fundamental fields and for the corresponding modular flows. The first seminal example, –1– JHEP03(2021)205 1 Introduction –2– JHEP03(2021)205 in generic spacetime dimensions, is the modular Hamiltonian of half space x > 0 for a Lorentz invariant quantum field theory in its vacuum. This modular Hamiltonian, found by Bisognano and Wichmann [3, 4], is given by the boost generator in the x-direction. In Conformal Field Theory, by combining the result of Bisognano and Wichmann with the conformal symmetry, some modular Hamiltonians can be written in explicit form [5–10]. All these modular Hamiltonians are local: they are written as an integral over A of a local operator multiplied by a proper weight function. The first example of non-local modular Hamiltonian has been found by Casini and Huerta [11] for the massless Dirac field in its ground state and on the infinite line, when the subsystem A is the union of disjoint intervals, by employing the lattice results for this operator obtained by Peschel [12] (see also the reviews [13, 14]). In [11] also the modular flow of the Dirac field has been found, while the two-point correlators along this evolution satisfying the Kubo-Martin-Schwinger (KMS) condition [1] have been written in [15]. Other modular Hamiltonians for the massless Dirac fermion containing non-local terms have been discussed in [16–19]. In the examples of modular Hamiltonians mentioned above, the underlying system is invariant under spatial translations. The simplest way to break this symmetry in 1 + 1 dimensions is to consider a quantum field theory on the half-line. For the massless Dirac field on the half line, the energy conservation imposed in any boundary conformal field theory [20–22] allows only two kinds of boundary conditions [23, 24]. Correspondingly, two inequivalent models are defined: the vector phase and the axial phase. Each phase is parameterised by an angle and characterised by specific conservation laws; indeed, either the charge or the helicity is preserved but not both of them [25]. Instead, for the massless Dirac field on the line both these symmetries are conserved. In these two inequivalent phases, the modular Hamiltonians of an interval and the corresponding modular flows for the Dirac field have been studied in [26]. These modular Hamiltonians contain bi-local terms and preserve the symmetry of the underlying phase. The invariance under spatial translations on the line is broken also by introducing a point-like defect. A basic difference between boundaries and defects (see [27] for a recent review) is that apart from reflection, the latter ones are able to transmit as well. In the theory of quantum transport [28–31], a defect is usually implemented by a one-body scattering matrix, which describes its interaction with the bulk particles. Such a scattering matrix can be introduced by adding to the bulk Hamiltonian an interaction term localised at the defect. This is for instance the conventional approach to the Kondo effect [32–35]. Another option, which works for point-like defects, is to impose boundary conditions on the bulk fields at the defect. This approach finds relevant applications to the transport in quantum wire junctions. The boundary conditions characterising the defect have an important physical impact. For quantum wires, where the universality in the bulk is described by a Luttinger liquid, the boundary conditions at the junction can give origin to rich phase diagrams [36–38], whose degree of universality has still not been fully (...truncated)