Extension of the Goldstone and the Englert-Brout-Higgs mechanisms to non-Hermitian theories

Abstract

We discuss the extension of the Goldstone and Englert-Brout-Higgs mechanisms to non-Hermitian Hamiltonians that possess an antilinear PT symmetry. We study a model due to Alexandre, Ellis, Millington and Seynaeve and show that for the spontaneous breakdown of a continuous global symmetry we obtain a massless Goldstone boson in all three of the antilinear symmetry realizations: eigenvalues real, eigenvalues in complex conjugate pairs, and eigenvalues real but eigenvectors incomplete. In this last case we show that it is possible for the Goldstone boson mode to be a zero-norm state. For the breakdown of a continuous local symmetry the gauge boson acquires a non-zero mass by the Englert-Brout-Higgs mechanism in all realizations of the antilinear symmetry, except the one where the Goldstone boson itself has zero norm, in which case, and despite the fact that the continuous local symmetry has been spontaneously broken, the gauge boson remains massless.

Electric conductivity in non-Hermitian holography

“…For future work, it will be interesting to investigate PT -symmetric non-Hermitian versions of superconductivity in holography [76], and compare to recent field theory studies [77][78][79][80][81][82][83], as well as to holographic hydrodynamics [71]. Another interesting route is to further investigate PT -symmetric deformations of SYK-type models, following [84][85][86][87][88][89].…”

“…time-independent. Also, the evolution in (77) will not be invariant under the complexified U(1) transformation, as e θQ H † e −θQ ̸ = (e θQ H e −θQ ) † for a general θ ∈ . To compute the expectation value (77) with ρ ∝ e −β H CFT and the non-Hermitian Hamiltonian H in (6) in holography, we could first prepare an Euclidean black hole at temperature 1/β without sources, next apply a quench on one side with H CFT + d d−1 x (M O † + M O) for time t and a quench on the conjugate side with H CFT + d d−1 x ( M * O † + M * O) for time t, then measure the observable O, and finally glue the two sides together following [101,102].…”