Functional renormalization group for non-Hermitian and $\mathcal{PT}$-symmetric systems

Abstract

We generalize the vertex expansion approach of the functional renormalization group to non-Hermitian systems. As certain anomalous expectation values might not vanish, additional terms as compared to the Hermitian case can appear in the flow equations. We investigate the merits and shortcomings of the vertex expansion for non-Hermitian systems by considering an exactly solvable \mathcal{PT}𝒫𝒯-symmetric non-linear toy-model and reveal, that in this model, the fidelity of the vertex expansion in a perturbatively motivated truncation schema is comparable with that of the Hermitian case. The vertex expansion appears to be a viable method for studying correlation effects in non-Hermitian systems.

PT -symmetric, non-Hermitian quantum many-body physics—a methodological perspective

“…However, the physical meaning of this operator is far from obvious. A reasoning similar to this plays an important role when setting up functional integrals for PT -symmetric and pseudo-Hermitian systems [59,71,72,74,75]. We will return to this in section 6.…”

“…This can also be used as a strategy for numerically determining the eigenvalues and eigenvectors. For an example see [59] in which the non-Hermitian but PTsymmetric Hamiltonian equation (1.9) is investigated. With this restriction the spectrum is also guaranteed to be discrete, which is an additional asset when it comes to avoiding mathematical subtleties [3,23,37,40,53].…”

“…The same holds for the continuum model with Hamiltonian H = (p 2 + x2 )/2 + igx 3 (see equation (1.9)). In this the position operator x is not an observable in the sense of PTsymmetric quantum mechanics [9,59]. Bender argues that there is no fundamental problem with the absence of a position operator in PT -symmetric quantum mechanics; see section 3.4 of [9].…”

“…Secondly, for all observables a formal equivalence to the algebraic expression for the expectation value known from Hermitian quantum mechanics is achieved, if the standard inner product is replaced by the biorthogonal one. This leads to a consistent formalism within the biorthogonal Hilbert space H. As we will discuss in section 6 PT -symmetric quantum mechanics can be extended to quantum field theory allowing to use standard concepts such as generating functionals, path integrals, Green and vertex functions, diagrammatic perturbation theory etc [9,59,71,72]. For these reasons the above definitions are used in mathematical physics [9,23] and non-Hermitian quantum field theory [9,[71][72][73][74][75].…”

“…The first concerns the PT -symmetric harmonic oscillator with imaginary, cubic anharmonicity equation (1.9). For this model one might be interested in the ground state or canonical PT -symmetric 'expectation value' of the product of two position operators at different times in the imaginary time Heisenberg picture [59]. Remember that a conventional Heisenberg picture exists only when using this 'expectation value'.…”