Chiral symmetry in non-Hermitian systems: product rule and Clifford algebra

Abstract

Chiral symmetry provides the symmetry protection for a large class of topological edge states. It exists in non-Hermitian systems as well, and the same anti-commutation relation between the Hamiltonian and a linear chiral operator, i.e., {H, Π} = 0, now warrants a symmetric spectrum about the origin of the complex energy plane. Here we show two general approaches to construct chiral symmetry in non-Hermitian systems, with an emphasis on lattices with detuned on-site potentials that can vary in both their real and imaginary parts. One approach relies on the simultaneous satisfaction of both non-Hermitian particle-hole symmetry and non-Hermitian bosonic anti-linear symmetry, while the other utilizes the Clifford algebra satisfied by the Dirac matrices. We also distinguish non-Hermitian chiral symmetry from pseudo-chirality, with the latter defined by ηH T η −1 = −H and belonging to a broadened definition of symmetry that maps between the left and right eigenspaces of a non-Hermitian Hamiltonian.

Perspective on topological states of non-Hermitian lattices

“…Indeed, symmetries such as particle hole symmetry fork into two since complex conjugation and transposition become distinct [52], while others get unified, as antiunitary symmetries which are distinct in the Hermitian case and now can be mapped onto each other [113]. There are also new symmetries which appear for the nonhermitian case, such as pseudo-Hermiticity [114], while others such as non-hermitian chiral symmetry [115,116] may remain hidden [117].…”

The connection between multiple prices of an Option at a given time with single prices defined at different times: The concept of weak-value in quantum finance

“…The Financial market is an open system. As a consequence of this, the Hamiltonian has to be non-Hermitian [8,12]. In an open system, it is the interaction between the system and the environment (flow of information) what reproduces the effect of loss of information which is effectively modeled by non-Hermitian Hamiltonians.…”

“…This is a consequence of the fact that the Financial market is not an isolated system, but it is rather an open system with permanent input and output of information. It is well-known that Open systems are modeled with non-Hermitian Hamiltonians and this does not violate any Quantum principle [8][9][10][11][12][13]. Given the non-Hermitian character of the system (without any imposed symmetry), the eigenvalues of the financial Hamiltonians are in general complex numbers and the evolution of the system is not necessarily unitary.…”

Non-Hermitian Physics

“…The symmetries in non-Hermitian systems are richer than those in Hermitian systems [949][950][951][952]. For example, the CS H = −ΓH † Γ † differs generally from the sublattice symmetry (SLS) H = −SHS † in non-Hermitian systems, whereas they are equivalent in the Hermitian case.…”