Locality and Exceptional Points in Pseudo-Hermitian Physics
- URL: http://arxiv.org/abs/2306.04044v1
- Date: Tue, 6 Jun 2023 22:19:05 GMT
- Title: Locality and Exceptional Points in Pseudo-Hermitian Physics
- Authors: Jacob L. Barnett
- Abstract summary: Pseudo-Hermitian operators generalize the concept of Hermiticity.
This thesis is devoted to the study of locality in quasi-Hermitian theory.
Chiral symmetry and representation theory are used to derive large classes of pseudo-Hermitian operators.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: Pseudo-Hermitian operators generalize the concept of Hermiticity. This class
of operators includes the quasi-Hermitian operators, which reformulate quantum
theory while retaining real-valued measurement outcomes and unitary time
evolution. This thesis is devoted to the study of locality in quasi-Hermitian
theory, the symmetries and conserved quantities associated with non-Hermitian
operators, and the perturbative features of pseudo-Hermitian matrices. In
addition to the presented original research, scholars will appreciate the
lengthy introduction to non-Hermitian physics.
Local quasi-Hermitian observable algebras are examined. Expectation values of
local quasi-Hermitian observables equal expectation values of local Hermitian
observables. Thus, quasi-Hermitian theories do not increase the values of
nonlocal games set by Hermitian theories. Furthermore, Bell's inequality
violations in quasi-Hermitian theories never exceed the Tsirelson bound of
Hermitian quantum theory.
Exceptional points, which are branch points in the spectrum, are a
perturbative feature unique to non-Hermitian operators. Cusp singularities of
algebraic curves are related to higher-order exceptional points.
To exemplify novelties of non-Hermiticity, one-dimensional lattice models
with a pair of non-Hermitian defect potentials with balanced loss and gain,
$\Delta \pm i \gamma$, are explored. When the defects are nearest neighbour,
the entire spectrum becomes complex when $\gamma$ is tuned past a second-order
exceptional point. When the defects are at the edges of the chain and the
hopping amplitudes are 2-periodic, as in the Su-Schrieffer-Heeger chain, the
$\mathcal{PT}$-phase transition is dictated by the topological phase.
Chiral symmetry and representation theory are used to derive large classes of
pseudo-Hermitian operators with closed-form intertwining operators.
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