Non-Hermitian Quantum Theory via Hermitian Operators: A Unified Framework with Application to Anyons

Abstract

Non-Hermitian (NH) operators are more prevalent than we often recognize across a variety of domains. They naturally arise in diverse open system settings such as photonic lattices, non- equilibrium statistical mechanics, etc. In closed systems, they also appear in disguise, such as chiral boundary Hamiltonians of topological insulators, complex order parameters for phase transitions, among others. However, describing systems with NH operators remains a challenge in quantum theory due to peculiarities, such as exceptional point singularities, decoherence effects, etc., arising from interactions with the environment. In this thesis, I introduce a theoretical framework that provides a well-defined treatment of NH operators. I define computational basis for representing the NH Hamiltonian eigenstates, in which singularities are shifted from the basis states to the expansion coefficients. This basis is derived from a suitably chosen Hermitian operator and serves to simplify the mathematical treatment of NH operators. Furthermore, I propose a local ‘space-time’ transformation on the computational basis that defines a generic dual space mapping. Interestingly, this transformation reveals a static/global symmetry for real/imaginary energy values, unveiling inherent conserved quantities in open quantum systems. I further demonstrate that these mapping and the resulting symmetries in the computational space, naturally lead to a classification of NH Hamiltonians within the generalized Bernard-LeClair framework. The formalism thus provides fresh insights into key aspects of NH systems, including the nature of exceptional points, dual space structures, symmetry-enforced real eigenvalues, and symmetry-based classification.

The theory is further extended to higher-dimensional Hilbert spaces. Here, I find that in odd dimensions, at least one complex energy eigenvalue exists whose absolute value becomes independent of system parameters, referred to as ‘flat energy’ or ‘flat band’. I further identify two interesting cases of degeneracies, termed circular and point degeneracies, that can be accessed through a spectral flattening procedure applied to the Hamiltonian. Through various examples and applications, I demonstrate how the formalism unravels the underlying interpretation of exceptional points, normal operators, topology, dual space, inner product structures, symmetry-enforced real eigenvalues, flat energy, and other features. These results illustrate that the proposed framework is broadly applicable across various areas of physics where NH operators appear in diverse roles such as, ladder operators, order parameters, self-energies, projectors, and other entities.

While the familiar NH ladder/creation-annihilation operator structures in standard quantum mechanics, such as those of the harmonic oscillator or of bosonic/fermionic modes, lead to an integer grading of the state space corresponding to an integer valued number operator, I discover that the NH operator ladder action can, in general, generate a fractional grading of the state space corresponding to the fractional eigenvalues of the computational basis states. The emergence of such fractional numbers is naturally indicative of anyons, which are quasiparticles that obey fractional exchange statistics. Based on this observation, in the final part of this thesis, I make use of the NH quantum theory to formulate a second quantization framework for abelian anyons in one-dimension (1D). Anyons are primarily obtained in fractional quantum Hall systems and in simulated 1D cold atom lattices with gauge fields. However, a consistent second quantization formalism for anyons remains a challenge. After identifying a NH aspect of the anyonic commutation algebra with exchange phase π/N , I formulate a finite dimensional Fock space consisting of N states. Moreover, for the π/3 anyon case, I obtain a modified Jordon-Winger transformation to spin S = 1 representation. With this setting, I study a tight-binding model of an anyon chain with periodic boundary condition and a contact potential that penalizes double occupancy. I use exact diagonalization for system sizes up to L = 19 sites, and scan through all possible filling fractions for π/3 anyon case. The study reveals interesting phase transitions as a function of filling fraction and tight-binding hopping, which are characterized by persistent current in ground state, fidelity, momentum quantum number, and correlations functions. This second quantization formalism, together with the resulting analysis, opens up promising new directions for the study and engineering of lattice based anyonic systems.