A Hermitian bypass to the non-Hermitian quantum theory

TL;DR

This work introduces a Hermitian bypass for non-Hermitian quantum theory by constructing a stable computational basis from the Hermitian operator $F=H^{\dagger}H$. It develops a full dual-space structure via discrete space-time transformations, with unitary and anti-unitary maps ($\mathcal{U}_i,\mathcal{A}_i$) and corresponding metrics ($\mathcal{C}_i$) to yield a positive-definite inner product; this leads to a weaker symmetry that can enforce real energies beyond standard Hermiticity or $\mathcal{PT}$ symmetry. The formalism clarifies exceptional points as vacua in the $F$-space, provides explicit two-level solutions, and extends to higher dimensions with degeneracy structures (circular and point). The approach unifies several NH phenomena—ladder-operator structure, dual space, and symmetry classifications—and has broad applicability in open quantum systems, photonics, and condensed matter, including non-reciprocal hopping and NH topologies.

Abstract

Describing systems with non-Hermitian (NH) operators remains a challenge in quantum theory due to instabilities (e.g., exceptional points and decoherence) arising from interactions with the environment. We propose a framework to express the energy states of NH Hamiltonians using a well-defined basis (dub computational basis) derived from a related Hermitian operator. This suitably shifts the singularities from the basis states to the expansion coefficients, allowing for their easier mathematical treatment and parametric control. Furthermore, we introduce a `space-time' transformation on the computational basis that defines a generic dual space map for the energy states. Interestingly, this transformation leads to a symmetry for real/imaginary energy values, revealing the existence of weaker condition than hermiticity or the $\mathcal{PT}$ symmetry. This leads to clearer understanding and novel interpretations of key features like exceptional points, dual space, and weaker symmetry-enforced real eigenvalues. The applicability of our framework extends to various branches of physics where NH operators manifest as ladder operators, order parameters, self-energies, projectors, and other entities.

Figures (6)

• Figure 1: The span of the computational eigenvalue space between the two exceptional points/contours at $f=0, 1$, and a normal point at $f=1/2$, and the cyclic ladder actions of $H$ and $H^\dagger$ across the normal point are schematically shown here.
• Figure 2: Energy eigenstates and the dual states are exhibited on the computational Bloch sphere parameterized by expressing the expansion parameter $a={\rm tan}\frac{\theta}{2} e^{i\phi}$. Here the exceptional points/contours are located at the two poles, while the normal point (contour) lies at the equator. The two energy eigenstates $|E_{\pm}\rangle$ lie in the same hemisphere, demonstrating that they are not orthogonal to each other, except at the equator (normal point), while collapsing to a single basis state at the poles (exceptional points/contours). The biorthogonal dual states $|\tilde{E}_{\pm}\rangle$ lie at the antipodal points to $|E_{\mp}\rangle$ in this computation space.
• Figure 3: Schematic representation of the splitting of the complex parameter space of the Hamiltonian into the computational Hilbert space $\mathbb{S}_\mathfrak{f}^{2}$, expansion parameter space $\mathbb{R}'_{|\mathfrak{f}|}$, $\mathbb{S}_{\phi}^1$, and the complex energy space $\mathbb{S}^1_{\gamma}$. The radius of the hypersphere $|h|$ corresponds to the scaling parameter $d=2|h|^2$. The blue thin dashed hypercycle is the $|\mathfrak{f}|=0$ contour of the normal point. The computational space "encircles" the normal point as shown by the thick blue dashed line and extends up to $|\mathfrak{f}|=1/2$ contour. $\gamma$ and $\phi$ are defined with respect to the exceptional point/contour.
• Figure 4: (a) Energy spectra in a higher dimensional Hamiltonian are schematically plotted as concentric circles in the complex energy plane for the case where energy eigenvalues do not cross. Each circle corresponds to a different $f_n$ with the eigenvalue pair $\pm |E_n|e^{i\gamma_n}$ lying on their diametrically opposite points. The dashed blue circle gives the maximum radius $|E_n|=1/\sqrt{2}$ corresponding to the normal point, where the center of the complex plane is the exceptional point. (b) Circular degeneracy: In this case, two or more energy levels coincide on the same circle $|E|$, but differ by the phases $\gamma_n$. (c) Point degeneracy: This occurs when two or more energy levels possess the same amplitude and phase, i.e., they correspond to a single point on the complex plane (with the 'particle-hole' component lying at the diametrically opposite point).
• Figure 5: The parameter space of the Hamiltonian defined in Sec. \ref{['Subsec:Example alpha beta']} gives a sphere of radius $d=2|h|^2=1$. The $\alpha=\pi/4$ and $3\pi/4$ circles give exceptional contours while $\alpha=\pi/2$ is the normal contour. The energy eigenvalues are real (imaginary) within (outside) the regions bounded by the exceptional points/contours. In the region with real eigenvalues, $\mathcal{S}_1$ becomes a static symmetry and coincides with the $\mathcal{PT}$-symmetry.