Non-Hermitian Hamiltonian Formalism

• Non-Hermitian Hamiltonian Formalism is a framework describing open quantum systems, characterized by energy-dependent effective Hamiltonians, complex eigenvalues, and nonunitary evolution.
• It utilizes the Feshbach projection method to partition the total Hilbert space into discrete and continuum subspaces, enabling detailed analysis of resonance positions and decay widths.
• Key phenomena such as exceptional points, reduced phase rigidity, and enhanced energy transport in systems like photosynthetic complexes underscore its broad applications and theoretical significance.

A non-Hermitian Hamiltonian formalism addresses quantum systems that interact with their environment, where the generator of time evolution is no longer Hermitian and thus does not generate unitary dynamics. Unlike standard Hermitian quantum mechanics, where observables and the evolution generator are strictly self-adjoint to guarantee the conservation of probability, the non-Hermitian approach naturally emerges in the theoretical description of open quantum systems, many-body resonances, dissipative processes, and scenarios featuring effective gains or losses. This framework rigorously connects resonance phenomena, the interplay between discrete and continuum states, eigenfunction fluctuations, exceptional points, and even near-unitary transport efficiency observed in biological processes such as photosynthesis (Rotter, 2017).

1. Feshbach Projection and the Emergence of Non-Hermitian Hamiltonians

The non-Hermitian formalism is structurally rooted in the projection-operator (Feshbach) method. The total Hilbert space ℋ (system plus environment) is partitioned into two orthogonal subspaces: a discrete subspace Q (typically describing bound or resonance states) and a continuum subspace P (the environment or scattering channels), with projection operators P and Q obeying $P+Q=1$. From the Schrödinger equation for the full Hermitian Hamiltonian H, projection yields the coupled equations:

• $$(H_{PP} - E) |\Psi_P\rangle = -H_{PQ} |\Psi_Q\rangle$$
• $$(H_{QQ} - E) |\Psi_Q\rangle = -H_{QP} |\Psi_P\rangle$$

Eliminating $|\Psi_Q\rangle$ gives an effective, generally energy-dependent, non-Hermitian Hamiltonian $H_\mathrm{eff}(E)$ acting in the P-subspace: $$H_\mathrm{eff}(E) = H_{PP} + H_{PQ}\frac{1}{E^+ - H_{QQ}} H_{QP}$$ Here, $E^+ = E + i0$ encodes boundary conditions in the continuum, and the non-Hermitian character arises from the coupling to the environment. The real part of the second term shifts resonance positions; the imaginary part (from the continuum poles’ residues) gives finite widths to resonance states (Rotter, 2017).

2. Eigenvalue Structure, Biorthogonality, and Phase Rigidity

The eigenvalue problem

$H_\mathrm{eff}(E) \phi_k = z_k \phi_k$

yields complex eigenvalues $z_k = E_k - \frac{i}{2} \Gamma_k$, where $E_k$ is a resonance energy and $\Gamma_k$ is its decay width, interpreting the imaginary component as an inverse lifetime. Since $H_\mathrm{eff} \ne H_\mathrm{eff}^\dagger$, its left and right eigenfunctions are distinct and must be treated in a biorthogonal basis:

• $$H_\mathrm{eff} |\phi_k^R\rangle = z_k |\phi_k^R\rangle$$
• $$\langle \phi_k^L| H_\mathrm{eff} = z_k \langle \phi_k^L|$$

The biorthogonality condition is $\langle \phi_k^L | \phi_j^R \rangle = \delta_{kj}$. The phase rigidity $$r_k = \frac{\langle \phi_k^L | \phi_k^R \rangle}{\langle \phi_k^R | \phi_k^R \rangle}$$ measures nonorthogonality; $r_k\rightarrow1$ in the Hermitian limit (well-separated resonances), and $r_k\rightarrow0$ near exceptional points (maximal eigenfunction mixing). A loss of phase rigidity signals strong continuum-induced interference, a key signature of non-Hermitian dynamics (Rotter, 2017).

3. Exceptional Points and Nonlinear Effects

An exceptional point (EP) in parameter space is encountered when two (or more) eigenvalues and their corresponding eigenvectors coalesce such that the set of eigenvectors becomes incomplete. For a minimal two-level system,

$$\det [ H_\mathrm{eff} - z I ] = (\varepsilon_1 - z)(\varepsilon_2 - z) - \omega^2 = 0$$

gives the EP condition $(\varepsilon_1-\varepsilon_2)^2 + 4\omega^2 = 0$. These manifest as branch-point singularities in the complex energy plane, with characteristic square-root eigenvalue splitting. Signatures of EPs include drastic reduction in phase rigidity, nontrivial Berry phases when encircling the EP, and enhanced sensitivity of scatterings observables. The formalism links the presence of EPs to intrinsic nonlinearities in open system amplitude equations—especially pronounced at low level density and overlapping resonance regimes (Rotter, 2017, Eleuch et al., 2018).

4. Eigenfunction Fluctuations and Coherent Energy Transport

The role of eigenfunctions—beyond just eigenvalues—is fundamental in describing phenomena exclusive to non-Hermitian systems. Specifically, strong fluctuations of biorthogonal eigenfunctions near EPs enable robust, coherent energy transport along dissipation-assisted pathways. This mechanism is prominent in biological exciton transfer networks, such as photosynthetic complexes, where non-Hermitian Hamiltonians (not merely the effective $H_\mathrm{eff}$ but genuine non-Hermitian $H$ with both gain and loss terms) yield highly efficient (>99%) energy transport. In such scenarios, efficient transfer is achieved not by populating individual resonances, but by exploiting fluctuating superpositions mediated by eigenfunction structure near EPs. Standard Hermitian or effective Hamiltonians fail to capture this phenomenon, as they lack the continuum-induced eigenfunction mixing essential for the process (Rotter, 2017).

5. Limitations, Challenges, and Many-Body Extensions

Application of the non-Hermitian Hamiltonian formalism to realistic many-body systems is nontrivial and faces several unsolved challenges:

• Correct incorporation of strong many-body correlations in the discrete part $H_B$ when coupling to the environment is nonperturbative.
• Handling multiple decay channels and the complex interplay of their respective EPs in large Hilbert spaces.
• The computational overhead of diagonalizing non-Hermitian many-body Hamiltonians.
• Accurate description of higher-order or anomalous singularities beyond conventional EPs (e.g., accidental orthogonalization).
• Development of systematic approximations that not only capture resonance positions and widths, but also dynamic phenomena related to eigenfunction fluctuations.

Despite major progress in nuclear, atomic, and mesoscopic systems, full exploitation of this machinery in strongly correlated many-body regimes and complex biological networks is ongoing (Rotter, 2017).

6. Physical Interpretation and Broader Significance

The non-Hermitian Hamiltonian formalism, originated in the systematic partitioning of open, interacting quantum systems, provides a comprehensive unifying framework for resonance scattering, decay, collective dynamical phenomena, and even almost-perfect quantum energy transport in disordered or fluctuating environments. It encompasses features such as complex spectrum, nonunitary evolution, biorthogonal bases, phase rigidity, continuum-state-induced mixing, and exceptional points. By emphasizing the pivotal role of eigenfunction structure alongside eigenvalues, this framework captures experimentally observed phenomena in diverse contexts, from resonance width bifurcation and unidirectional transparency in optics to near-unity quantum yields in organic light-harvesting complexes. Its continued extension to many-body and dynamical regimes remains at the forefront of theoretical research in quantum open systems (Rotter, 2017).