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International Journal of Theoretical Physics

Global-Local Algebraic Quantization of a Two-Dimensional Non-Hermitian Potential

D. Vrinceanu, Carl Marth, Carlos R. Handy

Published Dec 22, 2014• 1 citation
Publisher
DOI
10.1007/s10773-014-2434-9

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The well-established non-Bloch band theory predicts exponential localization of skin-mode eigenstates in one-dimensional (1D) non-Hermitian systems. Recent studies, however, have uncovered anomalous algebraic localization in higher dimensions. Here, we extend these ideas to Hermitian bosonic quadratic Hamiltonians incorporating quantum squeezing, offering a genuine quantum framework to explore non-Hermitian phenomena without external reservoirs. We construct a two-dimensional (2D) bosonic lattice model with two-mode squeezing and study its spectral properties of bosonic excitation within the Bogoliubov-de Gennes (BdG) formalism. We demonstrate an algebraic non-Hermitian skin effect (NHSE), characterized by quasi-long-range power-law localization of complex eigenstates. The system shows ultra spectral sensitivity to double infinitesimal on-site and long-range hopping impurities, while remaining insensitive to single impurities. Analytical treatment via the Green's function reveals that this sensitivity originates from the divergence of the nonlocal Green's function associated with the formation of nonlocal bound states between impurities. Our study establishes a framework for realizing novel higher-dimensional non-Hermitian physics in Hermitian bosonic platforms such as superconducting circuits, photonic lattices, and optomechanical arrays, with the demonstrated ultraspectral sensitivity enabling quantum sensing and amplification via bosonic squeezing.
We analyze a quantum walk on a bipartite one-dimensional lattice, in which the particle can decay whenever it visits one of the two sublattices. The corresponding non-Hermitian tight-binding problem with a complex potential for the decaying sites exhibits two different phases, distinguished by a winding number defined in terms of the Bloch eigenstates in the Brillouin zone. We find that the mean displacement of a particle initially localized on one of the nondecaying sites can be expressed in terms of the winding number, and is therefore quantized as an integer, changing from zero to one at the critical point. We show that the topological transition is relevant for a variety of experimental settings. The quantized behavior can be used to distinguish coherent from incoherent dynamics.
Quasi-Hermitian quantum systems, including PT\mathcal{PT}PT-symmetric ones, can be mapped to equivalent Hermitian systems via a similarity transformation that redefines the inner product with a positive-definite metric operator. Although an instantaneous algebraic Hermitianization can be obtained locally from a positive metric operator, a stronger requirement is needed for dynamical equivalence: the similarity transformation must be proper, globally single-valued, and compatible with the modified quasi-Hermitian Schrodinger equation. We identify two distinct obstructions: geometric obstructions arising from the curvature of a metric-induced connection, and topological obstructions originating from non-trivial holonomies around non-contractible loops in parameter space. We derive explicit criteria for these obstructions and illustrate them with concrete examples. Our results establish a geometric and topological foundation for the Hermitianization of quasi-Hermitian systems, clarifying when they can be globally reduced to Hermitian ones and when intrinsic non-Hermitian features persist.
Non-Hermitian physics has unveiled a realm of exotic phenomena absent in Hermitian systems, with the non-Hermitian skin effect (NHSE) showcasing boundary-localized eigenstates driven by non-reciprocal interactions. Here, we introduce a new class of non-Hermitian systems exhibiting pure decay modes-eigenstates with pure, smooth exponential decay, devoid of the oscillatory wave patterns typical of traditional NHSE. Modeled as directed graphs with nonreciprocal hopping, these systems reveal quantized decay charges, defined as the sum of decay constants along edges at each node, offering a novel topological invariant. We derive universal conditions for these modes, enabling versatile configurations from one-dimensional rings, directed graphs with complicated connectivity, to higher-dimensional lattices. Experimental validation using microwave resonant circuits confirms the predicted pure decay profiles. This discovery paves the way for potential applications in photonics, signal processing, and beyond, harnessing the unique topological properties of non-Hermitian networks.