Designing
non-Hermitian real spectra through electrostatics
Russell Yang, Jun Wei Tan, Tommy Tai, Jin Ming Koh, Linhu Li, Stefano
Longhi, Ching Hua Lee
Science Bulletin, 2022, 67(18): 1865–1873
doi: 10.1016/j.scib.2022.08.005
简介
非厄米性的研究蕴涵着诸多新物理机制与现象, 在激光与传感等多个方面有着重要的应用价值. 在非厄米系统中, 实本征能量意味着系统具有稳定性, 这一性质可通过奇偶时间(PT)对称性保护, 并在不同物理系统中得到广泛的研究. 本文利用一种新的动力学机制保证实能谱, 提出了一种普适的方案来设计具有实能谱的哈密顿量. 该方案基于对扩展非厄米体边对应的静电类比, 其中静电荷对应态密度, 电场则对应复能谱流. 通过这一方案, 可以重构出具有任意能谱及态局域性质的哈密顿量, 即便这些性质不具备任何对称性特征. 通过用泊松边值问题代替原本的非厄米哈密顿量对角化问题, 这一静电类比也避免了传统数值方法中因非厄米耗散/增益引起的浮点误差问题, 从而允许对更大尺寸的系统进行计算.
图文导读
Fig. 1 (a) Hermiticity and PT-symmetry are two well-known routes towards real energy spectra, but our work provides a new approach for designing generic real-spectrum Hamiltonians (pink) satisfying neither condition. (b) Conventionally, model Hamiltonian parameters have to be repeatedly optimized to yield the desired spectrum. By contrast, our electrostatics design approach directly outputs a parent Hamiltonian H possessing almost any desired eigenspectrum and eigenstate profiles. (c) Non-Hermitian eigenstates are characterized by their complex energies E and inverse spatial decay lengths, which together describe a landscape . In particular, PBC eigenstates lie along loops, while OBC eigenstates accumulate along ridges where is not smooth. (d) The landscape of a non-Hermitian system is mathematically equivalent to the corresponding to grounded conductors and lines of induced charges respectively.
Fig. 2 Exact agreement between the stipulated and reconstructed spectra in our design approach, for the two warm-up models. (a) The spectrum of the output Hamiltonian is reconstructed by setting energy intervals between equal momentum spacings to be inversely proportional to the density of states which is obtained from the induced charge density in the equivalent electrostatic problem. (b) Warm-up example I with stipulated OBC spectrum on the real line segmentwith constant eigenstate decay profile , corresponding to a constant electrical potential V. Its elliptical PBC spectral locus corresponds to a grounded conductor, giving rise to induced charges that enable the reconstruction of the full Hamiltonian . As a check, its PBC and OBC spectra fall exactly on the initially stipulated loci. Parameters used are , , and lattice sites. (c) Warm-up example II with two stipulated real OBC line segments, PBC locus of the form of Eq. (4) and constant . The corresponding electrical potential induces charges that allow the reconstruction of an dispersion, corresponding to the 2-component from Eq. (5) with parameters , , sites. Both the PBC and OBC spectra of HSSH display perfect agreement with their initially stipulated loci. Note that we have normalized the potential to along the real line segments in both (b) and (c).
Fig. 3 Illustrative examples demonstrating a search for real-spectra parent Hamiltonians in nontrivial settings without special symmetries. (a) PBC ellipse with two real OBC segments with unequal skin depths: the stipulated real OBC segments are of asymmetrically different skin depths and 1/2, corresponding to potentials and 1.6. Together with grounded conductors defined by the stipulated PBC spectral locus , they define an electrostatic problem, yielding a parent Hamiltonian in the form of Eq. (7). (b) PBC Reuleaux triangle with real OBC segment of length d and position offset , with inhomogeneous skin depths corresponding to a concave potential of amplitude A (Eq. (10)). Engineered (dotted) and stipulated (solid) spectra agree well, even though the engineered Hamiltonian is single-component with only up to next-nearest neighbor hoppings. (c) Values of for the OBC spectrum of our engineered Hamiltonian in the parameter space of A and d. There exists a large parameter region (blue) with almost real OBC, which is not expected given the very different symmetries of the Reuleaux triangle and the profile. Tolerance for reality is , and 20 sites are used.
Fig. 4 Designing beyond real spectra. (a) Almost perfect agreement between the stipulated and engineered spectra in the right lobe was achieved, with only up to next-nearest neighbor hoppings, by discarding solutions from the two other extraneous lobes. The stipulated PBC spectrum is given by the parametrization Eq. (12). (b) Excellent reconstruction can also be achieved by enlarging the number of components of the ansatz Hamiltonian and its solution branches. Here, the 3-component ansatz (Eq. (15)) with allows for an excellent local fitting of with minimal Fourier components. Stipulated OBC and PBC spectra are given by Eqs. (11) and (12), with , . We used 30 and 120 lattice sites for (a) and (b) respectively, the large number in the latter necessary for demonstrating excellent convergence in all the branches.
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