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本文研究了一维复相互作用调制的非厄米玻色子模型.通过数值计算能谱的实-复转变、Shannon熵、标准参与比率与拓扑缠绕数,发现当相互作用强度低于临界相互作用强度时,系统的能谱全为实数,处于扩展相,且系统是拓扑平庸的;而当相互作用强度超过临界相互作用强度时,系统开始出现复能谱,处于扩展态与局域态混合相,且此时系统是拓扑非平庸的.计算结果表明,能谱的实-复转变点、扩展-局域的转变点与拓扑转变点相一致.动力学演化结果可以验证系统的实-复转变与局域化转变.最后,提出利用二维光子波导阵列可以模拟这一复相互作用调制的一维玻色子模型.此项工作将对非厄米两体系统的局域性质提供很好的参考.In this work, We investigate a one-dimensional two-boson system with complex interaction modulation, described by the Hamiltonian: $$\hat{H}=-J\sum_{j}\left(\hat{c}_j^\dagger\hat{c}_{j+1}+h.c\right)+\sum_{j}\frac{U}{2}e^{2i\pi\alpha j}\hat{n}_j\left(\hat{n}_j-1\right),$$ where U is the interaction amplitude, and the modulation frequency α = (√5 - 1) is an irrational number. The interaction satisfies $U_{-j}=U^*_j$, ensuring the system possesses party-time (PT) reversal symmetry. Using the exact diagonalization method, we numerically calculated the real-to-complex transition of the energy spectrum, Shannon entropy, the normalized participation ration, and the topological winding number. For small U, all eigenvalues are real. However, as U increases, eigenvalues corresponding to two particles occupying the same site become complex, marking a PT symmetry-breaking transition at U = 2. This point signifies a real-to-complex transition in the spectrum. To characterize the localization properties of the system, we employ the Shannon entropy and the normalized participation ration (NPR). When U < 2, all the eigenstates are extended, exhibiting high Shannon entropy and NPR values. Conversely, for U > 2, states with complex eigenvalues showing low Shannon entropy and significantly reduced NPR, indicating localization. Meanwhile, in this regime, states with real eigenvalues remain extended. We further analyze the topological aspects of the system using the winding number. A topological phase transition occurs at U = 2, where the winding number changes from 0 to 1. This transition coincides with the onset of PT symmetry breaking and the localization transition. The dynamical evolution can be used to detect the localization properties and the real-to-complex transition with the initial state being two bosons at the center site of the chain. Finally, we propose an experimental realization using a two dimensional linear photonic waveguide array. The modulated interaction can be controlled by adjusting the changing the real and imaginary parts of the refractive index of diagonal waveguides. To simulate this non-Hermitian two-body problem, we numerically calculate the density distributions of the wave packets in a two-dimensional plane, which indirectly reflects the propagation of light in a two-dimensional waveguide array. We hope that our work can deepen the understanding of the interplay between interaction and disorder while stimulating further interest in two-body systems and non-Hermitian localization.
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Keywords:
- non-Hermitian /
- localization /
- topology /
- PT symmetry
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