近日，东南大学郭昊团队研究了一维自相似δ势垒阵列中的束缚态。相关论文于2025年7月29日发表在《物理评论A》杂志上。

研究组分析了具有δ函数势垒自相似排列的一维量子系统，该系统具有离散尺度不变性。奇异势在x=0处引起运动强制对称破缺，使正、负空间区域解耦，导致非对称零能态。他们证明了该系统支持一个独特的零能量波函数，尽管它不是平方可积的，但在无穷远处衰减为零，并在离散缩放变换下充当具有自相似性质的束缚态，类似于Efimov物理，但仅限于单一状态。

在动量空间中，该波函数在低动量下表现出阈值奇异性，其行为取决于标度指数α：当0<α<1时，呈现幂律发散和对数周期调制；当α=1时，呈现对数发散；当α>1时，呈现有限极限。这些特征可通过冷原子实验中的飞行时间或光谱测量来观测。该系统的连续谱从零能量开始，缺乏离散束缚态。这些发现凸显了奇异势在产生标度不变量子现象中的作用，并为研究离散标度对称性及其潜在的实验特征提供了一个最小框架。

附：英文原文

Title: Bound-like states in a one-dimensional self-similar δ-barrier array

Author: Jia-Chen Tang, Xu-Yang Hou, Yan He, Hao Guo

Issue&Volume: 2025/07/29

Abstract: We investigate a one-dimensional quantum system with a self-similar arrangement of δ-function potential barriers, exhibiting discrete scale invariance. The singular potential induces kinematically enforced symmetry breaking at x=0, decoupling the positive and negative spatial regions and leading to nonsymmetric zero-energy states. We demonstrate that the system supports a unique zero-energy wave function, which, although it is not square integrable, decays to zero at infinity and acts as a bound-like state with self-similar properties under discrete scaling transformations, akin to Efimov physics but limited to a single state. In momentum space, this wave function exhibits a threshold singularity at low momenta, with behavior depending on the scaling exponent α: power-law divergence and log-periodic modulations for 0<α<1, logarithmic divergence for α=1, and a finite limit for α>1, which may be observable through time-of-flight or spectroscopic measurements in cold-atom experiments. The system's continuous spectrum, starting at zero energy, lacks discrete bound states. These findings highlight the role of singular potentials in generating scale-invariant quantum phenomena and provide a minimal framework for studying discrete scale symmetry and its potential experimental signatures.

DOI: 10.1103/kbty-mxxb

Source: https://journals.aps.org/pra/abstract/10.1103/kbty-mxxb