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Acta Physica Sinica

Berry phase in time-dependent non-Hermitian SU(1, 1) and SU(2) systems

GU Yan, LU Zhanpeng

Published Jan 1, 2026• 0 citations
Publisher

Abstract

本文研究了由SU(1,1)和SU(2)李代数生成元所构成的含时非厄米系统的Berry相位, 其分别对应于周期驱动外场下的一维谐振子及任意参数下的单个自旋粒子模型. 具体来说, 基于广义规范变换方法,通过引入含时厄米的且非幺正的变换算符, 系统可变换为不显含时间的形式, 并可导出系统的精确解及其对应的非绝热Berry相位. 结果表明, 对于SU(2)系统, 存在奇异点Gc (ω) = Ω+ω/2, 在该点处等效本征值发散, Berry相位从实数变为复数, 对应赝厄米相的自发破缺. 而SU(1,1)系统的能谱始终保持实数,Berry相位为纯实数, 无奇异点出现. 该对比揭示了SU(1,1)非紧致群与SU(2) 紧致群结构对非厄米系统几何相位的根本性影响. 该研究结果为非厄米系统中Berry相位的理论分析提供了新的视角和理解.

DOI
10.7498/aps.75.20260270

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本文研究了由SU(1,1)和SU(2)李代数生成元所构成的含时非厄米系统的Berry相位, 其分别对应于周期驱动外场下的一维谐振子及任意参数下的单个自旋粒子模型. 具体来说, 基于广义规范变换方法,通过引入含时厄米的且非幺正的变换算符, 系统可变换为不显含时间的形式, 并可导出系统的精确解及其对应的非绝热Berry相位. 结果表明, 对于SU(2)系统, 存在奇异点Gc (ω) = Ω+ω/2, 在该点处等效本征值发散, Berry相位从实数变为复数, 对应赝厄米相的自发破缺. 而SU(1,1)系统的能谱始终保持实数,Berry相位为纯实数, 无奇异点出现. 该对比揭示了SU(1,1)非紧致群与SU(2) 紧致群结构对非厄米系统几何相位的根本性影响. 该研究结果为非厄米系统中Berry相位的理论分析提供了新的视角和理解.
在时间依赖者相互作用导致的一个合成系统的果浆阶段被讨论。我们作为合成系统选二为联合 spin-1/2 系统：分系统之一受到一个静态的磁场，并且在二旋转之间的联合参数及时是可控制的。我们证明时间依赖者相互作用能作为那以一个类似的方法导致果浆阶段一个 spin-1/2 系统(qubit ) 被一个有效时间依赖者磁场驾驶。而且，与静态的磁场以及反面的相反的方向使用二个连续周期联合参数签名，重要 two-qubit 单一的转变基于完全，果浆阶段能被构造。
Based on the generalized gauge transformation and invariant methods, we study in this work the rigorous solution for non-Hermitian Hamiltonians consisting of SU(1, 1) and SU(2) generators, which describe the harmonic oscillator and a single spin of arbitrary spin value, respectively, in periodically time-dependent fields. A non-Hermitian Hamiltonian with real eigenvalues is called pseudo-Hermitian. Such a Hamiltonian can be converted to a Hermitian one by a similarity transformation. We extend the concept of pseudo-Hermitian Hamiltonians to the time-dependent case. Under the generalized gauge transformation with a time-dependent Hermitian operator, the Hamiltonian becomes Hermitian and time-independent in the new gauge. The rigorous solution, as well as the Berry phase, is obtained by applying the inverse transformation to return to the original gauge. The time-dependent Schrödinger equation is also solved in terms of non-Hermitian invariant operators, which possess conserved real eigenvalues. We derive the time-dependent solutions and corresponding geometric phases by eigenstate expansion of the invariant operators, combined with the time-evolution operator. It is observed that the non-Hermitian coupling constant indeed has an exceptional point Gc(ω) for the SU(2) non-Hermitian Hamiltonian, where all eigenstates are degenerate with zero eigenvalue and the Berry phase diverges. The eigenvalues become complex beyond Gc(ω) . However, the eigenvalues are real within the entire region of coupling constant for the SU(1, 1) system, different from the common belief that the exceptional point must exist in the non-Hermitian Hamiltonian. The classical counterpart of the SU(1, 1) non-Hermitian Hamiltonian is, in fact, a complex function of position-momentum variables. It can be converted to a real function via the generalized gauge transformation, accompanied by a canonical-variable transformation. This process corresponds exactly to the Hermitization procedure of the pseudo-Hermitian Hamiltonian. We obtain the classical Hannay angle for the periodically time-dependent SU(1, 1) pseudo-Hermitian Hamiltonian. The non-adiabatic Hannay angle and the Berry phase satisfy precisely the quantum-classical correspondence, γn(T)=−(n+12)∆Θ .
Abstract We study in this paper both the stationary and time-dependent pseudo-Hermitian Hamiltonians consisting respectively of SU(1, 1), SU(2) generators. The pseudo-Hermitian Hamiltonians can be generated from kernel Hermitian-Hamiltonians by a generalized gauge transformation with a non-unitary but Hermitian operator. The metric operator of the biorthogonal sets of eigenstates is simply the square of the transformation operator, which is formulated explicitly. The exact solutions of pseudo-Hermitian Hamiltonians are obtained in terms of the eigenststates of the Hermitian counterparts. We observe a critical point G c of coupling constant, where all eigenstates of the stationary Hamiltonians are degenerate with a vanishing eigenvalue. This critical point is modified as G c (ω) in the time-dependent case including the frequency of external field. Returning to the original gauge we obtain analytically the wave functions and associated non-adiabatic Berry phase, which diverges at the critical point for the SU(2)Hamiltonian. Beyond the critical point Berry phase becomes a complex domain.