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arXiv:2008.13271 · Journal of Mathematical Physics

Algebraic approach and Berry phase of a Hamiltonian with a general SU (1, 1) symmetry

E. Choreño, R. Valencia, D. Ojeda-Guillén

Published Jul 1, 2021• 4 citations
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Abstract

In this paper, we study a general Hamiltonian with a linear structure given in terms of two different realizations of the SU(1, 1) group. We diagonalize this Hamiltonian by using the similarity transformations of the SU(1, 1) and SU(2) displacement operators performed to the su(1, 1) Lie algebra generators. Then, we compute the Berry phase of a general time-dependent Hamiltonian with this general SU(1, 1) linear structure.

DOI
10.1063/5.0027957

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In this paper, we study a general Hamiltonian with a linear structure given in terms of two different realizations of the SU(1, 1) group. We diagonalize this Hamiltonian by using the similarity transformations of the SU(1, 1) and SU(2) displacement operators performed to the su(1, 1) Lie algebra generators. Then, we compute the Berry phase of a general time-dependent Hamiltonian with this general SU(1, 1) linear structure.
We study the dynamics of a quantum system with a time-dependent Hamiltonian which is given by a linear combination of SU(1,1) generators and coupled with a heat bath. An effective Hamiltonian of this dissipative system is derived. With the help of a time-dependent gauge transformation, we obtain the exact solutions of the time-dependent Schrödinger equation, from which the time-evolution operator and the non-adiabatic and adiabatic Berry phases, which depend on time, are calculated explicitly. In the weak dissipation limit, an additional term besides the original Berry phase is found. The additional phase does not have a geometrical meaning due to the dissipation.
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.
We study the time evolution of quantum systems with a time-dependent Hamiltonian given by a linear combination of SU(1,1) and SU(2) generators. The invariant Hermitian operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. With the help of the invariant Hermitian operator we obtain not only the exact solutions of the Schr\"odinger equation but also the time-evolution operator. The adiabatic and nonadiabatic Berry phases are calculated with the exact solutions. \textcopyright{} 1996 The American Physical Society.
The methods of Kähler geometry are applied to generalize the results of Berry obtained for SU(2) (namely, the existence of a geometrical part in the adiabatic phase) to any compact Lie group. We obtain explicit expressions for Berry’s geometric phases, Berry’s connections, and Berry’s curvatures in terms of parameters of the corresponding Lie algebra valued Hamiltonian. It is demonstrated that the parameter space of the Hamiltonian in the general theory is essentially a homogeneous Kähler manifold. The fundamental Kähler potentials of this manifold completely determine Berry’s phase. A general approach is exemplified by the Lie algebra Hamiltonians corresponding to SU(2) and SU(3) evolution groups.