Berry Phases with Real Hamiltonians With and Without a Many-body System as a Background

We present both the gauge theoretic description and the numerical calculations of the Berry phases with the real eigenstates, involving one with a many-body system as a background and the other with no such background. We demonstrate that for the former the sign of the Berry phase factor for a spin 1 2 particle (hole) coupled to a slow subsystem (phonon) depends on both the strength of electron correlations and the characteristics of the closed paths, unlike the cases for the latter. PACS numbers: 03.65.Bz, 71.10.Fd, 71.38.+i, 71.27.+a Typeset using REVTEX 1 The real wave functions in association with real Hamiltonians are frequently encountered in numerous physical problems, not to speak of condensed matter physics. There exists still an unresolved problem in defining the Berry phase concerned with the real eigenfunctions in that the gauge potential cannot be defined. The Berry phase [1–4] arises from a U(1) gauge potential as a result of the adiabatic transition involving the single-valued, complex, non-degenerate eigenstates of a fast subsystem coupled to a slow subsystem. In the following we briefly draw an attention to the main issue of the present work. In a system described by the time-dependent Hamiltonian H(R(t)) through a slowly varying parameter R, H(R(t))|n;R(t)〉 = En(R(t))|n;R(t)〉 , (1) the total phase change of the state |ψ〉 round a closed loop for a time period of T is given by |ψ(T )〉 = exp[iγn(C)] exp { − i h̄ ∫ T 0 dtEn(R(t)) } |ψ(0)〉 , where the Berry phase γn(C) in the first factor is given by γn(C) = i ∮ 〈n;R|∇R|n;R〉 · dR , (2) and |ψ(0)〉 = |n;R(0)〉 at t = 0. To obtain a non-zero value of γn, the above eigenstate |n;R(t)〉 needs to be single-valued, complex and non-degenerate. Thus one cannot define the gauge potential A(R) = i〈n;R|∇R|n;R〉 (3) from the use of the real eigenfunctions. In order to allow for the case of multi-valuedness of the complex eigenstates |n;R〉 Berry [1] derived the following expression of the ‘magnetic field’ from (2), B(R) = −Im ∑ m6=n 〈n;R|(∇RH)|m;R〉 × 〈m;R|(∇RH)|n;R〉 [Em(R)−En(R)] . (4)