Adiabatic Curvature - Deduce the Local Adiabatic Curvature K

[lyylynn]

Hi,

I am working on the Laughlin model of Quantum Hall Effect, which relates to the concept ' adiabatic curvature'. The paper didn't include much details, and I knew little about berry phase. Could some one please give me some idea that how is the local adiabatic curvature deduced? ( the expression of K below)

Here is the problem: consider a quantum Hamiltonian H(psi, theta). Suppose the Hamiltonian has a non-degenerate ground state at energy zero. Ground state is e^ia|phi(psi,theta)>, a is the initial phase that is free to choose. The local adiabatic curvature K of the bundle of ground states in the parameter space is defined as the limit of the Berry phase mismatch divided by the loop area, turns out to be
K = 2 I am < d(psi) phi| d(theta) phi> (here d indicates partial differential operator)

Thanks a lot.:)

 

[azntoon]

</code>The expression for the local adiabatic curvature K is derived from the Berry phase formula, which is a measure of the geometric phase acquired by a quantum system when cycled through a closed loop in its parameter space. The Berry phase formula is given by B = ∫A F , where A is the area enclosed by the loop and F is the Berry curvature, which is given by F = -2 Im < d(psi) phi| d(theta) phi>. Therefore, the local adiabatic curvature K can be expressed as K = B/A = -2 Im < d(psi) phi| d(theta) phi>. This expression is valid for any non-degenerate ground state of the Hamiltonian H.