Searching for degeneracies of real Hamiltonians using homotopy classification of loops in SO (n)
Abstract
Topological tests to detect degeneracies of Hamiltonians have been put forward in the past. Here, we address the applicability of a recently proposed test [Phys. Rev. Lett. 92, 060406 (2004)] for degeneracies of real Hamiltonian matrices. This test relies on the existence of nontrivial loops in the space of eigenbases SO(n) . We develop necessary means to determine the homotopy class of a given loop in this space. Furthermore, in cases where the dimension of the relevant Hilbert space is large the application of the original test may not be immediate. To remedy this deficiency, we put forward a condition for when the test is applicable to a subspace of Hilbert space. Finally, we demonstrate that applying the methodology of Phys. Rev. Lett. 92, 060406 (2004) to the complex Hamiltonian case does not provide any new information.
- Publication:
-
Physical Review A
- Pub Date:
- January 2005
- DOI:
- 10.1103/PhysRevA.71.012106
- arXiv:
- arXiv:quant-ph/0406105
- Bibcode:
- 2005PhRvA..71a2106J
- Keywords:
-
- 03.65.Vf;
- 02.40.Re;
- 31.50.Gh;
- 41.20.Cv;
- Phases: geometric;
- dynamic or topological;
- Algebraic topology;
- Surface crossings non-adiabatic couplings;
- Electrostatics;
- Poisson and Laplace equations boundary-value problems;
- Quantum Physics
- E-Print:
- Minor changes, journal reference added