# Non-Hermitian Hamiltonian for electron conductance in electric field?

Electron conductance in a solid state is usually driven by electric field - making some direction of jumps more likely.

It makes (e.g. Hubbard's) Hamiltonian no longer self-adjoint, how to simulate QM with non-Hermitian Hamiltonian? How electron conductance in electric field is being modeled?

PS. I have just found non-Hermitian QM Wikipedia article, PI lecture by Carl Bender, where he says that PT symmetric Hamiltonian should still have real spectrum. However, I have checked MERW-based conductance simulator which uses Hubbard-like Hamiltonian, and turns out its spectrum has not only real eigenvalues if turning on the potential.

• Can you give some details about why the hamiltonian is non-self-adjoint? Is this a boundary condition problem? – mike stone May 11 at 15:08
• In Hubbard model you have a^+ a terms corresponding to jumps. To have self-adjoint Hamiltonian, both i -> j and j -> i jumps should have the same weights. However, electric field makes one of them more likely. – Jarek Duda May 11 at 15:14
• More on PT-symmetric QM: physics.stackexchange.com/q/161817/2451 – Qmechanic May 11 at 15:20
• @Jarek Duda. the diffrent weighting may break left right symmetry, but I should not violate self-adjointness. Are you sure it does? .After all the hamiltonian always has "+ cc" indicating that you add the hermitan to conjugate to eveything. Can you edit and write down you non-self adjoint Hamiltionian in the question so we can see if there is something missing? – mike stone May 11 at 16:38
• @mikestone, the "+cc" is exactly adding opposite edges with the same weights. To model conductance we need to add some asymmetry - distinguish the two directions. Maybe it can be introduced in a self-adjoint way? MERW is kind of QM in imaginary time - leads to ground state probability distribution, assumes Boltzmann distribution among paths. The linked conductance simulator adds potential by reducing/increasing energy of right/left step. It means asymmetry in adjacency matrix being minus Hamiltonian, MERW has no problem with that, but I don't know how to realize it in QM, QRW? – Jarek Duda May 11 at 17:50

Rather than keep up a discussion, I will just write down the hopping part of the Hamiltonian for a 1d Hubbard model in an electric field: $$\hat H= \sum_n \left(a^\dagger_{n+1} a_n e^{iA_n(t)} + a^\dagger_{n}a_{n+1} e^{-iA_n(t)}\right)$$ where $$\frac{\partial A_n}{\partial t}=-E_n(t)$$ is the electric field.
We can also make a time-dependent change gauge so that the field is expressed by a scalar potential: $$\hat H =\sum_n \left(a^\dagger_{n+1} a_n + a^\dagger_{n}a_{n+1} -(\sum_{m=0}^n E_m) a^\dagger_n a_n \right)$$ Both these expressions are self adjoint. In the absence of impurities neither will have a true electric conductivity in a steady field because the eigenstates will undergo Bloch oscillations. I think this will be true even when interactions are included. The usual way to get a physically meaningful conductivity is to allow the electric field to depend on space and time, and to take the limit $$k\to 0$$, and then $$\omega \to 0$$.