# Trivial phase in Kitaev chain, trivial phase in what sense?

I am reading 1D p-wave SC proposed by Kiteav---Kitaev chain recently.

This paramedic model somehow bothers me with the word "trivial."

In this SE post, the answer pointed out what we mean by "trivial" is "simple." This is not satisfying to me. I think the question should be revised in a more precise way---

Question1: What do we mean by topologically trivial phase? What is special of it for such choice of parameters over a huge range of parameter space?

20170601Edited Under Majorana fermion representation, let's consider the Hamiltonian for the lattice p-wave chain given as $$\frac{i}{4}\sum_{j=1}^{N-1}\bigg[\big(t+\Delta_0\big)a_{2j}a_{2j+1}+\big(-t+\Delta_0\big)a_{2j-1}a_{2j+2}\bigg]-i\frac{\mu}{2}\sum_{j=1}^{N}a_{2j-1}a_{2j},$$ where $\Delta_0$ is an induced pairing gap function that we by convention chose as real and positive.

In literature (Kitaev chain and Bernevig&Hughes p200), the trivial phase is chosen by turning off the first term, $$H_{\text{trivial}}=-i\frac{\mu}{2}\sum_{j=1}^{N}a_{2j-1}a_{2j},$$ which simply states that only on-site interactions contribute and there is no between-sites interactions. At the bottom of Bernevig&Hughes p200), it says

The Hamiltonian in the physical-site basis is in the atomic limit; thus the ground state is trivial.

Question2: I am not sure how to appreciate this "trivial" here. Topology is a mathematical concept to classify shapes. When we apply it to physics, we consider something like whether a quantum mechanical wavefunction is adiabatically connected to others. If yes, they are called topologically identical which is trivial to our interested.

For the trivial case, naively, I would imagine the situation is something like Hubbard model---the atomic orbital is very localized, and thus if one turns off the hopping term there's only on-site contributions. But I still can not see how trivial the ground state (wavefunction) is here.

Maybe this question is still not really organized in a concrete manner (I'll try to modify it later.) Thanks in advance!

• Trivial in this context means that the ground state can be adiabatically connected to a product state (without having to explicitly break any of the relevant symmetries). May 28 '17 at 23:09

Look at this cartoon of the Kitaev chain: 