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!

  • 1
    $\begingroup$ 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). $\endgroup$ May 28 '17 at 23:09

In condensed matter physics, "topologically trivial" means a product state. (Or more precisely, a state which can be adiabatically deformed into a product state by a local Hamiltonian or quantum circuit with an error that decreases exponentially with time or circuit depth.)

Look at this cartoon of the Kitaev chain: enter image description here

The blue balls represent the spinless fermions (or the spin-1's in the corresponding AKLT picture). The top chain is clearly a product state of spinless fermions because the entanglement between the Majoranas is all "internal" within single physical fermions. The bottom chain is clearly not a product state, because the entanglement reaches across Majoranas in different fermions. With some more work, you can show that any operation that tries to disentangle this chain must take a minimum time that grows linearly with the chain length - so for a macroscopically long chain, this is extremely hard to do.


From a phenomenological point of a view, the terms like "topologically trivial" or "topologically non-trial" don't make much sense. What one can immediately find out is that the ground state degeneracies of the "trivial" phase and "non-trivial" phase are different. That can be understood via a topological invariant, e.g. Chern number for topological insulator or Pfaffian invariant for Kitaev chain. The two distinct phases have different value of the topological invariant.

But if you use the language of classification of symmetry-protected topological orders, the terms such as "topologically trivial" or "topologically non-trial" make sense now. The trivial phase's ground state can be described by a product state, while the non-trivial phase cannot be described by a product state (indeed it can be written as a matrix product state). You can get a glance of this language through this paper: arXiv: 1008.3745. By the way, the paper is for gapped spin/bosonic chain. The fermionic case is a bit more complicated, and there are many papers to explain this issue.


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