Calculating the Z2 Invariant in Kane-Mele Model

It may just be me misunderstanding but I would like to see if I am doing something wrong.In summary, the speaker is working on a computational project involving topological insulators and is trying to calculate the Z2 invariant in a tight binding model of Kane-Mele Graphene. They have produced band structure and surface states consistent with literature, but are having trouble getting the correct answer when calculating the Z2 invariant. They have tried using a matrix for time reversal and following the method described in Fu and Kane's paper, but it is not giving the expected results. They are seeking advice on a more effective method for calculating the Z2 invariant. They mention that there is not much expertise at their institution and they are considering consulting Bernevig and
  • #1
Teek
11
0
Hi Everyone,

As part of my computational project in topological insulators, I wish to calculate the Z2 invariant in my tight binding model of Kane-Mele Graphene. I have so far produced band structure and surface states consistent with literature, and have been looking at the theory of the Z2 invariant, and in my attempt at writing code to produce the result in a QSH insulator, I cannot seem to get it to work. I simply produce a matrix for time reversal, which is [[0, 1],[-1,0]] tensor identity, and calculate the sewing matrix elements defined in Fu and Kane's paper (2006). However when finding the Pfaffian of the Matrix and dividing by the square route of the determinant for each Time-reversal Invariant momenta in Graphene's 2D BZ, their combined product does not give me a correct answer nor even a quantised one. There isn't really anyone with much knowledge at my institution, so I do not know where to go for advise on this. Is there an effect computational method of calculating the Z2 invariant from the Kane-Mele model in tight binding form?

I really appreciate the help.
Thanks.
 
  • #3
I would look at Bernevig and Hughes's book topological insulators and topological superconductors.
 

1. What is the Z2 invariant in the Kane-Mele model?

The Z2 invariant is a topological invariant that characterizes the electronic properties of materials in the presence of spin-orbit coupling. In the Kane-Mele model, it is a numerical value that can take on either 0 or 1 and is used to classify materials as either a trivial or a topological insulator.

2. How is the Z2 invariant calculated in the Kane-Mele model?

The Z2 invariant is calculated by evaluating the parity of the occupied electronic states in the Brillouin zone. This is done by constructing a Bloch wavefunction and using symmetry operations to determine its parity at specific points in the Brillouin zone. The Z2 invariant is then obtained by combining the parities at these points.

3. What factors affect the value of the Z2 invariant in the Kane-Mele model?

The main factors that affect the value of the Z2 invariant in the Kane-Mele model are the strength of the spin-orbit coupling and the crystal symmetry of the material. Additionally, the presence of disorder or external perturbations can also influence the Z2 invariant.

4. How does the Z2 invariant relate to the electronic properties of materials?

The Z2 invariant is a topological invariant that is related to the bulk-boundary correspondence in topological insulators. It determines the number and location of topologically protected gapless surface states, which are known as edge states, in a material. Materials with a non-zero Z2 invariant are characterized by the presence of these edge states, making them topological insulators.

5. What are the practical applications of calculating the Z2 invariant in the Kane-Mele model?

The Z2 invariant has important implications for the electronic and transport properties of materials. It can be used to predict and understand the behavior of topological insulators, which have potential applications in spintronics, quantum computing, and other emerging technologies. Additionally, the Z2 invariant can also be used as a tool for material design and discovery, as it allows for the identification of topological phases in new materials.

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