Berry phase in condensed matter

[semc]

TL;DR Summary

How is changing Hamiltonian related to the periodicity of the Brillouin zone

I am trying to understand how is topology used to characterize materials. So I understand that to calculate the Berry phase you will parameterize your Hamiltonian and change this parameter in some way and return to the initial value. What I do not understand is what does this changing of parameter correspond to? Are we changing the system somehow? I understand that the Hamiltonian describes the system so if we change the Hamiltonian then we are changing the system?

If we are really physically changing the system, why do everybody talk about the periodicity of the Brillouin zone and this satisfies the close loop evolution of the Hamiltonian? The Hamiltonian gives the band structure, shouldn't it change when we change the Hamiltonian? Why does the periodicity = this change?

Greatly appreciate if someone can help me with this

 

[Cthugha]

I am not an expert in topology, but I can share the few bits I know. Most of the stuff on band structures goes back to this paper:
Berry Phase for Band Structures

You take your Hamiltonian and investigate the parameter space. For a crystal you can have a look at the energy bands in momentum space. For a given band, you can calculate the Energy you will get for each value and direction of the crystal momentum k. This is the parameter you can vary. You can now apply a slight perturbation, which results only in small adiabatic changes. Consider, e.g., a weak electric field that accelerates a crystal electron initially at k=0, so that its k increases. At some point it will hit the border of the Brillouin zone at k=pi/a, will get reflected to k=-pi/a and will then again be accelerated to higher k, where the process will start from scratch. This is a simple version of Bloch oscillations. However, while doing so the electron went full circle on a closed path in parameter space (where k is the parameter) and may have picked up a Berry phase along the way that depends on the topology of the band.

This is very loose and probably oversimplifying. I am convinced that an expert on topology can explain this much better than I do.

 

[semc]

Thank you for your reply. I see that the paper you cited and your explanation revolves around a changing Hamiltonian. Does this mean that this close loop evolution is only present when we have some perturbation that continuously change the k? When we talk about topological materials, I thought we can decide whether it is a trivial insulator without any perturbation? We can't calculate the topological invariant without perturbation?

 

[Cthugha]

Maybe we should go back one step.
The very idea of the Berry phase relies on changing the Hamiltonian. The key concept one must keep in mind here is the adiabatic theorem. The adiabatic theorem states that a perturbation that acts very slowly on the system (thus: adiabatic) will result in the system staying in its instantaneous eigenstate (if there is an energy gap). So you do not change the state your system is in.

This translates to: into whatever set of base vectors |n> you decompose your state, their relative weights |c_n|^2 will remain the same. So if you use a cyclic perturbation, so that the conditions at the end and at the beginning are the same, not much happened. It was a great insight that this still opens up the possibility that the coefficients may pick up a phase shift along the way, which is the Berry phase. Maybe it is helpful to revise the adiabatic theorem now as it is very important to understand the concept of the Berry phase.

 

[DrDu]

Being an isolator or not refers to the response of a material to an applied electric field, namely whether a current is induced or not. So you always have to include the field as a "perturbation" to decide whether a material is an isolator or not.
The most convenient configuration is that of a ring made of the material which encircles a magnetic flux increasing linearly with time. As E=-dA/dt, this yields a constant electric field inside the material. The magnetic vector potential yields a k which changes constantly with t, thus encircling the whole Brillouin zone.

 

[semc]

Thank you both for the detailed explanation!

 

[semc]

Maybe we should go back one step.
The very idea of the Berry phase relies on changing the Hamiltonian

Yes and this is exactly my question. To calculate Berry phase means the Hamiltonian has to change but when characterizing a material using topology, why does the Hamiltonian changes? Can't I calculate the topological invariant without applying any field? If I can, then the Hamiltonian doesn't change which means no Berry phase...

Being an isolator or not refers to the response of a material to an applied electric field, namely whether a current is induced or not. So you always have to include the field as a "perturbation" to decide whether a material is an isolator or not.

I always thought it is similar to the classical band theory where we can determine whether the material is metal or insulator from purely theoretical calculation. So I guess we can't?

 

[DrDu]

I always thought it is similar to the classical band theory where we can determine whether the material is metal or insulator from purely theoretical calculation. So I guess we can't?

Exactly how does classical band theory tell you whether a material is a metal or an insulator? I mean, what are the logical steps behind this theory?

 

[semc]

We take the unperturbed Hamiltonian and calculate E(k) then look at the position of E_F? or can look at the number of electron(s) per primitive unit cell I guess?

 

[DrDu]

We take the unperturbed Hamiltonian and calculate E(k) then look at the position of E_F? or can look at the number of electron(s) per primitive unit cell I guess?

Ok, but this is only an automatism. There is a reasoning behind this automatism and why it really tells you how a material will behave when you apply an electric field.

 

[semc]

I found a video by prof. C. Kane and he said 'Berry phase arise whenever you have a Hamiltonian that depends continuously on some set of parameters'. So that means we can have Berry phase without any perturbation since the Hamiltonian depends on k.



I still don't see how Berry phase can arise without changing the Hamiltonian ...