Time reversal symmetry in Topological insulators of HgTe quantum Wells

In summary, the conversation is about trying to prove that the BHZ model used to describe HgTe quantum well topological insulators is time reversal symmetric. The effective Hamiltonian, a 4x4 matrix, can be written in a block diagonal form where the lower 2x2 block can be derived from the upper 2x2 block. The dispersion relations for upspin and down spin should intersect at k=0, according to Cramer's degeneracy. However, the speaker is having trouble getting this result and seeks help in finding the correct form of the time reversal symmetry operator or using the concept of CPT symmetry to prove the time reversal symmetry of the Hamiltonian.
  • #1
Minato
4
0
Hi everyone,

While reading about the BHZ model used to describe HgTe quantum well topological insulators, I read at many places that the effective Hamiltonian (which is a 4 x 4 matrix) can be written in block diagonal form and the lower 2x2 block can be derived from upper 2x2 block as follows:
[H(k)][/lower]=[H(-k)][/*]

This effective Hamiltonian is said to be Time reversal symmetric and then using Cramer's degeneracy, it is said that the dispersion relations for upspin and down spin should intersect at [k][/x]=0.

I want to just show this through simple mathematical steps, but I am unable to get this result. In order to show time reversal invariance, I tried the following equation:
[T][/-1]HT=H, where T is the Time reversal symmetry operator.
but I am not sure what form of T should be used. I tried to use the following form:
T=-i x [0 [σ][/y];[σ][/y] 0]K {K is complex conjugation which is a 4x4 matrix with [0][/2x2] in the diagonals and Pauli matrix in y as off diagonal elements.}

But this is not giving me that BHZ Hamiltonian is time reversal symmetric.
Can anybody help me where I am going wrong?

Thanks

Regards
Minato
 
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  • #2
Can you show us the explicit form of the Hamiltonian you start out with?

Time reversal inverts the sign of momentum k and of spin/magnetic moment s.

In the Schroedinger equation, complex conjugation of a wave function it will result in time reversal.

With that you practically have your relation.

T H(k) T psi = T H(k) psi* = H*(-k) psi

btw, I have trouble reading your notation with []. Can you try to use [itex]?
 
  • #3
I am sorry for the formatting in the previous post.

The original Hamiltonian for BHZ model used to describe HgTe quantum well Topological insulators is
[tex]H=\left(\begin{array}{cc}h_{+}(k)&0\\0&{h_{-}(k)}\end{array}\right)[/tex]

[tex]{h_{-}(k)}=h_{+}^{*}(-k)[/tex]

here the meaning of * is to take the complex conjugate of the matrix.
[tex]h_{+}(k)=\left(\begin{array}{cc}M-(B_{+})(k_{x}^{2}-\frac{\partial ^{2}}{\partial y^{2}}) & {A(k_x-\frac{\partial}{\partial y})}\\{A(k_x+\frac{\partial}{\partial y})} & {-M+B_{-}(k_{x}^{2}-\frac{\partial ^{2}}{\partial y^{2}}) }\end{array}\right)[/tex]

where [tex] M, A, B_{+},B_{-} [/tex] are various system parameters.
The form of Time reversal operator which I have used is:
[tex]T=-i\left(\begin{array}{cc}0&0&0&-i\\0&0&i&0\\0&-i&0&0\\i&0&0&0\end{array}\right)K[/tex]
where K is the conjugation operator
I am trying to prove the following equation to show that the above Hamiltonian is Time reversal symmetric:
[tex]H=T^{-1}HT[/tex]


Regards
Minato
 
  • #4
Should that not be [itex]A(k_x \pm i \frac{\partial}{\partial y})[/itex]?

Also, with the time reversal operator you write, I get [itex]T^2 = -1[/itex] instead of [itex]T^2=1[/itex], so there are too many "i"s.
 
  • #5
M Quack said:
Should that not be [itex]A(k_x \pm i \frac{\partial}{\partial y})[/itex]?

Also, with the time reversal operator you write, I get [itex]T^2 = -1[/itex] instead of [itex]T^2=1[/itex], so there are too many "i"s.

He wrote something about the system showing Cramers degeneracy. Then I would expect T^2=-1.
 
  • #6
M Quack said:
Should that not be [itex]A(k_x \pm i \frac{\partial}{\partial y})[/itex]?

Also, with the time reversal operator you write, I get [itex]T^2 = -1[/itex] instead of [itex]T^2=1[/itex], so there are too many "i"s.

Regarding the first point, it is [itex]A(k_x \pm i k_{y})[/itex] which will give the form I have earlier written.([itex]k_{y}=-i \frac{\partial}{\partial y}[/itex])

Regarding the second point, the system is fermionic. That is why, [itex]T^2=-1[/itex] is required.

Regards
Minato
 
  • #7
Thanks for clarifying that.

Going with the 2x2 block motif, let's write [itex]
T = -i \left( \begin{array}{cc} 0 & t \\ t & 0 \end{array} \right)K
[/itex] with [itex]
t = \left( \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right)
[/itex] such that [itex] t^\star t = -1 [/itex]

We already know that [itex] T^2 = -1[/itex] and therefore [itex]T^{-1} = -T[/itex]

Then

[itex] T^{-1} H T = i \left( \begin{array}{cc} 0 & t \\ t & 0 \end{array} \right) K
\left( \begin{array}{cc} h_+(k) & 0 \\ 0 & h_-(k) \end{array} \right)
(-i) \left( \begin{array}{cc} 0 & t \\ t & 0 \end{array} \right) K
=
-\left( \begin{array}{cc} t h^*_-(k) t^* & 0 \\ 0 & t h_+^*(k) t^* \end{array} \right)
[/itex]

We still have to show [itex] h_{\pm}(k) = -t h_{\mp}^*(k) t^*[/itex], but at least we're down to 2x2 matrices.
 
  • #8
[itex]
-t h_+^* t^*
[/itex] gives [itex]
\left(
\begin{array}{cc}
-M^* + B_-^* (k_x^2 - \frac{\partial^2}{\partial y^2})
&
-A^*(k_x + \frac{\partial}{\partial y})
\\
-A^*(k_x - \frac{\partial}{\partial y})
&
M^* - B_+^* (k_x^2 - \frac{\partial^2}{\partial y^2})
\end{array}
\right)
[/itex]
 
  • #9
M Quack said:
[itex]
-t h_+^* t^*
[/itex] gives [itex]
\left(
\begin{array}{cc}
-M^* + B_-^* (k_x^2 - \frac{\partial^2}{\partial y^2})
&
-A^*(k_x + \frac{\partial}{\partial y})
\\
-A^*(k_x - \frac{\partial}{\partial y})
&
M^* - B_+^* (k_x^2 - \frac{\partial^2}{\partial y^2})
\end{array}
\right)
[/itex]

You are right regarding this. I forgot to tell that all the parameters are real so you can remove the conjugation. But by no means, I have [itex]B_{+}=\pm B_{-}[/itex].

I have come to know 2 ways to solve this problem.
(1) First is, I am probably choosing wrong matrix for Time reversal transformations. As my equation is for massless Dirac fermions, I should use proper relativistic quantum mechanics to calculate the transformation matrix for time reversal.
(2)Second is to use CPT symmetry. The argument goes as : if I apply Parity operation, [itex]h(k)-> h(-k)[/itex] and applying Conjugation operation, it should go to [itex]h(-k)-> h^{*}(-k)[/itex] which is the lower 2 χ 2 matrix of the Hamiltonian. These 2 are equivalent to applying [itex]T^{-1}[/itex]. I know that there are some loop holes in this derivation also, but I just want to give a general idea on how it can be solved.

I am trying these methods if they work.

Regards
Minato
 

FAQ: Time reversal symmetry in Topological insulators of HgTe quantum Wells

What is time reversal symmetry?

Time reversal symmetry is a fundamental principle in physics that states that the laws of physics should be the same regardless of whether time is moving forward or backward. In other words, if we could reverse the direction of time, the behavior of physical systems should remain the same.

What are topological insulators?

Topological insulators are materials that exhibit unique electrical properties due to their topology, or arrangement of electrons. They are insulators in their interior but have conducting states on their surface, making them potential candidates for use in electronic devices.

How do HgTe quantum wells relate to time reversal symmetry and topological insulators?

HgTe quantum wells are a type of material that has been extensively studied for their potential as topological insulators. They have a unique band structure that allows for the presence of conducting surface states while still maintaining time reversal symmetry.

What is the importance of time reversal symmetry in topological insulators of HgTe quantum wells?

Time reversal symmetry is crucial in topological insulators of HgTe quantum wells because it allows for the presence of conducting surface states, which are responsible for their unique electrical properties. Without time reversal symmetry, these materials would not exhibit the same behavior and would not be suitable for use in electronic devices.

How is time reversal symmetry in topological insulators of HgTe quantum wells studied and measured?

Time reversal symmetry in topological insulators of HgTe quantum wells can be studied and measured through various experimental techniques, such as angle-resolved photoemission spectroscopy (ARPES) and tunneling microscopy. These techniques allow scientists to observe the electronic properties of the material and determine if time reversal symmetry is present.

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