I am reading the book "Topological insulator: Dirac equation in condensed matters" by Shun-Qing Sheng. I do not know much about this topic and this is the first time I am confronted with it, so this might be an easy question: On page 15, chapter 2, the Jackiw-Rebbi Solution in one dimension is presented. It is assumed that there are two regions: one with positive mass and one with negative mass.

What is the interpretation of this negative mass? Why is there such a thing as an negative mass? Is this related to the negative energy interpretation in the Dirac equation (Dirac sea/anti-particles)?

The concept of negative masses seems to be widely accepted in this part of physics, since there are a lot papers which assume similar things (for example here).

EDIT: Here is a scan of the mentioned section from Sheng's book:

Scan of the aforementioned book

EDIT 2: Maybe it would help, if someone could explain to me the term "topological mass" as I suppose this is what is meant.


1 Answer 1


You can find an excellent description of what a topological insulator is in this brief presentation from the Yazdani Group at Princeton: Topological Insulators.

To answer your question on the meaning of negative effective mass:

The effective mass is actually determined by the behavior of the energy levels $E({\bf k})$ as functions of the crystal wave vector ${\bf k}$ in the Brillouin cell. If an energy level has a local minimum, usually at the center of the cell, it can be approximated to second order in ${\bf k}$ by an expression of the form
$$ E({\bf k}) = E_0 + \frac{\hbar^2 {\bf k}^2}{2 m_{eff}} $$ where the expansion coefficient $\frac{\hbar^2}{2 m_{eff}}$ is written so that the entire expression becomes analogous to the kinetic energy formula ${\bf p}^2/2m$ for ${\bf p}= \hbar {\bf k}$. By identification, the scalar $m_{eff}$ plays the role of a "mass" and is therefore called the "effective mass". At first sight it simply tells how strongly does the energy vary with ${\bf k}$ in the Brillouin cell, but it does act like a mass in many calculations, so the name is justified. Note that the above expansion is the simplest possible. It often happens that $m_{eff}$ is a symmetric tensor, which after diagonalization supplies different effective masses along 3 orthogonal ${\bf k}$ directions (anisotropy of energy levels and effective mass).

In a normal insulator, the energy levels behave largely as above and exhibit a positive effective mass. In a topological insulator, conducting and valence energy levels become inverted in the center of the Brillouin cell (see reference for more details), such that the valence levels become concave, like a "sad parabola", and display a local maximum. Their expansion becomes
$$ E({\bf k}) = E_0 - \frac{\hbar^2 {\bf k}^2}{2 m_{eff}} $$

and exhibits "negative effective mass", with all the interesting consequences that follow.

  • $\begingroup$ Thank you for the link and the explanation! However, in semiconductors one also finds the same shape of the bands: the valence band is a "sad" parabola and the conducting band is a "happy" parabola (I like the way you wrote that ;) ). The negative mass in the v-band is there turned into a positive mass by introducing holes with positive charge. The bandstructure looks then very similar to image b) in the linked article. What is the big difference? $\endgroup$
    – Merlin1896
    Commented Aug 26, 2015 at 9:58
  • 1
    $\begingroup$ @Merlin1896 Your observation is correct. The difference is the topology (twists) of the energy surfaces, which determines the number of Kramer state pairs in edge modes. If the total number $Z_2$ of edge Kramers pairs is even (0,2,...), the crystal is a regular insulator, if $Z_2$ is odd (1) it is a topological insulator. In the latter case the edge states become symmetry protected for lack of other states to transition into (see their spin). You may be interested in science.uva.nl/research/cmp/TI.html and arxiv.org/pdf/1002.3895v2. $\endgroup$
    – udrv
    Commented Aug 26, 2015 at 20:56

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