# A Black hole as topological insulator

1. Apr 3, 2017

### shuijing

Hi, is anyone familiar with topological insulator? I read an interesting paper:
http://arxiv.org/abs/1703.09365,
Black hole as topological insulator

Abstract: Black holes are extraordinary massive objects which can be described
classically by general relativity, and topological insulators are new phases of
matter that could be use to built a topological quantum computer. They seem to
be different objects, but in this paper, we claim that the black hole can be
considered as a kind of topological insulator. For BTZ black hole in three
dimensional $AdS_3$ spacetime we give two evidences to support this claim: the
first evidence comes from the black hole "membrane paradigm", which says that
the horizon of black hole behaves like an electrical conductor. On the other
hand, the vacuum can be considered as an insulator. The second evidence comes
from the fact that the horizon of BTZ black hole can support two chiral
massless scalar field with opposite chirality. Those are two key properties of
2D topological insulator. For higher dimensional black hole the first evidence
is still valid. So we conjecture that the higher dimensional black hole can
also be considered as higher dimensional topological insulators. This
conjecture will have far-reaching influences on our understanding of quantum
black hole and the nature of gravity.

I am familiar with black hole but not topological insulator. Can anyone explain it to me?

2. Apr 3, 2017

### ZapperZ

Staff Emeritus
You're barking at the wrong tree.

Topological insulator is a topic in condensed matter physics, and a very active one too. There have been a few threads started in the condensed matter forum on topological insulators. So you may want to browse that forum first, or ask this question there (see if a Mentor will move this thread for you).

Zz.

3. Jun 7, 2017

### DeathbyGreen

Wow this sounds interesting! btw, Zapperz, this is the condensed matter forum, right? :D An insulator is simply a material which doesn't conduct electricity. In the system's band structure, the valence and conduction bands are separated by an energy gap which (assuming it is large enough) prevents electrons from jumping between bands. In condensed matter we use the Hamiltonian to describe a system; a few years ago the idea came around to classify these Hamiltonians according to their topology. This is usually done by considering $H = h(k)\tilde{\sigma}$, and using the $h(k)$ (or some other vector, it is somewhat up to the physicist) and finding it's map onto a unit sphere. The number of times this vector wraps around the unit sphere during the momentum shifting from $-\pi\rightarrow\pi$ in the Brillouin zone (band structure) is known as the winding number, or Chern number (similar to a genus in pure mathematics).

It was discovered that in certain systems, predicting the topology can correspond to protected edge states. So in some insulators which were previously thought to contain no conductance, two edge modes appear (under certain parameters) which are protected (not affected by disorder and giving conductance quantized similar to the quantum hall conductance) and move antiparallel to each other (chiral). A VERY nice review on the topic can be found in Topological Insulators, by Hasan and Kane.

Regarding the vacuum question, this is addressed directly in the Hasan and Kane review. The idea is that a vacuum is an insulator, with a gap separating the particle and antiparticle states. So if a black hole is a vacuum with chiral edge modes around the perimeter, the author's should be suggesting that this system can be classified topologically; it is an overall insulator but contains currents propagating in antiparallel directions on the edge. Whether these are protected states would interesting to see. Check out that review!

Last edited: Jun 7, 2017
4. Jun 7, 2017

### ZapperZ

Staff Emeritus
The thread was moved from where it was originally posted.

Zz.

5. Jul 12, 2017

### muscaria

Here's a nice 2 min video on topological insulators