Fractional topological insulators

1
Fractional topological insulators

• Michael Levin (Harvard), Ady Stern (Weizmann)

2
Topological vs. trivial insulators
(Zhang et al, Kane et al)

• Non-interacting fermions in a 2D gapped system
• (energy bands, Landau levels)
• An integer index the Chern number,
• Transition between different Chern numbers
  requires closing the energy gap.
• ( ) Gapless edge states on the
  interface with a vacuum.

(Thouless et al, Avron et al)
3
A useful toy model two copies of the IQHE Red
electrons experience a magnetic field B Green
electrons experience a magnetic field -B
nodd per each color a topological insulator
neven per each color a trivial insulator
4
With interactions, gapped systems are more
complicated and have more quantum
numbers. Several states may have the same sxy ,
but differ in other topological quantum numbers
(e.g., charge of quasi-particles, statistics,
thermal Hall conductivity).
A natural question does time reversal symmetry
introduce a distinction between trivial and
topological classes for interacting systems?
5
The modified toy model two copies of the FQHE

• n fractional per each color, artificial type of
  interaction.
• Two quantum numbers characterizing a fractional
  state
• the (spin) Hall conductivity
• e - the smallest charge allowed for an excitation

The question can the edge states be gapped out
without breaking time reversal symmetry ?
The answer is determined by the parity of
n/e Even Yes Odd - No
6

• Two ways to analyze this question
• A flux insertion argument
• A microscopic calculation
• Write down the edge theory.
• Look for perturbation(s) that gap(s) these 2n
  gapless modes without breaking time reversal
  symmetry.
• (Haldanes topological stability)

7
Flux insertion argument non-interacting case
(FuKane)

• Turn on a in the hole.
• A spin imbalance of n (integer number) is
  created on each edge.
• Two states that are degenerate in energy and
  time reversed of one another. Can they be
  coupled?
• Yes, if the imbalance is even.
• No, if it is odd. So, no edge perturbation will
  lift the degeneracy and open a gap without
  breaking time reversal symmetry.

8
For fractional quantum spin Hall states
spin-ups
spin-downs

• Half a flux quantum leaves the edges coupled.
• We need to insert the flux needed to bring each
  edge back to the topological sector from which it
  started.
• The number of flux quanta needed for that is
  1/2e.
• The imbalance is then n/e. It is the parity of
  this number that determines the protection of the
  gap.

9
A microscopic calculation The unperturbed edge,
two copies of the QHE
(Wen)
The Integer case
10
The double-FQHE unperturbed edge (N modes each
color)
11
Perturbations
The perturbation a. Charge conserving b.
Possibly strong (relevance is irrelevant) c.
So is the relevance to experiments (apologies!)
d. No conservation of momentum,
spin Symmetry with respect to time reversal a
crucial issue
12
Perturbations
l is an integer vector To conserve charge The
number of flipped spins
The time reversal operator
A time reversal symmetric perturbation
13
We look for a set of ls that will gap the edge
modes
Start from the double Laughlin state (N1)
Perturbation
j1 Zeeman field in the x-direction
j2 electron-electron interaction
14

• The perturbation gaps the two gapless modes, but
• For j1 it explicitly breaks time reversal
  symmetry
• For j2 it spontaneously breaks time reversal
  symmetry, giving an expectation value to

It is impossible to gap the two modes without
breaking symmetry to time reversal a
topological fractional insulator
For N1, always n/e1
15
Now for two edge modes in each direction (N2).
Most generally
b,s,u,v integers. u and v have no common
factors. The answer gapping is possible for
even (uv) impossible without TRS breaking
for odd (uv)
16
The method transform the problem into two
decoupled Luttinger liquids, and then gap them,
with two perturbations.
Even (uv) The two required vectors are
Each of the two vectors flips (uv)/2 spins.
Second order splits the degeneracy. No
spontaneous breaking of time reversal symmetry.
Odd (uv) flipping (uv) spins necessarily
break time reversal symmetry by giving an
expectation value to
17
Can a trivial insulator change to a topological
insulator without the gap being
closed, but with breaking of time
reversal symmetry?
For the Integer topological insulator yes For
the fractional more difficult, since the charge
e cannot change without the gap closing. The
answer If 1/e is odd yes.
If 1/e is even no. Why? a
relation between and
18

• q flux quanta (quanta of vorticity) in each
  topologically trivial
• p units of charge in each
• Statistics bosonic if p even,
  fermionic if p odd

The phase of interchange
So, parity of p must equal parity of pq
19
The phase of interchange
So, parity of p must equal parity of pq
p odd q odd q even
p even
only trivial insulators
(By passing this implies that a n5/2 state
must have quarter charged excitations in its
spectrum)
(Frohlich et al)
20

• Summary
• Gapless edge modes in fractional spin Hall
  insulators are protected in the case of odd
  n/e, as long as time reversal symmetry is
  preserved.
• These modes may be gapped by spin flipping
  perturbations when n/e is even.