1. The Helical Luttinger Liquid and the Edge of Quantum Spin Hall Systems
Congjun Wu
Kavli Institute for Theoretical Physics, UCSB
B. Andrei Bernevig, and Shou-Cheng Zhang
Physics Department, Stanford University
It is my great pleasure to introducing our work on the “……..”.
Let me thank my collaborator Andrei Bernevig, and Prof. S. C. Zhang.
Here is the reference.
Ref: C. Wu, B. A. Bernevig, and S. C. Zhang, Phys. Rev. Lett. 96, (2006).
March Meeting, 03/15/2006, 3:42pm.

2. Introduction
Spin Hall effect (SHE): Use electric field to generate transverse spin current in spin-orbit (SO) coupling systems.
S. Murakami et al., Science 301 (2003); J. Sinova et al. PRL 92, (2004).
Y. Kato et al., PRL 93, (2004); J. Wunderlich et al. PRL 94, (2005).
Quantum SHE systems: bulk is gapped; no charge current.
Gapless edge modes support spin transport.
F. D. M. Haldane, PRL 61, 2015 (1988); Kane et al., cond-mat/ , Phys. Rev. Lett. 95, (2005); B. A. Bernevig et al., cond-mat/ , to appear in PRL; X. L Qi et al., cond-mat/ ; L. Sheng et al., PRL 95, (2005); D. N. Sheng et al, cond-mat/
Stability of the gapless edge modes against impurity, disorder under strong interactions.
The motivation of this work is inspired by the rapid progress of spin Hall effect.
It was proposed to use electric fields to generate transverse spin current
in spin-orbit coupling systems.
However, the electric field still generates longitudinal charge current.
This causes dissipation. This issue motivated the proposal of QSHE effect.
Here are some references. In QSHE systems, the bulk is gapped, and thus
there is no charge current. On the other hand, the gapless edge modes support the spin transport.
Then the stability of the gapless edge modes in the presence of impurity, disorder, interactions
becomes an interesting problem. In the following, I will address on these questions.
C. Wu, B. A. Bernevig, and S. C. Zhang, cond-mat/ , to appear in Phys. Rev. Lett;
C. Xu and J. Moore, PRB, (2006).

3. QSHE edges: Helical Luttinger liquids (HLL)
Edge modes are characterized by helicity.
Right-movers with spin up, and left-movers with spin down:
n-component HLL: n-branches of time-reversal pairs (T2=-1).
upper edge
lower edge
HLL with an odd number of components are special.
The QSHE edges constitute a new class of 1D liquid. We dub it helical Luttinger liquid.
This is because the edge modes are characterized by helicity, say, right-movers are with
spin-up, and left-movers are with spin down. The two channels forms
one branch of time-reversal (TR) pairs which satisfy T2=-1.
We will use the terminology that an n-component HLL contains
n-branches of time-reversal pairs.
Helical liquids are very special, because they are TR invariant, and
can only contain an odd number of branches of TR pairs.
It is unlike the chiral LL of the QHE edges where break the TR symmetry.
It is unlike the spinless non-chiral LL whose TR symmetry satisfying T2=1 instead of -1.
It is unlike the ordinary LL which can only contain an ever number of TR branches.
chiral Luttinger liquids in quantum Hall edges break TR symmetry;
spinless non-chiral Luttinger liquids: T2=1;
non-chiral spinful Luttinger liquids have an even number of branches of TR pairs.
Kane et al., PRL 2005, Wu et al., PRL 2006.

4. The no-go theorem for helical Luttinger liquids
1D HLL with an odd number of components can NOT be constructed in a purely 1D lattice system. EF -p p E
Double degeneracy occurs at k=0 and p.
Periodicity of the Brillouin zone.
HLL with an odd number of components can appear as the edge states of a 2-D system.
Next, we present some properties of the helical liquid.
We can prove the helical liquid with an odd number of TR
pairs can NOT be constructed in a purely 1D lattice system.
We sketch the band structure in the 1D Brillouin zone.
There are two special points with k=0, pi, because they
transform to themselves under time-reversal transformation.
So these two points must be doubly degenerate.
Because of the periodicity of the Brillouin zone, we can
see no matter where the Fermi energy lies, there
Are always an even number of TR branches.
On the other hand, the HLL with single component can indeed appear
as edge states of a 2-D system. As edge states, they do not need to
Cover the entire Brillouin zone.
For example, this band structure goes the singlet component of HLL.
H. B. Nielsen et al., Nucl Phys. B 185, 20 (1981); C. Wu et al., Phys. Rev. Lett. 96, (2006).

5. Instability: the single-particle back-scattering
The non-interacting Hamiltonian.
not allowed
Kane and Mele : The non-interacting helical systems with an odd number of components remain gapless against disorder and impurity scatterings.
Single particle backscattering term breaks TR symmetry (T2=-1).
Now let us come to the instability problem. We consider the following
non-interacting Hamiltonian with right movers spin up and left movers
spin down.
As pointed by Kane and Mele, ‘………..’
However, with strong interactions, HLL can indeed open the gap from another mechanism.

6. Two-particle correlated back-scattering
TR symmetry allows two-particle correlated back-scattering. 1 1 2 2
Microscopically, this Umklapp process can be generated from anisotropic spin-spin interactions.
Although the single particle back-scattering is time-reversal odd,
the two-particle correlated back-scattering is time-reversal even,
thus is allowed by the theory.
This means two right-movers with spin up can be scattered
simultaneously to left-movers with spin-down.
Microscopically, this Umklapp process can be generated from anisotropic
spin-spin interactions. This can be described by this effective Hamiltonian.
This Umklapp term reduces the the U(1)
Effective Hamiltonian:
U(1) rotation symmetry  Z2.

7. Bosonization+Renormalization group
Sine-Gordon theory at kf=p/2.
If K<1/2 (strong repulsive interaction), the gap D opens. Order parameters 2kf SDW orders Nx (gu<0) or Ny at (gu>0) .
Following the standard 1D paradigm of Bosonization, we write the bosonic sine-Gordon theory.
At commensurate filling, the umklapp term is just the cosine term.
If the Luttinger parameter K is less than ½, the cosine term becomes relevant and thus gap opens.
The order parameter is just the 2kf SDW, thus TR is spontaneously broken in the ground state.
On the other hand, at any finite temperature, the TR symmetry must be restored by thermal fluctuations.
If the T is less than the gap, the gap remains.
TR symmetry is spontaneously broken in the ground state.
At , TR symmetry much be restored by thermal fluctuations and the gap remains.

8. Random two-particle back-scattering
Scattering amplitudes are quenched Gaussian variables.
Giamarchi, Quantum physics in one dimension, oxford press (2004).
If K<3/8, gap D opens. SDW order is spatially disordered but static in the time domain.
TR symmetry is spontaneously broken.
Now we consider the case of the random two-particle back-scattering,
Where the scattering strength and phase are Gaussian random variables.
Following the standard replica and RG, we conclude that at the Luttinger
Parameter K<3/8, the gap opens.
This means the ground state exhibits glassy SDW, thus TR is again broken.
Similarly, at small but finite temperatures, gap remains and TR is restored
By thermal fluctuations.
At small but finite temperatures, gap remains but TR is restored by thermal fluctuations.

9. Single impurity scattering
Boundary Sine-Gordon equation.
C. Kane and M. P. A. Fisher, PRB 46, (1992).
If K<1/4, gu term is relevant. 1D line is divided into two segments.
TR is restored by the instanton tunneling process.
Now let us look the single impurity scattering problem, which can be mapped to a
boundary Sine-Gordon form. This problem was was studied in detail by Kane and Fisher.
The RG result shows that at K<1/4, this impurity scattering is relevant.
The 1D chain is divided into two segments.
On the other hand, because it is a zero-dimensional problem, even at zero temperature,
the TR symmetry is dynamically restored by the instanton tunneling process.

10. Kondo problem: magnetic impurity scattering
Poor man RG: critical coupling Jz is shifted by interactions.
If K<1 (repulsive interaction), the Kondo singlet can form with ferromagnetic couplings.
At last, we consider the Kondo problem in such systems.
The sigma is the spin of the local moment. The spin flip
Process is combined with the back-scattering of the conducting
electron, while the spin non-flip process is combined by the
forward scattering process.
We did a poor man RG analysis, the RG flow looks similar to the
the 3D Kondo problem. However, the interaction effect in 1D is
also important, which causes an overall shift in the Jz axis.
In particular, the Kondo singlet can form with ferromagnetic coupling
if the interaction is repulsive.

11. Summary Helical Luttinger liquid (HLL) as edge states of QSHE systems.
No-go theorem: HLL with odd number of components can not be constructed in a purely 1D lattice system.
Instability problem: Two-particle correlated back-scattering is allowed by TR symmetry, and becomes relevant at:
Kc<1/2 for Umklapp scattering at commensurate fillings.
Kc<3/8 for random disorder scattering.
Kc<1/4 for a single impurity scattering.
Critical Kondo coupling Jz is shifted by interaction effects.