L. Sheng ( 盛利 )
Y. Y. Yang ( 杨运友 ), Z. Xu ( 徐中 ),
D. Y. Xing ( 邢定钰 ), B. G. Wang ( 王伯根 )
NLSSM and Dept. of Phys., Nanjing University, Nanjing
D. N. Sheng
Dept. of Phys. and Astro, California State University,
Northridge
E. Prodan
Dept. of Phys., Yeshiva University, New York

Outlines
 Motivations
 Spin Chern number theory of quantum spin-Hall (QSH) state without TR symmetry
[Phys. Rev. Lett. 107, 066602 (2011)]
 Topological delocalization in QSH systems without TR symmetry
[Preprint: cond-mat/arXiv:1108.2929 (2011)]
 Summary

Motivations
QSH state – a new state of matter with potential applications in spintronics devices
 A bulk band gap
 Gapless edge modes traversing the gap
 A new example of topologically ordered states
The Z2 invariant
[Kane & Mele, PRL 95, 146802 (2005)]
The spin Chern number
[D. N. Sheng et al., PRL 97, 036806, (2007);
E. Prodan, PRB 80,125327 (2009)]

Motivations
The TR symmetry protects the gapless edge modes as well as the Z2 invariant. In fact, the definition of
Z2 index relies on the presence of TR symmetry.

Motivations
Issues we are interested in:
• Will the topological order of the QSH state be destroyed immediately, when the TR symmetry is broken weakly?
(In usual, a topological invariant is purely a geometric effect, and should not be protected by any symmetries.)
• Can the topologically protected bulk extended states survive TR symmetry breaking?
A previous work [M. Onoda, et al., PRL 98, 076802 (2007)] has confirmed extended states in TR symmetric QSH systems. However, they concluded that the extended states will be destroyed immediately if the TR symmetry is broken. Their argument is that the QSH systems without TR symmetry belongs to the trivial unitary class, where all electron states must be localized .

TR Symmetry-Broken QSH State
Kane-Mele Model
Standard Kane-Mele model for QSH effect, which is defined on a honeycomb lattice: g – term: an exchange field, which breaks timereversal (TR) symmetry

TR Symmetry-Broken QSH State
Kane-Mele Model
In the momentum space, we can expand H near the two Dirac points K and K’. For each given momentum k, we obtain totally four eigenstates of H (The analytical expression is too lengthy to write out) Occupied bands
Unoccupied bands

TR Symmetry-Broken QSH State
Kane-Mele Model
General characteristics of the energy spectrum, in the presence of the exchange field (g≠0):
1. The middle band gap remains open for |g| < g c
2. The gap closes at |g| = g c
3. The gap then reopens for |g| > g c
For V
R
For V
R
<V
SO
>V
SO
, g c
, g c is given by
= 0
|g|/V
SO
A topological phase transition usually happens at the point where the band gap closes

TR Symmetry-Broken QSH State
Calculation of Spin-Chern Number
Smooth decomposition of the subspace of valence bands:
1. Diagonalize σ z in valence bands each k separately , as σ z
. This can be done commutes with momentum.
for
If the Rashba spin-orbit coupling V
R vanishes, σ z will be a conserved quantity. One can expect that the eigenvalues of σ z must be +1 or -1.
With turning on V
R
, which violates spin conservation, the eigenvalues of σ z deviate from +1 and -1, but usually still exists in the spectrum of σ z. a finite gap
-1 Spin down Spin up +1
A sketch of spin spectrum

TR Symmetry-Broken QSH State
Calculation of Spin-Chern Number
2. Linearly recombine and into eigenstates of
σ z
:
Here, + and – correspond to the two spin sectors.
A unitary transformation of the wave functions of the occupied electron states, which is a very useful way to find the relevant topological invariants in multi-band systems for different problems.

TR Symmetry-Broken QSH State
Calculation of Spin-Chern Number
3. Calculate the spin Chern numbers, i.e., the Chern numbers of the two spin sectors (use standard formula and summarize over two Dirac cones)
Note: It is more rigorous to calculate in the band (tightbinding) model. The continuum approximation does not always yield the correct result.

TR Symmetry-Broken QSH State
Calculation of Spin-Chern Number
Some comments:
The definition of the spin Chern numbers relies on the existence of the two spectrum gaps :
1. Middle band gap (valence and conduction bands are well separated)
2. Spin spectrum gap (the spin-up and down sectors are unambiguously distinguished)
The spin-Chern numbers are protected by the two gaps, rather than any symmetries, in contrast to Z2. They are topological invariants as long as the two gaps stay open.

TR Symmetry-Broken QSH State
Phase Diagram of KM Model with An Exchange Field
Resulting phase diagram:
1. |g| < g
C
, we have a QSHElike phase – The bulk topological order is intact when the TR symmetry is weakly broken.
2. |g| > g
C
, there is a quantum anomalous Hall (QAH) phase
3. The phase boundary is just at the place where the band gap closes.

Topological Delocalization
Kane-Model Model with Disorder
We have shown the topological invariants are intact when
TR symmetry is broken weakly. Since topological invariants are known to characterize extended states , now it is important to show the existence of extended states in the TR-symmetrybroken QSH state. Besides, delocalization in 2D is always an important topic of great theoretical and practical interest.
Kane-Mele model with an exchange field and on-site random disorder :

Topological Delocalization
• We carry out exact diagonalization for a finite system with 40 * 40 unit cells.
• To obtain the information for localization/delocalization, we perform level statistics analysis .
• We set nearest neighbor hopping integral t to be the unit of energy, for simplicity

Topological Delocalization
Level Statistics (Vertical Exchange Field)
A covariance equal to 0.178 indicating extended states
Still stay at 0.178
1. At weak disorder, extended states exist on two sides of the band gap.
2. The extended states are destroyed through pair-annihilation in both cases, i.e., closing of the energy mobility gap.

Topological Delocalization
Localization Length (Vertical Exchange Field)
Localization length calculation for essentially infinitely long ribbons with finite widths using Recursive Green’s Function method
The scaling behavior of the localization length further confirms the existence of extended states , and the pairannihilation scenario .

Topological Delocalization
Mapping of Phase Diagram (Vertical Exchange Field)
Theoretical Analysis: The existence of the extended states can be attributed to the spin-Chern numbers . The extended states are located near the phase boundary where the spin-Chern numbers change values.

Proposed Experiment
Mercury telluride (HgTe)
Bismuth selenride (Bi
2
Se
3
)
Bismuth telluride(Bi
2
Te
3
)
Insulator
Marginal metal o
Temperature
In bulk samples, effective size is controlled by inelastic scattering length. Inelastic length increases with decreasing temperature. So
Temperature dependence = Size dependence

Summary
 The
of the QSH state is
when the TR symmetry is broken weakly.
 As an important consequence, there exist
in disordered QSH systems without TR symmetry.

of the resistivity is proposed to verify the present theory experimentally.

Acknowledgements
1. State Key Program for Basic Researches of
China ( 中国重大基础研究发展 [973] 计划项目）
2. National Natural Science Foundation of
China ( 中国自然科学基金面上项目）
3. Partially by U.S. National Natural Science
Foundation
4. U.S. DOE Grants