Transcript Topological Delocalization in Quantum Spin

Topological Delocalization in Quantum Spin-Hall
Systems without Time-Reversal Symmetry
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
It is widely believed that the QSH state is
protected by the TR symmetry
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| < gc
2. The gap closes at |g| = gc
3. The gap then reopens for |g| > gc
For VR<VSO, gc is given by
For VR>VSO, gc = 0
|g|/VSO
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. Diagonalizeσz in valence bands. This can be done for
each k separately, as σz commutes with momentum.
If the Rashba spin-orbit coupling VR vanishes, σz will be a
conserved quantity. One can expect that the eigenvalues
of σz must be +1 or -1.
With turning on VR, which violates spin conservation, the
eigenvalues of σz deviate from +1 and -1, but a finite gap
usually still exists in the spectrum of σz.
-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 σz :
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| < gC, we have a QSHElike phase – The bulk
topological order is intact
when the TR symmetry is
weakly broken.
2. |g| > gC, 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
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
Still stay at 0.178 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.
Resistivity of bulk samples
Proposed Experiment
Mercury telluride (HgTe)
Bismuth selenride (Bi2Se3)
Bismuth telluride(Bi2Te3)
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 bulk topological order of the QSH
state is intact when the TR symmetry is
broken weakly.
As an important consequence, there exist
extended states in disordered QSH
systems without TR symmetry.
Marginal metallic behavior of the
resistivity is proposed to verify the present
theory experimentally.
Acknowledgements
Our work is supported by:
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
Thank you for your
attention !