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. ShengDept. of Phys. and Astro, California State University, Northrid

ge 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

Standard Kane-Mele model for QSH effect, which is defined on a honeycomb lattice:

g – term: an exchange field, which breaks time-reversal (TR) symmetry

Kane-Mele Model

TR Symmetry-Broken QSH State

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)

Kane-Mele Model

Occupied bands

Unoccupied bands

TR Symmetry-Broken QSH State

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

General characteristics of the energy spectrum, in the presenceof the exchange field (g≠0):

For VR>VSO, gc = 0

A topological phase transition usually happens at the point where the band gap closes

|g|/VSO

Kane-Mele Model

TR Symmetry-Broken QSH State

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 +1Spin upSpin downA sketch of spin spectrum

Calculation of Spin-Chern Number

TR Symmetry-Broken QSH State

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.

Calculation of Spin-Chern Number

TR Symmetry-Broken QSH State

3. Calculate the spin Chern numbers, i.e., the Chern numbersof the two spin sectors (use standard formula and summarize over two Dirac cones)

Note: It is more rigorous to calculate in the band (tight-binding) model. The continuum approximation does not always yield the correct result.

Calculation of Spin-Chern Number

TR Symmetry-Broken QSH State

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.

Calculation of Spin-Chern Number

TR Symmetry-Broken QSH State

Resulting phase diagram:1. |g| < gC, we have a QSHE-li

ke 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.

Phase Diagram of KM Model with An Exchange Field

Topological Delocalization

We have shown the topological invariants are intact when TR symmetry is broken weakly. Since topological invariantsare known to characterize extended states, now it is important to show the existence of extended states in the TR-symmetry-broken 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 randomdisorder:

Kane-Model Model with 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

A covariance equal to 0.178 indicatingextended 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 mobility gap.

Level Statistics (Vertical Exchange Field)

Still stay at 0.178

Topological DelocalizationLocalization Length (Vertical Exchange Field)

The scaling behavior of the localization length further confirms theexistence of extended states, and the pair-annihilation scenario.

Localization length calculation for essentially infinitely long ribbonswith finite widths using Recursive Green’s Function method

Topological DelocalizationMapping 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

Insulator

Marginalmetal

In bulk samples, effective size is controlled by inelastic scattering length. Inelastic length increases with decreasing temperature. SoTemperature dependence = Size dependence

o TemperatureR

esis

tivity

of

bulk

sam

ples Mercury telluride (HgTe)

Bismuth selenride (Bi2Se3)Bismuth telluride(Bi2Te3)

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.

1.1. State Key Program for Basic Researches of State Key Program for Basic Researches of China (China ( 中国重大基础研究发展中国重大基础研究发展 [973][973] 计划项计划项目）目）

2.2. National Natural Science Foundation of ChiNational Natural Science Foundation of China (na ( 中国自然科学基金面上项目）中国自然科学基金面上项目）

3.3. Partially by U.S. National Natural Science FPartially by U.S. National Natural Science Foundationoundation

4.4. U.S. DOE GrantsU.S. DOE Grants

Our work is supported by:

Acknowledgements

Thank you for your Thank you for your attention !attention !