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, (2011)]  Topological delocalization in QSH systems without TR symmetry [Preprint: cond-mat/arXiv: (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, (2005)] The spin Chern number [D. N. Sheng et al., PRL 97, , (2007); E. Prodan, PRB 80, (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, (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| < g c 2. The gap closes at |g| = g c 3. The gap then reopens for |g| > g c For V R <V SO, g c is given by General characteristics of the energy spectrum, in the presence of the exchange field (g≠0): For V R >V SO, g c = 0 A topological phase transition usually happens at the point where the band gap closes |g|/V SO Kane-Mele Model

TR Symmetry-Broken QSH State Smooth decomposition of the subspace of valence bands: 1. Diagonalizeσ z in valence bands. This can be done for each k separately, as σ z commutes with momentum. 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 a finite gap usually still exists in the spectrum of σ z. +1 Spin up Spin down A sketch of spin spectrum Calculation of Spin-Chern Number

TR Symmetry-Broken QSH State 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. Calculation of Spin-Chern Number

TR Symmetry-Broken QSH State 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 (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| < g C, we have a QSHE- like 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. 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 invariants are 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 random disorder: 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 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 mobility gap. Level Statistics (Vertical Exchange Field) Still stay at 0.178

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

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

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.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 Our work is supported by: Acknowledgements

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