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ABSTRACT: A field theory of the Anderson transition in two-dimensional disordered systems with spin-orbit interactions and time-reversal symmetry is developed, in which the proliferation of vortexlike topological defects is essential for localization. The sign of vortex fugacity determines the Z_{2} topological class of the localized phase. There are two distinct fixed points with the same critical exponents, corresponding to transitions from a metal to an insulator and a topological insulator, respectively. The critical conductivity and correlation length exponent of these transitions are computed in an N=1-ϵ expansion in the number of replicas, where for small ϵ the critical points are perturbatively connected to the Kosterlitz-Thouless critical point. Delocalized states, which arise at the surface of weak topological insulators and topological crystalline insulators, occur because vortex proliferation is forbidden due to the presence of symmetries that are violated by disorder, but are restored by disorder averaging.
Physical Review Letters 12/2012; 109(24):246605. · 7.37 Impact Factor
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Liang Fu
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ABSTRACT: The recent discovery of topological insulators has revived interest in the band topology of insulators. In this Letter, we extend the topological classification of band structures to include certain crystal point group symmetry. We find a class of three-dimensional "topological crystalline insulators" which have metallic surface states with quadratic band degeneracy on high symmetry crystal surfaces. These topological crystalline insulators are the counterpart of topological insulators in materials without spin-orbit coupling. Their band structures are characterized by new topological invariants. We hope this work will enlarge the family of topological phases in band insulators and stimulate the search for them in real materials.
Physical Review Letters 03/2011; 106(10):106802. · 7.37 Impact Factor
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ABSTRACT: Topological superconductors have a full pairing gap in the bulk and gapless surface Andreev bound states. In this Letter, we provide a sufficient criterion for realizing time-reversal-invariant topological superconductors in centrosymmetric superconductors with odd-parity pairing. We next study the pairing symmetry of the newly discovered superconductor CuxBi2Se3 within a two-orbital model, and find that a novel spin-triplet pairing with odd parity is favored by strong spin-orbit coupling. Based on our criterion, we propose that CuxBi2Se3 is a good candidate for a topological superconductor. We close by discussing experimental signatures of this new topological phase.
Physical Review Letters 08/2010; 105(9):097001. · 7.37 Impact Factor
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Liang Fu
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ABSTRACT: Zero-energy Majorana bound states in superconductors have been proposed to be potential building blocks of a topological quantum computer, because quantum information can be encoded nonlocally in the fermion occupation of a pair of spatially separated Majorana bound states. However, despite intensive efforts, nonlocal signatures of Majorana bound states have not been found in charge transport. In this work, we predict a striking nonlocal phase-coherent electron transfer process by virtue of tunneling in and out of a pair of Majorana bound states. This teleportation phenomenon only exists in a mesoscopic superconductor because of an all-important but previously overlooked charging energy. We propose an experimental setup to detect this phenomenon in a superconductor-quantum-spin-Hall-insulator-magnetic-insulator hybrid system.
Physical Review Letters 02/2010; 104(5):056402. · 7.37 Impact Factor
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ABSTRACT: Topological superconductors have been theoretically predicted as a new class of time-reversal-invariant superconductors which are fully gapped in the bulk but have protected gapless surface Andreev bound states. In this work, we provide a simple criterion that directly identifies this topological phase in \textit{odd-parity} superconductors. We next propose a two-orbital $U-V$ pairing model for the newly discovered superconductor Cu$_x$Bi$_2$Se$_3$%. Due to its peculiar three-dimensional Dirac band structure, we find that an inter-orbital triplet pairing with odd-parity is favored in a significant part of the phase diagram, and therefore gives rise to a topological superconductor phase. Finally we propose sharp experimental tests of such a pairing symmetry. Comment: 4.1 pages, 2 figures
12/2009;
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Liang Fu
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ABSTRACT: A single two-dimensional Dirac fermion state has been recently observed on the surface of the topological insulator Bi2Te3 by angle-resolved photoemission spectroscopy. We study the surface band structure using k x p theory and find an unconventional hexagonal warping term, which is the counterpart of cubic Dresselhaus spin-orbit coupling in rhombohedral structures. We show that this hexagonal warping term naturally explains the observed hexagonal snowflake Fermi surface. The strength of hexagonal warping is characterized by a single parameter, which is extracted from the size of the Fermi surface. We predict a number of testable signatures of hexagonal warping in spectroscopy experiments on Bi2Te3. We also explore the possibility of a spin-density wave due to strong nesting of the Fermi surface.
Physical Review Letters 12/2009; 103(26):266801. · 7.37 Impact Factor
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ABSTRACT: We propose two experiments to probe the Majorana fermion edge states that occur at a junction between a superconductor and a magnet deposited on the surface of a topological insulator. Combining two Majorana fermions into a single Dirac fermion on a magnetic domain wall allows the neutral Majorana fermions to be probed with charge transport. We will discuss a novel interferometer for Majorana fermions, which probes their Z2 phase. This setup also allows the transmission of neutral Majorana fermions through a point contact to be measured. We introduce a point contact formed by a superconducting junction and show that its transmission can be controlled by the phase difference across the junction. We discuss the feasibility of these experiments using the recently discovered topological insulator Bi2Se3.
Physical Review Letters 06/2009; 102(21):216403. · 7.37 Impact Factor
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ABSTRACT: We study the electronic surface states of the semiconducting alloy bismuth antimony (Bi1−xSbx). Using a phenomenological tight-binding model, we show that the Fermi surface for the 111 surface states encloses an odd number of time-reversal-invariant momenta (TRIM) in the surface Brillouin zone. This confirms that the alloy is a strong topological insulator in the (1;111) Z2 topological class. We go on to develop general arguments which show that spatial symmetries lead to additional topological structure of the bulk energy bands, and impose further constraints on the surface band structure. Inversion-symmetric band structures are characterized by eight Z2 “parity invariants,” which include the four Z2 invariants defined by time-reversal symmetry. The extra invariants determine the “surface fermion parity,” which specifies which surface TRIM are enclosed by an odd number of electron or hole pockets. We provide a simple proof of this result, which provides a direct link between the surface-state structure and the parity eigenvalues characterizing the bulk. Using this result, we make specific predictions for the surface-state structure for several faces of Bi1−xSbx. We next show that mirror-invariant band structures are characterized by an integer “mirror Chern number” nM, which further constrains the surface states. We show that the sign of nM in the topological insulator phase of Bi1−xSbx is related to a previously unexplored Z2 parameter in the L point k⋅p theory of pure bismuth, which we refer to as the “mirror chirality” η. The value of η predicted by the tight-binding model for bismuth disagrees with the value predicted by a more fundamental pseudopotential calculation. This explains a subtle disagreement between our tight-binding surface-state calculation and previous first-principles calculations of the surface states of bismuth. This suggests that the tight-binding parameters in the Liu-Allen model of bismuth need to be reconsidered. Implications for existing and future angle-resolved photoemission spectroscopy (ARPES) experiments and spin-polarized ARPES experiments will be discussed.
Phys. Rev. B. 04/2008; 78(4).
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ABSTRACT: We study the proximity effect between an s-wave superconductor and the surface states of a strong topological insulator. The resulting two-dimensional state resembles a spinless px+ipy superconductor, but does not break time reversal symmetry. This state supports Majorana bound states at vortices. We show that linear junctions between superconductors mediated by the topological insulator form a nonchiral one-dimensional wire for Majorana fermions, and that circuits formed from these junctions provide a method for creating, manipulating, and fusing Majorana bound states.
Physical Review Letters 04/2008; 100(9):096407. · 7.37 Impact Factor
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ABSTRACT: We study three-dimensional generalizations of the quantum spin Hall (QSH) effect. Unlike two dimensions, where a single Z2 topological invariant governs the effect, in three dimensions there are 4 invariants distinguishing 16 phases with two general classes: weak (WTI) and strong (STI) topological insulators. The WTI are like layered 2D QSH states, but are destroyed by disorder. The STI are robust and lead to novel "topological metal" surface states. We introduce a tight binding model which realizes the WTI and STI phases, and we discuss its relevance to real materials, including bismuth.
Physical Review Letters 04/2007; 98(10):106803. · 7.37 Impact Factor
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ABSTRACT: Topological insulators are materials with a bulk excitation gap generated by the spin orbit interaction, and which are different from conventional insulators. This distinction is characterized by Z_2 topological invariants, which characterize the groundstate. In two dimensions there is a single Z_2 invariant which distinguishes the ordinary insulator from the quantum spin Hall phase. In three dimensions there are four Z_2 invariants, which distinguish the ordinary insulator from "weak" and "strong" topological insulators. These phases are characterized by the presence of gapless surface (or edge) states. In the 2D quantum spin Hall phase and the 3D strong topological insulator these states are robust and are insensitive to weak disorder and interactions. In this paper we show that the presence of inversion symmetry greatly simplifies the problem of evaluating the Z_2 invariants. We show that the invariants can be determined from the knowledge of the parity of the occupied Bloch wavefunctions at the time reversal invariant points in the Brillouin zone. Using this approach, we predict a number of specific materials are strong topological insulators, including the semiconducting alloy Bi_{1-x} Sb_x as well as \alpha-Sn and HgTe under uniaxial strain. This paper also includes an expanded discussion of our formulation of the topological insulators in both two and three dimensions, as well as implications for experiments. Comment: 16 pages, 7 figures; published version
11/2006;
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ABSTRACT: We introduce and analyze a class of one dimensional insulating Hamiltonians which, when adiabatically varied in an appropriate closed cycle, define a "$Z_2$ pump". For an isolated system a single closed cycle of the pump changes the expectation value of the spin at each end even when spin orbit interactions violate the conservation of spin. A second cycle, however returns the system to its original state. When coupled to leads, we show that the $Z_2$ pump functions as a spin pump in a sense which we define, and transmits a finite, though non quantized spin in each cycle. We show that the $Z_2$ pump is characterized by a $Z_2$ topological invariant that is analogous to the Chern invariant that characterizes a topological charge pump. The $Z_2$ pump is closely related to the quantum spin Hall effect, which is characterized by a related $Z_2$ invariant. This work presents an alternative formulation which clarifies both the physical and mathematical meaning of that invariant. A crucial role is played by time reversal symmetry, and we introduce the concept of the time reversal polarization, which characterizes time reversal invariant Hamiltonians and signals the presence or absence of Kramers degenerate end states. For non interacting electrons we derive a formula for the time reversal polarization which is analogous to the Berry's phase formulation of the charge polarization. For interacting electrons, we show that abelian bosonization provides a simple formulation of the time reversal polarization. We discuss implications for the quantum spin Hall effect, and argue in particular that the $Z_2$ classification of the quantum spin Hall effect is valid in the presence of electron electron interactions. Comment: 14 pages, 6 figures
06/2006;
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ABSTRACT: We study junctions between superconductors mediated by the edge states of a quantum-spin-Hall insulator. We show that such junctions exhibit a fractional Josephson effect, in which the current phase relation has a 4π rather than a 2π periodicity. This effect is a consequence of the conservation of fermion parity—the number of electron mod 2—in a superconducting junction and is closely related to the Z2 topological structure of the quantum-spin-Hall insulator. Inelastic processes, which violate the conservation of fermion parity, lead to telegraph noise in the equilibrium supercurrent. We predict that the low-frequency noise due these processes diverges exponentially with temperature T as T→0. Possible experiments on HgCdTe quantum wells will be discussed.
Phys. Rev. B. 79(16).
