### Topological Insulators : A New State of Quantum Matter

**M. Zahid Hasan**

[This is an invited article based on a series of recent works by the author and his collaborators -- 2Physics.com]

**Author:**

**M. Zahid Hasan**

**Affiliation: Joseph Henry Laboratories of Physics,**

**Department of Physics,**

**Princeton University**

**, USA**

Most quantum states of condensed-matter systems or the fundamental forces are categorized by spontaneously broken symmetries. The remarkable discovery of quantum Hall effects (1980s) revealed that there exists an organizational principle of matter based not on the broken symmetry but only on the topological distinctions in the presence of time-reversal symmetry breaking [1,2]. In the past few years, theoretical developments suggest that new classes of topological states of quantum matter might exist in nature [3,4,5]. Such states are purely topological in nature in the sense that they do not break time-reversal symmetry, and hence can be realized without any applied magnetic field : "Quantum Hall-like effects without magnetic field".

**Research Team at Princeton University: [L to R] David Hsieh, Dong Qian, L. Andrew Wray, YuQi Xia**

This exotic phase of matter is a subject of intense research because it is predicted to give rise to dissipationless (energy saving) spin currents, quantum entanglements and novel macroscopic behavior that obeys axionic electrodynamics rather than Maxwell's equations [6]. Unlike ordinary quantum phases of matter such as superconductors, magnets or superfluids, topological insulators are not described by a local order parameter associated with a spontaneously broken symmetry but rather by a quantum entanglement of its wave function, dubbed topological order. In a topological insulator this quantum entanglement survives over the macroscopic dimensions of the crystal and leads to surface states that have unusual spin textures.

Topologically ordered phases of matter are extremely rare and are experimentally challenging to identify. The only known example was the quantum Hall effect discovered in the 1980s by von Klitzing (Nobel Prize 1985). It was identified by measuring a quantized magneto-transport in a two-dimensional electron system under a large external magnetic field at very low temperatures, which is characterized by robust conducting states localized along the one-dimensional edges of the sample. Two-dimensional topological insulators, on the other hand, are predicted to exhibit similar edge states even in the absence of a magnetic field because spin-orbit coupling can simulate its effect (Fig.1A) due to the relativistic terms added in a band insulator's Hamiltonian.

Remarkably, three-dimensional topological insulators, an entirely new state of matter with no charge quantum Hall analogue, are also postulated to exist. And its topological order or exotic quantum entanglement is predicted to give rise to unusual conducting two-dimensional surface states (Fig.1B) that have novel spin-selective energy-momentum dispersion relations. Utilizing state-of-the-art angle-resolved photoemission spectroscopy, an international collaboration led by scientists from Princeton University have studied the electronic structure of several bismuth based spin-orbit materials [7,8,9]. By systematic tuning of the incident photon energy, it was possible to isolate surface quantum states from the bulk states, which confirmed that these materials realized a three-dimensional topological insulator phase.

**Figure 1. (A) Schematic of the 1D edge states in a 2D topological insulator. The red and blue curves represent the edge current with opposite spin character. (B) Schematic of the 2D surface states in a 3D topological insulator. (C) Most elemental topological Insulators exhibit odd number of Dirac cones on their surface unlike the even numbers observed in graphene. Topological insulator Dirac cones are spin polarized where as Dirac cones in graphene are not.**

The remarkable property of the surface states of a 3D topological insulator is that its Fermi surface supports a geometrical quantum entanglement phase, which occurs when the spin-polarized Fermi surface encloses the Kramers' points and on the surface Brillouin zone an odd number of times in total (Fig.2B). ARPES intensity map of the (111) surface states of bulk insulating Bi1-xSbx (Fig.2A) shows that a single Fermi surface encloses . However, determination of the degeneracy of the additional Fermi surface around requires a detailed study of its energy-momentum dispersion. ARPES spectra along the - direction (Fig.2C) reveal that the Fermi surface enclosing is actually composed of two bands, therefore two Fermi surfaces enclose , leading to a total of seven and Fermi surface enclosures.

**Figure 2. (A) ARPES surface state (SS) Fermi surface of insulating Bi1-xSbx showing spin polarization directions as indicated by red and blue arrows. (B) Schematic of the SS Fermi surface of a 3D topological insulator. (C) ARPES energy-momentum dispersion of the surface states. The shaded**

**areas denote the bulk bands while the dashed white lines are guides to the eye for surface state dispersions. (D) A single Dirac cone is observed in Bi2Te3.**

These results constitute the first direct experimental evidence of a topological insulator in nature which is fully quantum entangled. The observed spin-texture in BiSb is consistent with a magnetic monopole image field beneath the surface. It shows that spin-orbit materials are a new family in which exotic topological order quantum phenomena, such as dissipationless spin currents and axion-like electrodynamics, may be found without the need for an external magnetic field. The results presented in this study also demonstrate a general measurement algorithm of identifying and characterizing topological insulator materials for future research which can be utilized to discover, observe and study other forms of topological order and quantum entanglements in nature. A detailed study of topological order and quantum entanglement can potentially pave the way for fault-tolerant (topological) quantum computing [10].

**Figure 3: A new type of quantum matter called a topological insulator contains only half an electron pair (represented by just one Dirac cone in schematic crystal structure at top left), which is observed in the form of a single ring (red) in the center of the electron-map (top right) with electron spin in only one direction. This highly unusual observation shows that if an electron is tagged "red" and then undergoes a full 360-degree revolution about the ring, it does not recover its initial face as an ordinary everyday object would, but instead acquires a different color "blue" (represented by the changing color of the arrows around the ring). This new quantum effect can be the basis for the realization of a rare quantum phase that had been a long-sought key ingredient for developing quantum computers that can be highly fault-tolerant.**

References:

References:

**[1]**K. von Klitzing, G. Dorda, M. Pepper, "New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance", Phys. Rev. Lett. 45, 494-497 (1980). Abstract.

**[2]**D.C. Tsui, H. Stormer, A.C. Gossard, "Two-dimensional magnetotransport in the extreme quantum limit", Phys. Rev. Lett. 48, 1559-1562 (1982). Abstract.

**[3]**L. Fu, C. L. Kane and E. J. Mele, "Topological insulators in three dimensions", Physical Review Letters 98, 106803 (2007). Abstract.

**[4]**J. E. Moore and L. Balents, "Topological invariants of time-reversal-invariant band structures", Physical Review B 75, 121306(R) (2007). Abstract.

**[5]**S.-C. Zhang, "Topological states of quantum matter", Physics 1, 6 (2008). Abstract.

**[6]**M. Franz, "High energy physics in a new guise", Physics 1, 36 (2008). Abstract.

**[7]**D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava and M. Z. Hasan, "A topological Dirac insulator in a quantum spin Hall phase", Nature 452, 970 (2008). Abstract.

**[8]**Y. Xia, D. Qian, L. Wray, D. Hsieh, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava and M. Z. Hasan, "Observation of a large-gap topological insulator class with single surface Dirac cone”, Nature Physics 5, 398 (2009). Abstract.

**[9]**D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. Cava and M. Z. Hasan, "Observation of Unconventional Quantum Spin Textures in Topological Insulators", Science 323, 919 (2009). Abstract.

**[10]**A. Akhmerov, J. Nilsson, C. Beenakker, “Electrically detected interferometry of Majorana fermions in a topological insulator”, Phys. Rev. Lett. 102, 216404 (2009). Abstract.

Labels: Complex System, Condensed Matter 2, Graphene, Quantum Computation and Communication 2

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