Topological Materials

Until recently, the standard paradigm for understanding phases of matter was through Landau’s symmetry breaking theory and order parameters. A classic example being a ferromagnet, which, after being cooled through its Curie temperature there is a reduction of rotational symmetry and the appearance of an order parameter, the magnetization.

Yet, not all phases of matter conform to this paradigm. There exist exotic states that exhibit no change in symmetry, no local order parameter, and yet are fundamentally distinct from trivial phases. The key to their identity lies in topology — a global property of their quantum ground state. In such systems, topological phase transitions involve a fundamental restructuring of the Hamiltonian, such that one phase cannot be smoothly deformed into another without closing the energy gap.

Some cornerstone examples of topological materials are

• Chern Insulator
• Topological Insulators
• Topological Semimetals
• Topological Crystalline Insulators
• Topological Superconductors

More recently, topology has emerged in diverse contexts — from magnetic skyrmions to topological spin liquids, revealing a rich interplay between geometry, entanglement, and quantum matter.

⏳ Timeline of Topological Quantum Matter ⌛️

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👾 1980s — Topology Enters Condensed Matter

Su-Schrieffer_Heeger Model

A one-dimensional model for polyacetylene. A predecessor of topological insulators.
📜 Su, Schrieffer, Heeger, PRL, 42, 1698 (1979)

Integer Quantum Hall Effect Observed

First experimental discovery of quantized Hall conductance.
📜 Klitzing et al., PRL 45, 494 (1980)

TKNN Invariant Introduced

Topology enters condensed matter via the Chern number and quantized Hall effect.
📜 Thouless et al., PRL 49, 405 (1982)

Berry Phase introduced

Michael Berry introduces a geometrical phase for adiabatically varying wavefunctions.
📜 M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984)

Haldane Model

Duncan Haldane proposed a tight-binding model exhibiting a quantized Hall conductance in the absence of an external magnetic field
📜 Haldane, PRL 61, 2015 (Oct. 31 🎃 1988)

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⚛️ 2000s — The Rise of Topological Insulators

Kane-Mele Model

Time-reversal invariant topological insulator in graphene.
📜 Kane & Mele, PRL 95, 146802 (2005)

Bernevig-Hughes-Zhang (BHZ) model

Quantum spin Hall effect predicted to be realized in HgTe-CdTe semiconductor quantum wells.
📜 Bernevig, Hughes, Zhang, Science 314, 1757 (2006)

Here is a clip from the popular show “The Big Bang Theory” where the CdTe/HgTe quantum well setup can be seen on the whiteboard in the background

${\mathbb{Z}}_2$ Topological Order in 3D

Classification of 3D topological insulators, appearance of “weak” and “strong” indices.
📜 Fu, Kane, Mele, PRL 98, 106803 (2007)

Fu-Kane Parity Criterion

Topological insulators with inversion symmetry have a simplified form of the ${\mathbb{Z}}_2$ invariant.
📜 Fu, Kane, PRB 76, 045302 (2007)
📜 Moore, Balents, PRB 75, 121306 (2007)

Observation of 2D TI in HgTe quantum wells

Experimental detection of topological surface states.
📜 M. König et al., Science 318, 766 (2007)

3D Topological Insulators Realized

Surface states experimentally observed in Bi₂Se₃ and similar materials.
📜Hsieh et al., Nature 452, 970 (2008)
📜 Xia et al., Nat. Phys. 5, 398 (2009)

Axion Insulator Theory Formulated

TR-breaking TIs with quantized $\theta =\pi$ response, gapped surface.
📜 Qi, Hughes, Zhang, PRB 78, 195424 (2008)
📜 Essin et al., PRL 102, 146805 (2009)
📜 Qi, Li, J. Zhang, S.C. Zhang, Science, 323, 5918 (2009)

Prediction of 3D TI in Bi₂Se₃, Bi₂Te₃

📜 H. Zhang et al., Nature Phys. 5, 438 (2009)

Observation of 3D TI surface state in Bi₂Te₃

ARPES confirmation of topological surface states.
📜 Y. Xia et al., Nature Phys. 5, 398 (2009)

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💎 Early 2010s — Weyl Semimetals, TBG, TCIs

Weyl Semimetals Proposed

Massless quasiparticles with topological charge; Fermi arcs predicted.
📜 Wan et al., PRB 83, 205101 (2011)

Bistritzer-MacDonald Model

A continuum Dirac model for twisted bilayer graphene appears.
📜 R. Bistritzer & A. H. MacDonald, PNAS 108, 12233 (2011)

Topological Crystalline Insulators

Topological protection via crystal symmetries.
📜 Fu, PRL 106, 106802 (2011)

Experimental TCI in SnTe

Tin telluride, a predicted mirror Chern insulator, shows metallic Dirac-cone surface band.
📜 Y. Tanaka et al., Nature Phys.8, 800 (2012)

Topological Superconductivity

Signatures of Majorana bound states in nanowires and hybrid systems.
📜 Mourik et al., Science 336, 1003 (2012)

Quantum Anomalous Hall Effect Observed in Cr doped Bi₂Te₃

📜 Chang et al., Science, 340, 6129 (2013)

Weyl semimetal observation in TaAs

📜 Xu et al., Science 349, 613 (2015)

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🌀Late 2010s — HOTIs, Moiré, Symmetry

Higher-Order Topological Insulators (HOTIs)

Topological modes localized to corners and hinges.
📜 Benalcazar, Bernevig, Hughes, Science 357, 61 (2017)

Symmetry Indicators of Band Topology

Generalization of Fu-Kane parity criterion to all 230 space groups.
📜 Po, Watanabe, Vishwanath, Nat. Commun. 8, 50 (2017)

Topological Quantum Chemistry

Elementary band representation classify band topology.
📜 B. Bradlyn et al., Nature 547, 298 (2017)

Unconventional Superconductivity in TBG

Intrinsic superconductivity not explained by electron-phonon interactions appears in magic angle TBG.
📜 Y. Cao et al., Nature 556, 43 (2018)

Fragile Topology

The emergence of topological bands that admit become trivialized upon the addition of trivial bands.
📜 Po, Watanabe, Vishwanath, PRL, 121, 126404 (2018)

Fragile Topology of Moiré Flat Bands

Tight-binding models for flat bands with fragile topology appear.
📜 Po, Zou, Senthil, Vishwanath, PRB., 99, 195455 (2019)

Topological Flat Bands in Moiré

Gapped set of bands at magic angles have non-trivial topology.
📜 Song, Wang, Benalcazar et al., PRL, 123, 036401 (2019)

🌌 2020s — Moiré, Quantum Geometry, and Flat Bands

Bulk-Boundary Correspondence of Fragile Topology

Invoking twisted boundary conditions, description fragile topology emerges.
📜 Song, Elcoro, Berevig, Science, 367, 6479 (2020)

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