Classification of Crystalline Topological Phases with Point Group Symmetries

Transcription

1 Classification of Crystalline Topological Phases with Point Group Symmetries Eyal Cornfeld - Tel Aviv University Adam Chapman - Tel Hai Academic College Upper Galilee Bad Honnef Physics School on Gauge theory and topological quantum matter 2018

2 Topological Phases & Crystalline Symmetry Extend periodic table of topological insulators and superconductors New phases protected by crystalline point group symmetry [Taken from Shinsei Ryu] [Taken from Ashcroft & Mermin]

3 Topological Phases & Crystalline Symmetry Extend periodic table of topological insulators and superconductors New phases protected by crystalline point group symmetry [Taken from Shinsei Ryu] [Taken from Ashcroft & Mermin]

4 What are all Possible (noninteracting) Hamiltonians? [Taken from Andreas P. Schnyder] [Taken from Ricardo Kennedy]

5 What are all Possible (noninteracting) Hamiltonians? [Taken from Andreas P. Schnyder] [Taken from Ricardo Kennedy]

6 2D Hamiltonians in Altland-Zirnbauer Class AII Dirac Hamiltonian mass and gamma matrices All possible Hamiltonians? All possible mass terms? [Taken from Akira Furusaki]

7 2D Hamiltonians in Altland-Zirnbauer Class AII Antiunitary with and Dirac Hamiltonian mass and gamma matrices All possible Hamiltonians? All possible mass terms? [Taken from Akira Furusaki]

8 (Clifford) Algebra Extensions The symmetries form an algebra acting on the Hilbert space. Hamiltonians Mass terms Actions of B that are compatible with A For AZ class AII

9 (Clifford) Algebra Extensions The symmetries form an algebra acting on the Hilbert space. Hamiltonians Mass terms Actions of B that are compatible with A For AZ class AII in 2D we find

10 Topological Phases & Crystalline Symmetry Extend periodic table of topological insulators and superconductors New phases protected by crystalline point group symmetry [Taken from Andreas P. Schnyder] [Taken from Ashcroft & Mermin]

11 Point Group Symmetry Generators Inversion Rotation Reflection Rotoreflection

12 Strasbourg Mineralogy Museum

13 Topological Phases & Crystalline Symmetry Extend periodic table of topological insulators and superconductors New phases protected by crystalline point group symmetry [Taken from Andreas P. Schnyder] [Taken from Ashcroft & Mermin]

14 Class AI with Tetragonal-Disphenoidal Symmetry Dirac Hamiltonian What is the new algebra extension Study? symmetry

15 Class AI with Tetragonal-Disphenoidal Symmetry Dirac Hamiltonian Generators of AZ Class AI in 3D What is the new algebra extension Study? symmetry

16 Class AI with Tetragonal-Disphenoidal Symmetry Dirac Hamiltonian Generators of AZ Class AI in 3D What is the new algebra extension Study Redefine symmetry symmetry generator generated by and?

17 Solution

18 Solution Find an subalgebra generated by which commutes with

19 Solution Find an Similarly subalgebra generated by which commutes with

20 Solution Find an subalgebra generated by Similarly Subalgebra (graded) structure We find topological classification of which commutes with instead of

21 Results Example: all Tetragonal Point Group Symmetry Crystals in all AZ Classes (7 10 out of 32 10) [See draft for all 32 point group crystals]

22 Results Example: all Tetragonal Point Group Symmetry Crystals in all AZ Classes (7 10 out of 32 10) [See draft for all 32 point group crystals]

23 Results Example: all Tetragonal Point Group Symmetry Crystals in all AZ Classes (7 10 out of 32 10) [See draft for all 32 point group crystals]

24 Outlook 1) Generalize [technically all in Freed & Moore (2013) using twisted K-theory] a) Magnetic point groups <122> - relatively easy. e.g. Schindler et. al. (Sci. Adv. 2018) b) Nonsymmorphic space groups <230> - harder. c) Nonsymmorphic magnetic space groups <1651> -?? 2) Connection to higher order topological insulators a) Shiozaki & Sato (PRB 2014) not all bulk invariants have distinct surface states. b) Schindler et. al. (Nat. Phys. 2018) experiments and theory for Bismuth in AZ class AII. 3) Find more materials which are crystalline topological insulators!