SCIENTIFIC PROGRAM:



Timetable:
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Morning session (3h 45'):
Lectures 08:15 - 10:15 10:35 - 12:20
Coffee break: 10:15 - 10:35
Lunch: 12:20 - 13:45
Afternoon session (4h 15'):
Lectures: 13:45 - 15:45 16:15 - 18:30
Coffee break: 15:45 - 16:15
Dinner: 18:30 - 20:15
Evening session (optional): 20:30 - 21:30
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1st Day (1 July 2019) morning session
08:15 - 08:30Opening of the school
08:30 - 10:15 (MIA) Crystallographic symmetry operations and their descriptions
Crystallographic point and space-group symmetry operations. Matrix-column presentation of symmetry operations. Geometric interpretation of the matrix-column pairs of symmetry operations. Symmetry elements: geometric elements and element sets.
10:35 - 12:20 (MN) Crystal pattern
Crystal lattice vs. crystal pattern and crystal structure. Symmetry directions in a lattice .Unit cells: primitive cells, multiple cells, conventional cells in 2D and 3D. Crystal families. Symmetry groups and types of symmetry in direct space: morphological symmetry; symmetry of physical properties; symmetry of lattices; symmetry of the unit cell content; symmetry of crystallographic patterns.
1st Day (1 July 2019) afternoon session
13:45 - 14:45 (MN) Crystallographic point groups
Stereographic projection and the morphology of crystals. Hermann-Mauguin symbols for point groups.
14:45 - 15:45 (BS) Crystallographic point groups (cont.)
Basic properties of groups (group axioms, order, multiplication tables, etc.). Subgroups, index, coset decompositions, Lagrange theorem.
16:15 - 17:40 (BS) Crystallographic point groups (cont.)
Group actions: conjugation and normalizers. Wyckoff positions for point groups. Relations of Wyckoff positions for a group-subgroup pair. Supergroups of point groups.
17:40 - 18:30 (MIA) Crystallographic point groups (cont.)
Overview of crystallographic point groups.
Evening session (optional)
Discussions, questions and comments on the lecture and exercise material of the day

2nd Day (2 July 2019) morning session
08:15 - 10:15 (BS) Space groups - general introduction:
Periodic structure of the crystalline matter: lattices and their basic properties. Structure of space groups: point groups of space groups. Coset decomposition of the space group with respect to its translation subgroup.
10:35 - 12:20 (MN) Space groups and their descriptions
Hermann-Mauguin symbols of space groups. Space-group diagrams. General and special positions, site-symmetry groups, crystallographic orbits.
2nd Day (2 July 2019) afternoon session
13:45 - 14:45 (BS) Space groups and their descriptions (cont.):
Symmorphic and non-symmorphic space groups. Classification of space groups.
14:45 - 15:45 (MN) Space groups and their descriptions (cont.):
Exercises on orthogonal projections of space groups 1.
16:15 - 18:30 (MN) Space groups and their descriptions (cont.):
Exercises on orthogonal projections of space groups 2.
Evening session (optional)
Discussions, questions and comments on the lecture and exercise material of the day

3rd Day (3 July 2019) morning session
08:15 - 10:15 (MIA) Space groups and their description in International Tables for Crystallography
Vol. A: Transformations of the coordinate systems: change of origin and orientation. Conventional and non-conventional descriptions of space groups; ITA-settings.
Hands-on session: Computer databases and access tools of the Bilbao Crystallographic Server(BCS) to crystallographic symmetry data for space: symmetry operations and their matrix-column presentation; geometric interpretation; general and special Wyckoff positions, site symmetry.
10:35 - 12:20 (BS) General remarks on representations
Representations of discrete groups. Equivalence of representations. Unitary representations. Invariant subspaces and reducibility. Theorem of orthogonality. Characters of representations and character tables. Subduced and direct-product representations.
3rd Day (3 July 2019) afternoon session
13:45 - 15: 45 (BS) Representations of point groups
Representations of Abelian groups: cyclic groups and direct products of cyclic groups. Character tables of representations of point groups.
16:15 - 18:30 (MIA) Representations of groups: further developments
Double point groups and their representations. Subduced and induced representations. Direct product representations. Representations of a group in terms of the representations of an invariant subgroup.
Evening session (optional)
Discussions, questions and comments on the lecture and exercise material of the day

4th Day (4 July 2019) morning session
08:15 - 10:15 (MIA) Representations of space groups
Representation of the translation group. Symmetry in reciprocal space: Brillouin zones and wave-vector symmetry types. Star of a representation. Little groups and small representations. Full-group representations of space groups
10:35 - 12:20 (MIA) Representations of space groups (cont.)
Double space groups and their representations. Subduced and direct-product representations of space groups. Compatibility relations and electronic energy bands. Site-symmetry approach and band representations.
13:45 - 14:45 (MIA) Hands-on session with the computer tools of BCS for point- and space-group representations
4th Day (4 July 2019) afternoon session
14:45 - 19:00 Excursion in Shanghai
19:00 - 22:00 dinner in the town

5th Day (5 July 2019) morning session
08:15 - 10:15 (AB) Topological insulators: general overview (strong/weak TI, TCI, Chern bands).
Bulk-boundary correspondence. First generation topological materials. Automated materials search via topological quantum chemistry.
10:35 -12:20 (JC) Bloch's theorem and energy bands.
Symmetry approach to tight-binding basis. Bands transform as space group irreps. Symmetry constraints. How to find little co-group irreps.
5th Day (5 July 2019) afternoon session
13:45 - 15:45 (BB) Berry phase and Zak phase.
Polarization. Wannier functions.
16:15 - 18:30 (BB, JC) Hands-on session: topological insulators with PythTB
(Band structure, edge states, hybrid Wannier centers.)
Poster session
18:30 - 21:00 Poster session (with buffet and drink) ICQMS Building E 1/F

6th Day (6 July 2019) morning session
08:15 - 10:15 (JC) Band representations are not topological.
Disconnected EBRs are topological. Quasi band representations. Symmetry indicators.
10:35 -12:20 (BB) Absence of localized Wannier functions, Symmetry constraints on Berry phase and Wilson loops.
Correspondence between Wilson loop spectrum and edge spectrum.
6th Day (6 July 2019) afternoon session
13:45 - 15:45 (JC, BB, LE) Hands-on session with the computer tools of the BCS
(BANDREP, Check Topological Materials)
16:15 - 17:00 (JC, BB, LE) Hands-on session (cont.)
17:00 - 18:30 (AB) High-order topological insulators
Evening session (optional)
Discussions, questions and comments on the lecture and exercise material of the day

7th Day (7 July 2019) morning session
08:15 - 10:15 (MIA) Magnetic symmetry.
Magnetic point operations and point groups, connection to magnetic tensor properties. Magnetic space groups. Types of magnetic groups. BNS and OG settings in type IV magnetic space groups. Wyckoff positions of magnetic space groups.
10:35 - 11:20 (LE) Representations analysis vs. magnetic symmetry description of magnetic structures. Magnetic degrees of freedom.
11:30 - 12:20 (LE) Symmetry-based computational tools for magnetic crystallography in the Bilbao Crystallographic Server.
Extinction rules for magnetic space groups (MAGNEXT). Possible magnetic groups for a given gray group and a propagation vector (k_SUBGROUPSMAG) and modelization/visualization ///(MAGMODELIZE,MVISUALIZE)//
7th Day (7 July 2019) afternoon session
13:45 - 16:15 (LE) Hand-onl session: The Bilbao Crystallographic Server.
MAXMAGN, k-SUBGROUPSMAG, MAGMODELIZE. Determine possible models for a given propagation vector.
16:45 - 17:30 (LE) Database of magnetic structures on the Bilbao Crystallographic Server.
18:00 - 18:30 Final farewell remarks: end of the school

Acknowledgements: