Math 4 Wisdom. "Mathematics for Wisdom" by Andrius Kulikauskas.

Bott periodicity, Bott periodicity concepts, Super division algebras, Condensed matter

Understand Bott periodicity by understanding the Topological Invariants

Is fractionalization of quanta (charge) a division of everything?
Look for connections between topological insulators, Stokes theorem and Noether's theorem.

读物

Periodic Table of Topological Invariants
M. Z. Hasan, C. L. Kane. Topological Insulators. Includes periodic table (the tenfold way).
Shinsei Ryu, Andreas Schnyder, Akira Furusaki, Andreas Ludwig. Topological insulators and superconductors: ten-fold way and dimensional hierarchy.
Michel Fruchart, David Carpentier. An Introduction to Topological Insulators
Michael Stone, Ching-Kai Chiu, Abhishek Roy. Symmetries, Dimensions, and Topological Insulators: the mechanism behind the face of the Bott clock.
Michael Stone. Gamma matrices, Majorana fermions, and discrete symmetries in Minkowski and Euclidean signature. Video
Alexei Kitaev. Periodic table for topological insulators and superconductors. This paper in 2009 showed the role of Bott periodicity in condensed matter research: Gapped phases of noninteracting fermions, with and without charge conservation and time-reversal symmetry, are classified using Bott periodicity. The symmetry and spatial dimension determines a general universality class, which corresponds to one of the 2 types of complex and 8 types of real Clifford algebras. The phases within a given class are further characterized by a topological invariant, an element of some Abelian group that can be {$\mathbb{0}$}, {$\mathbb{Z}$}, or {$\mathbb{Z}_2$}.
Shinsei Ryu, Andreas Schnyder, Akira Furusaki, Andreas Ludwig. Topological insulators and superconductors: ten-fold way and dimensional hierarchy. The paper on the tenfold way from 2010, which came after the paper in 2009 by Kitaev regarding Bott periodicity: We demonstrate how topological insulators (superconductors) in different dimensions and different classes can be related via dimensional reduction by compactifying one or more spatial dimensions (in Kaluza-Klein-like fashion). For Z-topological insulators (superconductors) this proceeds by descending by one dimension at a time into a different class. The Z_2-topological insulators (superconductors), on the other hand, are shown to be lower-dimensional descendants of parent Z-topological insulators in the same class, from which they inherit their topological properties. The 8-fold periodicity in dimension d that exists for topological insulators (superconductors) with Hamiltonians satisfying at least one reality condition (arising from time-reversal or charge-conjugation/particle-hole symmetries) is a reflection of the 8-fold periodicity of the spinor representations of the orthogonal groups SO(N) (a form of Bott periodicity).
See 29:11 Zirnbauer

影片

Expositions and videos

Periodic table of topological insulators

Topological insulators

C. Kane. Topological insulators I. 2016.
Charles Kane will give a talk in February, 2021 on topological insulators.

Altland–Zirnbauer classes of random matrices

Martin Zirnbauer. Bott periodicity and the "Periodic Table" of topological insulators and superconductors.

Condensed matter physics

School on Condensed Matter Physics. Schedule.
Condensed matter physics
Kondensuotų medžiagų fizika Alfonsas Grigonis, Ignas Požėla, Alfredas Balandis.

Concepts

Xiao-Gang Wen: Topological order is nothing but the pattern of long range entanglements. This led to a notion of symmetry protected topological (SPT) order (short-range entangled states with symmetry) and its description by group cohomology of the symmetry group (2011). The notion of SPT order generalizes the notion of topological insulator to interacting cases.
- Liang Kong, Tian Lan, Xiao-Gang Wen, Zhi-Hao Zhang, Hao Zheng. Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry
Wigner's Theorem
- Tobias Osborne. Proves Wigner's Theorem.
- Fundamental Theorem of Projective Geometry
- Unitary operators and anti-unitary operators.
Bulk-boundary correspondence. Chern number n = number of chiral edge nodes.
CPT symmetry
- Particle hole symmetry (Charge conjugation)
- Parity - axis matched with time. This acts like time - one dimension.
- Time reversal symmetry.
Chern number (TKNN) number is an integer how the state wraps around.
Topological order Topological order is an order in the zero-temperature phase of matter (also known as quantum matter).
Topological entropy in physics
维基百科: Second quantization (Canonical quantization) Important for the tenfold way in condensed matter research.
An insulator is a vector bundle of annihilation space of momentum space.
A pseudo-symmetry turns an annihilation operator into a creation operator. It acts on a ground state to yield an excited state. In that sense, it is not a true symmetry, which would take an annihilation operator to another annihilation operator.

Gapped phases, topological phases

Kitaev: Bott periodicity classifies gapped phases of noninteracting fermions, with and without charge conservation and time-reversal symmetry.
Kitaev: We do not look for analytic formulas for topological numbers, but rather enumerate all possible phases.

Hamiltonian

Kitaev: Two Hamiltonians belong to the same phase if they can be continuously transformed one to the otherwhile maintaining the energy gap or localization.

Mass term

A Hamiltonian around a given point may be represented (in some non-canonical way) by a mass term that anticommutes with a certain Dirac operator; the problem is thus reduced to the classification of such mass terms.

Ideas

Quantum matter (zero-temperature phase of matter) is like stopping the mind so that it does not undergo reflection. Topological order then describes the possible state, in particular, the pattern of long-range entanglement.
Insulators express the key way of figuring things out in physics. They function like Faraday's pail, they isolate a system.
Topological gap - padalinimais grindžia.
Information stored in two places. Quantum computing: How to keep a system from accidentally measuring itself. Relate this to preservation of the gap boundary as in a topological insulatior, a division of everything.
Spin. Kramer's theorem {$T^2=–1$}. Basis for topological symmetry? 40:00 in Lane video.
Measurement occurs when 8=0. This collapse is the collapse of the wave function, and also the collapse relevant for Bott periodicity.

Facts

Magnetic system breaks time reversal symmetry.

People

Xiao-Gang Wen. Research interests. Since 2005, Wen became interested in the mathematical foundations of topological order, which turns out to be higher category and group cohomology theories, and discovered a new class of topological states of matter – symmetry protected topological order in 2011. Wen is also applying the theory of topological order to obtain an unification of elementary particles and interactions in term of qubits (i.e. it from qubit).
Alexei Kitaev.
Conference: Modern Aspects of Symmetry.

Notes

Charlie Kane. Introduction to Topological Band Theory video: Topological Phase Theory - think of it as dividing the indivisible 39:00