Introduction E9F5FC

Questions FFFFC0

• View
• Edit
• History
• Print

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.
• Freeman J. Dyson. The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics. J. Math. Phys. 3, 1199–1215 (1962)
• 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 {$\mathbb{Z}_2$}.
  • Cameron Krulewski. The K-Theoretic Classification of Topological Materials Senior thesis explains low dimensional examples of Kitaev's 2009 paper. Provides mathematical and physical motivation for why it works. Discusses Kitaev's proposal:
    • The possible phases of gapped, free-fermion models in {$d$} dimensions and with {$p$} negative symmetries are classified by {$\widetilde{KO}^{−p+d+2}(pt) = \pi_0(R_{p−d−2\mod 8})$} or by {$\widetilde{K}^{−p+d+1}(pt) = \pi_0(C_{p−d−1 \mod 2})$} where {$R_q$} and {$C_q$} denote spaces of operators. The choice of {$R_q$} versus {$C_q$} is also determined by the symmetry properties of the system.
• 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

• John Baez and James Dolan on Bott Periodicity John Baez on the tenfold way.
• Greg Moore. Quantum Symmetries and the 10-Fold Ways (1 of 2), (2 of 2), Greg Moore
  • New Insights Into Quantum Matter
  • Robbert Dijkgraaf. Introduction to Topological and Conformal Field Theory (1 of 2)
• Martin Zirnbauer. Bott periodicity and the "Periodic Table" of topological insulators and superconductors
• Xiao-Gang Wen. YouTube channel
  • Xiao-Gang Wen 2020/11/17: Mathematics of topological order
  • Xiao-Gang Wen: Emergent symmetry -- a new way to understand gapless states
  • Xiao-Gang Wen 2020/11/16: Topological order -- pattern of many-body long range entanglement
  • 2018-11-06 Xiao-Gang Wen: Classification of 3+1D Topological orders
  • 2022/05/26 X-G Wen: A unification of elementary particles and forces by quantum information
  • Xiao-Gang Wen. The Second Quantum Revolution: A Unification of Matter and Information
  • Playlist: Introduction of Topological Order
  • Playlist: Mathematics of Topological Order
  • Playlist: Physics of Topological Order
  • Playlist: Foundation Theory of Topological Order
• Vijay Shenoy. The Tenfold Way

Expositions and videos

• Vijay Shenoy
  • 影片: Vijay Shenoy. Tenfold Way. (1 of 4) Slides
  • 影片: Vijay Shenoy. Tenfold Way. (2 of 4) Slides
  • 影片: Vijay Shenoy. Tenfold Way. (3 of 4) Slides
  • 影片: Vijay Shenoy. Tenfold Way. (4 of 4)
  • 影片: School on Current Frontiers in Condensed Matter Research
  • Untwisting twisted matter
  • Towards understanding topological insulators
  • The tenfold way redux: Fermionic systems with N-body interactions

Periodic table of topological insulators

• Course on topology in condensed matter
  • Shinsei Ryu. Periodic table of topological insulators and superconductors summary.
  • Shinsei Ryu. Periodic table of topological insulators and superconductors intro.
• Xiao-Gang Wen. Classification of topological matter.
• https://topocondmat.org/w8_general/classification.html
• Shinsei Ryu. General classification of topological insulators and superconductors.

Topological insulators

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

Homotopy classification and K-theory classification

• Physics Stack Exchange. Topological insulators. Why K-theory classification rather than homotopy classification?

Altland–Zirnbauer classes of random matrices

• There are ten discrete symmetry classes of topological insulators and superconductors, corresponding to the ten 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.

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.

• 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.

CPT symmetry

• CPT symmetry The CPT transformation turns our universe into its "mirror image" and vice versa. A "mirror-image" of our universe — with all objects having their positions reflected through an arbitrary point (corresponding to a parity inversion), all momenta reversed (corresponding to a time inversion) and with all matter replaced by antimatter (corresponding to a charge inversion) — would evolve under exactly our physical laws.

Charge conjugation

• This relates to simplexes.

Time reversal

Parity transformation

• This relates to the axes of coordinate systems.

Chern class

• Chern class is a characteristic class associated with complex vector bundles. Chern classes arise via homotopy theory which provides a mapping associated with a vector bundle to a classifying space, an infinite Grassmannian. The Chern classes of a complex vector bundle V can be defined as the pullback of the Chern classes of the universal bundle. In turn, these universal Chern classes can be explicitly written down in terms of Schubert cycles. The intuitive meaning of the Chern class concerns 'required zeroes' of a section of a vector bundle: for example the theorem saying one can't comb a hairy ball flat (hairy ball theorem).

Feynman–Stueckelberg interpretation

• Antiparticle An antiparticle is interpreted as the corresponding particle traveling backwards in time.

Principle of locality

Spin-statistics theorem

• Spin-statistics theorem An elementary explanation for the spin-statistics theorem cannot be given despite the fact that the theorem is so simple to state. In the Feynman Lectures on Physics, Richard Feynman said that this probably means that we do not have a complete understanding of the fundamental principle involved.

Klein group modifiers

• Klein four-group

Forbidden zone

Second quantization

Time evolution operator

Symmetric spaces

• Symmetric space

Kitaev's table

Grassmanian manifold

Reducible representations

Hamiltonian

Condensed matter research

Lattice

Orbitals

Base space

Compactification

Insulator

Ground state

Continuum problem

Lattice problem

Bands

Band dispersion`

Gap closing and Gap crossing

Gauge invariant

Wave function of Hamiltonian

Topological order

• Topological order Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized non-Abelian geometric phases of degenerate ground states. Microscopically, topological orders correspond to patterns of long-range quantum entanglement. States with different topological orders (or different patterns of long range entanglements) cannot change into each other without a phase transition.
• Guo Chuan Thiang: In many-body physics, one studies strongly interacting particle systems whose description requires techniques from quantum field theory, and which exhibit what is sometimes called topological order. These systems are less well-understood, and we will not say anything further about them.

Lattice gauge theory

Topological {$\mathbb{Z}_2$} invariant

• Ralph M. Kaufmann, Dan Li, Birgit Wehefritz–Kaufmann. Topological insulators and K-theory.

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