# Charlie Kane Introduction to Topological Band Theory

Topological Matter School2022-09-03

5K views|1 years ago

💫 Short Summary

The video delves into topological superconductivity, band theory, Berry phase, and topological states in quantum mechanics. It discusses symmetry, topology, and the application of these concepts in understanding electronic phases of matter. The importance of chiral symmetry, zero modes, and fractionalized charge is highlighted. The video explores the implications of topological band theory in one-dimensional models and the concept of quantized thermal and electrical conductance. It concludes with a discussion on chiral fermions and the chiral anomaly. The audience applauds the speaker for their insightful presentation.

✨ Highlights

📊 Transcript

✦

Topological superconductivity and its relation to electronic phases of matter.

00:29Symmetry and topology are crucial for understanding matter, with topology emphasizing what remains constant under continuous deformations.

Adiabatic continuity in quantum mechanics is utilized to classify gapped Quantum States and transitions between them.

Many particle Quantum states are complex, making it difficult to extract topological possibilities.

The speaker intends to concentrate on phases of matter describable using single-particle quantum mechanics.

✦

Application of Topology to Band Theory of Solids

04:27The segment introduces the concept of Hamiltonian as a function of momentum in the Broan Zone, posing a quantum mechanics problem.

Focus is on topological band phenomena such as quantum Hall states, topological insulators, and superconductivity.

Importance of expanding Band Theory to include bogalubav Degen Hamiltonians and classify band structures topologically is emphasized.

The segment lays the groundwork for further exploration of topological Band Theory and its applications in theoretical physics.

✦

Summary of Block's Theorem in lattice systems.

07:44Translation symmetry in a lattice leads to operators that commute with the Hamiltonian, allowing classification of eigenstates by eigenvalues.

Crystal momentum within a Brillouin Zone is defined by the Block Hamiltonian, parameterized by K, which has eigenvectors and eigenvalues forming a band structure.

A gauge transformation can change the phase of a wave function based on K values, introducing the concept of the berry phase.

The connection quantity, defined based on block wave functions, behaves like a vector potential under gauge transformations, and changes in phase can be observed through closed loops or interference.

✦

Discussion of Berry phase in quantum mechanics and its significance in Band Theory.

13:15Berry phase is defined as half of the solid angle swept out when going around a closed loop.

Crucial in understanding topological Band Theory and its physical implications.

Relationship between Berry phase and electric polarization emphasized.

Importance of considering periodic boundary conditions in Band Theory calculations.

✦

Calculating electric polarization in band structures using Berry phase.

17:04Localized states are needed to compute dipole moments and Fourier transforms define Wannier states.

Berry phase calculation around a circle in a one-dimensional Brillouin zone determines electric polarization.

Understanding gauge invariance is crucial for accurate measurements of vector potential and loop integrals.

✦

Ambiguity in electric polarization and its connection to the Berry phase.

19:06Electric polarization is only defined modulo the electric charge due to the ability to add electrons.

Berry phase undergoes a change with a gauge transformation, leading to an ambiguity in polarization.

Special loops around the Brillouin Zone affect the polarization change, and the Berry phase change is well-defined.

Understanding these concepts is important in physics.

✦

Overview of the Sue Schreffer-Hegar model and its implications on material properties.

22:55The model is a one-dimensional tight binding model with alternating hopping, resulting in either a metal or an insulator depending on bond strengths.

Analysis of the model in real space and momentum space, including defining a block hamiltonian and determining the Berry phase.

The Berry phase distinguishes between insulating states based on polarization, showcasing unique characteristics and their relationship to unit cell positioning.

✦

Overview of topological states and symmetries in a one-dimensional model.

26:50Chiral symmetry in the model leads to eigenstates coming in pairs at positive and negative energies.

This symmetry results in a reflection symmetric spectrum around zero energy, known as particle-hole symmetry.

The presence of chiral symmetry dictates that DZ must equal zero, resulting in an integer winding number and a topological invariant.

The model symmetry may not directly apply to physical systems like polyacetylene due to additional factors like second neighbor hopping and non-particle-hole symmetric spectra.

✦

Symmetry in physics, particularly in relation to polyacetylene and topological Band Theory.

29:32Reflection symmetry in polyacetylene imposes constraints on the system, affecting the D Vector and topological classes.

Topological Band Theory shows that boundaries between insulators have topological boundary modes, as seen in the SSH model.

Domain walls between phases in topological systems lead to unique atom behavior and zero modes.

Understanding symmetries and their effects is essential for studying the behavior of physical systems.

✦

Discussion on zero modes in a system with weak bonds turned up.

33:56The zero mode is topologically protected even when weak and strong bonds are almost equal.

The model has a chiral symmetry leading to a particle-hole symmetric spectrum.

Jakiv and Rebby identified the protected zero mode.

A homework problem suggests demonstrating a specific eigenstate as an exact zero energy eigenstate of the Hamiltonian by plugging it into the time-independent Schrodinger equation.

✦

Overview of topological band theory and fractionalized charge concept.

36:57Explanation of the relationship between single particle and many body levels in quantum mechanics with valence band and ground state.

Delving into the occupation of zero modes by electrons and symmetric spectrum.

Highlighting the quantization of electric charge in units of E and fractionalization of charge at e over 2.

Exploration of splitting indivisible entities through topological phase insertions, demonstrating charge fractionalization.

✦

The concept of energy gaps in one-dimensional conductors is discussed in the video segment.

41:53Coupling right and left movers on neighboring wires can open gaps, defining a churn insulator.

The boundary between insulators separates right and left movers, similar to the SSH model.

Understanding insulators involves single particle and many-body levels, with a focus on conduction and valence bands separated by an energy gap.

States connecting conduction and valence bands with positive group velocity are emphasized.

✦

The concept of a chiral Fermi liquid and its topologically protected features are discussed.

44:20A many-body picture of single-particle states filled up to the Fermi energy is defined, leading to a many-body ground state with low-energy excitations.

The chiral Fermi liquid is explained in terms of a conformal field theory, showcasing quantized properties like electrical conductance and central charge.

The segment explores the chiral anomaly and its connection to the conservation of electric charge in the presence of an electric field.

Emphasis is placed on the relevance of the chiral anomaly in the context of the quantum Hall effect.

✦

Discussion on quantized thermal conductance and its impact on defining units of measurement like the kilogram.

48:15Experiments measuring quantized hall conductance in the 1980s influenced the current definition of the kilogram.

Recent experiments on quantized thermal conductance in the last five years show advancements in the field.

Topological properties and Berry phase in two-dimensional structures are highlighted for their importance in understanding physical phenomena.

✦

Discussion on the calculation of Berry curvature and its physical significance in terms of Chern number and quantized Hall conductance.

51:09Explanation of the connection between threading flux, electric fields, and hall current flows leading to quantized polarization and the quantized Hall effect.

Highlighting the relationship between the Chern number and quantized hall conductance.

Mention of the Kubo formula for a more complex calculation.

Conclusion with a mention of continuing the topic the next day and a brief Q&A session.

✦

Discussion on chiral fermions, edge states, chiral anomaly, and chiral central charge.

56:41Exploring the concept of a chiral Fermi liquid and comparing it to conventional Fermi liquids and conformal field theory.

Delving into the low energy spectrum of free fermions and interacting systems, providing an alternative description through conformal field theory.

Bridging the gap between non-interacting and interacting cases for a generalized understanding of interacting ground states and low energy excitations.

Touching on parameter descriptions and theories that typically exhibit divergences.

✦

Conclusion of the segment.

58:45The speaker expressed gratitude towards the audience.

The audience responded with applause.

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