Condensed Matter Inspired Definition of K-Theory

Introduction

• Content: Part physics, part mathematics
• Focus: Work in progress, aims to generalize possibly beyond condensed matter
• Subject: Quantum Hall effect and related concepts in condensed matter physics using a mathematical framework

Graphene and Quantum Hall Effect

• Graphene: Sheets of carbon atoms in a honeycomb lattice
• Properties: Exhibits quantum Hall effect at low temperatures
• Setup:
  • Magnetic Field: Applied perpendicular to the plane
  • Electric Field: Causes current to flow in a perpendicular direction
• Hall Conductivity:
  • Quantized in units of e²/2πħ (Planck constant)
  • Integer Quantum Hall Effect: Integer quantization
  • Fractional Quantum Hall Effect: Rational numbers, more complex due to quasiparticles
• Edge Modes: Quasiparticles propagate along edges at low temperatures

Quantum Hamiltonians and Ground States

Goals

• Study: Ground states and their space and homotopy types
• Example: Transverse Field Ising Model
  • Hilbert space: C₂ⁿ
  • Parameters: H, J
  • Point 1:
    • H > 0, J = 0
    • Ground state: All spins up
    • Unique ground state, energy gap exists
  • Point 2:
    • H = 0, J > 0
    • Two degenerate ground states
    • Multidegenerate spectrum due to flipped spins

Local Energy Gap

• Definition: Hard to define for many-body Hamiltonians with interacting particles
• Fermions:
  • Non-interacting case solved, but interacting remains conjectural
  • Aims to define and solve for invertible phases

Free Fermions and K-Theory

• Operations:
  • Hopping from atom to atom
  • Consideration of non-conserved charge (Majorana fermions)
• Clifford Algebra: Clifford of 2n over complex numbers
• Hilbert Space Construction:
  • Use of creation and annihilation operators and their relationships