Higher gauge fields and fermions in lattice models

Introduction and summary
Basic notions
Geometric setup and field configurations
Degrees of freedom and holonomies
Gauge and electric transformations
Interesting field configurations — examples
Hamiltonian models
Construction
An explicit example
Symmetries
Vacuum states
A peek at dynamics
Kernel and cokernel of partial
Twisted cohomology
Classifying spaces
Classifying spaces of groups
Classifying spaces of crossed modules
Postnikov class
Homomorphisms and weak equivalences
Construction of classifying spaces
Introduction
Description of the model
Degrees of freedom, action and gauge freedom
Nonlocal order parameters and symmetries
Reduction of dynamics to simpler models
Phase diagram proposal for D = 4
Monte Carlo study
Simulation method
Numerical results for local observables
Numerical results for non-local observables
Summary and conclusions
Non-spherical Wilson surfaces
Comparison with continuous theories
Introduction
Geometric setup
Fermions — generators and relations
Gamma model
Definition of the model
Choice of a representation
Modified constraints and Z(2) gauge fields
Example: toroidal geometries
Example: quadratic fermionic hamiltonians
Deformed Z(2) gauge theories
Gauge invariant operators
Classification of Gauss' operators
Local formulations
Duality with higher gauge theory
Summary and outlook
Canonical transformations for Ising degrees of freedom
Graphs with vertices of odd degree
Bosonization of Majorana modes and edge states
Abstract
I Introduction
II The bosonization method
III Examples
A Honeycomb lattice and Kitaev's model
B Decagonal lattice
C Hubbard model
IV Boundary effects
A Rectangular lattice with a boundary
B Comment about topological phases
V Euclidean representation of unconstrained ``Majorana spins"
A Basic idea and the (1+1)-dimensional example
1 YY terms - the phases.
B (2+1)-dimensional system
C The sign problem
1 (1+1)-dimensions
2 (2+1)-dimensions
3 The sign problem - summary
VI Conclusions and outlook
Acknowledgments
References