Free fermion Hamiltonians, namely, Hamiltonians with quadratic fermion terms, are analytically solvable and their ground and thermal states fall into the class of states called Gaussian states. I introduce the basics of free fermion systems from the perspective of quantum information theory. This lecture is basically for quantum information researchers, so no knowledge on condensed matter theory is assumed. I'll derive (almost) all mathematical formulas so that the lecture is essentially self-contained. The lecture will be given on the blackboard.

Wed 28 Nov 10:30 ~ 12:00

Reference : cond-mat/0212631, cond-mat/0206196, Nucl. Phys. 15, 89 (1960)

• Definition of fermionic Gaussian states

• Examples of free fermion models

• Wick's theorem

Thu 29 Nov 10:30 ~ 12:00

Reference : quant-ph/0108010, quant-ph/0404180

• Majorana representation

• Grassmann algebra

• Entanglement entropy

Fri 30 Nov 9:00 ~ 10:30

Reference : quant-ph/0503219

• Logarithmic violation of the area law in gapless systems

• Entanglement area law in gapped systems

Fri 30 Nov 14:00 ~ 15:30

Reference : cond-mat/0010440, arXiv:0909.2654, arXiv:1706.01637

• Kitaev chain

• Band topology

• Entanglement spectrum

• Concurrence