Formation probabilities and statistics of observables as defect problems in the free fermions and the quantum spin chains

Understand

We show that the computation of formation probabilities (FP) in the configuration basis and the full counting statistics (FCS) of observables in the quadratic fermionic Hamiltonians are equivalent to the calculation of emptiness formation probability (EFP) in the Hamiltonian with a defect.

• In particular, we first show that the FP of finding a particular configuration in the ground state is equivalent to the EFP of the ground state of the quadratic Hamiltonian with a defect.
• Then, we show that the probability of finding a particular value for any quadratic observable is equivalent to a FP problem and ultimately leads to the calculation of EFP in the ground state of a Hamiltonian with a defect.
• We provide new exact determinant formulas for the FP in the generic quadratic fermionic Hamiltonians.