Formation probabilities and statistics of observables as defect problems in free fermions and quantum spin chains
M. N. Najafi, M. A. Rajabpour
Abstract
We show that the computation of formation probabilities (FP's) in the configuration basis and the full counting statistics of observables in 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 exact determinant formulas for the FP in generic quadratic fermionic Hamiltonians. For applications of our formalism we study the statistics of the number of particles and kinks. Our conclusions can be extended also to quantum spin chains, which can be mapped to free fermions via Jordan-Wigner transformation. In particular, we provide an exact solution to the problem of the transverse field $XY$ chain with a staggered line defect. We also study the distribution of magnetization and kinks in the transverse field $XY$ chain numerically and show how the dual nature of these quantities manifests itself in the distributions.