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

TL;DR

This paper establishes a formal connection between formation probabilities, full counting statistics, and emptiness formation probabilities in quadratic fermionic systems, providing exact formulas and applications to quantum spin chains with defects.

Contribution

It introduces new determinant formulas for formation probabilities in quadratic fermionic Hamiltonians and applies them to analyze defect effects in quantum spin chains.

Findings

Equivalent formulations of FP, FCS, and EFP in defect Hamiltonians

Exact determinant formulas for FP in quadratic fermionic systems

Analysis of magnetization and kink distributions in XY chains

Abstract

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. As applications of our formalism we study the statistics of the number of particles and kinks. Our conclusions can be extended also to the quantum spin chains that can be mapped to the free fermions via Jordan-Wigner (J-W) 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 and show how the dual nature of these quantities manifest itself in the distributions.