Scaling of the Formation Probabilities and Universal Boundary Entropies in the Quantum XY Spin Chain

We calculate exactly the probability to find the ground state of the XY chain in a given spin configuration in the transverse $\sigma^z$-basis. By determining finite-volume corrections to the probabilities for a wide variety of configurations, we obtain the universal Boundary Entropy at the critical point. The latter is a benchmark of the underlying Boundary Conformal Field Theory characterizing each quantum state. To determine the scaling of the probabilities, we prove a theorem that expresses, in a factorized form, the eigenvalues of a sub-matrix of a circulant matrix as functions of the eigenvalues of the original matrix. Finally, the Boundary Entropies are computed by exploiting a generalization of the Euler-MacLaurin formula to non-differentiable functions. It is shown that, in some cases, the spin configuration can flow to a linear superposition of Cardy states. Our methods and tools are rather generic and can be applied to all the periodic quantum chains which map to free-fermionic Hamiltonians.