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

Abstract

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.

v2: 17 pages, 5 figures. Minor revisions and clarifications, references added. Accepted for publication in JSTAT

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• Rényi entropy of a line in two-dimensional Ising models
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• Open topological defects and boundary RG flows
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• Gauging or extending bulk and boundary conformal field theories: Application to bulk and domain wall problem in topological matter and their descriptions by (mock) modular covariant
• Efficient Representation of Gaussian Fermionic Pure States in Non-Computational Bases
• Shannon entropy in quasiparticle states of quantum chains