Logarithmic corrections to the free energy from sharp corners with angle $2\pi$

Abstract

We study subleading corrections to the corner free energy in classical two-dimensional critical systems, focusing on a generic boundary perturbation by the stress-tensor of the underlying conformal field theory (CFT). In the particular case of an angle $2\pi$, we find that there is an unusual correction of the form $L^{-1}\log L$, where $L$ is a typical length scale in the system. This correction also affects the one-point function of an operator near the corner. The prefactor can be seen as semi-universal, in the sense that it depends on a \emph{single} non-universal quantity, the extrapolation length. Once this ultraviolet cutoff is known, the term is entirely fixed by the geometry of the system, and the central charge of the CFT. Such a corner appears for example in the bipartite fidelity of a one-dimensional quantum system at criticality, which allows for several numerical checks in free fermions systems. We also present an exact result in the XX and Ising chains that confirms this analysis. Finally, we consider applications to the time evolution of the (logarithmic) Loschmidt echo and the entanglement entropy following a local quantum quench. Due to subtle issues in analytic continuation, we find that the logarithmic term in imaginary time transforms into a time-dependent $L^{-2}$ correction for the entanglement entropy, and a $L^{-2}\log L$ term for the Loschmidt echo.

44 pages, 21 figures. v2: minor corrections

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• Entanglement asymmetry in CFT and its relation to non-topological defects
• Geometric mutual information at classical critical points
• Formation probabilities and statistics of observables as defect problems in the free fermions and the quantum spin chains
• Exact solution of the $2d$ dimer model: Corner free energy, correlation functions and combinatorics
• Entanglement asymmetry in the critical XXZ spin chain
• Entanglement entropy after selective measurements in quantum chains
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• Topological and Geometric Universal Thermodynamics in Conformal Field Theory
• Absence of logarithmic enhancement in the entanglement scaling of free fermions on folded cubes
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• Renormalization group flows in one-dimensional lattice models: impurity scaling, umklapp scattering and the orthogonality catastrophe
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• The open XXZ chain at $\Delta =-1/2$ and the boundary quantum Knizhnik-Zamolodchikov equations
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• Bubble quenches in the AdS/BCFT model