Recently, entropy corrections on nonorientable manifolds such as the Klein bottle are proposed as a universal characterization of critical systems with an emergent conformal field theory (CFT). We show that entropy correction on the Klein bottle can be interpreted as a boundary effect via transforming the Klein bottle into an orientable manifold with nonlocal boundary interactions. The interpretation reveals the conceptual connection of the Klein bottle entropy with the celebrated Affleck-Ludwig entropy in boundary CFT. We propose a generic scheme to extract these universal boundary entropies from quantum Monte Carlo calculation of partition function ratios in lattice models. Our numerical results on the Affleck-Ludwig entropy and Klein bottle entropy for the q-state quantum Potts chains with q = 2, 3 show excellent agreement with the CFT predictions. For the quantum Potts chain with q = 4, the Klein bottle entropy slightly deviates from the CFT prediction, which is possibly due to marginally irrelevant terms in the low-energy effective theory.
We extend the scope of the Klein bottle entropy, originally introduced by [Tu, Phys. Rev. Lett. 119, 261603 (2017)] in the rational conformal field theory (CFT), to the compactified boson CFT, which are relevant to the studies of Luttinger liquids. We first review the Klein bottle entropy in rational CFT and discuss details of how to extract the Klein bottle entropy from lattice models using the example of the transverse field Ising model. We then go beyond the scope of rational CFT and study the Klein bottle entropy ln g in the compactified boson CFT, which turns out to have a straightforward relation to the compactification radius R, ln g = ln R. This relation indicates a convenient and efficient method to extract the Luttinger parameter from lattice model calculations. Our numerical results on the Klein bottle entropy in the spin-1/2 XXZ chain show excellent agreement with the CFT predictions, up to some small deviations near the isotropic point, which we attribute to the marginally irrelevant terms. For the S = 1 XXZ chain that cannot be exactly solved, our numerical results provide an accurate numerical determination of the Luttinger parameter in this model. arXiv:1805.01300v2 [cond-mat.str-el]
Experimental study on the LF refining of aluminum killed cold heading steel shows that calcium content in the molten steel increased to about 0.0010% at the end of refining, and the aluminum deoxized products were transformed from Al 2 O 3 to the complex inclusions CaO-MgO-Al 2 O 3 with lower melting point by the high basicity, high Al 2 O 3 , and strong deoxidizing slag. The inclusions are in liquid state and can be easily floated up during LF refining and continuous casting. The total oxygen content of the steel falls to about 0.0020%. The experimental technology uses only 50 m calcium wire to the 80-t heat or even without calcium treatment. As compared to the traditional technology with higher amount of calcium for treatment, which forms CaS and CaO-MgO-Al 2 O 3 inclusions with high melting point, the experimental technology improves the castability and reduces the manufacturing cost.
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