Abstract
We introduce a method which based on Bell inequalities, to study quantum phase transitions. By using the non-linear programming, we compare two different kinds of Bell inequalities, the original Bell inequality and Clauser-Horne-Shimony-Holt (CHSH) inequality. And we find that the original Bell inequality is more accurate in detecting the Bell non-locality. By defining the maximal violation of Bell inequalities, we calculate two kinds of transitions, the one is magnetic transition in the spin- $\frac {1}{2}$ XX model and the other is topological transition in the Kitaev honeycomb model. The critical points are detected successfully. Compared with traditional methods, our method requires no prior knowledge of order parameters and it is base-free.
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