Violation of Bell inequality in perfect translation invariant systems

Bell inequalities and nonlocality have been widely studied in one-dimensional quantum systems. As a kind of quantum correlation, it is expected that bipartite nonlocaity should be present in quantum systems, just as bipartite entanglement does. Surprisingly, for various models, two-qubit states do not violate Bell inequalities, i.e., they are local. Recently, it is realized that the results are related to the monogamy trade-off obeyed by bipartite Bell correlations, thus it is believed that for general translation invariant systems, two-qubit states should not violate the Bell inequality(Oliveira, EPL 100, 60004 (2012)). In this report, we will demonstrate that in perfect translation invariant systems, the Bell inequality can be violated. A nontrivial model is constructed to confirm the conclusion.