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I am currently studying the following paper http://iopscience.iop.org/article/10.1088/1367-2630/14/6/065012/meta (Phase diagram of the extended Bose- Hubbard model). In the paper, they define two energy gaps namely the charge energy gap and the neutral energy gap. The charge energy gap is defined as $E_c=E^{0}(N+1)+E^{0}(N-1)-2E^{0}(N)$ where $N$ is the number of bosons and $E^{0}$ is the ground state energy of the system. The neutral energy gap is defined as $E_n = E^{1}(N)-E^{0}(N)$ where $E^{1}(N)$ is the energy of the first excited state in the system. The charge energy gap shows non-monotonic behavior only at the Mott insulator -Haldane insulator transition point whereas the neutral energy gap shows non-monotonic behavior at both the Mott insulator-Haldane insulator transition point and the Haldane insulator-Density wave transition point. I don't understand why this occurs?

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