The phase diagram of the one-dimensional quantum sine-Gordon system ($β^{2}=4π)$ with a linear spatial modulation

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The one-dimensional quantum sine-Gordon system with a linear spatial modulation is investigated in a special case, $β^{2}$ =4$π$. The model is tranformed into a massive Thirring model and then is exactly diagonalized, the energy spetrum of the model is obtained. Our result clearly demonstrates that cancelling the cosine term without any considering is unadvisable.
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