Tensor-network renormalization approach to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi></mml:math> -state clock model

We simulate the phase diagram and critical behavior of $q$-state clock model on square lattice by using state-of-the-art loop optimization for tensor-network renormalization (loop-TNR) algorithm. The two transition points $q\ensuremath{\ge}5$ are determined with very high accuracy. Furthermore, computing conformal scaling dimensions both points, we able to determine radius $R$ compactified boson theories at precision. In particular, higher temperature point is precisely same as one predicted Berezinskii-Kosterlitz-Thouless (BKT) transition. Moreover, find that fixed-point tensors also converge approximately large enough $q$ corresponding operator product expansion (OPE) coefficient theory can be read out directly from tensor.