We accurately simulate the phase diagram and critical behavior of the
q-state clock model on the square lattice by using the state-of-the-art loop optimization for tensor network renormalzation(loop-TNR) algorithm. The two phase transition points for
q≥5 are determined with very high accuracy. Furthermore, by computing the conformal scaling dimensions, we are able to accurately determine the compactification radius
R of the compactified boson theories at both phase transition points. In particular, the compactification radius
R at high-temperature critical point is precisely the same as the predicted
R for Berezinskii-Kosterlitz-Thouless (BKT) transition. Moreover, we find that the fixed point tensors at high-temperature critical point also converge(up to numerical errors) to the same one for large enough
q and the corresponding operator product expansion(OPE) coefficient of the compactified boson theory can also be read out directly from the fixed point tensor.
We simulate the phase diagram and critical behavior of the
q-state clock model on the square lattice by using the state-of-the-art loop optimization for tensor-network renormalization (loop-TNR) algorithm. The two phase transition points for $q\ensuremath{\ge}5$ are determined with very high accuracy. Furthermore, by computing the conformal scaling dimensions for both transition points, we are able to determine the radius
R of the compactified boson theories at both transition points with high precision. In particular, the radius
R at higher temperature phase transition point is precisely the same as the one predicted by Berezinskii-Kosterlitz-Thouless (BKT) transition. Moreover, we find that the fixed-point tensors at higher temperature transition point also converge to the same one approximately for large enough
q and the corresponding operator product expansion (OPE) coefficient of the compactified boson theory can also be read out directly from the fixed-point tensor.
The q-state clock model, sometimes called the discrete XY model, is known to show a second-order (symmetry-breaking) phase transition in two dimensions for q≤4 (q=2 corresponds to the Ising model). On the other hand, the q→∞ limit of the model corresponds to the XY model, which shows the infinite order (non-symmetry-breaking) Berezinskii-Kosterlitz-Thouless (BKT) phase transition in two dimensions. Interestingly, the two-dimensional clock model with q≥5 is predicted to show three different phases and two associated phase transitions. There are varying opinions about the actual characters of phases and the associated transitions. In this work, we develop the basic and higher-order mean-field (MF) theories to study the q-state clock model systematically. Our MF calculations reaffirm that, for large q, there are three phases: (broken) Z_{q} symmetric ferromagnetic phase at the low temperature, emergent U(1) symmetric BKT phase at the intermediate temperature, and paramagnetic (disordered) phase at the high temperature. The phase transition at the higher temperature is found to be of the BKT type, and the other transition at the lower temperature is argued to be a large-order spontaneous symmetry-breaking type (the largeness of transition order yields the possibility of having some of the numerical characteristics of a BKT transition). The higher-order MF theory developed here better characterizes phases by estimating the spin-spin correlation between two neighbors.
The two-dimensional
q-state clock model for $q\ensuremath{\ge}5$ undergoes two Berezinskii-Kosterlitz-Thouless (BKT) phase transitions as temperature decreases. Here we report an extensive worm-type simulation of the square-lattice clock model for
q=5--9 in a pair of flow representations, from high- and low-temperature expansions, respectively. By finite-size scaling analysis of susceptibilitylike quantities, we determine the critical points with a precision improving over the existing results. Due to the dual flow representations, each point in the critical region is observed to simultaneously exhibit a pair of anomalous dimensions, which are ${\ensuremath{\eta}}_{1}=1/4$ and ${\ensuremath{\eta}}_{2}=4/{q}^{2}$ at the two BKT transitions. Further, the approximate self-dual points ${\ensuremath{\beta}}_{\mathrm{sd}}(L)$, defined by the stringent condition that the susceptibilitylike quantities in both flow representations are identical, are found to be nearly independent of system size
L and behave as ${\ensuremath{\beta}}_{\mathrm{sd}}\ensuremath{\simeq}q/2\ensuremath{\pi}$ asymptotically at the large-
q limit. The exponent $\ensuremath{\eta}$ at ${\ensuremath{\beta}}_{\mathrm{sd}}$ is consistent with
1/q within statistical error as long as $q\ensuremath{\ge}5$. Based on this, we further conjecture that $\ensuremath{\eta}({\ensuremath{\beta}}_{\mathrm{sd}})=1/q$ holds exactly and is universal for systems in the
q-state clock universality class. Our work provides a vivid demonstration of rich phenomena associated with the duality and self-duality of the clock model in two dimensions.
In this paper, we revisit the q-states clock model for small systems. We present results for the thermodynamics of the q-states clock model from
q=2 to
q=20 for small square lattices
L×L, with L ranging from
L=3 to
L=64 with free-boundary conditions. Energy, specific heat, entropy and magnetization are measured. We found that the Berezinskii-Kosterlitz-Thouless (BKT)-like transition appears for
q>5 regardless of lattice size, while the transition at
q=5 is lost for
L<10; for
q≤4 the BKT transition is never present. We report the phase diagram in terms of
q showing the transition from the ferromagnetic (FM) to the paramagnetic (PM) phases at a critical temperature T
1 for small systems which turns into a transition from the FM to the BKT phase for larger systems, while a second phase transition between the BKT and the PM phases occurs at T
2. We also show that the magnetic phases are well characterized by the two dimensional (2D) distribution of the magnetization values. We make use of this opportunity to do an information theory analysis of the time series obtained from the Monte Carlo simulations. In particular, we calculate the phenomenological mutability and diversity functions. Diversity characterizes the phase transitions, but the phases are less detectable as
q increases. Free boundary conditions are used to better mimic the reality of small systems (far from any thermodynamic limit). The role of size is discussed.