Boundary-field-driven control of discontinuous phase transitions on hyperbolic lattices

The multistate Potts models on two-dimensional hyperbolic lattices are studied with respect to various boundary effects. The free energy is numerically calculated using the corner transfer matrix renormalization group method. We analyze phase transitions of the Potts models in the thermodynamic limit with respect to contracted boundary layers. A false phase transition is present even if a couple of the boundary layers are contracted. Its significance weakens, as the number of the contracted boundary layers increases, until the correct phase transition (deep inside the bulk) prevails over the false one. For this purpose, we derive a thermodynamic quantity, the so-called bulk excess free energy, which depends on the contracted boundary layers and memorizes additional boundary effects. In particular, the magnetic field is imposed on the outermost boundary layer. While the boundary magnetic field does not affect the second-order phase transition in the bulk if suppressing all the boundary effects on the hyperbolic lattices, the first-order (discontinuous) phase transition is significantly sensitive to the boundary magnetic field. Contrary to the phase transition on the Euclidean lattices, the discontinuous phase transition on the hyperbolic lattices can be continuously controlled (within a certain temperature coexistence region) by varying the boundary magnetic field.

Figure 1

The schematic picture of the hyperbolic lattice $\{5,4\}$ in the Poincaré disk representation. The entire hyperbolic lattice is composed of $L$ concentric layers, where $L$ is referred to as the lattice size (the radius), which enumerates the layers from the center outward. The inner sublattice, the bulk, has $n_{L,k}$ sites with the diameter $L-k$ of the layers. The outer layers, the boundary, are enumerated by $k$ and consist of $m_{L,k}$ sites.

Figure 2

The scaling ratio $R_k$ between the boundary and the bulk layers is calculated for the three selected hyperbolic lattices $\{20,4\}$, $\{3,8\}$, and $\{5,4\}$ in the thermodynamic limit ($L\to \infty$). The inset shows the detail for the outermost layer when $k=1$.

Figure 3

The hyperbolic lattice geometries $\{20,4\}$ (left) and $\{\infty ,4\}$ (right) are numerically indistinguishable if the bulk properties of the spin models are considered.

Figure 4

The Ising model: temperature dependence of the spontaneous magnetization $M_{\{p,4\}}$, the free energy per site ${\mathcal{F}}_{\{p,4\}}$, the internal energy $U_{\{p,4\}}$, and the specific heat $C_{\{p,4\}}$ calculated on the square lattice $\{4,4\}$ (black circles) and the Bethe lattice $\{\infty ,4\}$ (red triangles). The response of the model to the external magnetic fields $h=0$ (green symbol ‘$\times$') and $h=0.1$ (blue symbol ‘$+$') is shown. The vertical dot-dashed lines help to identify the correct phase-transition temperature $T_{\mathrm{pt}}^{\left(p\right)}$ including the false one located at $T_{\mathrm{f}}^{}\approx 0.339$.

Figure 5

The five-state Potts model: the spontaneous magnetization $M_{\{p,4\}}$, the free energy per site ${\mathcal{F}}_{\{p,4\}}$, the nearest-neighbor correlation energy $E_{\{p,4\}}$, and the normalized entropy per site $S_{\{p,4\}}$ are plotted with respect to temperature on the square and Bethe lattices for $h=0$ and 0.1. The false phase-transition temperature is located at $T_{\mathrm{f}}^{}\approx 0.296$ on the Bethe lattice at $h=0$.

Figure 6

The two top panels show the functional relations between the magnetization $M$ and the entropy $S$. The two bottom panels show the entropy $S$ vs the NN correlation energy $E$. The second-order phase transition is manifested by the continuous behavior of the thermodynamic functions, whereas the first-order phase transition opens the nonzero gaps in the thermodynamic functions.

Figure 7

The convex set of $\{E,S,M\}$ for the five-state Potts model on the Bethe lattice $\{\infty ,4\}$ forms a two-dimensional surface of a shell-like object. Each thick or thin surface curve corresponds to the particular flows of $\{E,S,M\}$ with respect to the varying temperature $T$ for given fixed values $J$ and $h$. The data shown are calculated for $h_b=k=0$ in the thermodynamic limit ($L\to \infty$).

Figure 8

The bulk excess free energy (the upper graph) and the “specific heat” (the lower graph) with respect to temperature in the thermodynamic limit calculated for the Ising model on the Bethe lattice when $h=0$, regardless of $h_b$. The two cases are shown: (1) $k=0,L\to \infty$ (the circles), i.e., no contracted boundary layers, and (2) $k\to \infty ,L\to \infty$ (the triangles), i.e., infinitely many boundary layers are contracted while the inner lattice size remains infinite, too. The two dot-dashed vertical lines denote the correct and the false phase-transition temperatures.

Figure 9

Temperature dependence of the bulk excess free energy per site for the Ising model on the Bethe lattice for a wide range of contracted boundary layers $k$. The specific region located around the false phase transition $T_{\mathrm{f}}$, where the bulk excess free energy grows, is denoted by the dot-dashed arrows.

Figure 10

The specific heat vs temperature exhibits unique behavior around the false phase-transition temperature $T_{\mathrm{f}}\approx 0.339$ after the boundary layers $k=0,1,2,3,\cdots ,20$ are gradually contracted. The inset shows the maximum for $k=0$ and 1.

Figure 11

The bulk excess free energy of the Ising model on the selected set of hyperbolic lattices $\{p,q\}$ when taking both the limits $L\to \infty$ and $k\to \infty$ for $h=0$. The bulk excess free energy remains unchanged for arbitrarily chosen $h_b\ne 0$.

Figure 12

The temperature dependence of the bulk excess free energy for the five-state Potts model on the Bethe lattice $\{\infty ,4\}$ after contracting an infinite number of boundary layers ($k\to \infty$). The phase transitions are shown as discontinuities in ${\mathcal{B}}_{\{\infty ,4\}}^{\left(\infty \right)}$ for the particular boundary fields $h_b$. The discontinuities are marked by dotted vertical lines. The graphs were constructed for $L\to \infty$, $k\to \infty$, and $h=0$.

Figure 13

The temperature dependence of the bulk excess free energy for the $Q$-state Potts model for the selected Potts states $Q\ge 3$ and lattices $\{p,q\}$. The vertical dotted lines border the discontinuous phase-coexistence regions $\mathrm{\Delta }T$ for $L\to \infty$, $k\to \infty$, and $h=0$.