Free-energy analysis of spin models on hyperbolic lattice geometries

We investigate relations between spatial properties of the free energy and the radius of Gaussian curvature of the underlying curved lattice geometries. For this purpose we derive recurrence relations for the analysis of the free energy normalized per lattice site of various multistate spin models in the thermal equilibrium on distinct non-Euclidean surface lattices of the infinite sizes. Whereas the free energy is calculated numerically by means of the corner transfer matrix renormalization group algorithm, the radius of curvature has an analytic expression. Two tasks are considered in this work. First, we search for such a lattice geometry, which minimizes the free energy per site. We conjecture that the only Euclidean flat geometry results in the minimal free energy per site regardless of the spin model. Second, the relations among the free energy, the radius of curvature, and the phase transition temperatures are analyzed. We found out that both the free energy and the phase transition temperature inherit the structure of the lattice geometry and asymptotically approach the profile of the Gaussian radius of curvature. This achievement opens new perspectives in the AdS-CFT correspondence theories.

Figure 1

The illustration of the three selected lattice geometries ($4,4$), ($5,4$), and ($4,5$). The first two CTMRG iteration steps $k=1$ (left) and $k=2$ (right) show the building process of the lattices by means of the $p$-gonal Boltzmann weight tessellation with the uniform coordination number $q$. The Boltzmann weights for given $k$ are represented by the shaded regular (congruent) $p$-gons.

Figure 2

The Poincaré disk representation of the three hyperbolic lattices chosen for the analysis of the thermodynamic functions of the spin model.

Figure 3

The spontaneous magnetization with respect to temperature for the Euclidean square lattice as well as for the three hyperbolic lattices depicted in Fig. 2.

Figure 4

The Poincaré representation of the asymptotic hyperbolic lattices ($\infty ,7$) on the left, ($7,\infty$) in the middle, and ($\infty ,\infty$) on the right.

Figure 5

The temperature dependence of the spontaneous magnetization toward the asymptotic lattice geometries plotted in Fig. 4. The insets of the top panel show the fast convergence of the magnetization, and the model on the lattice ($7,7$) exhibits almost identical behavior as the lattice ($\infty ,7$). The top and the bottom graphs, respectively, describe the linear increase of the phase transitions if the model is studied on the lattices ($7,q$) and ($q,q$) for $q=4,5,6,\cdots ,\infty$.

Figure 6

The top graph shows the temperature dependence of a few nonzero values of the squared magnetization approaching the phase transition from the ordered phase $T\le T_{\mathrm{pt}}^{(\infty ,q)}$ for the lattices $p=20$ and $4\le q\le 13$ obtained by CTMRG, which accurately reproduce the Bethe lattice. The inset shows the same data in detail, rescaled to temperatures $T-(q-1)$. In the bottom graph, the linearity of $T_{\mathrm{pt}}^{(\infty ,q)}$ on the Bethe lattice is satisfied with increasing $q$. The lower right inset shows the convergence of the effective exponent ${\alpha }_{\mathrm{eff}}^{(\infty ,q)}\to 1$ in the log-lin scale. The upper left inset displays the numerical accuracy by evaluating the relative error for the phase transition temperature on the Bethe lattices.

Figure 7

The graphical representation of the corner transfer tensor ${\mathcal{C}}_3$ for the square lattice ($4,4$) on the left and for the hyperbolic lattice ($5,4$) on the right, respectively, in accord with Eqs. (4) and (5) for $k=3$. We use the dark and the bright shaded $p$-gons in order to distinguish clearly between ${\mathcal{C}}_k$ and ${\mathcal{T}}_k$, respectively.

Figure 8

The exponential dependence of the total number of the spins on the iteration number $k$ (the least-squares fitting in the log-log plot) for the two lattices ($5,4$) and ($7,7$).

Figure 9

The free energy from Eq. (33) and the specific heat from Eq. (37) vs temperature on the selected lattices ($4,4$), ($4,7$), ($7,4$), and ($7,7$). The inset of the bottom graph shows the details of the broadened specific heat maxima.

Figure 10

The bulk free energy (the top graph) and the bulk specific heat (the bottom graph) vs temperature for the Ising model on the lattices ($4,4$), ($4,7$), ($7,4$), and ($7,7$). The vertical dotted lines accurately correspond to the phase transition temperatures we have obtained from the spontaneous magnetization in Fig. 3.

Figure 11

The free energy per site as a function of the lattice geometry ($p,q$) at the selected lower temperatures $T=1,2$, and 3.

Figure 12

The same as in Fig. 11 for $T=10$ and 20.

Figure 13

The high-temperature asymptotics of free energy per site (top graph) and the entropy (bottom graph) applied to the lattice ($7,7$). The full and the dashed lines correspond to the $M$-state clock and the $M$-state Potts models, respectively, where $M=2,3,\cdots ,7$.

Figure 14

The free energy for the seven-state clock and Potts models at $T=5$, which do not differ from the Ising model ($M=2$) in Figs. 11 and 12 qualitatively.

Figure 15

The functional dependence of the Gaussian radius of curvature ${\mathcal{R}}_{(q,p)}$ plotted in the dual lattice geometry ($q,p$).

Figure 16

The comparison of the free energy per site for the Ising model at $T=0.5$ with the Gaussian radius of curvature on the dual geometry. The top graph shows the case of the two fixed coordination numbers $q^{*}=4$ and $q^{*}=7$, whereas the bottom graph depicts the opposite case when fixing the $p$-gons to the sizes $p^{*}=4$ and $p^{*}=7$.

Figure 17

The asymptotic behavior of the free energy per site ${\mathcal{F}}_{(p,q^{*})}^{\left[\infty \right]}-{\mathcal{F}}_{(\infty ,q^{*})}^{\left[\infty \right]}$ for the Ising model at $T=0.1$ and the Gaussian radius of curvature ${\mathcal{R}}_{(q^{*},p)}$ for $q^{*}=4$ and $q^{*}=7$. The asymptotic fitting functions for the energy decay as $-\frac{1}{p}$, where the “+” symbols connected with the dashed line fit to the free-energy data (the circles and the triangles). The asymptotics $\propto 1/ln{(2p/\pi )}^{-2}$ of the radius of curvature (the squares and the diamonds) is plotted by the “$*$” symbols connected with the dashed line. The inset of the top graphs shows the temperature dependence of ${\mathcal{F}}_{(\infty ,q^{*})}^{\left[\infty \right]}$. The log-log plot in the bottom graph is used to enhance the asymptotics of the top graph.

Figure 18

The rescaled phase transition temperatures with respect to $p$ and $q$ are shown in the dual geometry ($q,p$) to emphasize the similarity with the radius of Gaussian curvature in Fig. 15.