Calculation of critical exponents on fractal lattice Ising model by higher-order tensor renormalization group method

The critical behavior of the Ising model on a fractal lattice, which has the Hausdorff dimension ${log}_{4}12\approx 1.792$, is investigated using a modified higher-order tensor renormalization group algorithm supplemented with automatic differentiation to compute relevant derivatives efficiently and accurately. The complete set of critical exponents characteristic of a second-order phase transition was obtained. Correlations near the critical temperature were analyzed through two impurity tensors inserted into the system, which allowed us to obtain the correlation lengths and calculate the critical exponent $\nu$. The critical exponent $\alpha$ was found to be negative, consistent with the observation that the specific heat does not diverge at the critical temperature. The extracted exponents satisfy the known relations given by various scaling assumptions within reasonable accuracy. Perhaps most interestingly, the hyperscaling relation, which contains the spatial dimension, is satisfied very well, assuming the Hausdorff dimension takes the place of the spatial dimension. Moreover, using automatic differentiation, we have extracted four critical exponents $(\alpha ,$ $\beta ,$ $\gamma , \mathrm{and} \delta )$ globally by differentiating the free energy. Surprisingly, the global exponents differ from those obtained locally by the technique of the impurity tensors; however, the scaling relations remain satisfied even in the case of the global exponents.

Figure 1

The layout of the fractal lattice after three extension steps, $n=3$. Tiny circles represent the two-state Ising spins. The horizontal and vertical lines represent the spin-spin interactions. The number of sites grows as ${12}^n$ with the number of extension steps $n$, whereas the number of outgoing bonds grows as $2^{n+2}$.

Figure 2

Specific heat $c\left(T\right)$ as a function of temperature calculated from the (local) bond energy $u$ in the close vicinity of $T_{\mathrm{c}}$. Specific heat $c$ was obtained by automatic differentiation as a first derivative of $u$ with respect to the temperature $T$. Data points with the bond dimension $D=16$ and $D=24$ are depicted as smaller blue circles and bigger black circles, respectively, whereas the fitting curve for $D=24$ is depicted as a red line. Fitting yielded $\alpha \approx -0.87$ and $T_{\mathrm{c}}\approx 1.3171715$ (with $D=24$). A vertical dotted line indicates the critical temperature $T_{\mathrm{c}}$.

Figure 3

The detail view of the linear dependence of $|c\left(T\right)-c\left(T_{\mathrm{c}}\right){|}^{-1/\alpha }$ with respect to the temperature near $T_{\mathrm{c}}$, where $c$ is the local specific heat. Three values of $\alpha$ are presented: (1) $\alpha =-0.88$ (smaller than our estimate of $\alpha$, shown as squares), (2) $\alpha =-0.87$ (corresponding to our estimate, show as circles), and (3) $\alpha =-0.86$ (larger than our estimate of $\alpha$, show as diamonds). Data points with the bond dimension $D=16$ and $D=24$ are depicted as smaller blue and bigger black shapes, respectively, whereas the fitting curve for $D=24$ is depicted as a red line.

Figure 4

Magnetic susceptibility $\chi \left(T\right)$ as a function of temperature $T$ in the vicinity of the critical temperature $T_{\mathrm{c}}$. Magnetic susceptibility $\chi$ was obtained by automatic differentiation as the first derivative of (local) magnetization $m$ with respect to the global field $h$. Data points with the bond dimension $D=16$ and $D=24$ are depicted as smaller blue circles and bigger black circles, respectively, whereas the fitting curve for $D=24$ is depicted as a red line. Inset: The linear dependence of $\chi {\left(T\right)}^{-1/\gamma }$ above $T_{\mathrm{c}}$.

Figure 5

The correlation $G\left(r\right)$ as a function of distance $r$. The correlation $G\left(r\right)$ for three different temperatures is shown: (1) $T=1.317$ (below $T_{\mathrm{c}}$), (2) $T=1.31718$ (very close to $T_{\mathrm{c}}$), and (3) $T=1.31736$ (above $T_{\mathrm{c}}$). Here, the bond dimension was implemented adaptively where the normalized singular values smaller than ${10}^{-14}$ were discarded. $D_{\max }$ denotes the maximal (unbounded) adaptive dimension (in practice $D_{\max }\le 23$). We present data points calculated with $D_{\max }$ and $D=16$, which we depict as black and blue symbols, respectively. The fitting curves Eq. (36) were obtained for $D_{\max }$ and are depicted by red lines. The values of the correlation lengths $\xi$ for $T=1.31718$ and $T=1.31736$ are indicated by full and dashed line, respectively. The numerical analysis did not include data points for $r<1024$.

Figure 6

The correlation length $\xi$ as a function of temperature $T$ above $T_{\mathrm{c}}$. The data points were obtained by fitting the correlation function $G\left(r\right)$ for $r\ge 1024$ at each temperature $T$ separately. The fitting curve was obtained for $D_{\max }$ and is depicted by a red line. The resulting value of exponent $\nu$ was found to be $\nu \approx 1.60$ and $T_{\mathrm{c}}\approx 1.3171723$. Inset: Critical scaling of $\xi {\left(T\right)}^{{\nu }^{-1}}$ above the critical temperature $T_{\mathrm{c}}$.

Figure 7

A detailed view of the exponent $E\left(T\right)$ and the correlation length $\xi \left(T\right)$ as functions of temperature $T$ in the vicinity of $T_{\mathrm{c}}$. The correlation length $\xi \left(T\right)$ is presented here to indicate the location of the critical temperature $T_{\mathrm{c}}$. The value of the exponent $E\left(T\right)$ seems to be nearly constant (indicated by a dotted horizontal line) in the right vicinity of $T_{\mathrm{c}}$, where we observed $E\left(T\right)\approx 0.02$.

Figure 8

A detailed view of the global specific heat $c\left(T\right)$ in the vicinity of the critical temperature $T_{\mathrm{c}}$. Global specific heat $c\left(T\right)$ is calculated as a second derivative of the free energy per site $f\left(T\right)$ with respect to the temperature by the automatic differentiation. Data points for the bond dimension $D=24$ are depicted as big black circles, whereas for $D=16$ as smaller blue circles. The fitting curve is obtained for $D=24$ and is shown as a thick red line. For $D=24$, the fitting yielded $\alpha =-0.824$ and $T_{\mathrm{c}}=1.3171725$. A vertical dotted line indicates the location of the critical temperature $T_{\mathrm{c}}$.

Figure 9

The detail view of the linear dependence of $|c\left(T\right)-c\left(T_{\mathrm{c}}\right){|}^{-1/\alpha }$ with respect to the temperature near $T_{\mathrm{c}}$, where $c$ is the global specific heat. Three values of $\alpha$ are compared: (1) $\alpha =-0.834$ (smaller than our estimate of $\alpha$, shown as squares), (2) $\alpha =-0.824$ (corresponding to our estimate, shown as circles), and (3) $\alpha =-0.814$ (larger than our estimate of $\alpha$, shown as diamonds). Data points with the bond dimension $D=16$ and $D=24$ are depicted as smaller blue and bigger black shapes, respectively, whereas the fitting curve for $D=24$ is depicted as a red line.

Figure 10

A detailed view of the global spontaneous magnetization per site $m\left(T\right)$ in the vicinity of the critical temperature $T_{\mathrm{c}}$. The global spontaneous magnetization per site $m\left(T\right)$ is calculated as a first derivative of the free energy $f(h,T)$ per site with respect to the external field $h$ using the automatic differentiation. The data points for the bond dimension $D=24$ and $D=16$ are depicted as big black circles and smaller blue circles, respectively. The fitting curve is obtained for $D=24$ and is shown as a thick red line here. For $D=24$, the fitting yields $\beta =0.0629$ and $T_{\mathrm{c}}=1.3171724$. Inset: Critical scaling of $m{\left(T\right)}^{{\beta }^{-1}}$ below the critical temperature $T_{\mathrm{c}}$.

Figure 11

A detailed view of the global magnetic susceptibility per site $\chi \left(T\right)$. The global magnetic susceptibility per site $\chi \left(T\right)$ is calculated as a second derivative of the free energy per site $f(h,T)$ with respect to the external field $h$ using automatic differentiation. The data points for the bond dimension $D=24$ and $D=16$ are depicted as big black circles and smaller blue circles, respectively. The fitting curve is obtained for $D=24$ and is shown as a thick red line here. For $D=24$, the fitting yields $T_{\mathrm{c}}=1.3171724$ and $\gamma =2.764$. A vertical dotted line indicates the location of the critical temperature $T_{\mathrm{c}}$. Inset: Critical scaling of $\chi {\left(T\right)}^{-{\gamma }^{-1}}$ above the critical temperature $T_{\mathrm{c}}$.

Figure 12

A detailed view of the global magnetization $m\left(h\right)$ as a function of the external field $h$ at fixed temperature $T=T_{\mathrm{c}}=1.3171724$. The critical exponent $\delta$ was found to be $\delta \approx 44.8$. Inset: Critical scaling of $m{\left(h\right)}^{\delta }$ at $T=T_{\mathrm{c}}=1.3171724$.