Sine-square deformation applied to classical Ising models

Sine-square deformation (SSD) is a treatment proposed in quantum systems, which spatially modifies a Hamiltonian, gradually decreasing the local energy scale from the center of the system toward the edges by a sine-squared envelope function. It is known to serve as a good boundary condition as well as to provide physical quantities reproducing those of the infinite-size systems. We apply the SSD to one- and two-dimensional classical Ising models. Based on the analytical calculations and Monte Carlo simulations, we find that the classical SSD system is regarded as an extended canonical ensemble of a local subsystem, each characterized by its own effective temperature. This effective temperature is defined by normalizing the system temperature by the deformed local energy scale. A single calculation for a given system temperature provides a set of physical quantities of various temperatures that quantitatively reproduces well those of the uniform system.

Figure 1

(a) Change in effective temperature $k_B{T}_{\mathrm{eff}}$ from the center ($r=0$) toward the edge ($r/R=1$) and the corresponding $f_{\mathrm{SSD}}\left(r\right)$. (b) Two-dimensional square lattice with interactions $J_{1,ij}$ and $J_{2,ij}$ on bonds running in the positive directions of the $x$ and $y$ axes from site $\gamma =(i,j)$. The transfer matrix ${\mathit{V}}_i$ is constructed in a unit of $L$ sites in the $y$ direction. (c) SSD along rows [case (i)] and columns [case (ii)].

Figure 2

Dispersion of fermions of a uniform Hamiltonian ${\epsilon }_q$ in Eq. (22) at several different temperatures.

Figure 3

Results of (a) and (b) the SSD Ising model in one dimension and (c) and (d) the uniform Ising model to be compared with (a) and (b). (a) Location (index-$i$)-dependent bond energy $-\langle {\sigma }_i{\sigma }_{i+1}\rangle$, given in Eq. (6), plotted against the effective temperature $k_B{T}_{\mathrm{eff}}$ for $L=4,...,100$. The solid line is the exact energy of a uniform Ising model $e_{\mathrm{bulk}}$ at $L=\infty$ as a function of temperature. (b) Deviations of bond energy in (a) from $e_{\mathrm{bulk}}$. Calculations are done for several choices of $k_{B}T=0.1n$ ($n$ is an integer), where each single choice of $k_{B}T$ generates $L/2$ data points. The inset shows how the results vary with $n$ for $L=10$ magnified from the main panel; better accuracy is observed for the bonds closer to the system center. (c) Exact bond energy for $L=4,...,100$ in the uniform 1D Ising model and (d) finite-size correction against $e_{\mathrm{bulk}}$. The insets of (a) and (c) show ${({\lambda }^{-}/{\lambda }^{+})}^L$ as a function of temperature.

Figure 4

Dispersions of fermions of a transfer matrix along the columns ($j$ direction) with $L=50$ in case (i): (a) $E_q/L$ in Eq. (35) for case (i) with $L=50$ (symbols) at $k_{B}T=0.5$, 1, $2.2692(=k_B{T}_c)$, and 3; (b) ${\epsilon }_q^{\left(u\right)}$ obtained using Eq. (22) with the temperature $k_B{T}_{\mathrm{eff}}$ that depends on the location of bond $i$, $i=1,...,25$; and (c) $i$-dependent $k_B{T}_{\mathrm{eff}}$ used to evaluate the dispersions in (b). The solid lines in (a) are the summation of ${\epsilon }_q^{\left(u\right)}$ throughout the system, $i=1,...,50$, to be compared with $E_q$ for the same system temperature $k_{B}T$. The right panel in (c) is the density plot of $k_B{T}_{\mathrm{eff}}$, together with the profile of $f_{\mathrm{SSD}}$ along the rows.

Figure 5

(a) Bond energy $-\langle {\sigma }_{\gamma }{\sigma }_{{\gamma }^{\prime }}\rangle$ evaluated for the 2D SSD Ising model plotted against $k_B{T}_{\mathrm{eff}}$ using Eqs. (49) and (50) for case (ii) with $L=10,...,500$ and $k_{B}T=0.5$. The solid line is the exact bulk energy ${\epsilon }_{\mathrm{bulk}}$. (b) SSD error $|e_{\mathrm{bulk}}+\langle {\sigma }_{\gamma }{\sigma }_{{\gamma }^{\prime }}\rangle |$ for the data in (a) with $k_{B}T=0.5$. (c) SSD error for a set of bond energies evaluated for $k_{B}T=2.2692=k_B{T}_c$. (d) Finite-size corrections of the bond energy evaluated for the uniform 2D system using Eqs. (49) and (50) as a function of $k_{B}T$.

Figure 6

(a) Dispersions ${\epsilon }_l$ of fermions of a transfer matrix along the columns ($j$ direction) with $L=50$ in Eq. (46) for case (ii) with $k_{B}T=0.5$, 1, $2.2692(=k_B{T}_c)$, and 3. (b) Spatial amplitude of the one-body eigenstate of index $l$ calculated at $L=50$ and $k_{B}T=0.5$ in (a). The bottom panel for the $l=25$ state is the zero-energy state.

Figure 7

Results obtained by the classical Monte Carlo simulations in the 2D Ising model with the SSD in one direction. The simulation temperature is denoted by $k_{B}T$, which corresponds to the lowest effective temperature. After discarding the initial 5000 steps, we measured the bond energy for $50000$ steps unless specified otherwise. The insets show enlarged views near the critical temperature. (a) Bond energy plotted against the effective temperature of each column. We compare our results of different lattice sizes with the exact bulk energy $e_{\mathrm{bulk}}$ and the result of Gaussian kernel regression (GKR). (b) The SSD error. The MC results of $L=150$ are consistent with the transfer-matrix (TM) results shown by the solid line for case (ii) with SSD of the same size. We also plotted the result of $L=15000$ measured for 500 MCSs to check the MCS dependences. (c) and (d) Results of the specific heat compared with the exact bulk value $c_{\mathrm{bulk}}$ and the GKR results.

Figure 8

(a) Functional form of Eq. (58) and SSD, and the corresponding effective temperature $k_B{T}_{\mathrm{eff}}$ as functions of $r/R$. (b) Deformation error $|e_{\mathrm{bulk}}+\langle {\sigma }_{\gamma }{\sigma }_{{\gamma }^{\prime }}\rangle |$ calculated using case (ii) by replacing the SSD functions with the three other functions given in Eq. (58). The bond energies for different $k_B{T}_{\mathrm{eff}}$ are obtained at different locations of the system of $L=200$ along the rows.