Size effects in finite systems with long-range interactions

Small systems consisting of particles interacting with long-range potentials exhibit enormous size effects. The Tsallis conjecture [Tsallis, Fractals 3, 541 (1995)], valid for translationally invariant systems with long-range interactions, states a well-known scaling relating different sizes. Here we propose to generalize this conjecture to systems with this symmetry broken, by adjusting one parameter that determines an effective distance to compute the strength of the interaction. We apply this proposal to the one-dimensional Ising model with ferromagnetic interactions that decay as $1/r^{1+\sigma }$ in the region where the model has a finite critical temperature. We demonstrate the convenience of using this generalization to study finite-size effects, and we compare this approach with the finite-size scaling theory.

Figure 1

(a) Evolution of the absolute magnetization starting from the ordered state ($m_0=1$) for $\sigma =0.3$ and different lattice sizes at temperature $T=5$. (b) Same as (a) at $T^{*}=5$ using $a_0=3.3$ and Eq. (17).

Figure 2

(a) Absolute magnetization vs $T$ and (b) susceptibility vs $T$ for $\sigma =0.3$ and the indicated sizes $100\le N\le 5000$. The black squares joined by the solid line are data obtained for the bulk, while dashed lines are a qualitative continuation of this curve. The vertical dashed lines represent the critical temperature of the system taken from Ref. [31].

Figure 3

Panels (a–c) The quantities corresponding to Eqs. (4, 5, 6), respectively, plotted against the system size $N$ in log-log plots. The fitted exponents are listed in Table 1.

Figure 4

(a) and (b) The collapses for magnetization and susceptibility, respectively, obtained from the data shown in Fig. 2 and using the theoretical exponents listed in Table 1.

Figure 5

Generalized Tsallis scaling for the magnetization using $T^{*}$ given by Eq. (17). (a) Absolute magnetization as a function of $T^{*}$ for different sizes as indicated. The black squares joined by the solid line are bulk results. (b) In the paramagnetic phase $\left|m\right|N^{1/2}$ is a function (size independent) of $T^{*}$ and corresponds to normal fluctuation. (c) Log-log plots of $\left|m\right|$ vs $|T^{*}-T_c|$. The dashed line is a power law given by Eq. (18) with exponent $\beta =1/2$.

Figure 6

Susceptibility ${\chi }_T$ as a function of the rescaled temperature $T^{*}$ for different sizes as indicated. The black squares joined by the solid line are bulk results as a function of $T$. The inset is a zoom of the critical region.

Figure 7

The same as Fig. 3 for the case of $\sigma =0.75$. The fitted exponents are listed in Table 2.

Figure 8

(a, b) The collapses for magnetization and susceptibility, respectively, using the exponents calculated in this work, for the case $\sigma =0.75$.

Figure 9

(a) Absolute magnetization against the temperature $T$ for $\sigma =0.75$. (b) The same data against the scaled temperature $T^{*}$ calculated using the generalized Tsallis conjecture with $a_0=20$. The arrows and the vertical dotted line indicate the critical temperature $T_c$ (see Table 2). The insets of Fig. (b) show the collapse obtained using the effective distances $R_0=7.0+N/20$.

Figure 10

Same as Fig. 9 for $\sigma =0.90$ with $a_0=48$. The inset of panel (b) shows the collapse obtained using the effective distances $R_0=6.0+N/48$.