Results in one and two dimensions

7 Results in one dimension

Figure 5 shows the energy, specific heat and susceptibility of a chain of 100 spins, simulated with the Wolff algorithm, compared with the exact transfer-matrix results. The agreement is perfect within the error bars. The finite size only matters at low temperature, where the correlation length becomes comparable to the length of the chain: the susceptibility then saturates at $\chi \simeq N/T$, the value for a fully aligned chain, instead of growing like $e^{2J/T}/T$. The correlation function (Figure 2) decays exponentially at every temperature, as expected in the absence of a phase transition.

[Image: Energy, specific heat and susceptibility per spin of the Ising chain. Points: Monte Carlo simulations of 100 spins with the Wolff algorithm (10⁵ sweeps per temperature). Lines: exact transfer-matrix results for 100 spins and for the infinite chain.]Figure 5: Energy, specific heat and susceptibility per spin of the Ising chain. Points: Monte Carlo simulations of 100 spins with the Wolff algorithm (10⁵ sweeps per temperature). Lines: exact transfer-matrix results for 100 spins and for the infinite chain.

Try it with the code

python3 ising_1D.py reproduces these results (options 1 and 3), shows the magnetization in a field (option 2) and the space–time picture of Figure 3 (option 4). Try increasing the number of spins at low temperature: how does the susceptibility change?

8 Results in two dimensions

8.1 Configurations

Figure 6 shows typical configurations of a $256\times256$ lattice. Below $T_c$, the lattice is almost fully magnetized, with small islands of reversed spins. Above $T_c$, the spins form small domains of typical size $\xi$, with no net magnetization. At $T_c$, domains of all sizes coexist, nested within each other: the configuration looks the same at all scales. This scale invariance is the hallmark of a critical point.

[Image: Equilibrium configurations of a 256×256 lattice (white: up, black: down), below, at and above the critical temperature.]Figure 6: Equilibrium configurations of a 256×256 lattice (white: up, black: down), below, at and above the critical temperature.

8.2 Thermodynamics

Figure 7 shows the energy, magnetization, specific heat and susceptibility for $L = 8$ to $64$. The simulated energy and specific heat agree with Kaufman's exact results for the same lattices, and approach Onsager's infinite-lattice curves as $L$ grows. Over all sizes and temperatures, the deviations from Kaufman's values have a spread of about one error bar, the largest being about three error bars, as expected for more than a hundred comparisons. The magnetization $\left\langle |m| \right\rangle$ follows Yang's formula below $T_c$, but does not vanish above it: in a finite lattice, $\left\langle |m| \right\rangle \sim 1/\sqrt{N}$ at high temperature. The specific heat and the susceptibility have peaks that grow with $L$ and approach $T_c$.

[Image: Thermodynamics of the 2D Ising model. Points: Wolff simulations (10⁵ sweeps per temperature) for L = 8, 16, 32 and 64. Dashed lines: Kaufman's exact results for the same lattices. Black lines: Onsager's and Yang's exact results for the infinite lattice. The dotted line marks Tc.]Figure 7: Thermodynamics of the 2D Ising model. Points: Wolff simulations (10⁵ sweeps per temperature) for L = 8, 16, 32 and 64. Dashed lines: Kaufman's exact results for the same lattices. Black lines: Onsager's and Yang's exact results for the infinite lattice. The dotted line marks Tc.

8.3 Correlations at the critical point

Away from $T_c$, correlations decay exponentially. At $T_c$, the correlation length is infinite and the correlation function decays as a power law, $G(r) \sim r^{-1/4}$ ($\eta = 1/4$). Figure 8 shows both behaviours in a $256 \times 256$ lattice.

[Image: Spin–spin correlation function of a 256×256 lattice (log–log scale). At Tc it decays as the power law r^(−1/4); above Tc it decays exponentially.]Figure 8: Spin–spin correlation function of a 256×256 lattice (log–log scale). At Tc it decays as the power law r^(−1/4); above Tc it decays exponentially.

8.4 Finite-size scaling and the Binder cumulant

How do the finite-lattice quantities depend on $L$ near $T_c$? The finite-size scaling hypothesis states that the only relevant lengths are $\xi \sim |t|^{-\nu}$ and $L$. For the susceptibility, for example,

where $\tilde\chi$ is a universal function. Consequences:

• at $T_c$, $\left\langle |m| \right\rangle \sim L^{-\beta/\nu} = L^{-1/8}$ and $\chi' \sim L^{\gamma/\nu} = L^{7/4}$;

• the maximum of $\chi'$ grows like $L^{7/4}$, and its position approaches $T_c$ like $L^{-1/\nu} = L^{-1}$;

• plotting $\chi' L^{-7/4}$ against $L\, t$ makes the curves of all sizes collapse onto one.

Figure 9 shows this data collapse. Figure 10 shows the power laws: the fits give $\beta/\nu = 0.125$ and $\gamma/\nu = 1.770$, close to the exact $1/8 = 0.125$ and $7/4 = 1.75$. The maxima of $\chi'$ are obtained from dedicated simulations around each peak. The small excess of $\gamma/\nu$ comes from corrections to scaling at small $L$: fitting only $L = 32$, 64 and 128 gives $1.759$.

A convenient way to locate $T_c$ is the Binder cumulant [13]

Deep in the ordered phase, $m \simeq \pm m_0$ and $U_L \to 2/3$. In the disordered phase, $m$ has Gaussian fluctuations, for which $\left\langle m^4 \right\rangle = 3\left\langle m^2 \right\rangle^2$, and $U_L \to 0$. At $T_c$, $U_L$ takes a universal value independent of $L$, so the curves for different sizes cross at $T_c$ (Figure 9a). Here the curves for $L = 64$ and $L = 128$ cross at $T = 2.266$ and $U = 0.615$, very close to $T_c = 2.269$.

[Image: (a) Binder cumulant for L = 8 to 128: the curves cross at the critical temperature (dotted line). (b) Finite-size scaling of the susceptibility, Eq. (24), with the exact exponents ν = 1 and γ = 7/4: the data for all sizes collapse onto a single curve.]Figure 9: (a) Binder cumulant for L = 8 to 128: the curves cross at the critical temperature (dotted line). (b) Finite-size scaling of the susceptibility, Eq. (24), with the exact exponents ν = 1 and γ = 7/4: the data for all sizes collapse onto a single curve.

[Image: (a) Magnetization at Tc and (b) maximum of the susceptibility as functions of L (log–log scale). The slopes give −β/ν and γ/ν.]Figure 10: (a) Magnetization at Tc and (b) maximum of the susceptibility as functions of L (log–log scale). The slopes give −β/ν and γ/ν.

Try it with the code

Option 5 of python3 ising_2D.py computes the Binder cumulant for $L = 8$ to 64. With these data, estimate $T_c$ from the crossing points, and $\gamma/\nu$ from the maxima of $\chi$.

8.5 Hysteresis

Below $T_c$, the system has two ordered states, $m \approx \pm m_0$. If the field is swept from positive to negative values, the lattice stays magnetized up while the field is slightly negative. The up state is still stable against small fluctuations, and a reversed domain must be large enough to grow. The magnetization flips only at a finite coercive field. Sweeping back gives the symmetric branch, and the magnetization traces a hysteresis loop (Figure 11). The loop shrinks as $T$ increases and disappears above $T_c$, where $m$ is a single-valued function of $h$: at $T = 3$ the two branches coincide within the statistical noise. Hysteresis is a non-equilibrium effect, and the width of the loop depends on how fast the field is swept. At $T = 2$, sweeping four times more slowly (160 instead of 40 sweeps per field value) reduces the coercive field from about $0.20$ to about $0.12$: the slower the sweep, the more time a reversed domain has to nucleate and grow. In true equilibrium, the magnetization of an infinite system would jump from $-m_0$ to $+m_0$ at $h = 0$.

[Image: Magnetization of a 64×64 lattice while the field is swept from +1.5 to −1.5 and back, in steps of 0.04. At each field value, 20 Metropolis sweeps let the lattice relax and the next 20 are used to measure m.]Figure 11: Magnetization of a 64×64 lattice while the field is swept from +1.5 to −1.5 and back, in steps of 0.04. At each field value, 20 Metropolis sweeps let the lattice relax and the next 20 are used to measure m.

8.6 Critical slowing down

Figure 12 compares the Metropolis and Wolff algorithms at $T_c$. With Metropolis, $|m|$ wanders slowly and its autocorrelation time grows like $L^{2.09}$, close to the known dynamic exponent $z \approx 2.17$. With Wolff, the autocorrelation time, measured in sweeps, stays of order one and grows only like $L^{0.40}$. For $L = 64$, the autocorrelation time is $718$ sweeps with Metropolis and $2.0$ sweeps with Wolff, about $362$ times shorter. A Wolff sweep costs $1.1$ times more computer time, so Wolff is still about $316$ times more efficient. This is why all the scans near $T_c$ in these notes use it.

[Image: (a) Time series of |m| for a 64×64 lattice at Tc with the Metropolis and Wolff algorithms. (b) Integrated autocorrelation time of |m| at Tc as a function of L.]Figure 12: (a) Time series of |m| for a 64×64 lattice at Tc with the Metropolis and Wolff algorithms. (b) Integrated autocorrelation time of |m| at Tc as a function of L.