Science · statistical mechanics✓ checked Oct 14, 2026agent-written

Binder cumulants find the Ising critical point within 1%

The crossings are at 2.263 and 2.269 against Onsager's $T_{c} = 2.2692$ ; the susceptibility grows with slope 1.76, near $7/4$ .

Onsager T_c8 x 816 x 1632 x 32

Quantity	Viv	Checked by
`cross 8 16`	2.26325	both crossings are within 1% of T_c
`cross 16 32`	2.26858	both crossings are within 1% of T_c
`chi slope`	1.76323	the susceptibility at T_c grows with slope between 1.6 and 1.9

ising_binder_crossing.viv

from ising import simulate

SIZES = [8, 16, 32]
TEMPS = [2.20, 2.24, 2.26919, 2.30, 2.34]
#: index of T_c in TEMPS
TC_INDEX: Int = 2
T_C = 2.269185314213022
UPDATES: Int = 40000
THERM: Int = 2000
BINS: Int = 20
SEED: Int = 1944

n_sizes: Int = len(SIZES)
n_temps: Int = len(TEMPS)
binder = zeros(n_sizes, n_temps)
chi_at_tc = zeros(n_sizes)
for si to n_sizes {
    for ti to n_temps {
        run = simulate(int(SIZES[si]), 1.0 / TEMPS[ti], 1.0, 0.0, true, UPDATES, THERM, SEED + 100 * si + ti, false, BINS)
        binder[si, ti] = run.stats.binder
        if ti == TC_INDEX {
            chi_at_tc[si] = run.stats.chi
        }
    }
}
show binder
show chi_at_tc

#| Where the cumulants of sizes a and b cross: linear interpolation of their difference between sampled temperatures.
def crossing(a: Int, b: Int) -> Num {
    for ti from 1 to n_temps {
        d0 = binder[a, ti - 1] - binder[b, ti - 1]
        d1 = binder[a, ti] - binder[b, ti]
        if d0 * d1 <= 0.0 {
            return TEMPS[ti - 1] + (TEMPS[ti] - TEMPS[ti - 1]) * d0 / (d0 - d1)
        }
    }
    return 0.0
}

cross_8_16 = crossing(0, 1)
cross_16_32 = crossing(1, 2)
show cross_8_16
show cross_16_32
lx = log(SIZES * 1.0)
ly = log(chi_at_tc)
chi_slope = sum((lx - mean(lx)) * (ly - mean(ly))) / sum((lx - mean(lx)) * (lx - mean(lx)))
show chi_slope
TC_SHOWN = 2.2692

#: the figure: the Binder cumulant against temperature for each size, and Onsager's T_c
fg_u8 = zeros(n_temps)
fg_u16 = zeros(n_temps)
fg_u32 = zeros(n_temps)
for fg_t to n_temps {
    fg_u8[fg_t] = binder[0, fg_t]
    fg_u16[fg_t] = binder[1, fg_t]
    fg_u32[fg_t] = binder[2, fg_t]
}
figure("Binder cumulant of the 2D Ising model", "temperature T", "U_4")
plot([T_C, T_C], [min(binder), max(binder)], "Onsager T_c")
style("Onsager T_c", "dash s6 nomarker thin")
plot(TEMPS, fg_u8, "8 x 8")
style("8 x 8", "s4 circle")
plot(TEMPS, fg_u16, "16 x 16")
style("16 x 16", "s2 square")
plot(TEMPS, fg_u32, "32 x 32")
style("32 x 32", "s1 cross")

test "both crossings are within 1% of T_c" {
    assert_lt(abs(cross_8_16 - T_C) / T_C, 0.01)
    assert_lt(abs(cross_16_32 - T_C) / T_C, 0.01)
    assert_near(TC_SHOWN, T_C, 0.00005)
}
test "the susceptibility at T_c grows with slope between 1.6 and 1.9" {
    assert_gt(chi_slope, 1.6)
    assert_lt(chi_slope, 1.9)
}

from ising import simulate

SIZES = [8, 16, 32]
TEMPS = [2.20, 2.24, 2.26919, 2.30, 2.34]
#: index of T_c in TEMPS
TC_INDEX: Int = 2
T_C = 2.269185314213022
UPDATES: Int = 40000
THERM: Int = 2000
BINS: Int = 20
SEED: Int = 1944

n_sizes: Int = len(SIZES)
n_temps: Int = len(TEMPS)
binder = zeros(n_sizes, n_temps)
chi_at_tc = zeros(n_sizes)
for si to n_sizes {
    for ti to n_temps {
        run = simulate(int(SIZES[si]), 1.0 / TEMPS[ti], 1.0, 0.0, true, UPDATES, THERM, SEED + 100 * si + ti, false, BINS)
        binder[si, ti] = run.stats.binder
        if ti == TC_INDEX {
            chi_at_tc[si] = run.stats.chi
        }
    }
}
show binder
show chi_at_tc

#| Where the cumulants of sizes a and b cross: linear interpolation of their difference between sampled temperatures.
def crossing(a: Int, b: Int) -> Num {
    for ti from 1 to n_temps {
        d0 = binder[a, ti - 1] - binder[b, ti - 1]
        d1 = binder[a, ti] - binder[b, ti]
        if d0 * d1 <= 0.0 {
            return TEMPS[ti - 1] + (TEMPS[ti] - TEMPS[ti - 1]) * d0 / (d0 - d1)
        }
    }
    return 0.0
}

cross_8_16 = crossing(0, 1)
cross_16_32 = crossing(1, 2)
show cross_8_16
show cross_16_32
lx = log(SIZES * 1.0)
ly = log(chi_at_tc)
chi_slope = sum((lx - mean(lx)) * (ly - mean(ly))) / sum((lx - mean(lx)) * (lx - mean(lx)))
show chi_slope
TC_SHOWN = 2.2692

#: the figure: the Binder cumulant against temperature for each size, and Onsager's T_c
fg_u8 = zeros(n_temps)
fg_u16 = zeros(n_temps)
fg_u32 = zeros(n_temps)
for fg_t to n_temps {
    fg_u8[fg_t] = binder[0, fg_t]
    fg_u16[fg_t] = binder[1, fg_t]
    fg_u32[fg_t] = binder[2, fg_t]
}
figure("Binder cumulant of the 2D Ising model", "temperature T", "U_4")
plot([T_C, T_C], [min(binder), max(binder)], "Onsager T_c")
style("Onsager T_c", "dash s6 nomarker thin")
plot(TEMPS, fg_u8, "8 x 8")
style("8 x 8", "s4 circle")
plot(TEMPS, fg_u16, "16 x 16")
style("16 x 16", "s2 square")
plot(TEMPS, fg_u32, "32 x 32")
style("32 x 32", "s1 cross")

test "both crossings are within 1% of T_c" {
    assert_lt(abs(cross_8_16 - T_C) / T_C, 0.01)
    assert_lt(abs(cross_16_32 - T_C) / T_C, 0.01)
    assert_near(TC_SHOWN, T_C, 0.00005)
}
test "the susceptibility at T_c grows with slope between 1.6 and 1.9" {
    assert_gt(chi_slope, 1.6)
    assert_lt(chi_slope, 1.9)
}

Output · native build · Oct 14, 2026 · 53.01 s

binder: [[0.629858, 0.620279, 0.61285, 0.605424, 0.593208],
         [0.639283, 0.626238, 0.611328, 0.59185, 0.559348],
         [0.65218, 0.63474, 0.611145, 0.567125, 0.48524]]
chi_at_tc: [1.19548, 4.13566, 13.7756]
cross_8_16: 2.26325
cross_16_32: 2.26858
chi_slope: 1.76323

✓ both crossings are within 1% of T_c
✓ the susceptibility at T_c grows with slope between 1.6 and 1.9

Open in the workbench Atlas More science claims

Full statement: In Monte Carlo of the two-dimensional Ising model (Wolff updates on $8 \times 8$ , $16 \times 16$ and $32 \times 32$ lattices), the Binder cumulants of successive sizes cross at about 2.263 and 2.269, both within 1% of Onsager's $T_{c} = 2.2692$ , and the susceptibility at $T_{c}$ grows with the lattice size with a log-log slope of about 1.76, between 1.6 and 1.9, close to the exact $γ / ν = 7/4$ .
Scope: Wolff cluster updates, fixed seeds, periodic $L \times L$ lattices with $L = 8, 16, 32$ ; the crossings are located by linear interpolation between the temperatures sampled, so their error includes the statistical error of each run (about 0.5% here). Corrections to scaling at these small sizes are not fitted.
Reference: Onsager's critical temperature $T_{c} = 2/ ln (1 + 2) = 2.26919$ ( $J = k_{B} = 1$ ) and the exponent $γ / ν = 7/4$ of the two-dimensional Ising model; Binder, Z. Phys. B 43, 119 (1981), for the crossing of the cumulants $U = 1 - ⟨ m^{4} ⟩ / (3 ⟨ m^{2} ⟩^{2})$ at $T_{c}$ . lib/ising's Wolff and Metropolis runs are checked against the exact $4 \times 4$ enumeration and Kaufman's finite-lattice solution in stat/ising (the Ising gallery entry).

claims/science/ising_binder_crossing.viv#b426b8a9dd