3D Ising Model
FMStripedPM
Phase:Paramagnetic
ε:ⓘEnergy per site in units of |J₁| (kB = 1; N ≡ L³ = total sites).
H = −J₁Σ⟨ij⟩sisj − J₂Σ⟪ij⟫sisj − hΣsi
ε = H / (N|J₁|).
Ground state (FM, J₁ > 0, J₂ = 0, h = 0): ε = −3 (6 NN bonds × ½ per site).
High-T limit: ε → 0.
Shown as mean ± 1σ once ≥ 20 sweeps are sampled.0.0000 |J1|
H = −J₁Σ⟨ij⟩sisj − J₂Σ⟪ij⟫sisj − hΣsi
ε = H / (N|J₁|).
Ground state (FM, J₁ > 0, J₂ = 0, h = 0): ε = −3 (6 NN bonds × ½ per site).
High-T limit: ε → 0.
Shown as mean ± 1σ once ≥ 20 sweeps are sampled.0.0000 |J1|
M:ⓘFerromagnetic order parameter.
M = (1/N) Σ si ∈ [−1, +1].
Below Tc (FM): |M| → 1 as T → 0 (spontaneous symmetry breaking at h = 0).
Above Tc: time-average ⟨M⟩ = 0.
A bimodal magnetisation histogram indicates ergodicity breaking in the ordered phase.0.0000
M = (1/N) Σ si ∈ [−1, +1].
Below Tc (FM): |M| → 1 as T → 0 (spontaneous symmetry breaking at h = 0).
Above Tc: time-average ⟨M⟩ = 0.
A bimodal magnetisation histogram indicates ergodicity breaking in the ordered phase.0.0000
MNéel:ⓘStaggered magnetisation — order parameter for Néel antiferromagnetism.
MNéel = (1/N) Σ si(−1)xi+yi+zi
→ ±1 in a perfect checkerboard; 0 in the paramagnetic phase.
Néel order is stable for J₁ < 0, |J₂/J₁| ≲ 1/2 (3D SC lattice).0.0000
MNéel = (1/N) Σ si(−1)xi+yi+zi
→ ±1 in a perfect checkerboard; 0 in the paramagnetic phase.
Néel order is stable for J₁ < 0, |J₂/J₁| ≲ 1/2 (3D SC lattice).0.0000
Mstripe:ⓘStripe order parameter.
Mstripe = maxα √(S(kα) / N),
kα ∈ {(π,0,0), (0,π,0), (0,0,π)} (X-points of the 3D SC Brillouin zone).
Detects layered phases where spins modulate along a single axis.
Ground-state boundaries (3D SC): stripe ↔ FM at |J₂/J₁| = 1/4 (FM J₁); stripe ↔ Néel at |J₂/J₁| = 1/2 (AFM J₁).0.0000
Mstripe = maxα √(S(kα) / N),
kα ∈ {(π,0,0), (0,π,0), (0,0,π)} (X-points of the 3D SC Brillouin zone).
Detects layered phases where spins modulate along a single axis.
Ground-state boundaries (3D SC): stripe ↔ FM at |J₂/J₁| = 1/4 (FM J₁); stripe ↔ Néel at |J₂/J₁| = 1/2 (AFM J₁).0.0000
Cv:ⓘSpecific heat per site (kB = 1).
Cv = N · Var(ε) / T*² — fluctuation-dissipation theorem.
Diverges at Tc in the thermodynamic limit (critical exponent α ≈ 0.11 for 3D Ising).
Finite-size peak scales as Lα/ν ≈ L0.17 — nearly flat for large L.
Requires ≥ 20 energy samples.waiting
Cv = N · Var(ε) / T*² — fluctuation-dissipation theorem.
Diverges at Tc in the thermodynamic limit (critical exponent α ≈ 0.11 for 3D Ising).
Finite-size peak scales as Lα/ν ≈ L0.17 — nearly flat for large L.
Requires ≥ 20 energy samples.waiting
χ:ⓘMagnetic susceptibility per site.
χ = N · Var(M) / T* — fluctuation-dissipation theorem.
Diverges at Tc (critical exponent γ ≈ 1.24).
Finite-system peak height scales as Lγ/ν ≈ L1.97.
Requires ≥ 20 magnetisation samples.waiting
χ = N · Var(M) / T* — fluctuation-dissipation theorem.
Diverges at Tc (critical exponent γ ≈ 1.24).
Finite-system peak height scales as Lγ/ν ≈ L1.97.
Requires ≥ 20 magnetisation samples.waiting
ξ:ⓘSecond-moment correlation length in lattice units.
ξ = (1/|δk|) · √[Sconn(k*) / Sconn(k*+δk) − 1],
where k* is the ordering wavevector (Γ for FM, R for Néel AFM).
Diverges at Tc in infinite systems (ν ≈ 0.63). In finite systems it saturates near L/2.
Near T*c, Metropolis dynamics slows critically: τ ~ Lz(z ≈ 2). For L = 128, ~10⁴ MCS are needed to reach equilibrium — ξ > L/2 after a few hundred sweeps reflects non-equilibrium coarsening, not a true divergence.
Display: "waiting" = collecting data · < a = below one lattice spacing · > L/2 = deeply ordered or coarsening.waiting
ξ = (1/|δk|) · √[Sconn(k*) / Sconn(k*+δk) − 1],
where k* is the ordering wavevector (Γ for FM, R for Néel AFM).
Diverges at Tc in infinite systems (ν ≈ 0.63). In finite systems it saturates near L/2.
Near T*c, Metropolis dynamics slows critically: τ ~ Lz(z ≈ 2). For L = 128, ~10⁴ MCS are needed to reach equilibrium — ξ > L/2 after a few hundred sweeps reflects non-equilibrium coarsening, not a true divergence.
Display: "waiting" = collecting data · < a = below one lattice spacing · > L/2 = deeply ordered or coarsening.waiting
Sweeps:0
3D Ising Model
FMStripedPM
Phase:Paramagnetic
ε:ⓘEnergy per site in units of |J₁| (kB = 1; N ≡ L³ = total sites).
H = −J₁Σ⟨ij⟩sisj − J₂Σ⟪ij⟫sisj − hΣsi
ε = H / (N|J₁|).
Ground state (FM, J₁ > 0, J₂ = 0, h = 0): ε = −3 (6 NN bonds × ½ per site).
High-T limit: ε → 0.
Shown as mean ± 1σ once ≥ 20 sweeps are sampled.0.0000 |J1|
H = −J₁Σ⟨ij⟩sisj − J₂Σ⟪ij⟫sisj − hΣsi
ε = H / (N|J₁|).
Ground state (FM, J₁ > 0, J₂ = 0, h = 0): ε = −3 (6 NN bonds × ½ per site).
High-T limit: ε → 0.
Shown as mean ± 1σ once ≥ 20 sweeps are sampled.0.0000 |J1|
M:ⓘFerromagnetic order parameter.
M = (1/N) Σ si ∈ [−1, +1].
Below Tc (FM): |M| → 1 as T → 0 (spontaneous symmetry breaking at h = 0).
Above Tc: time-average ⟨M⟩ = 0.
A bimodal magnetisation histogram indicates ergodicity breaking in the ordered phase.0.0000
M = (1/N) Σ si ∈ [−1, +1].
Below Tc (FM): |M| → 1 as T → 0 (spontaneous symmetry breaking at h = 0).
Above Tc: time-average ⟨M⟩ = 0.
A bimodal magnetisation histogram indicates ergodicity breaking in the ordered phase.0.0000
MNéel:ⓘStaggered magnetisation — order parameter for Néel antiferromagnetism.
MNéel = (1/N) Σ si(−1)xi+yi+zi
→ ±1 in a perfect checkerboard; 0 in the paramagnetic phase.
Néel order is stable for J₁ < 0, |J₂/J₁| ≲ 1/2 (3D SC lattice).0.0000
MNéel = (1/N) Σ si(−1)xi+yi+zi
→ ±1 in a perfect checkerboard; 0 in the paramagnetic phase.
Néel order is stable for J₁ < 0, |J₂/J₁| ≲ 1/2 (3D SC lattice).0.0000
Mstripe:ⓘStripe order parameter.
Mstripe = maxα √(S(kα) / N),
kα ∈ {(π,0,0), (0,π,0), (0,0,π)} (X-points of the 3D SC Brillouin zone).
Detects layered phases where spins modulate along a single axis.
Ground-state boundaries (3D SC): stripe ↔ FM at |J₂/J₁| = 1/4 (FM J₁); stripe ↔ Néel at |J₂/J₁| = 1/2 (AFM J₁).0.0000
Mstripe = maxα √(S(kα) / N),
kα ∈ {(π,0,0), (0,π,0), (0,0,π)} (X-points of the 3D SC Brillouin zone).
Detects layered phases where spins modulate along a single axis.
Ground-state boundaries (3D SC): stripe ↔ FM at |J₂/J₁| = 1/4 (FM J₁); stripe ↔ Néel at |J₂/J₁| = 1/2 (AFM J₁).0.0000
Cv:ⓘSpecific heat per site (kB = 1).
Cv = N · Var(ε) / T*² — fluctuation-dissipation theorem.
Diverges at Tc in the thermodynamic limit (critical exponent α ≈ 0.11 for 3D Ising).
Finite-size peak scales as Lα/ν ≈ L0.17 — nearly flat for large L.
Requires ≥ 20 energy samples.waiting
Cv = N · Var(ε) / T*² — fluctuation-dissipation theorem.
Diverges at Tc in the thermodynamic limit (critical exponent α ≈ 0.11 for 3D Ising).
Finite-size peak scales as Lα/ν ≈ L0.17 — nearly flat for large L.
Requires ≥ 20 energy samples.waiting
χ:ⓘMagnetic susceptibility per site.
χ = N · Var(M) / T* — fluctuation-dissipation theorem.
Diverges at Tc (critical exponent γ ≈ 1.24).
Finite-system peak height scales as Lγ/ν ≈ L1.97.
Requires ≥ 20 magnetisation samples.waiting
χ = N · Var(M) / T* — fluctuation-dissipation theorem.
Diverges at Tc (critical exponent γ ≈ 1.24).
Finite-system peak height scales as Lγ/ν ≈ L1.97.
Requires ≥ 20 magnetisation samples.waiting
ξ:ⓘSecond-moment correlation length in lattice units.
ξ = (1/|δk|) · √[Sconn(k*) / Sconn(k*+δk) − 1],
where k* is the ordering wavevector (Γ for FM, R for Néel AFM).
Diverges at Tc in infinite systems (ν ≈ 0.63). In finite systems it saturates near L/2.
Near T*c, Metropolis dynamics slows critically: τ ~ Lz(z ≈ 2). For L = 128, ~10⁴ MCS are needed to reach equilibrium — ξ > L/2 after a few hundred sweeps reflects non-equilibrium coarsening, not a true divergence.
Display: "waiting" = collecting data · < a = below one lattice spacing · > L/2 = deeply ordered or coarsening.waiting
ξ = (1/|δk|) · √[Sconn(k*) / Sconn(k*+δk) − 1],
where k* is the ordering wavevector (Γ for FM, R for Néel AFM).
Diverges at Tc in infinite systems (ν ≈ 0.63). In finite systems it saturates near L/2.
Near T*c, Metropolis dynamics slows critically: τ ~ Lz(z ≈ 2). For L = 128, ~10⁴ MCS are needed to reach equilibrium — ξ > L/2 after a few hundred sweeps reflects non-equilibrium coarsening, not a true divergence.
Display: "waiting" = collecting data · < a = below one lattice spacing · > L/2 = deeply ordered or coarsening.waiting
Sweeps:0