Compute lattice energy from spins, coupling, and field. Compare open and periodic boundaries easily. Export results for reporting and future simulations.
The calculator uses the nearest-neighbor Ising Hamiltonian with an external field:
E = −J Σ⟨ij⟩ (sᵢ sⱼ) − h Σᵢ sᵢ, where sᵢ ∈ {+1, −1}
| Case | Dimension | Spins | J | h | Boundary | Total Energy E | Magnetization M |
|---|---|---|---|---|---|---|---|
| A | 1D | +1 −1 +1 −1 +1 −1 +1 −1 | 1 | 0 | Open | +7 | 0 |
| B | 1D | +1 +1 +1 +1 +1 +1 | 1 | 0 | Periodic | −6 | +6 |
| C | 2D | 4×4 checkerboard | 1 | 0 | Periodic | +32 | 0 |
In spin-lattice studies, energy is the central score that ranks configurations. Lower energy states are more probable at low temperature, while higher energy states appear more often as thermal agitation grows. By turning a spin map into a single number, the calculator helps compare patterns, tune parameters, and sanity-check simulation output.
The nearest-neighbor coupling uses products sisj, which equal +1 for aligned spins and −1 for opposite spins. With positive J, aligned neighbors reduce energy, encouraging ferromagnetic domains. With negative J, alternating neighbors are preferred, producing antiferromagnetic order on bipartite lattices and a larger energy penalty for local alignment.
The field term depends on magnetization M = Σsi. A positive h favors up spins, shifting energy downward when M is positive. When h is zero, configurations with the same bond structure but opposite global orientation are energetically identical. Field control is useful for biasing initial states and exploring hysteresis-like behavior in parameter sweeps.
Open boundaries omit missing neighbors at edges, reducing the number of bonds relative to the bulk. Periodic boundaries wrap the lattice so edge spins interact across the boundary. This increases the bond count and reduces finite-size edge artifacts, which is valuable when comparing energies across different lattice sizes or when validating periodic Monte Carlo implementations.
Total energy grows with system size, so E/N is a cleaner comparison metric. For a large 2D lattice with periodic boundaries, each spin effectively participates in a fixed number of neighbor interactions. Reporting E per spin helps detect whether changes come from physics (J, h, structure) or simply from adding more spins to the model.
If T and kB are provided, the calculator returns ln w = −E/(kB·T) and w = exp(−E/(kB·T)). In practice, ln w is safer for large systems because exp can overflow. These values support quick checks of acceptance ratios and relative likelihoods between two candidate configurations at the same temperature.
Manual entry accepts +1/−1 (or U/D tokens) and verifies rectangular 2D grids. Random mode generates spins using a controllable p(up). Providing a seed reproduces the same configuration, which is helpful for benchmarking, debugging, and sharing examples in reports where identical inputs must yield identical energies.
A strongly negative bond sum typically indicates large aligned domains for J > 0, while a positive bond sum suggests frequent anti-alignment. If E changes unexpectedly, confirm the boundary setting and inspect the spin preview. For 2D inputs, checkerboard patterns should favor J < 0. For 1D, periodic closure adds one extra bond.
Use +1 or −1. You can also type UP/DOWN or U/D. Any other value will be rejected to keep the Hamiltonian consistent.
No. It sums each nearest-neighbor pair once by checking right and down neighbors in 2D, and forward neighbors in 1D.
Use periodic boundaries to reduce edge effects and approximate an infinite system. This is especially useful for comparing energies across lattice sizes.
Negative J favors anti-aligned neighbors. In 2D on a square lattice, this typically promotes a checkerboard (antiferromagnetic) pattern.
If T ≤ 0 or kB ≤ 0, the factor is undefined. Also, for very large |E|/(kB·T), exp may overflow, so the calculator prioritizes ln w.
M is total magnetization, the sum of all spins. m is magnetization per spin, m = M/N, which is easier to compare across different system sizes.
Yes. Manual 2D lattices can be large, but keep rows rectangular. The preview shows only the first 10×10 spins to remain readable.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.