Ising Heat Capacity Calculator

Formula used

Heat capacity is obtained from energy fluctuations or derivatives of the partition function. Using inverse temperature β = 1/(kB·T):

• 1D exact (periodic chain): partition function Z = λ+^N + λ−^N, where λ± = e^{βJ}cosh(βh) ± √(e^{2βJ}sinh²(βh) + e^{-2βJ}). Then U = −∂ ln Z/∂β and C = kB·β²·∂² ln Z/∂β².
• 2D mean-field: magnetization solves m = tanh(β(zJm + h)), with coordination z=4. Energy per spin u = −(zJ/2)m² − h m, and C = dU/dT is estimated numerically.
• 2D Monte Carlo: using Metropolis sampling, total heat capacity C = (⟨E²⟩ − ⟨E⟩²)/(kB·T²).

Numerical derivatives use a small central step to remain stable near sharp changes.

How to use this calculator

1. Select a method: exact 1D, mean-field 2D, or Monte Carlo 2D.
2. Enter T, J, and optional h. Keep T>0.
3. Set kB (use 1 for reduced units).
4. Provide system size: N for 1D, or L for 2D.
5. For Monte Carlo, tune sweeps, burn-in, and sample interval.
6. Click Calculate. The result appears above the form.
7. Use Download CSV or Download PDF for exports.

Example data table

For Monte Carlo, results vary slightly across seeds and run lengths.

Article: Technical notes and interpretation for the Ising heat capacity outputs.

Why heat capacity matters in the Ising model

Heat capacity quantifies how strongly internal energy responds to temperature. In lattice spin systems it highlights critical behavior through peaks and divergences. In reduced units (kB=J=1), the 2D square-lattice model has a well-known critical region near T≈2.269, where energy fluctuations grow rapidly and finite-size effects become important.

What this calculator computes

This tool estimates heat capacity per spin using three selectable methods: an exact 1D transfer-matrix expression, a 2D mean-field approximation, and a 2D Metropolis Monte Carlo estimator. Outputs include energy per spin, heat capacity, magnetization estimates, and uncertainty metrics when sampling is stochastic.

1D exact chain: smooth thermodynamics

For a periodic 1D chain, the transfer-matrix eigenvalues λ± determine the partition function Z=λ+^N+λ−^N. From Z, the calculator derives energy and C via β-derivatives. The 1D model has no finite-temperature phase transition, so C(T) remains finite and typically shows a broad, nondivergent maximum.

Mean-field 2D: fast insight with limitations

Mean-field replaces neighbors by an average magnetization m. The self-consistency equation m=tanh((zJm+h)/(kB T)) with coordination z=4 gives an energy estimate and C from numerical temperature derivatives. It predicts a critical temperature T_c^MF=zJ/kB, which overestimates the exact 2D value, but trends are useful for quick scans.

Monte Carlo 2D: fluctuation-based estimator

In Metropolis sampling, the heat capacity uses energy fluctuations: C = (⟨E^2⟩−⟨E⟩^2)/(kB T^2). The calculator reports C per spin by dividing by L^2. Near criticality, autocorrelation times increase, so longer runs and wider sampling intervals reduce bias in ⟨E⟩ and ⟨E^2⟩.

Finite-size scaling and peak sharpening

For finite L, the critical divergence is rounded into a peak whose height and location depend on size. As L grows, the peak becomes taller and shifts toward the thermodynamic critical point. Comparing L=16, 32, 64 across temperatures helps visualize scaling and distinguish physical peaks from statistical noise.

Practical parameter choices

For Monte Carlo, typical starting points are 5,000–20,000 burn-in sweeps and 20,000–200,000 measurement sweeps, with sampling every 5–20 sweeps. Use h=0 for symmetry and to emphasize critical behavior, or small |h| to smooth transitions. Keep T>0 and choose J>0 for ferromagnetic ordering.

Interpreting outputs and exporting results

Treat Monte Carlo uncertainty as an estimate, not a guarantee; critical slowing down and insufficient equilibration can underestimate errors. Cross-check by repeating runs with different seeds, increasing sweeps, and comparing against mean-field or 1D behavior. Export buttons generate CSV for data pipelines and PDF for reporting.