Spin ladder

A spin ladder is a system of coupled chains of spins, with interactions along the chains and between neighboring chains. The chains are called legs and their transverse couplings are called rungs; the model provides a controlled step between one-dimensional and two-dimensional magnetism.

In a two-leg spin-$1/2$ ladder with strongly antiferromagnetic rung coupling, pairs on each rung approximately form singlets. Exciting a rung to a triplet costs finite energy, illustrating a possible spin gap. Different couplings and leg numbers need not share that behavior. Magnetic response and excitation spectra test the model rather than the ladder geometry alone fixing the ground state.

A spin gap is a nonzero minimum energy required to reach a specified magnetic excitation above the ground state. One must specify which spin sector or measurement couples to that excitation; not every excitation gap is a spin gap.

Two antiferromagnetically coupled spin-$1/2$ particles illustrate it. For $H=J{\mathbf{S}}_1\cdot {\mathbf{S}}_2$ with dimensionless spin operators and $J>0$, the singlet has energy $-3J/4$ and the triplet has $J/4$, giving gap $J$. Thermal magnetic response is suppressed when $k_{B}T$ is well below that scale. Extended systems require their collective excitation spectrum rather than this isolated-pair argument alone.