Fig. 1: Realization of a supersymmetric t-J chain from a spin ladder in an applied magnetic field.

a Schematic representation of the structure of (C₅D₁₂N)₂CuBr₄ (BPCB) and (C₅D₁₂N)₂CuCl₄ (BPCC) projected on the ac plane, accompanied by a photograph of one BPCC crystal. Cu²⁺ ions (blue) form the ladder and halide (Br⁻ or Cl⁻) ions (green) the superexchange paths yielding the interactions J_⊥ (yellow) and J_∥ (blue). The piperidinium ions are omitted for clarity. b Evolution of triplet excitation branches in a strong-rung spin-1/2 ladder in a field, B^z. For B_c ≤ B^z ≤ B_s, the spin degrees of freedom fractionalize and the response becomes finite over a continuum of energies in each of the triplet branches t⁺, t⁰ and t⁻ (the illustration represents an integral over all q_h in panel c). c Many-body physics in the spin ladder at zero temperature and half-magnetization (m^z = 0.5). The t⁺ sector gives rise to the turquoise two-spinon continuum and the t⁰ sector the blue spinon-holon continuum, which is described by an adapted t-J model. Supersymmetry between the spinon and holon is manifest at wavevector q_h = 0 (green). d Schematic representations of (i) singlet \((\vert s\rangle=\frac{1}{\sqrt{2}}(\vert \uparrow \downarrow \rangle -\vert \downarrow \uparrow \rangle ))\) and triplet \((\vert {t}^{+}\rangle=\vert \uparrow \uparrow \rangle)\) states on the ladder rungs in the ground manifold created by the applied field; (ii) creation of a “middle triplon” \((\vert {t}^{0}\rangle=\frac{1}{\sqrt{2}}(\vert \uparrow \downarrow \rangle+\vert \downarrow \uparrow \rangle ))\) by a neutron-mediated spin excitation on a rung; (iii) propagation of the \(\vert {t}^{0}\rangle\) excitation and rearrangement of the \(\vert s\rangle\)-\(\vert {t}^{+}\rangle\) configuration. e Representation of these states in the basis of pseuospins (red arrows) \(\vert \tilde{\uparrow }\rangle \equiv \vert {t}^{+}\rangle\) and \(\vert \tilde{\downarrow }\rangle \equiv \vert s\rangle\) on a chain (i), where flipping one spin creates two freely mobile ___domain walls (spinons). Exciting a \(\vert {t}^{0}\rangle\) by INS is analogous to a photoemission process, where the injected “hole” is accompanied by one spinon (ii), both of which can propagate separately (iii). f Representation of supersymmetry as an extended symmetry connecting the three site-basis states of the pure t-J model at J = 2t: the additional duality between the fermionic spins (with SU(2) spin symmetry) and the bosonic hole (with U(1) charge symmetry), which makes them superpartners, is encoded in the supercharge operators Q_σ and \({Q}_{\sigma }^{{\dagger} }\).