Bound States and Field-Polarized Haldane Modes in a Quantum Spin Ladder

The challenge of one-dimensional systems is to understand their physics beyond the level of known elementary excitations. By high-resolution neutron spectroscopy in a quantum spin-ladder material, we probe the leading multiparticle excitation by characterizing the two-magnon bound state at zero field. By applying high magnetic fields, we create and select the singlet (longitudinal) and triplet (transverse) excitations of the fully spin-polarized ladder, which have not been observed previously and are close analogs of the modes anticipated in a polarized Haldane chain. Theoretical modeling of the dynamical response demonstrates our complete quantitative understanding of these states.

Figure 1

One- and two-magnon excitations in BPCC. (a) $S(\mathbf{Q},\omega )$ measured in the $q_y=\pi$ sector. The solid red line is the dispersion obtained from strong-coupling [Eq. (S5) of the SM 30] and bond-operator treatments [Eq. (S6) of the SM 30]. (b) $S(\mathbf{Q},\omega )$ measured for $q_y=0$. The green line indicates the dispersion [Eq. (S8) of the SM 30] and range [Eq. (1)] over which the triplet component of the two-magnon bound state is expected in BPCC. The white line marks the boundary of the two-magnon continuum [Eq. (S9) of the SM 30]. (c) $S(\mathbf{Q},\omega )$ for $q_y=\pi$ obtained by DMRG and SE calculations, whose results are indistinguishable. (d) $S(\mathbf{Q},\omega )$ from DMRG for $q_y=0$. (e) $I(\mathbf{Q})$ obtained by integration over $Q_k$ (denoted ${\overline{Q}}_k$) and over the energy range $0.19\le \omega \le 0.43\mathrm{meV}$, shown as red points and computed in the SMA (dashed black) and by DMRG (solid red lines); the solid black line denotes the background and $Q_{l_{\max }}$ is defined in Eq. (S4) of the SM [30]. (f) Bound-state (green) and continuum (gray) contributions to $I(\mathbf{Q})$, obtained by integrating over energy windows below and above the two-magnon continuum edge; measured intensities (points) are compared with DMRG (solid line) and SE (dashed line) calculations. The black line denotes the background and $Q_{l_{\min }}$ is defined in Eq. (S4) of the SM [30]. (g) Ground state of antisymmetric rung singlets in the two-leg ladder. (h) Ground state of the Haldane chain in the representation of auxiliary $S=1/2$ spin pairs; red boxes represent the operator projecting the two spins at each site to a real $S=1$ state. (i) One-magnon excited state, a single symmetric rung triplet. (j) Two-magnon excited state, a triplet pair.

Figure 2

Excitations in fully polarized BPCC. (a) Magnetization of BPCC (from DMRG) superimposed on the field-induced phase diagram; LL denotes the Luttinger-liquid and QD the quantum disordered phase. (b) $S(\mathbf{Q},\omega )$ measured at $H_1=4.067$ $\mathrm{T}\simeq H_s$ and (d) $H_2=6.141\mathrm{T}$, both in the $q_y=0$ sector. (c) $S(\mathbf{Q},\omega )$ measured at $H_2$ in the $q_y=\pi$ sector. Green and red solid lines are, respectively, fits to the triplet ($q_y=0$) and singlet ($q_y=\pi$) modes using Eq. (2). The dashed line in panel (c) indicates the position of the singlet at $H_s$ (not measured due to dominant nuclear incoherent scattering at low energies). (e) Schematic representation of a FP two-leg ladder with one rung-singlet excitation. (f) FP ladder with one rung-triplet ($t_0$) excitation. (g) FP Haldane chain in the auxiliary-spin representation containing one leg-singlet and (h) one leg-triplet excitation.