Direct Observation of Magnon Fractionalization in the Quantum Spin Ladder

We measure by inelastic neutron scattering the spin excitation spectra as a function of applied magnetic field in the quantum spin-ladder material $({\mathrm{C}}_5{\mathrm{H}}_{12}\mathrm{N}{)}_2{\mathrm{CuBr}}_4$. Discrete magnon modes at low fields in the quantum disordered phase and at high fields in the saturated phase contrast sharply with a spinon continuum at intermediate fields characteristic of the Luttinger-liquid phase. By tuning the magnetic field, we drive the fractionalization of magnons into spinons and, in this deconfined regime, observe both commensurate and incommensurate continua.

Figure 1

Spin dynamics in the QD phase ($B<B_c$) at $T=50\mathrm{mK}$. (a)–(c) Triplet dispersion at $B=0\mathrm{T}$ from constant-$Q$ scans along the ladder axis and in the two perpendicular directions. (d)–(e) Triplet excitations at $\mathit{Q}=[1.500]$ in fields of 0 and 4 T. (f) Zeeman splitting of the triplet modes at $\mathit{Q}=[1.500]$. Solid lines are fits explained in the text.

Figure 2

Excitation spectrum in the LL phase ($T_N<T=250\mathrm{mK}<T_{\mathrm{LL}}$) at $B=10.1\mathrm{T}$ ($m\approx 0.5$) after subtraction of the zero-field background. (a) Measured (left) and simulated (right) INS intensities. Solid lines mark the edges of the two-spinon continuum. (b),(d) Constant-$E$ scans taken along maxima of the transverse structure factor (insets: scan trajectories in white). (c) Constant-$Q$ scan at $\mathit{Q}=[0.500.61]$. Black dashed lines in (b)–(d) are based on a $\delta$-function spinon spectrum [solid lines in panel (a)], red solid lines and shading on a full continuum calculation.

Figure 3

Incommensurate excitations in the LL phase. (a),(b) Spinon continua calculated for an $S=1/2$ $XXZ$ chain with anisotropy $\delta =0.5$ (see text) at $m=0.25$, 0.75 (a) and $m=0.5$ [(b), data shown also in Fig. 2a convolved with instrumental resolution]; dashed white lines represent INS scans at $E=0.2$ and 0.4 meV. (c)–(f) Constant-$E$ scans (trajectories as in Fig. 2b inset) for $B=7.9$ and $12.3\mathrm{T}$. (g) Constant-$Q$ scans at the ZB for two chosen magnetizations $m$ between 0 and 0.5. (h) Summary of data for ZB excitation energy as a function of $m$. Red lines are predictions based on the $XXZ$ chain: solid from panel (a) and dashed (shown for comparison) from panel (b).

Figure 4

Spin dynamics in the FM phase. (a) Magnetic-field dependence of the ZB excitation energy. (b),(c) Magnon dispersions along $Q_l$ and $Q_h$. (d),(e) Constant-$E$ scans at $B=14.8\mathrm{T}$ and $T=50\mathrm{mK}$ with Gaussian fits. Solid lines in panels (a)–(c) are fits to the dispersion, while dashed lines allow extraction of $B_s$.