Higgs Mode of Planar Coupled Spin Ladders and its Observation in ${\mathrm{C}}_9{\mathrm{H}}_{18}{\mathrm{N}}_2{\mathrm{CuBr}}_4$

Polarized inelastic neutron scattering experiments recently identified the amplitude (Higgs) mode in ${\mathrm{C}}_9{\mathrm{H}}_{18}{\mathrm{N}}_2{\mathrm{CuBr}}_4$, a two-dimensional near-quantum-critical spin-$1/2$ two-leg ladder compound, which exhibits a weak easy-axis exchange anisotropy. Here, we theoretically examine the dynamic spin structure factor of such planar coupled spin-ladder systems using large-scale quantum Monte Carlo simulations. This allows us to provide a quantitative account of the experimental neutron scattering data within a consistent quantum spin model. Moreover, we trace the details of the continuous evolution of the amplitude mode from a two-particle bound state of coupled ladders in the classical Ising limit all the way to the quantum spin-$1/2$ Heisenberg limit with fully restored SU(2) symmetry, where it gets overdamped by the two-magnon continuum in neutron scattering.

Figure 1

Ground-state phase diagram of a 2D array of coupled spin-$1/2$ ladders with easy-axis anisotropy $\lambda$ and $J_{\mathrm{rung}}=J_{\mathrm{leg}}$. Inset: 2D array of coupled ladders ($L=4$ ladders, $L_r=8$ rungs per ladder), with a two-site unit cell as indicated.

Figure 2

DSF of a 2D array of coupled spin ladders, (a) $S_T(\mathit{k},\omega )$ and (b) $S_L(\mathit{k},\omega )$, for $J_{\mathrm{rung}}=J_{\mathrm{leg}}$, $J_{\text{inter}}=0.228J_{\mathrm{leg}}$, and $\lambda =0.6$, along the indicated path in momentum space, obtained by QMC simulations with $L=20$ at $T=0.02J_{\mathrm{leg}}$.

Figure 3

Excitation gaps ${\mathrm{\Delta }}_{\mathrm{TM}}$, $2{\mathrm{\Delta }}_{\mathrm{TM}}$, and ${\mathrm{\Delta }}_{\mathrm{LM}}$ as functions of $\lambda$ at $J_{\mathrm{rung}}=J_{\mathrm{leg}}$ for (a) $J_{\text{inter}}=0.228J_{\mathrm{leg}}$ (case I) and (b) $J_{\text{inter}}=0.4J_{\mathrm{leg}}$ (case II). Dashed lines show results from series expansions. The triangle in (b) shows the position of the Higgs mode for $\lambda =1$ from the dynamic singlet structure factor.

Figure 4

Excitation gap ${\mathrm{\Delta }}_{\mathrm{TM}}$ from QMC and PCUTs (bare order ${\lambda }^8$ series and Dlog-Padé approximants) as functions of $\lambda$ for $J_{\mathrm{leg}}=0.6\mathrm{meV}$, $J_{\mathrm{rung}}=0.64\mathrm{meV}$, and $J_{\text{inter}}=0.19\mathrm{meV}$. Horizontal lines set the margin of the INS estimate ${\mathrm{\Delta }}_{\mathrm{TM}}=0.33(3)\mathrm{meV}$ for DLCB from Ref. [11]. The inset shows ${\mathrm{\Delta }}_{\mathrm{TM}}$, ${\mathrm{\Delta }}_{\mathrm{LM}}$, and $2{\mathrm{\Delta }}_{\mathrm{TM}}$ as functions of $\lambda$ as obtained from QMC simulations.

Figure 5

Scattering spectra for DLCB as a function of energy and wave vector transfer in the transverse (a),(b) and longitudional (c),(d) configurations, exhibiting the TM and LM modes, respectively. Data based on QMC simulations ($L=20$) of the 2D model $H$ for the displayed parameters.