Multiple Magnon Modes and Consequences for the Bose-Einstein Condensed Phase in ${\mathrm{BaCuSi}}_2{\mathrm{O}}_6$

The compound ${\mathrm{BaCuSi}}_2{\mathrm{O}}_6$ is a quantum magnet with antiferromagnetic dimers of $S=1/2$ moments on a quasi-2D square lattice. We have investigated its spin dynamics by inelastic neutron scattering experiments on single crystals with an energy resolution considerably higher than in an earlier study. We observe multiple magnon modes, indicating clearly the presence of magnetically inequivalent dimer sites. The more complex spin Hamiltonian revealed in our study leads to a distinct form of magnon Bose-Einstein condensate phase with a spatially modulated condensate amplitude.

Figure 1

Physical properties at a field-induced QPT in dimer-based $S=1/2$ spin systems. (a) triplet dispersion at zero field, (b) uniform magnetization $M_z$, (c) dimer lattice with triplet condensate amplitude for $H>H_c$ represented by increasing gray scale. (I) 3D systems with uniform magnon BEC, e.g., ${\mathrm{TlCuCl}}_3$. (II) Shastry-Sutherland geometry with magnetization plateaus and strong contrast in condensate amplitude, e.g., ${\mathrm{SrCu}}_2({\mathrm{BO}}_3{)}_2$ [field in panel (c) yielding $M_z=1/3$]. (III) Same features for 3D system with strong translational symmetry-breaking and weak interdimer interactions, e.g., ${\mathrm{NH}}_4{\mathrm{CuCl}}_3$ at $H_{c2}<H<H_{s2}$. (IV) Quasi-2D system on square lattice with weakly broken translational symmetry showing no magnetization plateaus, e.g., schematic model for ${\mathrm{BaCuSi}}_2{\mathrm{O}}_6$ at $H_c<H<H_3$.

Figure 2

(a) Rocking curves across two Bragg peaks of a coaligned ${\mathrm{BaCuSi}}_2{\mathrm{O}}_6$ sample. Horizontal line indicates instrumental resolution (full width) of neutron diffractometer MORPHEUS. (b)–(d) INS spectra of ${\mathrm{BaCuSi}}_2{\mathrm{O}}_6$ at $T=1.8\mathrm{K}$ and $H=0\mathrm{T}$, experimental conditions as indicated. Momentum vectors in reciprocal lattice units of the tetragonal space group $I{4}_1/acd$ with $a=b=10.01\AA$ and $c=22.47\AA$ [17], i.e., $(\begin{array}{ccc}{q}_h& 0& 4\end{array})$ is the diagonal direction on the square lattice.

Figure 3

(a)–(b) Triplet dispersion in ${\mathrm{BaCuSi}}_2{\mathrm{O}}_6$ at zero field and $T=1.8\mathrm{K}$: (a) in-plane along $\mathit{Q}=(\begin{array}{ccc}{q}_h& 0& 4\end{array})$, (b) perpendicular to plane with $\mathit{Q}=(\begin{array}{ccc}0& 0& q_l\end{array})$ (see inset). (c)–(d) Momentum-dependence of (scaled) INS intensity, with solid lines as described in text: total intensity modes I–III (black), mode–I (red), and modes $\mathrm{II}+\mathrm{III}$ (blue) measured with $k_f=1.3{\AA }^{-1}$ (solid circles) and $1.5{\AA }^{-1}$ (open circles).

Figure 4

(a)–(c) INS spectra at three magnetic-field values in ${\mathrm{BaCuSi}}_2{\mathrm{O}}_6$ for $\mathit{Q}=(\begin{array}{ccc}0.6& 0& 4\end{array})$, $T=1.8\mathrm{K}$, and $k_f=1.3{\AA }^{-1}$. (d) Zeeman splitting of triplet modes following procedure described in text. (e) Schematic phase diagram around QPT; intensity of red shading indicates qualitative degree of $c$-axis modulation of BEC order parameter.