Field-controlled magnetic order in the quantum spin-ladder system ${\left(\text{Hpip}\right)}_2{\text{CuBr}}_4$

Neutron diffraction is used to investigate the field-induced, antiferromagnetically ordered state in the two-leg spin-ladder material ${\left(\text{Hpip}\right)}_2{\text{CuBr}}_4$. This “classical” phase, a consequence of weak interladder coupling, is nevertheless highly unconventional: its properties are influenced strongly by the spin Luttinger-liquid state of the ladder subunits. We determine directly the order parameter (transverse magnetization), the ordering temperature, the spin structure, and the critical exponents around the transition. We introduce a minimal microscopic model for the interladder coupling and calculate the quantum fluctuation corrections to the mean-field interaction.

Figure 1

(Color online) Low-$T$ phase diagram of ${\left(\text{Hpip}\right)}_2{\text{CuBr}}_4$. The crossover temperature to the spin-LL phase is derived from MCE measurements and the phase transition to the BEC (3D-$\mathrm{XY}$ magnetic order) from neutron diffraction. The contour plot is based on 18 individual field scans of the MCE (two shown as gray lines), using $\left(\delta Q/\delta B\right)/T=-\left(\partial m_z/\partial T\right){\mid }_B$ (Ref. 15). The red line is based on a theoretical fit (see text).

Figure 2

(Color online) Summary of neutron-diffraction data. (a) $Q$-scans across an AF Bragg peak after subtraction of a flat background measured in the QD phase at $B=6\text{T}$ and $T=63\text{mK}$. (b) $T$-dependence of the Bragg intensity, demonstrating the onset of 3D long-range order at $T_{\mathrm{N}}\left(B\right)$; solid lines are fits using the 3D-$\mathrm{XY}$ exponent. (c) $B$-dependence of $m_x^2$ measured at $Q=\left(1.500\right)$ for $T=54\text{mK}$ (blue) and $T=75\text{mK}$ (red). Solid and dashed lines are from the DMRG MFA for the given interladder interaction $J^{\prime }$. Error bars on data points are based on counting statistics, while the vertical black line indicates the systematic error of the calibration to absolute units. (d) Magnetic signal at $Q=\left(102\right)$, which is proportional to $m_z^2$: shown is the neutron intensity after subtraction of the nuclear contribution, which also corrects for magnetostriction effects. The red line is obtained from a QMC calculation.

Figure 3

(Color online) Magnetic structure (black arrows) in the 3D ordered phase of ${\left(\text{Dpip}\right)}_2{\text{CuBr}}_4$, determined by neutron diffraction at $B=8.6\text{T}$ and $T=63\text{mK}$. Only Cu atoms forming the ladders are shown (red and blue). The spin component along the field $\left(B\parallel b\right)$ is fixed by QMC calculations. Projections are shown (a) on the $bc$ plane and (b) on the $ac$ plane. Interladder bonds $J^{\prime }$ for one rung are shown in green.

Figure 4

(Color online) The 3D-$\mathrm{XY}$ phase boundary by DMRG MFA and QMC. (a) $T_{\mathrm{N}}\left(B\right)$ for several values of $J^{\prime }$, as indicated, $\alpha =0.74\left(1\right)$. (b) $T_{\mathrm{N}}$ as a function of $J^{\prime }$.