Field-induced criticality in a gapped quantum magnet with bond disorder

Neutron diffraction and calorimetric measurements are used to study the field-induced quantum phase transition in piperazinium-Cu${}_2$(Cl${}_{1-x}$Br${}_x$)${}_6$ ($x=0$, $3.5$$\%$, and $7.5$$\%$), a prototypical quantum antiferromagnet with random bonds. The critical indexes $\phi$ and $\beta$ are determined. The findings contradict some original predictions for Bose glass based on the assumption $z=d$, but are consistent with recent theoretical results implying $z⩽d$. Inelastic neutron experiments reveal that disorder has a profound effect on the lowest-energy magnetic gap excitation in the system.

Figure 1

Symbols: Heat capacity of PHCX with varying Br concentration $x$ measured (a) as a function of magnetic field at $T=0.5$ K and (b) as a function of temperature at $H=14$ T. The solid lines are fits to the data as described in the text.

Figure 2

(a) Phase boundaries of an ordered state for Br-free, 3.5$\%$ Br, and 7.5$\%$ Br substituted PHCX. Open and filled circles are from $C\left(H\right)$ and $C\left(T\right)$ measurements, respectively. Solid lines are power laws $H-H_c\sim T_c{\left(H\right)}^{1/\phi }$ with critical fields $H_c$ and crossover exponents $\phi$ derived from the data as described in the text. (b)–(d) Crossover exponent $\phi$ obtained from a least-squares fit as a function of a fitting window given by $T<T_{\text{max}}$. Filled symbols show the value quoted in Table  satisfiying our best-fit criteria described in the text.

Figure 3

Upper panels: Neutron diffraction intensity of the ($\frac{1}{2}0\overline{\frac{1}{2}}$) peak measured in PHCX, (a) as a function of magnetic field at $T=160$ mK, (b) as a function of temperature. Open and solid symbols correspond to zero field cooling and field cooling, respectively. Squares (circles) correspond to $x=3.5$$\%$ ($7.5$$\%$) Br content. Lower left corner: the trajectory in $(H,T)$ space for the measurements on the $x=7.5$$\%$ material. Inset: rocking curve measured at $T=0.5$ K (filled) and $T=2$ K (empty circles) in that sample. Lower panels: order-parameter exponents as obtained by shrinking the fitting interval analysis of the 3.5$\%$ (c) and 7.5$\%$ Br (d) sample data. Filled symbols show the best-fit values of $\beta$.

Figure 4

Inelastic neutron scattering intensity at ($0.5,0.5,-0.5$) in the energy transfer range from 0.5 to 2.5 meV for $x=3.5$$\%$ (green squares) and $x=7.5$$\%$ Br (red circles) content samples. Corresponding solid lines are the fitted Lorentzian energy profiles convoluted with the four-dimensional instrumental resolution function. For 0.0$\%$ Br sample the resolution convoluted simulation is shown for comparison using parameters from Ref. 15