Emergent critical phenomenon in spin-$\frac{1}{2}$ ferromagnetic-leg ladders: Quasi-one-dimensional Bose–Einstein condensate

We examine the magnetic-field-induced criticality of phase boundary near saturation field $H_{\mathrm{c}}$ in the spin-$\frac{1}{2}$ ferromagnetic (FM)-leg ladder 3-Cl-4-F-V [=3-(3-chloro-4-fluorophenyl)-1,5-diphenylverdazyl], the predominant interactions of which arise from FM chains (strong-leg type). Critical temperatures were precisely determined through DC magnetization, specific heat, and magnetocaloric effect measurements. The criticality of 3-Cl-4-F-V is characterized by a linear phase boundary with respect to $H_{\mathrm{c}}-H$ near $H=H_{\mathrm{c}}$. This behavior is similar to that of another strong-leg-type FM-leg ladder. The universal critical behavior in these strong-leg-type FM-leg ladders is expected to demonstrate the theoretically predicted quasi-one-dimensional Bose–Einstein condensation.

Figure 1

Temperature dependence of the magnetic susceptibility $\chi =M/H$ for magnetic fields ranging from 4.0 to 4.4 T in 0.05 T steps. Each curve is shifted by $+0.002$ emu/mol for clarity. The arrows pointing up (down) mark the cusp (kink) anomalies, indicating the upper (lower) critical temperatures (see text).

Figure 2

Temperature dependence of the magnetic susceptibility $\chi$ for magnetic fields ranging from 4.45 to 5.5 T in 0.05 T steps. Each curve is shifted by $+0.002$ emu/mol for clarity. The arrows show the cusp maximum, indicating the critical temperatures (see text).

Figure 3

Temperature dependence of the specific heat $C$ for magnetic fields ranging from 4.0 to 4.75 T in 0.05 T steps. Each curve is shifted by $+0.3$ J/mol K for clarity. The triangles show double-peak structures, indicating the upper and lower critical temperatures. The three dashed lines for each curve above 4.35 T present linear fits for defining the merged double-peak structures (see text). The squares indicate the onset of the upper anomalies.

Figure 4

Temperature dependence of the specific heat $C$ for magnetic fields ranging from 4.8 to 5.55 T in 0.05 T steps. Each curve is shifted by $+0.3$ J/mol K for clarity. The triangles show peak anomalies, indicating critical temperatures. The dashed lines at 4.8 and 4.85 T are the same as those in Fig. 3. The squares indicate the onset of the upper and single-peak anomalies.

Figure 5

MCE $dT/d\left({\mu }_{0}H\right)$ under quasiadiabatic conditions at a fixed bath temperature of (a) 0.3 K, (b) 0.6 K, and (c) 0.9 K. The red and blue symbols correspond to the up and down sweeps, respectively. The arrows show the sign change of $dT/d\left({\mu }_{0}H\right)$, indicating a phase transition. The error bars denote the fitting errors of the initial slope of $T\left(H\right)$ (see text).

Figure 6

$H-T$ phase boundaries determined from the present measurements. The circles show the critical temperatures determined from $\chi \left(T\right)$ (Figs. 1 and 2). The squares present temperatures determined the onset of the upper and single-peak anomalies for $C\left(T\right)$, and the triangles displays temperatures obtained from the lower anomalies of $C\left(T\right)$ (Figs. 3 and 4). The diamonds present values obtained from the MCE measurements (Fig. 5). The dashed line marks the magnetic field ($\sim 4.9$ T) at which the double phase transition appears to merge into a single phase transition.

Figure 7

(a) Enlarged plot of the $H-T$ phase boundary in the single phase boundary region. The solid lines present linear fits for the measurement data above 4.9 T. (b) Log-log plot of the critical temperature vs ${\mu }_0(H_{\mathrm{c}}-H)$, where ${\mu }_0{H}_{\mathrm{c}}$ is estimated from each fitted line in (a). The crosses indicate the critical temperatures obtained from $\chi \left(T\right)$ data for another spin-$\frac{1}{2}$ FM-leg ladder, 3-I-V, as previously reported [25]. The dotted line indicate $T\propto H_{\mathrm{c}}-H$.

Figure 8

(a) Enlarged plot of the $H-T$ phase boundary in the single phase boundary region and (b) log-log plot of the critical temperature vs ${\mu }_0(H_{\mathrm{c}}-H)$, in the same format as Fig. 7 for the upper peak temperatures of the specific heat (squares), indicated by the triangles in Fig. 4.

Figure 9

Temperature dependence of $(C-C_{\mathrm{nuc}})/T$ for magnetic fields ranging from 4.9 to 5.55 T in 0.05 T steps, where $C_{\mathrm{nuc}}$ is the estimated nuclear Schottky contributions for each magnetic field. Each curve is shifted by $+0.25$ J/mol ${\mathrm{K}}^2$ for clarity. The arrows show the midpoints of the two extrema (see text).

Figure 10

(a) Enlarged plot of the $H-T$ phase boundary in the single phase boundary region and (b) log-log plot of the critical temperature vs ${\mu }_0(H_{\mathrm{c}}-H)$, in the same format as Fig. 7 for the midpoint temperatures of $(C-C_{\mathrm{nuc}})/T$ (squares), indicated by the arrows in Fig. 9.