Three-dimensional Bose-Einstein condensation in the spin-$\frac{1}{2}$ ferromagnetic-leg ladder 3-Br-4-F-V

The critical exponent of the phase boundary has been examined on the three-dimensional incommensurate ordering phase in the spin-$\frac{1}{2}$ ferromagnetic-leg ladder 3-Br-4-F-V [=3-(3-bromo-4-fluorophenyl)-1,5-diphenylverdazyl]. Using the temperature-window fitting technique, we obtained the critical exponents which agreed with the three-dimensional (3D) Bose-Einstein condensation (BEC) universality class at both sides of the lower critical field and the saturation field. 3-Br-4-F-V thus becomes a new member of the quantum magnets which prove the universality of the 3D BEC exponent.

Figure 1

Schematic picture of a two-leg ladder described by Eq. (1).

Figure 2

Temperature dependence of the magnetic heat capacity in several magnetic fields near (a) the lower critical field $H_{\mathrm{c}1}$ and (b) the upper critical (saturation) field $H_{\mathrm{c}2}$. Each curve is shifted by $+1.2$ J/mol ${\mathrm{K}}^2$ for clarity. Open triangles denote the peak positions indicating the 3D ordering temperatures, $T_c$. The open arrow denotes the partial-ordering temperature at 7 T.

Figure 3

Temperature dependence of the magnetic susceptibility $\chi =M/H$ in several magnetic fields (a) from 5 T to 6.1 T and (b) from 8 T to 9 T in 0.1 T steps. Arrows indicate the temperature $T_{\mathrm{ex}}$ at which $\chi$ takes the nontrivial (a) minimum or (b) maximum.

Figure 4

Temperature derivative of $T\chi \left[d\right(T\chi )/dT]$ obtained from Figs. 3 and 3. Open arrows indicate the temperature at which $d\left(T\chi \right)/dT$ takes the nontrivial (a) minimum or (b) maximum.

Figure 5

Phase boundaries of the 3D ordering determined from the present measurements. Closed circles and squares show $T_c$ from the peak positions of $C/T$ (Fig. 2) and the dip or peak anomalies of $d\left(T\chi \right)/dT$ (Fig. 4), respectively. Open circles and squares denote previous (Ref. [17]) results of $T_c$ from $C/T$ and $d\chi /dT$, respectively. Crosses (stars) denote the temperature $T_{\mathrm{ex}}$ at which the magnetic susceptibility $\chi$ takes extremum in the present (previous) results, and solid lines are a linear fit of the present ones at below 5.3 T and above 8.6 T. All the dotted lines are guides for the eye.

Figure 6

Fitting results of the temperature-window technique with Eq. (4), applied to the 3D ordering phase boundary. As a function of the maximum temperature of the window, $t_{\max }$, for several fixed $\nu$'s, (a) $H_{\mathrm{c}1}$ and (b) $H_{\mathrm{c}2}$ from $C/T$ and (c) $H_{\mathrm{c}1}$ and (d) $H_{\mathrm{c}2}$ from $d\left(T\chi \right)/dT$ are shown. The dotted lines are linear fitting lines to a part of the data on each $\nu$, used to determine $H_{\mathrm{c}1,2}\left(0\right)$ (see text).

Figure 7

Critical exponent $\nu$ as a function of $t_{\max }$ with fixed $H_{\mathrm{c}1,2}\left(0\right)$ determined from Fig. 6. Down and up triangles denote the results from $C/T$ near $H_{\mathrm{c}1}$ and $H_{\mathrm{c}2}$, respectively. Left and right triangles denote the results from $d\left(T\chi \right)/dT$ near $H_{\mathrm{c}1}$ and $H_{\mathrm{c}2}$, respectively. The dotted line shows the critical exponent at the QCP that belongs to the 3D BEC universality class ($\nu =2/3$). All the error bars represent fitting errors.