Wilson Ratio of a Tomonaga-Luttinger Liquid in a Spin-$1/2$ Heisenberg Ladder

Using micromechanical force magnetometry, we have measured the magnetization of the strong-leg spin-$1/2$ ladder compound $({\mathrm{C}}_7{\mathrm{H}}_{10}\mathrm{N}{)}_2{\mathrm{CuBr}}_2$ at temperatures down to 45 mK. Low-temperature magnetic susceptibility as a function of field exhibits a maximum near the critical field $H_c$ at which the magnon gap vanishes, as expected for a gapped one-dimensional antiferromagnet. Above $H_c$ a clear minimum appears in the magnetization as a function of temperature, as predicted by theory. In this field region, the susceptibility in conjunction with our specific-heat data yields the Wilson ratio $R_W$. The result supports the relation $R_W=4K$, where $K$ is the Tomonaga-Luttinger-liquid parameter.

Figure 1

(a) Magnetization of DIMPY as a function of magnetic field at 300 mK and 4.3 K. The open circles represent DMRG results for $x=2.0$ and $g=2.2$. (b) Magnetic susceptibility as a function of magnetic field.

Figure 2

(a) Magnetization of DIMPY as a function of temperature. Symbols are experimental data, from which the magnetization of paramagnetic impurities has been subtracted; solid lines are quantum Monte Carlo simulations. Fields are, from bottom to top: 3 T, 4 T, 4.5 T, 5.5 T. (b) From bottom to top: 7.5 T, 9 T. (c) Position of the magnetization minimum from the data for $H\ge 4\mathrm{T}$ as a function of magnetic field. Dashed line is the universal behavior, Eq. (3), for free fermions.

Figure 3

(a) Magnetic specific heat, $C_m$, of fully deuterated DIMPY, shown as $C_m/T$. (b) Ordering temperature as a function of magnetic field. Solid circles are from specific-heat peaks. Open circles are from the magnetocaloric-effect data shown in the inset, where pink (medium gray) and blue (dark gray) lines represent the sample temperature during upward and downward field sweeps, respectively. Other lines are guides to the eye.

Figure 4

Dependence of the Wilson ratio $R_W$ of DIMPY on the normalized magnetization $m$. Open symbols are experimental data, assuming $g=2.2$ in Eq. (1); specific-heat data from Fig. 3a have been used for circles, and those from Ref. 13 for triangles. The error bars represent only the combined uncertainties of the magnetization and heat-capacity measurements; the uncertainty of the $g$ factor has not been included. Small filled circles are the TLL parameter $K$ for $x=2.0$, computed by the DMRG method and displayed as $R_W$ by assuming Eq. (2). The line is a guide to the eye.