Magnetic entropy landscape and Grüneisen parameter of a quantum spin ladder

We present measurements of the magnetic entropy landscape and Grüneisen parameter of the ${\mathrm{Cu}}^{2+}$ complex $({\mathrm{C}}_5{\mathrm{H}}_{12}\mathrm{N}{)}_2\mathrm{Cu}{\mathrm{Cl}}_4$ in a magnetic field. Our thermodynamic measurements are in very good agreement with a theoretical description by a $S=1/2$ Heisenberg ladder model. Due to its excellent experimental access, the compound crosses two quantum critical points in the applied range of the magnetic field, first from a gapped unpolarized state to a critical phase and then to a gapped fully polarized state. This behavior is reflected directly in the magnetic entropy map. Due to the remarkable properties of the magnetic Grüneisen parameter, we are able to discuss the validity of critical behavior with respect to temperature and magnetic field for this model quasi-one-dimensional system.

Figure 1

Magnetization (a), heat capacity (b), and magnetocaloric effect measurements (c) of (C${}_5$H${}_{12}$N)${}_2$CuCl${}_4$ (points) with comparison to calculations (blue line, exact diagonalization 10 rungs; dashed green line, XXZ model). C and O mark the closing and reopening of the spin gap, and LL marks the spin Luttinger liquid. Quantitative entropy data are obtained from the MCE (gray points in lower panel); see text.

Figure 2

Quantitative entropy landscape $S/T$ of (C${}_5$H${}_{12}$N)${}_2$CuCl${}_4$. The typical elements of the spin ladder phase diagram are observed: two QCPs, quantum disordered spin liquid (QD) for $B<B_{c1}$, spin Luttinger liquid (LL) for $B_{c1}<B<B_{c2}$, and full magnetic polarization (FP) for $B_{c2}<B$. For lower temperatures a Bose-Einstein condensate (BEC) is expected to set in due to the weak coupling between the spin ladders. QC denotes the quantum critical regime.

Figure 3

Magnetic Grüneisen parameter ${\Gamma }_{\text{mag}}$ of (C${}_5$H${}_{12}$N)${}_2$CuCl${}_4$ (experimental data) with comparison to calculations: blue line, exact diagonalization 10 rungs; dashed green line, XXZ model. Gray vertical lines mark the critical fields.

Figure 4

Temperature dependence of ${\Gamma }_{\text{mag}}$ at the critical magnetic fields, obtained by experiment (points) as well as DMRG (solid lines) and XX chain model (dashed line) calculations for (C${}_5$H${}_{12}$N)${}_2$CuBr${}_4$ (left) and (C${}_5$H${}_{12}$N)${}_2$CuCl${}_4$ (right). Plotted normalized to the rung coupling $J_{\text{rung}}$ of the two materials [9, 17, 19].