Diverging Thermal Expansion of the Spin-Ladder System $({\mathrm{C}}_5{\mathrm{H}}_{12}\mathrm{N}{)}_2{\mathrm{CuBr}}_4$

We present high-resolution measurements of the $c^{\star }$-axis thermal expansion and magnetostriction of piperidinium copper bromide $({\mathrm{C}}_5{\mathrm{H}}_{12}\mathrm{N}{)}_2{\mathrm{CuBr}}_4$. The experimental data at low temperatures are well accounted for by a two-leg spin-ladder Hamiltonian. The thermal expansion shows a complex behavior with various sign changes and approaches a $1/\sqrt{T}$ divergence at the critical fields. All low-temperature features are semiquantitatively explained within a free-fermion model; full quantitative agreement is obtained with quantum Monte Carlo simulations.

Figure 1

(a) Magnetostriction $\epsilon =\Delta L(H)/L$ and (b) $\lambda =\partial \epsilon /\partial H$ measured along $c^{\star }$. The solid line in (a) is a fit of $\epsilon$ at 335 mK within a free-fermion model.

Figure 2

Thermal expansion $\alpha =1/L\partial L/\partial T$ measured along the $c^{\star }$ direction in various magnetic fields (a) below $H_{c1}$, (b),(c) between $H_{c1}$ and $H_{c2}$, and (d) above $H_{c2}$. Note the logarithmic temperature scale. The dashed line in (a) is a fit with the Schottky formula Eq. (2). The arrows signal the order of the $\alpha (T)$ curves with increasing field.

Figure 3

Regions of positive and negative thermal expansion $\alpha$ along the $c^{\star }$ axis of $({\mathrm{C}}_5{\mathrm{H}}_{12}\mathrm{N}{)}_2{\mathrm{CuBr}}_4$. The triangles and circles denote the positions of vanishing $\alpha$ determined from an interpolation along the $T$ and $H$ axis, respectively. In the shaded area $\alpha$ is negative. The dashed line indicates the zeros of $\alpha$ as predicted by the Schottky formula (2). The solid lines show zeros of $\alpha$ derived from the free-fermion fitting formula (4), which for $T=0$ start at the two quantum critical points located at $H_{c1}=6.8\mathrm{T}$ and $H_{c2}=13.9\mathrm{T}$ and at $(H_{c1}+H_{c2})/2$. Note that a crossing point of lines of zeros as suggested by Eq. (4) is not realized as the regions of negative $\alpha$ are connected.

Figure 4

Panels (a)–(c) compare representative experimental data (symbols) of $\alpha (T)$ to theoretical curves (lines) obtained within the free-fermion model (4). All these curves are fully determined by $H_{c1/c2}$ and a factor $\gamma$ obtained from a fit to $\epsilon (H)$ data at $T=335\mathrm{mK}$ (see Fig. 1). Panel (d) shows the thermal expansion near $H_{c1/c2}$ on a double-logarithmic scale together with the limiting $1/\sqrt{T}$ behavior (dashed line).

Figure 5

Anisotropic magnetostriction coefficients $\lambda$ (symbols) measured along $c^{\star }$, $a$, and $b$. The lines are quantum Monte Carlo fits using the experimental values of $J_{\perp }$ and $J_{\parallel }$ and two additional parameters for each direction, which measure the uniaxial pressure dependencies of $J_{\perp }$ and $J_{\parallel }$ [22].