Low-temperature thermodynamics of the antiferromagnetic J 1 − J 2 model: Entropy, critical points, and spin gap

The antiferromagnetic ${J}_{1}\ensuremath{-}{J}_{2}$ model is a spin-1/2 chain with isotropic exchange ${J}_{1}g0$ between first neighbors and ${J}_{2}=\ensuremath{\alpha}{J}_{1}$ between second neighbors. The model supports both gapless quantum phases with nondegenerate ground states and gapped phases with $\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})g0$ and doubly degenerate ground states. Exact thermodynamics is limited to $\ensuremath{\alpha}=0$, the linear Heisenberg antiferromagnet (HAF). Exact diagonalization of small systems at frustration $\ensuremath{\alpha}$ followed by density matrix renormalization group calculations returns the entropy density $S(T,\ensuremath{\alpha},N)$ and magnetic susceptibility $\ensuremath{\chi}(T,\ensuremath{\alpha},N)$ of progressively larger systems up to $N=96$ or 152 spins. Convergence to the thermodynamic limit, $S(T,\ensuremath{\alpha})$ or $\ensuremath{\chi}(T,\ensuremath{\alpha})$, is demonstrated down to $T/J\ensuremath{\sim}0.01$ in the sectors $\ensuremath{\alpha}l1$ and $\ensuremath{\alpha}g1$. $S(T,\ensuremath{\alpha})$ yields the critical points between gapless phases with ${S}^{\ensuremath{'}}(0,\ensuremath{\alpha})g0$ and gapped phases with ${S}^{\ensuremath{'}}(0,\ensuremath{\alpha})=0$. The ${S}^{\ensuremath{'}}(T,\ensuremath{\alpha})$ maximum at ${T}^{*}(\ensuremath{\alpha})$ is obtained directly in chains with large $\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})$ and by extrapolation for small gaps. A phenomenological approximation for $S(T,\ensuremath{\alpha})$ down to $T=0$ indicates power-law deviations ${T}^{\ensuremath{-}\ensuremath{\gamma}(\ensuremath{\alpha})}$ from $exp[\ensuremath{-}\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})/T]$ with exponent $\ensuremath{\gamma}(\ensuremath{\alpha})$ that increases with $\ensuremath{\alpha}$. The $\ensuremath{\chi}(T,\ensuremath{\alpha})$ analysis also yields power-law deviations, but with exponent $\ensuremath{\eta}(\ensuremath{\alpha})$ that decreases with $\ensuremath{\alpha}$. Spin correlation functions account for $S(T,\ensuremath{\alpha})$ differences between frustration $\ensuremath{\alpha}l1$ within a chain and $\ensuremath{\alpha}g1$ between HAFs on sublattices. $S(T,\ensuremath{\alpha})$ and the spin density $\ensuremath{\rho}(T,\ensuremath{\alpha})=4T\ensuremath{\chi}(T,\ensuremath{\alpha})$ probe the thermal and magnetic fluctuations, respectively, of strongly correlated spin states. Gapless chains have constant $S(T,\ensuremath{\alpha})/\ensuremath{\rho}(T,\ensuremath{\alpha})$ for $Tl0.10$. Remarkably, the ratio decreases (increases) with $T$ in chains with large (small) $\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})$.