Low temperature thermodynamics of the antiferromagnetic $J_1-J_2$ model: Entropy, critical points and spin gap

Abstract

The antiferromagnetic $J_1-J_2$ model is a spin-1/2 chain with isotropic exchange $J_1>0$ between first neighbors and $J_2=\hat{I}\pm J_1$ between second neighbors. The model supports both gapless quantum phases with nondegenerate ground states and gapped phases with $\hat{I}”(\hat{I}\pm )>0$ and doubly degenerate ground states. Exact thermodynamics is limited to $\hat{I}\pm =0$, the linear Heisenberg antiferromagnet (HAF). Exact diagonalization of small systems at frustration $\hat{I}\pm$ followed by density matrix renormalization group (DMRG) calculations returns the entropy density $S(T,\hat{I}\pm ,N)$ and magnetic susceptibility $\ddot{I}‡(T,\hat{I}\pm ,N)$ of progressively larger systems up to $N=96$ or 152 spins. Convergence to the thermodynamics limit, $S(T,\hat{I}\pm )$ or $\ddot{I}‡(T,\hat{I}\pm )$, is demonstrated down to $T/J\sim 0.01$ in the sectors $\hat{I}\pm <1$ and $\hat{I}\pm >1$. $S(T,\hat{I}\pm )$ yields the critical points between gapless phases with $S^{\prime }(0,\hat{I}\pm )>0$ and gapped phases with $S^{\prime }(0,\hat{I}\pm )=0$. The $S^{\prime }(T,\hat{I}\pm )$ maximum at $T^{\ast }(\hat{I}\pm )$ is obtained directly in chains with large $\hat{I}”(\hat{I}\pm )$ and by extrapolation for small gaps. A phenomenological approximation for $S(T,\hat{I}\pm )$ down to $T=0$ indicates power-law deviations $T^{-{\hat{I}}^3(\hat{I}\pm )}$ from $\exp (-\hat{I}”(\hat{I}\pm )/T)$ with exponent ${\hat{I}}^3(\hat{I}\pm )$ that increases with $\hat{I}\pm$. The $\ddot{I}‡(T,\hat{I}\pm )$ analysis also yields power-law deviations, but with exponent $\hat{I}\cdot (\hat{I}\pm )$ that decreases with $\hat{I}\pm$. $S(T,\hat{I}\pm )$ and the spin density $\ddot{I}(T,\hat{I}\pm )=4T\ddot{I}‡(T,\hat{I}\pm )$ probe the thermal and magnetic fluctuations, respectively, of strongly correlated spin states. Gapless chains have constant $S(T,\hat{I}\pm )/\ddot{I}(T,\hat{I}\pm )$ for $T<0.10$. Remarkably, the ratio decreases (increases) with $T$ in chains with large (small) $\hat{I}”(\hat{I}\pm )$.

12 pages, 14 figures