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

arXiv comment12 pages, 14 figures

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Cited by in corpus (1)

• Spin Peierls transition of $J_1-J_2$ and extended models with ferromagnetic $J_1$.Sublattice dimerization and thermodynamics of zigzag chains in $\beta$-TeVO${}_4$