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تسجيل مستخدم جديدSpin-Peierls transition of the dimer phase of the $J_1-J_2$ model: Energy cusp and CuGeO$_3$ thermodynamics
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The spin-Peierls transition is modeled in the dimer phase of the spin-$1/2$ chain with exchanges $J_1$, $J_2 = alpha J_1$ between first and second neighbors. The degenerate ground state generates an energy cusp that qualitatively changes the dimerization $delta(T)$ compared to Peierls systems with nondegenerate ground states. The parameters $J_1 = 160$ K, $alpha = 0.35$ plus a lattice stiffness account for the magnetic susceptibility of CuGeO$_3$, its specific heat anomaly, and the $T$ dependence of the lowest gap.
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The spin-Peierls transition at $T_{SP}$ of spin-$1/2$ chains with isotropic exchange interactions has previously been modeled as correlated for $T > T_{SP}$ and mean field for $T < T_{SP}$. We use correlated states throughout in the $J_1-J_2$ model w
ith antiferromagnetic exchange $J_1$ and $J_2 = alpha J_1$ between first and second neighbors, respectively, and variable frustration $0 leq alpha leq 0.50$. The thermodynamic limit is reached at high $T$ by exact diagonalization of short chains and at low $T$ by density matrix renormalization group calculations of progressively longer chains. In contrast to mean field results, correlated states of 1D models with linear spin-phonon coupling and a harmonic adiabatic lattice provide an internally consistent description in which the parameter $T_{SP}$ yields both the stiffness and the lattice dimerization $delta(T)$. The relation between $T_{SP}$ and $Delta(delta,alpha)$, the $T = 0$ gap induced by dimerization, depends strongly on $alpha$ and deviates from the BCS gap relation that holds in uncorrelated spin chains. Correlated states account quantitatively for the magnetic susceptibility of TTF-CuS$_4$C$_4$(CF$_3$)$_4$ crystals ($J_1 = 79$ K, $alpha = 0$, $T_{SP} = 12$ K) and CuGeO$_3$ crystals ($J_1 = 160$ K, $alpha = 0.35$, $T_{SP} = 14$ K). The same parameters describe the specific heat anomaly of CuGeO$_3$ and inelastic neutron scattering. Modeling the spin-Peierls transition with correlated states exploits the fact that $delta(0)$ limits the range of spin correlations at $T = 0$ while $T > 0$ limits the range at $delta= 0$.
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 ga
pped 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 $T^*(alpha)$ 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 $rho(T,alpha) = 4Tchi(T,alpha)$ 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) $Delta(alpha)$.
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