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Exact diagonalization (ED) of small model systems gives the thermodynamics ofspin chains or quantum cell models at high temperature $T$. Density matrixrenormalization group (DMRG) calculations of progressively larger systems areused to obtain excitations up to a cutoff $W_C$ and the low-$T$ thermodynamics.The hybrid approach is applied to the magnetic susceptibility $\chi(T)$ andspecific heat $C(T)$ of spin-$1/2$ chains with isotropic exchange such as thelinear Heisenberg antiferromagnet (HAF) and the frustrated $J_1-J_2$ model withferromagnetic (F) $J_1 < 0$ and antiferromagnetic (AF) $J_2 > 0$. The hybridapproach is fully validated by comparison with HAF results. It extends$J_1-J_2$ thermodynamics down to $T \sim 0.01|J_1|$ for $J_2/|J_1| \geq\alpha_c = 1/4$ and is consistent with other methods. The criterion for thecutoff $W_C(N)$ in systems of $N$ spins is discussed. The cutoff leads tobounds for the thermodynamic limit that are best satisfied at a specific $T(N)$at system size $N$.

Hybrid ED/DMRG approach to the thermodynamics of 1D quantum models