Hybrid exact diagonalization and density matrix renormalization group approach to the thermodynamics of one-dimensional quantum models

Exact diagonalization of small model systems gives the thermodynamics of spin chains or quantum cell models at high temperatures $T$. Density matrix renormalization group calculations of progressively larger systems are used to obtain excitations up to a cutoff ${W}_C$ and the low-$T$ thermodynamics. The hybrid approach is applied to the magnetic susceptibility $\ensuremath{\chi}(T)$ and specific heat $C(T)$ of spin-$1/2$ chains with isotropic exchange such as the linear Heisenberg antiferromagnet and the frustrated ${J}₁\ensuremath{-}{J}₂$ model with ferromagnetic ${J}₁<0$ and antiferromagnetic ${J}₂>0$. The hybrid approach is fully validated by comparison with Heisenberg antiferromagnet results. It extends ${J}₁\ensuremath{-}{J}₂$ thermodynamics down to $T\ensuremath{\sim}0.01|{J}₁|$ for ${J}₂/|{J}₁|\ensuremath{\ge}{\ensuremath{\alpha}}_c=1/4$ and is consistent with other methods. The criterion for the cutoff ${W}_C(N)$ in systems of $N$ spins is discussed. The cutoff leads to bounds for the thermodynamic limit that are best satisfied at a specific $T(N)$ at system size $N$.