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 $\chi \left(T\right)$ and specific heat $C\left(T\right)$ of spin-$1/2$ chains with isotropic exchange such as the linear Heisenberg antiferromagnet and the frustrated $J_1-J_2$ 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 $J_1-J_2$ thermodynamics down to $T\sim 0.01|J_1|$ for $J_2/\left|J_1\right|\ge {\alpha }_c=1/4$ and is consistent with other methods. The criterion for the cutoff $W_C\left(N\right)$ in systems of $N$ spins is discussed. The cutoff leads to bounds for the thermodynamic limit that are best satisfied at a specific $T\left(N\right)$ at system size $N$.

Figure 1

Normalized density of excitations, $\rho (\varepsilon ,\alpha )$ with $\varepsilon =E(\alpha ,N)/N|J_1|$, for three models of $N$ spins. The ground state is a singlet $(S=0)$ at $\varepsilon =0$. The Néel state $(...\alpha \beta \alpha \beta ...)$ has excitation energy ${\varepsilon }_N=1/2$ for the ${\alpha }_c$ model, while the FM state $(...\alpha \alpha \alpha \alpha ...)$ has ${\varepsilon }_F=ln2$ for the HAF.

Figure 2

Convergence of the truncated $S_C(T,N)/T$ and ${\chi }_C(T,N)$ with an increasing number of states in the $M=0$ sector for $N=48$ and 64 spins at $\alpha =2/3$ in Eq. (3). The maxima are converged when the cutoff $W_C\left(N\right)$ leads to $R_C(0,N)=400$ states.

Figure 3

Convergence of the truncated $S_C(T,N)/T$ and ${\chi }_C(T,N)$ with an increasing number of states in the $S^z=0$ sector for $N=48$ and 64 spins at $\alpha =1/2$ in Eq. (3). The maxima are converged when the cutoff $W_C\left(N\right)$ leads to $R_C(0,N)=400$ states.

Figure 4

HAF results for $\chi \left(T\right)$ (upper panel) and $C\left(T\right)/T=S^{\prime }\left(T\right)$ (lower panel). Solid and dashed lines represent ED and the truncated DMRG, respectively, at the indicated system size $N$. The QMC calculations of $\chi \left(T\right)$ follow Ref. ([14]). Arrows are exact at $T=0$. The high-$T$ peak is fully converged in both panels. The DMRG lines for $C(T,N)/T$ terminate at $T^{\prime }\left(N\right)$, the maximum of $S_C(T,N)/T$, shown as open circles (lower panel) and filled squares (upper panel). Squares and extrapolation in the lower panel are discussed in the text.

Figure 5

$\chi (T,{\alpha }_c)$ and $S^{\prime }(T,{\alpha }_c)$ in reduced units at the critical point, ${\alpha }_c=1/4$. Open circles are at $T_n\left({\alpha }_c\right)=0.02$, where the thermodynamic limit is reached for ED up to 24 spins.

Figure 6

Upper panel: $\chi (T,N)$ for $\alpha =2/3$ and $N$ spins in Eq. (3). Solid and dashed lines represent the ED and DMRG. The bold dashed line is the estimated $\chi \left(T\right)$ in the thermodynamic limit. Lower panel: $C(T,N)/T$ for the same systems. Open circles represent $T^{\prime }\left(N\right)$, the maxima in Eq. (8). Squares show Eq. (12) results, the mean value in successive $T^{\prime }\left(N\right)$ intervals; the dashed line connecting them is a guide for the eye.

Figure 7

$4T\chi (T,\alpha )$ for 24 spins in Eq. (3). Open circles represent $T_n\left(\alpha \right)$, where the thermodynamic limit is reached. The dashed line corresponds to free spins. Deviations above or below indicate that the net interaction is FM or AF.

Figure 8

Upper panel: $\chi (T,N)$ for $\alpha =1/2$ and $N$ spins in Eq. (3). Solid and dashed lines represent the ED and DMRG. The bold dashed line is the estimated $\chi \left(T\right)$ in the thermodynamic limit. Lower panel: $C(T,N)/T$ for the same systems. Open circles represent $T^{\prime }\left(N\right)$, the maxima in Eq. (8). Squares show Eq. (12) results, the mean value in successive $T^{\prime }\left(N\right)$ intervals; the bold dashed line is a guide for the eye.

Figure 9

Upper panel: $\chi (T,N)$ for $\alpha =1/3$ in Eq. (3) and ED for $N=24$ spins and DMRG for 32 spins up to the open circle at $T^{\prime }\left(32\right)$. Lower panel: $C(T,N)$ for $\alpha =1/3$ and 1/4 for $N=24$. Filled squares are based on the DMRG for $N=32$ and Eq. (12) for three intervals based on $T^{\prime }\left(32\right), T^{\prime }\left(24\right),$ and the open circle at $T_n\left(24\right)=0.06$.