Grand canonical finite-size numerical approaches: A route to measuring bulk properties in an applied field

We exploit a prescription to observe directly the physical properties of the thermodynamic limit in a continuously applied field in one-dimensional quantum finite lattice systems. By systematically scaling down the energy of the Hamiltonian of the open system from center toward both ends, one could adopt the edge sites with a negligibly small energy scale as the grand canonical small particle bath, and equilibrium states with noninteger arbitrary conserved numbers, e.g., electron numbers or $s_z$, are realized in the main part of the system. This will enable the evaluation of response functions under a continuously varying external field in a small lattice without any fine-tuning or scaling of parameters while keeping the standard numerical accuracy. Demonstrations are given on quantum spin systems and on a Hubbard model by the density-matrix renormalization group.

Figure 1

(a) Magnetization curve of the $S=1/2$ Heisenberg ($\Delta =1$) and $XXZ$ ($\Delta =2$) spin chains as a function of magnetic field $h$, with $J=1$. The results are obtained by the DMRG with $m\le 200$ on ${\mathcal{H}}_{\mathrm{SSD}}$ by our analyses. The solid line is the exact solution. (b) Magnetization curve of the $S=1/2$ $J_1$-$J_2$ Heisenberg model with $J_2/J_1=0.5$ and 1 obtained in the same manner as (a) with $m\le 300$. Arrows in the main panel indicate the phase transition (see Ref. 15 for details). The insets of (a) and (b) show the magnified curves (indicated in the main panel) for several $L$.

Figure 2

Analyses given for the Heisenberg spin chain. (a) Site ($i$)-dependent local spin operator, $\langle S_i^z\rangle$, and (b) the average of $\langle S_i^z\rangle$ over $2r$ sites from the system center, $O\left(r\right)$ [Eq. (2)], at $h/J=0.15$ for $L=50$ and 100. Dashed lines in (b) are the fitting function $O\left(r\right)=m_0+c{r}^2$. (c) Site ($i$) dependence of $\langle S_i^z{S}_{i+1}^z\rangle$ and deformed bond energy $e\left(i\right)=3J{f}_0\left(i\right)\langle S_i^z{S}_{i+1}^z\rangle$ at $h/J=1$. The bottom panel shows $\langle S_i^z\rangle$ for several choices of $M_{\mathrm{given}}$. Excess/deficient $\langle S_i^z\rangle$ is absorbed at the edge bath. (d) $m_0-m_{\mathrm{exact}}$ ($m_{\mathrm{exact}}$ is the exact solution) and the corresponding total energy (bottom panel) as functions of $M_{\mathrm{given}}/L$ for various $L$. Those analyzed under OBC are compared. (e) $m_0$ and the logarithmic $|m_0-m_{\mathrm{exact}}|$ as a function of $h$ for various $M_{\mathrm{given}}/L$. Shaded ($L=50$) and meshed ($L=100$) regions are the typical accuracies of the bare OBC results (stepwise data on the top panel). Data for different truncation errors, for different fitting ranges $r$, and with power scaling functions [non-SSD $f_l\left(i\right)$] are shown together.

Figure 3

(a) Chemical potential $\mu$ as a function of electron density $\rho$ by the present analyses with DMRG on the SSD Hubbard Hamiltonian with $m\le 300$ at $L=64$ for $U/t=4$. Dashed line indicates the one-particle gap of the exact solution.[13] The inset shows the evaluation of the one-particle gap by the standard finite-size scaling procedure using the DMRG results for the usual OBC without deformation compared with the present SSD results with$L=32,64,\mathrm{and}80$.