Magnetic field dependent dynamics and field-driven metal-to-insulator transition of the half-filled Hubbard model: A DMFT+DMRG study

We study the magnetic field-driven metal-to-insulator transition in half-filled Hubbard model on the Bethe lattice, using the dynamical mean-field theory by solving the quantum impurity problem with density-matrix renormalization group algorithm. The method enables us to obtain a high-resolution spectral densities in the presence of a magnetic field. It is found that the Kondo resonance at the Fermi level splits at relatively high magnetic field: the spin-up and -down components move away from the Fermi level and finally form a spin-polarized band insulator. By calculating the magnetization and spin susceptibility, we clarify that an applied magnetic field drives a transition from a paramagnetic metallic phase to a band insulating phase. In the weak interaction regime, the nature of the transition is continuous and captured by the Stoner's description, while in the strong interaction regime the transition is very likely to be metamagnetic, evidenced by the hysteresis curve. Furthermore, we determine the phase boundary by tracking the kink in the magnetic susceptibility, and the steplike change of the entanglement entropy and the entanglement gap closing. Interestingly, the phase boundaries determined from these two different ways are largely consistent with each other.

Figure 1

(a) Single-impurity Anderson model with the bath as half-infinite chain. (b) Equivalent model with two impurities coupled with two half-infinite bath chain. Square and circle dot, respectively, represent the impurity site and bath site.

Figure 2

Spin-dependent spectral density of Bethe lattice for various values of magnetic field strength $h$: spin down ${\rho }_{\downarrow }\left(\omega \right)$ (red dashed line), spin up ${\rho }_{\uparrow }\left(\omega \right)$ (blue dashed line), and total density $\rho \left(\omega \right)={\rho }_{\downarrow }\left(\omega \right)+{\rho }_{\uparrow }\left(\omega \right)$ (green solid line). The left column corresponds to the interaction strength $U=D=2.0$ and the right column to $U=1.5D=3.0$. As to the DMFT impurity solver, we use a chain enclosing $L=80$ fermionic sites, which is solved by DMRG algorithm by keeping 128 states. The DMRG projection error in each variational step is less than ${10}^{-8}$. DMFT iterations are stopped by the condition $\mathrm{\Delta }\rho \left(\omega \right)={\rho }^{i+1}\left(\omega \right)-{\rho }^i\left(\omega \right)<0.01$ for every $\omega$ $\left[\mathrm{\Delta }\rho \right(\omega )$ is the spectral density difference between two consecutive DMFT iterations]. Before the use of deconvolution scheme, the smearing energy in calculating the Green's function is set by $\eta =0.2D$.

Figure 3

Impurity site spin-dependent occupation density $n_{\sigma }$ in a magnetic field for interaction strength $U=0.5D=1.0$ (left), $U=D=2.0$ (middle), and $U=1.5D=3.0$ (right), where the occupation density is defined by integrating the spectral density up to Fermi level: $n_{\sigma }={\int }_{-\infty }^{0}d\omega {\rho }_{\sigma }\left(\omega \right)$. Inset: the magnetization $m\left(h\right)=n_{\uparrow }-n_{\downarrow }$ as a function of magnetic field $h$ (black dotted line). The metal-to-insulator transition point is determined to be $h_c$, determined by a kink in magnetization curve and $\left|m\right(h>h_c\left)\right|>0.9$. The red dashed line shows the mean-field result.

Figure 4

Magnetization as a function of the magnetic field $h$ for various values of the interaction strength $U$. In the strong interaction regime, the insulator transition point is determined by a kink in magnetization curve. Here, the magnetization curve is obtained by sweeping the magnetic field $h$ upward. Inset: hysteresis curve ($U=1.5D=3.0$) by continuously increasing $h$ (black diamond) and by continuously decreasing $h$ (red cross).

Figure 5

Local magnetic susceptibility at the impurity site calculated for various magnetic field $h$ by setting $U=D=2.0$: (a) real part $\text{Re}{\chi }_{\mathrm{loc}}\left(\omega \right)$ and (b) imaginary part $\text{Im}{\chi }_{\mathrm{loc}}\left(\omega \right)$. (c) $\text{Re}{\chi }_{\mathrm{loc}}(\omega =0)$ as a function of $h$. Dotted line is the best fit to guide for eyes. The phase transition point is determined by the kink around $h_c$.

Figure 6

Quantum phase transition determined by the evolution of entropy and entanglement spectrum for $U=1.5D=3.0$. (a) Entanglement entropy (black dotted line) for the ground state of the single-impurity Anderson model versus magnetic field $h$. (b) Derivative of entropy (orange dotted line), where the dip indicates the phase boundary. (c) Low-lying entanglement spectrum for the ground state of the single-impurity Anderson model versus magnetic field $h$. We denote ${\delta }_1$ as the “entanglement gap” between the largest eigenvalue and the second largest eigenvalue of the reduced density matrix. The red circle marks the entanglement gap closing ${\delta }_1=0$. The dashed line is the guide for eyes. For the entanglement cut, we separate the whole chain into two blocks: left block enclosing the impurity site and one bath site, and the right block enclosing electron bath [see Fig. 1].

Figure 7

Quantum phase diagram versus interaction strength $U$ and magnetic field $h$. The red line represents the phase boundary determined by DMFT+DMRG calculations. The black dashed line represents the classical phase boundary: $2D-2h-U=0$, where $2D=4.0$ is the bandwidth of the Bethe lattice. The shaded area marks the regime without stable solution in current scheme.

Figure 8

Dynamical DMRG solution for one-dimensional single-impurity Anderson model [by setting $V=D/2$, ${\gamma }_i=\gamma =1.0$ in Eq. (A5)] for different interaction strength $U=0,D,2D$: (left) spectral densities (imaginary part of Green function by scaling a global constant $\pi D$) and (right) real part of Green functions as a function frequency $\omega$. For DMRG simulation, we choose a chain length $L=80$ fermionic sites (after mapping into two-component spinless model, we have $L^{\prime }=160$ lattice sites). We kept 128 states in each DMRG block and the resulting projection error is less than ${10}^{-10}$. In the dynamical DMRG calculation, we use a smearing energy $\eta =0.1D$ before deconvolution calculation.

Figure 9

Spectral densities of single-orbital Hubbard model on the Bethe lattice obtained by DMFT scheme. We choose the impurity model enclosing $L=80$ fermionic sites, which is solved by DMRG algorithm by limiting each DMRG block with dimension $M=128$. Before the deconvolution calculation, we select the broadening parameter as $\eta =0.1D$.

Figure 10

Comparison of different bath discretization schemes for spectral densities: (left) logarithmic discretization and (right) mixed discretization. For logarithmic discretization, we choose $\mathrm{\Lambda }=1.2$. For mixed scheme, we choose ${\omega }^{hyd}=0.45$ and $\mathrm{\Lambda }=1.4$. We choose single impurity of Anderson impurity model with chain length $L=80$ fermionic sites, solving DMRG using kept state $M=128$. We select parameters as $U=3.0$ and $h=0.2$. Here, we show spectral density for spin-up electrons.

Figure 11

Spectral densities of single-orbital Hubbard model on the Bethe lattice obtained by DMFT+DMRG scheme. We choose the impurity model enclosing $L=80$ fermionic sites (black), $L=120$ fermionic sites (red), and $L=160$ fermionic sites (green).

Figure 12

CPU times for a DMRG run performed with spinful Hubbard model and two-component spinless model. We set all parameters in single-impurity Anderson impurity model are the same. The time unit is the time cost of each DMFT loop for spinful Hubbard model by solving DMRG using kept state $M=256$.

Figure 13

Self-energy function of single-orbital Hubbard model on the Bethe lattice obtained by DMFT scheme. Interaction strength is set to be $U=3.0$. We choose the impurity model enclosing $L=80$ fermionic sites, which is solved by DMRG algorithm by limiting each DMRG block with dimension $M=128$.

Figure 14

Inverse of quasiparticle weight as a function of magnetic field $h$, by setting $U=3.0$.