Degenerate orbital effect in a three-orbital periodic Anderson model

The competition between the Ruderman-Kittel-Kasuya-Yosida effect and Kondo effect is a central subject of the periodic Anderson model. By using the density matrix embedding theory, we study a three-orbital periodic Anderson model, in which the effects of degenerate conduction orbitals, via the local magnetic moments, number of electrons, and spin-spin correlation functions, are investigated. From the phase diagram at half filling, we find there exist two different antiferromagnetic phases and one paramagnetic phase. To explore the difference between the two antiferromagnetic phases, the topology of the Fermi surface and the connection with the standard periodic Anderson model are considered. The spin-spin correlation functions yield insight into the competition between Ruderman-Kittel-Kasuya-Yosida interaction and Kondo interaction. We further find there exist “scaling transformations,” and by applying them to the data with different hybridization strength, all the data collapses. Our calculations agree with previous studies on the standard periodic Anderson model.

Figure 1

The noninteracting dispersion relations on the square lattice when $E_f=0$ and $V_1=V_2=V=1.0t$. (a) Standard PAM which has one conduction orbital and one localized orbital on each site; the hybridization between the two orbitals open a gap. (b) Three-orbital PAM, which has two conduction orbitals and one localized orbital on each site. The shape of the upper band (black line) and the lower band (red line) is the same as the standard PAM. The black dashed line is the Fermi level at half filling.

Figure 2

In the DMET the square lattice is first divided into clusters ($1\times 2$ here), and one of the clusters is chosen as impurity sites (the red sites); the rest of the blue sites of the lattice are considered environment sites. The bath orbitals, core orbitals, and virtual orbitals are linear combinations of orbital environment sites. The impurity orbitals and bath orbitals constitute active space.

Figure 3

Ground-state phase diagram when $U=8t$; it is symmetric with respect to $E_f=-U/2$. There are three different phases: the paramagnetic (PM) phase and two antiferromagnetic phases (AF1 and AF2). The phase transitions from the AF2 phase to the other two phases are first order, and marked as the red solid and blue solid lines. The phase transition between the AF1 phase and the PM phase are continuous, and marked as the blue dashed line. Within the AF2 phase there's a Kondo region, in which $n_f\approx 1.0$. The gray scale is $x=|n_f-1.0|$.

Figure 4

By varying $E_f$, the results when $V=0.1,V=0.5,V=1.0$, and $V=1.5$ are displayed in black, red, blue, and magenta. (a) The occupation number of $f$ orbitals. (b) The local magnetic moment of $f$ orbitals.

Figure 5

(a) Band structure in the AF1 phase. (b) Band structure in the AF2 phase. The dashed line is the Fermi level at half filling. (c) The first Brillouin zone of the square lattice, and the gray shaded region is the first Brillouin zone when antiferromagnetic order is present. (d) The upper right quarter of the Brillouin zone. The band structure are plotted from $M$ to $\mathrm{\Gamma },\mathrm{\Gamma }$ to $X$, and $X$ to $M$ as the arrows indicated. (e) The Fermi surface in the PM phase. (f) The Fermi surface in the AF1 phase. (g) The Fermi surface in the AF2 phase (slightly away from half filling).

Figure 6

The spin-spin correlation functions as a function of $E_f$ in left panels, and as a function of $V$ in the right panels. (a) and (b) The magnetic correlation between intrasite $f$ orbital and $c$ orbital. (c) and (d) The magnetic correlation between neighboring $f$ orbitals.

Figure 7

The hybridization parameter $V_u$ defined in Eq. (10) to characterize the Kondo screening. (a) $V_u$ as a function of $E_f$. (b) $V_u$ as a function of $V$; the dashed line is $1/3\approx 0.3333$.

Figure 8

With the transformation in Eqs. (12) and (13), the data of $m^{\alpha }$ and $n^{\alpha }$ when $E_f=-1,V_1=V$, and $V_2=\gamma V$. The insets are the data before the transformations. (a) Local magnetic moment of $c$ orbital. (b) Occupation number of electrons on $c$ orbital. (c) Local magnetic moment of $f$ orbital. (d) Occupation number of electrons on $f$ orbital.

Figure 9

In the standard PAM $V^{\prime }=1/\sqrt{2}$ such that in the corresponding three-orbital model $V_1=V_2=0.5$. (a) The ground-state energy of the standard PAM as a function of occupation number, and the lowest energy is marked with pentagons. (b) The $f$ orbital local magnetic moment as a function of $E_f$. (c) The number of electrons on the $g$ orbtial as a function of $E_f$.

Figure 10

The spin-spin correlation functions when $E_f=-1,V_1=V$, and $V_2=\gamma V$. The data before the transformations in Eq. (14) are displayed in the insets. (a) Magnetic correlation function between intrasite $c$ and $f$ orbitals. (b) Magnetic correlation function between $c$ and $f$ orbitals on neighboring sites.