Crossover from conventional to inverse indirect magnetic exchange in the depleted Anderson lattice

We investigate the finite-temperature properties of an Anderson lattice with regularly depleted impurities. The physics of this model is ruled by two different magnetic exchange mechanisms: conventional Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction at weak hybridization strength $V$ and an inverse indirect magnetic exchange (IIME) at strong $V$, both favoring a ferromagnetic ground state. The stability of ferromagnetic order against thermal fluctuations is systematically studied by static mean-field theory for an effective low-energy spin-only model emerging perturbatively in the strong-coupling limit as well as by dynamical mean-field theory for the full model. The Curie temperature is found at a maximum for a half-filled conduction band and at intermediate hybridization strengths in the crossover regime between RKKY and IIME.

Figure 1

Depleted periodic Anderson model with $R=L/2$ impurities on a $D=3$-dimensional simple-cubic lattice with $L$ sites $\left(L\to \infty \right)$. Correlated impurity sites (red) with onsite Hubbard interaction $U$ are coupled via a hybridization of strength $V$ to the B sites (blue) of the bipartite lattice. For strong $U,V\gg t$, local Kondo singlets are formed on the half-filled “dimers” consisting of impurity and B sites (if the total electron number $N$ satisfies $2R\le N\le 4R$) and strongly confine the motion of the excess conduction electrons on the A-sublattice sites (green). Virtual excitations of the local Kondo singlets induce an effective interaction of the conduction electrons on A sites.

Figure 2

Geometry of the one-dimensional diluted Anderson lattice. Red: “impurities” with finite Hubbard interaction. Green and blue: sites of the A and of the B sublattice, respectively.

Figure 3

The same as in Figs. 2 and  1 for the two-dimensional case.

Figure 4

Spin-dependent mean field [see Eq. (14)] as a function of the reduced temperature $T/T_{\mathrm{C}}$ for different dimensions $D$ (top, middle, and bottom panels) and different fillings $n_{\mathrm{A}}$ as indicated (see top panel) at and below half-filling $\left(n_{\mathrm{A}}=1\right)$. Solid lines: $\sigma =\uparrow$. Dashed lines: $\sigma =\downarrow$.

Figure 5

Order parameter $m_{\mathrm{A}}=n_{\mathrm{A}\uparrow }-n_{\mathrm{A}\downarrow }$ [see Eq. (15)] as a function of $T/T_{\mathrm{C}}$ for different dimensions $D$ and fillings $n_{\mathrm{A}}$.

Figure 6

Filling dependence of the Curie temperature for lattices with different dimensions as obtained from the static mean-field theory. Note that $T_{\mathrm{C}}$ is rescaled by $D^3$ and given in units of the coupling constant $\alpha$.

Figure 7

Ordered magnetic moments $m_{\mathrm{A}},m_{\mathrm{B}},m_{\mathrm{imp}}$ (circles) on the A sites, the B sites, and the impurity sites, respectively, and the inverse homogeneous static impurity magnetic susceptibility ${\chi }^{-1}$ (diamonds) as functions of temperature $T$ as obtained by DMFT for the $D=3$-dimensional depleted Anderson lattice (see Fig. 1). Hubbard interaction: $U=8$, hybridization strength: $V=2$. The line indicates a linear fit to the trend of ${\chi }^{-1}\left(T\right)$. The temperature and energy scales are fixed by the nearest-neighbor hopping $t=1$ [see Eq. (6)].

Figure 8

The same as in Fig. 7 but for $V=3$.

Figure 9

$m_{\mathrm{A}},m_{\mathrm{B}},m_{\mathrm{imp}}$ (circles) and ${\chi }^{-1}$ (diamonds) as functions of $T$, as in Fig. 7 but for the $D=1$-dimensional depleted Anderson lattice (see Fig. 2). Hubbard interaction: $U=8$, hybridization strength: $V=2$. DMRG data for $T=0$ (squares) from Ref. [9] are included for comparison.

Figure 10

The same as in Fig. 9 but for $V=3$.

Figure 11

Curie temperature $T_{\mathrm{C}}$ for the $D=3$-dimensional depleted Anderson lattice at $U=8$ and half-filling as a function of the hybridization strength $V$. Points are obtained via ${\chi }^{-1}\left(T_{\mathrm{C}}\right)=0$ by extrapolating the linear temperature trend of the inverse susceptibility ${\chi }^{-1}\left(T\right)$. Solid lines: a dependence of $T_{\mathrm{C}}\left(V\right)\propto V^4/t{U}^2$ is expected for $V\to 0$. For strong $V$, the data are consistent with $T_{\mathrm{C}}\left(V\right)\propto \alpha \left(V\right)$. Dashed line: $T_{\mathrm{C}}\left(V\right)\propto 2t^{4}U/V^4$ represents a good approximation to $\alpha \left(V\right)$ at $U=8$.