Exact ground states of the periodic Anderson model in $D=3$ dimensions

We construct a class of exact ground states of three-dimensional periodic Anderson models (PAMs), including the conventional PAM, on regular Bravais lattices at and above $3∕4$ filling, and discuss their physical properties. In general, the $f$ electrons can have a (weak) dispersion, and the hopping and the nonlocal hybridization of the $d$ and $f$ electrons extend over the unit cell. The construction is performed in two steps. First the Hamiltonian is cast into positive semidefinite form using composite operators in combination with coupled nonlinear matching conditions. This may be achieved in several ways, thus leading to solutions in different regions of the phase diagram. In a second step, a nonlocal product wave function in position space is constructed which allows one to identify various stability regions corresponding to insulating and conducting states. The compressibility of the insulating state is shown to diverge at the boundary of its stability regime. The metallic phase is a non-Fermi-liquid with one dispersing and one flat band. This state is also an exact ground state of the conventional PAM and has the following properties: (i) it is nonmagnetic with spin-spin correlations disappearing in the thermodynamic limit, (ii) density-density correlations are short ranged, and (iii) the momentum distributions of the interacting electrons are analytic functions, i.e., have no discontinuities even in their derivatives. The stability regions of the ground states extend through a large region of parameter space, e.g., from weak to strong on-site interaction $U$. Exact itinerant, ferromagnetic ground states are found at and below $1∕4$ filling.

Figure 1

A unit cell $I_{\mathbf{i}}$ connected to an arbitrary site $\mathbf{i}$ showing the primitive vectors ${\mathbf{x}}_{\tau }$ and indices $n$ of the sites in $I$. Arrows depict some of the hopping and hybridization amplitudes $(J=t,V)$ defined within $I_{\mathbf{i}}$.

Figure 2

Ground-state energy per lattice site, Eq. (25), in units of $U$ as a function of $y=\mid t_2^d∕t_1^d\mid$ for the localized solution in a simple cubic crystal for $\mid t_1^f∕t_1^d\mid =0.01$. As seen from the plot, $E_g∕N_{\Lambda }$ is finite at $y_c=1∕2$ but has infinite slope.

Figure 3

Surface in parameter space representing the stability region of the conducting ground state discussed in Sec. 4A.

Figure 4

Tilted unit cell located at site $\mathbf{i}$ discussed in Sec. 4B. Number represent the intracell numbering of lattice sites. Only those bonds are presented along which hopping of $d$ electrons occurs.

Figure 5

(Color online) Surface in parameter space representing the stability region of the itinerant solution derived in the case of the conventional PAM, Eq. (33). For $\mid t_2^d∕t_1^d\mid \to 0$, the surface asymptotically approaches the $\mathbf{(}\mid V_1∕t_1^d\mid ,(U+E_f)∕\mid t_1^d\mid \mathbf{)}$ plane.

Figure 6

Momentum distribution functions $n_{\mathbf{k},\sigma }^d,n_{\mathbf{k},\sigma }^f$ for the conducting solution discussed in Sec. 4B, for $\mid t_1^d\mid ∕\mid t_2^d\mid =5.0$, $\mid V_1\mid ∕\mid t_1^d\mid =0.5$, $t_1^d>0$, and $k_2=k_3=0$, where $k_{\tau }=\mathbf{k}{\mathbf{x}}_{\tau }$. The plot presents the behavior in the first Brillouin zone for $k_1∊[-\pi ,\pi ]$. For other $\mathbf{k}$ directions a similar behavior is found.

Figure 7

Density-density correlation function for $\mid t_1^d∕t_2^d\mid =4$, $\mid V_1∕t_1^d\mid =0.25$, $t_1^d>0$ for the itinerant case obtained from Eq. (49) in the thermodynamic limit. The nine-dimensional integration was performed by a Monte Carlo method using 69 points. The distance $\mathbf{r}$ was taken in the $\tau =1$ direction, and is expressed in units of the lattice constant $a$.

Figure 8

Octahedral cell defined at lattice site $\mathbf{i}$ as discussed in Sec. 5A. The $\tau =1,2,3$ axes are represented by arrows with broken lines, ${\mathbf{x}}_{\tau }$ are indicated by thick full line arrows, and $n$ represents the cell-independent notation of sites inside the octahedron.

Figure 9

(Color online) Surfaces in parameter space above which the conducting phase discussed in Sec. 5A. is stable. (a) $\mid V_1\mid ∕\mid t_1^d\mid <0.5$; $(U+E_f)∕\mid t_1^d\mid$ is seen to diverge as $\mid t_2^d\mid ∕\mid t_1^d\mid$ approaches zero. (b) $\mid V_1\mid ∕\mid t_1^d\mid >0.5$. In contrast to Fig. 5 where $V_1$ is imaginary, $V_1$ is real here.

Figure 10

Four neighboring unit cells in $D=3$. Each cell (e.g., $I_i$) is donoted by one of the lattice sites at which it is located (e.g., $\mathbf{i}$); see Sec. 2B1. For simplicity, orthorhombic unit cells are used.