Coherent and incoherent pairing instabilities and spin-charge separation in bipartite and nonbipartite nanoclusters: Exact results

Electron pairing and formation of various types of magnetic correlations for ensembles of small clusters of different geometries are studied with emphasis on tetrahedrons and square pyramids under variation of interaction strength, electron doping and temperature. These exact calculations of charge and spin collective excitations and pseudogaps yield intriguing insights into level crossing degeneracies, phase separation and condensation. Obtained coherent and incoherent pairings provide a route for possible superconductivity different from the conventional BCS theory. Criteria for spin-charge separation, reconciliation and recombination driven by interaction strength, next-nearest coupling and temperature are found. Resulting phase diagrams resemble a number of inhomogeneous, coherent and incoherent nanoscale phases seen recently in high-$T_c$ cuprates, manganites and colossal magnetoresistive (CMR) nanomaterials.

Figure 1

(Color online) Charge ${\Delta }^c$ and spin ${\Delta }^s$ gaps versus $U$ in an ensemble of squares at $⟨N⟩\approx 3$ and $T=0$. Phase A: Charge and spin pairing gaps of equal amplitude at $U\le U_c$ describe Bose condensation of electrons similar to BCS-type coherent pairing with a single energy gap. Phase B: Mott-Hubbard-type insulator at $U_c<U<U_F$ leads to $S=\frac{1}{2}$ spin liquid behavior. Phase C: Parallel (triplet) spin pairing $\left({\Delta }^s<0\right)$ at $U>U_F$ displays $S=\frac{3}{2}$ saturated ferromagnetism (see Sec. 3A). The spin gap in Phase A for $U\le U_c$ region has been derived in grand canonical approach for very low temperatures ($T\to 0$).

Figure 2

(Color online) $⟨N⟩$ and $⟨s^z⟩$ versus $\mu$ in grand canonical ensemble of tetrahedrons and fcs at $U=4.0$, $T=0.01$ and $h=0.1$. Mott-Hubbard-type ferromagnetism for $⟨s_z⟩=\frac{3}{2}$ at $⟨N⟩=3$ in tetrahedron occurs for $t=-1$, while absence of charge pseudogap near $⟨N⟩\approx 3$ metallic state with spin rigidity manifests level crossing degeneracy related to pairing (see inset).

Figure 3

(Color online) ${\Delta }^c$ versus $U$ for one hole off half filling in tetrahedrons and fcs at $T=0$. In tetrahedron, ${\Delta }^c<0$ at $t=1$ implies phase separation and coherent pairing with ${\Delta }^s\equiv {\Delta }^P$, while ${\Delta }^c>0$ for $S=\frac{3}{2}$ at $t=-1$ leads to a ferromagnetic insulator $\left({\Delta }^s<0\right)$ for all $U$. In fcs, ${\Delta }^c>0$ at $⟨N⟩\approx 4$ for $t=\pm 1$ describes Mott-Hubbard-type antiferromagnetism $\left({\Delta }^s>0\right)$.

Figure 4

(Color online) ${\Delta }^c$ versus $U$ at $⟨N⟩=3$ and $T=0.01$ in tetrahedron $\left(t=1\right)$ and deformed tetrahedral clusters ($c=0.96$ and 1.1). Charge and spin pairing gaps of equal amplitude at $t=1$ imply coherent pairing, while ${\Delta }^c>0$ and ${\Delta }^s<0$ at $c=1.1$ correspond to an unsaturated ferromagnetic insulator for $S=\frac{1}{2}$. Coherent pairing is retained in a narrow range near $c\approx 1$.

Figure 5

(Color online) The $T\text{−}\mu$ phase diagram of tetrahedrons without electron-hole symmetry at optimally doped $⟨N⟩\approx 3$ regime near ${\mu }_P=1.593$ at $U=4$ and $t=1$ illustrates the condensation of electron charge and onset of phase separation for charge degrees below $T_c^P$. The incoherent phase of preformed pairs with unpaired opposite spins exists above $T_s^P$. Below $T_s^P$, the paired spin and charge coexist in a coherent pairing phase. The charge and spin susceptibility peaks, denoted by $T^{\ast }$ and $T_c$, define pseudogaps calculated in the grand canonical ensemble, while ${\mu }_{+}\left(T\right)$ and ${\mu }_{-}\left(T\right)$ are evaluated in canonical ensemble. Charge and spin peaks reconcile at $T^{\prime }\ge T\ge T_s^P$, while ${\chi }^c$ peak below $T_s^P$ signifies metallic (charge) liquid (see inset for square cluster and Ref. 16).