High-temperature expansion method for calculating paramagnetic exchange interactions

The method for calculating the isotropic exchange interactions in the paramagnetic phase is proposed. It is based on the mapping of the high-temperature expansion of the spin-spin correlation function calculated for the Heisenberg model onto the Hubbard Hamiltonian one. The resulting expression for the exchange interaction has a compact and transparent formulation. The quality of the calculated exchange interactions is estimated by comparing the eigenvalue spectra of the Heisenberg model and low-energy magnetic part of the Hubbard model. By the example of quantum rings at half-filling with different hopping setups, we analyze contributions from different parts of the Hubbard model spectrum to the resulting exchange interaction. The developed method can be applied for simulating magnetic couplings in correlated materials and surface nanostructures.

Figure 1

Comparison of the excitation spectra of the Hubbard model (solid and dashed lines) and Heisenberg models with exchange interaction calculated by using the developed method (green triangles) and superexchange Anderson approach (blue rhombus) for isolated (a) dimer and (b) triangle.

Figure 2

(a) Comparison of the eigenvalue spectra obtained from the solution the Hubbard model (black solid line) and the Heisenberg models with parameters calculated by the developed approach (9) (green triangles), local force theorem method (13) (red circles), and Anderson's superexchange theory (blue rhombus). (b) Magnetization as a function of the localization.

Figure 3

Schematic representation of the dimer adsorbed on a substrate. Here, $t$ and $U$ are impurity kinetic and Coulomb repulsion terms whereas $\varepsilon$ and $V$ denote energy of bath states and hybridization terms, respectively.

Figure 4

(a) Contributions of $d^0, d^1$, and $d^2$ atomic states of the impurity to the ground state, the average value of square of the spin operator for impurity and bath orbitals. (b) Corresponding exchange constant derived with (9) for Anderson model (14) with two bath states at each impurity level (see Fig. 3).

Figure 5

Low-energy magnetic part of the Hubbard model eigenspectrum for trimer as a function of ratio $\frac{t_{\text{nnn}}}{t_{\text{nn}}}$. Numbers around the lines denote degeneracy of the corresponding energy levels. The spectra were calculated with $t_{\text{nn}}=0.03$ eV and $\mu =\frac{U}{2}=1.5$ eV.

Figure 6

(a) The calculated exchange interactions between nearest neighbors $J_{\text{nn}}$ and next-nearest neighbors $J_{\text{nnn}}$ in the trimer. (b) Comparison of the eigenvalue spectra for the trimer. Solid and dashed lines denote the Hubbard model spectrum. The symbols correspond to the Heisenberg model solutions with different sets of the exchange interactions and presented by blue rhombus (the Anderson's superexchange) and green triangles [the developed HTE method (9)].

Figure 7

Schematic representation of the hopping setups for the Hubbard Hamiltonian simulations. (a) The ring model with the nearest-neighbor hoppings. (b) All-to-all configuration in which all the hoppings between sites are equal.

Figure 8

(a) The calculated exchange interactions between nearest neighbors $J_{\text{nn}}$ and next-nearest neighbors ${\mathrm{J}}_{\text{nnn}}$ in the five-site ring. (b) Comparison of the eigenvalue spectra for the five-site ring. Solid and dashed lines denote the Hubbard model spectrum. Blue rhombus and green triangles correspond to the Heisenberg model solutions with different sets of the exchange interactions: blue rhombus (Anderson's superexchange) and green triangles [developed method (9)].

Figure 9

Contributions from the different eigenstates of the Hubbard model to (a) the nearest-neighbor and (b) next-nearest-neighbor exchange interactions calculated for four-site ring with $t/U=0.1$.

Figure 10

Excitation spectra of all-to-all systems. Solid and dashed lines denote the Hubbard model spectrum. Green triangles correspond to the Heisenberg model solutions with the exchange interactions calculated by the high-temperature expansion method (9).

Figure 11

The calculated ratio $\frac{J_{\text{nn}}^{\text{ring}}}{J_{\text{nn}}^{\text{all-to-all}}}$ demonstrating the contribution of the indirect hopping processes to the two-spin interaction.

Figure 12

Comparison of the eigenvalue spectra for the five-site ring. Solid and dashed lines denote the Hubbard model spectrum. Green triangles and red pluses denote Heisenberg model solutions obtained by using the developed method as described in Appendix pp3.