Nonlocal exchange interactions in strongly correlated electron systems

We study the influence of ferromagnetic nonlocal exchange on correlated electrons in terms of an SU(2)-Hubbard-Heisenberg model and address the interplay of on-site interaction induced local moment formation and the competition of ferromagnetic direct and antiferromagnetic kinetic exchange interactions. In order to simulate thermodynamic properties of the system in a way that largely accounts for the on-site interaction driven correlations in the system, we advance the correlated variational scheme introduced in [M. Schüler et al., Phys. Rev. Lett. 111, 036601 (2013)] to account for explicitly symmetry broken electronic phases by introducing an auxiliary magnetic field. After benchmarking the method against exact solutions of a finite system, we study the SU(2) Hubbard-Heisenberg model on a square lattice. We obtain the $U-J$ finite temperature phase diagram of a SU(2)-Hubbard-Heisenberg model within the correlated variational approach and compare it to static mean-field theory. While the generalized variational principle and static mean-field theory yield transitions from dominant ferromagnetic to antiferromagnetic correlations in similar regions of the phase diagram, we find that the nature of the associated phase transitions differs between the two approaches. The fluctuations accounted for in the generalized variational approach render the transitions continuous, while static mean-field theory predicts discontinuous transitions between ferro- and antiferromagnetically ordered states.

Figure 1

Spin-spin correlation between next neighbors on a four-site model. The $4t^2/U$ line is where the transition is expected analytically in the strong-$U$ limit. (a) Exact diagonalization. (b) Mean-field treatment which allows ferromagnetic and antiferromagnetic ordering. (c) Variational approach.

Figure 2

Double occupancy on a four-site model. (a) Exact diagonalization. (b) Hartree-Fock mean-field treatment which allows ferromagnetic and antiferromagnetic ordering. (c) Variational approach.

Figure 3

Correlation functions for two different, fixed values $U$, depending on $J$. The black line shows the exact solution. The red dotted line stems from the mean-field solution, while the blue dashed lines come from our variational approach. [(a) and (b)] Next-neighbor spin-spin correlation. [(c) and (d)] Double occupancy.

Figure 4

Spin-spin correlation of the Hubbard-Heisenberg model between next neighbors on a half-filled square lattice, obtained from extrapolated data ($N\to \infty$) at $\beta t=10$. The $4t^2/U$ line is where the transition is expected analytically in the strong-$U$ limit. (a) Hartree-Fock mean-field treatment which allows ferromagnetic and antiferromagnetic ordering. (b) Variational approach.

Figure 5

Double occupancy of the Hubbard-Heisenberg model on a half-filled square lattice, obtained from extrapolated data ($N\to \infty$) at $\beta t=10$. (a) Hartree-Fock mean-field treatment which allows ferromagnetic and antiferromagnetic ordering. (b) Variational approach.

Figure 6

Double occupancy of the Hubbard-Heisenberg model as function of nearest-neighbor Heisenberg exchange $J$ at a fixed $U/t=3$. The shaded, grey area is meant to mark uncertainties resulting from finite size effects in the DQMC data.

Figure 7

Mean-field solutions for a half filled square lattice. (a) Next-neighbor spin-spin correlation with the $4t^2/U$ line. (b) Double occupancy.

Figure 8

(a) Double occupancy of a four-site Hubbard model at half filling, obtained from ED, DQMC for a set of Trotter steps. (b) Total error in the double occupancy after extrapolation $\mathrm{\Delta }\tau \to 0$. One extrapolation was performed with the two data points used throughout this work, the other included all Trotter steps from (a). The temperature is set to $\beta t=10$

Figure 9

Absolute errors which stem from the integration procedure and the remaining Trotter error. (a) Double occupancy. (b) Spin-spin correlation. (c) On-site interaction. (d) Magnetic field.

Figure 10

(a) Analytical (black line) and extrapolated (blue, dashed line) grand potential $\mathrm{\Phi }$ of $\tilde{H}$ with $\tilde{U}=0$. (b) Magnetization, computed as the derivative of $\mathrm{\Phi }$ with respect to $\tilde{B}$. (c) Double occupancy. (d) Next-neighbor spin-spin correlation.

Figure 11

Functional $\tilde{\mathrm{\Phi }}$, along the line connecting the two minima at ${(\tilde{U},\tilde{B})}_1\approx (1.11,0.34)$ and ${(\tilde{U},\tilde{B})}_2\approx (1.02,0.88)$, for fixed $(U,J)=(2.0,0.9525)$.