Symmetry breaking in occupation number based slave-particle methods

We describe a theoretical approach to finding spontaneously symmetry-broken electronic phases due to strong electronic interactions when using recently developed slave-particle (slave-boson) approaches based on occupation numbers. We describe why, to date, spontaneous symmetry breaking has proven difficult to achieve in such approaches. We then provide a total energy based approach for introducing auxiliary symmetry-breaking fields into the solution of the slave-particle problem that leads to lowered total energies for symmetry-broken phases. We point out that not all slave-particle approaches yield energy lowering: the slave-particle model being used must explicitly describe the degrees of freedom that break symmetry. Finally, our total energy approach permits us to greatly simplify the formalism used to achieve a self-consistent solution between spinon and slave modes while increasing the numerical stability and greatly speeding up the calculations.

Figure 1

$\mathrm{\Delta }n=n_{\uparrow }-n_{\downarrow }$ as a function of $\mathrm{\Delta }h=h_{\uparrow }-h_{\downarrow }$ at one site of the 1D, half-filled, single-band Hubbard model with $U=2$ and $t=1$. Top: FM phase. Bottom: AFM phase. The $\mathrm{\Delta }h$ dependences of the spinon and slave occupancies are shown separately. Self-consistency between the two requires zero occupancy difference.

Figure 2

Total energy per site and quasiparticle weight $Z$ (renormalization factor) versus symmetry-breaking perturbation field strength $b$ based on the slave-rotor method for the half-filled, single-band 1D Hubbard model with $U=2$ and $t=1$.

Figure 3

Total energy per site and $Z$ versus field strength $b$ for the number-slave method for the single-band 1D Hubbard model at half-filling with $U=2$ and $t=1$.

Figure 4

Total energy per site and $Z$ versus field $b$ for the spin + orbital-slave approach for the single-band 1D Hubbard model at half-filling with $U=2$ and $t=1$. Unlike with the number-slave and slave-rotor approaches, correlations decrease with increasing $b$ for the AFM phase and slowly increase with $b$ for the FM phase.

Figure 5

Individual components of the total energy are shown versus $b$ for the spin + orbital-slave approach for the single-band 1D Hubbard model at half-filling with U = 2 and t = 1.

Figure 6

Comparison of the total ground-state energies (in units of $t$) for the single-band 1D Hubbard model at half-filling based on the AFM Hartree-Fock solution, the PM slave-spin solution, the symmetry-broken (AFM) slave-spin ground-state solution, and the exact Bethe ansatz (AFM) solution as calculated using the method in Ref. [36].