Mean field theory of short-range order in strongly correlated low dimensional electronic systems

Mean field approach, although a generally reliable tool that captures major short-range correlations, often fails in symmetric low dimensional strongly correlated electronic systems like those described by the Hubbard model. In these situations a symmetry is almost broken. The problem is linked to the restoration of the symmetry due to strong fluctuations (both quantum and thermal) on all scales. The restoration of symmetry in statistical models of scalar order parameter fields was treated recently successfully on the Gaussian approximation level by symmetrization of the correlators. Here the idea is extended to fermionic systems in which the order parameter is composite. Furthermore, the precision of the correlators can be improved perturbatively. Such a scheme (based on covariant Gaussian approximation) is demonstrated on the one dimensional (1D) and 2D one band Hubbard models by comparison of the correlator with exact diagonalization and MC simulations, respectively.

Figure 1

Imaginary part of the correlator for quantum dot at half filling in a wide range of couplings $u=U/T$. Matsubara frequencies are $\omega =\pi T$ and $\omega =3\pi T$ with $T=1$ in (a) and (b), respectively. The red line is the exact result, the green line is the Hartree Fock result, the darker green line is the symmetrized Green function Eq. (32) for the magnetic phase, and the purple line is the perturbative correction to Gaussian approximation (PCGA) Eqs. (57) and (58).

Figure 2

Comparison of the exact correlator of a short Hubbard chain at temperature $T=0.2$ with approximations in a wide range of couplings for $\omega =\pi T,k=0$. The approximations include the CGA [green lines, the parasolution from Eq. (36); darker green lines from Eq. (48)] that is symmetrized above the spurious transition at $U_c=1.33$ and perturbative correction to Gaussian approximation (PCGA, purple lines).

Figure 3

Comparison of the exact correlator of a short Hubbard chain at temperature $T=0.2$ with approximations in a wide range of couplings for $\omega =\pi T,k=\pi /2$ at $\delta \mu =0,0.25,1$ from top to bottom. The approximations include the CGA [green lines, the parasolution from Eq. (36); darker green lines from Eq. (48)] that is symmetrized above the spurious transition at $U_c=1.33$, and perturbative correction to Gaussian approximation in (a) (PCGA, purple lines). The inset of (a) is an enlarged figure near the “critical coupling” region. In (b) and (c) $(\delta \mu =0.25,1$, respectively), the results for the CGA (green lines for parasolution and darker green lines from the symmetrized correlator) are present; the exact results are plotted as red lines.

Figure 4

Setting sun diagrams that contribute to PCGA. The directed lines are Gaussian correlators, while the vertices are “perturbative.”

Figure 5

Quasimomentum distribution function $n_k=2T{\sum }_{m}g\left(m,k\right)$ in a half filled 1D Hubbard model $\left(N_s=24\right)$. Factor 2 is due to spin summation. Horizontal axis as $k$ is quasimomentum and quantized as in unit $2\pi /N_s$, and the plot range of $k$ is from zero to $\pi$. The results for CGA (green dots) and PCGA (purple dots) are plotted along with MC results (red lines).

Figure 6

Charge density correlator ${\chi }^{\rho }={\chi }_{n,k}^{\rho }$ dependence of $U$ at $T=1$, for frequency $n=0$ and momentum at $k=0$. The green curve is the Lindhard diagram result and the purple curve contains in addition the next order correction to the Lindhard diagram. The green curve and the purple curve in the inset are the ratios between the Lindhard result and the result including the next order correction to the exact value, respectively.

Figure 7

Spin correlator ${\chi }_n^s=〈{\vec{S}}_{n,x}·{\vec{S}}_{-n,x}〉$ dependence of $U$ at $T=1$ for frequency $n=0$. The green curve is the Lindhard diagram result and the purple curve contains in addition the next order correction to the Lindhard diagram.

Figure 8

Imaginary part of the Green's function $G\left(\tau ,\mathbf{k}\right)$ of a half filled 2D Hubbard model at $k=\left(\pi ,0\right)$. The results for CGA (green dots) and PCGA (purple dots) are plotted along with MC results (red lines).