Mott insulator: Tenth-order perturbation theory extended to infinite order using a quantum Monte Carlo scheme

We present a method based on the combination of analytical and numerical techniques within the framework of the dynamical mean-field theory. Building upon numerically exact results obtained in an improved quantum Monte Carlo scheme, tenth-order strong-coupling perturbation theory for the Hubbard model on the Bethe lattice is extrapolated to infinite order. We obtain continuous estimates of energy $E$ and double occupancy $D$ with unprecedented precision $\mathcal{O}\left({10}^{-5}\right)$ for the Mott insulator above its stability edge $U_{c1}\approx 4.78$ as well as critical exponents. The relevance for recent experiments on Cr-doped ${\mathrm{V}}_2{\mathrm{O}}_3$ is pointed out.

Figure 1

Low-order diagrams in PT: A first hopping process creates an empty site (circle). After a total of $2n$ hoppings, impacting $n$ other sites (spheres), the ground state is restored.

Figure 2

(Color online) Ground-state energy $E$ and double occupancy $D$: PT according to Eqs. (2, 3) (solid lines) in comparison with low-order PT (dashed and dotted lines) and QMC (circles).

Figure 3

(Color online) Construction of ePT:PT values for $U_c\left[n\right]$ (squares) are extrapolated to $1∕n\to 0$ using a quadratic fit (solid line) restricted to $\tau =3.5$. Values at smaller $1∕n$ (circles) define ePT coefficients to all orders. Inset: restricted and unrestricted fit (solid and dashed lines), PT “data,” and range compatible with QMC (shaded) after subtraction of a linear term.

Figure 4

(Color online) Main panel: energy from QMC (circles), ePT (solid line), and truncated ePT (dashed and dotted lines); in all cases, the PT result [Eq. (2) and solid line in Fig. 2] has been subtracted. Inset: energy correction in ePT beyond noncritical terms (solid line) in comparison with the leading critical term.

Figure 5

(Color online) Double occupancy from QMC (circles) and ePT (solid lines) in complete analogy with Fig. 4.