Effective classical correspondence of the Mott transition

We derive an effective classical model to describe the Mott transition of the half-filled one-band Hubbard model in the framework of the dynamical mean-field theory with hybridization expansion of the continuous time quantum Monte Carlo. We find a simple two-body interaction of exponential form and reveal a classical correspondence of the Mott transition driven by a logarithmically divergent interaction length. Our paper provides an alternative angle to view the Mott physics and suggests a renewed possibility to extend the application of the quantum-to-classical mapping in understanding condensed-matter physics.

Figure 1

A typical phase diagram of the half-filled one-band Hubbard model on the Bethe lattice. The point marks the critical end point (CEP) of the first-order Mott transition. $U_{c1}$ and $U_{c2}$ are the critical $U$ at zero temperature from the insulating and metallic sides, respectively. The shadow area is the coexisting (hysteresis) region of the two phases. The inset shows the structure of the Bethe lattice and its DMFT mapping to an impurity model coupled to a bath. The densities of states of the two phases are also plotted for comparison.

Figure 2

Mapping between the CT-HYB configuration and the classical configuration of charged particles. For CT-HYB, the full and empty circles at the imaginary times ${\tau }_i\left({\tilde{\tau }}_i\right)$ and ${\tau }_i^{\prime }\left({\tilde{\tau }}_i^{\prime }\right)$ represent the creation and annihilation operators of spin up (down), respectively. The shaded areas indicate the overlap between the segments (the solid lines) of two spin channels. In the classic model, the circles represent charged particles on a unit circle at $x_i^{+}={\tau }_i/\beta ,x_i^{-}={\tau }_i^{\prime }/\beta ,{\tilde{x}}_i^{+}={\tilde{\tau }}_i/\beta$, and ${\tilde{x}}_i^{-}={\tilde{\tau }}_i^{\prime }/\beta$.

Figure 3

Construction of the classical model for CT-HYB configurations. (a) Comparison of the logarithmic weight of test samples and the linear regression results. (b) Comparison of the logarithmic hybridization weight and the effective classical energy $-E_{\mathrm{eff}}$. Each point represents a test sample, and the dotted line is a guide to the eye. (c) The derived Legendre coefficients $V_{q_i{q}_j}^l$ for the two-body interactions. (d) The two-body interactions $V_{q_i{q}_j}\left(x\right)$ as functions of the distance $x$ on the unit circle. The parameters are $U=3$ and $\beta =30$.

Figure 4

(a) Exponential fit of the effective potential for different values of $U$ and $\beta$, showing excellent agreement in all three regimes. (b) The fitting values of $V_0$ and $\xi$ in the classical model after DMFT convergence with gradually increasing $(⊳)$ or decreasing $(⊲)U$.

Figure 5

(a) Variations of $V_0$ as functions of $\beta$ and $U$, showing a rapid change across the Mott transition and a slight change in the other parameter regime. The arrows indicate the direction of decreasing (the solid line) or increasing (the dashed line) $U$ for DMFT calculations. The dotted lines mark the region of numerical hysteresis of the first-order transition. (b) The pair number $\langle k\rangle$ as a function of the inverse temperature $\beta$ for different values of $U$, showing the slope change across the Mott transition. The inset compares the calculated pair density $\langle k\rangle /\beta$ and the double occupancy $\langle D\rangle$ for increasing $U$.

Figure 6

(a) Illustration of the parameter flow on the $V_0\text{−}\xi$ space with increasing (the circle) or decreasing (the square) $U$ at $\beta =50$, showing almost constant $V_0$ and an instablity with varying $\xi$. (b) Logarithmic fit (the solid lines) of the effective length $\xi$ from both sides of the Mott transition for $\beta =50$. $U_{c2}$ and $U_{c1}$ are the critical values of $U$ from the metallic and insulating sides, respectively.