Mott metal-insulator transition in a modified periodic Anderson model: Insights from entanglement entropy and role of short-range spatial correlations

TL;DR

The paper analyzes the Mott metal-insulator transition in a three-orbital modified periodic Anderson model, identifying a zero-temperature quantum critical point between a Fermi liquid and a Mott insulator. It employs a two-site (linearized) DMFT to obtain analytical expressions for the critical interaction $U_c$, and extends to a minimal cluster (LCDMFT) to assess short-range spatial correlations. A key result is the analytical relation $U_c = 6\sqrt{M_2}$ with $M_2$ defined by the lattice DOS and hybridization, complemented by a closed-form near-critical expression for the local entanglement entropy $S$ that tracks the transition from $S=\ln 4$ to $S=\ln 2$ as the system localizes. The study demonstrates that short-range correlations modestly modify the transition near $t_{\perp}/D \approx 1$ and that entanglement entropy provides a robust, symmetry-breaking-free marker of the quantum critical point, with potential extensions to larger clusters and finite-temperature analyses.

Abstract

The Mott transition is a paradigmatic phenomenon where Coulomb interactions between electrons drive a metal-insulator phase transition. It is extensively studied within the Hubbard model, where a quantum critical transition occurs at a finite temperature second-order critical point. This work investigates the Mott transition in a modified periodic Anderson model that may be viewed as a three-orbital lattice model including an interacting, localized orbital coupled to a delocalized conduction orbital via a second conduction orbital. Within the dynamical mean field theory, this model possesses a strictly zero temperature quantum critical point separating a Fermi liquid and a Mott insulating phase. By employing a simplified version of the dynamical mean field theory, namely, the two-site or linearized dynamical mean field theory, an analytical estimate is provided for the critical parameter strengths at which the transition occurs at zero temperature. An analytical estimate of the single-site von Neumann entanglement entropy is also provided. This measure can be used as a robust identifier for the phase transition. These calculations are extended to their cluster version to incorporate short-range, spatial correlations and discuss their effects on the transition observed in this model.

Figures (8)

• Figure 1: A schematic of the model Hamiltonian studied in this work. Each site ($i$) on the lattice consists of three orbitals: a highly localized $f$ orbital (depicted as the blue layer) that hybridizes with a delocalized $c$ orbital (yellow layer) via a hybridization energy $V$. The $c$ orbital is further coupled to a second set of delocalized orbitals ($c_M$, brown layer) via the hopping integral $t_\perp$. The electrons in the delocalized $c$ and $c_M$ orbitals hop around intra-orbitally with hopping energy $t$ between nearest neighbours and are non-interacting. The electrons on the localized $f$ orbitals interact with each other via a repulsive (Hubbard) interaction $U$.
• Figure 2: Schematic of the cluster framework used in this work. (a) The model Hamiltonian is represented on a 2-dimensional square lattice, with three orbitals on each lattice site, namely, (green) delocalized conduction orbital, $c_M$, locally coupled to a (orange) conduction orbital, $c$. The latter conduction orbital hybridizes with a localized $f$-orbital (blue). This lattice is then mapped within the cellular DMFT framework (CDMFT), considering a pair ($2\times1$) cluster as shown in panel (b). This correlated "impurity" pair (blue shaded sites) is, in principle, embedded in a non-interacting host that needs to be obtained self-consistently within the CDMFT framework. (c) In this work, we further simplify this mapping by considering a linearized-CDMFT (LCDMFT) framework, where the entire host is approximated as two sites (brown shaded sites) coupled to the impurity cluster. Note the hopping scheme used between the impurity cluster and host atoms. ster. Note the hopping scheme used between the impurity cluster and host atoms.
• Figure 3: Phase diagram of the modified periodic Anderson model for hybridization energy $V/D=0.4$ (connecting $c$ and interacting $f$-orbitals) separating a Fermi liquid (FL) phase from a Mott insulating (MI) phase in the $U-t_\perp$ plane, where $D$ is the half-bandwidth of the conduction orbitals. This phase diagram is evaluated using the expression for $U_c=6\sqrt{M_2}$ (given in equation \ref{['eq:U_c']}), calculated within linearized dynamical mean field theory framework. Here the information about the lattice, the $c$ electron bare density of states (here chosen 2D square lattice density of state) and the parameter $t_\perp$ (hybridization energy connecting two conduction orbitals $c$ ad $c_M$) enters through $M_2$.
• Figure 4: The quasi-particle weight, $Z$, calculated within the linearized dynamical mean field theory (LDMFT) and linearized cellular dynamical mean filed theory (LCDMFT) framework, is plotted as a function of the interaction strength, $U$ for $t_\perp/D=1.005,\,1.02\,, 1.31$ in panel (A) and as a function of $t_\perp$ in panel (B), for $U/D=0.8\,, 1.2$. All parameters are scaled by half-bandwidth, $D$. In panel (A), for $t_\perp/D\to 1$, the non-local correlations show a relative increase in the critical interaction strength relative to the local theory. However, as the $t_\perp$ is gradually increased, the local and cluster theories overlap with each other. This could be suggesting that the pair cluster used here is not sufficiently large to capture the non-local correlations due to the increasing $t_\perp$ parameters. The difference is quantified in panel (C), where we plot the quantity $(\Delta Z)^2=(Z_{LDMFT}-Z_{LCDMFT})^2$, where $Z_{LDMFT(LCDMFT)}$ is the $Z$ calculated within the LDMFT (LCDMFT) framework. In Panel(B), the LDMFT and LCDMFT results overlap as a function of $t_\perp$ for the relatively low $U$ parameters shown.
• Figure 5: (A) The local (von Neumann) entanglement entropy $S$ evaluated using equation \ref{['eq:entanglement_en']}, with the double occupancy, $d$, calculated using linearized DMFT (LDMFT: solid line) and linearized cellular DMFT(LCDMFT: dashed line) are plotted as a function of $U$ for $t_\perp=1.005,\, 1.02,\, 1.31$. (B) The same is plotted as a function of $t_\perp$ for $U=0.8,\, 1.2$. The respective double occupancy plots are shown in the insets (C, E). All parameters are scaled by half-bandwidth, $D$. The free or the non-interacting limit is marked by $S=\ln 4$, and the local moment limit of the model is marked as $S=\ln 2$. (D) $(\Delta S)^2=(S_{LDMFT}-S_{LCDMFT})^2$, where, $S_{LDMFT(LCDMFT)}$ refers to the entanglement entropy evaluated using $d$ calculated within LDMFT(LCDMFT) framework. $(\Delta S)^2$ is plotted as a function of $U$ for the respective $t_\perp$ parameters, highlighting the increasing prominence of non-local effects for decreasing $t_\perp$. Similar observations are shown in Figure \ref{['fig:z']}. The $U_c$ values mentioned in (A, B) are obtained using the relation, $U_c=6\sqrt{M_2}$, and the $t_{\perp c}$ values are respectively interpolated.