Mott Quantum Critical Points at Finite Doping

Maria Chatzieleftheriou, Alexander Kowalski, Maja Berović, Adriano Amaricci, Massimo Capone, Lorenzo De Leo, Giorgio Sangiovanni, and Luca de’ Medici

Phys. Rev. Lett. 130, 066401 – Published 7 February 2023

Abstract

We demonstrate that a finite-doping quantum critical point (QCP) naturally descends from the existence of a first-order Mott transition in the phase diagram of a strongly correlated material. In a prototypical case of a first-order Mott transition the surface associated with the equation of state for the homogeneous system is “folded” so that in a range of parameters stable metallic and insulating phases exist and are connected by an unstable metallic branch. Here we show that tuning the chemical potential, the zero-temperature equation of state gradually unfolds. Under general conditions, we find that the Mott transition evolves into a first-order transition between two metals, associated with a phase separation region ending in the finite-doping QCP. This scenario is here demonstrated solving a minimal multiorbital Hubbard model relevant for the iron-based superconductors, but its origin—the splitting of the atomic ground state multiplet by a small energy scale, here Hund’s coupling—is much more general. A strong analogy with cuprate superconductors is traced.

Received 2 April 2022
Accepted 22 December 2022

DOI:https://doi.org/10.1103/PhysRevLett.130.066401

Physics Subject Headings (PhySH)

Metal-insulator transition

Strongly correlated systems

Dynamical mean field theory Slave bosons

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Maria Chatzieleftheriou ¹, Alexander Kowalski ², Maja Berović³, Adriano Amaricci ⁴, Massimo Capone ^3,4, Lorenzo De Leo¹, Giorgio Sangiovanni ², and Luca de’ Medici ^1,*

¹Laboratoire de Physique et Etude des Matériaux, UMR8213 CNRS/ESPCI/UPMC, 75005 Paris, France
²Institut für Theoretische Physik und Astrophysik and Würzburg-Dresden Cluster of Excellence ct.qmat, Universität Würzburg, 97074 Würzburg, Germany
³International School for Advanced Studies (SISSA), Via Bonomea 265, I-34136 Trieste, Italy
⁴CNR-IOM DEMOCRITOS, Istituto Officina dei Materiali, Consiglio Nazionale delle Ricerche, Via Bonomea 265, I-34136 Trieste, Italy

^*Corresponding author. luca.demedici@espci.fr

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Issue

Vol. 130, Iss. 6 — 10 February 2023

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Images

Figure 1
Multiple solutions close to the Mott transition at $T = 0$ : interaction-driven transition. Its first-order character is due to the finite value of Hund’s coupling $J$ and is embodied by the sigmoidal shape of the $Z (U)$ curve in the range of interaction strengths $U_{c 1} < U < U_{c 2}$ where two stable solutions—one metallic at finite $Z$ (blue) and one insulating at $Z = 0$ (red)—coexist, and are connected by a third unstable metallic branch (light blue dashed line). Inset: change of spectral function between the metallic and insulating stable solutions when passing $U_{c 2} = 1.5 D$ .
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Figure 2
Multiple solutions close to the Mott transition at $T = 0$ : density-driven transition. We plot the density (measured as doping from half-filling) vs chemical potential curves for several values of the interaction strength $U / D$ . The Mott insulator is incompressible and is thus indicated by the horizontal plateau at half-filling, while the doped solutions are metallic. The points are calculated within NRG DMFT, the dashed lines sketch the unstable branch connecting the two stable ones, as deduced from calculations within exact-diagonalization(ED)-DMFT (see Supplemental Material [21]). The adiabatic connection of the solutions implies the crossing of the energy between the two stable branches at some $μ_{c} (U)$ in the coexistence zone, and thus a discontinuous jump from one to the other (corresponding to a Maxwell construction).
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Figure 3
Folding of the equation of state. (a) Illustration of the dependence of the quasiparticle weight $Z$ as a surface plot, as a function of the position in the $U − μ$ plane. The golden line at $μ = 0$ corresponds to the interaction-driven Mott transition at half-filling (Fig. 1). (b) Corresponding surface plot of the density (doping from half-filling). Curves are plotted corresponding roughly to those of Fig. 2. The vertical light blue surface corresponds to the Maxwell construction (and physically defines the zone of phase separation) and illustrates the continuity, and common first-order nature, between the interaction-driven Mott transition, the density-driven one in the range near $U_{c 1}$ and $U_{c 2}$ , and the transition between two metals ending in a QCP. This whole scenario is entailed by the folding of the equation of state surface as a multivalued function on the $U − μ$ plane (thin golden lines).
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Figure 4
Zero-temperature phase diagram in the interaction strength-doping plane. The thick black line at half-filling is the Mott insulating phase. The dashed black line signals its coexistence with a metal at half-filling. The density-driven Mott transition is first order for $U_{c 1} < U ≲ U_{c 2}$ , due to the first-order nature of the interaction-driven Mott transition, and is accompanied by a zone of phase separation (in this zone, for any given value of $U$ , the stable phase of the system is a mixture of the two homogeneous phases located at the border of the zone for the same $U$ , in proportions needed to obtain the system’s doping—its frontier starts at $U_{c 1}$ and terminates near, but not exactly at, $U_{c 2}$ , see Supplemental Material [21]). For larger interaction strengths the first-order transition is realized between two metals, whereas the density-driven Mott transition becomes second order.
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Figure 5
Landau theory of the first-order Mott transition. Landau energy function calculated within the slave-spin mean field (SSMF), where the extrema indicate the stable (minima) and unstable (maxima) equilibrium solutions for the order parameter $m_{x} / M = \sqrt{Z}$ (where $M$ is the number of orbitals, here $M = 2$ , and $Z$ is the quasiparticle weight) for various values of $U$ at half-filling (see Fig. 1): below $U_{c 1}$ (one minimum at large $Z$ , corresponding to a metal), in the coexistence zone $U_{c 1} < U < U_{c 2}$ (two minima—one of which at $Z = 0$ , corresponding to the Mott insulator—and one maximum in between), and above $U_{c 2}$ (one minimum in $Z = 0$ ).
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Figure 6
Landau theory of the first-order Mott transition. Sketch of the main features of the atomic spectrum (see Fig. S9 in the Supplemental Material [21] for the complete spectrum) and hopping processes involved in the strong-coupling perturbation theory giving the fourth-order correction to the ground-state energy $e_{4}$ , which determines the order of the interaction-driven Mott transition. In our model at half-filling the spectrum is symmetric with respect to the half-filled sector (number of electrons $N = M$ ). The connected process “ $a − b − b − a$ ” is negative in sign and its denominator involves a small energy difference $\sim O (J)$ due to the visiting of an excited state in the $N = M$ sector, and thus dominates over the positive lower-order processes (back and forth on the “ $a$ ” arrows) whose denominators are $\sim O (U)$ . This entails a negative $e_{4}$ and in turn a negative fourth order coefficient $γ_{4}$ in the Landau energy (see text), thus a first-order Mott transition.
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