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arXiv:2509.18317 · Physical Review Research

Frenkel line and the pseudogap: An analogy between classical and electronic fluids

Jérôme Fournier, P.-O. Downey, Olivier Gingras, Charles-David Hébert, M. Charlebois, A.–M. S. Tremblay

Published May 12, 2026• 0 citations
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Abstract

Asymptotically close to critical end points of first-order transitions, maxima in thermodynamic quantities occur along a line called the Widom line, a concept first introduced in classical fluids. This concept has been extended to strongly correlated electronic fluids in the context of the Mott transition. Namely, upon increasing interaction strength in the Hubbard model at half filling, one finds a first-order Mott metal-insulator transition with a critical end point at high temperature, above which several crossover lines are observable. Using the dynamical cluster approximation for the triangular-lattice Hubbard model, we compute the Frenkel line, a concept borrowed from classical fluids, which is used to define a sharp crossover between the pseudogap and the correlated Fermi liquid. The Frenkel line in the electron fluid is defined by the appearance of backscattering upon entering the pseudogap. The signature of backscattering is the existence of a negative value in the time-domain optical conductivity. The Frenkel line extends to high temperatures.

DOI
10.1103/j8fc-jrmt

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Asymptotically close to critical end points of first-order transitions, maxima in thermodynamic quantities occur along a line called the Widom line, a concept first introduced in classical fluids. This concept has been extended to strongly correlated electronic fluids in the context of the Mott transition. Namely, upon increasing interaction strength in the Hubbard model at half filling, one finds a first-order Mott metal-insulator transition with a critical end point at high temperature, above which several crossover lines are observable. Using the dynamical cluster approximation for the triangular-lattice Hubbard model, we compute the Frenkel line, a concept borrowed from classical fluids, which is used to define a sharp crossover between the pseudogap and the correlated Fermi liquid. The Frenkel line in the electron fluid is defined by the appearance of backscattering upon entering the pseudogap. The signature of backscattering is the existence of a negative value in the time-domain optical conductivity. The Frenkel line extends to high temperatures.
The phase diagram of hole-doped high-temperature superconductors remains puzzling. A state of matter with unusual physical properties, dubbed pseudogap, is found below a characteristic temperature T∗T^*T∗ in a doped Mott insulator. Since the superconducting state is born out of the pseudogap over much of the phase diagram, the nature of the pseudogap is a fundamental issue in the field and it is under intense theoretical and experimental scrutiny. In many cases, it is not even clear whether T∗T^*T∗ is a crossover or a phase transition. Here we show, by constructing the normal-state phase diagram of the archetypal model of strongly correlated electron systems, that T∗T^*T∗ is a crossover line above a critical endpoint, while below that point it is a first-order phase transition between two metallic phases, one with a pseudogap and one without. Thus T∗T^*T∗ appears in a new light: it is an unexpected example of a phenomenon observed in fluids, namely a sharp crossover between different dynamical regimes along a line of thermodynamic anomalies that appears above a first-order phase transition, the Widom line. Our findings thus suggest that the critical point of a first-order transition can be the organizing principle for the rich behavior of the normal state of the cuprates.
The Mott transition is observed experimentally in materials that are magnetically frustrated so that long-range order does not hide the Mott transition at finite temperature. The Hubbard model on the triangular lattice at half filling is a paradigmatic model to study the interplay of interactions and frustration on the normal-state phase diagram. We use the dynamical cluster approximation with continuous-time auxiliary-field quantum Monte Carlo to solve this model for 1-, 4-, 6-, 12-, and 16-site clusters with detailed analysis performed for the 6-site cluster. We show that (a) for every cluster there is an inflection point in the double occupancy as a function of interaction, defining a Widom line that extends above the critical point of the first-order Mott transition; (b) the presence of this line and the cluster size dependence argue for the observability of the Mott transition at finite temperature in the thermodynamic limit; and (c) the loss of spectral weight in the metal-to-Mott-insulator transition as a function of temperature and for strong interactions is momentum dependent, the hallmark of a pseudogap. That pseudogap spans a large region of the phase diagram near the Mott transition.
The interaction-driven Mott transition in the half-filled Hubbard model is a first-order phase transition that terminates at a critical point (Tc,Uc)(T_\mathrm{c},U_\mathrm{c})(Tc​,Uc​) in the temperature-interaction plane T−UT-UT−U. A number of crossovers occur along lines that extend for some range above (Tc,Uc)(T_\mathrm{c},U_\mathrm{c})(Tc​,Uc​). Asymptotically close to (Tc,Uc)(T_\mathrm{c},U_\mathrm{c})(Tc​,Uc​), these lines coalesce into the so-called Widom line. The existence of (Tc,Uc)(T_\mathrm{c},U_\mathrm{c})(Tc​,Uc​) and of the associated crossovers becomes unclear when long-wavelength fluctuations or long-range order occur above (Tc,Uc)(T_\mathrm{c},U_\mathrm{c})(Tc​,Uc​). We study this problem using continuous-time quantum Monte Carlo methods as impurity solvers for both Dynamical Mean-Field Theory (DMFT) and Cellular Dynamical Mean-Field Theory (CDMFT). We contrast the cases of the square lattice, where antiferromagnetic fluctuations dominate in the vicinity of the Mott transition, and the triangular lattice where they do not. The inflexion points and maxima found near the Widom line for the square lattice can serve as proxy for the triangular lattice case. But the only crossover observable in all cases at sufficiently high temperature is that associated with the opening of the Mott gap. The same physics also controls an analog crossover in the resistivity called the "Quantum Widom line".
The pseudogap refers to an enigmatic state of matter with unusual physical properties found below a characteristic temperature T* in hole-doped high-temperature superconductors. Determining T* is critical for understanding this state. Here we study the simplest model of correlated electron systems, the Hubbard model, with cluster dynamical mean-field theory to find out whether the pseudogap can occur solely because of strong coupling physics and short nonlocal correlations. We find that the pseudogap characteristic temperature T* is a sharp crossover between different dynamical regimes along a line of thermodynamic anomalies that appears above a first-order phase transition, the Widom line. The Widom line emanating from the critical endpoint of a first-order transition is thus the organizing principle for the pseudogap phase diagram of the cuprates. No additional broken symmetry is necessary to explain the phenomenon. Broken symmetry states appear in the pseudogap and not the other way around.
The experimentally established phase diagram of the half-filled Hubbard model features the existence of three distinct finite-temperature regimes, separated by extended crossover regions. A number of crossover lines can be defined to span those regions, which we explore in quantitative detail within the framework of dynamical mean-field theory. Most significantly, the high-temperature crossover between the bad metal and Mott-insulator regimes displays a number of phenomena marking the gradual development of the Mott insulating state. We discuss the quantum critical scaling behavior found in this regime, and propose methods to facilitate its possible experimental observation. We also introduce the concept of quantum Widom lines and present a detailed discussion that highlights its physical meaning when used in the context of quantum-phase transitions.
For doped two-dimensional Mott insulators in their normal state, the challenge is to understand the evolution from a conventional metal at high doping to a strongly correlated metal near the Mott insulator at zero doping. To this end, we solve the cellular dynamical mean-field equations for the two-dimensional Hubbard model using a plaquette as the reference quantum impurity model and continuous-time quantum Monte Carlo method as impurity solver. The normal-state phase diagram as a function of interaction strength UUU, temperature TTT, and filling nnn shows that, upon increasing nnn toward the Mott insulator, there is a surface of first-order transition between two metals at nonzero doping. That surface ends at a finite temperature critical line originating at the half-filled Mott critical point. Associated with this transition, there is a maximum in scattering rate as well as thermodynamic signatures. These findings suggest a new scenario for the normal-state phase diagram of the high temperature superconductors. The criticality surmised in these systems can originate not from a T === 0 quantum critical point, nor from the proximity of a long-range ordered phase, but from a low temperature transition between two types of metals at finite doping. The influence of Mott physics therefore extends well beyond half-filling.