The Frenkel line and the pseudogap: an analogy between classical and electronic fluids

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 endpoint at high temperature, above which several crossover lines are observable. Using the dynamical cluster approximation for the triangular-lattice Hubbard model, we compute a new crossover line, the Frenkel line, a concept borrowed from classical fluids that is useful for defining a sharp crossover between the pseudogap and the correlated Fermi liquid. The Frenkel line in the electron fluid is defined by the appearance of back-scattering upon entering the pseudogap. The signature of back-scattering is the existence of a negative value in the time-domain optical conductivity. The Frenkel line extends to high temperatures.

Figures (9)

• Figure 1: $T-U$ phase diagram of the isotropic triangular-lattice Hubbard model at half-filling. The Frenkel line and the dynamic line $U_{\sigma(\omega)}$ have been added to the phase diagram reported in Ref. downey_mott_2023, which already included the other crossover and transition lines. The dashed and the dotted lines represent crossover lines in the supercritical regime. The inflection in double occupancy $U_D$, the Frenkel line $U_{\sigma(t)}$, and the dynamic lines $U_{\sigma(\omega)}$ and $U_{inc}$ are respectively black, purple, green, and blue. The markers for the different crossover lines between the pseudogap and correlated Fermi liquid are, respectively, inverted triangles, squares, diamonds, and circles. The crossover lines $U_D$ ($U_W$ in the earlier notation) and $U_{inc}$ have been introduced in Ref. downey_mott_2023 along with the red dotted line $U_\text{gap}$ defined by the formation of a gap at $\omega=0$. The correlated Fermi liquid (cFL), the pseudogap, and the Mott insulator regime are colored cyan, blue, and red. The red star at $U = 8.258$ and $T = 2/17$ identifies the Mott critical point. The solid lines represent the first-order phase transition. The hatched region between $U_{c1}$ and $U_{c2}$ represents a region of phase coexistence between the correlated Fermi liquid and the pseudogap or Mott insulating phase. The tiny grey circles are the data points.
• Figure 2: a) Six-site cluster with hopping terms The coordinate axes $\mathbf{x}$ and $\mathbf{y}$ are shown in black. The primitive vectors $\mathbf{a}_1$ and $\mathbf{a}_2$ of the lattice are indicated in blue. The superlattice vectors $\mathbf{R}_1$ and $\mathbf{R}_2$ illustrate the periodic boundary conditions. They are chosen such that if $t_2$ were zero, the resulting lattice would be bipartite. b) Brillouin zone. The Fermi surface in orange is for $U = 0$ and $n = 1$ at $T = 0.1$ on the isotropic triangular lattice. It is a hole Fermi surface. The primitive reciprocal-lattice vectors $\mathbf{b}_1$ and $\mathbf{b}_2$ are shown in cyan. The patches for the DCA are outlined with their associated labels. The reciprocal lattice vectors of the superlattice determine the choice of the patches in the Brillouin zone. The tilt of the patches arises because of a change of basis. In the computational reciprocal-lattice basis that we use, the patches are rectangular.
• Figure 3: Definition of the Frenkel line and other crossover lines presented in Fig. \ref{['fig:phasediag']}. All data in these figures are at $T=1/6$. a) Derivative of the double occupancy with respect to the interaction $U$ as a function of $U$. The line of inflection of the double occupancy $U_D$ is given by the extrema of $dD/dU$. b) Density of states as a function of $\omega$ for the different values of $U$ found in the legend of Fig. \ref{['fig:differentlines']}d). A dip at $\omega=0$ appears at $U_{inc}$. It should be noted that under a sign flip of $t_2$, the density of states must be modified as follows $A(\omega)\to A(-\omega)$downey_leffet_2022. c) Optical conductivity as a function of $\omega$ for different values of $U$. The Drude peak disappears at $U_{\sigma(\omega)}$. d) Conductivity as a function of time $t$ for different $U$. That conductivity, $\sigma_1(t)$ can become negative at $U_{\sigma(t)}$. This defines the Frenkel line. The black triangle, blue circle, purple diamond, and red square represent the markers used to display the corresponding lines in Fig \ref{['fig:phasediag']}.
• Figure 4: a) Frequency-dependent optical conductivity obtained from analytical continuation with the maximum entropy method OmegaMaxEnt OmegaMaxEnt; and b) with Padé approximants. c) Time-dependent optical conductivity obtained from Fourier transformation of $\sigma_1(\omega)$ in a); and d) Fourier transformation of $\sigma_1(\omega)$ in b). All these curves are for $T=1/6$.
• Figure 5: a) Frequency-dependent optical conductivity obtained with the DCA Green's function, and b) with the lattice Green's function. c), d) Time-dependent optical conductivity obtained from Fourier transformation of $\sigma_1(\omega)$ in a); and d) Fourier transformation of $\sigma_1(\omega)$ in b). All these curves are for $T=1/6$.