Nonperturbative landscape of the Mott-Hubbard transition: Multiple divergence lines around the critical endpoint

We analyze the highly nonperturbative regime surrounding the Mott-Hubbard metal-to-insulator transition (MIT) by means of dynamical mean field theory (DMFT) calculations at the two-particle level. By extending the results of Schäfer et al. [Phys. Rev. Lett. 110, 246405 (2013)] we show the existence of infinitely many lines in the phase diagram of the Hubbard model where the local Bethe-Salpeter equations, and the related irreducible vertex functions, become singular in the charge as well as the particle-particle channel. By comparing our numerical data for the Hubbard model with analytical calculations for exactly solvable systems of increasing complexity [disordered binary mixture (BM), Falicov-Kimball (FK), and atomic limit (AL)], we have (i) identified two different kinds of divergence lines; (ii) classified them in terms of the frequency structure of the associated singular eigenvectors; and (iii) investigated their relation to the emergence of multiple branches in the Luttinger-Ward functional. In this way, we could distinguish the situations where the multiple divergences simply reflect the emergence of an underlying, single energy scale ${\nu }^{*}$ below which perturbation theory is no longer applicable, from those where the breakdown of perturbation theory affects, not trivially, different energy regimes. Finally, we discuss the implications of our results on the theoretical understanding of the nonperturbative physics around the MIT and for future developments of many-body algorithms applicable in this regime.

Figure 1

Dyson equation (upper rows) and Bethe-Salpeter equation (lower rows) represented in terms of Feynman diagrams and simplified, symbolic notation. The blue double lines with arrows represent interacting Green's functions $\left(G\right)$, whereas single blue lines with arrows are noninteracting Green's functions $\left(G_0\right)$. The self-energy is denoted by $\mathrm{\Sigma }$. At the two-particle level, $\chi$ denotes the interacting generalized susceptibility and $\mathrm{\Gamma }$ is the irreducible vertex in the selected channel of the Bethe-Salpeter equation.

Figure 2

Phase diagram of the BM model (with a Bethe-lattice DOS), showing the divergence lines for ${\mathrm{\Gamma }}_{c,\pm }$. The index $n$ corresponds to the index of the Matsubara frequency $\nu =\pi T(2n+1)$ for which the divergence appears at a given temperature $T$ and disorder strength $W$. Inset: The energy scale ${\nu }^{*}\left(W\right)$ at which the divergence appears for a given $W$. Note that the same phase diagram is also applicable to the frequency-localized divergences of the FK model.

Figure 3

Imaginary parts of ${\mathrm{\Sigma }}^{\pm }\left(i\nu \right)$ and $\mathrm{\Sigma }\left(i\nu \right)={\mathrm{\Sigma }}_{\text{phys}}\left(i\nu \right)$ for the half-filled BM (and FK) model on the Bethe lattice at $T=0.002$. Left panel: $W=0.65<\tilde{W}$. Right panel: $W=0.8>\tilde{W}$. For $W>\tilde{W}$, at the frequency ${\nu }^{*}\left(W\right)$, the physical self-energy changes from ${\mathrm{\Sigma }}^{-}$ to ${\mathrm{\Sigma }}^{+}$. Insets: Imaginary parts of the corresponding (physical) Green's functions $G\left(i\nu \right)$. For $W<\tilde{W},G\left(i\nu \right)$ is monotonic (for $\nu >0$) while for $W>\tilde{W}$ it exhibits a well-defined minimum [or maximum in absolute values $\left|G\right(i\nu \left)\right|]$ at $\nu ={\nu }^{*}$.

Figure 4

Colored lines: $\mathrm{\Sigma }\left[G\right]$ as given in Eq. (23) for $T=0.01$ and $W=1$. Different colors correspond to different Matsubara frequencies $\nu =\pi T(2n+1)$, from $n=1$ (brick red) to $n=11$ (dark blue). Black (full and dashed) line: ${\mathrm{\Sigma }}^{\pm }\left[G\right]$ as given in Eq. (16). The intersection between the colored and the black lines determines the physical self-energy and Green's function at the given frequency $\nu$. The dashed yellow curve is that at $\nu ={\nu }^{*}$.

Figure 5

Irreducible vertex in the charge channel ${\mathrm{\Gamma }}_{\text{c}}^{\nu \nu (\mathrm{\Omega }=0)}$ of the BM model for different values of $W=0.7,1.0,1.5,2.0$ at $T=0.5$, plotted as a function of the fermionic Matsubara frequencies $\nu$.

Figure 6

Real (left panels) and imaginary (right panels) parts of the DMFT self-energy of the BM model on the real frequency axis for $W=0.65<\tilde{W}$ (top panels) and for $W=0.8>\tilde{W}$ (bottom panels).

Figure 7

Comparison between the energy scales on the imaginary axis ${\nu }^{*}\left(W\right)$ and the real frequency axis ${\omega }^{*}\left(W\right)$, respectively, obtained from the DMFT solution of the BM model on a Bethe lattice.

Figure 8

Main panel: Phase diagram for the atomic limit: The divergence lines for ${\mathrm{\Gamma }}_c$ associated with the energy scale ${\nu }^{*}\left(U\right)$ are shown in red and classified according to the number of the Matsubara frequency at which the divergence takes place. The divergence lines of the second kind, appearing in ${\mathrm{\Gamma }}_{\text{si}}$, are depicted in orange. Inset: Energy scale ${\nu }^{*}$ for the AL.

Figure 9

Upper panel: Matsubara frequency density plot of the full (local) scattering amplitude $F^{\nu {\nu }^{\prime }(\mathrm{\Omega }=0)}$ of the DMFT solution of the Hubbard model, on both sides of the first divergence (red) line of Fig. 10 at $T=0.1$. Middle/lower panels: The same for 2PI functions in the charge ${\mathrm{\Gamma }}_{\text{c}}^{\nu {\nu }^{\prime }(\mathrm{\Omega }=0)}$ and in all channels ${\mathrm{\Lambda }}^{\nu {\nu }^{\prime }(\mathrm{\Omega }=0)}$, respectively. Note that, for a better readability, in all cases the bare interaction term $U$ has been subtracted.

Figure 10

Left panel: DMFT phase diagram of the irreducible vertex divergences (of the first kind) of the BM/FK model (see Sec. 3), replotted for a comparison against the Hubbard model results discussed in this section. Right panel: DMFT phase diagram showing the landscape of irreducible vertex divergences surrounding the Mott-Hubbard MIT in the half-filled unfrustrated Hubbard model. The blue line indicates the MIT [42]. The red and orange lines show the points where ${\mathrm{\Gamma }}_{\text{c}}^{\nu {\nu }^{\prime }(\mathrm{\Omega }=0)}$ diverges at low frequencies, whereas an additional divergence of ${\mathrm{\Gamma }}_{\text{pp},\uparrow \downarrow }^{\nu {\nu }^{\prime }(\mathrm{\Omega }=0)}$ takes place simultaneously to the one of ${\mathrm{\Gamma }}_{\text{c}}^{\nu {\nu }^{\prime }(\mathrm{\Omega }=0)}$ (only) at the orange lines. On the right-hand side, the (exact) values of the ratio $T/U$ for the atomic limit $\left(t=0\right)$ are listed.

Figure 11

Evolution of the frequency structure of the eigenvector of ${\left({\chi }_{\text{c}}/{\chi }_0\right)}^{\nu {\nu }^{\prime }(\mathrm{\Omega }=0)}$ (see also Ref. [78]) associated with the first zero eigenvalue for different temperatures along the first divergence line of Fig. 10.

Figure 12

Low temperature dependence of the first seven divergence lines in the Hubbard model for the 2D square lattice. The blue line represents the Mott-Hubbard transition.

Figure 13

Low temperature dependence of the first divergence line in the Hubbard model for the 2D square lattice (solid line) and the Bethe lattice (dashed line).

Figure 14

Distributed disorder with gap $W$.

Figure 15

Phase diagram of the disorder model. Dashed line: Vertex divergence, solid line: MIT.