Interplay of Correlations and Kohn Anomalies in Three Dimensions: Quantum Criticality with a Twist

A general understanding of quantum phase transitions in strongly correlated materials is still lacking. By exploiting a cutting-edge quantum many-body approach, the dynamical vertex approximation, we make important progress, determining the quantum critical properties of the antiferromagnetic transition in the fundamental model for correlated electrons, the Hubbard model in three dimensions. In particular, we demonstrate that—in contradiction to the conventional Hertz-Millis-Moriya theory—its quantum critical behavior is driven by the Kohn anomalies of the Fermi surface, even when electronic correlations become strong.

Figure 1

Phase diagram of the 3D Hubbard model at $U=2.0$, showing the leading magnetic instability as a function of the density $n$ in both DMFT and $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$. Inset: Evolution of the magnetic ordering vector along the instability line of $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$, showing a transition from an commensurate AFM with $Q_z=\pi$ (open triangles in the main panel) to incommensurate SDW with $Q_z<\pi$ (full triangles in the main panel). The dashed red line indicates the presumptive crossover between AFM and SDW.

Figure 2

Inverse correlation length (${\xi }^{-1}$, upper panels) and susceptibility (${\chi }_{\mathbf{Q}}^{-1}$, lower panels) computed in $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ as a function of $T$ for different $n$. The solid lines show the fits for extracting the critical exponents $\nu$ and $\gamma$ (using the respective green points). The insets show enlargements of the four respective lowest temperature points.

Figure 3

(a),(b) Magnetic correlation length $\xi$ and magnetic susceptibility ${\chi }_{\mathbf{Q}}$ vs $T$ comparing the critical exponents $\nu$ and $\gamma$ for a classical finite-temperature phase transition in (a) mean-field theory and (b) for the 3D Heisenberg model. (c),(d) Quantum critical behavior comparing (c) standard HMM theory and (d) our scenario with Kohn line anomaly. (e) Visualization of (one out of four pairs of) Kohn lines in the 3D FS of the simple cubic lattice with nearest-neighbor hopping and the connecting SDW vector ${\mathbf{Q}}_0$. (f) 2D cut with the Kohn line of (e) and the corresponding (opposite) Fermi velocities.