Critical Fermi surfaces and non-Fermi liquid metals

At certain quantum critical points in metals, an entire Fermi surface may disappear. A crucial question is the nature of the electronic excitations at the critical point. Here we provide arguments showing that at such quantum critical points, the Fermi surface remains sharply defined even though the Landau quasiparticle is absent. The presence of such a critical Fermi surface has a number of consequences for the universal phenomena near the quantum critical point which are discussed. In particular, the structure of scaling of the universal critical singularities can be significantly modified from more familiar criticality. Scaling hypotheses appropriate to a critical Fermi surface are proposed. Implications for experiments on heavy fermion critical points are discussed. Various phenomena in the normal state of the cuprates are also examined from this perspective. We suggest that a phase transition that involves a dramatic reconstruction of the Fermi surface might underlie a number of strange observations in the metallic states above the superconducting dome.

Figure 1

Possible schematic zero temperature phase diagram showing the onset of magnetism in a heavy fermion metal. The magnetic phase is a “local moment magnetic metal” where the local moments are not part of the Fermi surface unlike in the heavy Fermi liquid. The Fermi surface needs to reconstruct across such a quantum phase transition.

Figure 2

Possible schematic zero-temperature phase diagram for a half-filled single band repulsive Hubbard model on a nonbipartite lattice. $U$ is the Hubbard interaction strength and $t$ is the hopping amplitude. The Fermi surface of the metal needs to disappear at the Mott transition.

Figure 3

Possible schematic zero-temperature phase diagram for the cuprate materials showing the evolution of the underlying normal ground state as a function of doping. The large Fermi surface of the overdoped metal is presumed to disappear at a critical doping $x_c$ to an underdoped metal with a qualitatively different small Fermi surface.

Figure 4

Evolution of the ground state momentum distribution $n\left(k\right)$ across a second-order phase transition where the Fermi surface disappears, such as the Mott transition of Fig. 2. (a) $n\left(k\right)$ in the Fermi liquid with a discontinuity $Z$ at the Fermi surface. (b) $n\left(k\right)$ in the Mott insulator which is smooth as a function of $k$. (c) $n\left(k\right)$ at the critical point—the discontinuity of (a) has just vanished and is replaced by a kink singularity.

Figure 5

Finite temperature crossovers near a second-order Mott transition with angle dependent exponents on the critical Fermi surface.

Figure 6

Various possible evolutions between a small Fermi surface underdoped metal (such as the staggered flux/ddw state) and a large Fermi surface overdoped Fermi liquid. In (a) there is a direct first-order transition. In (b) the transition proceeds through an intermediate phase which has staggered flux order but has a folded version of the large Fermi surface. This state has both hole and electron pockets. The transition at $x_{c2}$ is a Lifshitz transition while the one at $x_{c1}$ involves the development of the broken symmetry but no extra reconstruction of the large Fermi surface. This transition will be described by a theory along the lines of Ref. 6 and will only have weak departures from Fermi-liquid theory. In (c) there is a direct second-order transition where the Fermi-surface reconstruction from large to small accompanies the development of the broken symmetry. The critical point at $x_c$ will be strongly non-Fermi liquid like.

Figure 7

(Color online) Finite temperature crossovers near the proposed quantum critical point between the overdoped “large Fermi-surface” metal and an underdoped “small Fermi-surface” metal. The black dashed lines mark the crossovers associated with the large Fermi surface and the blue dash-dot lines those of the small Fermi surface. It is suggested that the critical singularities in the spectral function associated with the small Fermi surface are too weak to be resolved in photoemission experiments.