Exact Critical Exponents for the Antiferromagnetic Quantum Critical Metal in Two Dimensions

Unconventional metallic states which do not support well-defined single-particle excitations can arise near quantum phase transitions as strong quantum fluctuations of incipient order parameters prevent electrons from forming coherent quasiparticles. Although antiferromagnetic phase transitions occur commonly in correlated metals, understanding the nature of the strange metal realized at the critical point in layered systems has been hampered by a lack of reliable theoretical methods that take into account strong quantum fluctuations. We present a nonperturbative solution to the low-energy theory for the antiferromagnetic quantum critical metal in two spatial dimensions. Being a strongly coupled theory, it can still be solved reliably in the low-energy limit as quantum fluctuations are organized by a new control parameter that emerges dynamically. We predict the exact critical exponents that govern the universal scaling of physical observables at low temperatures.

Popular Summary

Strange metals are an enigma in physics. Their physical properties do not change with external perturbations such as temperature in the same way as conventional metals. In typical metals, electrons behave as largely independent particles—the electrostatic repulsion of like charges is screened by the electrons themselves. This behavior changes, however, when a metal is on the verge of becoming a magnet: Microscopic magnets are about to be arranged in a regular pattern, and the electrons start interacting with each other over long distances by exchanging waves made of nearly arranged magnets. The electrons no longer move independently, and they form a strongly interacting fluid, often called a strange metal. Understanding the nature of strange metals that arise near these magnetic quantum phase transitions is a long-standing problem, hindered by a lack of theoretical methods that take into account strong interactions among electrons. Our work provides a quantitative understanding of a two-dimensional strange metal near a magnetic critical point.

We solved the theory that describes the behavior of electrons and magnetic fluctuations in a two-dimensional strange metal. Our analysis shows that as temperature is lowered, electrons and magnetic excitations propagate at increasingly different velocities. This result allows us to make exact predictions about equations that describe many observable properties, such as the amount of heat needed to raise the temperature by a unit degree and how the magnetic correlation length depends on the distance from the critical point.

While our predictions are in qualitative agreement with currently available experiments, more experiments are needed to make detailed comparisons. Our work might serve as a stepping stone toward more systematic studies of other strongly interacting metals.

Figure 1

A Fermi surface with fourfold rotational symmetry. The (red) dots represent the hot spots connected by the AFM wave vector ${\overrightarrow{Q}}_{\mathrm{AFM}}$.

Figure 2

The exact boson self-energy. The double line is the fully dressed fermion propagator. The triangle represents the fully dressed vertex.

Figure 3

The leading order diagrams for the boson self-energy in the small $v$ limit. Solid lines are the bare fermion propagators. The wiggly double line represents the boson propagator consistently dressed with the self-energy in (a) and (b). The dressed boson propagator includes an infinite series of nested self-energies with a fractal structure.

Figure 4

(a) A four-loop diagram with one fermion loop. The numbers next to the fermion lines represent the patch indices. (b) The four exclusive propagators are denoted as dashed lines. The remaining propagators represent the connected tree diagram. Loops (closed solid colored lines) are chosen such that each loop momentum goes through only one of the exclusive propagators. (c) The seven internal fermion propagators whose energies are denoted as $E_l$ with $1\le l\le 7$. $E_1,E_2,\dots ,E_5$ are used as new integration variables along with $p_i^{\prime }=c{p}_{i,x}$, with $i=1$, 2, 3, as discussed in the text.

Figure 5

(a) The vertex that describes the process where a boson is absorbed by a fermion. (b) For a boson momentum $\overrightarrow{q}$, there exists a unique $\overrightarrow{k}$ such that ${ϵ}_1(\overrightarrow{k})={ϵ}_3(\overrightarrow{k}+\overrightarrow{q})=0$ for $v\ne 0$.

Figure 6

The one-loop diagram for the fermion self-energy.

Figure 7

Two-loop diagrams for the fermion self-energy. While (a) is subleading in the small $v$ limit, (b) is of the same order as Fig. 6.

Figure 8

The one-loop diagram for the vertex correction.