How non-Fermi liquids cure their infrared divergences

Non-Fermi liquids in $d=2$ spatial dimensions can arise from coupling a Fermi surface to a gapless boson. At finite temperature, however, the perturbative quantum field theory description breaks down due to infrared divergences. These are caused by virtual static bosonic modes and affect both fermionic and bosonic correlators. We show how these divergences are resolved by self-consistent boson and fermion self-energies that resum an infinite class of diagrams and correct the standard Eliashberg equations. Extending a previous approach in $d=3-\epsilon$ dimensions, we find a new “thermal non-Fermi liquid” regime that violates the scaling laws of the zero-temperature fixed point and dominates over a wide range of scales. We conclude that basic properties of quantum phase transitions and quantum-classical crossovers at finite temperature are modified in crucial ways in systems with soft bosonic fluctuations, and we begin a study of some of the phenomenological consequences.

Figure 1

Depiction of phase diagram at finite temperature. Here $\mathrm{\Lambda }$ is a dynamical scale that controls the flow to the fixed point and $g$ is the Yukawa coupling; we take $\mathrm{\Lambda }\ll g^2$. The left figure ignores thermal effects and represents the traditional quantum critical region; the right figure includes ${\mathrm{\Sigma }}_T$. In this case there appears a thermal NFL regime, which dominates over a wide range of scales, ${\mathrm{\Lambda }}^2/g^2<T<g^2$, and modifies the quantum-classical crossover.

Figure 2

One loop boson and fermion self-energies. The wiggly line represents a boson, and the straight line is a fermion; frequencies are omitted to avoid cluttering the diagrams.

Figure 3

Rainbow diagrams for the fermion self-energy. They all have the same leading in $N$ behavior.

Figure 4

Schwinger-Dyson equations for boson and fermion self-energies.

Figure 5

Quantum corrections from boson bubble diagrams.

Figure 6

Log-log plot of the numerical solution for ${\mathrm{\Sigma }}_T\left({\omega }_n\right)$, for $T/M_D={10}^{-13},g/M_D=1,\text{and}{\lambda }_{\varphi }/M_D={10}^{-2}$. The thermal and NFL regimes of (3.22) are also shown.

Figure 7

Behavior of the fermionic $A\left({\omega }_n\right)$ in $(T,{\omega }_n)$. The two curves delimiting the thermal NFL region from above are ${\mathrm{\Lambda }}_T\left(T\right)/\mathrm{\Lambda }$ and ${\mathrm{\Lambda }}_T^{\prime }\left(T\right)/\mathrm{\Lambda }$.