Normal State Properties of Quantum Critical Metals at Finite Temperature

Avraham Klein,Erez Berg,Yoni Schattner,Andrey V Chubukov

doi:10.1103/physrevx.10.031053

Abstract

We study the effects of finite temperature on normal state properties of a metal near a quantum critical point to an antiferromagnetic or Ising-nematic state. At $T = 0$ bosonic and fermionic self-energies are traditionally computed within Eliashberg theory and obey scaling relations with characteristic power-laws. Quantum Monte Carlo (QMC) simulations have shown strong systematic deviations from these predictions, casting doubt on the validity of the theoretical analysis. We extend Eliashberg theory to finite $T$ and argue that for the $T$ range accessible in the QMC simulations, the scaling forms for both fermionic and bosonic self energies are quite different from those at $T = 0$. We compare finite $T$ results with QMC data and find good agreement for both systems. This, we argue, resolves the key apparent contradiction between the theory and the QMC simulations.

Highlights

Electron-boson models [1,2,3,4] have long been used to study the behavior of interacting fermions near a metallic quantum critical point (QCP)
We studied the effect of thermal fluctuations on metals near a QCP, either to a spin-density-wave state or to an Ising nematic state
We calculated the deviation from the scaling behavior predicted by Eliashberg theory (ET), in the regime where the thermal contributions do not permit a separation of scales between fermionic and bosonic degrees of freedom

Summary

INTRODUCTION

Electron-boson models [1,2,3,4] have long been used to study the behavior of interacting fermions near a metallic quantum critical point (QCP). Other properties showed systematic deviations from ET For both the spin-fermion and Ising-nematic models, fermionic self-energies in the normal state, extracted from QMC, do not show the power-law forms, expected from the theory, and appear to saturate at a finite value even at the smallest fermionic Matsubara frequency ω 1⁄4 πT. We argue that at finite temperature, one has to go beyond perturbation theory and compute fermionic ΣðωÞ and bosonic ΠðΩÞ self-consistently and without factorization of momentum integration. We compute ΣTðωmÞ and ΣQðωmÞ using the same equations as in the MET, but we integrate over the internal fermionic momenta in the full Brillouin zone (i.e., compute the self-energy without linearizing the fermionic dispersion near the FS) We call this the lattice theory (LT).

Models

Review of diagrammatic theory

SELF-ENERGIES AT FINITE TEMPERATURES

LATTICE THEORY

COMPARISON TO LATTICE THEORY AND TO QUANTUM MONTE CARLO SIMULATIONS

SUMMARY