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 they obey scaling relations with characteristic power laws. Corrections to Eliashberg theory break these power laws but only at very small frequencies. Quantum Monte Carlo (QMC) simulations have shown that, already at much larger frequencies, there are 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 in the T range accessible in the QMC simulations above the superconducting transition, 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. We argue that this agreement resolves the key apparent contradiction between the theory and the QMC simulations.
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We thank S. Lederer, M. H. Christensen, X. Wang, R. M. Fernandes, X.-Y. Xu, K. Sun, Z. Y. Meng, and L. Classen for helpful conversations. This work was supported by the US-Israel Binational Science Foundation (BSF) Grant No. 2018217. E. B. acknowledges support from the European Research Council (ERC) under the HQMAT grant (Grant No. 817799) and from the Minerva Foundation. Y. S. was supported by the Department of Energy, Office of Basic Energy Sciences, under Contract No. DE-AC02-76SF00515 at Stanford, and by the Zuckerman STEM Leadership Program.
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