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arXiv:1706.07471 · Physical review. B./Physical review. B

Non-Fermi liquids at finite temperature: Normal-state and infrared singularities

Huajia Wang, Gonzalo Torroba

Published Oct 16, 2017• 15 citations
arXivHTMLPDF

Abstract

We analyze quantum criticality at finite temperature for a class of non-Fermi liquids with massless bosons. Finite temperature gives rise to new infrared singularities that invalidate standard perturbative treatments. We show how such divergences are resolved at a nonperturbative level, and obtain the resulting fermion self-energy. This leads to a new ``thermal'' non-Fermi liquid regime that extends over a wide range of frequencies, and which violates finite temperature scaling laws near quantum critical points. We analyze the resulting quantum critical region and properties of the retarded Green's function. More generally, such effects dominate in the nearly static limit and are expected to have a nontrivial impact on superconductivity and transport.

DOI
10.1103/physrevb.96.144508

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We analyze quantum criticality at finite temperature for a class of non-Fermi liquids with massless bosons. Finite temperature gives rise to new infrared singularities that invalidate standard perturbative treatments. We show how such divergences are resolved at a nonperturbative level, and obtain the resulting fermion self-energy. This leads to a new ``thermal'' non-Fermi liquid regime that extends over a wide range of frequencies, and which violates finite temperature scaling laws near quantum critical points. We analyze the resulting quantum critical region and properties of the retarded Green's function. More generally, such effects dominate in the nearly static limit and are expected to have a nontrivial impact on superconductivity and transport.
Non-Fermi liquids in d=2d=2d=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 $d=3\ensuremath{-}\ensuremath{\epsilon}$ 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.
We study the interplay between superconductivity and non-Fermi liquid behavior of a Fermi surface coupled to a massless SU(N)\text{SU}(N)SU(N) matrix boson near the quantum critical point. The presence of thermal infrared singularities in both the fermionic self-energy and the gap equation invalidates the Eliashberg approximation, and makes the quantum-critical pairing problem qualitatively different from that at zero temperature. Taking the large NNN limit, we solve the gap equation beyond the Eliashberg approximation, and obtain the superconducting temperature Tc{T}_{c}Tc​ as a function of NNN. Our results show an anomalous scaling between the zero-temperature gap and Tc{T}_{c}Tc​. For NNN greater than a critical value, we find that Tc{T}_{c}Tc​ vanishes with a Berezinskii-Kosterlitz-Thouless scaling behavior, and the system retains non-Fermi liquid behavior down to zero temperature. This confirms and extends previous renormalization-group analyses done at T=0T=0T=0, and provides a controlled example of a naked quantum critical point. We discuss the crucial role of thermal fluctuations in relating our results with earlier work where superconductivity always develops due to the special role of the first Matsubara frequency.
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 ¼ 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 ¼ 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.
Abstract Quantum Monte Carlo (QMC) simulations of correlated electron systems provide unbiased information about system behavior at a quantum critical point (QCP) and can verify or disprove the existing theories of non-Fermi liquid (NFL) behavior at a QCP. However, simulations are carried out at a finite temperature, where quantum critical features are masked by finite-temperature effects. Here, we present a theoretical framework within which it is possible to separate thermal and quantum effects and extract the information about NFL physics at T = 0. We demonstrate our method for a specific example of 2D fermions near an Ising ferromagnetic QCP. We show that one can extract from QMC data the zero-temperature form of fermionic self-energy Σ( ω ) even though the leading contribution to the self-energy comes from thermal effects. We find that the frequency dependence of Σ( ω ) agrees well with the analytic form obtained within the Eliashberg theory of dynamical quantum criticality, and obeys ω 2/3 scaling at low frequencies. Our results open up an avenue for QMC studies of quantum critical metals.
Landau theory of Fermi liquids is the foundational concept behind our understanding of conventional metals. Its basic postulate is that the low-energy physics can be described by weakly interacting quasiparticles --- a dressed but effectively free version of electrons formally represented by poles of the single-particle Green's function. However, this concept notoriously fails for strongly interacting systems such as high-Tc{T}_{c}Tc​ cuprate superconductors, and our understanding of properties of these ``non-Fermi'' quantum liquids is far from complete. One well-known candidate theory for non-Fermi liquids is Hertz-Millis theory: a finite-density fermion system coupled to a massless critical boson. In (2+1) dimensions, this theory is strongly interacting at low energies, but as of yet there is no controlled insight in this regime; conventional perturbation theory fails. In this article, the authors show that in the so-called quenched approximation the fermionic spectrum can be computed exactly to all orders in the coupling constant. They show extensive detailed analysis that, in this quenched limit, the nonperturbative low-energy dynamics are indeed that of a non-Fermi liquid. Instead of coherent quasiparticles one finds the emergence of critical fermionic excitations described by branch cuts in the single particle Green's function.
Since the mid-eighties there has been an accumulation of metallic materials whose thermodynamic and transport properties differ significantly from those predicted by Fermi liquid theory. Examples of these so-called non-Fermi liquids include the strange metal phase of high transition temperature cuprates, and heavy fermion systems near a quantum phase transition. We report on a class of non-Fermi liquids discovered using gauge/gravity duality. The low energy behavior of these non-Fermi liquids is shown to be governed by a nontrivial infrared (IR) fixed point which exhibits nonanalytic scaling behavior only in the temporal direction. Within this class we find examples whose single-particle spectral function and transport behavior resemble those of strange metals. In particular, the contribution from the Fermi surface to the conductivity is inversely proportional to the temperature. In our treatment these properties can be understood as being controlled by the scaling dimension of the fermion operator in the emergent IR fixed point.