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arXiv:cond-mat/0605306 · Physical Review B

Quantum critical behavior in itinerant electron systems: Eliashberg theory and instability of a ferromagnetic quantum critical point

Jérôme Rech, C. Pépin, Andrey V. Chubukov

Published Nov 29, 2006• 196 citations
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Abstract

We consider the problem of fermions interacting with gapless long-wavelength collective bosonic modes. The theory describes, among other cases, a ferromagnetic quantum-critical point (QCP) and a QCP towards nematic ordering. We construct a controllable expansion at the QCP in two steps: we first create a non-Fermi-liquid ``zero-order'' Eliashberg-type theory, and then demonstrate that the residual interaction effects are small. We prove that this approach is justified under two conditions: the interaction should be smaller than the fermionic bandwidth, and either the band mass ${m}_{B}$ should be much smaller than $m={k}_{F}∕{v}_{F}$, or the number of fermionic flavors $N$ should be large. For an SU(2) symmetric ferromagnetic QCP, we find that the Eliashberg theory itself includes a set of singular renormalizations which can be understood as a consequence of an effective long-range dynamic interaction between quasiparticles, generated by the Landau damping term. These singular renormalizations give rise to a negative nonanalytic ${q}^{3∕2}$ correction to the static spin susceptibility, and destroy a ferromagnetic QCP. We demonstrate that this effect can be understood in the framework of the ${\ensuremath{\phi}}^{4}$ theory of quantum criticality. We also show that the nonanalytic ${q}^{3∕2}$ correction to the bosonic propagator is specific to the SU(2) symmetric case. For systems with a scalar order parameter, the ${q}^{3∕2}$ contributions from individual diagrams cancel out in the full expression of the susceptibility, and the QCP remains stable.

DOI
10.1103/physrevb.74.195126

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We develop a theoretical framework based on the nonperturvative renormalization group (RG) in the one-particle irreducible (Wetterich) formulation to tackle the interplay of coupled fermionic and order-parameter fluctuations at metallic quantum critical points (QCPs) with ordering wavevector Q⃗=0⃗\vec{Q}=\vec{0}Q​=0. We consistently treat the dynamical emergence of the Landau damping of the bosonic mode and non-Fermi liquid scaling of fermions upon lowering the cutoff scale. The loop integrals of the present theory involve only contributions from fluctuations above the cutoff scale, which protects the system from developing singular bosonic interactions. We emphasize the importance of the nontrivial relative scaling of the bosonic and fermionic cutoffs ΛΛΛ and ΛfΛ_fΛf​, which we fix by analyzing the RG flow of the scale-dependent ordering wave-vector Q⃗Λ\vec{Q}_ΛQ​Λ​. Upon neglecting Fermi self-energy in the loop integrals of the functional RG flow, we identify a non-Fermi liquid RG fixed point and recover the features obtained earlier within RPA-type approaches. In a subsequent step, we self-consistently include the scaling of the self-energy and the Yukawa coupling. We find a generic instability of the non-Fermi liquid RG fixed point. This implies, at least at this truncation level, absence of the QCP with Q⃗=0⃗\vec{Q}=\vec{0}Q​=0 and development of a first-order phase transition or a phase characterized by Q⃗≠0⃗\vec{Q}\neq\vec{0}Q​=0.
The applicability of Eliashberg theory, for fermions interacting with fluctuations of a soft collective boson near a quantum-critical point (QCP) in two dimensions, remains a subject of debate. The velocity of the soft boson is comparable to the Fermi velocity. Hence, Migdal's justification of Eliashberg theory is formally not applicable. The authors argue here that some elements of Migdal's reasoning remain valid because the collective boson is overdamped, yet vertex corrections to Eliashberg theory are OOO(1). At the same time, they show that these corrections are numerically small, at least near an Ising-nematic QCP.
We study the dynamic response of a two-dimensional system of itinerant fermions in the vicinity of a uniform (Q=0)(\mathrm{Q}=0)(Q=0) Ising nematic quantum critical point of $d\text{\ensuremath{-}}\text{wave}$ symmetry. The nematic order parameter is not a conserved quantity, and this permits a nonzero value of the fermionic polarization in the $d\text{\ensuremath{-}}\text{wave}$ channel even for vanishing momentum and finite frequency: $\mathrm{\ensuremath{\Pi}}(\mathrm{q}=0,{\mathrm{\ensuremath{\Omega}}}_{m})\ensuremath{\ne}0$. For weak coupling between the fermions and the nematic order parameter (i.e., the coupling is small compared to the Fermi energy), we perturbatively compute $\mathrm{\ensuremath{\Pi}}(\mathrm{q}=0,{\mathrm{\ensuremath{\Omega}}}_{m})\ensuremath{\ne}0$ over a parametrically broad range of frequencies where the fermionic self-energy $\mathrm{\ensuremath{\Sigma}}(\ensuremath{\omega})$ is irrelevant, and use Eliashberg theory to compute $\mathrm{\ensuremath{\Pi}}(\mathrm{q}=0,{\mathrm{\ensuremath{\Omega}}}_{m})$ in the non-Fermi-liquid regime at smaller frequencies, where $\mathrm{\ensuremath{\Sigma}}(\ensuremath{\omega})>\ensuremath{\omega}$. We find that $\mathrm{\ensuremath{\Pi}}(\mathrm{q}=0,\mathrm{\ensuremath{\Omega}})$ is a constant, plus a frequency-dependent correction that goes as $|\mathrm{\ensuremath{\Omega}}|$ at high frequencies, crossing over to ${|\mathrm{\ensuremath{\Omega}}|}^{1/3}$ at lower frequencies. The ${|\mathrm{\ensuremath{\Omega}}|}^{1/3}$ scaling holds also in a non-Fermi-liquid regime. The nonvanishing of $\mathrm{\ensuremath{\Pi}}(q=0,\mathrm{\ensuremath{\Omega}})$ gives rise to additional structure in the imaginary part of the nematic susceptibility ${\ensuremath{\chi}}^{\ensuremath{''}}(q,\mathrm{\ensuremath{\Omega}})$ at $\mathrm{\ensuremath{\Omega}}>{v}_{F}q$, in marked contrast to the behavior of the susceptibility for a conserved order parameter. This additional structure may be detected in Raman scattering experiments in the $d\text{\ensuremath{-}}\text{wave}$ geometry.
Quantum phase transition arises in general as second order phase transition at zero temperature, tuned by a non-thermal parameter such as pressure, doping or a magnetic field. The point in the phase diagram of the material in which different phases meet is called a quantum critical point (QCP). Physics around QCPs are of extensive current interest because the critical quantum fluctuations influence the physical properties in a wide temperature range (quantum criticality), and are believed to be responsible for many emergent physical properties such as non-Fermi liquids and unconventional superconductivity. In this research we explore dynamics and thermodynamics near QCPs via investigating three classes of models, which all have real material correspondence. Specifically first, we study local dynamics in a perturbed quantum critical Ising chain with E8 symmetry, where we show the local dynamical spin susceptibility has a singular dependence on frequency, but differs from the diffusion form. The nuclear magnetic resonance (NMR) relaxation rate at low temperatures depends exponentially on the inverse temperature, whose prefactor we also determine. We propose NMR experiments as a means to further test the applicability of the E8 description for CoNb2O6. Second, we investigate the thermodynamic properties of itinerant ferromagnets near quantum critical points, described by the quantum Landau-Ginzburg theory. We provide a regularized perturbative renormalization group procedure to calculate the free energy. We further carry out numerical calculations on thermodynamic quantities, capturing not only the leading critical behaviors, but also the subleading and nonsingular contributions. We demonstrate various thermodynamic signatures of quantum criticality, including the entropy accumulation effect and the divergence of the specific heat coefficient. A detailed comparison to the recent experimental results on an itinerant ferromagnet Sr3Ru2O7 is also presented. Third, we explore Ising-nematic and magnetic phases and their transitions in iso-electronically doped iron pnictides by carrying out a large-N study of an effective low-energy Ginzburg-Landau model for these systems. We demonstrate that the magnetic and Ising orders transitions are concurrent at zero temperature, and both transitions are weakly first-order, which is consistent with RG-based prediction and experimental observations.
We analyze the form of the fermionic propagator for two-dimensional fermions interacting with massless overdamped bosons. Examples include a nematic and Ising ferromagnetic quantum-critical points and fermions at a half-filled Landau level. Fermi-liquid behavior in these systems is broken at criticality by a singular self-energy, but the Fermi surface remains well defined. These are strong-coupling problems with no expansion parameter other than the number of fermionic species, NNN. The two known limits, N⪢1N⪢1N⪢1 and N=0N=0N=0, show qualitatively different behavior of the fermionic propagator $G({ϵ}_{k},\ensuremath{\omega})$. In the first limit, $G({ϵ}_{k},\ensuremath{\omega})$ has a pole at some ϵk{ϵ}_{k}ϵk​; in the other it is analytic. We analyze the crossover between the two limits. We show that the pole survives for all NNN, but at small NNN it only exists in a range O(N2)O({N}^{2})O(N2) near the mass shell. At larger distances from the mass shell, the system evolves and $G({ϵ}_{k},\ensuremath{\omega})$ becomes regular. At N=0N=0N=0, the range where the pole exists collapses and $G({ϵ}_{k},\ensuremath{\omega})$ becomes regular everywhere.
We analyze the behavior of an itinerant two-dimensional Fermi system near a charge nematic (n=2)(n=2)(n=2) Pomeranchuk instability in terms of the Landau Fermi-liquid (FL) theory. A key object of our study is the fully renormalized vertex function ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$, related to the Landau interaction function. We derive ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$ for a model case of the long-range interaction in the nematic channel. Already within the random-phase approximation (RPA), the vertex is singular near the instability. The full vertex, obtained by resumming the ladder series composed of the RPA vertices, differs from the RPA result by a multiplicative renormalization factor ${Z}_{\ensuremath{\Gamma}}$, related to the single-particle residue ZZZ and effective-mass renormalization ${m}^{\ensuremath{\ast}}/m$. We employ the Pitaevski-Landau identities, which express the derivatives of the self-energy in terms of ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$, to obtain and solve a set of coupled nonlinear equations for ${Z}_{\ensuremath{\Gamma}}$, ZZZ, and ${m}^{\ensuremath{\ast}}/m$. We show that near the transition the system enters a critical FL regime, where ${Z}_{\ensuremath{\Gamma}}\ensuremath{\sim}Z\ensuremath{\propto}{(1+{g}_{c,2})}^{1/2}$ and ${m}^{\ensuremath{\ast}}/m\ensuremath{\approx}1/Z$, where gc,2{g}_{c,2}gc,2​ is the n=2n=2n=2 charge Landau component which approaches $\ensuremath{-}1$ at the instability. We construct the Landau function of the critical FL and show that all but gc,2{g}_{c,2}gc,2​ Landau components diverge at the critical point. We also show that in the critical regime the one-loop result for the self-energy $\ensuremath{\Sigma}(K)\ensuremath{\propto}\ensuremath{\int}dPG(P)D(K\ensuremath{-}P)$ is asymptotically exact if one identifies the effective interaction DDD with the RPA form of ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$.