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arXiv:2403.00007 · Physical review. B./Physical review. B

Marginal Fermi liquid behavior at the onset of 2 k F density wave order in two-dimensional metals with flat hot spots

Lukas Debbeler, Walter Metzner

Published Jun 5, 2024• 1 citation
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Abstract

We analyze quantum fluctuation effects at the onset of incommensurate 2 k F charge- or spin-density-wave order in two-dimensional metals for a model in which the ordering wave vector Q connects a single pair of hot spots on the Fermi surface with a vanishing Fermi surface curvature. The tangential momentum dependence of the bare dispersion near the hot spots is proportional to | k t | α with α > 2 . We first compute the order parameter susceptibility and the fermion self-energy in a random phase approximation (RPA). Logarithmic divergences are subsequently treated by a renormalization-group analysis. The coupling between the order parameter fluctuations and the fermions vanishes logarithmically in the low-energy limit. As a consequence, the logarithmic divergences found in the RPA do not sum up to anomalous power laws. Instead, only logarithmic corrections to Fermi liquid behavior are obtained. In particular, the quasiparticle weight and the Fermi velocity vanish logarithmically at the hot spots. Published by the American Physical Society 2024

DOI
10.1103/physrevb.109.235112

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We analyze quantum fluctuation effects at the onset of incommensurate 2 k F charge- or spin-density-wave order in two-dimensional metals for a model in which the ordering wave vector Q connects a single pair of hot spots on the Fermi surface with a vanishing Fermi surface curvature. The tangential momentum dependence of the bare dispersion near the hot spots is proportional to | k t | α with α > 2 . We first compute the order parameter susceptibility and the fermion self-energy in a random phase approximation (RPA). Logarithmic divergences are subsequently treated by a renormalization-group analysis. The coupling between the order parameter fluctuations and the fermions vanishes logarithmically in the low-energy limit. As a consequence, the logarithmic divergences found in the RPA do not sum up to anomalous power laws. Instead, only logarithmic corrections to Fermi liquid behavior are obtained. In particular, the quasiparticle weight and the Fermi velocity vanish logarithmically at the hot spots. Published by the American Physical Society 2024
We analyze quantum fluctuation effects at the onset of charge- or spin-density wave order with a 2kF2{k}_{F}2kF​ wave vector Q\mathbf{Q}Q in two-dimensional metals, for the special case where Q\mathbf{Q}Q connects a pair of hot spots situated at high symmetry points of the Fermi surface with a vanishing Fermi surface curvature. We compute the order parameter susceptibility and the fermion self-energy in one-loop approximation. The susceptibility has a pronounced peak at Q\mathbf{Q}Q, and the self-energy displays non-Fermi liquid behavior at the hot spots, with a linear frequency dependence of its imaginary part. The real part of the one-loop self-energy exhibits logarithmic divergencies with universal prefactors as a function of both frequency and momentum, which may be interpreted as perturbative signatures of power laws with universal anomalous dimensions. As a result, one obtains a non-Fermi liquid metal with a vanishing quasiparticle weight at the hot spots, and a renormalized dispersion relation with anomalous algebraic momentum dependencies near the hot spots.
We analyze quantum fluctuation effects at the onset of charge or spin density wave order in two-dimensional metals with an incommensurate 2kF2{k}_{F}2kF​ wave vector connecting a single pair of hot spots on the Fermi surface. We compute the momentum and frequency dependence of the fermion self-energy near the hot spots to leading order in a fluctuation expansion (one loop). Non-Fermi liquid behavior with anomalous frequency scaling and a vanishing quasiparticle weight is obtained. The momentum dependence yields a divergent renormalization of the Fermi velocity and a flattening of the Fermi surface near the hot spots. Going beyond the leading-order calculation, we find that the one-loop result is not self-consistent. We show that any momentum-independent self-energy with a non-Fermi liquid frequency exponent wipes out the peak of the polarization function at the 2kF2{k}_{F}2kF​ wave vector, and thus destroys the mechanism favoring 2kF2{k}_{F}2kF​ density waves over those with generic wave vectors. However, a 2kF2{k}_{F}2kF​ density wave quantum critical point might survive in presence of a sufficiently flat renormalized Fermi surface.
We analyze quantum fluctuation effects at the onset of charge or spin-density wave order in two-dimensional metals with an incommensurate nesting (2kF2{k}_{F}2kF​) wave vector connecting two pairs of hot spots on the Fermi surface. We first compute the momentum and frequency dependence of the fermion self-energy near the hot spots to leading order in a perturbation expansion (one loop). Non-Fermi liquid behavior with a linear (in energy) quasiparticle decay rate and a logarithmically vanishing quasiparticle weight is obtained. The momentum dependence of the self-energy entails only finite renormalizations of the Fermi velocity and the Fermi-surface curvature at the hot spots. The perturbative one-loop result is not self-consistent and casts doubt on the stability of the 2kF2{k}_{F}2kF​ quantum critical point. We construct a self-consistent solution of the one-loop equations with self-energy feedback, where the quantum critical point is stabilized rather than being destroyed by fluctuations, while the non-Fermi liquid behavior as found in the perturbative one-loop calculation is confirmed.
We analyze the influence of quantum critical fluctuations on single-particle excitations at the onset of incommensurate 2kF2{k}_{F}2kF​ charge- or spin-density wave order in two-dimensional metals. The case of a single pair of hot spots at high symmetry positions on the Fermi surface needs to be distinguished from the case of two hot spot pairs. We compute the fluctuation propagator and the electronic self-energy perturbatively in leading order. The energy dependence of the single-particle decay rate at the hot spots obeys non-Fermi-liquid power laws, with an exponent 2/3 in the case of a single hot spot pair, and exponent one for two hot spot pairs. The prefactors of the linear behavior obtained in the latter case are not particle-hole symmetric.
We present a field-theoretic renormalization group analysis of Abanov and Chubukov's model of the spin density wave transition in two dimensional metals. We identify the independent field scale and coupling constant renormalizations in a local field theory and argue that the damping constant of spin density wave fluctuations tracks the renormalization of the local couplings. The divergences at two-loop order overdetermine the renormalization constants and are shown to be consistent with our renormalization scheme. We describe the physical consequences of our renormalization-group equations, including the breakdown of Fermi liquid behavior near the ``hot spots'' on the Fermi surface. In particular, we find that the dynamical critical exponent zzz receives corrections to its mean-field value z=2z=2z=2. At higher orders in the loop expansion, we find infrared singularities similar to those found by Lee [Phys. Rev. B 80, 165102 (2009)] for the problem of a Fermi surface coupled to a gauge field. A treatment of these singularities implies that an expansion in 1∕N1∕N1∕N (where NNN is the number of fermion flavors) fails for the present problem. We also discuss the renormalization of the pairing vertex and find an enhancement which scales as logarithm squared of the energy scale. A similar enhancement is also found for a modulated bond order which is locally an Ising-nematic order.
Many correlated materials display a quantum-critical point between a paramagnetic and a spin-density wave (SDW) state. The SDW wave vector connects points, so-called hot spots, on opposite sides of the Fermi surface. The Fermi velocities at these pairs of points are in general not parallel. Here, we consider the case where pairs of hot spots coalesce, and the wave vector $(\ensuremath{\pi},\ensuremath{\pi})$ of the SDW connects hot spots with parallel Fermi velocities. Using the specific example of electron-doped cuprates, we first show that Kanamori screening and generic features of the Lindhard function make this case experimentally relevant. The temperature dependence of the correlation length, the spin susceptibility, and the self-energy at the hot spots are found using the two-particle self-consistent theory and specific numerical examples worked out for band and interaction parameters characteristic of the electron-doped cuprates. While the curvature of the Fermi surface at the hot spots leads to deviations from perfect nesting, the pseudonesting conditions lead to drastic modifications of the temperature dependence of these physical observables: Neglecting logarithmic corrections, the correlation length $\ensuremath{\xi}$ scales like 1/T1/T1/T, namely, z=1z=1z=1 instead of the naive z=2z=2z=2, the $(\ensuremath{\pi},\ensuremath{\pi})$ static spin susceptibility $\ensuremath{\chi}$ like 1/T1/\sqrt{T}1/T​, and the imaginary part of the self-energy at the hot spots like T3/2{T}^{3/2}T3/2. The correction ${T}_{1}^{\ensuremath{-}1}\ensuremath{\sim}{T}^{3/2}$ to the Korringa NMR relaxation rate is subdominant. We also consider this problem at zero temperature, or for frequencies larger than temperature, using a field-theoretical model of gapless collective bosonic modes (SDW fluctuations) interacting with fermions. The imaginary part of the retarded fermionic self-energy close to the hot spots scales as $\ensuremath{-}{\ensuremath{\omega}}^{3/2}\mathrm{ln}\ensuremath{\omega}$. This is less singular than earlier predictions of the form $\ensuremath{-}\ensuremath{\omega}\mathrm{ln}\ensuremath{\omega}$. The difference arises from the effects of umklapp terms that were not included in previous studies.