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Physical review. B, Condensed matter

Susceptibility scaling and vertex corrections for a nested Fermi liquid

Attila Virosztek, J. Ruvalds

Published Jan 1, 1999• 21 citations
Publisher

Abstract

An unusual scaling of the spin susceptibility, as a function of frequency/temperature, was discovered for noninteracting electrons on a nested Fermi surface. This scaling has been confirmed by neutron-scattering experiments on high-temperature superconductors, and it can explain the anomalous quasiparticle damping in cuprates if electron collisions are dominant. The present work proves that self-energy and vertex corrections preserve the scaling features of the susceptibility to leading order in the Hubbard on-site Coulomb repulsion U. Analytic results for the static susceptibility show how self-energy and vertex terms modify the traditional random-phase approximation results for a spin-density-wave instability and suppress the charge susceptibility. These results are relevant to d-wave superconductors, organic metals, and chromium.

DOI
10.1103/physrevb.59.1324

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An unusual scaling of the spin susceptibility, as a function of frequency/temperature, was discovered for noninteracting electrons on a nested Fermi surface. This scaling has been confirmed by neutron-scattering experiments on high-temperature superconductors, and it can explain the anomalous quasiparticle damping in cuprates if electron collisions are dominant. The present work proves that self-energy and vertex corrections preserve the scaling features of the susceptibility to leading order in the Hubbard on-site Coulomb repulsion U. Analytic results for the static susceptibility show how self-energy and vertex terms modify the traditional random-phase approximation results for a spin-density-wave instability and suppress the charge susceptibility. These results are relevant to d-wave superconductors, organic metals, and chromium.
We investigate the response of an electron system which exhibits ideal nesting features. Using the standard Matsubara formalism we derive analytic expressions for the imaginary and real parts of the bare particle-hole susceptibility. The imaginary part has sharp peaks whose maxima at the nesting momenta approximately scale with frequency divided by temperature. The peak lineshapes resemble neutron scattering data on chromium and some copper oxide superconductors. The real part of the bare susceptibility at the nesting vectors diverges logarithmically at low temperatures. Analytic formulas for the first vertex correction to the susceptibility are derived for a Hubbard interaction, and its momentum and temperature variations are calculated numerically. This term detracts substantially from the ordinary RPA terms for intermediate values of the Coulomb repulsion. Exact cancellation of a certain class of diagrams at half filling is shown to result from particle-hole symmetry. We discuss the consequences of these results for spin fluctuation theories of high temperature superconductors and spin density wave instabilities.
We investigate the response of an electron system which exhibits ideal nesting features. Using the standard Matsubara formalism we derive analytic expressions for the imaginary and real parts of the bare particle–hole susceptibility. The imaginary part has sharp peaks whose maxima at the nesting momenta approximately scale with (ω/T). The peak lineshapes resemble neutron scattering data on chromium and some copper oxide superconductors. The real part of the bare susceptibility at the nesting vectors diverges logarithmically at low temperatures. Analytic formulas for the first vertex correction to the susceptibility are derived for a Hubbard interaction and its momentum and temperature variations are calculated numerically. This term detracts substantially from the ordinary RPA terms for intermediate values of the Coulomb repulsion. Exact cancellation of a certain class of diagrams at half filling is shown to result from particle–hole symmetry. We discuss the consequences of these results for spin fluctuation theories of high temperature superconductors and spin density wave instabilities.
We investigate the response of an electron system which exhibits ideal nesting features. Using the standard Matsubara formalism we derive analytic expressions for the imaginary and real parts of the bare particle–hole susceptibility. The imaginary part has sharp peaks whose maxima at the nesting momenta approximately scale with (ω/T). The peak lineshapes resemble neutron scattering data on chromium and some copper oxide superconductors. The real part of the bare susceptibility at the nesting vectors diverges logarithmically at low temperatures. Analytic formulas for the first vertex correction to the susceptibility are derived for a Hubbard interaction and its momentum and temperature variations are calculated numerically. This term detracts substantially from the ordinary RPA terms for intermediate values of the Coulomb repulsion. Exact cancellation of a certain class of diagrams at half filling is shown to result from particle–hole symmetry. We discuss the consequences of these results for spin fluctuation theories of high temperature superconductors and spin density wave instabilities.
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.
Identification of the electronic state that appears upon doping a Mott insulator is important to understand the physics of cuprate high-temperature superconductors. Recent scanning tunneling microscopy of cuprates provides evidence that a charge-ordered state emerges before the superconducting state upon doping the parent compound. We study this phenomenon by computing the charge response function of the Hubbard model including frequency-dependent local vertex corrections that satisfy the compressibility sum rule. We find that upon approaching the Mott phase from the overdoped side, the charge fluctuations at wave vectors connecting hot spots are suppressed much faster than at the other wave vectors. It leads to a momentum dependence of the dressed charge susceptibility that is very different from either the bare susceptibility or from the susceptibility obtained from the random phase approximation. We also find that the paramagnetic lightly hole-doped Mott phase at finite temperature is unstable to charge ordering only at zero wave vector, confirming the results previously obtained from the compressibility. Charge order is driven by the frequency-dependent scattering processes that induce an attractive particle-hole interaction at large interaction strength and small doping.
Recent photoemission spectroscopy measurements (T. J. Reber et al., arXiv:1509.01611) on cuprate superconductors have inferred that over a wide range of doping, the imaginary part of the electron self-energy scales as ${\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{''}}\ensuremath{\sim}{({\ensuremath{\omega}}^{2}+{\ensuremath{\pi}}^{2}{T}^{2})}^{a}$ with a=1a=1a=1 in the overdoped Fermi-liquid state and a<0.5a<0.5a<0.5 in the optimal to underdoped regime. We show that this non-Fermi-liquid scaling behavior can naturally be explained by the presence of a scale-invariant state of matter known as unparticles. We evaluate analytically the electron self-energy due to interactions with fermionic unparticles. We find that, in agreement with experiments, the imaginary part of the self-energy scales with respect to temperature and energy as ${\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{''}}\ensuremath{\sim}{T}^{2+2\ensuremath{\alpha}}$ and ${\ensuremath{\omega}}^{2+2\ensuremath{\alpha}}$, where $\ensuremath{\alpha}$ is the anomalous dimension of the unparticle propagator. In addition, the calculated occupancy and susceptibility of fermionic unparticles, unlike those of normal fermions, have significant spectral weights even at high energies. This unconventional behavior is attributed to the branch cut in the unparticle propagator which broadens the unparticle spectral function over a wide energy range and nontrivially alters the scattering phase space by enhancing (suppressing) the intrinsic susceptibility at low energies for negative (positive) $\ensuremath{\alpha}$. Our work presents evidence suggesting that unparticles might be important low-energy degrees of freedom in strongly coupled systems such as the cuprate superconductors.