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arXiv:1106.0001 · Physical Review B

Quantum critical response at the onset of spin-density-wave order in two-dimensional metals

Sean A. Hartnoll, Diego M. Hofman A, Max A. Metlitski, Subir Sachdev A

Published Sep 9, 2011• 80 citations
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Abstract

We study the frequency dependence of the electron self-energy and the optical conductivity in a recently developed field theory of the spin-density-wave quantum phase transition in two-dimensional metals. We focus on the interplay between the Fermi surface ``hot spots'' and the remainder of the ``cold'' Fermi surface. Scattering of electrons off the fluctuations of the spin-density-wave order parameter, $\ensuremath{\varphi}$, is strongest at the hot spots; we compute the conductivity due to this scattering in a rainbow approximation. We point out the importance of composite operators, built of products of the primary electron or $\ensuremath{\varphi}$ fields: These have important effects also away from the hot spots. The simplest composite operator, ${\ensuremath{\varphi}}^{2}$, leads to deviations from Landau Fermi-liquid behavior on the entire Fermi surface. We also find an intermediate frequency window in which the cold electrons lose their quasiparticle form due to effectively one-dimensional scattering processes. The latter processes are part of umklapp scattering, which leads to singular contributions to the optical conductivity at the lowest frequencies at zero temperature.

DOI
10.1103/physrevb.84.125115

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We study the frequency dependence of the electron self-energy and the optical conductivity in a recently developed field theory of the spin-density-wave quantum phase transition in two-dimensional metals. We focus on the interplay between the Fermi surface hot spots'' and the remainder of the cold'' Fermi surface. Scattering of electrons off the fluctuations of the spin-density-wave order parameter, \ensuremathφ\ensuremath{\varphi}\ensuremathφ, is strongest at the hot spots; we compute the conductivity due to this scattering in a rainbow approximation. We point out the importance of composite operators, built of products of the primary electron or \ensuremathφ\ensuremath{\varphi}\ensuremathφ fields: These have important effects also away from the hot spots. The simplest composite operator, \ensuremathφ2{\ensuremath{\varphi}}^{2}\ensuremathφ2, leads to deviations from Landau Fermi-liquid behavior on the entire Fermi surface. We also find an intermediate frequency window in which the cold electrons lose their quasiparticle form due to effectively one-dimensional scattering processes. The latter processes are part of umklapp scattering, which leads to singular contributions to the optical conductivity at the lowest frequencies at zero temperature.
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 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 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 report on numerically exact determinantal quantum Monte Carlo simulations of the onset of spin-density-wave (SDW) order in itinerant electron systems captured by a sign-problem-free two-dimensional lattice model. Extensive measurements of the SDW correlations in the vicinity of the phase transition reveal that the critical dynamics of the bosonic order parameter are well described by a dynamical critical exponent z=2z=2z=2, consistent with Hertz-Millis theory, but are found to follow a finite-temperature dependence that does not fit the predicted behavior of the same theory. The presence of critical SDW fluctuations is found to have a strong impact on the fermionic quasiparticles, giving rise to a dome-shaped superconducting phase near the quantum critical point. In the superconducting state we find a gap function that has an opposite sign between the two bands of the model and is nearly constant along the Fermi surface of each band. Above the superconducting Tc{T}_{c}Tc​, our numerical simulations reveal a nearly temperature and frequency independent self-energy causing a strong suppression of the low-energy quasiparticle weight in the vicinity of the hot spots on the Fermi surface. This indicates a clear breakdown of Fermi liquid theory around these points.
The hyperscaling property implies that spatially isotropic critical quantum states in ddd spatial dimensions have a specific heat, which scales with temperature as Td/z{T}^{d/z}Td/z, and an optical conductivity, which scales with frequency as ${\ensuremath{\omega}}^{(d\ensuremath{-}2)/z}$ for $\ensuremath{\omega}\ensuremath{\gg}T$, where zzz is the dynamic critical exponent. We examine the spin density wave critical fixed point of metals in d=2d=2d=2 found by Sur and Lee [Phys. Rev. B 91, 125136 (2015)] in an expansion in $\ensuremath{\epsilon}=3\ensuremath{-}d$. We find that the contributions of the ``hot spots'' on the Fermi surface to the optical conductivity and specific heat obey hyperscaling (up to logarithms), and agree with the results of the large NNN analysis of the optical conductivity by Hartnoll et al. [ Phys. Rev. B 84, 125115 (2011)]. With a small bare velocity of the boson associated with the spin density wave order, there is an intermediate energy regime where hyperscaling is violated with $d\ensuremath{\rightarrow}{d}_{t}$, where dt=1{d}_{t}=1dt​=1 is the number of dimensions transverse to the Fermi surface. We also present a Boltzmann equation analysis which indicates that the hot-spot contribution to the dc conductivity has the same scaling as the optical conductivity, with TTT replacing $\ensuremath{\omega}$.
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.