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arXiv:1708.05308 · Physical review. B./Physical review. B

Dynamical susceptibility near a long-wavelength critical point with a nonconserved order parameter

Avraham Klein, Samuel Lederer, Debanjan Chowdhury, Erez Berg, Andrey V. Chubukov

Published Apr 9, 2018• 31 citations
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Abstract

We study the dynamic response of a two-dimensional system of itinerant fermions in the vicinity of a uniform $(\mathrm{Q}=0)$ 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.

DOI
10.1103/physrevb.97.155115

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ω/Tω/Tω/T scaling and IR/UV-mixing in Ising-nematic quantum critical metals
The instability of a Fermi surface against Ising nematic order destroys the quasiparticle character of the low-energy degrees of freedom. Therefore, observables exhibit deviations from Fermi liquid behavior which gives rise to the term Ising nematic quantum critical metal. To obtain a theoretical description we use a finite-temperature version of Eliashberg theory which allows to treat the strong coupling between quantum and thermal fluctuations in the absence of well-defined quasiparticles. Here, we use this self-consistent, diagrammatic approach to compute, in particular, the nematic susceptibility and the non-Fermi liquid correlations. Upon decreasing the temperature, the susceptibility crosses over from a (Tlog⁡T)−1(T \log T)^{-1}(TlogT)−1 to T−2/3T^{-2/3}T−2/3 behavior, which is induced by the absence of quasiparticles and restores the Ising nematic critical scaling. Correspondingly, the fermions obey a simple ω/Tω/Tω/T scaling law at low enough temperatures. However, this regime is characterized by strong IR-UV mixing since the proportionality factors exhibit a dependence on the spectral width of the non-quasiparticle excitations and on the underlying lattice. Tuning the parameters of the model, therefore, gives rise to several scenarios for the breakdown of the scaling theory. We discuss them within the Eliashberg approach and estimate the related crossover scales. We also show that the leading order vertex corrections do not change the scaling with temperature or coupling constants.
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 mB{m}_{B}mB​ should be much smaller than m=kF∕vFm={k}_{F}∕{v}_{F}m=kF​∕vF​, or the number of fermionic flavors NNN 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 q3∕2{q}^{3∕2}q3∕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 q3∕2{q}^{3∕2}q3∕2 correction to the bosonic propagator is specific to the SU(2) symmetric case. For systems with a scalar order parameter, the q3∕2{q}^{3∕2}q3∕2 contributions from individual diagrams cancel out in the full expression of the susceptibility, and the QCP remains stable.
We study dynamic response of a Fermi liquid in the spin, charge, and nematic channels beyond the random phase approximation for the dynamically screened Coulomb potential. In all the channels, one-loop order corrections to the irreducible susceptibility result in a nonzero spectral weight of the corresponding fluctuations above the particle-hole continuum boundary. It is shown that the imaginary part of the spin susceptibility, $\mathrm{Im}{\ensuremath{\chi}}_{s}(\mathbf{q},\ensuremath{\omega})$, falls off as ${q}^{2}/\ensuremath{\omega}$ for frequencies above the continuum boundary ($\ensuremath{\omega}\ensuremath{\gg}{v}_{\mathrm{F}}q$) and below the model-dependent cutoff frequency, whereas the imaginary part of the charge susceptibility, $\mathrm{Im}{\ensuremath{\chi}}_{c}(\mathbf{q},\ensuremath{\omega})$, falls off as ${(q/{k}_{F})}^{2}{q}^{2}/\ensuremath{\omega}$ for frequencies above the plasma frequency. An extra factor of (q/kF)2{(q/{k}_{F})}^{2}(q/kF​)2 in $\mathrm{Im}{\ensuremath{\chi}}_{c}(\mathbf{q},\ensuremath{\omega})$ as compared to $\mathrm{Im}{\ensuremath{\chi}}_{s}(\mathbf{q},\ensuremath{\omega})$ is a direct consequence of Galilean invariance. The imaginary part of the nematic susceptibility increases linearly with $\ensuremath{\omega}$ up to a peak at the ultraviolet energy scale---the plasma frequency and/or Fermi energy---and then decreases with $\ensuremath{\omega}$. We also obtain explicit forms of the spin susceptibility from the kinetic equation in the collisionless limit and for the Landau function that contains up to the first three harmonics.
We study finite-temperature properties of metals close to an Ising-nematic quantum critical point in two spatial dimensions. In particular we show that at any finite temperature there is a regime where order parameter fluctuations are characterized by a dynamical critical exponent z=2z=2z=2, in contrast to z=3z=3z=3 found at zero temperature. Our results are based on a simple Eliashberg-type approach, which gives rise to a boson self-energy proportional to $\mathrm{\ensuremath{\Omega}}/\ensuremath{\gamma}(T)$ at small momenta, where $\ensuremath{\gamma}(T)$ is the temperature dependent fermion scattering rate. These findings might shed some light on recent Monte Carlo simulations at finite temperature, where results consistent with z=2z=2z=2 were found.
We consider two-dimensional metals near a Pomeranchuk instability which breaks ${90}^{\ensuremath{\circ}}$ lattice rotation symmetry. Such metals realize strongly coupled non-Fermi liquids with critical fluctuations of an Ising-nematic order. At low temperatures, impurity scattering provides the dominant source of momentum relaxation and, hence, a nonzero electrical resistivity. We use the memory matrix method to compute the resistivity of this non-Fermi liquid to second order in the impurity potential, without assuming the existence of quasiparticles. Impurity scattering in the ddd-wave channel acts as a random ``field'' on the Ising-nematic order. We find contributions to the resistivity with a nearly linear temperature dependence, along with more singular terms; the most singular is the random-field contribution which diverges in the limit of zero temperature.
We discuss shape (Pomeranchuk) instabilities of the Fermi surface of a two-dimensional Fermi system using bosonization. We consider in detail the quantum critical behavior of the transition of a two-dimensional Fermi fluid to a nematic state which breaks spontaneously the rotational invariance of the Fermi liquid. We show that higher dimensional bosonization reproduces the quantum critical behavior expected from the Hertz-Millis analysis, and verify that this theory has dynamic critical exponent z=3z=3z=3. Going beyond this framework, we study the behavior of the fermion degrees of freedom directly, and show that at quantum criticality as well as in the quantum nematic phase (except along a set of measure zero of symmetry-dictated directions) the quasiparticles of the normal Fermi liquid are generally wiped out. Instead, they exhibit short-ranged spatial correlations that decay faster than any power law, with the law $\ensuremath{\mid}x{\ensuremath{\mid}}^{\ensuremath{-}1}\mathrm{exp}(\ensuremath{-}\mathrm{const}\phantom{\rule{0.3em}{0ex}}\ensuremath{\mid}x{\ensuremath{\mid}}^{1∕3})$ and we verify explicitly the vanishing of the fermion residue utilizing this expression. In contrast, the fermion autocorrelation function has the behavior $\ensuremath{\mid}t{\ensuremath{\mid}}^{\ensuremath{-}1}\mathrm{exp}(\ensuremath{-}\mathrm{const}\phantom{\rule{0.3em}{0ex}}\ensuremath{\mid}t{\ensuremath{\mid}}^{\ensuremath{-}2∕3})$. In this regime we also find that, at low frequency, the single-particle fermion density of states behaves as ${N}^{*}(\ensuremath{\omega})={N}^{*}(0)+B{\ensuremath{\omega}}^{2∕3}\mathrm{ln}\ensuremath{\omega}+\ensuremath{\cdots}$, where N∗(0){N}^{*}(0)N∗(0) is larger than the free Fermi value, N(0)N(0)N(0), and BBB is a constant. These results confirm the non-Fermi liquid nature of both the quantum critical theory and of the nematic phase.