Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1806.10159 · Physical review. B./Physical review. B

Dynamical susceptibility of a Fermi liquid

Vladimir A. Zyuzin, Prachi Sharma, Dmitrii L. Maslov

Published Sep 21, 2018• 13 citations
arXivHTMLPDF

Abstract

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 ${(q/{k}_{F})}^{2}$ 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.

DOI
10.1103/physrevb.98.115139

Related papers

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

Avraham Klein, Samuel Lederer, Debanjan Chowdhury +2 more

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.

arXiv:1708.05308•Physical review. B./Physical review. B•Apr 9, 2018•31 citations

Dynamical spin structure factor of one-dimensional interacting fermions

Vladimir A. Zyuzin, Dmitrii L. Maslov

We revisit the dynamic spin susceptibility $\ensuremath{\chi}(q,\ensuremath{\omega})$ of one-dimensional interacting fermions. To second order in the interaction, backscattering results in a logarithmic correction to $\ensuremath{\chi}(q,\ensuremath{\omega})$ at $q\ensuremath{\ll}{k}_{F}$, even if the single-particle spectrum is linearized near the Fermi points. Consequently, the dynamic spin structure factor $\mathrm{Im}\ensuremath{\chi}(q,\ensuremath{\omega})$ is nonzero at frequencies above the single-particle continuum. In the boson language, this effect results from the marginally irrelevant backscattering operator of the sine-Gordon model. Away from the threshold, the high-frequency tail of $\mathrm{Im}\ensuremath{\chi}(q,\ensuremath{\omega})$ due to backscattering is larger than that due to finite mass by a factor of ${k}_{F}/q$. We derive the renormalization group equations for the coupling constants of the $g\ensuremath{-}\text{ology}$ model at finite $\ensuremath{\omega}$ and $q$ and find the corresponding expression for $\ensuremath{\chi}(q,\ensuremath{\omega})$, valid to all orders in the interaction but not in the immediate vicinity of the continuum boundary, where the finite-mass effects become dominant.

arXiv:1411.4993•Physical Review B•Feb 4, 2015•3 citations

Fermi liquid near Pomeranchuk quantum criticality

Dmitrii L. Maslov, Andrey V. Chubukov

We analyze the behavior of an itinerant two-dimensional Fermi system near a charge nematic $(n=2)$ 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 $Z$ 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 ${Z}_{\ensuremath{\Gamma}}$, $Z$, 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 ${g}_{c,2}$ is the $n=2$ charge Landau component which approaches $\ensuremath{-}1$ at the instability. We construct the Landau function of the critical FL and show that all but ${g}_{c,2}$ 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 $D$ with the RPA form of ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$.

arXiv:0911.1251•Physical Review B•Jan 11, 2010•75 citations

Conductivity of a non-Galilean-invariant Fermi liquid: Exact solution of the kinetic equation

Tatia Kiliptari, Vladimir I. Yudson, Dmitrii L. Maslov

We obtain an exact expression for the conductivity of a disordered, non-Galilean-invariant Fermi liquid by solving the kinetic equation with both screened Coulomb and $z=3$ Pomeranchuk critical interactions. While consistent with previous asymptotic results, our solution shows that electron-electron interactions enter the conductivity solely via the quasiparticle scattering time, $τ_\mathrm{ee}$. Accordingly, the crossovers between the collisionless and hydrodynamic regimes occur when $1/τ_\mathrm{ee}$ becomes comparable to the larger of the impurity scattering rate and the probe frequency, $Ω$. In addition, the exact solution yields the optical response in the hydrodynamic regime, $Ω\ll 1/τ_\mathrm{ee}$, which is inaccessible within perturbation theory. Near a $z=3$ Pomeranchuk quantum critical point, consistency between the kinetic-equation and Kubo approaches requires proper inclusion of mass renormalization within the Eliashberg approximation, which also ensures that the crossover between the collisionless and hydrodynamic regimes in the optical conductivity occurs at the Planckian scale $Ω\sim T$.

arXiv:2605.21774•Physical review. B./Physical review. B•Oct 14, 2026•0 citations

Dynamic structure function of some singular Fermi liquids

Chandra Mohan Varma

The density correlations of some singular Fermi liquids with anomalous properties such as resistivity varying linearly with $T$ at low temperatures and a $TlogT$ contribution to the entropy and thermopower are expected to be quite different from that in Landau Fermi liquids. A possible statistical mechanical model for the quantum-critical fluctuations in diverse systems with such properties is the 2D dissipative quantum XY model. Exact relations between the density correlations and singular irreducible vertices due to coupling of fermions to the topological excitations of the 2D dissipative quantum XY model are used to derive results which have become measurable only recently because of advances in experimental techniques. The density correlations are unusual at all momenta $\mathbf{q}$ and energy $\ensuremath{\omega}$, from the hydrodynamic limit to that for large momenta and energy. The hydrodynamic limit together with the continuity equation gives the linear in $T$ resistivity. The density correlations are almost independent of frequency up to a high-frequency cutoff for ${q}_{ZB}\ensuremath{\gtrsim}q\ensuremath{\gg}\ensuremath{\omega}/{v}_{F}$; ${q}_{ZB}$ is the Brillouin zone boundary and ${v}_{F}$ is the Fermi velocity. The results should be applicable to loop-current quantum criticality in cuprates, and to 2D Fe-based compounds near their antiferromagnetic quantum criticality. The relation of the results to the temperature and frequency dependent conductivity and to Raman response is also discussed.

arXiv:1704.05914•Physical review. B./Physical review. B•Aug 11, 2017•11 citations

Non-Fermi liquid behavior at flat hot spots from quantum critical fluctuations at the onset of charge- or spin-density wave order

Lukas Debbeler, Walter Metzner

We analyze quantum fluctuation effects at the onset of charge- or spin-density wave order with a $2{k}_{F}$ wave vector $\mathbf{Q}$ in two-dimensional metals, for the special case where $\mathbf{Q}$ 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 $\mathbf{Q}$, 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.

arXiv:2302.04172•Physical review. B./Physical review. B•Apr 26, 2023•5 citations
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, Im\ensuremathχs(q,\ensuremathω)\mathrm{Im}{\ensuremath{\chi}}_{s}(\mathbf{q},\ensuremath{\omega})Im\ensuremathχs​(q,\ensuremathω), falls off as q2/\ensuremathω{q}^{2}/\ensuremath{\omega}q2/\ensuremathω for frequencies above the continuum boundary (\ensuremathω\ensuremath≫vFq\ensuremath{\omega}\ensuremath{\gg}{v}_{\mathrm{F}}q\ensuremathω\ensuremath≫vF​q) and below the model-dependent cutoff frequency, whereas the imaginary part of the charge susceptibility, Im\ensuremathχc(q,\ensuremathω)\mathrm{Im}{\ensuremath{\chi}}_{c}(\mathbf{q},\ensuremath{\omega})Im\ensuremathχc​(q,\ensuremathω), falls off as (q/kF)2q2/\ensuremathω{(q/{k}_{F})}^{2}{q}^{2}/\ensuremath{\omega}(q/kF​)2q2/\ensuremathω for frequencies above the plasma frequency. An extra factor of (q/kF)2{(q/{k}_{F})}^{2}(q/kF​)2 in Im\ensuremathχc(q,\ensuremathω)\mathrm{Im}{\ensuremath{\chi}}_{c}(\mathbf{q},\ensuremath{\omega})Im\ensuremathχc​(q,\ensuremathω) as compared to Im\ensuremathχs(q,\ensuremathω)\mathrm{Im}{\ensuremath{\chi}}_{s}(\mathbf{q},\ensuremath{\omega})Im\ensuremathχs​(q,\ensuremathω) is a direct consequence of Galilean invariance. The imaginary part of the nematic susceptibility increases linearly with \ensuremathω\ensuremath{\omega}\ensuremathω up to a peak at the ultraviolet energy scale---the plasma frequency and/or Fermi energy---and then decreases with \ensuremathω\ensuremath{\omega}\ensuremathω. 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 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.
We revisit the dynamic spin susceptibility $\ensuremath{\chi}(q,\ensuremath{\omega})$ of one-dimensional interacting fermions. To second order in the interaction, backscattering results in a logarithmic correction to $\ensuremath{\chi}(q,\ensuremath{\omega})$ at $q\ensuremath{\ll}{k}_{F}$, even if the single-particle spectrum is linearized near the Fermi points. Consequently, the dynamic spin structure factor $\mathrm{Im}\ensuremath{\chi}(q,\ensuremath{\omega})$ is nonzero at frequencies above the single-particle continuum. In the boson language, this effect results from the marginally irrelevant backscattering operator of the sine-Gordon model. Away from the threshold, the high-frequency tail of $\mathrm{Im}\ensuremath{\chi}(q,\ensuremath{\omega})$ due to backscattering is larger than that due to finite mass by a factor of kF/q{k}_{F}/qkF​/q. We derive the renormalization group equations for the coupling constants of the $g\ensuremath{-}\text{ology}$ model at finite $\ensuremath{\omega}$ and qqq and find the corresponding expression for $\ensuremath{\chi}(q,\ensuremath{\omega})$, valid to all orders in the interaction but not in the immediate vicinity of the continuum boundary, where the finite-mass effects become dominant.
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 ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$.
We obtain an exact expression for the conductivity of a disordered, non-Galilean-invariant Fermi liquid by solving the kinetic equation with both screened Coulomb and z=3z=3z=3 Pomeranchuk critical interactions. While consistent with previous asymptotic results, our solution shows that electron-electron interactions enter the conductivity solely via the quasiparticle scattering time, τeeτ_\mathrm{ee}τee​. Accordingly, the crossovers between the collisionless and hydrodynamic regimes occur when 1/τee1/τ_\mathrm{ee}1/τee​ becomes comparable to the larger of the impurity scattering rate and the probe frequency, ΩΩΩ. In addition, the exact solution yields the optical response in the hydrodynamic regime, Ω≪1/τeeΩ\ll 1/τ_\mathrm{ee}Ω≪1/τee​, which is inaccessible within perturbation theory. Near a z=3z=3z=3 Pomeranchuk quantum critical point, consistency between the kinetic-equation and Kubo approaches requires proper inclusion of mass renormalization within the Eliashberg approximation, which also ensures that the crossover between the collisionless and hydrodynamic regimes in the optical conductivity occurs at the Planckian scale Ω∼TΩ\sim TΩ∼T.
The density correlations of some singular Fermi liquids with anomalous properties such as resistivity varying linearly with TTT at low temperatures and a TlogTTlogTTlogT contribution to the entropy and thermopower are expected to be quite different from that in Landau Fermi liquids. A possible statistical mechanical model for the quantum-critical fluctuations in diverse systems with such properties is the 2D dissipative quantum XY model. Exact relations between the density correlations and singular irreducible vertices due to coupling of fermions to the topological excitations of the 2D dissipative quantum XY model are used to derive results which have become measurable only recently because of advances in experimental techniques. The density correlations are unusual at all momenta q\mathbf{q}q and energy $\ensuremath{\omega}$, from the hydrodynamic limit to that for large momenta and energy. The hydrodynamic limit together with the continuity equation gives the linear in TTT resistivity. The density correlations are almost independent of frequency up to a high-frequency cutoff for ${q}_{ZB}\ensuremath{\gtrsim}q\ensuremath{\gg}\ensuremath{\omega}/{v}_{F}$; qZB{q}_{ZB}qZB​ is the Brillouin zone boundary and vF{v}_{F}vF​ is the Fermi velocity. The results should be applicable to loop-current quantum criticality in cuprates, and to 2D Fe-based compounds near their antiferromagnetic quantum criticality. The relation of the results to the temperature and frequency dependent conductivity and to Raman response is also discussed.
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.