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:2207.11982 · Physical review. B./Physical review. B

Quantum-critical transport in marginal Fermi liquids

Hideaki Maebashi, Chandra M. Varma

Published Dec 10, 2025• 5 citations
arXivHTMLPDF

Abstract

We use the Kubo response functions to calculate the electrical and thermal conductivity and Seebeck coefficient at low temperatures and frequencies in the quantum-critical region for fermions on a lattice. The theory uses scattering of the fermions with the previously derived collective fluctuations due to topological defects of the quantum XY model coupled to fermions. The microscopic model is applicable to the fluctuations of the loop-current order in cuprates as well as to a class of quasi-two-dimensional heavy-fermion and other metallic antiferromagnets, and proposed recently also for the possible loop-current order in Moiré twisted bi-layer graphene and bilayer WSe$_2$. All these metals have a linear-in-temperature electrical resistivity in the quantum-critical region of their phase diagrams, often termed ``Planckian" resistivity. The solution of the Kubo equation for transport shows that vertex renormalizations to the external fields, beside those caused by Aslamazov-Larkin (A-L) processes, are absent. A-L appears as an Umklapp scattering matrix, which gives a temperature-independent multiplicative factor for the electrical resistivity but does not affect the thermal conductivity. We also show that the mass renormalization which gives a logarithmic enhancement of the marginal Fermi-liquid specific heat does not appear in the electrical resistivity and, more remarkably, in the thermal conductivity. On the other hand the mass renormalization $\propto \ln ω_c/T$ appears in the Seebeck coefficient. We also discuss in detail the conservation laws which play a crucial role in all transport properties. We calculate exactly, the numerical coefficients of the transport properties for a circular Fermi surface. The leading temperature dependences is shown to remain the same for a general Fermi surface, but it is too messy to calculate the numerical coefficient.

DOI
10.1103/cxjv-1wn6

Related papers

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

Long-wavelength correlations and transport in a marginal Fermi liquid

Arkady Shekhter, Chandra Mohan Varma

Marginal Fermi liquid was originally introduced as a phenomenological description of the cuprates in a part of the metallic doping range which appears to be governed by fluctuations due to a quantum-critical point. An essential result due to the form of the assumed fluctuation spectra is that the large inelastic quasiparticle relaxation rate near the Fermi surface is proportional to the energy measured from the chemical potential, ${\ensuremath{\tau}}_{i}^{\ensuremath{-}1}\ensuremath{\propto}\ensuremath{\epsilon}$. We present a microscopic long-wavelength derivation of the hydrodynamic properties in such a situation by an extension of the procedure that Eliashberg used for the derivation of the hydrodynamic properties of a Landau-Fermi liquid. In particular, the density-density and the current-current correlations and the relation between the two are derived, and the connection to microscopic calculations of the frequency dependence of the optical conductivity with an additional Fermi-liquid correction factor shown to follow. The method used here may be necessary, quite generally, for the correct hydrodynamic theory for any problem of quantum-critical fluctuations in fermions.

arXiv:0809.1104•Physical Review B•Jan 23, 2009•20 citations

Universal Theory of Incoherent Metals

Aaron Kleger, N. V. Gnezdilov, Rufus Boyack

Numerous unconventional superconductors such as cuprates, heavy-fermions, and twisted-bilayer graphene exhibit incoherent metallic transport above the superconducting critical temperature. This phenomenon cannot be described with Fermi-liquid theory and has presented a significant theoretical challenge to overcome. We utilize the two-dimensional Yukawa-SYK model of fermions with spatially random coupling to quantum-critical bosons to study transport in a manner which is non-perturbative in the coupling strength. Our work provides a microscopic model of quantum-critical incoherent metals and their concomitant properties, including a non-Boltzmann transport formula between resistivity and quasi-particle lifetime, violation of the Mott-Ioffe-Regel resistivity bound, and violation of the Kovtun-Son-Starinets shear viscosity to entropy density bound.

arXiv:2605.03013•May 4, 2026•0 citations

Unconventional scaling of resistivity in two-dimensional Fermi liquids

Jonathan M. Buhmann

We study the temperature dependence of the electrical resistivity of interacting two-dimensional Fermi liquids. We perform a numerical simulation of the nonequilibrium state based on semiclassical Boltzmann transport theory. A two-dimensional system of quasiparticles on a square lattice serves as a model system. We use a single-orbital tight-binding model of the dispersion. For conceptual purposes, we choose a simple repulsive onsite interaction that leads to quasiparticle scattering and delta-potential scatterers as a model of nonmagnetic impurity scattering. Through our simulation, we demonstrate that deviations from the predictions of standard Fermi-liquid theory can arise due to the nontrivial scattering geometry of umklapp processes, in special cases even in the ultralow-temperature limit. We show through qualitative arguments how these unconventional scaling properties of the electrical resistivity, which are often interpreted as indication of a non-Fermi-liquid state, can arise due to special geometric conditions of the Fermi surface. The appearance of robust deviations from the predictions of Fermi-liquid theory within our simple model presents a viewpoint in order to interpret unconventional transport properties in electron-electron scattering dominated metallic systems.

arXiv:1309.1470•Physical Review B•Dec 23, 2013•11 citations

Low-temperature T -linear resistivity due to umklapp scattering from a critical mode

Patrick A. Lee

We consider the transport properties of a model of fermions scattered by a critical bosonic mode. The mode is overdamped and scattering is mainly in the forward direction. Such a mode appears at the quantum critical point for an electronic nematic phase transition and in gauge theories for a U(1) spin liquid. It leads to a short fermion lifetime, violating Landau's criterion for a Fermi liquid. In spite of this, transport can be described by a Boltzmann equation. We include momentum relaxation by umklapp scattering, supplemented by weak impurity scattering. We find that above a very low temperature which scales with ${\mathrm{\ensuremath{\Delta}}}_{q}^{3}$, where ${\mathrm{\ensuremath{\Delta}}}_{q}$ is the minimum umklapp scattering vector, the resistivity is linear in $T$ with a coefficient which is independent of the amount of disorder. This picture holds until the temperature dependent part of the resistivity exceeds that due to impurity scattering. We compare the relaxation time approximation with an exact numerical solution of the Boltzmann equation. Surprisingly we find that, unlike the resistivity, the Hall coefficient strongly deviates from the relaxation time approximation and shows a strong reduction with increasing temperature. We comment on possible comparisons with experiments on high ${T}_{c}$ cuprates.

arXiv:2012.09339•Physical review. B./Physical review. B•Jul 19, 2021•32 citations

Scattering mechanisms and electrical transport near an Ising nematic quantum critical point

Xiaoyu Wang, Erez Berg

Electrical transport properties near an electronic Ising-nematic quantum critical point in two dimensions are of both theoretical and experimental interest. In this work, we derive a kinetic equation valid in a broad regime near the quantum critical point using the memory matrix approach. The formalism is applied to study the effect of the critical fluctuations on the dc resistivity through different scattering mechanisms, including umklapp, impurity scattering, and electron-hole scattering in a compensated metal. We show that electrical transport in the quantum critical regime exhibits a rich behavior that depends sensitively on the scattering mechanism and the band structure. In the case of a single large Fermi surface, the resistivity due to umklapp scattering crosses over from $\ensuremath{\rho}\ensuremath{\sim}{T}^{2}$ at low temperature to sublinear at high temperature. The crossover temperature scales as ${q}_{0}^{3}$, where ${q}_{0}$ is the minimal wave vector for umklapp scattering. Impurity scattering leads to $\ensuremath{\rho}\ensuremath{-}{\ensuremath{\rho}}_{0}\ensuremath{\sim}{T}^{\ensuremath{\alpha}}$ (${\ensuremath{\rho}}_{0}$ being the residual resistivity), where $\ensuremath{\alpha}$ is either larger than 2 if there is only a single Fermi sheet present, or $4/3$ in the case of multiple Fermi sheets. Finally, in a perfectly compensated metal with an equal density of electrons and holes, the low-temperature behavior depends strongly on the structure of ``cold spots'' on the Fermi surface, where the coupling between the quasiparticles and order parameter fluctuations vanishes by symmetry. In particular, for a system where cold spots are present on some (but not all) Fermi sheets, $\ensuremath{\rho}\ensuremath{\sim}{T}^{5/3}$. At higher temperatures, there is a broad crossover regime where $\ensuremath{\rho}$ either saturates or $\ensuremath{\rho}\ensuremath{\sim}T$, depending on microscopic details. We discuss these results in the context of recent quantum Monte Carlo simulations of a metallic Ising nematic critical point, and experiments in certain iron-based superconductors.

arXiv:1902.04590•Physical review. B./Physical review. B•Jun 17, 2019•50 citations
We use the Kubo response functions to calculate the electrical and thermal conductivity and Seebeck coefficient at low temperatures and frequencies in the quantum-critical region for fermions on a lattice. The theory uses scattering of the fermions with the previously derived collective fluctuations due to topological defects of the quantum XY model coupled to fermions. The microscopic model is applicable to the fluctuations of the loop-current order in cuprates as well as to a class of quasi-two-dimensional heavy-fermion and other metallic antiferromagnets, and proposed recently also for the possible loop-current order in Moiré twisted bi-layer graphene and bilayer WSe2_22​. All these metals have a linear-in-temperature electrical resistivity in the quantum-critical region of their phase diagrams, often termed ``Planckian" resistivity. The solution of the Kubo equation for transport shows that vertex renormalizations to the external fields, beside those caused by Aslamazov-Larkin (A-L) processes, are absent. A-L appears as an Umklapp scattering matrix, which gives a temperature-independent multiplicative factor for the electrical resistivity but does not affect the thermal conductivity. We also show that the mass renormalization which gives a logarithmic enhancement of the marginal Fermi-liquid specific heat does not appear in the electrical resistivity and, more remarkably, in the thermal conductivity. On the other hand the mass renormalization ∝ln⁡ωc/T\propto \ln ω_c/T∝lnωc​/T appears in the Seebeck coefficient. We also discuss in detail the conservation laws which play a crucial role in all transport properties. We calculate exactly, the numerical coefficients of the transport properties for a circular Fermi surface. The leading temperature dependences is shown to remain the same for a general Fermi surface, but it is too messy to calculate the numerical coefficient.
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.
Marginal Fermi liquid was originally introduced as a phenomenological description of the cuprates in a part of the metallic doping range which appears to be governed by fluctuations due to a quantum-critical point. An essential result due to the form of the assumed fluctuation spectra is that the large inelastic quasiparticle relaxation rate near the Fermi surface is proportional to the energy measured from the chemical potential, ${\ensuremath{\tau}}_{i}^{\ensuremath{-}1}\ensuremath{\propto}\ensuremath{\epsilon}$. We present a microscopic long-wavelength derivation of the hydrodynamic properties in such a situation by an extension of the procedure that Eliashberg used for the derivation of the hydrodynamic properties of a Landau-Fermi liquid. In particular, the density-density and the current-current correlations and the relation between the two are derived, and the connection to microscopic calculations of the frequency dependence of the optical conductivity with an additional Fermi-liquid correction factor shown to follow. The method used here may be necessary, quite generally, for the correct hydrodynamic theory for any problem of quantum-critical fluctuations in fermions.
Numerous unconventional superconductors such as cuprates, heavy-fermions, and twisted-bilayer graphene exhibit incoherent metallic transport above the superconducting critical temperature. This phenomenon cannot be described with Fermi-liquid theory and has presented a significant theoretical challenge to overcome. We utilize the two-dimensional Yukawa-SYK model of fermions with spatially random coupling to quantum-critical bosons to study transport in a manner which is non-perturbative in the coupling strength. Our work provides a microscopic model of quantum-critical incoherent metals and their concomitant properties, including a non-Boltzmann transport formula between resistivity and quasi-particle lifetime, violation of the Mott-Ioffe-Regel resistivity bound, and violation of the Kovtun-Son-Starinets shear viscosity to entropy density bound.
We study the temperature dependence of the electrical resistivity of interacting two-dimensional Fermi liquids. We perform a numerical simulation of the nonequilibrium state based on semiclassical Boltzmann transport theory. A two-dimensional system of quasiparticles on a square lattice serves as a model system. We use a single-orbital tight-binding model of the dispersion. For conceptual purposes, we choose a simple repulsive onsite interaction that leads to quasiparticle scattering and delta-potential scatterers as a model of nonmagnetic impurity scattering. Through our simulation, we demonstrate that deviations from the predictions of standard Fermi-liquid theory can arise due to the nontrivial scattering geometry of umklapp processes, in special cases even in the ultralow-temperature limit. We show through qualitative arguments how these unconventional scaling properties of the electrical resistivity, which are often interpreted as indication of a non-Fermi-liquid state, can arise due to special geometric conditions of the Fermi surface. The appearance of robust deviations from the predictions of Fermi-liquid theory within our simple model presents a viewpoint in order to interpret unconventional transport properties in electron-electron scattering dominated metallic systems.
We consider the transport properties of a model of fermions scattered by a critical bosonic mode. The mode is overdamped and scattering is mainly in the forward direction. Such a mode appears at the quantum critical point for an electronic nematic phase transition and in gauge theories for a U(1) spin liquid. It leads to a short fermion lifetime, violating Landau's criterion for a Fermi liquid. In spite of this, transport can be described by a Boltzmann equation. We include momentum relaxation by umklapp scattering, supplemented by weak impurity scattering. We find that above a very low temperature which scales with ${\mathrm{\ensuremath{\Delta}}}_{q}^{3}$, where ${\mathrm{\ensuremath{\Delta}}}_{q}$ is the minimum umklapp scattering vector, the resistivity is linear in TTT with a coefficient which is independent of the amount of disorder. This picture holds until the temperature dependent part of the resistivity exceeds that due to impurity scattering. We compare the relaxation time approximation with an exact numerical solution of the Boltzmann equation. Surprisingly we find that, unlike the resistivity, the Hall coefficient strongly deviates from the relaxation time approximation and shows a strong reduction with increasing temperature. We comment on possible comparisons with experiments on high Tc{T}_{c}Tc​ cuprates.
Electrical transport properties near an electronic Ising-nematic quantum critical point in two dimensions are of both theoretical and experimental interest. In this work, we derive a kinetic equation valid in a broad regime near the quantum critical point using the memory matrix approach. The formalism is applied to study the effect of the critical fluctuations on the dc resistivity through different scattering mechanisms, including umklapp, impurity scattering, and electron-hole scattering in a compensated metal. We show that electrical transport in the quantum critical regime exhibits a rich behavior that depends sensitively on the scattering mechanism and the band structure. In the case of a single large Fermi surface, the resistivity due to umklapp scattering crosses over from $\ensuremath{\rho}\ensuremath{\sim}{T}^{2}$ at low temperature to sublinear at high temperature. The crossover temperature scales as q03{q}_{0}^{3}q03​, where q0{q}_{0}q0​ is the minimal wave vector for umklapp scattering. Impurity scattering leads to $\ensuremath{\rho}\ensuremath{-}{\ensuremath{\rho}}_{0}\ensuremath{\sim}{T}^{\ensuremath{\alpha}}$ (${\ensuremath{\rho}}_{0}$ being the residual resistivity), where $\ensuremath{\alpha}$ is either larger than 2 if there is only a single Fermi sheet present, or 4/34/34/3 in the case of multiple Fermi sheets. Finally, in a perfectly compensated metal with an equal density of electrons and holes, the low-temperature behavior depends strongly on the structure of ``cold spots'' on the Fermi surface, where the coupling between the quasiparticles and order parameter fluctuations vanishes by symmetry. In particular, for a system where cold spots are present on some (but not all) Fermi sheets, $\ensuremath{\rho}\ensuremath{\sim}{T}^{5/3}$. At higher temperatures, there is a broad crossover regime where $\ensuremath{\rho}$ either saturates or $\ensuremath{\rho}\ensuremath{\sim}T$, depending on microscopic details. We discuss these results in the context of recent quantum Monte Carlo simulations of a metallic Ising nematic critical point, and experiments in certain iron-based superconductors.