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:2011.07076

$ω/T$ scaling and IR/UV-mixing in Ising-nematic quantum critical metals

Frank, Bernhard, Piazza, Francesco

Published Nov 13, 2020• 3 citations
arXivHTMLPDF

Abstract

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 $(T \log T)^{-1}$ to $T^{-2/3}$ behavior, which is induced by the absence of quasiparticles and restores the Ising nematic critical scaling. Correspondingly, the fermions obey a simple $ω/T$ 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.

DOI
10.48550/arxiv.2011.07076

Related papers

Finite-temperature scaling close to Ising-nematic quantum critical points in two-dimensional metals

Matthias Punk

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 $z=2$, in contrast to $z=3$ 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 $z=2$ were found.

arXiv:1606.00918•Physical review. B./Physical review. B•Nov 7, 2016•13 citations

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

Quantum criticality in a two-component two-dimensional nematic metal

Yuhang Xiao, Ning Hao

Nematic fluctuations alter the low-energy electronic behavior and reshape the mean-field phase diagram. We study a two-dimensional nematic metal with a two-component internal degree of freedom. Near the quantum critical point (QCP), longitudinal nematic fluctuations generate the non-Fermi-liquid self-energy $\propto|ω|^{2/3}$, whereas transverse fluctuations give a subleading logarithmic, marginal-Fermi-liquid correction. Nevertheless, the transverse sector dominates the correction to the phase diagram through a quantum order-by-disorder mechanism: the emergent nematic background reshapes the electronic spectrum, alters the decay of nematic bosons, and produces a negative singular free-energy correction. This correction reconstructs the phase diagram, drives the transition first order, and is described by a scaling function whose two asymptotic limits define two critical regimes with free-energy corrections scaling as $R^{8/3}$ and $R^3$, respectively, where $R$ is the order-parameter magnitude; their powers follow from the competition between the characteristic momentum of normal-state nematic fluctuations and that of the ordered-state background. These two regimes are experimentally distinguishable through nematic-susceptibility hysteresis measurements. The corrected phase diagram shows that the enlarged nodal $α$ phase masks the otherwise bare QCP, making it inaccessible; consequently, the logarithmic self-energy correction is by no means negligible at low energies.

arXiv:2610.29745•Oct 21, 2026•0 citations

Normal State Properties of Quantum Critical Metals at Finite Temperature

Avraham Klein, Andrey V. Chubukov, Yoni Schattner +1 more

We study the effects of finite temperature on normal state properties of a metal near a quantum critical point to an antiferromagnetic or Ising-nematic state.At T ¼ 0, bosonic and fermionic self-energies are traditionally computed within Eliashberg theory, and they obey scaling relations with characteristic power laws.Corrections to Eliashberg theory break these power laws but only at very small frequencies.Quantum Monte Carlo (QMC) simulations have shown that, already at much larger frequencies, there are strong systematic deviations from these predictions, casting doubt on the validity of the theoretical analysis.We extend Eliashberg theory to finite T and argue that in the T range accessible in the QMC simulations above the superconducting transition, the scaling forms for both fermionic and bosonic self-energies are quite different from those at T ¼ 0. We compare finite T results with QMC data and find good agreement for both systems.We argue that this agreement resolves the key apparent contradiction between the theory and the QMC simulations.

arXiv:2003.09431•Physical Review X•Sep 8, 2020•44 citations

Fluctuating stripes in strongly correlated electron systems and the nematic-smectic quantum phase transition

Kai Sun, Benjamin M. Fregoso, Michael J. Lawler +1 more

We discuss the quantum phase transition between a quantum nematic metallic state to an electron metallic smectic state in terms of an order-parameter theory coupled to fermionic quasiparticles. Both commensurate and incommensurate smectic (or stripe) cases are studied. Close to the quantum critical point (QCP), the spectrum of fluctuations of the nematic phase has low-energy ``fluctuating stripes.'' We study the quantum critical behavior and find evidence that, contrary to the classical case, the gauge-type of coupling between the nematic and smectic is irrelevant at this QCP. The collective modes of the electron smectic (or stripe) phase are also investigated. The effects of the low-energy bosonic modes on the fermionic quasiparticles are studied perturbatively, for both a model with full rotational symmetry and for a system with an underlying lattice, which has a discrete point group symmetry. We find that at the nematic-smectic critical point, due to the critical smectic fluctuations, the dynamics of the fermionic quasiparticles near several points on the Fermi surface, around which it is reconstructed, are not governed by a Landau Fermi liquid theory. Surprisingly, the quasiparticles in the smectic phase also exhibit non-Fermi liquid behavior. We also present a detailed analysis of the dynamical susceptibilities in the electron nematic phase close to this QCP (the fluctuating stripe regime) and in the electronic smectic phase.

arXiv:0805.3526•Physical Review B•Aug 20, 2008•54 citations

Translationally Invariant Non-Fermi-Liquid Metals with Critical Fermi Surfaces: Solvable Models

Debanjan Chowdhury, Yochai Werman, Erez Berg +1 more

We construct examples of translationally invariant solvable models of strongly correlated metals, composed of lattices of Sachdev-Ye-Kitaev dots with identical local interactions. These models display crossovers as a function of temperature into regimes with local quantum criticality and marginal-Fermi-liquid behavior. In the marginal-Fermi-liquid regime, the dc resistivity increases linearly with temperature over a broad range of temperatures. By generalizing the form of interactions, we also construct examples of non-Fermi liquids with critical Fermi surfaces. The self-energy has a singular frequency dependence but lacks momentum dependence, reminiscent of a dynamical mean-field-theory-like behavior but in dimensions d < ∞. In the low-temperature and strong-coupling limit, a heavy Fermi liquid is formed. The critical Fermi surface in the non-Fermi-liquid regime gives rise to quantum oscillations in the magnetization as a function of an external magnetic field in the absence of quasiparticle excitations. We discuss the implications of these results for local quantum criticality and for fundamental bounds on relaxation rates. Drawing on the lessons from these models, we formulate conjectures on coarse-grained descriptions of a class of intermediate-scale non-Fermi-liquid behavior in generic correlated metals.

Physical Review Letters•Jul 25, 2018•123 citations
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 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 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.
Nematic fluctuations alter the low-energy electronic behavior and reshape the mean-field phase diagram. We study a two-dimensional nematic metal with a two-component internal degree of freedom. Near the quantum critical point (QCP), longitudinal nematic fluctuations generate the non-Fermi-liquid self-energy ∝∣ω∣2/3\propto|ω|^{2/3}∝∣ω∣2/3, whereas transverse fluctuations give a subleading logarithmic, marginal-Fermi-liquid correction. Nevertheless, the transverse sector dominates the correction to the phase diagram through a quantum order-by-disorder mechanism: the emergent nematic background reshapes the electronic spectrum, alters the decay of nematic bosons, and produces a negative singular free-energy correction. This correction reconstructs the phase diagram, drives the transition first order, and is described by a scaling function whose two asymptotic limits define two critical regimes with free-energy corrections scaling as R8/3R^{8/3}R8/3 and R3R^3R3, respectively, where RRR is the order-parameter magnitude; their powers follow from the competition between the characteristic momentum of normal-state nematic fluctuations and that of the ordered-state background. These two regimes are experimentally distinguishable through nematic-susceptibility hysteresis measurements. The corrected phase diagram shows that the enlarged nodal ααα phase masks the otherwise bare QCP, making it inaccessible; consequently, the logarithmic self-energy correction is by no means negligible at low energies.
We study the effects of finite temperature on normal state properties of a metal near a quantum critical point to an antiferromagnetic or Ising-nematic state.At T ¼ 0, bosonic and fermionic self-energies are traditionally computed within Eliashberg theory, and they obey scaling relations with characteristic power laws.Corrections to Eliashberg theory break these power laws but only at very small frequencies.Quantum Monte Carlo (QMC) simulations have shown that, already at much larger frequencies, there are strong systematic deviations from these predictions, casting doubt on the validity of the theoretical analysis.We extend Eliashberg theory to finite T and argue that in the T range accessible in the QMC simulations above the superconducting transition, the scaling forms for both fermionic and bosonic self-energies are quite different from those at T ¼ 0. We compare finite T results with QMC data and find good agreement for both systems.We argue that this agreement resolves the key apparent contradiction between the theory and the QMC simulations.
We discuss the quantum phase transition between a quantum nematic metallic state to an electron metallic smectic state in terms of an order-parameter theory coupled to fermionic quasiparticles. Both commensurate and incommensurate smectic (or stripe) cases are studied. Close to the quantum critical point (QCP), the spectrum of fluctuations of the nematic phase has low-energy ``fluctuating stripes.'' We study the quantum critical behavior and find evidence that, contrary to the classical case, the gauge-type of coupling between the nematic and smectic is irrelevant at this QCP. The collective modes of the electron smectic (or stripe) phase are also investigated. The effects of the low-energy bosonic modes on the fermionic quasiparticles are studied perturbatively, for both a model with full rotational symmetry and for a system with an underlying lattice, which has a discrete point group symmetry. We find that at the nematic-smectic critical point, due to the critical smectic fluctuations, the dynamics of the fermionic quasiparticles near several points on the Fermi surface, around which it is reconstructed, are not governed by a Landau Fermi liquid theory. Surprisingly, the quasiparticles in the smectic phase also exhibit non-Fermi liquid behavior. We also present a detailed analysis of the dynamical susceptibilities in the electron nematic phase close to this QCP (the fluctuating stripe regime) and in the electronic smectic phase.
We construct examples of translationally invariant solvable models of strongly correlated metals, composed of lattices of Sachdev-Ye-Kitaev dots with identical local interactions. These models display crossovers as a function of temperature into regimes with local quantum criticality and marginal-Fermi-liquid behavior. In the marginal-Fermi-liquid regime, the dc resistivity increases linearly with temperature over a broad range of temperatures. By generalizing the form of interactions, we also construct examples of non-Fermi liquids with critical Fermi surfaces. The self-energy has a singular frequency dependence but lacks momentum dependence, reminiscent of a dynamical mean-field-theory-like behavior but in dimensions d < ∞. In the low-temperature and strong-coupling limit, a heavy Fermi liquid is formed. The critical Fermi surface in the non-Fermi-liquid regime gives rise to quantum oscillations in the magnetization as a function of an external magnetic field in the absence of quasiparticle excitations. We discuss the implications of these results for local quantum criticality and for fundamental bounds on relaxation rates. Drawing on the lessons from these models, we formulate conjectures on coarse-grained descriptions of a class of intermediate-scale non-Fermi-liquid behavior in generic correlated metals.