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:2104.13802 · Physical review. B./Physical review. B

One-dimensional scattering of two-dimensional fermions near quantum criticality

Dimitri Pimenov, Alex Kamenev, Andrey V. Chubukov

Published Jun 30, 2021• 4 citations
arXivHTMLPDF

Abstract

Forward scattering and backscattering play an exceptional role in the physics of two-dimensional interacting fermions. In a Fermi liquid, both give rise to a nonanalytic ${\ensuremath{\omega}}^{2}ln(\ensuremath{\omega})$ form of the fermionic scattering rate at second order in the interaction. Here we argue that higher powers of $ln(\ensuremath{\omega})$ appear in the backscattering contribution at higher orders. We show that these terms come from ``planar'' processes, which are effectively one-dimensional. This is explicitly demonstrated by extending a Fermi liquid to the limit of $N\ensuremath{\gg}1$ fermionic flavors, when only planar processes survive. We sum the leading logarithms for the case of a two-dimensional Fermi liquid near a nematic transition, and we obtain an expression for the scattering rate at $T=0$ to all orders in the interaction. For a repulsive interaction, the resulting rate is logarithmically suppressed, and the result is valid down to $\ensuremath{\omega}=0$. For an attractive interaction, the ground state is an $s$-wave superconductor with a gap ${\mathrm{\ensuremath{\Delta}}}_{0}$. We show that in this case the scattering rate increases as $\ensuremath{\omega}$ is reduced toward ${\mathrm{\ensuremath{\Delta}}}_{0}$. At $\ensuremath{\omega}\ensuremath{\ge}{\mathrm{\ensuremath{\Delta}}}_{0}$, the behavior of the scattering rate is rather unconventional as many pairing channels compete near a nematic critical point, and the $s$-wave wins only by a narrow margin. We take superconductivity into consideration and obtain the scattering rate also at smaller $\ensuremath{\omega}\ensuremath{\simeq}{\mathrm{\ensuremath{\Delta}}}_{0}$.

DOI
10.1103/physrevb.103.214519

Related papers

Fermi liquids beyond the forward-scattering limit: The role of nonforward scattering for scale invariance and instabilities

Han Ma, Sung-Sik Lee

Landau Fermi liquid theory is a fixed-point theory of metals that includes the forward-scattering amplitudes as exact marginal couplings. However, the fixed-point theory that only includes the strict forward scatterings is nonlocal in real space. In this paper, we revisit the Fermi liquid theory for charge-neutral fermions using the field-theoretic functional renormalization group formalism and show how the scale-invariant fixed point emerges as a local theory, which includes not only the forward scatterings but also nonforward scatterings with small but nonzero momentum transfers. In the low-energy limit, the near-forward scattering and pairing interactions take scale-invariant forms if the momentum transfer and the center-of-mass momentum of Cooper pairs, respectively, are comparable to the energy. The coupling functions fully capture the universal low-energy dynamics of the collective modes and instabilities of Fermi liquids. A runaway flow of the coupling function in the particle-hole channel beyond a critical interaction suggests an instability toward an ordered phase with a wave vector that depends on the interaction strength. At the critical interaction, the instability corresponds to the uniform Pomeranchuk or Stoner instability, but the momentum of the leading instability becomes nonzero for stronger attractive interaction. In the particle-particle channel, the coupling function reveals the dynamics of the unstable mode associated with the BCS instability. When an unstable normal metal evolves into the superconducting state, there exists a period in which a superconducting state with spatially nonuniform phase appears due to the presence of unstable Cooperon modes with nonzero momenta.

arXiv:2302.06828•Physical review. B./Physical review. B•Jan 23, 2024•7 citations

Kohn-Luttinger effect and the instability of a two-dimensional repulsive Fermi liquid atT=0

Andrey V. Chubukov

We consider the possibility for a pairing in a two-dimensional (2D) repulsive Fermi liquid due to the singularity in the scattering amplitude \ensuremath{\Gamma}(q) at the momentum transfer q\ensuremath{\le}2${\mathit{p}}_{\mathit{F}}$ (Kohn-Luttinger effect). A common belief based on perturbative calculations up to second order in the s-wave scattering amplitude is that this effect is absent in two dimensions. I show that this is not the case. For an arbitrary Fermi liquid, \ensuremath{\Gamma}(q) is found to have a singular part, ${\mathrm{\ensuremath{\Gamma}}}^{\mathrm{sing}}$(q)\ensuremath{\sim} \ensuremath{\surd}1-${\mathit{q}}^{2}$/(2${\mathit{p}}_{\mathit{F}}$${)}^{2}$ , for q\ensuremath{\le}2${\mathit{p}}_{\mathit{F}}$. For large 2D orbital momentum l, this term gives a dominant attractive contribution to the scattering amplitude and leads to a pairing instability in a 2D Fermi liquid with arbitrary short-range repulsion. In the dilute limit, numerical studies show that the effect survives down to l=1 and gives rise to a p-wave pairing. The relevance of these results to experiments on $^{3}\mathrm{He}$ adsorbed on the free surface of $^{4}\mathrm{He}$ is discussed.

Physical review. B, Condensed matter•Jul 1, 1993•159 citations

Theory of non-Fermi liquid near a diagonal electronic nematic state on a square lattice

Ying-Jer Kao, Hae‐Young Kee

We study the effects of Fermi-surface fluctuations on the single-particle lifetime near the diagonal electronic nematic phase on a two-dimensional square lattice. It has been shown that there exists a quantum critical point between the diagonal nematic and isotropic phases [H. Doh et al., Phys. Rev. B 73, 125117 (2006)]. We study the longitudinal fluctuations of the order parameter near the critical point, where the singular forward scattering leads to a non-Fermi-liquid behavior over the whole Fermi surface except along the ${k}_{x}$ and ${k}_{y}$ directions. We will also discuss the temperature and chemical potential dependences of the single-particle decay rate.

arXiv:cond-mat/0701636•Physical Review B•Jul 13, 2007•9 citations

Nonperturbative behavior of the quantum phase transition to a nematic Fermi fluid

Michael J. Lawler, Daniel Gustavo Barci, Victoria Fernández +2 more

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 $z=3$. 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 ${N}^{*}(0)$ is larger than the free Fermi value, $N(0)$, and $B$ is a constant. These results confirm the non-Fermi liquid nature of both the quantum critical theory and of the nematic phase.

arXiv:cond-mat/0508747•Physical Review B•Feb 1, 2006•115 citations

Model of a Two-Dimensional Marginal Fermi Liquid Superconductor

Yu-Liang Liu, Tai Kai Ng

In this paper we study a model of an $s$-wave marginal Fermi liquid superconductor in two dimensions. Besides the usual Bardeen-Cooper-Schrieffer (BCS) point attractive interaction, the fermions in our system also interact via a long-range repulsive interaction $v(q){\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}g}_{D}/{q}^{2}$. We find that although superconductivity is not destroyed by the long-ranged repulsive interaction (demonstrated by the response to electromagnetic magnetic fields), the BCS pairing function ${\ensuremath{\Delta}}_{\mathrm{BCS}}$ is strongly suppressed, resulting in a power-law correlation $〈{\ensuremath{\Delta}}_{\mathrm{BCS}}(\stackrel{\ensuremath{\rightarrow}}{x}){\ensuremath{\Delta}}_{\mathrm{BCS}}(0)〉\ensuremath{\sim}|\stackrel{\ensuremath{\rightarrow}}{x}{|}^{\ensuremath{-}2\ensuremath{\gamma}}$. This observation is explained within a general picture of the marginal Fermi liquid.

Physical Review Letters•Dec 27, 1999•9 citations

Interplay between superconductivity and non-Fermi liquid at a quantum critical point in a metal. I. The γ model and its phase diagram at T = 0 : The case 0 < γ < 1

Ar. Abanov, Andrey V. Chubukov

Near a quantum critical point in a metal, a strong fermion-fermion interaction, mediated by a soft boson, acts in two different directions: it destroys fermionic coherence and it gives rise to an attraction in one or more pairing channels. The two tendencies compete with each other. We analyze a class of quantum critical models, in which momentum integration and the selection of a particular pairing symmetry can be done explicitly, and the competition between non-Fermi liquid and pairing can be analyzed within an effective model with dynamical electron-electron interaction $V({\mathrm{\ensuremath{\Omega}}}_{m})\ensuremath{\propto}1/{|{\mathrm{\ensuremath{\Omega}}}_{m}|}^{\ensuremath{\gamma}}$ (the $\ensuremath{\gamma}$ model). In this paper, the first in the series, we consider the case $T=0$ and $0<\ensuremath{\gamma}<1$. We argue that tendency to pairing is stronger, and the ground state is a superconductor. We argue, however, that a superconducting state is highly nontrivial as there exists a discrete set of topologically distinct solutions for the pairing gap ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ ($n=0,1,2,...,\ensuremath{\infty}$). All solutions have the same spatial pairing symmetry, but differ in the time domain: ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ changes sign $n$ times as a function of Matsubara frequency ${\ensuremath{\omega}}_{m}$. The $n=0$ solution ${\mathrm{\ensuremath{\Delta}}}_{0}({\ensuremath{\omega}}_{m})$ is sign preserving and tends to a finite value at ${\ensuremath{\omega}}_{m}=0$, like in BCS theory. The $n=\ensuremath{\infty}$ solution corresponds to an infinitesimally small $\mathrm{\ensuremath{\Delta}}({\ensuremath{\omega}}_{m})$, which oscillates down to the lowest frequencies as $\mathrm{\ensuremath{\Delta}}({\ensuremath{\omega}}_{m})\ensuremath{\propto}|{\ensuremath{\omega}}_{m}{|}^{\ensuremath{\gamma}/2}cos[2\ensuremath{\beta}\phantom{\rule{0.16em}{0ex}}log(|{\ensuremath{\omega}}_{m}|/{\ensuremath{\omega}}_{0})]$, where $\ensuremath{\beta}=O(1)$ and ${\ensuremath{\omega}}_{0}$ is of order of fermion-boson coupling. As a proof, we obtain the exact solution of the linearized gap equation at $T=0$ on the entire frequency axis for all $0<\ensuremath{\gamma}<1$, and an approximate solution of the nonlinear gap equation. We argue that the presence of an infinite set of solutions opens up a new channel of gap fluctuations. We extend the analysis to the case where the pairing component of the interaction has additional factor $1/N$ and show that there exists a critical ${N}_{cr}>1$, above which superconductivity disappears, and the ground state becomes a non-Fermi liquid. We show that all solutions develop simultaneously once $N$ gets smaller than ${N}_{cr}$.

arXiv:2004.13220•Physical review. B./Physical review. B•Jul 30, 2020•58 citations
Forward scattering and backscattering play an exceptional role in the physics of two-dimensional interacting fermions. In a Fermi liquid, both give rise to a nonanalytic \ensuremathω2ln(\ensuremathω){\ensuremath{\omega}}^{2}ln(\ensuremath{\omega})\ensuremathω2ln(\ensuremathω) form of the fermionic scattering rate at second order in the interaction. Here we argue that higher powers of ln(\ensuremathω)ln(\ensuremath{\omega})ln(\ensuremathω) appear in the backscattering contribution at higher orders. We show that these terms come from ``planar'' processes, which are effectively one-dimensional. This is explicitly demonstrated by extending a Fermi liquid to the limit of N\ensuremath≫1N\ensuremath{\gg}1N\ensuremath≫1 fermionic flavors, when only planar processes survive. We sum the leading logarithms for the case of a two-dimensional Fermi liquid near a nematic transition, and we obtain an expression for the scattering rate at T=0T=0T=0 to all orders in the interaction. For a repulsive interaction, the resulting rate is logarithmically suppressed, and the result is valid down to \ensuremathω=0\ensuremath{\omega}=0\ensuremathω=0. For an attractive interaction, the ground state is an sss-wave superconductor with a gap \ensuremathΔ0{\mathrm{\ensuremath{\Delta}}}_{0}\ensuremathΔ0​. We show that in this case the scattering rate increases as \ensuremathω\ensuremath{\omega}\ensuremathω is reduced toward \ensuremathΔ0{\mathrm{\ensuremath{\Delta}}}_{0}\ensuremathΔ0​. At \ensuremathω\ensuremath≥\ensuremathΔ0\ensuremath{\omega}\ensuremath{\ge}{\mathrm{\ensuremath{\Delta}}}_{0}\ensuremathω\ensuremath≥\ensuremathΔ0​, the behavior of the scattering rate is rather unconventional as many pairing channels compete near a nematic critical point, and the sss-wave wins only by a narrow margin. We take superconductivity into consideration and obtain the scattering rate also at smaller \ensuremathω\ensuremath≃\ensuremathΔ0\ensuremath{\omega}\ensuremath{\simeq}{\mathrm{\ensuremath{\Delta}}}_{0}\ensuremathω\ensuremath≃\ensuremathΔ0​.
Landau Fermi liquid theory is a fixed-point theory of metals that includes the forward-scattering amplitudes as exact marginal couplings. However, the fixed-point theory that only includes the strict forward scatterings is nonlocal in real space. In this paper, we revisit the Fermi liquid theory for charge-neutral fermions using the field-theoretic functional renormalization group formalism and show how the scale-invariant fixed point emerges as a local theory, which includes not only the forward scatterings but also nonforward scatterings with small but nonzero momentum transfers. In the low-energy limit, the near-forward scattering and pairing interactions take scale-invariant forms if the momentum transfer and the center-of-mass momentum of Cooper pairs, respectively, are comparable to the energy. The coupling functions fully capture the universal low-energy dynamics of the collective modes and instabilities of Fermi liquids. A runaway flow of the coupling function in the particle-hole channel beyond a critical interaction suggests an instability toward an ordered phase with a wave vector that depends on the interaction strength. At the critical interaction, the instability corresponds to the uniform Pomeranchuk or Stoner instability, but the momentum of the leading instability becomes nonzero for stronger attractive interaction. In the particle-particle channel, the coupling function reveals the dynamics of the unstable mode associated with the BCS instability. When an unstable normal metal evolves into the superconducting state, there exists a period in which a superconducting state with spatially nonuniform phase appears due to the presence of unstable Cooperon modes with nonzero momenta.
We consider the possibility for a pairing in a two-dimensional (2D) repulsive Fermi liquid due to the singularity in the scattering amplitude \ensuremath{\Gamma}(q) at the momentum transfer q\ensuremath{\le}2pF{\mathit{p}}_{\mathit{F}}pF​ (Kohn-Luttinger effect). A common belief based on perturbative calculations up to second order in the s-wave scattering amplitude is that this effect is absent in two dimensions. I show that this is not the case. For an arbitrary Fermi liquid, \ensuremath{\Gamma}(q) is found to have a singular part, ${\mathrm{\ensuremath{\Gamma}}}^{\mathrm{sing}}$(q)\ensuremath{\sim} \ensuremath{\surd}1-q2{\mathit{q}}^{2}q2/(2pF{\mathit{p}}_{\mathit{F}}pF​)2{)}^{2})2 , for q\ensuremath{\le}2pF{\mathit{p}}_{\mathit{F}}pF​. For large 2D orbital momentum l, this term gives a dominant attractive contribution to the scattering amplitude and leads to a pairing instability in a 2D Fermi liquid with arbitrary short-range repulsion. In the dilute limit, numerical studies show that the effect survives down to l=1 and gives rise to a p-wave pairing. The relevance of these results to experiments on 3He^{3}\mathrm{He}3He adsorbed on the free surface of 4He^{4}\mathrm{He}4He is discussed.
We study the effects of Fermi-surface fluctuations on the single-particle lifetime near the diagonal electronic nematic phase on a two-dimensional square lattice. It has been shown that there exists a quantum critical point between the diagonal nematic and isotropic phases [H. Doh et al., Phys. Rev. B 73, 125117 (2006)]. We study the longitudinal fluctuations of the order parameter near the critical point, where the singular forward scattering leads to a non-Fermi-liquid behavior over the whole Fermi surface except along the kx{k}_{x}kx​ and ky{k}_{y}ky​ directions. We will also discuss the temperature and chemical potential dependences of the single-particle decay rate.
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.
In this paper we study a model of an sss-wave marginal Fermi liquid superconductor in two dimensions. Besides the usual Bardeen-Cooper-Schrieffer (BCS) point attractive interaction, the fermions in our system also interact via a long-range repulsive interaction v(q)=gD/q2v(q){\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}g}_{D}/{q}^{2}v(q)=gD​/q2. We find that although superconductivity is not destroyed by the long-ranged repulsive interaction (demonstrated by the response to electromagnetic magnetic fields), the BCS pairing function ${\ensuremath{\Delta}}_{\mathrm{BCS}}$ is strongly suppressed, resulting in a power-law correlation $〈{\ensuremath{\Delta}}_{\mathrm{BCS}}(\stackrel{\ensuremath{\rightarrow}}{x}){\ensuremath{\Delta}}_{\mathrm{BCS}}(0)〉\ensuremath{\sim}|\stackrel{\ensuremath{\rightarrow}}{x}{|}^{\ensuremath{-}2\ensuremath{\gamma}}$. This observation is explained within a general picture of the marginal Fermi liquid.
Near a quantum critical point in a metal, a strong fermion-fermion interaction, mediated by a soft boson, acts in two different directions: it destroys fermionic coherence and it gives rise to an attraction in one or more pairing channels. The two tendencies compete with each other. We analyze a class of quantum critical models, in which momentum integration and the selection of a particular pairing symmetry can be done explicitly, and the competition between non-Fermi liquid and pairing can be analyzed within an effective model with dynamical electron-electron interaction $V({\mathrm{\ensuremath{\Omega}}}_{m})\ensuremath{\propto}1/{|{\mathrm{\ensuremath{\Omega}}}_{m}|}^{\ensuremath{\gamma}}$ (the $\ensuremath{\gamma}$ model). In this paper, the first in the series, we consider the case T=0T=0T=0 and $0<\ensuremath{\gamma}<1$. We argue that tendency to pairing is stronger, and the ground state is a superconductor. We argue, however, that a superconducting state is highly nontrivial as there exists a discrete set of topologically distinct solutions for the pairing gap ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ ($n=0,1,2,...,\ensuremath{\infty}$). All solutions have the same spatial pairing symmetry, but differ in the time domain: ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ changes sign nnn times as a function of Matsubara frequency ${\ensuremath{\omega}}_{m}$. The n=0n=0n=0 solution ${\mathrm{\ensuremath{\Delta}}}_{0}({\ensuremath{\omega}}_{m})$ is sign preserving and tends to a finite value at ${\ensuremath{\omega}}_{m}=0$, like in BCS theory. The $n=\ensuremath{\infty}$ solution corresponds to an infinitesimally small $\mathrm{\ensuremath{\Delta}}({\ensuremath{\omega}}_{m})$, which oscillates down to the lowest frequencies as $\mathrm{\ensuremath{\Delta}}({\ensuremath{\omega}}_{m})\ensuremath{\propto}|{\ensuremath{\omega}}_{m}{|}^{\ensuremath{\gamma}/2}cos[2\ensuremath{\beta}\phantom{\rule{0.16em}{0ex}}log(|{\ensuremath{\omega}}_{m}|/{\ensuremath{\omega}}_{0})]$, where $\ensuremath{\beta}=O(1)$ and ${\ensuremath{\omega}}_{0}$ is of order of fermion-boson coupling. As a proof, we obtain the exact solution of the linearized gap equation at T=0T=0T=0 on the entire frequency axis for all $0<\ensuremath{\gamma}<1$, and an approximate solution of the nonlinear gap equation. We argue that the presence of an infinite set of solutions opens up a new channel of gap fluctuations. We extend the analysis to the case where the pairing component of the interaction has additional factor 1/N1/N1/N and show that there exists a critical Ncr>1{N}_{cr}>1Ncr​>1, above which superconductivity disappears, and the ground state becomes a non-Fermi liquid. We show that all solutions develop simultaneously once NNN gets smaller than Ncr{N}_{cr}Ncr​.