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

AIP conference proceedings

Singular or Non-Fermi Liquids

C. M. Varma l

Published Jan 1, 2002• 14 citations
Publisher

Abstract

An introductory survey of the theoretical ideas and calculations and experimental results which depart from Landau Fermi liquids is presented. The common themes and possible routes to the singularities leading to breakdown of Landau Fermi liquids are categorized following an elementary discussion of the theory. Soluble examples of singular or Non Fermi‐liquids include models of impurities in metals with special symmetries and one‐dimensional interacting fermions. A review of these is followed by a discussion of singular Fermi liquids in a wide variety of experimental situations and theoretical models. These include the effects of low energy collective fluctuations, gauge fields, fluctuations around quantum critical points, the normal state of high temperature superconductors and the two dimensional metallic state.

DOI
10.1063/1.1509144

Related papers

LUTTINGER LIQUID, SINGULAR INTERACTION AND QUANTUM CRITICALITY IN CUPRATE MATERIALS

C. Di Castro, S. Caprara

With particular reference to the role of the renormalization group (RG) approach and Ward identities (WI's), we start by recalling some old features of the one-dimensional Luttinger liquid as the prototype of non-Fermi-liquid behavior. Its dimensional crossover to the Landau normal Fermi liquid implies that a non-Fermi liquid, as, e.g., the normal phase of the cuprate high temperature superconductors, can be maintained in d>1 only in the presence of a sufficiently singular effective interaction among the charge carriers. This is the case when, nearby an instability, the interaction is mediated by critical fluctuations. We are then led to introduce the specific case of superconductivity in cuprates as an example of avoided quantum criticality. We will disentangle the fluctuations which act as mediators of singular electron–electron interaction, enlightening the possible order competing with superconductivity and a mechanism for the non-Fermi-liquid behavior of the metallic phase. This paper is not meant to be a comprehensive review. Many important contributions will not be considered. We will also avoid using extensive technicalities and making full calculations for which we refer to the original papers and to the many good available reviews. We will here only follow one line of reasoning which guided our research activity in this field.

arXiv:1206.5962•International Journal of Modern Physics B•Jul 26, 2012•0 citations

LUTTINGER LIQUID, SINGULAR INTERACTION AND QUANTUM CRITICALITY IN CUPRATE MATERIALS

C. Di Castro, S. Caprara

With particular reference to the role of the renormalization group (RG) approach and Ward identities (WI's), we start by recalling some old features of the one-dimensional Luttinger liquid as the prototype of non-Fermi-liquid behavior. Its dimensional crossover to the Landau normal Fermi liquid implies that a non-Fermi liquid, as, e.g., the normal phase of the cuprate high temperature superconductors, can be maintained in d>1 only in the presence of a sufficiently singular effective interaction among the charge carriers. This is the case when, nearby an instability, the interaction is mediated by critical fluctuations. We are then led to introduce the specific case of superconductivity in cuprates as an example of avoided quantum criticality. We will disentangle the fluctuations which act as mediators of singular electron–electron interaction, enlightening the possible order competing with superconductivity and a mechanism for the non-Fermi-liquid behavior of the metallic phase. This paper is not meant to be a comprehensive review. Many important contributions will not be considered. We will also avoid using extensive technicalities and making full calculations for which we refer to the original papers and to the many good available reviews. We will here only follow one line of reasoning which guided our research activity in this field.

Series on directions in condensed matter physics•Oct 24, 2013•0 citations

Non-Fermi-liquid metals

L. B. Ioffe, Andrew J. Millis

The experimental discovery of a number of 'strange metals' has reopened the question of the low temperature behavior of interacting Fermi systems. Here we provide a subjective overview of some aspects of the resulting theoretical work. It seems to us that from a theoretical standpoint Landau's Fermi-liquid theory has proven to be a remarkably robust description of clean Fermi systems. The only well documented theoretical examples of non-Fermi-liquid behavior are metals subject to gauge interactions or at quantum critical points. The experimental anomalies which prompted the reexamination of Fermi-liquid theory remain in many cases mysterious.

Physics-Uspekhi•Jun 30, 1998•15 citations

Marginal Fermi Liquids

Vikram Tripathi

The Landau Fermi liquid theory of metals has been so successful that exceptions to it draw a fair amount of attention. The theory rests on the assumption that the low‐energy properties of interacting electrons in a metal in three dimensions may be described in terms of weakly repulsive charged spin 1 2 $ \frac{1}{2} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351121996/259a876e-1ad1-4598-839f-e9587e88d783/content/inline-math13_1.tif"/> fermionic excitations— the quasiparticles. This is of course not generally expected to hold following a phase transition, for example, in the form of a Pomeranchuk instability or a superconducting transition. In Chapter 2 we showed that Pomeranchuk instabilities occur whenever some Landau F function satisfies the condition F l s ( a ) / ( 2 l + 1 ) = - 1 $ F_{l}^{s(a)} /(2l + 1) = - 1 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351121996/259a876e-1ad1-4598-839f-e9587e88d783/content/inline-math13_2.tif"/> , where l = 0, 1, 2, . . . denotes the relevant angular momentum channel where the instability occurs. The l = 0 instability in the spin‐symmetric channel is associated with a divergence of compressibility that leads to phase separation and consequent breakdown of the Fermi liquid. In the spin‐antisymmetric channel, the l = 0 instability corresponds to ferromagnetism. In the ferromagnetic quantum critical region (“classical” Gaussian regime in Hertz theory), we found in Sec. 9.6.2.3 that spin fluctuations lead to the destruction of the Landau Fermi liquid and this is evident, for example, in the ln ( 1 / T ) $ {\text{ln }}(1/T) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351121996/259a876e-1ad1-4598-839f-e9587e88d783/content/inline-math13_3.tif"/> divergence of the specific heat coefficient γ ( T ) . Proximity to an electronic instability is however not a necessary condition for the breakdown of the Fermi liquid. In Sec. 10.2 we found that transverse electromagnetic fluctuations are only weakly screened by conduction electrons and that results in a γ ( T ) ∼ ln ( 1 / T ) $ \gamma (T) \sim {\text{ln }}(1/T) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351121996/259a876e-1ad1-4598-839f-e9587e88d783/content/inline-math13_4.tif"/> divergence for the specific heat coefficient without needing to be in the vicinity of a ferromagnetic quantum critical point 1 . The lack of screening of such electromagnetic fluctuations is a feature shared with low energy spin fluctuations in the Hertz theory of quantum critical ferromagnets. Another route to non Fermi‐liquid behavior, not reliant on the existence of long‐range interactions, appeared in Sec. 11.4 where we analyzed non‐optimally screened Kondo models. It was found that singularities in properties such as the susceptibility and specific heat coefficient may be expected in the event that residual low‐energy degeneracies associated with spin‐flip scattering with conduction electrons persist down to the lowest temperatures. Finally, weak potential disorder was found to preserve the Fermi liquid (in three dimensions) down to the lowest temperatures (see Chapter 8 ).

CRC Press eBooks•Feb 2, 2018•0 citations

Recent Developments in Non-Fermi Liquid Theory

Sung-Sik Lee

Non-Fermi liquids are unconventional metals whose physical properties deviate qualitatively from those of noninteracting fermions due to strong quantum fluctuations near Fermi surfaces. They arise when metals are subject to singular interactions mediated by soft collective modes. In the absence of well-defined quasiparticles, universal physics of non-Fermi liquids is captured by interacting field theories which replace Landau Fermi liquid theory. However, it has been difficult to understand their universal low-energy physics due to a lack of theoretical methods that take into account strong quantum fluctuations in the presence of abundant low-energy degrees of freedom. In this review, we discuss two approaches that have been recently developed for non-Fermi liquid theory with emphasis on two space dimensions. The first is a perturbative scheme based on a dimensional regularization, which achieves a controlled access to the low-energy physics by tuning the codimension of Fermi surface. The second is a nonperturbative approach which treats the interaction ahead of the kinetic term through a non-Gaussian scaling called interaction-driven scaling. Examples of strongly coupled non-Fermi liquids amenable to exact treatments through the interaction-driven scaling are discussed.

arXiv:1703.08172•Annual Review of Condensed Matter Physics•Dec 8, 2017•211 citations

Some aspects of singular interactions in condensed Fermi systems

Philip C. E. Stamp

This article gives a fairly detailed survey of some of the problems raised when the interaction energy fσσ'kk' between 2 fermionic quasiparticles (in 2 dimensions) is singular when $|\underline{k}-\underline{k}'|\to 0$. Before dealing with singular interactions, it is shown how a non-singular fσσ'kk' leads to a 2-dimensional Fermi liquid theory, which is internally consistent, at least as far as its infrared properties are concerned. The quasiparticle properties are calculated in detail. The question of whether singular interactions arise for the dilute Fermi gas, with short-range repulsive interactions, is investigated perturbatively. One finds a weak singularity in fσσ'kk', when the dimensionality D = 2, but it does not destabilize the Fermi liquid. A more sophisticated analysis is then given, to all orders in the interaction, using the Lippman-Schwinger equation as well as a phase shift analysis for a finite box. The conclusion is that any breakdown of Fermi liquid theory must come from non-perturbative effects. An examination is then made of some of the consequences arising if a singular interaction is introduced — the form proposed by Anderson is used as an example. A hierarchy of singular terms arise in all quantities — this is shown for the self-energy, and also the 3–point and 4–point scattering functions. These may be summed in a perfectly consistent manner. Most attention is given to the particle-hole channel, since it appears to lead to results different from those of Anderson. Nevertheless it appears that it is possible to derive a sensible theory starting from a singular effective Hamiltonian — although Fermi Liquid theory breaks down, all fermionic quantities may be calculated consistently. Finally, the effect of a magnetic field (which cuts off the infrared divergences) is investigated, and the de Haas-van Alphen amplitude calculated, for such a singular Fermionic system.

Journal de Physique I•Feb 1, 1993•16 citations
An introductory survey of the theoretical ideas and calculations and experimental results which depart from Landau Fermi liquids is presented. The common themes and possible routes to the singularities leading to breakdown of Landau Fermi liquids are categorized following an elementary discussion of the theory. Soluble examples of singular or Non Fermi‐liquids include models of impurities in metals with special symmetries and one‐dimensional interacting fermions. A review of these is followed by a discussion of singular Fermi liquids in a wide variety of experimental situations and theoretical models. These include the effects of low energy collective fluctuations, gauge fields, fluctuations around quantum critical points, the normal state of high temperature superconductors and the two dimensional metallic state.
With particular reference to the role of the renormalization group (RG) approach and Ward identities (WI's), we start by recalling some old features of the one-dimensional Luttinger liquid as the prototype of non-Fermi-liquid behavior. Its dimensional crossover to the Landau normal Fermi liquid implies that a non-Fermi liquid, as, e.g., the normal phase of the cuprate high temperature superconductors, can be maintained in d>1 only in the presence of a sufficiently singular effective interaction among the charge carriers. This is the case when, nearby an instability, the interaction is mediated by critical fluctuations. We are then led to introduce the specific case of superconductivity in cuprates as an example of avoided quantum criticality. We will disentangle the fluctuations which act as mediators of singular electron–electron interaction, enlightening the possible order competing with superconductivity and a mechanism for the non-Fermi-liquid behavior of the metallic phase. This paper is not meant to be a comprehensive review. Many important contributions will not be considered. We will also avoid using extensive technicalities and making full calculations for which we refer to the original papers and to the many good available reviews. We will here only follow one line of reasoning which guided our research activity in this field.
With particular reference to the role of the renormalization group (RG) approach and Ward identities (WI's), we start by recalling some old features of the one-dimensional Luttinger liquid as the prototype of non-Fermi-liquid behavior. Its dimensional crossover to the Landau normal Fermi liquid implies that a non-Fermi liquid, as, e.g., the normal phase of the cuprate high temperature superconductors, can be maintained in d>1 only in the presence of a sufficiently singular effective interaction among the charge carriers. This is the case when, nearby an instability, the interaction is mediated by critical fluctuations. We are then led to introduce the specific case of superconductivity in cuprates as an example of avoided quantum criticality. We will disentangle the fluctuations which act as mediators of singular electron–electron interaction, enlightening the possible order competing with superconductivity and a mechanism for the non-Fermi-liquid behavior of the metallic phase. This paper is not meant to be a comprehensive review. Many important contributions will not be considered. We will also avoid using extensive technicalities and making full calculations for which we refer to the original papers and to the many good available reviews. We will here only follow one line of reasoning which guided our research activity in this field.
The experimental discovery of a number of 'strange metals' has reopened the question of the low temperature behavior of interacting Fermi systems. Here we provide a subjective overview of some aspects of the resulting theoretical work. It seems to us that from a theoretical standpoint Landau's Fermi-liquid theory has proven to be a remarkably robust description of clean Fermi systems. The only well documented theoretical examples of non-Fermi-liquid behavior are metals subject to gauge interactions or at quantum critical points. The experimental anomalies which prompted the reexamination of Fermi-liquid theory remain in many cases mysterious.
The Landau Fermi liquid theory of metals has been so successful that exceptions to it draw a fair amount of attention. The theory rests on the assumption that the low‐energy properties of interacting electrons in a metal in three dimensions may be described in terms of weakly repulsive charged spin 1 2 $ \frac{1}{2} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351121996/259a876e-1ad1-4598-839f-e9587e88d783/content/inline-math13_1.tif"/> fermionic excitations— the quasiparticles. This is of course not generally expected to hold following a phase transition, for example, in the form of a Pomeranchuk instability or a superconducting transition. In Chapter 2 we showed that Pomeranchuk instabilities occur whenever some Landau F function satisfies the condition F l s ( a ) / ( 2 l + 1 ) = - 1 $ F_{l}^{s(a)} /(2l + 1) = - 1 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351121996/259a876e-1ad1-4598-839f-e9587e88d783/content/inline-math13_2.tif"/> , where l = 0, 1, 2, . . . denotes the relevant angular momentum channel where the instability occurs. The l = 0 instability in the spin‐symmetric channel is associated with a divergence of compressibility that leads to phase separation and consequent breakdown of the Fermi liquid. In the spin‐antisymmetric channel, the l = 0 instability corresponds to ferromagnetism. In the ferromagnetic quantum critical region (“classical” Gaussian regime in Hertz theory), we found in Sec. 9.6.2.3 that spin fluctuations lead to the destruction of the Landau Fermi liquid and this is evident, for example, in the ln ( 1 / T ) $ {\text{ln }}(1/T) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351121996/259a876e-1ad1-4598-839f-e9587e88d783/content/inline-math13_3.tif"/> divergence of the specific heat coefficient γ ( T ) . Proximity to an electronic instability is however not a necessary condition for the breakdown of the Fermi liquid. In Sec. 10.2 we found that transverse electromagnetic fluctuations are only weakly screened by conduction electrons and that results in a γ ( T ) ∼ ln ( 1 / T ) $ \gamma (T) \sim {\text{ln }}(1/T) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351121996/259a876e-1ad1-4598-839f-e9587e88d783/content/inline-math13_4.tif"/> divergence for the specific heat coefficient without needing to be in the vicinity of a ferromagnetic quantum critical point 1 . The lack of screening of such electromagnetic fluctuations is a feature shared with low energy spin fluctuations in the Hertz theory of quantum critical ferromagnets. Another route to non Fermi‐liquid behavior, not reliant on the existence of long‐range interactions, appeared in Sec. 11.4 where we analyzed non‐optimally screened Kondo models. It was found that singularities in properties such as the susceptibility and specific heat coefficient may be expected in the event that residual low‐energy degeneracies associated with spin‐flip scattering with conduction electrons persist down to the lowest temperatures. Finally, weak potential disorder was found to preserve the Fermi liquid (in three dimensions) down to the lowest temperatures (see Chapter 8 ).
Non-Fermi liquids are unconventional metals whose physical properties deviate qualitatively from those of noninteracting fermions due to strong quantum fluctuations near Fermi surfaces. They arise when metals are subject to singular interactions mediated by soft collective modes. In the absence of well-defined quasiparticles, universal physics of non-Fermi liquids is captured by interacting field theories which replace Landau Fermi liquid theory. However, it has been difficult to understand their universal low-energy physics due to a lack of theoretical methods that take into account strong quantum fluctuations in the presence of abundant low-energy degrees of freedom. In this review, we discuss two approaches that have been recently developed for non-Fermi liquid theory with emphasis on two space dimensions. The first is a perturbative scheme based on a dimensional regularization, which achieves a controlled access to the low-energy physics by tuning the codimension of Fermi surface. The second is a nonperturbative approach which treats the interaction ahead of the kinetic term through a non-Gaussian scaling called interaction-driven scaling. Examples of strongly coupled non-Fermi liquids amenable to exact treatments through the interaction-driven scaling are discussed.
This article gives a fairly detailed survey of some of the problems raised when the interaction energy fσσ'kk' between 2 fermionic quasiparticles (in 2 dimensions) is singular when ∣k‾−k‾′∣→0|\underline{k}-\underline{k}'|\to 0∣k​−k​′∣→0. Before dealing with singular interactions, it is shown how a non-singular fσσ'kk' leads to a 2-dimensional Fermi liquid theory, which is internally consistent, at least as far as its infrared properties are concerned. The quasiparticle properties are calculated in detail. The question of whether singular interactions arise for the dilute Fermi gas, with short-range repulsive interactions, is investigated perturbatively. One finds a weak singularity in fσσ'kk', when the dimensionality D = 2, but it does not destabilize the Fermi liquid. A more sophisticated analysis is then given, to all orders in the interaction, using the Lippman-Schwinger equation as well as a phase shift analysis for a finite box. The conclusion is that any breakdown of Fermi liquid theory must come from non-perturbative effects. An examination is then made of some of the consequences arising if a singular interaction is introduced — the form proposed by Anderson is used as an example. A hierarchy of singular terms arise in all quantities — this is shown for the self-energy, and also the 3–point and 4–point scattering functions. These may be summed in a perfectly consistent manner. Most attention is given to the particle-hole channel, since it appears to lead to results different from those of Anderson. Nevertheless it appears that it is possible to derive a sensible theory starting from a singular effective Hamiltonian — although Fermi Liquid theory breaks down, all fermionic quantities may be calculated consistently. Finally, the effect of a magnetic field (which cuts off the infrared divergences) is investigated, and the de Haas-van Alphen amplitude calculated, for such a singular Fermionic system.