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.

Series on directions in condensed matter physicsOct 24, 20130 citations

Singular or Non-Fermi Liquids

C. M. Varma l

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.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.

AIP conference proceedingsJan 1, 200214 citations

Quantum criticality and non-Fermi liquids: The Wilsonian renormalization group perspective

Anonymous, Paweł Jakubczyk, Hiroyuki Yamase

We develop a theoretical framework based on the nonperturvative renormalization group (RG) in the one-particle irreducible (Wetterich) formulation to tackle the interplay of coupled fermionic and order-parameter fluctuations at metallic quantum critical points (QCPs) with ordering wavevector $\vec{Q}=\vec{0}$ . We consistently treat the dynamical emergence of the Landau damping of the bosonic mode and non-Fermi liquid scaling of fermions upon lowering the cutoff scale. The loop integrals of the present theory involve only contributions from fluctuations above the cutoff scale, which protects the system from developing singular bosonic interactions. We emphasize the importance of the nontrivial relative scaling of the bosonic and fermionic cutoffs $Λ$ and $Λ_f$ , which we fix by analyzing the RG flow of the scale-dependent ordering wave-vector $\vec{Q}_Λ$ . Upon neglecting Fermi self-energy in the loop integrals of the functional RG flow, we identify a non-Fermi liquid RG fixed point and recover the features obtained earlier within RPA-type approaches. In a subsequent step, we self-consistently include the scaling of the self-energy and the Yukawa coupling. We find a generic instability of the non-Fermi liquid RG fixed point. This implies, at least at this truncation level, absence of the QCP with $\vec{Q}=\vec{0}$ and development of a first-order phase transition or a phase characterized by $\vec{Q}\neq\vec{0}$ .We develop a theoretical framework based on the nonperturvative renormalization group (RG) in the one-particle irreducible (Wetterich) formulation to tackle the interplay of coupled fermionic and order-parameter fluctuations at metallic quantum critical points (QCPs) with ordering wavevector $\vec{Q}=\vec{0}$. We consistently treat the dynamical emergence of the Landau damping of the bosonic mode and non-Fermi liquid scaling of fermions upon lowering the cutoff scale. The loop integrals of the present theory involve only contributions from fluctuations above the cutoff scale, which protects the system from developing singular bosonic interactions. We emphasize the importance of the nontrivial relative scaling of the bosonic and fermionic cutoffs $Λ$ and $Λ_f$, which we fix by analyzing the RG flow of the scale-dependent ordering wave-vector $\vec{Q}_Λ$. Upon neglecting Fermi self-energy in the loop integrals of the functional RG flow, we identify a non-Fermi liquid RG fixed point and recover the features obtained earlier within RPA-type approaches. In a subsequent step, we self-consistently include the scaling of the self-energy and the Yukawa coupling. We find a generic instability of the non-Fermi liquid RG fixed point. This implies, at least at this truncation level, absence of the QCP with $\vec{Q}=\vec{0}$ and development of a first-order phase transition or a phase characterized by $\vec{Q}\neq\vec{0}$.

arXiv:2505.10140Physical review. B./Physical review. BJul 14, 20260 citations

A Critique of Two Metals

R. B. Laughlin

I argue that the conflict between the fermi-liquid and non-fermi-liquid metallic states viewed by Anderson as the central intellectual issue of cuprate superconductivity, and which motivates the recent criticism by Baskaran and Anderson [cond-mat/9706076] of the work of Zhang [cond-mat/9610140], is a fundamentally wrong concept. All experimental evidence points to adiabatic continuability of the strange metal into a conventional one, and thus to one metallic phase rather than two, and all attempts to account theoretically for the existence of a luttinger-liquid at zero temperature in spatial dimension greater than 1 have failed. I discuss the underlying reasons for this failure and then argue that the true higher-dimensional generalization of the luttinger-liquid behavior is a propensity of the system to order. I speculate about how the conflict between antiferromagnetism and superconductivity, the two principal kinds of order in this problem, might result in both the observed zero-temperature phase diagram of the cuprates and the luttinger-liquid phenomenology, i.e. the breakup of the electron into spinons and holons in certain regimes of doping and energy. The key idea is a quantum critical point regulating a first-order transition between these phases and toward which one is first attracted under renormalization before bifurcating between the two phases.I argue that the conflict between the fermi-liquid and non-fermi-liquid metallic states viewed by Anderson as the central intellectual issue of cuprate superconductivity, and which motivates the recent criticism by Baskaran and Anderson [cond-mat/9706076] of the work of Zhang [cond-mat/9610140], is a fundamentally wrong concept. All experimental evidence points to adiabatic continuability of the strange metal into a conventional one, and thus to one metallic phase rather than two, and all attempts to account theoretically for the existence of a luttinger-liquid at zero temperature in spatial dimension greater than 1 have failed. I discuss the underlying reasons for this failure and then argue that the true higher-dimensional generalization of the luttinger-liquid behavior is a propensity of the system to order. I speculate about how the conflict between antiferromagnetism and superconductivity, the two principal kinds of order in this problem, might result in both the observed zero-temperature phase diagram of the cuprates and the luttinger-liquid phenomenology, i.e. the breakup of the electron into spinons and holons in certain regimes of doping and energy. The key idea is a quantum critical point regulating a first-order transition between these phases and toward which one is first attracted under renormalization before bifurcating between the two phases.

arXiv:cond-mat/9709195Sep 17, 19972 citations

Perspectives A critique of two metals

R. B. Laughlin

I argue that Anderson's identification of the conflict between the fermi-liquid and non-fermi-liquid metallic states as the central issue of cuprate superconductivity is fundamentally wrong. All experimental evidence points to adiabatic continuability of the strange metal into a conventional one, and thus to one metallic phase rather than two, and all attempts to account theoretically for the existence of a luttinger-liquid at zero temperature in spatial dimension greater than one have failed. I discuss the underlying reasons for this failure and then argue that the true higher-dimensional generalization of the luttinger-liquid behavior is a propensity of the system to order. This implies that the central issue is actually the conflict between different kinds of order, i.e. exactly the idea implicit in Zhang's paper. I then speculate about how the conflict between antiferromagnetism and superconductivity, the two principal kinds of order in this problem, might result in both the observed zero-temperature phase diagram of the cuprates and the luttinger-liquid phenomenology, i.e. the breakup of the electron into spinons and holons in certain regimes of doping and energy. The key idea is a quantum critical point regulating a first-order transition between these phases, and toward which one is first attracted under renormalization before bifurcating between the two phases. I speculate that this critical point lies on the insulating line, and that the difference between the Mott-insulator and fermi-liquid approaches to the high TC problem comes down to whether or not the superconducting states made by nand p-type doping can be continued into each other. A candidate for the second fixed point required for distinct superconducting phases is the P- and T-violating chiral spin liquid state invented by me.I argue that Anderson's identification of the conflict between the fermi-liquid and non-fermi-liquid metallic states as the central issue of cuprate superconductivity is fundamentally wrong. All experimental evidence points to adiabatic continuability of the strange metal into a conventional one, and thus to one metallic phase rather than two, and all attempts to account theoretically for the existence of a luttinger-liquid at zero temperature in spatial dimension greater than one have failed. I discuss the underlying reasons for this failure and then argue that the true higher-dimensional generalization of the luttinger-liquid behavior is a propensity of the system to order. This implies that the central issue is actually the conflict between different kinds of order, i.e. exactly the idea implicit in Zhang's paper. I then speculate about how the conflict between antiferromagnetism and superconductivity, the two principal kinds of order in this problem, might result in both the observed zero-temperature phase diagram of the cuprates and the luttinger-liquid phenomenology, i.e. the breakup of the electron into spinons and holons in certain regimes of doping and energy. The key idea is a quantum critical point regulating a first-order transition between these phases, and toward which one is first attracted under renormalization before bifurcating between the two phases. I speculate that this critical point lies on the insulating line, and that the difference between the Mott-insulator and fermi-liquid approaches to the high TC problem comes down to whether or not the superconducting states made by nand p-type doping can be continued into each other. A candidate for the second fixed point required for distinct superconducting phases is the P- and T-violating chiral spin liquid state invented by me.

Advances In PhysicsNov 1, 199857 citations

Unified theory of strongly correlated electron systems

Yu-Liang Liu

In framework of eigen-functional bosonization method, we introduce an imaginary phase field to uniquely represent electron correlation, and demonstrate that the Landau Fermi liquid theory and the Tomonaga-Luttinger liquid theory can be unified. It is very clear in this framework that the Tomonaga-Luttinger liquid behavior of one-dimensional interacting electron gases originates from their Fermi structure, and the non-Landau-Fermi liquid behavior of 2D interacting electron gases is induced by the long-range electron interaction, while 3D interacting electron gases generally show the Landau Fermi liquid behavior. Typeset using REVTEX 1 Since the discovery of high Tc cuprate superconductors [1], the strongly correlated electron systems has been extensively studied theoretically [2–10]. Now a common consensus is reached that the low energy physics properties of cuprate superconductors, such as anomalous normal state behavior and high superconducting transition temperature, are determinedIn framework of eigen-functional bosonization method, we introduce an imaginary phase field to uniquely represent electron correlation, and demonstrate that the Landau Fermi liquid theory and the Tomonaga-Luttinger liquid theory can be unified. It is very clear in this framework that the Tomonaga-Luttinger liquid behavior of one-dimensional interacting electron gases originates from their Fermi structure, and the non-Landau-Fermi liquid behavior of 2D interacting electron gases is induced by the long-range electron interaction, while 3D interacting electron gases generally show the Landau Fermi liquid behavior. Typeset using REVTEX 1 Since the discovery of high Tc cuprate superconductors [1], the strongly correlated electron systems has been extensively studied theoretically [2–10]. Now a common consensus is reached that the low energy physics properties of cuprate superconductors, such as anomalous normal state behavior and high superconducting transition temperature, are determined

Nov 15, 20002 citations

\vec{Q}=\vec{0}

. We consistently treat the dynamical emergence of the Landau damping of the bosonic mode and non-Fermi liquid scaling of fermions upon lowering the cutoff scale. The loop integrals of the present theory involve only contributions from fluctuations above the cutoff scale, which protects the system from developing singular bosonic interactions. We emphasize the importance of the nontrivial relative scaling of the bosonic and fermionic cutoffs

Λ

and

Λ_f

, which we fix by analyzing the RG flow of the scale-dependent ordering wave-vector

\vec{Q}_Λ

. Upon neglecting Fermi self-energy in the loop integrals of the functional RG flow, we identify a non-Fermi liquid RG fixed point and recover the features obtained earlier within RPA-type approaches. In a subsequent step, we self-consistently include the scaling of the self-energy and the Yukawa coupling. We find a generic instability of the non-Fermi liquid RG fixed point. This implies, at least at this truncation level, absence of the QCP with

\vec{Q}=\vec{0}

and development of a first-order phase transition or a phase characterized by

\vec{Q}\neq\vec{0}