arXiv:1402.5413

Slow Fermions in Quantum Critical Metals

A. Liam Fitzpatrick, Shamit Kachru A, Jared Kaplan A, Steven Allan Kivelson, S. Raghu

Published Feb 21, 2014• 3 citations

Abstract

We study the low-energy behavior of metals coupled to gapless bosons. This problem arises in several contexts in modern condensed matter physics; we focus on the theory of metals near continuous quantum phase transitions (where the boson is the order parameter). In the vicinity of d=3 spatial dimensions, the upper critical dimension of the theory, the ratio of fermion and boson speeds, v/c, acts as an additional control parameter, enabling us to access IR fixed points where this ratio vanishes. This limit corresponds to a non-Fermi liquid coupled to bosons with critical exponents governed by the Wilson-Fisher fixed point.

DOI: 10.48550/arxiv.1402.5413

Related papers

Non-Fermi-liquid fixed point in a Wilsonian theory of quantum critical metals

A. Liam Fitzpatrick, Shamit Kachru A, Jared Kaplan A +1 more

arXiv:1307.0004Physical Review BSep 10, 2013102 citations

Non-Fermi-liquid behavior of large- NB quantum critical metals

A. Liam Fitzpatrick, Shamit Kachru A, Jared Kaplan A +1 more

arXiv:1312.3321Physical Review BApr 11, 201493 citations

Controlled expansion for certain non-Fermi-liquid metals

David F. Mross, John McGreevy, Hong Liu +1 more

arXiv:1003.0894Physical Review BJul 26, 2010297 citations

Quantum critical metals in 4−ε dimensions

Gonzalo Torroba, Huajia Wang

arXiv:1406.3029Physical Review BOct 31, 201444 citations

Two-Dimensional Non-Fermi-Liquid Metals: A Solvable Large- N Limit

Jeremías Aguilera Damia, Shamit Kachru A, S. Raghu +1 more

arXiv:1905.08256Physical Review LettersAug 29, 201956 citations

Quantum critical metals in d=3+1 dimensions

Raghu Mahajan, David M. Ramirez, Shamit Kachru A +1 more

arXiv:1303.1587Physical Review BSep 10, 201350 citations

arXiv:1402.5413

Slow Fermions in Quantum Critical Metals

A. Liam Fitzpatrick, Shamit Kachru A, Jared Kaplan A, Steven Allan Kivelson, S. Raghu

Published Feb 21, 2014• 3 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.1402.5413

Related papers

Non-Fermi-liquid fixed point in a Wilsonian theory of quantum critical metals

A. Liam Fitzpatrick, Shamit Kachru A, Jared Kaplan A +1 more

We study the problem of disorder-free metals near a continuous quantum critical point. We depart from the standard paradigm [Hertz, Phys. Rev. B 14, 1165 (1976) and Millis, Phys. Rev. B 48, 7183 (1993)] and treat both fermions and bosons (i.e., order parameter fields) on equal footing. We construct a Wilsonian effective field theory that integrates out only high-energy boson and fermion modes. Below the upper critical dimension of the theory ( $d=3$ spatial dimensions), we find new fixed points in which the bosons are described by the Wilson-Fisher fixed point and are coupled to a non-Fermi-liquid metal. We describe subtleties with the renormalization group flow of four-Fermi interactions, which can be surmounted in a controlled large- $N$ limit. In this limit, we find that the theory has no superconducting instability.We study the problem of disorder-free metals near a continuous quantum critical point. We depart from the standard paradigm [Hertz, Phys. Rev. B 14, 1165 (1976) and Millis, Phys. Rev. B 48, 7183 (1993)] and treat both fermions and bosons (i.e., order parameter fields) on equal footing. We construct a Wilsonian effective field theory that integrates out only high-energy boson and fermion modes. Below the upper critical dimension of the theory ($d=3$ spatial dimensions), we find new fixed points in which the bosons are described by the Wilson-Fisher fixed point and are coupled to a non-Fermi-liquid metal. We describe subtleties with the renormalization group flow of four-Fermi interactions, which can be surmounted in a controlled large-$N$ limit. In this limit, we find that the theory has no superconducting instability.

arXiv:1307.0004Physical Review BSep 10, 2013102 citations

Non-Fermi-liquid behavior of large- NB quantum critical metals

A. Liam Fitzpatrick, Shamit Kachru A, Jared Kaplan A +1 more

The problem of continuous quantum phase transitions in metals involves critical bosons coupled to a Fermi surface. We solve the theory in the limit of a large number, ${N}_{B}$ , of bosonic flavors, where the bosons transform in the adjoint representation (a matrix representation), while the fermions are in the fundamental representation (a vector representation) of a global $SU({N}_{B})$ flavor symmetry group. The leading large ${N}_{B}$ solution corresponds to a non-Fermi liquid coupled to Wilson-Fisher bosons. In a certain energy range, the fermion velocity vanishes---resulting in the destruction of the Fermi surface. Subleading $1/{N}_{B}$ corrections correspond to a qualitatively different form of Landau damping of the bosonic critical fluctuations. We discuss the model in $d=3\ensuremath{-}\ensuremath{\epsilon}$ but because of the additional control afforded by large ${N}_{B}$ , our results are valid down to $d=2$ . In the limit $\ensuremath{\epsilon}\ensuremath{\ll}1$, the large ${N}_{B}$ solution is consistent with the renormalization group analysis of Fitzpatrick et al. [Phys. Rev. B 88, 125116 (2013)].The problem of continuous quantum phase transitions in metals involves critical bosons coupled to a Fermi surface. We solve the theory in the limit of a large number, ${N}_{B}$, of bosonic flavors, where the bosons transform in the adjoint representation (a matrix representation), while the fermions are in the fundamental representation (a vector representation) of a global $SU({N}_{B})$ flavor symmetry group. The leading large ${N}_{B}$ solution corresponds to a non-Fermi liquid coupled to Wilson-Fisher bosons. In a certain energy range, the fermion velocity vanishes---resulting in the destruction of the Fermi surface. Subleading $1/{N}_{B}$ corrections correspond to a qualitatively different form of Landau damping of the bosonic critical fluctuations. We discuss the model in $d=3\ensuremath{-}\ensuremath{\epsilon}$ but because of the additional control afforded by large ${N}_{B}$, our results are valid down to $d=2$. In the limit $\ensuremath{\epsilon}\ensuremath{\ll}1$, the large ${N}_{B}$ solution is consistent with the renormalization group analysis of Fitzpatrick et al. [Phys. Rev. B 88, 125116 (2013)].

arXiv:1312.3321Physical Review BApr 11, 201493 citations

Controlled expansion for certain non-Fermi-liquid metals

David F. Mross, John McGreevy, Hong Liu +1 more

The destruction of Fermi-liquid behavior when a gapless Fermi surface is coupled to a fluctuating gapless boson field is studied theoretically. This problem arises in a number of different contexts in quantum many-body physics. Examples include fermions coupled to a fluctuating transverse gauge field pertinent to quantum spin-liquid Mott insulators, and quantum critical metals near a Pomeranchuk transition. We develop a controlled theoretical approach to determine the low-energy physics. Our approach relies on combining an expansion in the inverse number $(N)$ of fermion species with a further expansion in the parameter $ϵ={z}_{b}\ensuremath{-}2$, where ${z}_{b}$ is the dynamical critical exponent of the boson field. We show how this limit allows a systematic calculation of the universal low-energy physics of these problems. The method is illustrated by studying spinon Fermi-surface spin liquids, and a quantum critical metal at a second-order electronic nematic phase transition. We calculate the low-energy single-particle spectra, and various interesting two-particle correlation functions. In some cases, deviations from the popular random-phase approximation results are found. Some of the same universal singularities are also calculated to leading nonvanishing order using a perturbative renormalization-group calculation at small $N$ extending previous results of Nayak and Wilczek. Implications for quantum spin liquids and for Pomeranchuk transitions are discussed. For quantum critical metals at a nematic transition, we show that the tunneling density of states has a power-law suppression at low energies.The destruction of Fermi-liquid behavior when a gapless Fermi surface is coupled to a fluctuating gapless boson field is studied theoretically. This problem arises in a number of different contexts in quantum many-body physics. Examples include fermions coupled to a fluctuating transverse gauge field pertinent to quantum spin-liquid Mott insulators, and quantum critical metals near a Pomeranchuk transition. We develop a controlled theoretical approach to determine the low-energy physics. Our approach relies on combining an expansion in the inverse number $(N)$ of fermion species with a further expansion in the parameter $ϵ={z}_{b}\ensuremath{-}2$, where ${z}_{b}$ is the dynamical critical exponent of the boson field. We show how this limit allows a systematic calculation of the universal low-energy physics of these problems. The method is illustrated by studying spinon Fermi-surface spin liquids, and a quantum critical metal at a second-order electronic nematic phase transition. We calculate the low-energy single-particle spectra, and various interesting two-particle correlation functions. In some cases, deviations from the popular random-phase approximation results are found. Some of the same universal singularities are also calculated to leading nonvanishing order using a perturbative renormalization-group calculation at small $N$ extending previous results of Nayak and Wilczek. Implications for quantum spin liquids and for Pomeranchuk transitions are discussed. For quantum critical metals at a nematic transition, we show that the tunneling density of states has a power-law suppression at low energies.

arXiv:1003.0894Physical Review BJul 26, 2010297 citations

Quantum critical metals in 4−ε dimensions

Gonzalo Torroba, Huajia Wang

We study the quantum theory of a Fermi surface coupled to a gapless boson scalar in $D=4\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions as a simple model for non-Fermi liquids (NFL) near a quantum phase transition. Our analysis takes into account the full backreaction from Landau damping of the boson, and obtains an RG flow that proceeds through three distinct stages. Above the scale of Landau damping, the Fermi velocity flows to zero, while the coupling evolves according to its classical dimension. Once damping becomes important, its backreaction leads to a crossover regime where dynamic and static damping effects compete and the fermion self-energy does not respect scaling. Below this crossover and having tuned the boson to criticality, the theory flows to a $z=3$ scalar interacting with an NFL. We finally analyze the IR phases of the theory with arbitrary number of flavors ${N}_{c}$ . When ${N}_{c}$ is small, the superconducting dome covers the NFL behavior; strikingly, for moderately large ${N}_{c}$ , we find that NFL effects become important first, before the onset of superconductivity. A generic prediction of the theory is that the Fermi velocity and quasiparticle residue vanish with a power law ${\ensuremath{\omega}}^{\ensuremath{\epsilon}}$ as the fixed point is approached. These features may be useful for understanding some of the phenomenology of high- ${T}_{c}$ materials in a systematic $\ensuremath{\epsilon}$ expansion.We study the quantum theory of a Fermi surface coupled to a gapless boson scalar in $D=4\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions as a simple model for non-Fermi liquids (NFL) near a quantum phase transition. Our analysis takes into account the full backreaction from Landau damping of the boson, and obtains an RG flow that proceeds through three distinct stages. Above the scale of Landau damping, the Fermi velocity flows to zero, while the coupling evolves according to its classical dimension. Once damping becomes important, its backreaction leads to a crossover regime where dynamic and static damping effects compete and the fermion self-energy does not respect scaling. Below this crossover and having tuned the boson to criticality, the theory flows to a $z=3$ scalar interacting with an NFL. We finally analyze the IR phases of the theory with arbitrary number of flavors ${N}_{c}$. When ${N}_{c}$ is small, the superconducting dome covers the NFL behavior; strikingly, for moderately large ${N}_{c}$, we find that NFL effects become important first, before the onset of superconductivity. A generic prediction of the theory is that the Fermi velocity and quasiparticle residue vanish with a power law ${\ensuremath{\omega}}^{\ensuremath{\epsilon}}$ as the fixed point is approached. These features may be useful for understanding some of the phenomenology of high-${T}_{c}$ materials in a systematic $\ensuremath{\epsilon}$ expansion.

arXiv:1406.3029Physical Review BOct 31, 201444 citations

Two-Dimensional Non-Fermi-Liquid Metals: A Solvable Large- N Limit

Jeremías Aguilera Damia, Shamit Kachru A, S. Raghu +1 more

Significant effort has been devoted to the study of "non-Fermi-liquid" (NFL) metals: gapless conducting systems that lack a quasiparticle description. One class of NFL metals involves a finite density of fermions interacting with soft order parameter fluctuations near a quantum critical point. The problem has been extensively studied in a large-N limit (N corresponding to the number of fermion flavors) where universal behavior can be obtained by solving a set of coupled saddle-point equations. However, a remarkable study by Lee revealed the breakdown of such approximations in two spatial dimensions. We show that an alternate approach, in which the fermions belong to the fundamental representation of a global SU(N) flavor symmetry, while the order parameter fields transform under the adjoint representation (a "matrix large-N" theory), yields a tractable large N limit. At low energies, the system consists of an overdamped boson with dynamical exponent z=3 coupled to a non-Fermi-liquid with self-energy Σ(ω)∼ω^{2/3}, consistent with previous studies.Significant effort has been devoted to the study of "non-Fermi-liquid" (NFL) metals: gapless conducting systems that lack a quasiparticle description. One class of NFL metals involves a finite density of fermions interacting with soft order parameter fluctuations near a quantum critical point. The problem has been extensively studied in a large-N limit (N corresponding to the number of fermion flavors) where universal behavior can be obtained by solving a set of coupled saddle-point equations. However, a remarkable study by Lee revealed the breakdown of such approximations in two spatial dimensions. We show that an alternate approach, in which the fermions belong to the fundamental representation of a global SU(N) flavor symmetry, while the order parameter fields transform under the adjoint representation (a "matrix large-N" theory), yields a tractable large N limit. At low energies, the system consists of an overdamped boson with dynamical exponent z=3 coupled to a non-Fermi-liquid with self-energy Σ(ω)∼ω^{2/3}, consistent with previous studies.

arXiv:1905.08256Physical Review LettersAug 29, 201956 citations

Quantum critical metals in d=3+1 dimensions

Raghu Mahajan, David M. Ramirez, Shamit Kachru A +1 more

We study the problem of disorder-free metals in the vicinity of a quantum critical point in $d=3+1$ dimensions. We begin with perturbation theory in the Yukawa coupling between the electrons and undamped bosons (order-parameter fluctuations) and show that the perturbation expansion breaks down below energy scales where the bosons get substantially Landau damped. Above this scale, however, we find a regime in which low-energy fermions obtain an imaginary self-energy that varies linearly with frequency, realizing the ``marginal Fermi liquid'' phenomenology [Varma et al., Phys. Rev. Lett. 63, 1996 (1989)]. We discuss a large- $N$ theory in which the marginal Fermi liquid behavior is enhanced while the role of Landau damping is suppressed, and show that quasiparticles obtain a decay rate parametrically larger than their energy.We study the problem of disorder-free metals in the vicinity of a quantum critical point in $d=3+1$ dimensions. We begin with perturbation theory in the Yukawa coupling between the electrons and undamped bosons (order-parameter fluctuations) and show that the perturbation expansion breaks down below energy scales where the bosons get substantially Landau damped. Above this scale, however, we find a regime in which low-energy fermions obtain an imaginary self-energy that varies linearly with frequency, realizing the ``marginal Fermi liquid'' phenomenology [Varma et al., Phys. Rev. Lett. 63, 1996 (1989)]. We discuss a large-$N$ theory in which the marginal Fermi liquid behavior is enhanced while the role of Landau damping is suppressed, and show that quasiparticles obtain a decay rate parametrically larger than their energy.

arXiv:1303.1587Physical Review BSep 10, 201350 citations

d=3

spatial dimensions), we find new fixed points in which the bosons are described by the Wilson-Fisher fixed point and are coupled to a non-Fermi-liquid metal. We describe subtleties with the renormalization group flow of four-Fermi interactions, which can be surmounted in a controlled large-

N

limit. In this limit, we find that the theory has no superconducting instability.

The problem of continuous quantum phase transitions in metals involves critical bosons coupled to a Fermi surface. We solve the theory in the limit of a large number,

{N}_{B}

, of bosonic flavors, where the bosons transform in the adjoint representation (a matrix representation), while the fermions are in the fundamental representation (a vector representation) of a global

SU({N}_{B})

flavor symmetry group. The leading large

{N}_{B}

solution corresponds to a non-Fermi liquid coupled to Wilson-Fisher bosons. In a certain energy range, the fermion velocity vanishes---resulting in the destruction of the Fermi surface. Subleading

1/{N}_{B}

corrections correspond to a qualitatively different form of Landau damping of the bosonic critical fluctuations. We discuss the model in $d=3\ensuremath{-}\ensuremath{\epsilon}$ but because of the additional control afforded by large

{N}_{B}

, our results are valid down to

d=2

. In the limit $\ensuremath{\epsilon}\ensuremath{\ll}1$, the large

{N}_{B}

solution is consistent with the renormalization group analysis of Fitzpatrick et al. [Phys. Rev. B 88, 125116 (2013)].

(N)

of fermion species with a further expansion in the parameter $ϵ={z}_{b}\ensuremath{-}2$, where

{z}_{b}

is the dynamical critical exponent of the boson field. We show how this limit allows a systematic calculation of the universal low-energy physics of these problems. The method is illustrated by studying spinon Fermi-surface spin liquids, and a quantum critical metal at a second-order electronic nematic phase transition. We calculate the low-energy single-particle spectra, and various interesting two-particle correlation functions. In some cases, deviations from the popular random-phase approximation results are found. Some of the same universal singularities are also calculated to leading nonvanishing order using a perturbative renormalization-group calculation at small

N

extending previous results of Nayak and Wilczek. Implications for quantum spin liquids and for Pomeranchuk transitions are discussed. For quantum critical metals at a nematic transition, we show that the tunneling density of states has a power-law suppression at low energies.

z=3

scalar interacting with an NFL. We finally analyze the IR phases of the theory with arbitrary number of flavors

{N}_{c}

. When

{N}_{c}

is small, the superconducting dome covers the NFL behavior; strikingly, for moderately large

{N}_{c}

, we find that NFL effects become important first, before the onset of superconductivity. A generic prediction of the theory is that the Fermi velocity and quasiparticle residue vanish with a power law ${\ensuremath{\omega}}^{\ensuremath{\epsilon}}$ as the fixed point is approached. These features may be useful for understanding some of the phenomenology of high-

{T}_{c}

materials in a systematic $\ensuremath{\epsilon}$ expansion.

We study the problem of disorder-free metals in the vicinity of a quantum critical point in

d=3+1

dimensions. We begin with perturbation theory in the Yukawa coupling between the electrons and undamped bosons (order-parameter fluctuations) and show that the perturbation expansion breaks down below energy scales where the bosons get substantially Landau damped. Above this scale, however, we find a regime in which low-energy fermions obtain an imaginary self-energy that varies linearly with frequency, realizing the ``marginal Fermi liquid'' phenomenology [Varma et al., Phys. Rev. Lett. 63, 1996 (1989)]. We discuss a large-

N

theory in which the marginal Fermi liquid behavior is enhanced while the role of Landau damping is suppressed, and show that quasiparticles obtain a decay rate parametrically larger than their energy.