DOCSLIB.ORG

Quantum Phase Transitions of Insulators, Superconductors and Metals in Two Dimensions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quantum Phase Transitions of Insulators, Superconductors and Metals in Two Dimensions Quantum phase transitions of insulators, superconductors and metals in two dimensions Talk online: sachdev.physics.harvard.edu HARVARD Friday, April 13, 2012 Outline 1. Phenomenology of the cuprate superconductors (and other compounds) 2. QPT of antiferromagnetic insulators (and bosons at rational filling) 3. QPT of d-wave superconductors: Fermi points of massless Dirac fermions 4. QPT of Fermi surfaces: A. Finite wavevector ordering (SDW/CDW): “Hot spots” on Fermi surfaces B. Zero wavevector ordering (Nematic): “Hot Fermi surfaces” Friday, April 13, 2012 Outline 1. Phenomenology of the cuprate superconductors (and other compounds) 2. QPT of antiferromagnetic insulators (and bosons at rational filling) 3. QPT of d-wave superconductors: Fermi points of massless Dirac fermions 4. QPT of Fermi surfaces: A. Finite wavevector ordering (SDW/CDW): “Hot spots” on Fermi surfaces B. Zero wavevector ordering (Nematic): “Hot Fermi surfaces” Friday, April 13, 2012 Max Metlitski Friday, April 13, 2012 Strategy 1. Write down local field theory for order parameter and fermions 2. Apply renormalization group to field theory Friday, April 13, 2012 Strategy 1. Write down local field theory for order parameter and fermions 2. Apply renormalization group to field theory Friday, April 13, 2012 Order parameter at a non- zero wavevector: “Hot spots” on the Fermi surface. Friday, April 13, 2012 T Strange Fluctuating,Small Fermi Metal pocketspaired Fermi with Large pairingpockets fluctuations Fermi surface d-wave Magnetic quantum SC criticality Thermally fluctuating Spin gap SDW SC+ SDW SC SDW (Small Fermi pockets) M "Normal" (Large Fermi H surface) Friday, April 13, 2012 T Strange Fluctuating,Small Fermi Metal pocketspaired Fermi with Large pairingpockets fluctuations Fermi surface d-wave Theory of SDW Magnetic quantum SC criticality quantum phase Thermally fluctuating Spin gap transition in SDW metal SC+ SDW SC SDW (Small Fermi pockets) M "Normal" (Large Fermi H surface) Friday, April 13, 2012 T Strange Fluctuating,Small Fermi Metal pocketspaired Fermi with Large pairingpockets fluctuations Fermi surface d-wave Theory of SDW Magnetic quantum SC criticality quantum phase Thermally fluctuating Spin gap transition in SDW metal SC+ SDW SC SDW (Small Fermi pockets) M "Normal" (Large Fermi H surface) Friday, April 13, 2012 Hole-doped cuprates Increasing SDW order Γ Γ Γ Γ Hole Electron pockets pockets Large Fermi surface breaks up into electron and hole pockets S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997). Friday, April 13, 2012 Hole-doped cuprates Increasing SDW order Γ Γ Γ ϕ Γ Hole Electron pockets pockets ϕ fluctuations act on the large Fermi surface S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997). Friday, April 13, 2012 Start from the “spin-fermion” model = c ϕ exp ( ) Z D αD −S ∂ = dτ c† ε c S kα ∂τ − k kα k iK r λ dτ c† ϕ σ c e · i − iα i · αβ iβ i 2 1 2 ζ 2 s 2 u 4 + dτd r ( r ϕ ) + (∂τ ϕ ) + ϕ + ϕ 2 ∇ 2 2 4 Friday, April 13, 2012 =3 =2 =4 =1 Low energy fermions ψ1α, ψ2α =1,...,4 = ψ † ζ∂ iv ψ + ψ † ζ∂ iv ψ Lf 1α τ − 1 · ∇r 1α 2α τ − 2 · ∇r 2α v=1 =(v ,v ), v=1 =( v ,v ) 1 x y 2 − x y Friday, April 13, 2012 = ψ † ζ∂ iv ψ + ψ † ζ∂ iv ψ Lf 1α τ − 1 · ∇r 1α 2α τ − 2 · ∇r 2α v1 v2 ψ1 fermions ψ2 fermions occupied occupied Friday, April 13, 2012 = ψ † ζ∂ iv ψ + ψ † ζ∂ iv ψ Lf 1α τ − 1 · ∇r 1α 2α τ − 2 · ∇r 2α v1 v2 “Hot spot” “Cold” Fermi surfaces Friday, April 13, 2012 = ψ † ζ∂ iv ψ + ψ † ζ∂ iv ψ Lf 1α τ − 1 · ∇r 1α 2α τ − 2 · ∇r 2α 1 ζ s u Order parameter: = ( ϕ )2 + (∂ ϕ )2 + ϕ 2 + ϕ 4 Lϕ 2 ∇r 2 τ 2 4 Friday, April 13, 2012 = ψ † ζ∂ iv ψ + ψ † ζ∂ iv ψ Lf 1α τ − 1 · ∇r 1α 2α τ − 2 · ∇r 2α 1 ζ s u Order parameter: = ( ϕ )2 + (∂ ϕ )2 + ϕ 2 + ϕ 4 Lϕ 2 ∇r 2 τ 2 4 “Yukawa” coupling: = λϕ ψ † σ ψ + ψ † σ ψ Lc − · 1α αβ 2β 2α αβ 1β Friday, April 13, 2012 = ψ † ζ∂ iv ψ + ψ † ζ∂ iv ψ Lf 1α τ − 1 · ∇r 1α 2α τ − 2 · ∇r 2α 1 ζ s u Order parameter: = ( ϕ )2 + (∂ ϕ )2 + ϕ 2 + ϕ 4 Lϕ 2 ∇r 2 τ 2 4 “Yukawa” coupling: = λϕ ψ † σ ψ + ψ † σ ψ Lc − · 1α αβ 2β 2α αβ 1β Hertz-Moriya-Millis (HMM) theory Integrate out fermions and obtain non-local corrections to Lϕ 1 2 2 2 ϕ = ϕ q + γ ω /2; γ = L 2 | | πvxvy Exponent z = 2 and mean-field criticality (upto logarithms) Friday, April 13, 2012 = ψ † ζ∂ iv ψ + ψ † ζ∂ iv ψ Lf 1α τ − 1 · ∇r 1α 2α τ − 2 · ∇r 2α 1 ζ s u Order parameter: = ( ϕ )2 + (∂ ϕ )2 + ϕ 2 + ϕ 4 Lϕ 2 ∇r 2 τ 2 4 “Yukawa” coupling: = λϕ ψ † σ ψ + ψ † σ ψ Lc − · 1α αβ 2β 2α αβ 1β Hertz-Moriya-Millis (HMM) theory Integrate out fermions and obtain non-local corrections to Lϕ 1 2 2 2 ϕ = ϕ q + γ ω /2; γ = L 2 | | πvxvy Exponent z = 2 and mean-field criticality (upto logarithms) But, higher order terms contain an infinite number of marginal couplings . Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004). Friday, April 13, 2012 = ψ † ζ∂ iv ψ + ψ † ζ∂ iv ψ Lf 1α τ − 1 · ∇r 1α 2α τ − 2 · ∇r 2α 1 ζ s u Order parameter: = ( ϕ )2 + (∂ ϕ )2 + ϕ 2 + ϕ 4 Lϕ 2 ∇r 2 τ 2 4 “Yukawa” coupling: = λϕ ψ † σ ψ + ψ † σ ψ Lc − · 1α αβ 2β 2α αβ 1β Perform RG on both fermions and ϕ , using a local field theory. Friday, April 13, 2012 Order parameter at zero wavevector: “Hot Fermi surface”. Friday, April 13, 2012 T T* Strange Fluctuating,Fluctuating, Metal pairedpaired FermiFermi Large pocketspockets Fermi surface Tc d-wave Tsdw SC VBS and/or Fluctuating, Ising nematic order paired Fermi pockets SC+ SDW SC Hc2 SDW M Insulator SDW (Small Fermi "Normal" pockets) (Large Fermi H 22 surface) Friday, April 13, 2012 Broken rotational symmetry in the pseudogap phase of a high-Tc superconductor R. Daou, J. Chang, David LeBoeuf, Olivier Cyr- Choiniere, Francis Laliberte, Nicolas Doiron- Leyraud, B. J. Ramshaw, Ruixing Liang, D. A. Bonn, W. N. Hardy, and Louis Taillefer arXiv: 0909.4430, Nature, in press Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability y x Fermi surface with full square lattice symmetry Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability y x Spontaneous elongation along x direction: Ising order parameter φ > 0. Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability y x Spontaneous elongation along y direction: Ising order parameter φ < 0. Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability φ =0 φ =0 λ λc Pomeranchuk instability as a function of coupling λ Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability T Quantum critical Tc φ =0 φ =0 λ λc Phase diagram as a function of T and λ Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability T Quantum critical Tc Classical d=2 Ising criticality φ =0 φ =0 λ λc Phase diagram as a function of T and λ Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability T Quantum critical Tc φ =0 φ =0 D=2+1 quantum λ criticalityλc ? Phase diagram as a function of T and λ Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability Effective action for Ising order parameter = d2rdτ (∂ φ)2 + c2( φ)2 +(λ λ )φ2 + uφ4 Sφ τ ∇ − c Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability Effective action for Ising order parameter = d2rdτ (∂ φ)2 + c2( φ)2 +(λ λ )φ2 + uφ4 Sφ τ ∇ − c Effective action for electrons: Nf = dτ c† ∂ c t c† c Sc iα τ iα − ij iα iα α=1 i i<j Nf dτc† (∂ + ε ) c ≡ kα τ k kα α=1 k Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability Coupling between Ising order and electrons Nf = γ dτφ (cos k cos k )c† c Sφc − x − y kα kα α=1 k for spatially independent φ φ > 0 φ < 0 Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability Coupling between Ising order and electrons Nf φc = γ dτ φq (cos kx cos ky)c† ck q/2,α S − − k+q/2,α − α=1 k,q for spatially dependent φ φ > 0 φ < 0 Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability = d2rdτ (∂ φ)2 + c2( φ)2 +(λ λ )φ2 + uφ4 Sφ τ ∇ − c Nf = dτc† (∂ + ε ) c Sc kα τ k kα α=1 k Nf φc = γ dτ φq (cos kx cos ky)c† ck q/2,α S − − k+q/2,α − α=1 k,q Friday, April 13, 2012 A φ fluctuation at wavevector q couples most efficiently to fermions near k . ± 0 Expand fermion kinetic energy at wavevectors about k0 Friday, April 13, 2012 2 2 = ψ+† ζ∂τ i∂x ∂y ψ+ + ψ† ζ∂τ + i∂x ∂y ψ L − − − − − 1 2 r 2 λφ ψ+† ψ+ + ψ† ψ + (∂yφ) + φ − − − 2g 2 Friday, April 13, 2012 Emergent “Galilean invariance” at low energy (s = ): ± 2 is( θ y+ θ x) φ(x, y) φ(x, y + θx), ψ (x, y) e− 2 4 ψ (x, y + θx) → s → s which implies for the fermion Green’s function 2 G(qx,qy)=G(sqx + qy).
Load more

Recommended publications
  • The University of Chicago Subgap Collective Modes In
    THE UNIVERSITY OF CHICAGO SUBGAP COLLECTIVE MODES IN TWO-DIMENSIONAL SUPERFLUID NEAR THE POMERANCHUK INSTABILITIES A DISSERTATION SUBMITTED TO THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF PHYSICS BY WEI-HAN HSIAO CHICAGO, ILLINOIS JUNE 2020 Copyright c 2020 by Wei-Han Hsiao All Rights Reserved TABLE OF CONTENTS LIST OF FIGURES . v ACKNOWLEDGMENTS . vi ABSTRACT . viii 1 INTRODUCTION . 1 1.1 The Statement of the Problem . 1 1.2 The Background . 2 1.2.1 The Fractional Quantum Hall Effect . 2 1.2.2 Superfluids of Paired Composite Fermions . 5 1.2.3 The Pomeranchuk Instability . 9 1.3 Main Contribution . 12 1.4 Plan of the thesis . 13 2 KINETIC THEORY AND COLLECTIVE MODES . 14 2.1 Introduction . 14 2.2 General kinetic theory . 14 2.2.1 Spin-singlet pairing . 19 2.2.2 spin-triplet pairing . 20 2.3 Collective Modes . 21 2.3.1 s-wave pairing . 22 2.3.2 p-wave pairing . 22 2.3.3 d-wave pairing . 26 2.3.4 Higher L chiral ground states . 26 2.4 Conclusion . 27 3 FERMI LIQUID CORRECTIONS . 28 3.1 Introduction . 28 3.2 Mass Renormalization by the Landau Parameters . 28 3.2.1 Massless modes . 28 3.2.2 Massive subgap Modes . 30 3.3 Conclusion . 34 4 FIELD-THEORY MODEL FOR CHIRAL P -WAVE SUPERFLUID . 35 4.1 Introduction . 35 4.2 Method of Effective Action . 36 4.3 The Model for the p-Wave Superfluid near the Nematic Critical Point .
    [Show full text]
  • Approaching Pomeranchuk Instabilities from Ordered Phase: a Crossing-Symmetric Equation Method
    Approaching Pomeranchuk Instabilities from Ordered Phase: A Crossing-symmetric Equation Method Kelly Reidya, Khandker Quadera,∗, Kevin Bedellb aDepartment of Physics, Kent State University, Kent, OH 44242, USA bDepartment of Physics, Boston College, Cestnut Hill, MA, USA Abstract We explore features of a 3D Fermi liquid near generalized Pomeranchuk insta- bilities using a tractable crossing symmetric equation method. We approach the instabilities from the ordered ferromagnetic phase. We find \quantum multi-criticality" as approach to the ferromagnetic instability drives instabil- ity in other channel(s). It is found that a charge nematic instability precedes and is driven by Pomeranchuk instabilities in both the ` = 0 spin and density channels. Keywords: Crossing-symmetric equation, Pomeranchuk instabilities, ordered phase, multi-critical behavior, nematic instability 1. Dedication (by Khandker Quader) \We are like dwarfs on the shoulders of giants. So that we can see more than they, And things at a greater distance, Not by virtue of any sharpness of sight on our part, Or any physical distinction, But because we are carried high, And raised up by their giant size" (Bernard of Chartres/John Salisbury (12th century); Isaac Newton (17th century)) arXiv:1404.5655v1 [cond-mat.str-el] 22 Apr 2014 Much of the many-fermion physics that treats short-range underlying interac- tion and longer-range quantum fluctuations on the same footing are rooted in the ∗Corresponding author: Khandker Quader, [email protected] Preprint submitted to Nuclear Physics A September 11, 2018 bold, seminal ideas of Gerry that some of us, as his students, had the great for- tune of learning first-hand from him.
    [Show full text]
  • Pomeranchuk Instability of Composite Fermi Liquids
    PHYSICAL REVIEW LETTERS 121, 147601 (2018) Editors' Suggestion Pomeranchuk Instability of Composite Fermi Liquids Kyungmin Lee,1 Junping Shao,2 Eun-Ah Kim,3,* F. D. M. Haldane,4 and Edward H. Rezayi5 1Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA 2Department of Physics, Binghamton University, Binghamton, New York 13902, USA 3Department of Physics, Cornell University, Ithaca, New York 14853, USA 4Department of Physics, Princeton University, Princeton, New Jersey, USA 5Department of Physics, California State University Los Angeles, Los Angeles, California 90032, USA (Received 26 February 2018; published 3 October 2018) Nematicity in quantum Hall systems has been experimentally well established at excited Landau levels. The mechanism of the symmetry breaking, however, is still unknown. Pomeranchuk instability of Fermi liquid parameter Fl ≤−1 in the angular momentum l ¼ 2 channel has been argued to be the relevant mechanism, yet there are no definitive theoretical proofs. Here we calculate, using the variational Monte Carlo technique, Fermi liquid parameters Fl of the composite fermion Fermi liquid with a finite layer width. We consider Fl in different Landau levels n ¼ 0, 1, 2 as a function of layer width parameter η. We find that unlike the lowest Landau level, which shows no sign of Pomeranchuk instability, higher Landau levels show nematic instability below critical values of η. Furthermore, the critical value ηc is higher for the n ¼ 2 Landau level, which is consistent with observation of nematic order in ambient conditions only in the n ¼ 2 Landau levels. The picture emerging from our work is that approaching the true 2D limit brings half-filled higher Landau-level systems to the brink of nematic Pomeranchuk instability.
    [Show full text]
  • Pomeranchuk Instability and Bose Condensation of Scalar Quanta in A
    Pomeranchuk instability and Bose condensation of scalar quanta in a Fermi liquid E. E. Kolomeitsev1 and D. N. Voskresensky2 1Matej Bel University, SK-97401 Banska Bystrica, Slovakia 2 National Research Nuclear University (MEPhI), 115409 Moscow, Russia (Dated: October 14, 2018) We study excitations in a normal Fermi liquid with a local scalar interaction. Spectrum of bosonic scalar-mode excitations is investigated for various values and momentum dependence of the scalar Landau parameter f0 in the particle-hole channel. For f0 > 0 the conditions are found when the phase velocity on the spectrum of the zero sound acquires a minimum at a non-zero momentum. For −1 < f0 < 0 there are only damped excitations, and for f0 < −1 the spectrum becomes unstable against a growth of scalar-mode excitations (a Pomeranchuk instability). An effective Lagrangian for the scalar excitation modes is derived after performing a bosonization procedure. We demonstrate that the Pomeranchuk instability may be tamed by the formation of a static Bose condensate of the scalar modes. The condensation may occur in a homogeneous or inhomogeneous state relying on the momentum dependence of the scalar Landau parameter. Then we consider a possibility of the condensation of the zero-sound-like excitations in a state with a non-zero momentum in Fermi liquids moving with overcritical velocities, provided an appropriate momentum dependence of the Landau parameter f0(k) > 0. PACS numbers: 21.65.-f, 71.10.Ay, 71.45.-d Keywords: Nuclear matter, Fermi liquid, sound-like excitations, Pomeranchuk instability, Bose condensation I. INTRODUCTION density fluctuations. In hydrodynamic terms the condi- tion f < 1 implies that the speed of the first sound 0 − The theory of normal Fermi liquids was built up by becomes imaginary.
    [Show full text]
  • Pomeranchuk Instability and Response Beyond the Quasiparticle Regime
    Pomeranchuk instabilities: Topic for Journal club discussion. Prachi Sharma. Outline • Introduction:- Fermi liquid theory and Pomeranchuk instabilities. • Motivation. • Ward Identities/conservation principles and continuity equations to determine the response function for generic order parameter. • Implication on various conserved and non conserved order • Conclusions. Fermi liquid theory(brief idea) • In fermi liquid theory, the energy change due to quasiparticle excitations is given as, with the single particle energy is, where the interaction function for the spin-isotropic fermi liquid is Pomeranchuk instability. • Pomeranchuk defined a condition of stability in FL as • Instabilities comes due to the deformation of the fermi surface as when we approach this threshold value of • The energy change is given as, So therefore if becomes negative one can find a energetically favored state with finite order parameter, by breaking some symmetry in that particular l channel. • Also the instabilities can be shown with quasiparticle susceptibility being divergent at the threshold value • For l=0 channel we get the ferromagnetic instability. Motivation. The pomeranchuk criteria only incorporated the contributions to susceptibility from quasiparticles, but this is not the full susceptibility. Main idea of my talk is to address the instabilities in l=1 channel for spin symmetric and spin antisymmetric sectors based on the conservation laws and continuity equation. • l=1 spin-symmetric (charge) Pomeranchuk instability is forbidden more generally by gauge- invariance for Galilean invariant system. 1. But what about instability in the spin-antisymmetric channel for l=1 in a non-rel case? 2. If there is no instability in spin sector for l=1, how can this be implied from the fermi liquid picture? 3.
    [Show full text]
  • Fermi Liquid Instabilities in the Spin Channel
    SLAC-PUB-13919 Fermi liquid instabilities in the spin channel Congjun Wu,1 Kai Sun,2 Eduardo Fradkin,2 and Shou-Cheng Zhang3 1Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106-4030 2Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080 3Department of Physics, McCullough Building, Stanford University, Stanford, California 94305-4045 (Dated: February 6, 2008) We study the Fermi surface instabilities of the Pomeranchuk type in the spin triplet channel with a high orbital partial waves (Fl (l > 0)). The ordered phases are classified into two classes, dubbed the α and β-phases by analogy to the superfluid 3He-A and B-phases. The Fermi surfaces in the α-phases exhibit spontaneous anisotropic distortions, while those in the β-phases remain circular or spherical with topologically non-trivial spin configurations in momentum space. In the α-phase, the Goldstone modes in the density channel exhibit anisotropic overdamping. The Goldstone modes in the spin channel have nearly isotropic underdamped dispersion relation at small propagating wavevectors. Due to the coupling to the Goldstone modes, the spin wave spectrum develops resonance peaks in both the α and β-phases, which can be detected in inelastic neutron scattering experiments. In the p-wave channel β-phase, a chiral ground state inhomogeneity is spontaneously generated due to a Lifshitz-like instability in the originally nonchiral systems. Possible experiments to detect these phases are discussed. PACS numbers: 71.10.Ay, 71.10.Ca, 05.30.Fk I. INTRODUCTION the electron fluid behaves, from the point of view of its symmetries and of their breaking, very much like an elec- tronic analog of liquid crystal phases [5, 6].
    [Show full text]
  • Nematic Quantum Phase Transition of Composite Fermi Liquids in Half-Filled
    Nematic quantum phase transition of composite Fermi liquids in half-filled Landau levels and their geometric response Yizhi You,1, 2 Gil Young Cho,3, 1 and Eduardo Fradkin1 1Department of Physics and Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, Illinois 61801-3080, USA 2Kavli Institute for Theoretical Physics, University of California Santa Barbara, Santa Barbara, California 93106 3Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea (Dated: May 4, 2016) We present a theory of the isotropic-nematic quantum phase transition in the composite Fermi liquid arising in half-filled Landau levels. We show that the quantum phase transition between the isotropic and the nematic phase is triggered by an attractive quadrupolar interaction between electrons, as in the case of conventional Fermi liquids. We derive the theory of the nematic state and of the phase transition. This theory is based on the flux attachment procedure which maps an electron liquid in half-filled Landau levels into the composite Fermi liquid close to a nematic transition. We show that the local fluctuations of the nematic order parameters act as an effective dynamical metric interplaying with the underlying Chern-Simons gauge fields associated with the flux attachment. Both the fluctuations of the Chern-Simons gauge field and the nematic order parameter can destroy the composite fermion quasiparticles and drive the system into a non-Fermi liquid state. The effective field theory for the isotropic-nematic phase transition is shown to have z = 3 dynamical exponent due to the Landau damping of the dense Fermi system.
    [Show full text]
  • Pomeranchuk Instability in a Non-Fermi Liquid from Holography
    UTTG-24-11 Pomeranchuk Instability in a non-Fermi Liquid from Holography Mohammad Edalati,1 Ka Wai Lo,2 and Philip W. Phillips2 1Weinberg Theory Group, University of Texas at Austin, Austin TX 78712, USA 2Department of Physics, University of Illinois at Urbana-Champaign, Urbana IL 61801, USA The Pomeranchuk instability, in which an isotropic Fermi surface distorts and becomes anisotropic due to strong interactions, is a possible mechanism for the growing number of experimental systems which display transport properties that differ along the x and y axes. We show here that the gauge- gravity duality can be used to describe such an instability in fermionic systems. Our holographic model consists of fermions in a background which describes the causal propagation of a massive neutral spin-two field in an asymptotically AdS spacetime. The Fermi surfaces in the boundary the- ory distort spontaneously and become anisotropic once the neutral massive spin-two field develops a normalizable mode in the bulk. Analysis of the fermionic correlators reveals that the low-lying fermionic excitations are non-Fermi liquid-like both before and after the Fermi surface shape distor- tion. Further, the spectral weight along the Fermi surface is angularly dependent and can be made to vanish along certain directions. I. INTRODUCTION To a surprising extent, the electronic properties of most metals are well described by Landau Fermi liq- uid theory. The key building blocks of this theory are quasi-particles and a Fermi surface, the surface in mo- mentum space that demarcates filled from empty states. For a system of weakly interacting electrons, the shape (a) (b) (c) of the Fermi surface is isotropic.
    [Show full text]
  • Collective Modes Near a Pomeranchuk Instability in Two Dimensions
    PHYSICAL REVIEW RESEARCH 1, 033134 (2019) Collective modes near a Pomeranchuk instability in two dimensions Avraham Klein ,1 Dmitrii L. Maslov,2 Lev P. Pitaevskii,3,4 and Andrey V. Chubukov1 1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA 2Department of Physics, University of Florida, P.O. Box 118440, Gainesville, Florida 32611-8440, USA 3INO-CNR BEC Center and Dipartimento di Fisica, Università degli Studi di Trento, 38123 Povo, Italy 4Kapitza Institute for Physical Problems, Russian Academy of Sciences, Moscow, 119334, Russia (Received 15 August 2019; revised manuscript received 3 October 2019; published 27 November 2019) We consider zero-sound collective excitations of a two-dimensional Fermi liquid. For each value of the angular momentum l, we study the evolution of longitudinal and transverse collective modes in the charge (c)andspin c(s) c(s) (s) channels with the Landau parameter Fl , starting from positive Fl and all the way to the Pomeranchuk c(s) =− transition at Fl 1. In each case, we identify a critical zero-sound mode, whose velocity vanishes at the c(s) < − Pomeranchuk instability. For Fl 1, this mode is located in the upper frequency half-plane, which signals an instability of the ground state. In a clean Fermi liquid, the critical mode may be either purely relaxational or almost propagating, depending on the parity of l and on whether the response function is longitudinal or transverse. These differences lead to qualitatively different types of time evolution of the order parameter following an initial perturbation. A special situation occurs for the l = 1 order parameter that coincides with the spin or charge current.
    [Show full text]
  • Fermi-Liquid Theory and Pomeranchuk Instabilities: Fundamentals and New Developments
    Fermi-liquid theory and Pomeranchuk instabilities: fundamentals and new developments Andrey V. Chubukov1, Avraham Klein1, and Dmitrii L. Maslov2 1Department of Physics, University of Minnesota, 116 Church Street, Minneapolis, MN 55455 2Department of Physics, University of Florida, P. O. Box 118440, Gainesville, FL 32611-8440 (Dated: May 7, 2019) This paper is a short review on the foundations and recent advances in the microscopic Fermi- liquid (FL) theory. We demonstrate that this theory is built on five identities, which follow from conservation of total charge (particle number), spin, and momentum in a translationally and SU(2)- invariant FL. These identities allows one to express the effective mass and quasiparticle residue in terms of an exact vertex function and also impose constraints on the \quasiparticle" and "incoherent" (or \low-energy" and \high-energy") contributions to the observable quantities. Such constraints forbid certain Pomeranchuk instabilities of a FL, e.g., towards phases with order parameters that coincide with charge and spin currents. We provide diagrammatic derivations of these constraints and of the general (Leggett) formula for the susceptibility in arbitrary angular momentum channel, and illustrate the general relations through simple examples treated in the perturbation theory. It is our great pleasure to write this article for the excitations vF is replaced by the effective Fermi velocity ∗ special volume in celebration of the 85th birthday of Lev vF (or, equivalently, the fermionic mass m = pF =vF is ∗ ∗ Petrovich Pitaevskii, who, in our view, is one the greatest replaced by the effective mass m = pF =vF ) and ii) the physicist of his generation and the model of a scientist wave function of a state near the FS in an interacting and citizen.
    [Show full text]
  • Nematic Quantum Criticality
    Nematic quantum criticality Walter Metzner 1. Introduction: Nematic quantum criticality and non-Fermi liquid π repulsive 2. Nematic lattice model 0 attractive 0.16 1st order transition −π 2nd order transition −π 0 π 0.14 1st order (MFT) 2nd order (MFT) 0.12 0.1 3. Turning a first order transition continuous T 0.08 0.06 0.04 0.02 0 -0.85 -0.8 -0.75 -0.7 -0.65 -0.6 -0.55 -0.5 -0.45 µ (0,π) T=0.15 1.5 ky T=0.20 T=0.30 (0,0) kx (π,0) T=0.40 ,ω) 1 Tc=0.14 k 4. Fermi surface truncation from thermal fluctuations Α( 0.5 0 q -1.5 -1 -0.5 0 0.5 1 1.5 1 ω k−p k−p 1 2 q q N 2 5. Validity of Hertz action at QCP? k−p N k−p3 q 3 k−p 4 Coworkers: Luca Dell’Anna (Padua) Hiroyuki Yamase (Tsukuba) Pawel Jakubczyk (Warsaw) Stephan Thier (Mainz) 1. Introduction Quantum phase transitions: Phase transition at T = 0 driven by control parameter δ (pressure, density, ...) T If order parameter vanishes quantum continuously at transition: critical quantum critical point. Quantum fluctuations lead to ordered quantum disordered unusual physical properties. QCP δ Quantum phase transitions in metals: non-Fermi liquid behavior in quantum critical regime d-wave Pomeranchuk instability – nematic transition π π Spontaneous breaking of tetragonal symmetry free free generated by d-wave 0 deformed 0 deformed forward scattering −π −π Halboth, wm ’00 −π π−π π 0 0 Yamase, Kohno ’00 (a) (b) P Order parameter nd = k dkhnki where dk = cos kx − cos ky Realization of ”nematic” electron liquid (→ Kivelson et al.
    [Show full text]
  • The Conditions for $ L= 1$ Pomeranchuk Instability in a Fermi
    The Conditions for l = 1 Pomeranchuk Instability in a Fermi Liquid Yi-Ming Wu, Avraham Klein, and Andrey V. Chubukov School of Physics and Astronomy, University of Minnesota, MN (Dated: January 23, 2018) We perform a microscropic analysis of how the constraints imposed by conservation laws affect q = 0 Pomeranchuk instabilities in a Fermi liquid. The conventional view is that these instabilities are determined by the static interaction between low-energy quasiparticles near the Fermi surface, in the limit of vanishing momentum transfer q. The condition for a Pomeranchuk instability is set c(s) c(s) by Fl = −1, where Fl (a Landau parameter) is a properly normalized partial component of the anti-symmetrized static interaction F (k; k + q; p; p − q) in a charge (c) or spin (s) sub-channel with angular momentum l. However, it is known that conservation laws for total spin and charge prevent Pomeranchuk instabilities for l = 1 spin- and charge- current order parameters. Our study aims to understand whether this holds only for these special forms of l = 1 order parameters, or is a more generic result. To this end we perform a diagrammatic analysis of spin and charge susceptibilities for charge and spin density order parameters, as well as perturbative calculations to second order in the Hubbard U. We argue that for l = 1 spin-current and charge-current order parameters, certain c(s) vertex functions, which are determined by high-energy fermions, vanish at Fl=1 = −1, preventing a Pomeranchuk instability from taking place. For an order parameter with a generic l = 1 form- c(s) factor, the vertex function is not expressed in terms of Fl=1 , and a Pomeranchuk instability does c(s) occur when F1 = −1.
    [Show full text]