Wilsonian and large N theories of quantum critical metals Srinivas Raghu (Stanford)
Collaborators and References R. Mahajan, D. Ramirez, S. Kachru, and SR, PRB 88, 115116 (2013). A. Liam Fitzpatrick, S. Kachru, J. Kaplan, and SR, PRB 88, 125116 (2013). A. Liam Fitzpatrick, S. Kachru, J. Kaplan, and SR, PRB (2014). A. Liam Fitzpatrick, S. Kachru, J. Kaplan, and SR, to appear. With Liam Fitzpatrick, Jared Kaplan, Shamit Kachru
Breakdown of fermion quasiparticles A recurring theme: Fermi liquid theory breaks down at a quantum phase transition. NFL emanates from a critical point at T=0. NFL can give way to higher Tc superconductivity. QCPs in metals: wide-open problem especially in d=2+1. NFL
BaFe2(As1-xPx)2 Example: Iron pnictides EVOLUTION FROM NON-FERMI- TO FERMI-LIQUID ρ xx ( µω cm) 200 100 R H (10-3 cm 3 /C) 3 2.5 2 x=0.33 1.5 0 50 100 T (K) ~ T 1.0 ~ T 1.2 ~ T 1.7 ~ T 1.9 =0.71 2.0x ~ T 0 0 50 100 150 T (K) x =0.33 x =0.41 x =0.56 x =0.64 (a) ρ xx (H)/ρ xx 0.015 0.01 0.005 40 K 50 K 60 K 80 K 100 K ρ xx (H)/ρ xx 0.01 0 0 0.001 tan (b) 2 Θ 0 H 0 0.1 0.2 (µ 0 H/ρ xx ) 2 (T 2 /(µω cm) 2 ) FIG. 3. Color online a Normal-state xx T for x=0.33, 0.41, 0.56, 0.64, and 0.71 at low temperatures can be fitted by the power Maximum superconducting Tc law Eq. 1. The inset shows T dependence of R H T for x below the NFL. =0.33. Solid line is a fit to the data by R H T =C 1 /T+C 2 with C 1 =0.048 Kcm 3 /C and C 2 =1.5 10 3 cm 3 /C. b Magnetoresistanceout xx of H / a xx plotted as a function of 0 H/ xx 2 for x=0.33. Superconductivity forms non-fermi liquid. The inset shows xx H / xx plotted as a function tan 2 H. FIG. 5. Evidence for the QCP in the superconducting dome in Talk by Prof. Shibauchi in the symposium. BaFe 2 (As 1 x P x ) 2. (a) Temperature dependence of in-plane resistivity ρ xx for 0.33 0.71 [50]. [PNAS The2009, red lines PRB 2010] are the fit of normal-state ρ xx (T )topower-lawdependenceρ 0 + AT α FIG. 5. Evidence for the QCP in the superconducting dome in t r o g n a h K d w s t c w c N
Fermi liquid Concrete model system Via Pomeranchuk Via meltingbreaking of point group symmetry. Ising nematic transition: instability stripes 2 possible ground states spin up spin down Nematic phases Figure 1 Nematic phases Via Pomeranchuk instability Fermi liquid Pomeranchuk instability Fermi liquid Fermi liquids: Analogy with classical liquid crystals Two different mechanisms for producing a nematic phase with point particles: The gentle melting of a Via Pomeranchuk t Alternatively, a nematic Fermi fluid can arise through the distortion of the Fermi surface of a metal via stripe phase can restore long-range translational symmetry while preserving orientational order (8). instability a Pomeranchuk instability (27, 28). (After, in part, Reference 8.) order Smectic or stripe phase c e Smectic or stripe phase parameter: t+ Via melting stripes thermal melting of a stripe state to form a nematic fluid is readily understood theoretically and, indeed, the resulting description is similar to the theory of the nearly Nematic phases (9, 18, 29 31)Nematic Isotropic smectic nematic fluid that has been developed in the context of complex classical fluids (2, 32). Within this perspective, the nematic state arises from the proliferation of dislocations, the topological defects of the stripe state. This can take place either via a thermal The nematic state preserves lattice translation symmetry. phase transition (as in the standard classical case) or as a quantum phase transition. Whereas the thermal phase transition is well understood (2, 32), the theory of the quantum smectic-nematic phase transition by a dislocation proliferation mechanism is largely an
Effective theory: Fermion-boson problem Starting UV action: S = S + S + S S S S Landau Fermi liquid Landau-Ginzburg-Wilson theory for order parameter. Fermion-boson Yukawa coupling Obtaining such an action: Start with electrons strongly interacting ( Hubbard model ). Integrate out high energy modes from lattice scale down to a new UV cutoff << E F. = Scale below which we can linearize the fermion Kinetic energy.
Effective theory: Fermion-boson problem Starting UV action (in imaginary time): een bosons and fermions: S = d d d x L = S + S + S L = [ + µ (i )] + 2 L = m 2 2 +( ) 2 + c 2 + 4! 4 d d+1 kd d+1 q S, = (2 ) g(k, q) (k) (k + q) (q), 2(d+1) Fermions bosons Yukawa coupling Ising nematic theory: g(k, q) =g (cos k x cos k y ). g=0: decoupled limit (Fermi liquid + ordinary critical point). non-zero g: complex tug-of-war between bosons and fermions.
Tug-of-war between bosons and fermions Non-zero g: Bosons can decay into particle-hole continuum -> overdamped bosons. a ab ab + b Non-zero g: Quasiparticle scattering enhanced due to bosons. ab a b a + q.p. Scattering rate can exceed its energy. Fermion propagators: poles become branch cuts. Result: breakdown of Landau quasiparticle. How to proceed???
Large N limits
Large N limits Essence of the problem: dissipative coupling between bosons and fermions. Large N limits: particles with many (N) flavors act as a dissipative bath while remaining degrees of freedom become overdamped. e.g. Large number of fermion flavors (Nf). Boson can decay in many channels -> Overdamped bosons (NFL is subdominant). Mainstream (Hertz) theory captures the IR behavior in this regime. e.g. Large number of boson flavors (Nb). Fermion can decay in many channels -> NFL is strongest effect (boson damping is subdominant).
Large N limits Large NF: O(1/N F ): Large NB: O(1/N B ):
Implementation of large N limits! i! i =1 N F i, j =1 N B! j i g! g i j j i (repeated indices summed). I will consider the case: N F =1,N B!1.
Large NB action L = i [@ + µ (ir)] i + i NB L =tr m 2 2 2 +(@ ) 2 + c 2 r ~ i j j (1) + tr( 4 )+ 8N B L, = g p NB i j j i (2) 8N 2 B (tr( 2 )) 2 i, j =1 N B Impose an SO(NB 2 ) symmetry: (1) =0 This symmetry is softly broken: i.e., only at O(1/N 2 B).
N B!1: = Large NB solution Properties of the solution: G(k,!) = 1) Fermi velocity vanishes at infinite NB. 2) Green function has branch cut spectrum. 3) Damping of order parameter is a 1/NB effect. =3 d 1! 1 /2 f! k ; N B! f k ; N B!1 =1 The solution matches on to perturbation theory in the UV. The theory can smoothly be extended to d=2. The theory describes infinitely heavy, incoherent fermionic quasiparticles. We are currently investigating the strong CDW and superconducting instabilities of this system.
Scaling landscape N B Our theory?? Real materials Hertz N F Moral of the story: there may be several distinct asymptotic limits with different scaling behaviors, dynamic crossovers in this problem.
Wilsonian RG analysis
Scaling near the upper-critical dimension UV theory: decoupled Fermi liquid+ nearly free bosons (g=0). (a) q y (b) empty states q y Scaling must contend with vastly different kinematics of bosons and fermions. (c) q x q k k + q filled states empty states filled states q x Fermions: low energy = Fermi surface. -> anisotropic scaling. Bosons: low energy = point in k-space. -> isotropic scaling. }Scale to preserve kinetic terms. [g] = 1 (3 d) 2 Result: d=3 is the upper critical dimension.
Renormalization group analysis Integrate out modes with energy Integrate out modes with momenta e t <E< k e t <k< k k / = UV cutoff: scale below which fermion dispersion can be linearized (with a well-defined Fermi velocity). Following Wilson, we will integrate out only highenergy modes to obtain RG flows. This is a radical departure from the standard approach to this problem. K. G. Wilson
Renormalization group analysis RG flows at one-loop: =3 d 4 term : d dt = a 2 a>0 g term : Fermi velocity: dg dt = 2 g dv bg3 b>0 dt = cg2 S(v) S(v) sgn(v) Naive fixed point: = O( ) g = O( p ) v =0
Properties of the naive fixed point v/c: vanishes before the system reaches the fixed point! This feature shuts down Landau damping. Fermion 2-pt function takes the form: G(!,k)= 1! 1 2 f! k? f(x) = scaling function Consistent with the large NB solution. Is this too much of a good thing?? Infinitely heavy, incoherent fermions + non-mean-field critical exponents!
Introducing leading irrelevant couplings (k) µ = v` + w`2 + w ~ band curvature RG flow equations dv dt = cg2 S(v) S(v) sgn(v) dw dt = w w cannot be neglected below an emergent energy scale: µ e v 0/g 2 0, O(1) w is dangerously irrelevant We don t know what happens below this scale (Lifshitz transition?)
Current and future work There are log-squared divergences in the Cooper channel in the vicinity of the quantum critical point. This reflects a much stronger superconducting tendency! Break inversion and time-reversal symmetry: these effects are gone. More detailed, systematic treatment is in progress. Related work: Metlitski et al. 1403.3694. Goal: to demonstrate enhanced superconductivity out of a non-fermi liquid.
Summary and outlook We studied a metal near a nematic quantum critical point and found non-fermi liquid phenomena via 1) large N and 2)RG methods. Both methods produce consistent results. The fixed point corresponds to an infinitely heavy incoherent soup of fermions + order parameter fluctuations. This fixed point is unstable, but it governs scaling laws over a broad range of energy/temperature scales. We are currently investigating experimental properties (eg. heat capacity) and superconducting/cdw instabilities in this regime.
Appendix
Scaling analysis Fermions: only momentum component normal to Fermi surface scale with energy:! 0 = e t!, k 0 F = k F, `0 = e t` ~ k ~` Fermi surface BCS coupling: 0 = ~ kf Bosons: all components of momentum scale with energy: ~q! 0 = e t!, k 0 = e t k quartic term: 0 = e (3 d)t
Scaling analysis Boson-fermion coupling: For small momentum transfer, the coupling is marginal in d=3. ~q Fermi surface ~ k ~k + ~q This coupling becomes relevant when d < 3, as is true for boson interactions: g 0 = e 3 d 2 t 0 g = e (3 d)t This results in non-trivial fixed points in d<3.