Quantum critical metals and their instabilities. Srinivas Raghu (Stanford)

Transcription

1 Quantum critical metals and their instabilities Srinivas Raghu (Stanford)

2 Collaborators and References R. Mahajan, D. Ramirez, S. Kachru, and SR, PRB 88, (2013). A. Liam Fitzpatrick, S. Kachru, J. Kaplan, and SR, PRB 88, (2013). A. Liam Fitzpatrick, S. Kachru, J. Kaplan, and SR, PRB (2014). A. Liam Fitzpatrick, S. Kachru, J. Kaplan, and SR, G. Torroba, H. Wang, to appear. With Liam Fitzpatrick, Jared Kaplan, Shamit Kachru Gonzalo Torroba, Huajia Wang

3 Basic themes to be explored Basic issues related to metallic quantum criticality. Hertz: metallic quasiparticles + order parameter fluctuations. Current frontier: treatment of the interplay between incoherent fermions and order parameter fluctuations. Instabilities towards conventional ground states? Goal: treat fermions and order parameter fields on equal footing, find controllable limits and associated scaling laws.

4 Ordinary metals Free electrons k(t) =e i (k)t ~ k (0) plane-wave eigenstates switch on short-range interactions quasiparticles k(t) e i (k)t ~ e (k)t ~ k (0) q.p. scattering rate Quasiparticle description is viable when Near Fermi surface, this is increasingly well-satisfied: Fairly robustly in the simplest setting,! fermions 0, contribute: 2 Consequences: Universal thermodynamics e.g. C V T T 2

5 Effective field theory of ordinary metals (Shankar 1991, Polchinski 1992) Z Non-interacting Fixed point action, S: Landau Fermi liquid theory S =!,k Z k,! Z (i! d! 2 Z v`) d`dd 1 k k (2 ) d k ` : perpendicular to F. S. empty states ) q y filled states q x Most interactions are irrelevant. BCS -> marginal. Fermi surface Clean metal - unstable only to BCS (at exponentially small energy scales).

6 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. NFL QCPs in metals: wide-open problem especially in d=2+1.

7 How do Fermi liquids break down? Shankar/Polchinski s theory cannot capture such behavior. New IR effects can occur if additional low energy modes couple to electrons: e.g. order parameter fluctuations near criticality. This occurs at quantum critical points in metals. Two types of quantum critical points:! 1) transition preserves translation symmetry (e.g. Ferromagnetism, nematic ordering) 2) transition breaks translation symmetry (e.g. CDW, SDW ordering) I will focus on the class 1) of critical points above.

8 Concrete model system Ising nematic transition: breaking of point group symmetry. Fermi liquids: Pomeranchuk instability Fermi liquid 2 possible ground states Analogy with classical liquid crystals spin up spin down t Isotropic Nematic order parameter: t + At the critical point is a massless field.

9 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.

10 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.

11 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???

12 Mainstream view: Hertz (1976) This approach takes the viewpoint that damping of bosons due to fermions is the most significant effect. Idea: integrating out all fermions results in a non-local theory of nearly free, overdamped bosons: S eff = Z k,!! 2 c 2 k 2 + i (!, k) 2 (!,k)=g 2 kd 1 F v! k Landau damping -> bosons governed by z=3 dynamic scaling. Interactions among bosons are irrelevant but singular: ignoring them could be dangerous!

13 A long line of works building along this direction exists: Hertz 1976 Millis 1993 Polchinski 1994 Altshuler, Ioffe, Millis, 1994 Nayak, Wilczek, 1994 Oganesyan, Fradkin, Kivelson, 2001 Chubukov et al, 2006 Sung-Sik Lee, 2009 Metlitski, Sachdev 2010 Mross, Mcgreevy, Liu, Senthil (2010) Davidovic, Sung-Sik Lee (2014). We go in a different direction

14 Outline of the rest of the talk Ignore SC here and address normal state { I. Large N theory II. Renormalization group analysis III. Superconducting domes

15 I. Large N limits

16 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. Large N limits present us with sharp separation of energy scales. 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).

17 Large N limits Large NF: O(1/N F ): Large NB: O(1/N B ):

18 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. Focus today on SO(N) global symmetry

19 Large NB action L = i [@ + µ (ir)] i + i NB L =tr m (@ ) 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 Emergent SO(NB 2 ) symmetry when (1) =0 This symmetry is softly broken: i.e., only at O(1/N 2 B).

20 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.

21 Large N limits The theory can be solved at large N even in d=2+1. First 1/NB correction: Landau damping of the boson due to incoherent particle-hole fluctuations. The form of Landau damping here very different than in the standard approach: (k,!) =! log! 2 v 2 F k 2 Landau damping due to ill-defined quasiparticles is weaker. This leads to a broad energy regime governed by a z=2 boson.

22 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.

23 II. RG analysis

24 Scaling near the upper-critical dimension UV theory: decoupled Fermi liquid + nearly free bosons (g=0). Scaling must contend with vastly different kinematics of bosons and fermions. Fermions: low energy = Fermi surface. -> anisotropic scaling. Bosons: low energy = point in k-space. -> isotropic scaling. (a) } [g] q y (c) q x k q (b) empty states k + q q y filled states empty states filled states = 1 2 (3 d) d=3 is the upper critical dimension. q x Note for experts: this can also be seen readily in z=1 patch scaling.

25 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 I will present RG results at large Nb.

26 Renormalization group analysis RG flows at one-loop: =3 d N B 1 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

27 A few heretical remarks a Landau damping, which plays a central role in Hertz s theory never contributes to RG running! ab ab b The same is not true for the fermion. It obtains non-trivial wave-function renormalization: a ab b a t+dt (!, k) t (k) =i!g 2 a g dt a positive constant. This effect gives rise to anomalous dimensions for the fermions (i.e. Green functions have branch cuts, no poles) and to velocity running. This effect (and vertex corrections) produce the NFL.

28 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!

29 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?)

30 Summary so far 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.

31 Summary so far E F Wilson-Fisher + dressed non-fermi liquid! LD ge F 1 p N Scale where Landau damping sets in???

32 III. Superconducting domes

33 Effect of QCP on pairing The order parameter has two effects on the fermions: 1) It destroys fermion quasiparticle. Bad for pairing. a ab b a 2) It enhances the pairing interaction: (like a critical optical phonon). Good for pairing. v v v v Which of these effects dominates??

34 Perturbation theory near d=3 There are log-squared divergences in the Cooper channel in the vicinity of the quantum critical point. v v v v k v v v + + v eff (!) + g 2 log 2 apple! + Naively summing these up in the spirit of BCS, we find a parametrically higher instability scale:! exp ( 1/ g )

35 Log 2 divergences and universality Exponentially enhanced pairing at QCP is encouraging. However, log 2 divergences pose a significant challenge for the RG. RG time: t = log [ /!] When log 2 divergences are present, RG flows are explicitly t-dependent: d dt (coupling) / t Notions of universality and fixed points could be destroyed. How to proceed?

36 RG Analysis Key point: ordinary log-divergence is hidden in a tree-level interaction: k v v v v k 0 k k 0 = g 2 (k 0 k 0 0 )2 + c 2 k k 0 2 To see this: consider the angular momentum basis: V (`) = 2` +1 2 Z 1 1 d(cos ) ( )P`(cos )

37 RG Analysis S-wave (` = 0) case for simplicity: V (0) = g2 2 Z 1 1 d(cos ) (k 0 k 0 0 )2 + k k 0 2 Momentum transfer: q 2 k k 0 2 ' 2k 2 F (1 cos ) d(cos ) = d(q 2 )/2k 2 F Note that kf as a scale is crucial here: it converts an angle to a momentum transfer q imparted by the boson: Z 2k 2 g2 F V (0) = 4k 2 F 2k 2 F d(q 2 ) (k 0 k 0 0 )2 + q 2

38 RG Analysis Idea: treat V(0) in a Wilsonian spirit, decimating only fast q modes. The contribution from decimating a thin shell d <q< : V (0) = g 2 d Adding the Fermi liquid contribution, which is present even when g=0, we obtain the RG flow for the BCS pairing amplitude: dv (`) dt = g 2 + V (`) 2 [dt = d log ] Note that flows are not explicitly t-dependent. It s solution contains the exponential enhancement of pairing that we guessed using perturbation theory.

39 Accounting for anomalous dimensions Last step in the RG: after integrating out a thin shell of modes, we must rescale fields. Fermion fields acquire anomalous dimension in the presence of the boson. This will affect the flow of BCS couplings after field rescaling. ab a b a Final result: competition between 2 effects. dv (`) dt = g 2 + V (`) 2 a g g 2 V (`) { enhancement of pairing by scalar { suppression due to qp destruction. a g = (1 + v /c)

40 Previous work on dense QCD D.T. Son considered superconductivity of quarks due to gluon exchange in finite density QCD. Similar log 2 divergences occur here. In a remarkable paper, Son concluded (based on intuitive reasoning): d = (g ) dt Our result agrees (upto anomalous dimensions) with Son s. Similar recent work: Metlitski et al.,

41 Phase diagrams at large Nb Consider the large Fermion velocity limit: c/v ->0. Taking g, v large but with fixed (and small) = g 2 /v we can still control the theory. BCS k F e 1/p NFL k F e 1/ p kf LD p v N In the large v limit, with moderately large N, we can arrange for BCS instability to occur above the scale where our theory breaks down (due to Landau damping). Note: there is a subdominant CDW instability also present in this theory.

42 Conclusions and outlook Near d=3+1, superconductivity forms at higher energy scales than the formation of a non-fermi liquid in the large Nb limit. Anomalous dimension contribution is small. Also Landau damping is unimportant. Near d=2+1, other possibilities may occur.! With both overdamped fermions and bosons, pairing can occur out of a non-fermi liquid with a large anomalous dimension (work in progress). Outstanding goal: to demonstrate enhanced superconductivity out of a non-fermi liquid.