Unaprticles: MEELS, ARPES and the Strange Metal

Transcription

1 Unaprticles: MEELS, ARPES and the Strange Metal K. Limtragool Chandan Setty Chandan Setty Zhidong Leong Zhidong Leong Bikash Padhi

2 strange metal: experimental facts T-linear resistivity (!) =C! 2 3 L xy = apple xy /T xy 6=#/ T Hall Angle cot H xx xy T 2

3 Phase Space Restriction T 2

4 Planckian dissipation Interactions??? / T

5 M-EELS (Momentum resolved electron energy loss spectroscopy) d 2 d de = 0[V e (k z i,k z s, q)] 2 S(q,!) S(q,!) = e ~!/k BT = (q,!) = 1 (q,!) susceptibility? 5

6 Transverse vs Longitudinal q? (q,!) (q,!) M-EELS E q E optics: Ellipsometry? (q! 0,!) = k (q! 0,!) The two dielectric functions must be the same in the zero momentum limit The Bible: Nozieres and Pines (1999) 6

7 H 0 (t) = Z drn(r, t)w(r, t) h n(k,!)i = (k,!)w(q,!) density-density response marginal Fermi liquids 00 (k,!) = N( F )!/T! T 00 (k,!) = N( F )(sgn!), T!! c is this seen experimentally?

8 1.) separable polarizability = (k,!) 1 V (k) (k,!) 00 = f(q)g(!) 2.) scale-invariance 00 (q,!) =q 2,! <! c 00 (q,!) = q2! 2,! >! c not MFL

9 Drude metal New Data (Abbamonte, et al. strange metal

10 doping (in)dependence no momentum dispersion!!

11 " (q,!) =1 V(k) (k,!) are the dielectric functions equal? " opt? (q,!) =" i (q,!)/!! 5/3

12 Cuprates " (q = 0,!) ="? (q = 0,!) / const. /! 5/3 " opt 6= " This equality doesn t hold even in limit of zero q in BSCCO!! What s the discrepancy in Cuprates? 12

13 Does this discrepancy offer a window into the strange metal? 13

14 14 Yes

15 Order of limits q E q E lim q?!0 anish. limqk!0, es not va limqk!0, es not va lim q?!0 anish. answer to this question lies in order in which the momentum sfer q is taken to zero, i.e., ther the parallel component is n to zero first or the pendicular. he case of optics: parallel first then perpendicular and osite for MEELS ically these two are the same i.e., commutator is zero. limqk!0, lim q?!0? k Ki (q?,q k,!) = 0. 15??

16 Normal metals: Mermin to Drude (q,!) = i! 4 V q (q,!) M (~q,!) = 0(~q,! + i ) 1+(1 i! ) 1 0(~q,!+ i ) 1 0(~q,0) 0(q,!) = nq2 M! 2 (q = 0,!) = ne2 m 1! + i N.D. Mermin PRB (1970) "? (~q! 0,!) =" (~q! 0,!) (Normal metals) 16

17 Drude case Z h 1) We can quickly check to see if this holds in the case of normal metals microscopically. 2) ithis is the bubble diagram that contributes to the conductivity, with vertices given by the momenta 3) For finite q and T this is the form of the conductivity. Describe the equation. 4) Typically q is simply set to zero here and the order doesn t matter.. show that in words. Z (q,!) = 1 Z +1 1 d F (,!) Z dp (2p + q)2 i (2m) 2 G p ( )G + p+q ( +!). retarded and advanced [G ± p ( )] 1 = p ± i 2! Finite q and T 17

18 Reconsidering interactions: Angular Vertex (a) (b) p p - s (c) (d) Z s L = L 0 + ds (p, s) 2 (p s) (s) (p) s the electron-boson p, p s = ˆp.(ˆp ŝ) scale-invariant interaction 18

19 Angular Vertex V i p,q,s = 4(ˆq ŝ) 2 (p) 2 i +2s i (p) i (ˆp ŝ)+(p) 2 i (ˆp ŝ) 2 (ˆq.ŝ) 2 = (q.s)2 q 2 s 2 = q ks k + q?.s? qk 2 + q 2 s? 2 2 apple lim, lim q!0 q?!0 K i(q?,q,!) =! 2 + O 1! 4 19

20 scale-invariant interaction " (q! 0,!) 6= "? (q! 0,!) unparticles (p 2 ) d U 20

21 massive free theory mass L = µ + m µ x! x/ (x)! (x) m 2 2 no scale invariance

22 unparticles Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 dm 2 theory with all possible mass!! (x, m 2 / 2 ) x! x/ m 2 / 2! m 2 L! 4 L scale invariance is restored!! not particles

23 propagator Z 1 0 dm 2 m 2 p 2 i m 2 + i 1 / p 2 d U 2 continuous mass (x, m 2 ) flavors e 2 (m) Karch, 2005 multi-bands

24 Does the `self-energy exhibit a power law? / (! 2 + T 2 ) 24

25 `Self-energy 2 (k,!) = X q V 2 (q) 2 (q,! " k q ) [n F ( " k q )+n B (! " k q )], Note that the susceptibility is exact since we are obtaining it from experiment, and this goes into the self-energy calculation 25

26 Modeling the data (a) (b) 26

27 Self-energy: Optimal doping V(q)= exp[-q z]/q V(q)= const V(q)=1/q (a) (b) (c) Require both phonons and an interaction with a scale for the kink and peak like feature 27

28 Self energy: doping dependence f(!) =! 00 2(!) +1 (!) 0 2 (a) (b) There exist important similarities and some differences with ARPES (Dessau s power-law liquid) 28

29 Dessau s Power-law liquid OD 00 (!) = 0 +!2 + 2 T 2! 2 1 N UD OP QC region 0 < < 1/2 29

30 Real part of 0 (!) = 1 P Z d! 0 00! 0! 0! T =0 0 (x! N )= 1 Z!N 0 P d!! N! 0 x! N 0 0 +! 0 2! 2 1 N 1 C A = + x + 30

31 0 k = k + 0 apple (2 1) 1, > 1 2 Z = 1 d 0 d!!=0! 1 0 apple 1 2 non-analytic at! =0 31

32 Absence of particlelike behavior Velocity Vishik, et al., PRL 104, (2010). ARPES doping Quantum oscillation measurements S.E. Sebastian, et al, PNAS 107, 6175 (2009) Metal-insulator transition 32

33 spectral function A (k,!) =N 00 (!) [! k 0 (!)] 2 +[ 00 (!)] 2, critical FS (Z=0) 33

34 superconductivity 1 g = X k Z d!d! tanh! 2T c + tanh!0 2T c! +! 0 A (k,!) A ( k,! 0 ) 34

35 minimal coupling 1 = g min Z d Z d!d! 0 1 2, 1! +! 0 A (,!) A! 0 35

36 Why? A 0 =0 A (r, r) r A ( = r,! = r) r 2 (r + r 2 ) 2 + r 4 > 1/2 < 1/2 r 2 2 r 2

37 origin of dome 1 g min Z dr r =1/2 log divergence g min 0 37

38 unparticles are unstable to superconductivity unparticles are unstable to gap formation 38

39 what kind of scale-invariant theories?

40 single-parameter scaling / z 2 ln Z! A µ A µ 1 / T (2 d)/z (!, T) /! (d 2)/z 2/3 C v / T d/z

41 d A 6= 1?? extra degree of freedom / T (2 d)/z 2d A [A µ ]=d A =1 A µ! A µ µ fixes dimension of current [d d xja] =0 I A d` =# [J] =d 1

42 strange metal: strange E&M [J µ ]=d + + z 1 [A µ ]=1 = 2/3 [E] =1+z [B] =2 BG,EK,SH,AK note r 2 B 6= flux

43 loophole

44 S! S + Z d d xj Noether s 1st µ J µ =0 what if [@ µ, Ŷ ]=0 new µ ŶJ µ µ J µ =0 [ J] =d 1 D Y A µ! A µ µ +@ G + Noether s 2nd theorem

45 arxiv:

46 possible gauge transformations Z S = 1 d d xf 2 4 S = 1 2 Z d d k 2 d A µ(k)[k 2 µ M µ { k µ k ]A (k) k =0 zero eigenvector ik µ A µ! A µ µ

47 family of zero eigenvalues M µ fk =0 { generator of gauge symmetry 1.) rotational invariance 2.) A is still a 1-form 3.) [f,k µ ]=0

48 only choice f f(k 2 ) ( ) A µ! A µ +( ) ( 1) µ [A µ ]= what kind of E&M has such gauge transformations?

49 S = Z dv d dy y a F 2 + eom d(y a? da) =0 y 6= 0 A! A A? = J +GL in CIMP

50 family of zero eigenvalues M µ fk =0 most fundamental conservation µ ( r 2 ) ( 1)/2 J µ =0

51 scale invariance in strange metal MEELS, ARPES, Optics,STM non-local actions anomalous dimensions running charge (no charge quantization)