A non-fermi liquid: Quantum criticality of metals near the Pomeranchuk instability

Transcription

1 A non-fermi liquid: Quantum criticality of metals near the Pomeranchuk instability Subir Sachdev sachdev.physics.harvard.edu HARVARD

2 y x Fermi surface with full square lattice symmetry

3 y x Spontaneous elongation along x direction: Ising order parameter > 0.

4 y x Spontaneous elongation along y direction: Ising order parameter < 0.

5 Ising-nematic order parameter Z d 2 k (cos k x cos k y ) c k c k Measures spontaneous breaking of square lattice point-group symmetry of underlying Hamiltonian

6 or =0 =0 r r c Pomeranchuk instability as a function of coupling r

7 T Quantum critical TI-n =0 r c =0 r Phase diagram as a function of T and r

8 T Quantum critical TI-n Classical d=2 Ising criticality =0 r c =0 r Phase diagram as a function of T and r

9 T Quantum critical TI-n Classical d=2 Ising criticality =0 D=2+1 rising criticality c? =0 r Phase diagram as a function of T and r

10 T Quantum critical TI-n Classical d=2 Ising criticality =0 D=2+1 rising criticality c? =0 r Phase diagram as a function of T and r

11 T T? Quantum critical TI-n =0 r c =0 r Phase diagram as a function of T and r

12 T T? Strange Quantum Metal critical? TI-n =0 r c =0 r Phase diagram as a function of T and r

13 E ective action for Ising order parameter S = d 2 rd ( ) 2 + c 2 ( ) 2 +( c) 2 + u 4

14 E ective action for Ising order parameter S = d 2 rd ( ) 2 + c 2 ( ) 2 +( c) 2 + u 4 E ective action for electrons: S c = d N f c i c i t ij c i c i =1 i i<j N f d c k ( + k ) c k =1 k

15 Coupling between Ising order and electrons S c = g Z d N f X =1 X k,q q (cos k x cos k y )c k+q/2, c k q/2, for spatially dependent > 0 < 0

16 S = d 2 rd ( ) 2 + c 2 ( ) 2 +( c) 2 + u 4 S c = N f d c k ( + k ) c k S c = =1 k Z g d N f X =1 X k,q q (cos k x cos k y )c k+q/2, c k q/2,

17 fluctuation at wavevector ~q couples most e ciently to fermions near ± ~ k 0. Expand fermion kinetic energy at wavevectors about ± ~ k 0 and boson ( ) kinetic energy about ~q = 0.

18 fluctuation at wavevector ~q couples most e ciently to fermions near ± ~ k 0. Expand fermion kinetic energy at wavevectors about ± ~ k 0 and boson ( ) kinetic energy about ~q = 0.

19 L[ ±, ]= i@ y i@ y g 2 (@ y ) 2 M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, (2010)

20 L = i@ y i@ y g 2 (@ y ) 2 One loop self-energy with N f fermion flavors: Z d 2 k d (~q,!) = N f = N f 4! q y 1 [ i( +!)+k x + q x +(k y + q y ) 2 ] i k x + ky 2 Landau-damping

21 L = i@ y i@ y g 2 (@ y ) 2 Electron self-energy at order 1/N f : ( ~ k, ) = 1 N f Z d 2 q d! [ i(! + )+k x + q x +(k y + q y ) 2 ] 1 " # q 2 y g 2 +! q y = i 2 g 2 2/3 p sgn( ) 2/3 3Nf 4

22 L = i@ y i@ y g 2 (@ y ) 2 Electron self-energy at order 1/N f : ( ~ k, ) = 1 N f Z d 2 q d! [ i(! + )+k x + q x +(k y + q y ) 2 ] 1 " # q 2 y g 2 +! q y = i 2 g 2 2/3 p sgn( ) 2/3 3Nf 4 d/3 in dimension d.

23 L = i@ y i@ y g 2 (@ y ) 2 Schematic form of and fermion Green s functions in d dimensions D(~q,!) = 1/N f q 2? +! q?, G f (~q,!) = q x + q 2? 1 isgn(!)! d/3 /N f In the boson case, q 2?!1/z b with z b =3/2. In the fermion case, q x q 2?!1/z f with z f =3/d. Note z f < z b for d > 2 ) Fermions have higher energy than bosons, and perturbation theory in g is OK. Strongly-coupled theory in d = 2.

24 L = i@ y i@ y g 2 (@ y ) 2 Schematic form of and fermion Green s functions in d = 2 D(~q,!) = 1/N f q 2 y +! q y, G f (~q,!) = q x + q 2 y 1 isgn(!)! 2/3 /N f In both cases q x qy 2! 1/z,withz =3/2. Note that the bare term! in G 1 f is irrelevant. Strongly-coupled theory without quasiparticles.

25 L = i@ y i@ y g 2 (@ y ) 2 Simple scaling argument for z =3/2.

26 L scaling = + i@ y i@ y 2 g + + +(@ y ) 2 Simple scaling argument for z =3/2.

27 L scaling = + i@ y i@ y 2 g + + +(@ y ) 2 Simple scaling argument for z =3/2. Under the rescaling x! x/s, y! y/s 1/2, and! /s z,we find invariance provided! s (2z+1)/4! s (2z+1)/4 g! gs (3 2z)/4 So the action is invariant provided z =3/2.