3. Quantum matter without quasiparticles

Transcription

1 1. Review of Fermi liquid theory Topological argument for the Luttinger theorem 2. Fractionalized Fermi liquid A Fermi liquid co-existing with topological order for the pseudogap metal 3. Quantum matter without quasiparticles (A) A mean-field model of a non-fermi liquid, and charged black holes (B) Field theory of a non-fermi liquid (Ising-nematic quantum critical point) (C) Theory of transport in strange metals: application to the (less) strange metal in graphene

2 y Occupied states x Empty states A metal with a Fermi surface with full square lattice symmetry

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

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

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

8 or =0 =0 c Pomeranchuk instability as a function of coupling

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

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

11 T Quantum critical TI-n =0 D=2+1 rising criticality c? =0 Phase diagram as a function of T and

12 T Quantum critical TI-n =0 D=2+1 rising criticality c? =0 Phase diagram as a function of T and

13 T Quantum critical TI-n =0 Strongly-coupled non-fermi liquid metal rwith no c quasiparticles =0 Phase diagram as a function of T and

14 T Quantum critical TI-n Fermi liquid =0 Strongly-coupled non-fermi liquid metal rwith no c quasiparticles =0 Fermi liquid Phase diagram as a function of T and

15 T Strange Metal TI-n Fermi liquid =0 Strongly-coupled non-fermi liquid metal rwith no c quasiparticles =0 Fermi liquid Phase diagram as a function of T and

16 The Fermi liquid Occupied L = f r 2 2m µ + uf f f f f k F! Empty states

17 The Fermi liquid: RG L = r 2 2m µ f + uf f f f Expand fermion kinetic energy at wavevectors about ~ k 0,by writing f ( ~ k 0 + ~q) = (~q)

18 The Fermi liquid: RG L = r 2 2m µ f + uf f f f Expand fermion kinetic energy at wavevectors about ~ k 0,by writing f ( ~ k 0 + ~q) = (~q) L[ i@ 2 y + u

19 The Fermi liquid: RG S[ ]= Z d d 1 ydxd i@ 2 y + u i

20 The Fermi liquid: RG S[ ]= Z d d 1 ydxd i@ 2 y + u i The kinetic energy is invariant under the rescaling x! x/s, y! y/s 1/2, and! /s z,providedz = 1 and! s (d+1)/4. Then we find u! us (1 d)/2, and so we have the RG flow du d` = (1 d) 2 u Interactions are irrelevant in d =2!

21 The Fermi liquid: RG S[ ]= Z d d 1 ydxd i@ 2 y + u i The kinetic energy is invariant under the rescaling x! x/s, y! y/s 1/2, and! /s z,providedz = 1 and! s (d+1)/4. Then we find u! us (1 d)/2, and so we have the RG flow du d` = (1 d) 2 u Interactions are irrelevant in d =2!

22 The Fermi liquid: RG S[ ]= Z d d 1 ydxd i@ 2 y + u i The fermion Green s function to order u 2 has the form (upto logs) G(~q,!) = A! q x qy 2 + ic! 2 So the quasiparticle pole is sharp. And fermion momentum distribution function n( ~ D k)= f ( ~ k)f ( ~ E k) had the following form:

23 The Fermi liquid: RG S[ ]= Z d d 1 ydxd i@ 2 y + u i The fermion Green s function to order u 2 has the form (upto logs) G(~q,!) = A! q x qy 2 + ic! 2 So the quasiparticle pole is sharp. And fermion momentum distribution function n( ~ D k)= f ( ~ k)f ( ~ E k) had the following form: n(k) A k F k

24 L = f r 2m µ +4Fermiterms The Fermi liquid f Occupied states k F! Empty states Fermi wavevector obeys the Luttinger relation kf d fermion density Q, the Sharp particle and hole of excitations near the Fermi surface with energy! q z, with dynamic exponent z =1. The phase space density of fermions is e ectively one-dimensional, so the entropy density S T. It is useful to write this is as S T (d )/z, with violation of hyperscaling exponent = d 1.

25 L = f r 2m µ +4Fermiterms The Fermi liquid f Occupied states q k F! Empty states Fermi wavevector obeys the Luttinger relation kf d fermion density Q, the Sharp particle and hole of excitations near the Fermi surface with energy! q z, with dynamic exponent z =1. The phase space density of fermions is e ectively one-dimensional, so the entropy density S T. It is useful to write this is as S T (d )/z, with violation of hyperscaling exponent = d 1.

26 L = f r 2m µ +4Fermiterms The Fermi liquid f Occupied states q k F! Empty states Fermi wavevector obeys the Luttinger relation kf d fermion density Q, the Sharp particle and hole of excitations near the Fermi surface with energy! q z, with dynamic exponent z =1. The phase space density of fermions is e ectively one-dimensional, so the entropy density S T. It is useful to write this is as S T (d )/z, with violation of hyperscaling exponent = d 1.

27 or =0 =0 c Pomeranchuk instability as a function of coupling

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

29 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

30 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

31 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,

32 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.

33 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.

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

35 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

36 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

37 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.

38 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.

39 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.

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

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

42 L = X i@ y 2 + X + i@ y g 2 (@ 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! s (2z+1)/4 g! gs (3 2z)/4 So the action is invariant provided z =3/2.

43 FL Fermi liquid q k F! k d F Q, the fermion density Sharp fermionic excitations near Fermi surface with q z, and z =1. Entropy density S T (d )/z with violation of hyperscaling exponent = d 1. Entanglement entropy S E k d 1 F P ln P.

44 FL Fermi liquid q kf! kfd Q, the fermion density NFL Nematic QCP kf! Fermi surface with kfd Q. Sharp fermionic excitations near Fermi surface with q z, and z = 1. Di use fermionic excitations with z = 3/2 n(k) to three loops. Entropy density S T (d )/z with violation of hyperscaling exponent = d 1. S T (d )/z with = d 1. Entanglement entropy SE kfd 1 P ln P. kf SE d 1 kf P ln P. k

45 FL Fermi liquid q NFL Nematic k F! k F! QCP q k d F Q, the fermion density Fermi surface with k d F Q. Sharp fermionic excitations near Fermi surface with q z, and z =1. Entropy density S T (d )/z with violation of hyperscaling exponent = d 1. Entanglement entropy S E k d 1 F P ln P. Di use fermionic excitations with z =3/2 to three loops. S T (d )/z with = d 1. S E k d 1 F P ln P. M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, (2010)

46 FL Fermi liquid q k F! NFL Nematic QCP q k F! k d F Q, the fermion density Fermi surface with k d F Q. Sharp fermionic excitations near Fermi surface with q z, and z =1. Entropy density S T (d )/z with violation of hyperscaling exponent = d 1. Entanglement entropy S E k d 1 F P ln P. Di use fermionic excitations with z =3/2 to three loops. S T (d )/z with = d 1. S E k d 1 F P ln P.

47 T Strange Metal TI-n Fermi liquid =0 Strongly-coupled non-fermi liquid metal rwith no c quasiparticles =0 Fermi liquid Phase diagram as a function of T and