Theory of metallic transport in strongly coupled matter. 2. Memory matrix formalism. Andrew Lucas

Transcription

1 Theory of metallic transport in strongly coupled matter 2. Memory matrix formalism Andrew Lucas Stanford Physics Geometry and Holography for Quantum Criticality; Asia-Pacific Center for Theoretical Physics August 18-19, 2017

2 Drude Peak: A Review 27 recall definition of σ (up to contact terms): σ(ω) = GR J xj x (ω). iω

3 Drude Peak: A Review 27 recall definition of σ (up to contact terms): σ(ω) = GR J xj x (ω). iω in the previous lecture we saw that for small ω, σ(ω) = ρ2 M 1 1 τ iω. where τ 1 is the momentum relaxation rate

4 Drude Peak: A Review 27 recall definition of σ (up to contact terms): σ(ω) = GR J xj x (ω). iω in the previous lecture we saw that for small ω, σ(ω) = ρ2 M 1. 1 τ iω where τ 1 is the momentum relaxation rate this lecture: with mild assumptions, we prove that this result is exact for any QFT, to leading order in a perturbatively weak amount of disorder. a (mostly complete) proof: [Hartnoll, Hofman; ], but a few subtleties only addressed later...

5 Summary of the Result 28 The Momentum Relaxation Time consider a many-body quantum system with H = H 0 + εh imp, [H 0, P x ] = 0, [H imp, P x ] 0.

6 Summary of the Result 28 The Momentum Relaxation Time consider a many-body quantum system with H = H 0 + εh imp, [H 0, P x ] = 0, [H imp, P x ] 0. example: random potential disorder for fermions: H imp = d d x V imp (x) }{{} not constant ψ (x)ψ(x).

7 Summary of the Result 28 The Momentum Relaxation Time consider a many-body quantum system with H = H 0 + εh imp, [H 0, P x ] = 0, [H imp, P x ] 0. example: random potential disorder for fermions: H imp = d d x V imp (x) }{{} not constant ψ (x)ψ(x). we will show that (here = 1) 1 τ 1 M lim 1 ( ) ω 0 ω Im G Ṙ P x P (ω) + O(ε 3 ), P x x = i[h, P x ]. the momentum relaxation time is given by the spectral weight of [H, P x ]

8 Memory Function Formalism 29 An Operator Hilbert Space we now embark on a rather technical derivation:

9 Memory Function Formalism 29 An Operator Hilbert Space we now embark on a rather technical derivation: define an operator Hilbert space spanned by operators A), B) etc., with inner product (A B) T 1/T 0 dλ A B(iλ) T.

10 Memory Function Formalism 29 An Operator Hilbert Space we now embark on a rather technical derivation: define an operator Hilbert space spanned by operators A), B) etc., with inner product (A B) T 1/T 0 dλ A B(iλ). T define (A(t) B) C AB (t). basic manipulations give Θ(t) t C AB (t) = it Θ(t) [A(t), B] T = T G R AB(t).

11 Memory Function Formalism 29 An Operator Hilbert Space we now embark on a rather technical derivation: define an operator Hilbert space spanned by operators A), B) etc., with inner product (A B) T 1/T 0 dλ A B(iλ). T define (A(t) B) C AB (t). basic manipulations give Θ(t) t C AB (t) = it Θ(t) [A(t), B] T = T G R AB(t). integrate above from t = 0 to t = : C AB (t = 0) = T G R AB(ω = 0) = T χ AB }{{} static susceptibility

12 Memory Function Formalism 30 An Operator Hilbert Space Laplace transform related to conductivity: C AB (z) = dte izt C AB (t) = T ( G R iz AB (z) G R AB(0) ) 0 T σ AB (z) (up to contact terms)

13 Memory Function Formalism 30 An Operator Hilbert Space Laplace transform related to conductivity: C AB (z) = dte izt C AB (t) = T ( G R iz AB (z) G R AB(0) ) 0 T σ AB (z) (up to contact terms) define the Liouvillian L: il A) = A), e ilt A) = A(t)) so that C AB (z) = (A i(z L) 1 B)

14 Memory Function Formalism 30 An Operator Hilbert Space Laplace transform related to conductivity: C AB (z) = dte izt C AB (t) = T ( G R iz AB (z) G R AB(0) ) 0 T σ AB (z) (up to contact terms) define the Liouvillian L: il A) = A), e ilt A) = A(t)) so that C AB (z) = (A i(z L) 1 B) our goal is to compute σ(ω) = 1 T C J xj x (z = ω + i0 + )

15 Memory Function Formalism 31 Conserved Quantities suppose that P ) is conserved i.e., L P ) = 0. then consider (A i(z L) 1 B) (A P )(P i(z L) 1 P )(P B) (P P ) 2 (A P )(P B) (P P ) i z

16 Memory Function Formalism 31 Conserved Quantities suppose that P ) is conserved i.e., L P ) = 0. then consider (A i(z L) 1 B) (A P )(P i(z L) 1 P )(P B) (P P ) 2 (A P )(P B) (P P ) i z these divergences are Drude divergences when τ 1 = 0!

17 Memory Function Formalism 31 Conserved Quantities suppose that P ) is conserved i.e., L P ) = 0. then consider (A i(z L) 1 B) (A P )(P i(z L) 1 P )(P B) (P P ) 2 (A P )(P B) (P P ) i z these divergences are Drude divergences when τ 1 = 0! long lived quantities will lead to nearly singular Green s functions as ω 0

18 Memory Function Formalism 32 Outline of the Rest of the Computation assumption: momentum P x is the only (almost) conserved operator where (P x J x ) 0.

19 Memory Function Formalism 32 Outline of the Rest of the Computation assumption: momentum P x is the only (almost) conserved operator where (P x J x ) formal re-writing of σ AB (matrix indices only include J x, P x ): σ AB = χ AC (M(ω) + N iωχ) 1 CD χ DB a component of the memory matrix MP P τ 1 N = 0 until last lecture

20 Memory Function Formalism 32 Outline of the Rest of the Computation assumption: momentum P x is the only (almost) conserved operator where (P x J x ) formal re-writing of σ AB (matrix indices only include J x, P x ): σ AB = χ AC (M(ω) + N iωχ) 1 CD χ DB a component of the memory matrix MP P τ 1 N = 0 until last lecture 2. show that perturbatively σ JxJ x χ 2 J xp x M PxP x (ω = 0) iωχ PxP x and relate M PxP x (ω = 0) to spectral weight of P x

21 Memory Function Formalism 32 Outline of the Rest of the Computation assumption: momentum P x is the only (almost) conserved operator where (P x J x ) formal re-writing of σ AB (matrix indices only include J x, P x ): σ AB = χ AC (M(ω) + N iωχ) 1 CD χ DB a component of the memory matrix MP P τ 1 N = 0 until last lecture 2. show that perturbatively σ JxJ x χ 2 J xp x M PxP x (ω = 0) iωχ PxP x and relate M PxP x (ω = 0) to spectral weight of P x 3. give more useful expressions for χ JxP x, χ PxP x, M PxP x

22 Memory Function Formalism 33 A Projection Matrix we wish to separate degrees of freedom into: slow (A, B {J x, P x }) p = 1 A)χ 1 AB T (B AB fast (all others) q = 1 p e.g. p J x ) = J x ), and q J x ) = 0.

23 Memory Function Formalism 33 A Projection Matrix we wish to separate degrees of freedom into: slow (A, B {J x, P x }) p = 1 A)χ 1 AB T (B AB fast (all others) q = 1 p e.g. p J x ) = J x ), and q J x ) = 0. this separation is arbitrary. good identification of fast and slow operators makes it possible to evaluate formal expressions later

24 Memory Function Formalism 33 A Projection Matrix we wish to separate degrees of freedom into: slow (A, B {J x, P x }) p = 1 A)χ 1 AB T (B AB fast (all others) q = 1 p e.g. p J x ) = J x ), and q J x ) = 0. this separation is arbitrary. good identification of fast and slow operators makes it possible to evaluate formal expressions later if we choose J x ) to be slow: C JxJ x (z) = (J x p(il iz) 1 p J x ).

25 Memory Function Formalism 33 A Projection Matrix we wish to separate degrees of freedom into: slow (A, B {J x, P x }) p = 1 A)χ 1 AB T (B AB fast (all others) q = 1 p e.g. p J x ) = J x ), and q J x ) = 0. this separation is arbitrary. good identification of fast and slow operators makes it possible to evaluate formal expressions later if we choose J x ) to be slow: C JxJ x (z) = (J x p(il iz) 1 p J x ). schematically: perform block matrix inversion and integrating out fast degrees of freedom

26 Memory Function Formalism 34 Integrating Out the Fast Modes since our basis J x ) is not orthogonal, it is easier to proceed differently. note the identity (z L) 1 = (z Lp Lq) 1 = (z Lq) 1 (1 + Lp(z L) 1 ) (to prove: multiply RHS by (z L))

27 Memory Function Formalism 34 Integrating Out the Fast Modes since our basis J x ) is not orthogonal, it is easier to proceed differently. note the identity (z L) 1 = (z Lp Lq) 1 = (z Lq) 1 (1 + Lp(z L) 1 ) (to prove: multiply RHS by (z L)) now we need some algebra: p(z Lq) 1 p = p z 1 n (Lq) n p = z 1 p n=0

28 Memory Function Formalism 34 Integrating Out the Fast Modes since our basis J x ) is not orthogonal, it is easier to proceed differently. note the identity (z L) 1 = (z Lp Lq) 1 = (z Lq) 1 (1 + Lp(z L) 1 ) (to prove: multiply RHS by (z L)) now we need some algebra: p(z Lq) 1 p = p z 1 n (Lq) n p = z 1 p σ }{{} AB iχab z slow only n=0 = i T (A (z Lq) 1 Lp(z L) 1 B) = i (A (z Lq) 1 L C)χ 1 T CDC DB CD (A L + Lq(z Lq) 1 L C)χ 1 = i T z CD CDC DB

29 Memory Function Formalism 35 The Memory Matrix the antisymmetric matrix N AB = i T (A L B) = 1 T (A Ḃ) = 1 T ( A B) describes how slow operators mix among themselves

30 Memory Function Formalism 35 The Memory Matrix the antisymmetric matrix N AB = i T (A L B) = 1 T (A Ḃ) = 1 T ( A B) describes how slow operators mix among themselves the (symmetric) memory matrix M AB (z) = i T (A Lq(z qlq) 1 ql B) describes coupling of slow and fast operators

31 Memory Function Formalism 35 The Memory Matrix the antisymmetric matrix N AB = i T (A L B) = 1 T (A Ḃ) = 1 T ( A B) describes how slow operators mix among themselves the (symmetric) memory matrix M AB (z) = i T (A Lq(z qlq) 1 ql B) describes coupling of slow and fast operators rearranging our results we obtain σ AB (z) = χ AC (M(z) + N iωχ) 1 CD χ DB

32 Memory Function Formalism 35 The Memory Matrix the antisymmetric matrix N AB = i T (A L B) = 1 T (A Ḃ) = 1 T ( A B) describes how slow operators mix among themselves the (symmetric) memory matrix M AB (z) = i T (A Lq(z qlq) 1 ql B) describes coupling of slow and fast operators rearranging our results we obtain σ AB (z) = χ AC (M(z) + N iωχ) 1 CD χ DB N AB = 0 for us: J x, P x both time reversal odd

33 Long Lived Momentum 36 Simplifications so far, manipulations are exact

34 Long Lived Momentum 36 Simplifications so far, manipulations are exact if L P x ) = 0 (exactly conserved), then M PxB(z) = 0. σ AB (ω = 0) = χ AC M CD (0) 1 χ DB ill-posed

35 Long Lived Momentum 36 Simplifications so far, manipulations are exact if L P x ) = 0 (exactly conserved), then M PxB(z) = 0. σ AB (ω = 0) = χ AC M CD (0) 1 χ DB ill-posed what we will show: if P x ɛ, for small ɛ: ( ) ( MJxJx M JxPx ɛ 0 ɛ 2 ) M PxJx M PxPx ɛ 2 ɛ 2 ) 1 ( ɛ 0 ɛ 0 ) M PxJ x ( MJxJ x M JxP x M PxP x ɛ 0 ɛ 2

36 Long Lived Momentum 36 Simplifications so far, manipulations are exact if L P x ) = 0 (exactly conserved), then M PxB(z) = 0. σ AB (ω = 0) = χ AC M CD (0) 1 χ DB ill-posed what we will show: if P x ɛ, for small ɛ: ( ) ( MJxJx M JxPx ɛ 0 ɛ 2 ) M PxJx M PxPx ɛ 2 ɛ 2 ) 1 ( ɛ 0 ɛ 0 ) ( MJxJ x M JxP x M PxJ x M PxP x ɛ 0 ɛ 2 taking ω ɛ 2 small: σ JxJ x = χ 2 J xp x M PxP x iωχ PxP x 1 ɛ 2

37 Long Lived Momentum 37 M PxP x to compute M PxP x, recall H = H 0 + εh imp, [H 0, P x ] = 0, [H imp, P x ] 0.

38 Long Lived Momentum 37 M PxP x to compute M PxPx, recall H = H 0 + εh imp, [H 0, P x ] = 0, [H imp, P x ] 0. the general form of H imp : H imp = α d d x h α (x)o α (x) and as P x generates translations: P x = i[εh imp, P x ] = i α d d x h α (x) x O α (x)

39 Long Lived Momentum 37 M PxP x to compute M PxPx, recall H = H 0 + εh imp, [H 0, P x ] = 0, [H imp, P x ] 0. the general form of H imp : H imp = α d d x h α (x)o α (x) and as P x generates translations: P x = i[εh imp, P x ] = i α d d x h α (x) x O α (x) thus we write P x ) = ɛ α d d k (2π) d h α( k)k x α(k))

40 Long Lived Momentum 38 M PxP x and M PxJ x translation invariance implies (A(k) B(q)) δ(k + q)(a(k) B(q)) and from above, P x ) consists of k 0 operators thus we find: M P P = ɛ 2 i T ( P x (ω qlq) 1 P x ) ɛ 2 i d d k T (2π) d k2 xh α (k)(α( k) (ω L) 1 β(k))h β ( k) αβ = ɛ d 2 d k 1 (2π) d k2 xh α (k)h β ( k) lim ω 0 ω Im ( G R αβ (k, ω)). αβ

41 Long Lived Momentum 38 M PxP x and M PxJ x translation invariance implies (A(k) B(q)) δ(k + q)(a(k) B(q)) and from above, P x ) consists of k 0 operators thus we find: M P P = ɛ 2 i T ( P x (ω qlq) 1 P x ) ɛ 2 i d d k T (2π) d k2 xh α (k)(α( k) (ω L) 1 β(k))h β ( k) αβ = ɛ d 2 d k 1 (2π) d k2 xh α (k)h β ( k) lim ω 0 ω Im ( G R αβ (k, ω)). αβ similarly, using translation invariance we find: M PxJx ɛ d d k (2π) d (α(k) (ω qlq) 1 J x ) = ɛ 0+O(ɛ 2 ) α

42 Long Lived Momentum 39 χ JxP x and χ PxP x from a deformed thermal density matrix ρ v = exp[ β(h 0 µq v P)] we may define susceptibilities via linear response: tr[ρ v J x (x)] = χ JxPx v x +, tr[ρ v P x (x)] = χ PxPx v x +.

43 Long Lived Momentum 39 χ JxP x and χ PxP x from a deformed thermal density matrix ρ v = exp[ β(h 0 µq v P)] we may define susceptibilities via linear response: tr[ρ v J x (x)] = χ JxP x v x +, tr[ρ v P x (x)] = χ PxP x v x +. from a hydrodynamic limit, we identify χ JxP x = thus we have derived ρ }{{}, χ PxPx = }{{} M charge density generalized mass density σ(ω) = ρ2 M 1 1 τ iω. for any QFT where only almost conserved operator that overlaps with J x is P x

44 Long Lived Momentum 40 Straightforward Generalizations replace charge current J x with heat current Q x : χ QxP x = ( ) σ T α = T α T κ T s }{{} entropy density ( ρ 2 T sρ T sρ (T s) 2 ) M τ 1 iωm

45 Long Lived Momentum 40 Straightforward Generalizations replace charge current J x with heat current Q x : χ QxP x = ( ) σ T α = T α T κ T s }{{} entropy density ( ρ 2 T sρ T sρ (T s) 2 ) M τ 1 iωm other long-lived conservation laws, e.g., supercurrent: [Davison, Delacrétaz, Goutéraux, Hartnoll; ]

46 Long Lived Momentum 40 Straightforward Generalizations replace charge current J x with heat current Q x : χ QxP x = ( ) σ T α = T α T κ T s }{{} entropy density ( ρ 2 T sρ T sρ (T s) 2 ) M τ 1 iωm other long-lived conservation laws, e.g., supercurrent: [Davison, Delacrétaz, Goutéraux, Hartnoll; ] broken time-reversal symmetry (last lecture)

47 The Spectral Weight 41 QFT Deformed by One Operator conformal field theory at finite T, deformed by scalar operator O of dimension coupled to random field : H = H CFT d d x h(x)o(x). h(x) dis = 0, h(x)h(y) dis = ε 2 δ(x y).

48 The Spectral Weight 41 QFT Deformed by One Operator conformal field theory at finite T, deformed by scalar operator O of dimension coupled to random field : H = H CFT d d x h(x)o(x). h(x) dis = 0, h(x)h(y) dis = ε 2 δ(x y). momentum relaxation: [Lucas, Sachdev, Schalm; ] M τ = ɛ + P τ ε 2 d d k kx 2 1 lim ω 0 ω Im ( G R OO(k, ω) ) ε 2 T 2 T d+2 1 T T 2 d 1

49 The Spectral Weight 42 The Harris Criterion in a CFT, we have T s = ɛ + P T d+1

50 The Spectral Weight 42 The Harris Criterion in a CFT, we have T s = ɛ + P T d+1 hence κ sτ ε 2 T 2d+1 2

51 The Spectral Weight 42 The Harris Criterion in a CFT, we have T s = ɛ + P T d+1 hence κ sτ ε 2 T 2d+1 2 this computation breaks down at low T if 1 τ > T, or 1 ε2 T 2 d 2

52 The Spectral Weight 42 The Harris Criterion in a CFT, we have T s = ɛ + P T d+1 hence κ sτ ε 2 T 2d+1 2 this computation breaks down at low T if 1 τ > T, or 1 ε2 T 2 d 2 Harris criterion: disorder is relevant if < d 2 + 1

53 The Spectral Weight 43 Realistic Quantum Critical Points many quantum critical points are not CFTs: Fermi surface: ρ M T 0

54 The Spectral Weight 43 Realistic Quantum Critical Points many quantum critical points are not CFTs: Fermi surface: ρ M T 0 Ising-nematic: [Hartnoll et al; ] h i =0 h i6=0 y)) c k c k Z 2 ρ 1 τ T 1/2

55 The Spectral Weight 43 Realistic Quantum Critical Points many quantum critical points are not CFTs: Fermi surface: ρ M T 0 Ising-nematic: spin density wave: [Hartnoll et al; ] [Patel, Sachdev; ] h i =0 h i6=0 y)) c k c k Z 2 ρ 1 τ T 1/2 ρ 1 τ V 2 imp + m 2 impt

56 The Spectral Weight 44 Kinetic Theory: A Single Fermi Surface empty filled ky kx empty filled ky filled kx

57 The Spectral Weight 44 Kinetic Theory: A Single Fermi Surface empty filled ky kx assume disorder couples to density operator (random Coulomb impurities) empty filled ky filled kx

58 The Spectral Weight 44 Kinetic Theory: A Single Fermi Surface empty filled ky kx assume disorder couples to density operator (random Coulomb impurities) empty use semiclassical kinetic filled theory to compute G R ρρ(ω) [Lucas, Hartnoll; ky ] filled kx

59 The Spectral Weight 44 Kinetic Theory: A Single Fermi Surface empty interactions always decrease ρ: ky filled k TFξ 1 k TFξ 1 kx assume disorder couples to density operator (random Coulomb impurities) empty use semiclassical kinetic filled theory to compute G R ρρ(ω) [Lucas, Hartnoll; ky ] filled ρ/ρres ξ/l ee why? (next lecture) toy model of single FS kinetics from: [Guo et al; PNAS, ] kx

60 The Spectral k x Weight 45 Kinetic Theory: Two Fermi Surfaces empty filled k y filled k x

61 The Spectral k x Weight 45 Kinetic Theory: Two Fermi Surfaces empty filled k y filled k x pockets have different v F

62 The Spectral k x Weight 45 Kinetic Theory: Two Fermi Surfaces empty filled k y filled k x pockets have different v F conservation laws: charge in pocket 1 charge in pocket 2 total momentum [Lucas, Hartnoll; ]

63 The Spectral k x Weight 45 Kinetic Theory: Two Fermi Surfaces empty numerical computation gives: filled k TF 1 k y filled k x / res 6 4 v F,1 /v F,2 =0.3 v F,1 /v F,2 =0.5 v F,1 /v F,2 =0.7 v F,1 /v F,2 =1 pockets have different v F conservation laws: charge in pocket 1 charge in pocket 2 total momentum [Lucas, Hartnoll; ] /`ee why does this happen? (next lecture)

64 The Spectral Weight 46 Holographic Models a brief holographic aside: [Lucas; ] consider scalar operator O dual to a field Φ in the bulk of AdS a planar black hole in the bulk, with horizon r = r + the regular solution Φ(k, r) to linearized bulk equations of motion

65 The Spectral Weight 46 Holographic Models a brief holographic aside: [Lucas; ] consider scalar operator O dual to a field Φ in the bulk of AdS a planar black hole in the bulk, with horizon r = r + the regular solution Φ(k, r) to linearized bulk equations of motion the spectral weight of O is linked to the physics of Φ at the horizon: 1 lim ω 0 ω Im ( G R OO(k, ω) ) s 4π Φ(k, r +) 2 with s the entropy density

66 The Spectral Weight 46 Holographic Models a brief holographic aside: [Lucas; ] consider scalar operator O dual to a field Φ in the bulk of AdS a planar black hole in the bulk, with horizon r = r + the regular solution Φ(k, r) to linearized bulk equations of motion the spectral weight of O is linked to the physics of Φ at the horizon: 1 lim ω 0 ω Im ( G R OO(k, ω) ) s 4π Φ(k, r +) 2 with s the entropy density early holographic derivations of conductivity: [Blake, Tong, Vegh; ] σ(ω = 0) Φ(k, r + ) 2 and are equivalent to more general formalism

67 Outlook 47 Drude Peak: A Summary we perturbatively derived universal Drude peak: σ(ω) = ρ2 M 1. 1 τ iω and showed τ is the momentum relaxation time: M τ = d d k ɛ2 (2π) d h 1 α(k)h β ( k) lim ω 0 ω Im ( G R αβ (k, ω)). αβ

68 Outlook 47 Drude Peak: A Summary we perturbatively derived universal Drude peak: σ(ω) = ρ2 M 1. 1 τ iω and showed τ is the momentum relaxation time: M τ = d d k ɛ2 (2π) d h 1 α(k)h β ( k) lim ω 0 ω Im ( G R αβ (k, ω)). αβ τ 1 sensitive to microscopic details: non-universal T -scaling, possibly non-monotonic

69 Outlook 47 Drude Peak: A Summary we perturbatively derived universal Drude peak: σ(ω) = ρ2 M 1. 1 τ iω and showed τ is the momentum relaxation time: M τ = d d k ɛ2 (2π) d h 1 α(k)h β ( k) lim ω 0 ω Im ( G R αβ (k, ω)). αβ τ 1 sensitive to microscopic details: non-universal T -scaling, possibly non-monotonic controlled (and useful!) but ultimately must go beyond