Theory of metallic transport in strongly coupled matter. 4. Magnetotransport. Andrew Lucas

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1 Theory of metallic transport in strongly coupled matter 4. Magnetotransport Andrew Lucas Stanford Physics Geometry and Holography for Quantum Criticality; Asia-Pacific Center for Theoretical Physics August 18-19, 2017

2 Drude Magnetotransport 77 A Ward Identity what are the conductivities in a magnetic field? magnetic field breaks time reversal symmetry; possibly rotational symmetry

3 Drude Magnetotransport 77 A Ward Identity what are the conductivities in a magnetic field? magnetic field breaks time reversal symmetry; possibly rotational symmetry do we learn anything new? easy experimentally

4 Drude Magnetotransport 77 A Ward Identity what are the conductivities in a magnetic field? magnetic field breaks time reversal symmetry; possibly rotational symmetry do we learn anything new? easy experimentally Ward identity for momentum conservation, in a general QFT t T ti + j T ji = F iµ J µ = ρe i + B ij J j.

5 Drude Magnetotransport 77 A Ward Identity what are the conductivities in a magnetic field? magnetic field breaks time reversal symmetry; possibly rotational symmetry do we learn anything new? easy experimentally Ward identity for momentum conservation, in a general QFT t T ti + j T ji = F iµ J µ = ρe i + B ij J j. in d = 2 spatial dimensions, B ij = Bɛ ij ; as ω 0: which gives d d x d j T ji d x [ = 0 = ρe i ] + B ij J j V d V d J i = B 1 ij ρe j = ρ B ɛ ije j.

6 Drude Magnetotransport 78 The Hall Conductivity universal Hall conductivity for all d = 2 QFTs: σ ij = ρ B ɛ ij

7 Drude Magnetotransport 78 The Hall Conductivity universal Hall conductivity for all d = 2 QFTs: σ ij = ρ B ɛ ij the Hall conductivity is antisymmetric σ ij = σ ji and so is non-dissipative: T ṡ = E i σ ij E j = 0.

8 Drude Magnetotransport 78 The Hall Conductivity universal Hall conductivity for all d = 2 QFTs: σ ij = ρ B ɛ ij the Hall conductivity is antisymmetric σ ij = σ ji and so is non-dissipative: T ṡ = E i σ ij E j = 0. modified Onsager reciprocity: σ ij (B) = σ ji ( B)

9 Drude Magnetotransport 78 The Hall Conductivity universal Hall conductivity for all d = 2 QFTs: σ ij = ρ B ɛ ij the Hall conductivity is antisymmetric σ ij = σ ji and so is non-dissipative: T ṡ = E i σ ij E j = 0. modified Onsager reciprocity: σ ij (B) = σ ji ( B) questions for this lecture: what happens at weak vs. strong magnetic fields?

10 Drude Magnetotransport 78 The Hall Conductivity universal Hall conductivity for all d = 2 QFTs: σ ij = ρ B ɛ ij the Hall conductivity is antisymmetric σ ij = σ ji and so is non-dissipative: T ṡ = E i σ ij E j = 0. modified Onsager reciprocity: σ ij (B) = σ ji ( B) questions for this lecture: what happens at weak vs. strong magnetic fields? interplay of magnetic fields and disorder? dissipative magnetotransport?

11 Drude Magnetotransport 78 The Hall Conductivity universal Hall conductivity for all d = 2 QFTs: σ ij = ρ B ɛ ij the Hall conductivity is antisymmetric σ ij = σ ji and so is non-dissipative: T ṡ = E i σ ij E j = 0. modified Onsager reciprocity: σ ij (B) = σ ji ( B) questions for this lecture: what happens at weak vs. strong magnetic fields? interplay of magnetic fields and disorder? dissipative magnetotransport? generalize memory matrix formalism, hydrodynamics?

12 Drude Magnetotransport 79 Drude Conductivity let s return to our toy Drude model (d = 2): iωmv i = ρe i M τ v i + Bɛ ij J j, with J i ρv i.

13 Drude Magnetotransport 79 Drude Conductivity let s return to our toy Drude model (d = 2): with J i ρv i. conductivity matrix is iωmv i = ρe i M τ v i + Bɛ ij J j, σ xx = ρ2 τ 1 iω ρ 3 B M (τ 1 iω) 2 + ωc 2, σ xy = M 2 [(τ 1 iω) 2 + ωc] 2, with σ yx = σ xy, σ yy = σ xx, and cyclotron frequency ω c ρb M.

14 Drude Magnetotransport 79 Drude Conductivity let s return to our toy Drude model (d = 2): with J i ρv i. conductivity matrix is iωmv i = ρe i M τ v i + Bɛ ij J j, σ xx = ρ2 τ 1 iω ρ 3 B M (τ 1 iω) 2 + ωc 2, σ xy = M 2 [(τ 1 iω) 2 + ωc] 2, with σ yx = σ xy, σ yy = σ xx, and cyclotron frequency ω c ρb M. what about d = 3? σ xx, σ xy etc. unchanged, and σ zz = ρ2 τ M, σ xz = 0,...

15 Drude Magnetotransport 80 Drude Resistivity the Drude model gives simpler formula for resistivity: ρ xx = ρ yy = M ρ 2 ( 1 τ iω ), ρ xy = ρ yx = B ρ.

16 Drude Magnetotransport 80 Drude Resistivity the Drude model gives simpler formula for resistivity: ρ xx = ρ yy = M ρ 2 ( 1 τ iω ), ρ xy = ρ yx = B ρ. ω = 0: if ω c τ 1, we have σ xx = M τb 2 = ρ2 B 2 ρ xx and so σ xx and ρ xx are proportional.

17 Some Experimental Puzzles 81 The Hall Angle the Drude model makes a very simple prediction: tan θ H σ xy Bσ xx = ρτ ω 0,B 0 M = σ xx ρ. B=0,ω=0 where θ H is called the Hall angle J θ H E

18 Some Experimental Puzzles 81 The Hall Angle the Drude model makes a very simple prediction: tan θ H σ xy = ρτ ω 0,B 0 M = σ xx ρ. B=0,ω=0 Bσ xx where θ H is called the Hall angle J θ H E this relation is violated in the strange metal phase of cuprates [Chien, Wang, Ong (1991)] among other materials: σ xx 1 T, tan θ H 1 T 2

19 Some Experimental Puzzles 82 Universal Dissipative Transport? the magnetic field may give universal corrections to dissipative transport in certain strange metals: [Hayes et al; ] ρ xx = M ρ 2 τ Drude ( T T 0 ) 2 ( ) B 2 +. B 0

20 The Memory Matrix Formalism 83 Review of the Memory Matrix Approach recall: Drude formula becomes rigorous in weak disorder limit

21 The Memory Matrix Formalism 83 Review of the Memory Matrix Approach recall: Drude formula becomes rigorous in weak disorder limit for slow operators A, B, the conductivity is σ AB = χ AC (M(ω) + N iωχ) 1 CD χ DB, with M the memory matrix (we ve discussed), and N AB = 1 (A Ḃ) = χ T AḂ.

22 The Memory Matrix Formalism 83 Review of the Memory Matrix Approach recall: Drude formula becomes rigorous in weak disorder limit for slow operators A, B, the conductivity is σ AB = χ AC (M(ω) + N iωχ) 1 CD χ DB, with M the memory matrix (we ve discussed), and N AB = 1 (A Ḃ) = χ T AḂ. as magnetic fields break time reversal symmetry, N AB 0

23 The Memory Matrix Formalism 83 Review of the Memory Matrix Approach recall: Drude formula becomes rigorous in weak disorder limit for slow operators A, B, the conductivity is σ AB = χ AC (M(ω) + N iωχ) 1 CD χ DB, with M the memory matrix (we ve discussed), and N AB = 1 (A Ḃ) = χ T AḂ. as magnetic fields break time reversal symmetry, N AB 0 with long lived momentum: 4 slow operators (in d = 2): P x, P y, J x, J y

24 The Memory Matrix Formalism 84 The N Matrix the most important elements of N are N PyP x = N PxP y = 1 T (P x P y ) = B T (P x J x ) = Bρ

25 The Memory Matrix Formalism 84 The N Matrix the most important elements of N are N PyP x = N PxP y = 1 T (P x P y ) = B T (P x J x ) = Bρ recall that if H = H 0 + ɛh imp with [H 0, P i ] = 0, then M Pi P j ɛ 2 ; and if B ω ɛ 2 : or σ Ji J j = χ Ji P k (M(ω) + N iωχ) 1 P k P l χ Pl J j + O(ɛ 0 ), ( σ ij = ρ 2 M(τ 1 iωχ) Bρ Bρ M(τ 1 iωχ) ) 1

26 The Memory Matrix Formalism 84 The N Matrix the most important elements of N are N PyP x = N PxP y = 1 T (P x P y ) = B T (P x J x ) = Bρ recall that if H = H 0 + ɛh imp with [H 0, P i ] = 0, then M Pi P j ɛ 2 ; and if B ω ɛ 2 : or σ Ji J j = χ Ji P k (M(ω) + N iωχ) 1 P k P l χ Pl J j + O(ɛ 0 ), ( σ ij = ρ 2 M(τ 1 iωχ) Bρ Bρ M(τ 1 iωχ) ) 1 recover Drude formula when τ 1 B ω ɛ 2

27 The Memory Matrix Formalism 84 The N Matrix the most important elements of N are N PyP x = N PxP y = 1 T (P x P y ) = B T (P x J x ) = Bρ recall that if H = H 0 + ɛh imp with [H 0, P i ] = 0, then M Pi P j ɛ 2 ; and if B ω ɛ 2 : or σ Ji J j = χ Ji P k (M(ω) + N iωχ) 1 P k P l χ Pl J j + O(ɛ 0 ), ( σ ij = ρ 2 M(τ 1 iωχ) Bρ Bρ M(τ 1 iωχ) ) 1 recover Drude formula when τ 1 B ω ɛ 2 M Pi P j, and thus τ, not affected by B (to leading order in ɛ) [Lucas, Sachdev; ]

28 The Memory Matrix Formalism 85 Exact Derivation of the Hall Conductivity the Drude limit is increasingly constrained (triple perturbative expansion)

29 The Memory Matrix Formalism 85 Exact Derivation of the Hall Conductivity the Drude limit is increasingly constrained (triple perturbative expansion) thus let us specialize to ω = 0

30 The Memory Matrix Formalism 85 Exact Derivation of the Hall Conductivity the Drude limit is increasingly constrained (triple perturbative expansion) thus let us specialize to ω = 0 treat B non-perturbatively

31 The Memory Matrix Formalism 85 Exact Derivation of the Hall Conductivity the Drude limit is increasingly constrained (triple perturbative expansion) thus let us specialize to ω = 0 treat B non-perturbatively assume only weak momentum relaxation (due to disorder)

32 The Memory Matrix Formalism 85 Exact Derivation of the Hall Conductivity the Drude limit is increasingly constrained (triple perturbative expansion) thus let us specialize to ω = 0 treat B non-perturbatively assume only weak momentum relaxation (due to disorder) if no disorder, then M APi = i T ( A q(z qlq) 1 q P i ) = i T Bɛ ij( A q(z qlq) 1 q J j ) = 0 for any A, because J j ) is a slow operator!

33 The Memory Matrix Formalism 85 Exact Derivation of the Hall Conductivity the Drude limit is increasingly constrained (triple perturbative expansion) thus let us specialize to ω = 0 treat B non-perturbatively assume only weak momentum relaxation (due to disorder) if no disorder, then M APi = i T ( A q(z qlq) 1 q P i ) = i T Bɛ ij( A q(z qlq) 1 q J j ) = 0 for any A, because J j ) is a slow operator! we also have N Ji P j = (J i P j ) T [Lucas; in progress] = ɛ jk χ Ji J k, N Pi P j = (P i P j ) = ɛ jk χ Pi J T k

34 The Memory Matrix Formalism 86 Exact Derivation of the Hall Conductivity (assuming isotropy) the electrical conductivity is σ = ( ) ( ) 1 ( Bχ χ JP χ JP ɛ Bχ JJ ɛ χjp JJ Bχ JJ ɛ M JJ + N JJ χ JJ ) where ɛ is a matrix corresponding to ɛ ij

35 The Memory Matrix Formalism 86 Exact Derivation of the Hall Conductivity (assuming isotropy) the electrical conductivity is σ = ( ) ( ) 1 ( Bχ χ JP χ JP ɛ Bχ JJ ɛ χjp JJ Bχ JJ ɛ M JJ + N JJ χ JJ ) where ɛ is a matrix corresponding to ɛ ij use block matrix inversion identities: σ = χ JP ( Bχ JP ɛ) 1 χ JP + X[M JJ + N JJ Bχ JJ ɛ( Bχ JP ɛ) 1 Bχ JJ ɛ] 1 X where we find X = χ JJ ( Bχ JP ɛ) 1 ( Bχ JJ ɛ)χ JP = 0.

36 The Memory Matrix Formalism 86 Exact Derivation of the Hall Conductivity (assuming isotropy) the electrical conductivity is σ = ( ) ( ) 1 ( Bχ χ JP χ JP ɛ Bχ JJ ɛ χjp JJ Bχ JJ ɛ M JJ + N JJ χ JJ ) where ɛ is a matrix corresponding to ɛ ij use block matrix inversion identities: σ = χ JP ( Bχ JP ɛ) 1 χ JP + X[M JJ + N JJ Bχ JJ ɛ( Bχ JP ɛ) 1 Bχ JJ ɛ] 1 X where we find X = χ JJ ( Bχ JP ɛ) 1 ( Bχ JJ ɛ)χ JP = 0. thus we find the Hall conductivity [Lucas; in progress] σ ij = χ JP B ɛ ij = ρ B ɛ ij.

37 The Memory Matrix Formalism 87 Adding Weak Momentum Relaxation what if we have weak inhomogeneity, so M P P 0?

38 The Memory Matrix Formalism 87 Adding Weak Momentum Relaxation what if we have weak inhomogeneity, so M P P 0? a straightforward generalization of above argument gives σ = χ JP (M P P + δn P P Bχ JP ɛ) 1 χ JP + O(ɛ 4 ) with δn P P arising from disorder corrections to (P x P y ).

39 The Memory Matrix Formalism 87 Adding Weak Momentum Relaxation what if we have weak inhomogeneity, so M P P 0? a straightforward generalization of above argument gives σ = χ JP (M P P + δn P P Bχ JP ɛ) 1 χ JP + O(ɛ 4 ) with δn P P arising from disorder corrections to (P x P y ). dissipative magnetotransport given by (e.g.) σ xx M yy B 2, and with H H 0 = ɛh imp ɛ d d xh(x)o(x): [Lucas; in progress] M ij = i T ( P i q(z qlq) 1 q P j ) d d k (2π) d k ik j h(k) 2 1 lim ω 0 ω Im ( G R OO(k, ω) ) }{{} evalauted at B 0

40 Hydrodynamic Magnetotransport 88 The Long Wavelength Limit thus we find that for weak inhomogeneity the Drude picture qualitatively holds, with universal formulas for τ 1

41 Hydrodynamic Magnetotransport 88 The Long Wavelength Limit thus we find that for weak inhomogeneity the Drude picture qualitatively holds, with universal formulas for τ 1 for stronger inhomogeneity, let us again resort to a hydrodynamic picture: l ee s(x) > 0 n(x) > 0 n(x) < 0 n x

42 Hydrodynamic Magnetotransport 89 Magnetic Fields in Generalized Hydrodynamics recall: linearized generalized hydrodynamics: i Ji A ( = 0 = i n A v i + Σ AB ( Ei B i µ B)) 0 = n A ( i µ A Ei A ) j (η ijkl k v l ).

43 Hydrodynamic Magnetotransport 89 Magnetic Fields in Generalized Hydrodynamics recall: linearized generalized hydrodynamics: i J A i = 0 = i ( n A v i + Σ AB ( E B i i µ B)) 0 = n A ( i µ A E A i ) j (η ijkl k v l ). in the absence of disorder, require consistency with Hall conductivity: 0 = n c E i + B ij J c j

44 Hydrodynamic Magnetotransport 89 Magnetic Fields in Generalized Hydrodynamics recall: linearized generalized hydrodynamics: i J A i = 0 = i ( n A v i + Σ AB ( E B i i µ B)) 0 = n A ( i µ A E A i ) j (η ijkl k v l ). in the absence of disorder, require consistency with Hall conductivity: 0 = n c E i + B ij J c j consistency with above equations: i J A i = 0 = i ( n A v i + Σ AB ( E B i + δ Bc B ij v j i µ B)) 0 = n A ( i µ A E A i ) B ij J c j j (η ijkl k v l ).

45 Hydrodynamic Magnetotransport 89 Magnetic Fields in Generalized Hydrodynamics recall: linearized generalized hydrodynamics: i J A i = 0 = i ( n A v i + Σ AB ( E B i i µ B)) 0 = n A ( i µ A E A i ) j (η ijkl k v l ). in the absence of disorder, require consistency with Hall conductivity: 0 = n c E i + B ij J c j consistency with above equations: i J A i = 0 = i ( n A v i + Σ AB ( E B i + δ Bc B ij v j i µ B)) 0 = n A ( i µ A E A i ) B ij J c j j (η ijkl k v l ). solve equations in inhomogeneous background, compute d d x J A i

46 Hydrodynamic Magnetotransport 90 Two Dimensions in d = 2: current conservation implies k ɛ ki (Bɛ ij J j ) = 0: ɛ ki k n A ( i µ A Ei A ) = ɛ ki i j (η ijmn m v n ) η 2 ɛ ki i v k

47 Hydrodynamic Magnetotransport 90 Two Dimensions in d = 2: current conservation implies k ɛ ki (Bɛ ij J j ) = 0: ɛ ki k n A ( i µ A E A i ) = ɛ ki i j (η ijmn m v n ) η 2 ɛ ki i v k transport problem ill-posed without viscosity! M d 2 τ k k 2 1 ( ) (2π) 2 2 µ(k) 2 lim ω 0 ω Im G Ṙ P P (k, ω) d 2 k B2 µ(k) 2 (2π) 2 ηk 2

48 Hydrodynamic Magnetotransport 90 Two Dimensions in d = 2: current conservation implies k ɛ ki (Bɛ ij J j ) = 0: ɛ ki k n A ( i µ A E A i ) = ɛ ki i j (η ijmn m v n ) η 2 ɛ ki i v k transport problem ill-posed without viscosity! M d 2 τ k k 2 1 ( ) (2π) 2 2 µ(k) 2 lim ω 0 ω Im G Ṙ P P (k, ω) d 2 k B2 µ(k) 2 (2π) 2 ηk 2 conductivity may provide a good viscometer? σ xx 1 d 2 k µ(k) 2 η (2π) 2 k 2 1 log L η ξ }{{} IR divergent! [Patel, Davison, Levchenko; ] [Lucas; in progress]

49 Hydrodynamic Magnetotransport 91 Three Dimensions in d = 3, the current constraint is less severe: k ɛ nki (ɛ ijm J j B m ) = B k k J n 0 and transport problem is well-posed without viscosity

50 Hydrodynamic Magnetotransport 91 Three Dimensions in d = 3, the current constraint is less severe: k ɛ nki (ɛ ijm J j B m ) = B k k J n 0 and transport problem is well-posed without viscosity for viscous dominated transport: σ ij 1 d 3 k (3 + η (2π) 3 µ(k) 2 cos2 θ)b 2 ρ 2 k i k j B 2 ρ 2 cos 2 θ + k 4 }{{} very anisotropic σ ij!

51 Hydrodynamic Magnetotransport 91 Three Dimensions in d = 3, the current constraint is less severe: k ɛ nki (ɛ ijm J j B m ) = B k k J n 0 and transport problem is well-posed without viscosity for viscous dominated transport: σ ij 1 d 3 k (3 + η (2π) 3 µ(k) 2 cos2 θ)b 2 ρ 2 k i k j B 2 ρ 2 cos 2 θ + k 4 }{{} very anisotropic σ ij! for diffusion dominated transport (generic situation): d 3 k k i k j σ ij Σ (2π) 3 µ(k) 2 k 2 (b 1 b 2 cos 2 θ) }{{} minor anisotropy in σ ij [Baumgartner, Karch, Lucas; ] [Lucas; in progress]

52 Hydrodynamic Magnetotransport 92 Accounting for σ q less rigorous: employ relaxation time approximation J i = ρv i + σ q (E i + Bɛ ij v j ), M τ v i = ρe i + Bɛ ij J j

53 Hydrodynamic Magnetotransport 92 Accounting for σ q less rigorous: employ relaxation time approximation J i = ρv i + σ q (E i + Bɛ ij v j ), M τ v i = ρe i + Bɛ ij J j the conductivity matrix (d = 2): σ xx = τ 1 Mσ q + ρ 2 + B 2 σ 2 q ρ 2 B 2 + [τ 1 M + B 2 σ q ] 2 M τ, σ xy = 2τ 1 Mσ q + ρ 2 + B 2 σ 2 q ρ 2 B 2 + [τ 1 M + B 2 σ q ] 2 Bρ.

54 Hydrodynamic Magnetotransport 92 Accounting for σ q less rigorous: employ relaxation time approximation J i = ρv i + σ q (E i + Bɛ ij v j ), M τ v i = ρe i + Bɛ ij J j the conductivity matrix (d = 2): σ xx = τ 1 Mσ q + ρ 2 + B 2 σ 2 q ρ 2 B 2 + [τ 1 M + B 2 σ q ] 2 M τ, σ xy = 2τ 1 Mσ q + ρ 2 + B 2 σ 2 q ρ 2 B 2 + [τ 1 M + B 2 σ q ] 2 Bρ. poles in the conductivity at ω = ± ρb M }{{} i B2 σ q M }{{} cyclotron pole violation of Kohn s theorem [Hartnoll, Kovtun, Müller, Sachdev; ] i τ

55 Hydrodynamic Magnetotransport 93 Revisiting the Hall Angle this relaxation time model resolves our Hall angle puzzle if tan θ H σ xy ρτ B 0 M 1 T 2, τ 1 T 2 Bσ xx σ xx B=0 = σ q + ρ2 τ M σ q, σ q 1 T

56 Hydrodynamic Magnetotransport 93 Revisiting the Hall Angle this relaxation time model resolves our Hall angle puzzle if tan θ H σ xy ρτ B 0 M 1 T 2, τ 1 T 2 Bσ xx σ xx B=0 = σ q + ρ2 τ M σ q, σ q 1 T unverified prediction: as T 0 (in strange metal), σ 1/T 2?

57 Hydrodynamic Magnetotransport 93 Revisiting the Hall Angle this relaxation time model resolves our Hall angle puzzle if tan θ H σ xy ρτ B 0 M 1 T 2, τ 1 T 2 Bσ xx σ xx B=0 = σ q + ρ2 τ M σ q, σ q 1 T unverified prediction: as T 0 (in strange metal), σ 1/T 2? similar formulas hold in holographic models, even beyond the hydrodynamic regime [Blake, Donos; ]

58 Thermoelectric Magnetotransport 94 Hydrodynamics with Conserved Charge/Heat incoherent conductivities ( σq T α Σ = q T α q T κ q ) together with relaxation time approximation give: α xx = J x x T = (τ 1 M + B 2 σ q )α q + ρs M ρ 2 B 2 + [τ 1 M + B 2 σ q ] 2 τ α xy = J x y T = τ 1 Mρα q + (ρ 2 + B 2 σq 2 + σ q Mτ 1 )s ρ 2 B 2 + [τ 1 M + B 2 σ q ] 2 B κ xx = Q x x T = κ q + (s2 B 2 αq)(b 2 2 σ q + τ 1 M) 2sρα q B 2 ρ 2 B 2 + [τ 1 M + B 2 σ q ] 2 T, κ xy = Q x y T = ρs2 B 2 ραq 2 + 2sα q (B 2 σ q + τ 1 M) ρ 2 B 2 + [τ 1 M + B 2 σ q ] 2 BT,

59 Thermoelectric Magnetotransport 95 Magnetization Currents: Review of Bound Currents to compute J i and Q i microscopically, we must subtract out magnetization currents: M ij J i mag

60 Thermoelectric Magnetotransport 95 Magnetization Currents: Review of Bound Currents to compute J i and Q i microscopically, we must subtract out magnetization currents: M ij J i mag as in textbook electromagnetism: Jmag i = j M ij, Kbound i = M ijn j with M ij 0 on the (inhomogeneous?) background

61 Thermoelectric Magnetotransport 95 Magnetization Currents: Review of Bound Currents to compute J i and Q i microscopically, we must subtract out magnetization currents: M ij J i mag as in textbook electromagnetism: J i mag = j M ij, K i bound = M ijn j with M ij 0 on the (inhomogeneous?) background on first glance: J mag does not contribute to transport? d Javg i = σ ij d x E j j M ij = 0. V d

62 Thermoelectric Magnetotransport 95 Magnetization Currents: Review of Bound Currents to compute J i and Q i microscopically, we must subtract out magnetization currents: M ij J i mag as in textbook electromagnetism: J i mag = j M ij, K i bound = M ijn j with M ij 0 on the (inhomogeneous?) background on first glance: J mag does not contribute to transport? d Javg i = σ ij d x E j j M ij = 0. V d for α ij, κ ij : cannot ignore magnetization currents

63 Thermoelectric Magnetotransport 96 Magnetization from Thermodynamics a thermodynamic definition of M ij : (in d = 3) M ij = P, P = P (µ, T, X) B ij X 1 ) 8 (ɛµνρσ ν u ν F ρσ ) (ɛ ρ σ µ u ν F ρ σ 1 2 B2

64 Thermoelectric Magnetotransport 96 Magnetization from Thermodynamics a thermodynamic definition of M ij : (in d = 3) M ij = P, P = P (µ, T, X) B ij X 1 ) 8 (ɛµνρσ ν u ν F ρσ ) (ɛ ρ σ µ u ν F ρ σ 1 2 B2 M ij = B ij P X

65 Thermoelectric Magnetotransport 96 Magnetization from Thermodynamics a thermodynamic definition of M ij : (in d = 3) M ij = P, P = P (µ, T, X) B ij X 1 ) 8 (ɛµνρσ ν u ν F ρσ ) (ɛ ρ σ µ u ν F ρ σ 1 2 B2 M ij ij P = B X as in first lecture, impose a thermal drive ζ i : ds 2 = η µν dx µ dx ν 2tζ i dtdx i.

66 Thermoelectric Magnetotransport 96 Magnetization from Thermodynamics a thermodynamic definition of M ij : (in d = 3) M ij = P, P = P (µ, T, X) B ij X 1 ) 8 (ɛµνρσ ν u ν F ρσ ) (ɛ ρ σ µ u ν F ρ σ 1 2 B2 M ij ij P = B X as in first lecture, impose a thermal drive ζ i : ds 2 = η µν dx µ dx ν 2tζ i dtdx i. X shifts in background gauge field F µν + ζ i : δx B ij ζ i t t δa j

67 Thermoelectric Magnetotransport 97 Magnetization Currents for α the thermodynamic generating functional is [ Z = exp[w [g µν, A µ ]] = exp i d d x P ] T

68 Thermoelectric Magnetotransport 97 Magnetization Currents for α the thermodynamic generating functional is [ Z = exp[w [g µν, A µ ]] = exp i d d x P ] T the current is J i = it δz δa i t ( ) tb ij P ζ j X

69 Thermoelectric Magnetotransport 97 Magnetization Currents for α the thermodynamic generating functional is [ Z = exp[w [g µν, A µ ]] = exp i d d x P ] T the current is J i = it δz δa i t ( ) tb ij P ζ j X one arrives at [Cooper, Halperin, Ruzin; cond-mat/ ] J i = M ij ζ j + and a similar equation for Q i : Q i = M ij E j M ij th ζ j +

70 Thermoelectric Magnetotransport 97 Magnetization Currents for α the thermodynamic generating functional is [ Z = exp[w [g µν, A µ ]] = exp i d d x P ] T the current is J i = it δz δa i t ( ) tb ij P ζ j X one arrives at [Cooper, Halperin, Ruzin; cond-mat/ ] J i = M ij ζ j + and a similar equation for Q i : Q i = M ij E j M ij th ζ j + these magnetization currents are thermodynamic; need to be subtracted out

71 Summary of the Lectures 98 What Have we Learned? A Rigorous Drude Model the past 5 years have seen large advances in transport theory: formal Drude theory for weakly disordered systems: σ(ω) = ρ2 M 1 1 τ iω.

72 Summary of the Lectures 98 What Have we Learned? A Rigorous Drude Model the past 5 years have seen large advances in transport theory: formal Drude theory for weakly disordered systems: σ(ω) = ρ2 M 1. 1 τ iω precise formula for Drude relaxation time: M τ lim 1 ω 0 ω Im ( G Ṙ P P (ω)) (also valid in background magnetic field)

73 Summary of the Lectures 98 What Have we Learned? A Rigorous Drude Model the past 5 years have seen large advances in transport theory: formal Drude theory for weakly disordered systems: σ(ω) = ρ2 M 1. 1 τ iω precise formula for Drude relaxation time: M τ lim 1 ω 0 ω Im ( G Ṙ P P (ω)) (also valid in background magnetic field) kinetic and hydrodynamic treatment of spectral weight in quasiparticle systems

74 Summary of the Lectures 98 What Have we Learned? A Rigorous Drude Model the past 5 years have seen large advances in transport theory: formal Drude theory for weakly disordered systems: σ(ω) = ρ2 M 1. 1 τ iω precise formula for Drude relaxation time: M τ lim 1 ω 0 ω Im ( G Ṙ P P (ω)) (also valid in background magnetic field) kinetic and hydrodynamic treatment of spectral weight in quasiparticle systems scaling theories (many holographic) in non-quasiparticle theories

75 Summary of the Lectures 99 What Have we Learned? Hydrodynamics and Conductivity Bounds when the inhomogeneity length scale ξ is large compared to l ee, use hydrodynamics to compute transport: i J A i = 0 = i ( n A v i + Σ AB ( E B i i µ B)) 0 = n A ( i µ A E A i ) j (η ijkl k v l ).

76 Summary of the Lectures 99 What Have we Learned? Hydrodynamics and Conductivity Bounds when the inhomogeneity length scale ξ is large compared to l ee, use hydrodynamics to compute transport: i J A i = 0 = i ( n A v i + Σ AB ( E B i i µ B)) 0 = n A ( i µ A E A i ) j (η ijkl k v l ). rigorous conductivity bounds! ρ xx T ṡ avg J 2 x J A =0.

77 Summary of the Lectures 99 What Have we Learned? Hydrodynamics and Conductivity Bounds when the inhomogeneity length scale ξ is large compared to l ee, use hydrodynamics to compute transport: i J A i = 0 = i ( n A v i + Σ AB ( E B i i µ B)) 0 = n A ( i µ A E A i ) j (η ijkl k v l ). rigorous conductivity bounds! ρ xx T ṡ avg J 2 x J A =0. universal semiclassical transport bound in kinetic theory [Lucas, Hartnoll; ]

78 Summary of the Lectures 100 What is Left to Do? generic magnetotransport theory at weak disorder [Lucas; in progress]

79 Summary of the Lectures 100 What is Left to Do? generic magnetotransport theory at weak disorder [Lucas; in progress] resistivity bounds beyond the semiclassical limit?

80 Summary of the Lectures 100 What is Left to Do? generic magnetotransport theory at weak disorder [Lucas; in progress] resistivity bounds beyond the semiclassical limit? how is transport limited/bounded by quantum chaos?

81 Summary of the Lectures 100 What is Left to Do? generic magnetotransport theory at weak disorder [Lucas; in progress] resistivity bounds beyond the semiclassical limit? how is transport limited/bounded by quantum chaos? experimental observations of electronic hydrodynamics: origin of the T -linear resistivity? practical applications: good viscous conductors