Strongly coupled metals and insulators Sean Hartnoll (Stanford) - PowerPoint PPT Presentation

Strongly coupled metals and insulators Sean Hartnoll (Stanford) Gauge/gravity duality 2013 @ MPI, Munich Saturday, July 27, 13

Take home buzzwords Memory matrix. Wiedemann-Franz law. Insulators. Saturday, July 27, 13

Charge transport at strong coupling • Most computations of conductivities etc. use the Boltzmann equation: E · @ f k − ~ = − I ei [ f k ] − I ee [ f k ] @~ k • Assumes long lived ‘quasiparticles’, not useful at strong coupling. • First objective: effective field theory framework for strongly coupled transport. Saturday, July 27, 13

• Theorem (1960s, easy): If there exists a conserved quantity P that overlaps with the electrical current operator J, i.e. χ P J 6 = 0 Then the d.c. conductivity is infinite: σ ∼ χ 2 P J δ ( ω ) χ P P • Example: absence of lattice and impurities ⇒� momentum conserved Saturday, July 27, 13

• Consequence: Suppose conservation of P is violated only by an irrelevant operator O in the low energy effective theory. Then the d.c. conductivity is large: σ = χ 2 Relaxation rate 1 P J Γ χ P P • At low temperatures, dominant T dependence is from Γ . Thus, resistivity: ρ ∼ Γ Saturday, July 27, 13

• The small scale Γ furthermore gives a Drude peak: Γ ≪ E F σ 1 Γ ω E F Z ∞ There is a sum rule that: Re σ ( ω ) d ω 0 is given by fixed UV data. Saturday, July 27, 13

Almost conserved quantities • (From Andrei-Shimshoni-Rosch ’03, studying Luttinger liquids) χ 2 D = 1 JP 2 χ P P Saturday, July 27, 13

Memory matrix • It is a theorem that X χ JA M − 1 σ = lim AB ( ω ) χ BJ ω → 0 AB • With Almost-conserved quantities Z 1 /T ⌧ � i ˙ ω − Q L QQ ˙ M AB ( ω ) = A (0) Q B ( i λ ) d λ 0 • Where L is the `Liouville operator’ LA =[H,A] and Q projects onto the space of operators orthogonal to the set A,B,... we kept. Saturday, July 27, 13

• Remnant of UV lattice in IR is a momentum-carrying operator O(k L ). • If O is irrelevant, Γ can be computed perturbatively in the IR coupling g of O: Γ = g 2 k 2 Im G R � OO ( ω , k L ) L � lim � ω → 0 χ P P ω � g =0 Hartnoll-Hofman @ 1201.3917 (case of impurities: Hartnoll-Kovtun-Muller-Sachdev @ 0706.3215) Saturday, July 27, 13

• Results quoted so far are all derived using the ‘‘memory matrix formalism’’. • This formalism builds around almost- conserved quantities, and is the correct way to think about charge transport in strongly correlated metallic systems. Suggested reading: Hartnoll-Hofman @ 1201.3917 Mahajan-Barkeshli-Hartnoll @ 1304.4249 Andrei-Shimshoni-Rosch @ cond-mat/0307578 Saturday, July 27, 13

Fermi liquids: The physics is at nonzero momentum • Famously (1930s!): a clean Fermi liquid has a low T electrical resistivity ρ ∼ T 2 • An effective field theory derivation of this result reveals nontrivial physics. Saturday, July 27, 13

• Recall the formula for relaxation rate: Γ = g 2 k 2 Im G R � OO ( ω , k L ) L � lim � ω → 0 χ P P ω � g =0 • Significant relaxation requires low energy spectral weight (i.e. on shell excitations) at nonzero momentum k L . • Clearly, such excitations do not exist in e.g. a Lorentz invariant theory: Γ ∼ e − k L /T ω ∼ k ⇒ Saturday, July 27, 13

• In a Fermi Liquid, low energy excitations live on the Fermi surface. • Leading irrelevant operator with finite momentum is the umklapp operator: Z 4 ! Y ψ † ( k 1 ) ψ † ( k 2 ) ψ ( k 3 ) ψ ( k 4 ) δ ( k 1 + k 2 − k 3 − k 4 − k L ) d ω i d 2 k i O ( k L ) = i =1 Saturday, July 27, 13

• Using the RG flow for Fermi surfaces of Polchinski (hep-th/9210046), the umklapp operator O(w,k) has scaling dimension 𝚬 =1. It is irrelevant. • Dimensional analysis then gives ρ ∼ Γ ∼ T 2 [Hartnoll-Hofman (1201.3917)] • Lesson: resistivity of a Fermi liquid depends upon the interplay of two momentum scales: k L and k F . Saturday, July 27, 13

Holography at nonzero density 101 • Density ⇒� Electric flux at boundary. ����� ���������� ����� ������� ��������� ����� ���������������� �������� • IR physics determined by near horizon geometry. Saturday, July 27, 13

• An interesting class of IR geometries is AdS 2 x R 2 or conformal to AdS 2 x R 2 (space does not scale!) • Via formula for Γ , power law resistivity: Im G R J t J t ( ω , k L ) ∼ T 2 ∆ ( k L ) ρ ∼ lim ω → 0 ω Hartnoll-Hofman (1201.3917), Hartnoll-Shaghoulian (1203.4236) Anantua-Hartnoll-Martin-Ramirez (1210.1590) Verified with numerical lattice by Horowitz-Santos-Tong (1204.0519) Saturday, July 27, 13

Wiedemann-Franz law • Electrical and thermal conductivity: ✓ J x ◆ ✓ ◆ ✓ ◆ α T E x σ = Q x α T κ T ¯ � ( r x T ) /T • Ratio of conductivities in a Fermi liquid: d ✏✏ 2 f 0 � T = ⇡ 2 R FD ( ✏ ) 3 ≡ L 0 = 1 L ≡  R T 2 FD ( ✏ ) d ✏ f 0 • The WF law requires: (i) long lived electronic quasiparticles. (ii) no additional heat carriers. (iii) elastic scattering. Saturday, July 27, 13

Lots of conserved quantities in a Fermi liquid • Low energy description: Patchwise excitations of a Fermi surface (Shankar, Polchinski). • Each patch theory is a free fermion: k θ Z k F ( θ ) dk ✏ θ k c † H θ = θ k c θ k • Infinitely many conserved quantities: δ n θ k ≡ c † θ k c θ k Saturday, July 27, 13

Ratio in Non-Fermi liquids • Strong interactions to arbitrarily low energies. • Simplest expectation: only almost-conserved quantity in effective low energy theory is total momentum P. κ = 1 σ = χ 2 M − 1 � T χ 2 M − 1 � � � P P , JP QP P P • Implying that: Universal ratio of thermodynamic susceptilibites χ 2 σ T = 1 κ QP χ 2 T 2 JP (Mahajan-Barkeshli-Hartnoll @ 1304.4249) Saturday, July 27, 13

Some NFLs violate WF • All suggestive • YbRh 2 Si 2 , of strong YbAgGe, interactions? c-axis CeCoIn 5 , YBCO. Saturday, July 27, 13

Some “NFLs” satisfy WF a-axis CeCoIn 5 , CeRhIn 5 , Sr 3 Ru 2 O 7 . Saturday, July 27, 13

Quick lessons from data • Not all linear in temperature resistivities are the same. • Some are suggestive of strong coupling physics (violate WF). • Others necessarily have a quasiparticle description (satisfy WF). Mahajan-Barkeshli-Hartnoll @ 1304.4249 Saturday, July 27, 13

Holographic insulators Momentum space becomes relevant • So far, have described good metals: momentum non-conservation described by irrelevant operators in IR. Physics captured by (i) formula for Γ and (ii) knowledge of IR kinematics. • If the lattice operators becomes relevant in the IR, we might expect to obtain insulators or perhaps incoherent metals. (Donos-Hartnoll @ 1212.2998) Saturday, July 27, 13

Localization transitions 101 • Metal-insulator transitions are dramatic phenomena: re-arrangement of degrees of freedom from itinerant to localized. • Spectral weight transfer from the Drude peak to the UV scale. Saturday, July 27, 13

Theories of localization review: Dobrosavljevic @ 1112.6166 • Band insulators • Anderson localization (impurities): Free electrons in random potential have localized wavefunctions. • Mott transition (charge commensurability): Electrons ‘jam’ at half filling. Low energy excitations particle-hole symmetric: χ P J = 0 Saturday, July 27, 13

Holographic insulator • Objective: realize a new type of localization. Main input from holography is that operators get O(1) anomalous dimensions. • Bulk action: 4 W ab W ab − m 2 ✓ ◆ Z Z d 5 x √− g R + 12 − 1 4 F ab F ab − 1 − κ 2 B a B a S = B ∧ F ∧ W . 2 • Used a helical lattice to avoid solving PDEs: ω 2 + i ω 3 = e ipx 1 ( dx 2 + idx 3 ) B (0) = λ ω 2 cf. Ooguri-Park (1007.3737), Donos-Gauntlett (1109.3866), Iizuka-Kachru-Kundu-Narayan-Sircar-Trivedi (1201.4861) Saturday, July 27, 13

�� �� • As a function of UV parameters, IR geometry undergoes a phase transition when lattice becomes relevant. �������� � �������� � ds 2 = − cr 2 dt 2 + dr 2 cr 2 + dx 2 r 1 / 3 + r 2 / 3 ω 2 2 + r 1 / 3 ω 2 1 3 , A = 0 , B = b ω 2 . �������� �������� ��������� ��������� � �������� ����������� � Zero temperature IR geometry Saturday, July 27, 13

• Spectral weight transfer! (Donos-Hartnoll @ 1212.2998) 10 2.0 8 1.5 Re H Σ L H A.U. L Re H Σ L H A.U. L 6 1.0 4 0.5 2 0.0 0 0.0 0.1 0.2 0.3 0.4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Ω ê Μ Ω ê Μ Metal Insulator σ ( ω ) ∼ ω 4 / 3 at T = 0 Saturday, July 27, 13

Conclusions • Strongly coupled metallic transport should be organized around almost conserved quantities: Memory matrix. • The Wiedemann-Franz law differentiates non-Fermi liquids with and without quasiparticles. • Holography enables the realization of a new, strong coupling, mechanism of charge localization -- lattice scattering becomes relevant in the IR. Saturday, July 27, 13