Transport w/o quasiparticles

Transcription

1 Transport w/o quasiparticles Good metals, bad metals and insulators Sean Hartnoll (Stanford) -- Caneel Bay, Jan Based on: w/ Diego Hofman w/ Aristos Donos (also work in progress with Barkeshli & Mahajan)

2 Black holes and dissipation The single most important reason that holography is useful is the unique way that black holes act as classical dissipative systems. E.g. Witczak-Krempa and Sachdev: holographic transport at quantum criticality satisfies sum rules that are obscured in a Boltzmann approach. This talk will be about transport in strongly correlated finite density systems: hj t i6=0

3 Finite density transport If the total momentum (or any other operator that overlaps with the total current) is conserved, the d.c. conductivity is infinite. The optical conductivity of a perfect metal: s 1 1 w HeVL Only alternative to breaking momentum conservation is to dilute the charge carriers. Makes spectral weight of delta function small: JJ 2 JP PP 1

4 Observed behaviors Conventional metals (sharp Drude peak) Strange metals (unconventional scalings) Bad metals (no Drude peak, violate MIR bound) Insulators (vanishing dc conductivity)

5 A conventional metal Optical conductivity in graphene by Li et al

6 Metal-insulator transitions Dramatic spectral weight transfer from Drude peak to interband scales: Itinerant to localized charge Uchida et al. 91 Nicoletti et al 11

7 Gapless insulators Quantum spin liquid candidates show a power law soft gap in the optical conductivity Herbertsmithite by Pilon et al Elsässer et al

8 A bad metal La 1.9 Sr 0.1 CuO 4 by Takenaka et al. 03, from Hussey et al. 04

9 Theory of sharp Drude peaks O(k L )= Sharp Drude peak Momentum relaxation rate! small Can treat momentum-nonconserving operators (remnant of UV lattice) as perturbations of a translationally invariant effective IR theory. E.g. Umklapp scattering in a Fermi liquid: Z 4Y i=1 d! i d 2 k i! In holographic models, often least irrelevant operator is: (k 1 ) (k 2 ) (k 3 ) (k 4 ) (k 1 + k 2 k 3 k 4 k L ) J t (k L )

10 Theory of sharp Drude peaks Framework to treat! perturbatively: Memory matrix formalism. (cf. Rosch and Andrei, 1+1) 1 (!) =, i! + M(!) 1 M(!) = Z 1/T 0 For case of scattering by a lattice d A(0)Q i! QLQ QḂ(i ). = g2 k 2 L ~P ~ P lim!!0 Im G R OO (!,k L)! g=0. (Hartnoll and Hofman, )

11 Semi-local criticality (cf. Iqbal, Liu, Mezei) Common in holography that IR geometries have z = (with or without ground state entropy). Scaling of time but not space efficient low energy dissipation in momentum-violating processes. Find e.g. dc resistivity r(t ) T 2 (k L) (Hartnoll and Hofman, ) (Confirmed numerically by Horowitz, Santos, Tong) Theories with z < do not dissipate efficiently in momentum-violating processes.

12 Making the lattice relevant Claim: (at least some) metal-insulator transitions are described by momentum-nonconserving operators becoming relevant in the effective low energy theory. (cf. Emery, Luther, Peschel, 1+1) We found a holographic realization of this mechanism. (Donos and Hartnoll, ) Simple theory S = Z d 5 x p g R F abf ab 1 4 W abw ab m 2 2 B ab a apple 2 Z B ^ F ^ W. (Chern-Simons term not essential but helps to find the IR geometries)

13 RG flow scenarios The theory has (T=0) IR geometries both with and without translation invariance (Donos and Hartnoll, )

14 A technical simplification To capture the physics without solving PDEs we use a lattice that breaks translation invariance while retaining homogeneity: B (0) =! 2! 2 + i! 3 = e ipx 1 (dx 2 + idx 3 ) (Invariant under Bianchi VII 0 algebra, cf. Nakamura-Ooguri-Park, Donos-Gauntlett, Kachru-Trivedi-...) Beyond a simplification, realization of smectic metal phases due to strong yet anisotropic lattice scattering in the IR. (cf. Emery, Fradkin, Kivelson, Lubensky)

15 Metal-insulator transition (Donos and Hartnoll, ) The metallic and insulating IR geometries are: ds 2 = r 2 dt 2 + dr2 r 2 + dx2 R 3, A =2 p 6 rdt, B =0. ds 2 = cr 2 dt 2 + dr2 cr 2 + dx2 1 r 1/3 + r2/3! r 1/3! 2 3, A =0, B = b! 2. (cf. D Hoker and Kraus) WêHVm 4 L -20 sêh4pm 3 L pêm pêm

16 Spectral weight transfer (Donos and Hartnoll, ) ReHsL HA.U.L 6 4 ReHsL HA.U.L wêm wê m Metal Insulator (!)! 4/3 at T =0

17 Further comments Can compute d.c. conductivities analytically: metal: (T ) T 2 (k L), insulator: (T ) T 4/3. In between the metallic and insulating phases we found bad metals with no Drude peak and large resistivities. Mid-infrared peak in insulating phase. No immediate connection to commensurability. Insulators without underlying Mott physics?

18 Take home messages Objective: non-quasiparticle language for transport. Good metal: Effective low energy theory translation invariant up to perturbative effects of momentumnonconserving operators. Effects become relevant: metal-insulator transition. Holography precisely realizes this scenario. Simple model exhibits experimental features that are difficult to otherwise describe in a controlled way: major spectral weight transfer, bad metals, insulators with power law gaps.