From Critical Phenomena to Holographic Duality in Quantum Matter

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1 From Critical Phenomena to Holographic Duality in Quantum Matter Joe Bhaseen TSCM Group King s College London 2013 Arnold Sommerfeld School Gauge-Gravity Duality and Condensed Matter Physics Arnold Sommerfeld Center for Theoretical Physics, Munich 5-9 August

2 Lecture 2 Relativisitic Quantum Transport in 2+1 Dimensions Accompanying Slides

3 Useful Theory References M. Fisher, P. Weichman, G. Grinstein and D. Fisher Boson localization and the superfluid-insulator transition, Phys. Rev. B, 40, 546, (1989). K. Damle and S. Sachdev Nonzero temperature transport near quantum critical points, Phys. Rev. B, 56, 8714 (1997). S. Sachdev Quantum Phase Transitions, Cambridge. Chapter 8 Physics close to and above the UCD. Chapter 9 Transport in d = 2. Chapter 10 Boson Hubbard model.

4 Nernst Effect in the Cuprates Xu, Ong, Wang, Kakeshita and Uchida, Nature 406, 486 (2000) ν E y ( T) x B ν = 1 B α xy σ xx α xx σ xy σ 2 xx +σ2 xy B z T E y Nernst AFM SC ( T) x x

5 Ong s Data PRB 73, (2010) Numbers indicate ν in nv/kt La 2-x Sr x CuO T T* onset T (K) T c Sr content x

6 The Bose Hubbard Model H = t ij (a i a j +h.c.)+ U 2 µ 3U i n i(n i 1) µ i n i 2U SF U MI 0 t/u Fisher, Weichman, Grinstein and Fisher, PRB 40, 546 (1989) L = d D x µ Φ 2 m 2 Φ 2 u 0 3 Φ 4

7 Effective Field Theory Set up path integral representation for Z = Tr(e βh ) Using a Hubbard Stratonovich transformation to decouple the hopping term e W β 0 dτ ij (b i b j+b j b i) DΦ DΦexp( 0 dτ( Φ i b i Φ ib i )+Φ iw 1 ij Φ j) and integrating over the b i one may ultimately obtain Z = DΦDΦ e β 0 dτ d d xl B where L B = K 1 Φ Φ τ +K 2 Φ τ 2 +K 3 Φ 2 +K 4 Φ 2 +K 5 Φ 4

8 Currents and Normal Modes Rather than imaginary time diagrams, include temperature via a transport equation for normal modes of complex bosonic field: d d k 1 [ Φ(x, t) = (2π) d a + (k,t)e ik.x +a 2εk (k,t)e ik.x] d d k εk Π(x, t) = i [a (2π) d (k,t)e ik.x a 2 +(k,t)e ik.x] where ε k = k 2 +m 2. The (uniform) electric current reads J(t) = Q d d k (2π) d v k [f + (k,t) f (k,t)] where Q = 2e, v k = k/ǫ k and f ± (k,t) = a ±(k,t)a ± (k,t) Dropped anomalous terms aa and a a for ω < 2m.

9 Quantum Boltzmann Equation Real time & finite T Damle & Sachdev, PRB 56, 8714 ( 97) Opposite charge particles + applied field + interactions Φ 4 ( t ±QE(t). ) k f± (k,t) = 2u2 9 dµf± dµ dd k 1 (2π) d d d k 2 (2π) d d d k 3 (2π) d 1 16ε k ε k1 ε k2 ε k3 (2π) d δ d (k+k 1 k 2 k 3 )(2π)δ(ε k +ε k1 ε k2 ε k3 ). F ± (k,k 1,k 2,k 3 ) 2f ± (k)f (k 1 )[1+f ± (k 2 )][1+f (k 3 )] +f ± (k)f ± (k 1 )[1+f ± (k 2 )][1+f ± (k 3 )] 2[1+f ± (k)][1+f (k 1 )]f ± (k 2 )f (k 3 ) [1+f ± (k)][1+f ± (k 1 )]f ± (k 2 )f ± (k 3 ) We have suppressed the t dependence ε k = k 2 +m 2 ǫ = 3 d u = 24π2 ǫ 5 m 2 = 4ǫT2 15

10 Universal Transport Damle and Sachdev, PRB 56, 8714 (1997) σ(ω) = (2e)2 T d 2 ǫ 2 Σ ( ω ǫ 2 T ) d = 3 ǫ 0.15 Σ' I ω ~ Two spatial dimensions σ(0) ( ) 4e 2 h Crossover between hydrodynamic and collisionless regimes

11 Transport near Quantum Critical Points Linear response for SF-MI transition in Bose Hubbard σ(ω) = (2e)2 T d 2 ǫ 2 Damle and Sachdev, PRB 56, 8714 (1997) Σ ( ω ǫ 2 T ) d = 3 ǫ σ(0) Bhaseen, Green and Sondhi, PRL 98, (2007) ( ) 4e 2 h α xy = S B κ xx gǫ 2 Td+3 (2e) 2 B 2 g 5.5 Hartnoll, Kovtun, Müller and Sachdev, PRB 76, (2007) κ xx (B) = TS2 B 2 σ xx (0) Relativistic hydrodynamics & AdS/CFT link all coeffs QBE Müller et al, PRB (2008) Bhaseen et al, PRB (2009) Viscosity/Entropy Quark Gluon Plasma Graphene

12 Linearized Equations In the absence of E f ± (k,t) = n(ε k ) = 1/(e βε k 1) In the presence of E one may parameterise f ± (k,ω) = 2πδ(ω)n(ε k )±k.e(ω)ψ(k,ω) To linear order in E, and after three angular integrals and two radial (in d = 3 to leading order) one eventually obtains... iωψ(k,ω)+ 1 k n(k) k = ǫ 2 0 dk 1 [F 1 (k,k 1 )ψ(k,ω)+f 2 (k,k 1 )ψ(k 1,ω)] Integral equation for the departure from equilibrium

13 Kernels After a considerable slog, the kernels F i (k,k 1 ) can be explicitly evaluated. For example, F 1 = 6π 25 n(k 1 )n(k k 1 ) k 2 n(k) [Θ(k k 1 )µ 2 (k,k 1 ) Θ(k 1 k)µ 2 (k 1,k)] where µ n (x,y) β n[ ] Li n (1)+Li n (e βx ) Li n (e βy ) Li n (e β(x y) ). Li p (z) is the polylogarithm of order p, n(k) is the Bose distribution: Li p (z) = n=1 z n n p, n(k) = 1 e βk 1. Note that the mass has been set equal to zero in these expressions.

14 F 2 F a 2 +F b 2 F a 2 = 2π 75 F b 2 = 4π 75 [1+n(k)]n(k +k 1 ) k 4 L a n(k 1 ) 2(k,k 1 ) n(k)n(k 1 k) [ Θ(k k1 k 4 )L b n(k 1 ) 2(k,k 1 ) (k k 1 ) ] L a 2 = 24λ 4 +12[kη 3 +(k k 1 )] 6kk 1 λ + 2 L b 2 = 3[4µ 4 +2(k k 1 )µ 3 kk 1 µ 2 +4k 1 ν 3 +2kk 1 ν 2 ] λ ± n β n[ ] Li n (e βx )+Li n (e βy )±Li n (e β(x+y) )±Li n (1) η n β n[ ] Li n (e βx ) Li n (e βy ) Li n (e β(x+y) )+Li n (1) ν n β n[ Li n (e βx ) Li n (e βy ) ] Note that the kernels have integrable singularities when k = k 1.

15 Electrical Conductivity J(t) = Q d d k (2π) d v k [f + (k,t) f (k,t)] Substituting in the parameterization of f ± (k,ω) into the current J(ω) = 2Q 2 d d k (2π) dv kk.e(ω)ψ(k,ω) It follows that electrical conductivty is given by σ xx (ω) = 2Q 2 d d k ( k 2 (2π) d x ε k )ψ(k,ω) Introducing rescaled variables k k T ω = ω ǫ 2 T ψ(k,ω) = Ψ( k, ω) ǫ 2 T 3 σ(ω) = Q2 T d 2 ǫ 2 Σ( ω) Σ( ω) = 1 (3π) 2 0 d k k 3 Ψ( k, ω)

16 Universal Transport Damle and Sachdev, PRB 56, 8714 (1997) σ(ω) = (2e)2 T d 2 ǫ 2 Σ ( ω ǫ 2 T ) d = 3 ǫ 0.15 Σ' I ω ~ Two spatial dimensions σ(0) ( ) 4e 2 h

17 Nernst Coefficient Ong, Ussishkin, Sondhi, Oganesyan, Huse,... ν 1 B E y ( T) x Defining relations of the transport coefficients Jtr e = σ α E Tα κ T J tr h Set J y = 0 ν = 1 B α xy σ xx α xx σ xy σ 2 xx +σ2 xy Particle-hole symmetry σ xy = 0 ν = 1 B α xy σ xx Need α xy : Thermal response to electric field with no temp gradient

18 QBE in a Magnetic Field Bhaseen, Green and Sondhi, Magnetothermolectric Response at a Superfluid Mott-insulator Transition, PRL 98, (2007) To compute Tα xy we may equivalently consider the transverse heat current which flows in response to an electric field f ± t ±Q(E(t)+v k B). f ± k = I [f +,f ] In linear response we may consider d d k J h (ω) = (2π) d ǫ kv k [f + (k,ω)+f (k,ω)] We need to solve QBE for distribution functions Turns out to be useful to recall the relativistic kinematics of a charged particle in crossed E and B fields

19 Solution in Drift Regime E < cb Move to frame moving with drift velocity where E vanishes V D = E B B 2 Particle in pure magnetic field which doesn t affect its energy so we expect a Bose-Distribution in moving frame. Solution in lab frame: ( ) f ± (k) = f 0 (ε ε k v D.k k) = f 0 1 v 2 D /c 2 Solution of QBE even with leading order collision term α xy = 2c2 d d ( k dbt (2π ) d k2 f ) 0 ε k α xy = 2 S B S is the entropy density of a scalar field Heat flow by entropy drift: J h = 2(TS)v D = 2TS E B and J h = TαE

20 Temperature Gradient f ± t +v k. f ± x ±Q(v k B). f ± k = I ±[f +,f ] Absence of any material inhomogenity ( ) ( ) f ± x = T f± T = T ε k f 0 T ε k Longitudinal and transverse shifts of distribution function f ± (k) = f 0 (ε k )+k.uψ(k)±qk.(u B)ψ (k) U T T To lowest order in epsilon expansion ( ) ψ(k) = 0 ψ (k) = ε k Q 2 B f 2 0 ε k Reproduces previous result Onsager satisfied without magnetization subtractions

21 Thermal Transport Coefficient Bhaseen, Green and Sondhi, PRL 98, (2007) To lowest order in epsilon ( ) ψ(k) = 0 ψ (k) = ε k Q 2 B f 2 0 ε k To next order in the epsilon expansion ψ(k) = ǫ 2 ε k dk1 [ψ (k)f 1 (k,k 1 )+ψ (k 1 )F 2 (k,k 1 )] Yields finite thermal transport coefficient κ = gǫ 2 Td+3 Q 2 B 2 g 5.5 ( = c = k B = 1) In contrast to zero field case where response is infinite Dependence on epsilon is inverse to zero field conductivity

22 Hydrodynamics and AdS/CFT Hartnoll, Kovtun, Müller and Sachdev, Theory of the Nernst effect near quantum phase transitions in condensed matter and in dyonic black holes, Phys. Rev. B 76, (2007) κ xx (B) = TS2 B 2 σ xx (0) Wiedemann Franz like Thermal conductivity inversely related to conductivity All transport coefficients are related to electrical conductivity and a thermodynamic variables Exact duality relation may be obtained by QBE and ǫ-expansion Bhaseen, Green and Sondhi, Magnetothermoelectric Response near Quantum Critical Points, PRB 79, (2009)

23 Crossover of Thermal Coefficient Bhaseen, Green and Sondhi, Magnetothermoelectric Response near Quantum Critical Points, PRB 79, (2009) r B 2 /(ǫt) 4 Within accuracy of 3D Monte Carlo integrations g 5.55 g 0 = 8π(2π 2 /45) 2 /(1.037) 4.66 κ xx (B) = TS2 B 2 σ xx (0) Can also be extracted analytically from Boltzmann

24 Implications for Nernst Hartnoll, Kovtun, Müller and Sachdev, PRB 76, (2007) Impurities and chemical potential Divergences regulated ( )( )[ 2ekB S/kB γ 2 +ω 2 ] c +γ/τ(1 µρ/ts) α xy = h B/φ 0 (γ +1/τ) 2 +ωc 2 φ 0 = h 2e ω c 2eBρ ε+p γ σ QB 2 ε+p Generalizations exist for all other transport coefficients ν = 1 ( )( )[ ] kb ε+p ω c /τ B 2e k B Tρ (ωc/γ 2 +1/τ) 2 +ωc 2 Diverges in clean PH limit: ν τ/t Graphene Müller, Fritz and Sachdev, PRB 78, (2008)