Strange metal from local quantum chaos

Transcription

1 Strange metal from local quantum chaos John McGreevy (UCSD) hello based on work with Daniel Ben-Zion (UCSD)

2 Compressible states of fermions at finite density The metallic states that we understand well are Fermi liquids. Landau quasiparticles poles in single-fermion Green function G R at k Z k k F = 0, ω = ω (k ) 0: G R ω v F k + iγ Measurable by angle-resolved photoemission: ω in k in ω = ω ω out in k out = k in k e ω k Intensity spectral density : A(ω, k) ImG R (ω, k) k 0 Zδ(ω v F k ) quasiparticles are long-lived: width is Γ ω 2, residue Z (overlap with external e ) is finite on Fermi surface.

3 Non-Fermi liquids exist but are mysterious There are other states with a fermi surface, but no pole at ω = 0. e.g.: normal phase of optimally-doped cuprates: ( strange metal ) ω in k in ω = ω ω out in k out = k in k e ω k = among other anomalies: ARPES shows gapless modes at finite k (FS!) with width Γ(ω ) ω, vanishing residue Z k 0 0. Working defintion of NFL: Still a sharp Fermi surface but no long-lived quasiparticles. More prominent mystery of the strange metal phase: e-e scattering: ρ T 2, phonons: ρ T 5,... no known robust effective theory: ρ T. T

4 Non-Fermi Liquid from non-holography Luttinger liquid in 1+1 dims. G R (k, ω) (k ω) α loophole in RG argument for ubiquity of FL: couple a Landau FL perturbatively to a bosonic mode (e.g.: magnetic photon, emergent gauge field, critical order parameter...) k q k q k nonanalytic behavior in G R 1 (ω) at FS: v F k +cω 2ν NFL. [Huge literature: Hertz, Millis, Nayak-Wilczek, S-S Lee, Metlitski-Sachdev, Mross-JM-Liu-Senthil, Kachru, Torroba, Raghu...] Not strange enough: These NFLs are not strange metals in terms of transport. ρ T 2ν+2 If the quasiparticle is killed by a boson with ω q z, z 1, small-angle scattering dominates = transport lifetime single-particle lifetime

5 Frameworks for non-fermi liquid in d > 1 a Fermi surface coupled to a critical boson field L = ψ (ω v F k ) ψ + ψψa + L(a) k q k q k small-angle scattering dominates = transport is not that of strange metal. a Fermi surface mixing with a bath of critical fermionic fluctuations with large dynamical exponent z 1 Discovered with AdS/CFT [Faulkner-Liu-JM-Vegh , Faulkner-Polchinski , FLMV+Iqbal ] L = ψ (ω v F k ) ψ + ψχ + ψ χ + χg 1 χ χ: fermionic IR CFT operator = + ψψ = 1 ω v F k G + G = χχ = c(k)ω 2ν +...

6 Charge transport and momentum sinks instead of this: with z 1 ω q z this: with z 1 The contribution to the conductivity from the Fermi surface [Faulkner-Iqbal-Liu-JM-Vegh, and ]: is ρ FS T 2ν when Σ ω 2ν. Dissipation of current is controlled by the decay of the fermions into the χ DoFs. = single-particle lifetime controls transport. ( marginal Fermi liquid: ν = [Varma et al] = ρ FS T. ) T

7 A word about the holographic construction The near-horizon region of the geometry AdS 2 R d describes a z = fixed point at large N: many critical dofs which are localized. charged black hole horizon r=rh R 3,1 Shortcomings: The Fermi surface degrees of freedom are a small part (o(n 0 )) of a large system (o(n 2 )). Here N 2 is the control parameter which makes gravity classical (and holography useful). Understanding their effects on the black hole requires quantum gravity. [Some attempts: Suh-Allais-JM 2012, Allais-JM 2013] All we need is a z = fixed point (with fermions, and with U(1) symmetry). r UV

8 SYK with conserved U(1) A solvable z = fixed point [Sachdev, Ye, Kitaev]: H SYK = N ijkl J ijklχ i χ j χ k χ l. J ijkl = 0, J 2 ijkl = J2 2N 3 {χ i, χj } = δ ij, {χ i, χ j } = 0 G = Schwinger-Dyson equations: G 1 (ω) = (iω) 1 Σ(ω) ω J G(ω)Σ(ω) 1 Σ(τ) = = J 2 G 2 (τ)g( τ) = G(ω) (iω) 1/2, ( χ) = 1 4. Also useful is the bath field : χ i J ijkl χ j χ k χ l, which has χ χ (iω) + 1 2, ( χ) = Duality: this model has many properties in common with gravity (plus electromagnetism) in AdS 2.

9 Using SYK clusters to kill the quasiparticles and take their momentum [D. Ben-Zion, JM in progress] One SYK cluster: AdS 2 : To mimic AdS 2 R d, consider a d-dim l lattice of SYK models: H 0 = xy lattice ( ) ψ xψ y + hc + x lattice AdS 2 R d H SYK (χ xi, J x ijkl ) Couple SYK clusters to Fermi surface by random gs (g ix = 0, g ix g jy = δ ij δ xy g 2 /N): H int = x,i g ix ψ xχ xi + h.c.

10 Advertisement for related work: [Gu-Qi-Stanford]: (no hybridization) a chain of SYK clusters with 4-fermion couplings [Banerjee-Altman]: (no locality) add all-to-all quadratic fermions to SYK [Song-Jian-Balents]: (no Fermi surface) a chain of SYK clusters with quadratic couplings

11 Large-N analysis = 1 ω v F k. = χ x χ y. = disorder contraction. G ψ (ω, k) small = ω 1 ω v F k G(ω) Full ψ propagator: This has ν = 1 4 : G(ω) ω 1 2. } {{ } O(N 1 ) The ψ self-energy is Σ(ω, k) = G(ω) (just as in the holographic model). (ν(q) = 2 q 2q ). (Coupling to bath field: ν = )

12 Does the Fermi surface destroy the clusters? Q: Does this destroy the clusters? g ix = 0, g ix g jy = δ ij δ xy g 2 /N. The SYK-on propagator G looks like: +... Dominant new contributions to G xy are still down by N: = z = behavior to survives parametrically low energies! Replica analysis reproduces diagrammatic results: Z n = [dgdσdρdσ]e NS[G,Σ,ρ,σ] Σ = J 2 G 2 G, But: δs δ{g, Σ, ρ, σ} = 0 = 1 G = t Σ G ψ /N, lim lim? = lim N ω 0 ω 0 G 1 ψ = G 1 ψ0 G. lim N

13 RG analysis of impurity problem Weak coupling: Consider a single SYK cluster coupled to FS. Following Kondo literature [Affleck] only s-wave couples: H FS = v F 2π 0 ) dr (ψ L r ψ L ψ R r ψ R = [ψ L/R ] = 1 2. H = gψ L (0)χ, H = gψ L (0) χ. χ i J ijkl χ j χ k χ l. χ g iχ i /g. Coupling to bath field: [ψ χ] = = 5 4 is irrelevant. Coupling to χ: [ψ χ] = = 1 4 is relevant. Strong coupling: pair up. At large enough g (g t, J), ψ x and χ x 1 g i g iχ ix H = g x ψ x χ x + h.c. Anti-Kondo phase: the impurity absorbs the conduction electrons!

14 Topology of coupling space 1) Possibilities for beta function (arrows toward IR): Consequences for entanglement entropy of half-chain at small g 0: 1) 2) 3) 3) Expect: L crossover g

15 Numerical results Half-chain entanglement entropy grows faster with L than free-fermion answer! This doesn t happen for quadratic clusters (SYK 2) Coupling to bath field is irrelevant same as free fermion answer. At large g, entanglement is destroyed.

16 Truncation to help the DMRG What matters is the low-lying level density. Keep just the lowest χ levels of each cluster. Future: variational tensor product state with the structure of the above graph.

17 Correlation functions i χ x,i χ L/2,i ψ x ψ L/2 sin 2k F (x L/2) x L/2 α α < 1: exponent is not free fermion value. χ are still localized. At large g, everybody is localized (anti-kondo phase).

18 Conclusions an interesting NFL fixed point. It s not Lorentz invariant. Numerical evidence is in 1d, but it s not a Luttinger liquid: c 1. Maybe we can access it perturbatively by q = 2 + ɛ (H = J i1 i q c i 1 c iq ). Cartoon map of phases: (Warning: this is a cartoon.)

19 The end. Thank you for listening. Thanks to Open Science Grid for computer time.