The bosonic Kondo effect:

Transcription

1 The bosonic Kondo effect: probing spin liquids and multicomponent cold gases Serge Florens Institut für Theorie der Kondensierten Materie (Karlsruhe) with: Lars Fritz, ITKM (Karlsruhe) Matthias Vojta, ITKM (Karlsruhe)

2 Summary Moment in metals: screening Moment in HTc SC: impurity quantum phase transition Moment in spin gapped magnets: survival of the spin Moment in 1D magnet: screening by fermionic spinons Moment in 2D spin liquids Bulk: Cs 2 CuCl 4 and theory Bosonic Kondo effect Application to cold gases

3 The Kondo model Impurity spin in metallic Fermi sea: D µ J > S H = kσ D ɛ k c kσ c kσ + J S Limit J = : local moment S(T) = log 2 χ(t) = 1/(4T) σσ c σ() τ σσ 2 c σ () Limit J = : singlet state (screening) S(T) = χ(t = ) does not diverge

4 Kondo effect in a nutshell Weak coupling RG: log(t/d) divergent J(Λ) log Λ = J2 (Λ) J(Λ) = 1 log(λ/t K ) Kondo scale: T K = D exp( D/J) Universal crossover: T K J Tχ(T,J,D).2.1 S(T,J,D) e-3 1e-2 1e-1 T 1e-3 1e-2 1e-1 T Universality: Tχ(T,J,D) = Φ(T/T K )

5 Pseudogap Kondo model Origin: ρ(ɛ) ɛ r ɛ k µ Metals: r = k y k x ɛ k µ Semiconductors: r = 1 k y k x E k = [(ɛ k µ) k ]1/2 HTc SC: r = 1 k = (cosk x cosk y ) k x k y Impurity spin in semi-metallic Fermi sea D µ J > S D

6 Impurity quantum phase transition Weak coupling RG: J(Λ) log Λ = rj(λ) J2 (Λ) J(Λ) = rj(λ/d) r r J + (Λ/D) r J c = r J Quantum critical point: fractional spin χ(t,j = J c ) 1 4T Universal crossover: (1 r) S(S + 1) 3T.25 NRG Theory Tχ(T,J < J c ).2 1e-4 1e-2 1 T/D

7 Ground states of unfrustrated 2D QAF Systems: Néel ordered: La 2 CuO 4,... Paramagnetic: Sr 2 Cu 4 O 6, TlCuCl 3,... Models: H = ij J ij S i S j J J = 2 Excitations: J J J J dispersive S=1 triplon confined S=1/2 spinon

8 Bulk quantum phase transition Phase diagram: quantum critical point φ z r c r J/J Effective theory: symmetry O(3) (α = x,y,z) S bulk = β dτ r < r c : φ z (rc r) β r c < r: (r r c ) ν d d x ( τ φ α ) 2 + ( φ α ) 2 + rφ 2 α + g(φ 2 α) 2

9 Coupling to a magnetic impurity Interaction: S = S bulk [ φ] + S Berry [ S] + β dτ γ S φ(x = ) Paramagnetic phase: impurity preserved χ imp = S(S + 1) 3T [ 1 + γ 2 e /T] S imp (T = ) = ln(2s+1) χ (1-3 emu/ni mol) Pb(Ni 1-x Mg x ) 2 V 2 O 8 x =.12 x =.8 x =.5 x =.3 x =.2 x =.1 x =. 8 6 Pb(Ni 1.97 Mg.3 ) 2 V 2 O 8 4 H =.1 T aligned samples T (K) H c H c Néel phase: spin polarization T (K) Critical point: Partial screening! [Sachdev-Vojta] Fixed point (g,γ ) controlled by ɛ = 3 d γ log Λ = ɛ 2 γ + γ gγ2 χ imp = S(S + 1) 3T [ 1 + (γ ) 2] γ γ

10 In one dimension Heisenberg chain: no order at T = deconfined 1D spinons With one magnetic impurity: [Clarke et al.] mapping on Kondo model in Luttinger liquid J 1 J 2 J J 1 J 1 = J 2 2CK χ(t) ln(t K /T) J 2 1CK χ(t) 1/T K Conclusion: screening in a paramagnetic insulator is possible!

11 Cs 2 CuCl 4 : neutron data Spiral order at T <.6K: [Coldea et al.] c a b Broad continuum: origin? T(K)

12 Status of the theory (Free) magnon theory: χ(τ,x) = S(τ,x) S(, ) χ(ω, k) = 1 ω 2 k 2 2 Fails! 1D non applicable: J b =.37meV, J c =.12meV 2D extension: [Chubukov, Senthil, Sachdev] τ σσ α 2 z σ Deconfined bosonic spinons φ α = z σ S bulk [z σ ] = 1 β dτ d d x µ z g σ 2 + λ [ z σ 2 1 ] χ(τ,x) = z (τ,x)z(, ) 2 χ(ω, k) = 1 ω2 k 2 2 η eff = 1

13 Impurity in 2D deconfined magnet Motivation: No screening in standard 2D magnets Different response in 1D deconfined magnets What happens in 2D deconfined magnets? Effective action: S = S bulk [z σ ]+S Berry [ S]+ β dτj S [ z σ() τ ] σσ 2 z σ () Remarks: Bosonic Kondo problem Interacting bulk bosons Questions: Phase diagram? Screening?

14 Weak coupling analysis Diagrammatics: H V = (V/4) σ z σz σ generated! J R = + = V R = RG flow: for free massless spinons ɛ = d 2 J log Λ = ɛj V J 1 2 J3 + O(J 5 ) V log Λ = ɛv 1 2 V J2 + O(J 5 ) V ɛ > V ɛ < BS LM QCP J LM BS QCP J

15 How to control bulk interaction? Question: what happens in d = 2? Answer: Free spinons: no quantum phase transition Interacting spinons: anomalous dimension quantum phase transition likely Technical problem: Bulk interaction g marginal in d = 3 Kondo interaction J marginal in d = 2 Trick: inverse Schrieffer-Wolff transformation S bulk = S imp = β β dτ d d x µ z σ 2 + g z σz σ 2 dτ ɛ b σb σ + +V b σaz σ () + h.c. [g] = 3 d and [V ] = 3 d 2 Controlled expansion possible near d = 3

16 Strong coupling analysis Toy model 1: canonical bosons H 1 = U( a σa σ ) 2 + JS σ Needs U σσ a σ Screening if S < S c J/(2U J) Underscreening otherwise τ σσ 2 a σ Toy model 2: anisotropic bosonic spinons O(4) O(2) O(2) z σ 2 = 1 z σ 2 = 1 z σ = e iφ σ σ J S σσ z σ τ σσ 2 z σ J 2 S+ e iφ iφ + h.c. Diagonalized by e i(φ φ )S z Unique GS: screening possible Question: non-perturbative approach?

17 Large N limit SU(N) generalization: S b σb σ (σ = 1...N) Q = z σ()b σ G b (iν) 1 = iν + µ + Q 2 ν d 2 3 phases: Q =, µ T free spin χ S(S + 1)/3T Q, µ screened spin χ cst. Q, µ T underscreened spin χ C/T Numerical solution:.8 S ɛ > Q(T) T K Tχ(T) S c Under Screened Local Exactly Screened MomentJ c J Conclusion: overall picture OK

18 Other systems Deconfined critical magnets: [Senthil et al.] Neel order VBS order or g c g Itinerant systems: Kondo lattice, doped Mott insulators... Description d σ = f b σ to be pursued... Cold gases: Two-component Bose gas in optical lattices Atomic quantum dot [Recati et al.] Dispersion ω k = k 2 : ɛ = (d 2)/2 Quantum phase transition One-loop exact! Role of BEC?

19 Conclusion Bosonic spinons can lead to a screening of magnetic impurities Peculiar "bosonic" Kondo effect Impurity quantum phase transition Screening of large spins possible Relevance for compound Cs 2 CuCl 4? Relevance for numerical simulations? Relevance for cold bosonic gases?