Serge Florens. ITKM - Karlsruhe. with: Lars Fritz and Matthias Vojta

Transcription

1 0.5 setgray0 0.5 setgray1 Universal crossovers and critical dynamics for quantum phase transitions in impurity models Serge Florens ITKM - Karlsruhe with: Lars Fritz and Matthias Vojta p. 1

2 Summary The Kondo problem in a pseudogap RG strategy and CS equations Computation of universal crossovers Critical dynamics Influence of p-h asymmetry Spin S = 1 case: NFL and QPT p. 2

3 Where it started Diluted magnetic impurities in a metal: Resistance T K D µ J > 0 S Temperature D Temperature dependent impurity scattering p. 3

4 Kondo model: basics Model: Spin 1/2 with AF coupling to a Fermi sea H = kσ ɛ k c kσ c kσ + J S σσ c σ(0) τ σσ 2 c σ (0) Physics: smooth crossover between two simple limits Limit J = 0: local moment S(T) = log 2 and χ imp (T) = 1/(4T) Limit J = : singlet state (screening) S(T) = 0 and χ imp (T) finite p. 4

5 RG for Kondo problem Perturbation theory (j 0 = J/D): badly convergent! χ imp (T) = 1 [1 j 0 j 20 4T log DT ] +... = 1 4T [ 1 j 0 + j0 2 log λ }{{} j2 0 log λd T j 0 (λ) Scaling form: χ imp (T,j 0,D) = χ imp (T,j 0 (λ),λd) Iterate RG relation: β(j 0 ) j 0(λ) log λ = j2 0 (λ) 1 j 0 (λ) = log(λd/t K ) with T K = D exp( 1/j 0 ) ] p. 5

6 RG flow Trick: now we can choose λ = T/D χ imp (T) = 1 4T [1 j 0(T/D)] = 1 4T [ 1 ] 1 log(t/t K ) 0.2 Tχ imp (T) 0.1 NRG 2nd order Pert. RG j 0 (T/D) LM 0 1e-20 1e-10 1 T/D SC 0 T K J p. 6

7 Universality RG Tχ imp (T,j 0,D) = Φ(T/T K ) 0.2 Tχ imp (T,J,D) 0.1 S(T,J,D) e-30 1e-20 1e-10 1 T/D 0 1e-30 1e-20 1e-10 T/D 0.2 Tχ imp (T,J,D) 0.1 Collapse at T D 0 1e e+10 T/T K p. 7

8 Another view on RG Universality take limit D Procedure: set χ imp (T = µ) finite at given scale µ χ imp (T = µ) = 1 [ 1 j 0 j0 2 log D ] 4µ µ χ imp (T) = 1 4T [ 1 j R j 2 R log µ T ] 1 4µ [1 j R] Scaling form: χ imp (T,j R,µ) = χ imp (T,j R (λ),λµ) RG: j R (λ) = log 1 (λµ/t K ) and T K = µ exp( 1/j R ) Origin: Choice of µ arbitrary dχ imp /dµ = 0 [ µ µ + β(j R) ] χ imp (T,j R,µ) = 0 (CS eq.) j R p. 8

9 Pseudogap Kondo model Gapless host DOS: ρ(ɛ) ɛ r ɛ k µ Metals: r = 0 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 y k = (cos k x cos k y ) k x p. 9

10 RG for pseudogap Kondo Weak coupling RG: easy guess as [J] = r β(j R ) = j R(λ) log λ = rj R(λ) j 2 R (λ) Proper method: Set j 0 = (D/µ) r Z j j R Consider interaction vertex Γ R (ω) = Z j j R [1 ω/µ r j R /r +...] Impose Γ R (ω = µ) j R = Z j j R [1 j R /r] Absorb poles with Z j = 1 + j R /r But one has dj 0 /dµ = 0 β(j R ) = rj R j R β(j R ) dlog Z j /dj R p. 10

11 Quantum Phase Transition Analyze flow equation: vary µ λµ Solve j R(λ) log λ = rj R(λ) j 2 R (λ) with j R(1) = j R j R (λ) = rj R λ r r j R + j R λ r = r 1 + Sgn(r j R ) λµ/t r with T µ r j R j R 1/r T, ω T 0 j c = r j j R < r j R (λ) λ r for λµ < T (flow to LM) j R = r j R (λ) = r for all λ (fixed point) j R > r j R (λ) diverges at T (flow to SC) p. 11

12 Impurity susceptibility At lowest order: χ imp = 1/(4T)[1 j R (T/µ)] Quantum critical point: fractional spin χ imp (T,j R = j c ) 1 4T (1 r) S(S + 1) 3T Universal crossover: from CR to LM (j < j c ) 0.25 NRG Pert. RG 0.25 NRG Pert. RG Tχ imp (T) Tχ imp (T) r = 0.1 j < jc 1e-40 1e-20 1 T/D r = 0.3 j < jc e-40 1e-20 1 T/D p. 12

13 Local susceptibility The direct computation is still divergent: Tχ 0 loc (T) = 1 j2 R T/µ 2r (1/2r + Cst.) Needs extra renormalization factor: χ R loc (T) = Z χχ 0 loc (T) with Z χ = 1 + j 2 R /2r Tχ R loc (T) 1 j2 R log(t/µ) Scaling analysis: χ 0 loc (T) is independent of µ [ µ µ + β(j R) ] + η χ (j R ) χ R j loc(t,j R,µ) = 0 R with η χ (j R ) = β(j R ) logz χ j R = j 2 R p. 13

14 Solution: χ R loc (T,j R,µ) = exp Choose λ = T/µ: Tχ R loc (T,j R,µ) Interpretation: ( jr (λ) j R [ r j R + η χ β ) T µ r ] r χ R loc (T,j R(λ),λµ) j R < r: Tχ R loc (r j R) r (order parameter) j R = r: χ R loc (T) 1/T 1 r2 (anomalous behavior) limtχ T 0 imp (T) 1/4 C imp limtχ loc (T) T 0 Two vastly different r/8 0 Jc J susceptibilities p. 14

15 Zero temperature dynamics Strategy: Compute in renormalized perturbation theory Choose RG scale λ to control the expansion Apply CS equation Example: for r = 0.1 and j < j c ωχ (ω,t = 0) loc NRG Pert. RG ω r T (ω,t = 0) 4 NRG Pert. RG ω r ω r e-40 1e e-06 ω 2r 2 1e-40 1e-20 1 ω/d 0 1e-40 1e-20 1 ω/d p. 15

16 Finite temperature dynamics Problem: The ɛ-expansion and the analytic continuation do NOT commute for ω T (Sachdev) Let s try: we apply CS in this regime to the T-matrix T ω r (ω) (not too bad) [(T ) r ± (T + ω ) r ] 2 T 1 (ω) r 0 [log((t + ω )/T K )] (BAD!) 2 1 T (ω,t) 0.01 T = 0 T = T T = 100T e e+08 ω/t Still a challenge for the NRG at ω T p. 16

17 On the other side of the street Question: Can we capture the crossover to SC? LM CR SC 0 j c = r j Answer: not with Kondo, but Anderson is good! H = kσ ɛ kc kσ c kσ +Un d n d + σ V [d σc σ (0)+H.c.] V SC CR LM 0 U Scaling analysis: [U] = 2r 1 ɛ Perform ɛ-expansion near r = 1/2!! p. 17

18 RG for pseudogap Anderson Flow equation: β(u) = ɛu 3(π 2ln4) π 2 u 3 Critical point at u c ɛ Poor convergence Example: for r = 0.47 and j > j c 1e e+05 Tχ(T) imp0.08 χ (ω,t = 0) loc e-40 1e-20 1 T/D 1e-05 1e-40 1e-20 1 ω/d p. 18

19 With particle-hole asymmetry NRG result: for < r < 1 E 0 ASC LM ACR CR SC 0 J c J Effective theory: U = Anderson model H = ɛ k c kσ c kσ +ɛ f kσ σ f σf σ +V [ f σ b c kσ +h.c.] kσ RG: V marginal at r = 1 expansion in ɛ = 1 r p. 19

20 Interpretation: Kondo and Anderson: same universality class Origin of the asymmetric QCP: competition doublet (spinon) vs. singlet (holon) states { S ACR 8 ln2 imp = ln 3 9 (1 r) Tχ ACR imp = ln (1 r) S imp CR ACR C imp = T χ imp CR ACR r r Open problem: transition between CR and SCR p. 20

21 Interesting generalization Spin S=1 Kondo model ACR persists with S imp = ln 5 at r = 1 Effective theory: crossing triplet-doublet H = kσ +V k ɛ k c kσ c kσ + ɛ f m f m f m [ f 0 b c k + f 0 b c k ] + h.c. + 2V [ f 1 b c k + f 1 b c k ] + h.c. Reproduces Kondo for ɛ f Crucial to introduce the good degrees of freedom!! p. 21

22 Check Results: properties of the ACR critical point { S ACR 24 ln2 imp = ln 5 25 (1 r) Tχ ACR imp = ln (1 r) CR ACR CR ACR S imp Tχ imp NFL NFL r r p. 22

23 S = 1 PG Kondo at p-h sym Strong coupling limit: effective spin S eff = 1/2 effective FERRO coupling j eff = 1/j < 0 S S eff effective bath exponent r eff = r < 0 c σ (0,iω)c σ (0,ω) 1 = iω t 2 01 c σ (1,iω)c σ (1,ω) p. 23

24 Extra NFL fixed point RG: usual Kondo scaling equation j eff lnλ = r effj eff j 2 eff j eff = r eff = r NFL LM 0 j eff j = 1/r Critical properties: LM CR NFL SC 0 j c = r j = 1/r j ACR CR CR ACR S imp Tχ imp NFL NFL S NFL imp = ln 2 + 2r ln 2 TχNFL r r imp = 1/4 + 3r/8 p. 24

25 Summary S = 1 PG Kondo E 0 ASC 0 < r < r LM SCR NFL SSC 0 J c J J E 0 ASC r < r E 0c ACR LM SCR NFL SSC J c 0 J J c J p. 25

26 Conclusion Quantum phase transition in pseudogap Kondo: well under control! Computation of universal crossover possible Quantitative comparison between NRG and perturbative RG Low ω dynamics not fully controlled apply functional RG? Transition between critical points not understood p. 26