Introduction to the RG approach to interacting fermion systems

Transcription

1 Introduction to the RG approach to interacting fermion systems Nicolas Dupuis Laboratoire de Physique Théorique de la Matière Condensée Université Pierre et Marie Curie, CNRS, Paris

2 Outline Fermi liquid theory 1D electron gas Breakdown of perturbation theory and Parquet solution Kadanoff-Wilson RG approach Fermionic RG in higher dimension Weakly-coupled 1D systems D system (Hubbard model) Beyond the fermionic RG: collective bosonic field BCS-BEC crossover Antiferromagnetism and superfluidity in the D Hubbard model Conclusion

3 Non-interacting electrons H = ϵ k c +k, σ c k, σ [c k, σ, c +k ',σ ' ]+=δ k, k ' δσ, σ ' k,σ Ground state: Fermi sphere σ=, FS> = Π k k Πσ c +k, σ vac> F Elementary excitations: particles and holes c +k, σ FS> ( k >k F ) c k, σ FS> ( k <k F ) ky particle hole kx C V =(m k F /3)T, 1 χ= M / H = g μ B D (ϵ F ), n e τ σ=, m etc. Why is the non-interacting limit a good starting point to understand a metal?

4 Fermi liquid theory Decay of a particle excitation: k (k+q) + ph pair (k ', k ' q ' ) ϵ k+q ϵ k = ϵk ' q +ϵk ' <0 ky k k+q Particle-hole excitation kx Small phase-space for ph pair excitation scattering rate 1 (ϵ μ) +π T k τk

5 1 R Spectral function: A (k,ω) = π Im G (k, ω) Quasi-particle peak ξ k =ϵk μ Fermi liquid Non-interacting electron gas Landau's quasi-particle: * C V =(m k F /3)T, χ= χ0 s 0 spectral weight Z k effective mass m * life-time τ k 1/ξk, etc. 1+F {F ls, F al } Landau parameters (interaction between QP's)

6 Breakdown of Fermi-liquid theory Fermi surface instabilities (e.g. superconducting instability) χ0 + = 1+g χ 0 RPA χ pairing = ω0 χ 0 D(ϵ F ) ln T T 1 If attractive interaction (g<0): SC instability T c ω0 exp g D(ϵ F ) ( Fermi liquid Tc (critical regime usually narrow in 3D) SC (or CDW, SDW) )

7 Electrons with Coulomb interaction and uniform positively charged background (jellium model) High-density : metal Low-density : Wigner crystal Electrons on a lattice with strong on-site repulsion : Mott insulator (commensurate density) 1D electron gas: metallic state but not a Fermi liquid

8 1D electrons : breakdown of perturbation theory ξk =ϵk μ=v F ( k k F ) linearized dispersion k F kf k = (i ω v F (k k F )) 1 right-moving electrons = (i ω v F ( k k F )) 1 left-moving electrons forward scattering Logarithmic singularities: backward scattering ξk =ξ k (TRI) ξk = ξk +k (nesting) F

9 A separate treatment of pp and ph channels (usually a good approximation in D or 3D) would lead to a finite-temperature phase transition (SC or CDW/SDW) Interferences between channels are of the same order O(g ln ( E 0 /T )) and RPA not included in χ pp

10 Diagrammatic (Parquet) solution [Y.A. Bychkov, L.P. Gor'kov and I. Dzyaloshinskii, JETP'66] Effective interaction γ = g [ O(g n ln n (E 0 /T ))+O(gn ln n 1 (E 0 /T ))+ ] Parquet diagrams Solution in differential form: sub-leading diagrams l = ln ( E0 /T ) and g i = gi /π v F d g 1 = g 1 dl d ( g g 1 ) = 0 dl g1 No finite-temperature phase transition γ χ T ν ν = SSC,TSC,CDW, SDW TS SDW 0 g

11 Multiplicative RG [J. Solyom, Adv. Phys.'79] Logarithmic structure of perturbation theory : vertices renormalize multiplicatively upon a change of the bandwidth l E0 E 0 (l) = E 0 e Similarities with Gell-Mann Low RG in QED One-loop : Parquet solution Two-loop (self-energy corrections) and three-loop order

12 Kadanoff-Wilson RG to 1D interacting fermions [C. Bourbonnais'85, C. Bourbonnais & L. Caron, Int. J. Mod. Phys. B'91] ψ theory: Z = D [ψ, ψ] e 4 * * S[ ψ, ψ] β S = d τ 0 * * dx ψ ( μ) ψ +H ( ψ, ψ) } σ τ σ {σ Integrate out ''high-energy'' fermion states k F kf Z = D[ ψ*<, ψ< ] D[ ψ*>, ψ> ] e S k

13 Dimensional analysis: S = k, ω n, σ,r ψ*r σ (k,i ω n ) [ i ω n v F (r k k F ) ] ψr σ (k,i ω n )+S int [ξ k =v F ( k k F )] = 1 [T ] = 1 (z=1) [ ψ(k,i ω n )] = 1/ [ gi ] = 0 (marginal) d=1 is the upper critical dimension (logarithmic corrections in PT) One-loop: resummation of Parquet diagrams Two-loop: anomalous dimension 1 [ψ(k,i ω n)] = (1 η) G(x, τ) 1 (x±i v F τ)1 η/

14 Luttinger liquid (Haldane'81) metallic (gapless) system no Landau's QP's (η 0) power-law diverging response functions χ ν T spin-charge separation (when g4 0 ) Other approaches to 1D fermions Bosonization Ward identities Bethe Ansatz etc. γ ν

15 RG in higher dimension (D or 3D) Weakly-coupled chain systems ξk =v F ( k k F ) t cos(k ) t ' cos( k ) t t ' + interactions g 1, g ''high-energy'' states to be integrated out Fermi surface ξk =0

16 D Hubbard model t t' ξk = t (cos k x +cos k y ) 4 t ' cos k x cos k y μ + local repulsion U Fermi surface for t'=0 at half filling

17 4 RG approach to the ψ theory Low-energy modes live near the Fermi surface ( points in 1D, line in D, etc.). Functional RG 4-point vertex: g ({k i,ω i }) marginal ( d) projection onto the Fermi surface: k=(ξ k,ω) k F =(0, Ω) g ({k i,ω i }) g({ω i }) discretize variable Ω D Hubbard model [Picture: Honerkamp et al. '04] The fermion field is not an order parameter: ψσ =0 order parameters: ψσ ψσ ' singlet/triplet SC * ψσ ψσ ' CDW/SDW The fermionic RG flows to strong coupling whenever there is a phase transition

18 Practical implementation Bourbonnais & Caron Z = D[ ψ*<, ψ< ] D[ ψ*>, ψ> ] e S [Int. J. Mod. Phys. B '91] Wick-ordered scheme [Salmhofer, Comm. Math. Phys. '98] 1PI scheme {ψα }={ψ*σ (r, τ), ψσ (r, τ)} Z Λ [J ] = D[ ψ]exp 1 ψα C Λ, αβ ψβ S int [ψ]+ J α ψα { } α α,β Θ(Λ ξ k ) with e.g. C Λ (k,i ω n )= (momentum cutoff) i ω n ξk δ ln Z Λ [J ] ϕα = ψα = (Grassmannian) δ Jα Γ Λ [ϕ] = ln Z Λ [J ]+ J α ϕα (effective action) α d () Γ Λ [ϕ] = ϕα C 1 ϕ Tr { C Λ Γ Λ [ϕ] } (exact flow equation) Λ,α β β dλ α,β

19 ϕ is a Grassmann variable Local Potential Approximation (LPA) not possible Blaizot Mendez-Galain Wschebor approximation not possible Expand Γ Λ [ ϕ] = β ΩΛ + Γ α β, Λ ϕα ϕβ+ Γ α β γ δ, Λ ϕα ϕβ ϕ γ ϕδ+ () α,β (4) α δ Standard truncation 1 (4) Ω, Γ() =G, Γ αβ αβ αβ γ δ (all higher-order vertices set to 0) one-loop RG equations (functional wrt momenta) d Σ = dλ Λ Γ S d (4) ΓΛ = dλ Γ G S Γ (4) (4) 1 S = G C G ( 4) Weak-coupling (functional) approach but no a priori assumption about leading collective fluctuations (SC, DW, etc.)

20 Phase diagram of quasi-1d conductors Q ξk =v F ( k k F ) t cos(k ) t ' cos( k ) + interactions g 1, g if t ' =0 : perfect nesting: ξk +Q = ξ k, Q=(k F, π) SDW expected if t ' 0 : imperfect nesting: ξ k+q ξ k ground state?

21 phase diagram quasi-1d organic conductors SDW SCd R. Duprat & C. Bourbonnais, EPJB '01 C. Nickel, R. Duprat, C. Bourbonnais & ND PRL'05, PRB'06 SC d x y : ψ (k) ψ (k) Δ cos k Experimental phase diagram [Review : ND, C. Nickel & C. Bourbonnais, J. Low. Temp. Phys. '07]

22 Susceptibilities χ SDW SDW phase χ SCd χ SDW SCd phase

23 (4) Two-particle vertex (diverges at the transition) Γ ( k ', k ', k, k ) Near the SDW phase (nesting): singularities at ( k ' ) k =±π Near the SCd phase: d-wave form factor cos k cos k ' [C. Nickel et al.'06]

24 Physical picture (4) Γ pp = SC induced by the exchange of spin fluctuations V.J. Emery, Synt. Metals '83 (Q1D organic conductors) D.J. Scalapino, Phys. Rep. '95 (high-tc superconductors)

25 Interchain interactions g i (k 1 ', k ', k, k 1 ) = g i + gi cos(k 1 ' k 1 ) Both spin and charge fluctuations are important Ground states : SDW, CDW, singlet SCd, triplet SCf C. Nickel, R. Duprat, C. Bourbonnais & ND PRL'05, PRB'06

26 Phase diagram of D Hubbard model Schulz '87, Lederer, Montambaux, Poilblanc '87, Zanchi & Schulz '97, Halboth & Metzner '00, Honerkamp, Salmhofer, Rice '01, Katanin & Kampf '03, Binz, Baeriswyl, Douçot '0, etc. H = t i, j, σ ψ+i σ ψ j σ t ' i, j, σ [Picture: Metzner et al. '10] ψi+σ ψ j σ +U ψi+ ψ+i ψi ψi μ ψ+i σ ψi σ i i,σ

27 (U =3 t, t '= 0.3 t ) [Honerkamp et al. '01] (van Hove filling: μ=4 t ' ) [Honerkamp & Salmhofer '01]

28 AF phase driven by Fermi surface nesting ξk +Q ξ k with Q=(π, π) Bad nesting : short-range AF and SCd instability Ferromagnetism enhanced by logarithmic divergence of the density of state (van-hove singularity for certain fillings) ''Naive'' picture Stoner criterion: 1 U D(μ) = 0 1 t Density of states: D(ϵ) ln (van Hove filling) ϵ μ t ( RG: includes Kanamori screening (screening of the interaction by particle-particle fluctuations) U U eff (q, i ων) = ) 1+U Π pp (q,i ω ν ) Ferromagnetism cannot be studied with momentum cutoff χ 0 ferro n F (ξk ) = involves only states near the Fermi surface ξ N k k Temperature flow [Honerkamp & Salmhofer '01] : T regularizes G 0 (k,i ωn ) = 1 i π ( n+1)t ξ k d ( ) dt

29 Fermionic RG infinite resummations of Feynman diagrams with no a priori assumption about leading collective fluctuations deals with fermionic degrees of freedom with no explicit treatment of collective (bosonic) fluctuations flows to strong coupling whenever there is a phase transition (4) ( χ and Γ diverge) no access to broken-symmetry phases How to improve the fermionic RG? Hubbard-Stratonovich fields Symmetry-breaking fields [Salmhofer et al. Prog. Theor. Phys. '04, Gersch et al. EPJB'05] Bosonic parametrization of vertices [Husemann & Salmhofer PRB'09] RG in the PI formalism [ND EPJB'05]

30 How to study collective (bosonic) fluctuations in a fermion system? β S = S 0 +λ d τ d r ψ ψ ψ ψ d * * 0 Introduce Hubbard-Stratonovich (bosonic) field (e.g. SC fluctuations) β { Δ * S = S 0 + d τ d r +( Δ ψ ψ +c.c. ) λ 0 d } Integrate out fermions S eff [Δ*, Δ] * Mean-field (BCS) theory : δ S eff /δ Δ =0 Gaussian fluctuations : collective SC fluctuations χ (r, τ ; r ', τ ' ) = Δ(r, τ) Δ * (r ', τ ') Goldstone mode if spontaneously broken symmetry Interaction of fermions with collective fluctuations

31 Difficulties with Hubbard-Stratonovich transformations Infinite number of possible transformations (Fierz ambiguity) * * + + z + λ ψ ψ ψ ψ = λ [ ( ψ ψ) ( ψ σ ψ) ] = λ ( ψ σ ψ) 4 6 pairing field Δ charge- and spin-density fields ρ, S z ψ= ψ ψ ( ) spin-density field S If only one HS field: a priori assumption about leading fluctuations Difficult to consider competing instabilities on equal footing

32 RG flows with (bosonic) HS field only Effective bosonic action (integrate out fermions): field and derivative expansion [Hertz'76, Millis'93] β ω 1 d 4 n S [ϕ] = ϕ( q, i ω n ) A q +Z z +r ϕ(q,i ω n )+u d τ d r ϕ β V q,ω q 0 ( n ) (z: dynamical exponent) Singular contribution (gapless fermions) Quantum phase transition controlled by non-thermal parameter r Quantum critical point r=r0 T T Tc or LRO r0 r LRO r0 r

33 (Bosonic) RG approach (LPA' approximation) Γ Λ [ϕ] = 1 βv ( ϕ( q, i ωn) A Λ q +Z Λ q,ω n ωn z q ) β d ϕ(q, i ωn )+ d τ d r U Λ (ϕ) 0 Is it possible to use a bosonic approach in the presence of gapless (fermionic) degrees of freedom? see D. Belitz, T.R. Kirkpatrick, and T. Vojta, RMP'05 Jakubczyk, Strack, Katanin, Metzner PRB'08 Bauer, Jakubczyk, Metzner PRB'10 PRB'11

34 RG flows with fermions and Hubbard-Stratonovich field AF in Hubbard model: Baier, Bick, Wetterich '04 BCS-BEC crossover: Birse, Krippa, McGovern, Wallet '05 Diehl, Gies, Pawlowski, Wetterich '07 Strack, Gersch, Metzner '08 Floerchinger, Scherer, Diehl, Wetterich '08 Bartosch, Kopietz, Ferraz '09 Scherer, Floerchinger, Gies '10 etc. β Fermions with attractive interaction: S int = λ d τ d r ψ ψ ψ ψ (λ>0) d * * 0 -body problem in vacuum: s-wave scattering length a T-matrix: = 4πa m λ>λ c -body bound state (a>0) 1 E b ma

35 Finite density : superfluidity BEC of composite bosons Tc ϵf BCS instability 1/ k F a Unitary limit: a = (no small parameter) k F =(3 π n)1/ 3 QCP at T=n=0 [Nikolic & Sachdev '07, Diehl et al.'07] Universal physics: μ (T =0) μ (T c ) T c P (μ, T ) ϵf, ϵ F, ϵ F, P (μ,t ) 0 (3 π n)/3 ϵf = m Experimentally, a (B) is controlled by a Feshbach resonance

36 ''Standard'' approach: Nozières Schmitt-Rink (1985) S eff [Δ *, Δ] : mean-field + Gaussian fluctuations many features are qualitatively correct not quantitatively accurate

37 NPRG flow for the BCS-BEC crossover Action S= r, τ {( ) ( fermionic atoms ) ψ τ μ ψ+δ * τ μ+ν Δ h ( Δ* ψ ψ +c.c. ) m 4m + bosonic molecules } atom-molecule conversion Effective action ( ) ( ) * Γ Λ [ψ, Δ] = ψ τ μ ψ+δ Z Λ τ A Λ Δ m 4m r,τ + +U Λ ( Δ ) hλ ( Δ* ψ ψ +c.c. ) Exact flow equation () () Λ Γ Λ [ψ, Δ] = Tr Λ R Δ, Λ ( Γ Δ, Λ [ψ, Δ]+R Δ, Λ ) + Tr Λ R ψ, Λ ( Γ ψ, Λ [ψ, Δ]+R ψ,λ )

38 Generation of fermionic interactions in the flow can be eliminated by ''dynamical'' bosonization [Gies & Wetterich '0] describes particle-hole pair fluctuation [Floerchinger et al. '08] Phase diagram [Picture: Boettcher, Pawlowski, Diehl, Nucl. Phys. B '1]

39 Critical behavior: Wilson-Fisher fixed point of the O() model BCS limit e C π/ k a T BCS = 8 e c π BCS T c T Gorkov (particle-hole pair fluctuations, Gorkov et al. '61) c. NPRG Gorkov Tc Tc F BEC limit dimer-dimer scattering length (vacuum): a dd 0.59 a [Scherer et al.'10] exact a dd = 0.6 a [Petrov et al.'04] [See discussion by Birse, Krippa, Wallet PRA'11] T c = T BEC (1+C n1/3 B add ) C 1.4, C exact 1.3 [but requires momentum dependence of bosonic propagator, see Baym et al. PRL'99] Unitary limit T NPRG c = 0.48 ϵf [Scherer et al.'10] T exp c ϵ F = T BDMC c = ϵf (Bold diagrammatic MC)

40 PI diagrammatics [Haussmann et al.'07] Bold diagrammatic MC: random resummation of millions of diagrams [K. Van Houcke et al. '1] Equation of state of the unitary Fermi gas MIT (normal phase) ENS (Paris) [Picture: K. Van Houcke et al. '1]

41 Hubbard-Stratonovich fields and competing instabilities Baier, Bick, Wetterich '04 Krahl, Friederich, Muller, Wetterich '07 '09 '11 [Related approach: Husemann & Salmhofer '09] Rewrite 4-point vertex in charge/spin ph and singlet/triplet pp channels Γ int, Λ [ ψ+, ψ] = U local, Λ ψ (k ) ψ(k ) ψ (k 3 ) ψ(k 1 k +k 3 ) ] [ ] [ 1 k 1 k 3 λ sp, Λ (k 1 k ) [ ψ (k 1 )σ ψ(k ) ] [ ψ (k 3 )σ ψ(k 1 k +k 3 ) ] λ ch, Λ (k 1 k ) [ ψ (k 1 ) ψ(k ) ] [ ψ (k 3 ) ψ(k 1 k +k 3 ) ] k 1 k 3 k 1 k (+ pp channel) Flowing bosonization (Gies & Wetterich '0): Hubbard-Stratonovich transformation at each step of the RG procedure to eliminate fermionfermion interaction bosons in charge/spin ph and singlet/triplet pp channels Phase transition: boson condensation, no divergence of the flow Phase diagram: qualitative agreement with fermionic RG

42 Phase diagram of D Hubbard model: U/t=3 and t'/t=-0.1 AF SCd [Friederich, Krahl & Wetterich PRB '11]

43 Conclusion Fermi-Dirac statistics (Fermi surface) makes the field theory of interacting fermion systems very peculiar low-energy modes lie near the Fermi surface the fermion field is not an order parameter Collective bosonic fluctuations drive phase transitions Fermionic RG well understood theoretically, efficient technique to deal with competing instabilities (low dimensions) functional RG weak-coupling RG (usually 1 loop) no description of broken-symmetry phases Beyond the fermionic RG some sort of bosonization has to be used (Hubbard-Stratonovich fields, etc.) towards a quantitative description of strongly-correlated fermion systems?

44 What has been left out of this talk Other fermion systems iron-based superconductors graphene etc. Quantum dots (Kondo physics, transport) etc. See review: Metzner et al. Rev. Mod. Phys. '1

45 Reviews C. Bourbonnais & L. Caron Renormalization group approach to quasi-one-dimensional conductors Int. J. Mod. Phys. B 5, 1033 (1991) W. Metzner, M. Salmhofer, C. Honerkamp, V. Meden, & K. Schönhammer Functional renormalization group approach to correlated fermion systems Rev. Mod. Phys. 84, 99 (01)

46 Acknowledgments C. Bourbonnais (Sherbrooke), R. Duprat (Sherbrooke) & C. Nickel (Orsay/Sherbrooke) LPTMC (Paris): B. Delamotte, D. Mouhanna, A. Rançon, G. Tarjus & M. Tissier