Spontaneous symmetry breaking in fermion systems with functional RG

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1 Spontaneous symmetry breaking in fermion systems with functional RG Andreas Eberlein and Walter Metzner MPI for Solid State Research, Stuttgart Lefkada, September 24 A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 1

2 Outline Introduction Coupled fermion-boson flow for a fermionic superfluid Fermionic functional RG for a fermionic superfluid Fusion of functional RG and mean-field theory A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 2

3 Introduction Spontaneous symmetry breaking in condensed matter Quantum liquids: Superfluid 4 He (bosons) Superfluid 3 He (fermions) Solids: Crystallization Electrons in solids: Magnetism: ferro-, ferri-, antiferro-,... Charge order: density waves, stripes,... Orbital order Superconductivity: singlet (s-wave, d-wave), triplet (p-wave) A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 3

4 Introduction Famous example: CuO 2 high temperature superconductors antiferromagnetism in undoped compounds Temperature AF pseudo gap S Doping x d-wave superconductivity at sufficient doping Pseudo gap, non-fermi liquid in normal phase at finite T energy [ev] Vast hierarchy of energy scales: Magnetic interaction and superconductivity generated from kinetic energy and Coulomb interaction U (Coulomb repulsion) t J (magnetic interaction) k B T c (kinetic energy, hopping) (transition temperature for SC) A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 4

5 Introduction Prototype: Hubbard model Effective single-band model for CuO 2 -planes in HTSC: (Anderson 87, Zhang & Rice 88) t t U Hamiltonian H = H kin + H I H kin = t ij c iσ c jσ = ɛ k n kσ i,j σ k,σ H I = U n j n j j Antiferromagnetism at/near half-filling for sufficiently large U d-wave superconductivity away from half-filling (perturbation theory, RG, cluster DMFT, variational MC, some QMC) A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 5

6 Fermionic flow equations Introduction Fermi surface singularity at ω = 0, ξ k = ɛ k µ = Flow parameter: Infrared cutoff Λ > k y Momentum cutoff: G0 Λ (k, iω) = Θ( ξ k Λ) iω ξ k Frequency cutoff: G0 Λ Θ( ω Λ) (k, iω) = iω ξ k π π π 0 k x π Many other choices: mixed momentum-frequency, smooth cutoff etc. Initial condition: Λ 0 = band width (momentum) or (frequency) A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 6

7 Fermionic flow equations Introduction Exact functional flow equation for effective action Γ Λ [ψ, ψ] (Wetterich 1993; Salmhofer & Honerkamp 20) Exact hierarchy of flow equations for m-particle vertex functions d Σ Λ dλ = d Γ Λ = dλ d Γ Λ = dλ S Λ S Λ G Λ 3 + Γ Λ Γ Λ S G Λ Γ Λ Λ G Λ Λ Γ Γ Λ + Γ Λ Γ3 Λ Γ Λ S Λ S Λ G Λ Γ3 Λ + Γ4 Λ G Λ = [(G 0 Λ ) 1 Σ Λ] 1 S Λ S Λ = d dλ G Λ Σ Λ fixed A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 7

8 Introduction Effective 2-particle interaction at 1-loop Λ = bare G Λ 0 Σ Λ = 0 All channels (particle-particle, particle-hole) captured on equal footing. Flow equations for susceptibilities (a) (b) Λ Λ = = A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 8

9 Introduction Effective interaction and susceptibilities in Hubbard model: π ky 0 π π 1-loop flow of interactions π kx 1-loop flow of susceptibilities n = U = t t = 0 Γ s (i 1,i 2 ;i 3 )/t χ/χ Γ s (1,1;5) Γ s (1,5;1) Γ s (1,1;9) Γ s (1,9;1) Γ s (1,9;5) Γ s (3,3;11) Γ s (3,11;3) Γ s (2,10;2) Γ s (2,10;4) Λ/t sdw (π,π) sdw (π δ,π δ) sdw (π,π δ) cdw (π,π) sc d x 2 -y 2 sc d xy sc s, sc xs Λ/t Singlet vertex Γ Λ s (k 1, k 2 ; k 1, k 2 ) for various choices of k 1, k 2, k 1 Divergence at critical scale Λ c indicates instability Zanchi & Schulz Halboth & wm 00 Honerkamp et al. A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 9

10 Introduction Routes to symmetry breaking Divergence of effective interaction at scale Λ c signals instability order parameter generated Routes to spontaneous symmetry breaking in functional RG: Issues: Hubbard Stratonovich bosonization (Baier, Bick, Wetterich 2004) Fermionic flow with order parameter (Salmhofer et al. 2004) Andreas Eberlein Accurate order parameters Order parameter fluctuations Ward identities Goldstone mode A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 10

11 Introduction Outline Introduction Coupled fermion-boson flow for a fermionic superfluid Fermionic functional RG for a fermionic superfluid Fusion of functional RG and mean-field theory Superfluid state prototype for continuous symmetry breaking with Goldstone mode This lecture: Focus on ground state (T = 0) A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 11

12 Model for fermionic superfluid Fermionic action with attractive contact interaction U < 0 S[ψ, ψ] = k,σ ψ kσ (ik 0 ξ k )ψ kσ + U k,k,q ψ q k ψ k ψ k ψ q k ψ kσ, ψ kσ Grassmann fields, k = (k 0, k), k 0 Matsubara frequency Spin-singlet pairing with U(1)-symmetry breaking Complex order parameter ψ k ψ k A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 12

13 Hubbard-Stratonovich bosonization Decouple two-fermion interaction by Hubbard-Stratonovich transformation, introducing a bosonic order parameter field S[ψ, ψ, φ] = + k,q ψ kσ (ik 0 ξ k )ψ kσ 1 U φ q φ q k,σ q ( ψ q k ψ k φ q + ψ k ψ q k φq) U(1) symmetry: ψ e iϕ ψ, ψ e iϕ ψ, φ e 2iϕ ψ, φ e 2iϕ ψ A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 13

14 Exact flow equation for effective action Add regulator functions R Λ f and R Λ b for fermions and bosons Scale dependent effective action Γ Λ [ψ, ψ, φ] Exact flow equation (Wetterich 1993) d dλ ΓΛ [ψ, ψ, φ] = 1 2 Str Λ R Λ Γ (2)Λ [ψ, ψ, φ] + R Λ Γ (2)Λ [ψ, ψ, φ] matrix of second derivatives w.r.t. fields A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 14

15 Ansatz for effective action Γ Λ [ψ, ψ, φ] = Γ Λ b [φ] + ΓΛ f [ψ, ψ] + Γ Λ bf [ψ, ψ, φ] bosons fermions mixed Two distinct regimes: Λ > Λ c : symmetric regime, no anomalous terms Λ < Λ c : symmetry broken regime, anomalous terms (ψψ, ψψφ etc.) Conditio sine qua non: respect symmetry A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 15

16 Ansatz for effective action: fermions Only quadratic terms: Γf Λ [ψ, ψ] = + k,σ k,σ ψ kσ (iz Λ f k 0 A Λ f ξ k ) ψ kσ ( Λ ψ k ψ k + Λ ψ k ψ k ) Second term with pairing gap Λ only in symmetry-broken regime Z Λ f and A Λ f finite renormalizations usually discarded (Z Λ f = A Λ f = 1) Quartic terms (generated by flow) may be decoupled during the flow by dynamical bosonization (Gies & Wetterich 2002, 2004; Floerchinger et al. 2008, 2009) A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 16

17 Ansatz for effective action: fermion-boson interaction ( Γbf Λ [ψ, ψ, φ] = g Λ ψ q k ψ k φ q + ψ k ψ q k φq) + g Λ k,q k,q ( ψ q k ψ k φ q + ψ k ψ q k φ q ) Second (anomalous) term only in symmetry-broken regime g Λ small, often neglected ( g Λ = 0) Renormalization of g Λ small, often neglected (g Λ = 1) Other terms (e.g. ψψφ, ψψφ φ) usually discarded A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 17

18 Ansatz for effective action: bosons General structure: Γb Λ [φ] = dx Uloc[(φ(x)] Λ + gradient terms Simplest (quartic) ansatz for local part: U Λ loc (φ) = 1 2 (mλ b )2 φ uλ φ 4 U Λ loc (φ) = 1 8 uλ [ φ 2 α Λ 2 ] 2 for Λ > Λ c for Λ < Λ c (mexican hat) A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 18

19 Ansatz for effective action: boson gradient terms Simplest (quadratic) ansatz for gradient terms: 1 dx [W Λ φ (x) x0 φ(x) Zb Λ φ(x) 2] 2 Birse et al Diehl et al Krippa 2007 The simplest ansatz, with quartic local and quadratic gradient terms yields decent results for and T c in three dimensions, not only for weak, but also for strong interactions (from BCS to BEC). A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 19

20 Example: T c in three dimensions Critical temperature T c versus inverse scattering length for fermions in 3D continuum, obtained from relatively simple ansatz for Γ Λ : Diehl, Gies, Pawlowski, Wetterich 2007 Decent approximation from weak to strong attraction! A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 20

21 Ansatz for effective action: boson gradient terms Simplest (quadratic) ansatz for gradient terms: 1 dx [W Λ φ (x) x0 φ(x) Zb Λ φ(x) 2] 2 Birse et al Diehl et al Krippa 2007 The simplest ansatz, with quartic local and quadratic gradient terms yields decent results for and T c in three dimensions, not only for weak, but also for strong interactions (from BCS to BEC). But: u Λ, W Λ, (Z Λ b ) 1 scale to zero for Λ 0 in dimensions d 3! Strong renormalization of longitudinal order parameter fluctuations expected, but transverse fluctuations (Goldstone) should be protected! Mission: Save the Goldstone mode! A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 21

22 Ansatz for effective action: boson gradient terms Goldstone mode protected by hand in Strack, Gersch, wm 2008 Goldstone mode protection by symmetry can be achieved by adding quartic gradient term 1 8 dx Y Λ ( φ(x) 2 ) 2 Strack, PhD thesis 2009 Obert, Husemann, wm 23 Previously introduced to treat Goldstone mode in O(N) models by Tetradis & Wetterich 1994 A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 22

23 Decomposition in longitudinal and transverse fluctuations Choose bosonic order parameter α Λ real and positive Decompose φ(x) = α Λ + σ(x) + iπ(x), φ q = α Λ δ q0 + σ q + iπ q with longitudinal (σ) and transverse (π) fluctuations Γb Λ [φ] = 1 [ mσ 2 + Z σ (q0 2 + ω 2 q) ] 2 σ q σ q q + 1 Z π (q0 2 + ω 2 2 q)π q π q + W q 0 π q σ q q q + 1 p σ q σ q p q,pu(p)ασ 1 U(p)σ q σ p q π q π p q 4 q,q,p with m 2 σ = uα 2, Z π = Z b, Z σ = Z b + Y α 2, and U(p) = u + Y (p ω2 p) ω 2 q q 2 for small q A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 23

24 Decomposition in longitudinal and transverse fluctuations Γ Λ bf [ψ, ψ, φ] = g σ + ig π k,q k,q with g σ = g + g and g π = g g ( ψ q k ψ k σ q + ψ k ψ q k σ q ) ( ψ q k ψ k π q ψ k ψ q k π q ) A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 24

25 Flows for attractive 2D Hubbard model Nearest neighbor hopping t, dispersion ɛ k = 2t(cos k x + cos k y ) Moderate attractive interaction U = 4t Quarter filling (Fermi surface nearly circular) T = 0 (ground state) Regulator functions: R f (k) = R f (k 0 ) = [iλsgn(k 0 ) ik 0 ] Θ(Λ k 0 ) R b (q) = R b (q 0 ) = Z b (Λ 2 q0) 2 Θ(Λ q 0 ) A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 25

26 Flows for attractive 2D Hubbard model Flow of gap and bosonic order parameter α: Δ α Λ slightly larger than α A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 26

27 Flows for attractive 2D Hubbard model Flow of bosonic masses m 2 b and m2 σ: m b 2 m σ Λ m 2 σ vanishes for Λ 0 A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 27

28 Flows for attractive 2D Hubbard model Flow of bosonic Z-factors: Z b Z σ Z π Λ Z σ diverges, Z π saturates for Λ 0 A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 28

29 Flows for attractive 2D Hubbard model Flow of interaction parameters u and Y : u Y Λ u vanishes, Y diverges for Λ 0 A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 29

30 Flows for attractive 2D Hubbard model Flow of the σ-π mixing coefficient W : W Λ W vanishes for Λ 0 A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 30

31 Goldstone mode cancellations from symmetry Vanishing Goldstone mass m π = 0 conserved by flow? At first sight, many contributions: d dλ m2 π = uα dα [ dλ 2g π 2 D Λ G f (k) 2 + Ff 2 (k) ] k {[u + 2U(q)] D Λ G ππ (q) + D Λ G σσ (q)} q [U(q)] 2 [ α 2 D Λ G σσ (q)g ππ (q) + Gσπ(q) ] 2 q G f (k) = ψ kσ ψ kσ, F f (k) = ψ k ψ k G σσ (q) = σ q σ q, G ππ (q) = π q π q, G πσ (q) = π q σ q Derivative D Λ acts only on regulator function in propagators A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 31

32 Goldstone mode cancellations from symmetry Bosonic fluctuation contributions cancel ( d gσ dλ m2 π = 2 α g π 2 ) D Λ F f (k) k Ward identity from U(1)-symmetry: = gα gα = g π α for real α case g = 0: Bartosch, Kopietz, Ferraz 2009 d dλ m2 π = 2 α (g σ g π ) D Λ F f (k) k d dλ m2 π = 0 iff g σ = g π Goldstone mass vanishes A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 32

33 Goldstone mode cancellations from symmetry Setting g σ = g π yields further cancellations in other quantities such that Ward identity = g π α consistent with flow of, g π and α Z π finite for Λ 0 W /m 2 σ C finite for Λ 0 G ππ (q) 1 Z π (q ω2 q) G πσ (q) Cq 0 Z π (q ω2 q) m 2 σ, W, and Z 1 σ vanish for Λ 0 (proportional to Λ in 2D) G σσ (q) exhibits anomalous scaling Full agreement with IR behavior known for interacting bosons (e.g. Pistolesi et al. 2004) A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 33

34 Goldstone mode cancellations from symmetry Not yet happy? Exact flow would yield g σ = g π Consistent truncation with g σ = g π requires inclusion of two-fermion-two-boson vertices (Ward identity) A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 34

35 Summary (symmetry-breaking via bosonization) frg with Hubbard-Stratonovich bosonization ideal framework to treat order parameter fluctuation effects A simple truncation of the effective action captures all singularities associated with the Goldstone boson in fermionic superfluids (Obert, Husemann, wm 23) Many other problems of symmetry-breaking treated by frg with HS-fields, mostly by Heidelberg group: antiferromagnetism, d-wave superconductivity, Kosterlitz-Thouless transition,... A. Eberlein and W. Metzner Spontaneous symmetry breaking in fermion systems 35