Ferromagnetic Luttinger liquids and: An exact integral equation for the renormalized Fermi surface Peter Kopietz Institut fur¨ Theoretische Physik, Universit¨at Frankfurt am Main Lorenz Bartosch, Marcus Kollar, and Peter Kopietz, Phys. Rev. B 67, 092403 (2003). Sascha Ledowski and Peter Kopietz, J.Phys.: Condens. Matter 15, 4779 (2003). 1/22 Introduction Can itinerant electrons in 1d form a ferromagnetic ground state? • Experiments and numerical studies indicate: • ferromagnetic ground state is possible in 1d! here: use perturbation theory and bosonization • to calculate correlation functions (spin correlations, single-particle Green function) Symmetry-broken Luttinger liquid (analogous to weak ferromagnets in 3d) • what happens to spin-waves in symmetry broken phase? • anomalous dimension close to quantum phase transition can be large! • 2/22 Lieb-Mattis theorem, (1962) There is no ferromagnetic ground state in a 1d conductor!? valid under fairly general conditions: spin and velocity independent forces • N p^2 H^ = + V (x^ ; : : : ; x^ ) • 2m∗ 1 N Xi=1 lattice model with nearest neighbor hopping • example: N=2: Helium atom (in any dimension) • E(S = 0) < E(S = 1) 3/22 Evidence for 1d itinerant ferromagnetism \0.7 structure" of quantum • point contacts has been linked to spontaneous ferromagnetism (Thomas et al., 1996) Extended Hubbard model can show 1d ferromagnetism (Daul and Noack, • PRB 1998) DFT calculation predicts band ferromagnetism for purely organic polymers • (Arita et al., PRL 2002). 4/22 Interacting electrons in 1d 1 Hamiltonian H^ = c^y c^ + [f ρ^ ( q)ρ^ (q) + f ρ^ ( q)ρ^ (q)] k kσ kσ 2L n n − n m n − n Xkσ Xq ρ^ (q) = c^y c^ (charge density) • n kσ kσ k+qσ P ρ^ (q) = σc^y c^ (spin density) • m kσ kσ k+qσ for weak Pferromagnets (m k /π): expand energy dispersion close to • F Fermi points: (k k )2 λ = + v (k k ) + − F + (k k )3 k kF F − F 2m∗ 6 − F cubic parameter λ will be necessary to stabilize ferromagnetic ground state • H^ is spin-rotationally invariant • 5/22 A first step: Hartree-Fock theory write Hamiltonian as H^ = H^0 + H^1 Hartree-Fock part: L H^ µN^ = ξ c^y c^ [f n2 + f m2] ; ξ = µ + σ∆ ; ∆ = f m 0 − kσ kσ kσ − 2 n m kσ k − − m Xkσ fluctuation part: 1 H^ = f δρ^ ( q)δρ^ (q) ; δρ^ (q) = ρ^ (q) δ ρ^ (0) 1 2L i i − i i i − q;0h i i Xq;i self-consistent determination of k"; k#; m; δn = n(m) n(0) at constant µ: − + f δn = σ∆ kσ − kF n δn = π−1(k + k 2k ) 9 k ; k ; δn; m " # F > " # −1 − = ) m = π (k" k#) − 6/22 ;> Change in ground-state energy at constant m Hartree Fock ground state energy Ω(m) = H^0 µN^ Landau expansion for weak ferromagnetismhπm− k i F Lv A Ω(m) Ω(0) = F I (I 1)(πm)2 + I4(πm)4 + : : : − 4π − 0 0 − 4 0 0 < I 0 < 1 m) δΩ( I 0 = 1 I 0 > 1 -m0 m m0 Stoner-parameter: I = 2f /πv 0 − m F quartic coefficient: A = 1 [λ 3λ2=(1 + F )] ; λ = (m∗v )−1 ; λ = λ/v 12 2 − 1 0 1 F 2 F non-trivial solution at πm = [2(I 1)=(I3A)]1=2 if I > 1. 0 0 − 0 0 7/22 Longitudinal spin fluctuations in RPA RPA for longitudinal spin- and charge susceptibilities (f = f = f > 0): n − m 0 χRPA(q; i!) = [χ0 + χ0 4f χ0 χ0 ]=D ; nn "" ## − 0 "" ## RPA 0 0 0 0 χmm (q; i!) = [χ"" + χ## + 4f0χ""χ##]=D ; where D(q; i!) = 1 4f 2χ0 χ0 and − 0 "" ## 0 1 f(ξk+q=2,σ0 ) f(ξk−q=2,σ) χσσ0 (q; i!) = − −L ξ 0 ξ i! Xk k+q=2,σ − k−q=2,σ − For small q and !: 2 0 vσ q χσσ(q; i!) 2 2 ≈ π (vσq) + ! 2 1 λ1 2 vσ=vF = 1 + σλ1∆=vF + λ2 (∆=vF ) + : : : 2 − 1 + F0 8/22 ballistic longitudinal spin fluctuations Wanted: correlation functions: 1 S (x; t) = [ ρ^ (x; t)ρ^ (0; 0) ρ^ (x; t) ρ^ (0; 0) ] i 2πV h i i i − h i ih i i −1 Fourier-transform: dynamic structure factor: Si(q; !) = π Imχii(q; ! + i0) Within RPA: longitudinal spin fluctuations propagate ballistically! SRPA(q; !) = Z q δ(! v q ) no Landau damping in 1d! • m mj j − mj j weight: Z = 1/πpδ 1 • m 0 velocity v = v pδ v • m F 0 F δ 2(I 1)=I 1 measures distance to zero temperature critical point • 0 ≡ 0 − 0 (quantum phase transition) susceptibility sum rule: 2 lim 1 d! SRPA(q; !) = χ • q!0 0 ! m m spin susceptibility: χ−1 = L−1 @2RΩ(m)=@m2 m m0 9/22 transverse suscrptibility: ladder approximation ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢£¤¥ ¦§ LAD LAD¡ ¡ ¡ ¡ ¡ 0 −1 −1 χ (q; i!) ¡ ¡ ¡ ¡ ¡ [χ (q; i!) 2f ] "# ¡ ¡ ¡ ¡ ¡ "# 0 ¡ ¡ ¡ ¡ ¡ ≡ ¡ ¡ ¡ ¡ ¡ − ¡ ¡ ¡ ¡ ¡ for small q and !: m i! χ0 (q; i!) 0 1 + Bq2 "# ≈ 2∆ 2∆ − −1 −1 k" 2 1 2 B = [4∆(k" k#)] [v" + v# ∆ dkv ] = [λ2 λ ] − − k# k 12 − 1 Spin-wave pole (Goldstone mode): R LAD m0 2 χ (q; i!) ; S"#(q; !) = m δ(! 2∆Bq ) "# ≈ −i! 2∆Bq2 0 − − However: Can perturbation theory be trusted? 10/22 ferromagnetic Luttinger liquid in ferromagnetic phase: marginal couplings: dominant forward scattering effective low-energy theory in symmetry broken phase ) Λ dq αy α 1 H^eff = αvσq ^ (q) ^ (q) + fijρ^i( q)ρ^j (q) Z 2π σ σ 2L − Xασ −Λ Xq;ij with renormalized couplings and densities Λ dq0 ^αy 0 ^α 0 ρ^n(q) = ασ −Λ 2π σ (q ) σ (q + q). P R need Wilsonian RG to arrive at this model • irrelevant couplings stablize ferromagnetism! calculate correlation functions using bosonization • closed loop theorem (Dzyaloshinskii, Larkin, 1974) : • all corrections beyond RPA for χmm(q; !) cancel! what happens to transverse spin waves in Luttinger liquid? • 11/22 Single particle Green function 2 ηn=2 2 ηm=2 1 r0 r0 Gσ(x; τ) = 2 2 2 2 2 2 2πi x + vnτ x + vmτ eiαkσx × [αx + iv τ]1=2[αx + iv τ]1=2 Xα n m Luttinger liquid behavior • anomalous dimensions η = 1 (K + K−1 2) • i 4 i i − with K = [I + 1]−1=2 and K = [2(I 1)]−1=2 n m − η diverges for I 1 • m ! Luttinger liquid relation: K = πv χ =2 • m m m at quantum phase transition: χ and v 0 m ! 1 m ! 12/22 13/22 ¦§ ¢£¤¥ osonization b 2 x ] ) τ # k m − iv " k + ( iα αx e [ from α symmetry y ersion P en m m η η disp 2 rok 2 b i 2 # τ to k 2 m 2 0 v onent r energy − + 2 due " x exp k is h ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ rized 2 ≈ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ) 1 ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ q π ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ − susceptibilit ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ r (2 BOS linea scaling fo contribution a = d o rge on ≡ go contribution ) cha τ x; no spin anomalous based only ( • • • • • BOS "# ransverse χ T Low energy spin excitations BOS S (q; !) = C Θ(! v q k" + k# ) "# m − mjj j − j 2ηm−1 [! v ( q k" + k#)] × − m j j − 2ηm+1 [! + v ( q k" + k#)] × m j j − −1 4ηm with Cm = [4πvmΓ(2ηm)Γ(2 + 2ηm)] (r0=vm) ¡£¢ ¤ 1d Stoner ¢ • continuum ¢ no Landau ¦ © • damping ¦ © ¥§¦ ¥§© ¤ ¤ ¨ ¢ 14/22 Conclusions and outlook: Effective low-energy theory of weak ferromagnetic Luttinger liquids • Stoner parameter δ 2(I 1) measures distance to quantum critical point • ≈ − longitudinal susceptibility: SRPA(q; !) = Z q δ(! v q ) • m mj j − mj j v δ1=2, Z δ−1=2 χ Z =v δ−1 m / m / ) m / m m / Luttinger liquid relation χ = 2K /πv • m m m anomalous dimension: η = 1 (K + K−1 2) at quantum ) m 4 m m − ! 1 phase transition! Open problems: study ferromagnetic instability beyond Hartree-Fock theory • (numerics, Onsager correction) use Wilsonian renormalization group to study instability: • how do irrelevant operators (band curvature) stabilize ferromagnetism? formulation of problem as self-consistent determination of the Fermi • surface at the RG fixed point. 15/22 Beyond Hartree-Fock: functional RG Strategy: determine Fermi-surface (i.e. k"; k#) self-consistently as fixed point of renormalization group! RG for fermions: 1d: g-ology [Solyom, 1979] • one-loop, using naive momentum-shell [Shankar (1994)] • functional RG, one-loop, using Polchinski equation: [Zanchi, Schulz • (1996); Salmhofer, Honerkamp (1998); Halboth, Metzner (1999)] using irreducible vertices [Honerkamp, Salmhofer (2001); P.K., Busche • (2001)] What is the problem with Fermions? Propagator singular on entire surface in k-space: Zk G(k; !) F ≈ ! v (k k ) iδ − F · − F 16/22 Fermi surface = RG fixed point Nozi`eres, 1964: • \In practice, we shall never try to calculate the Fermi surface, which is much too difficult. What is more important to us is to know that it exists." Here: need renormalized Fermi surface k ; k . • " # Strategy (S. Ledowski, P.K., J.Phys. Condens.Matter, 2003): • Requirement that RG flow approaches fixed point yields self-consistency condition for the renormalized Fermi surface! Implementation via functional RG for irreducible vertices • exact flow equation of two-point vertex Γ(2)(K; K) = [Σ (K) Σ(k ; i0)] Λ − Λ − F (2) δ( k0 kF Λ) (4) 0 0 @ΛΓΛ (K; K) = j − j − 0 ΓΛ (K; K ; K ; K) Z 0 i! 0 k0 + µ Σ (K ) K n − − Λ 17/22 Flow of four- and six-point vertices 1 1 1 1 1 1 4 6 4 4 2 2 2 2 2 2 BCS 1 1 1 1 4 4 4 4 2 2 ZS 2 2 2 1 2 1 4 4 4 4 2 1 ZS 2 1 1 1 1 1 1 1 2 6 2 2 8 2 3 2 6 2 4 3 3 3 3 3 3 1 1 1 1 1 1 2 6 2 2 6 2 3 4 2 6 2 9 3 3 4 4 3 3 3 3 1 1 1 1 4 4 2 2 2 2 4 4 4 4 9 3 3 3 3 1 1 1 1 4 4 2 2 2 3 4 4 3 3 36 4 4 2 3 18/22 Scaling toward the Fermi surface Field rescaling consider RG flow of renormalized vertices: () ~(2n) 0 0 Γl (Q1; : : : ; Qn; Qn; : : : ; Q1) = n0 n0 1=2 n−1 n−2 ^1 ^n n^n n^1 (2n) 0 0 ν0 Λ Zl Zl Zl Zl ΓΛ (K1; : : : ; Kn; Kn; : : : ; K1) : · · · · · · label momenta by unit vector n^ and distance p from Fermi surface: k = kF + v^F p v p scaling variables: Q = (n^; pξ; i!τ) k F kF length: ξ = vF =Λ ; time: τ = 1=Λ n infrared cutoff Λ (units of energy) The renormalized vertices can be viewed as dimensionless dynamic scaling functions! ~(2) example: 2-point vertex Γl (Q; Q) = Φl(n^; pξ; !τ) 19/22 flow equation for FS shift Zn^ rescaled two-point vertex Γ~(2)(Q) = l [Σ (K) Σ(k ; i0)] satisfies l − Λ Λ − F ~(2) n^ ~(2) _ 0 ~(4) 0 0 @lΓl (Q) = (1 ηl q@q ∂)Γl (Q) Gl(Q )Γl (Q; Q ; Q ; Q) − − − − ZQ0 relevant couplings µ~n^ Γ~(2)(σ; n^; q = 0; i = i0) = Γ~(2)(Q ) satisfy l ≡ l l 0 n n n @ µ~^ = (1 η^ )µ~^ + Γ_ (2)(n^) l l − l l l (2) 0 (4) 0 0 with Γ_ (n^) = 0 G_ (Q )Γ~ (Q ; Q ; Q ; Q ).