Disorder and Quantum Fluctuations in Effective Field Theories for Highly Correlated Materials

Transcription

1 Disorder and Quantum Fluctuations in Effective Field Theories for Highly Correlated Materials T. Nattermann and S. Scheidl Insitut für Theoretische Physik Universität zu öln

2 Introduction motivation: central aspect of many transition metal compounds: phase transitions controled by doping disorder effects on large length scales change of microscopic system parameters so far: Classical Systems flux-line lattice charge density waves H amorphous vortex glass (pinned liquid?) Bragg Glass liquid Meissner unpinned liquid H c H c Tc critical point normal T (a) (b) D * D D c U D (v) U * U F * c(v) C (c) v + v c C C I δc δ D δ c I δ [Emig & Nattermann, PRL 79, 5090 (997); Emig & Nattermann, PRL 8, 469 (998)] Results: new phases and phase transitions, e.g. Bragg-Glass phase with QLRO and non-universal critical exponents 0 ; 0 ;4 RF RM Bragg ADVANCES IN PHYSICS, 000, VOL. 49, NO. 5, 607±704 Vortex-glass phases in type-ii superconductors T. NATTERMANN{ and S. SCHEIDL{ 0 ;6 0 ;8 = a Institut fuèr Theoretische Physik, UniversitaÈt zuoèln ZuÈlpicher Straûe 77, D oèln, Germany [Received September 999; revised 3 March 000; accepted 3 March 000] 0 ;0 0 ln(l) ln(la) ln(l) Abstract A review is given on the theory of vortex-glass phases in impure type-ii superconductors in an external eld. We begin with a brief discussion of the e ects of thermal uctuations on the spontaneously broken U() and translation symmetries, on the global phase diagram and on the critical behaviour. Introducing disorder we restrict ourselves to the experimentally most relevant case of weak uncorrelated randomness which is known to destroy the long-ranged translational order of the Abrikosov lattice in three dimensions. Elucidating possible residual glassy ordered phases, we distinguish between positional and phase-coherent vortex glasses. The study of the behaviour of isolated vortex lines and their generalizationðdirected elastic manifoldsðin a random potential introduces further important concepts for the characterization of glasses. The discussion of elastic vortex glasses, i.e. topologically ordered dislocation-free positional glasses in two and three dimensions occupy the main part of our review. In particular, in three dimensions there exists an elastic vortex-glass phase which still shows quasi-long-range translational order: the `Bragg glass. It is shown that this phase is stable with respect to the formation of dislocations for intermediate elds. Preliminary results suggest that the Bragg-glass phase may not show phasecoherent vortex-glass order. The latter is expected to occur in systems with weak disorder only in higher dimensions (or for strong disorder, as the example of unscreened gauge glasses shows). We further demonstrate that the linear resistivity vanishes in the vortex-glass phase. The vortex-glass transition is studied in detail for a superconducting lm in a parallel eld. Finally, we review some recent developments concerning driven vortex-line lattices moving in a random environment. Contents PAGE. Introduction 608. Ginzburg±Landau description 6.. The Ginzburg±Landau model 6.. Mean- eld theory 6.3. Thermal uctuations Zero external eld Finite external eld The in uence of disorder 60 u y=a 3. Directed elastic manifolds in a random potential Equilibrium properties 65 u x=a { natter@thp.uni-koeln.de { sts@thp.uni-koeln.de Advances in Physics ISSN 000±873 print/issn 460±6976 online # 000 Taylor & Francis Ltd [Emig, Bogner & Nattermann, PRL 83, 400 (999); Bogner, Emig & Nattermann, PRB 63, 7450 (00)]

3 Future: Highly Correlated Fermi-Systems with Disorder Universal properties depend mainly on symmetries and dimensions detailed knowledge of microscopic mechanisms and parameters is not necessary Starting point: effective field theories allow to classify the mechanisms describe fluctuations of collective modes allow to deal with disorder in an appropriate way For the present we concentrate on effective theories for stripe phases. Effective Field Theories: an Example Ĥ Hubbard = t i,j,σ ĉ iσĉjσ + U i ˆn i ˆn i for t U mapping on t J model with J t U antiferromagnetism of the Cu layers is described by a quantum nonlinear σ model S eff / = ρ0 s β 0 dτ [ d d x Ω + Ω c τ ] S(k,ω)ofLa CuO 4 g = g/g 0 g 0 = c/ρ 0 s ρ 0 s J d = [Chakravarty, Halperin & Nelson, PRL 60, 057 (988)] 3

4 Experimental Researches Generic Phase-diagram: Experimental Evidence for Stripe Phases: La -xsr xcuo 4 (a) k tet δ 0. h tet δ (r.l.u.) 0.05 k tet (b) δ [] Hole concentration h tet [] Nickelates: La NiO 4 diagonal static SDW (Tranquada et al. 94, Shirane et al. 95) La x Sr x NiO 4 for 0 <x</ diagonal static CDW and SDW, T CDW >T SDW (Yoshinari, Hammel, Cheong 99) Manganates: La 0.5 Ca 0.5 MnO 3 static CDW (Mori/Chen/Cheong 98, Uehara/Mori/Chen/Cheong 99) La x Sr +x MnO 4 static CDW for 0.5 <x<0.65 (Larochelle et al. ) La x Sr +x Mn O 7 short range charge stripe order for x 0.40 (Vasiliu-Doloc et al. ) La 0.9 Sr 0. Mn0 3 static CDW (Vigliante et al. ) Cuprates: La.6 x Nd 0.4 Sr x CuO 4 static SDW (Tranquada et al. 95, Imai, Hunt, Singer), most clear at x =/8; static CDW for x<0. (Niemöller et al.); strong competition with SC La x y Eu y Sr x CuO 4 static stripes for 0.08 <x<0.7 (ataev, Validov, Büchner, Hücker, Berg) La x Sr x CuO 4 dynamic SDW and CDW (Cheong, Aeppli, Mason, Mook 9, Yamada et al.); coexistence with SC YBa Cu 3 O 6+x dynamic SDW and CDW (Mook et al. 98, Egami); coexistence with SC Phenomenological stripe picture: [3] [4] [5] 4

5 Preparatory Work Structure skip competetive pinning of stripes by disorder and crystal potential CDW- and SDW-period depend on doping: z.b. -problem lock-in phenomenon? 8 /8 j 4 3 FIG. 7. Sr-doping dependence of the incommensurability the spin fluctuations. of Y(x) [6] x [7] Model: Stripes as lattice of fluctuating quantum strings with disorder (induced by doping) and crystal potential ρ(r,τ) { a m S = /T { dτ 0 H el = d r γ ( u), H U = d rρ(r)u(x), H V = d rρ(r)v (x, y) } e iqm[x u(r,τ)] x u(r,τ) d r µ } ( τu) + H. Results of RG-analysis: disorder relevant, dominates quantum fluctuations implies glassy dynamics in D: disorder destroys translational order lock-in only possible with 3D coupling [Bogner & Scheidl, Phys. Rev. B 64, (00)] 5

6 Magnetism Motivation: Focus on the low doping regime. Investigation of the stability of AFM against doping; Description of the AFM spin glass transition. Model: Holes are located on bonds, AFM FM exchange couplings frustration, Heisenberg spin glass [8] H = J i (r) [ is(r)] J i (r) = [ i (r)]j { r,i with probability p, i (r) = 0 with probability p. RG approach (symmetries are preserved, quenched nature of defects) d dl j l = (d )j l N Λ /d t 4 j l d dl ˆR l (φ) = d ˆR l (φ) ( + N t )φ Λ /d j ˆR l(φ) l + φ[ ˆR l (φ) ˆR l (0)]. j l ˆR l (0), t = T/J φ = n [ is α (r)] t α= j 0 = p ˆR 0 (φ) = ln [ p + pe φ/] pφ Results: D phase diagram at T =0 Calculation of the D correlation-length ξ(t,x = p) Include D/3D crossover phase-boundary in qualitative agreement to experiments ordered ordered disordered p disordered p ξ [A o ] x=0.00 x=0.0 x=0.03 x=0.04 theory T [] T N (x)/t N (0) x [rüger & Scheidl, cond-mat 0006 (00)] 6

7 Localization in coupled Luttinger liquids with impurities striped phases in Mott insulators: electronic liquid crystal phases, smectic metal (SM) D non-fermi liquid [7] without disorder: Coulomb Josephson λ + + Ψ i Ψ i Ψ j Ψj + + Ψ Ψ Ψ Ψ L,i R,i R,j L,j SM phase strongly limited by interstripe CDW and SC [9]. i j CDW + + Ψ Ψ Ψ Ψ R,i L,i L,j R,j with disorder: weak CDW couplings between stripes irrelevant, relevance of SC couplings reduced renormalized D system delocalization transition possible even for repulsive interaction new phase: delocalized smectic metal, DSM, Luttinger-like longitudinal transport, short-ranged longitudinal and transversal order S 0 = π j xτ [ v J ( x Θ j ) + v N ( x Φ j ) ] i τ Φ j x Θ j + γv J ( τφ j ) S fw = η j (x) x Φ j, π j xτ S bw = { } ξ j (x)e i( Φ j k F x) +h.c.. πα S V = π S CDW = i j S SC = i j j i j xτ C i j xτ J i j x Φ i V i j x Φ j. xτ xτ [ ] cos (Φi Φ j ), [ ] cos (Θi Θ j ). q 0=ß 0.9 0:9 D > 0 D =0 + D= :7 localized 0.5 0: :3 0. 0: SC DSM localized CDW SLL DSM SC :035 0: 0: [Bogner, Emig & Scheidl, cond-mat 0065 (00)] 7

8 Low temperature behavior of one-dimensional quantum systems CDWs, superfluids, Luttinger Liquids Ĥ = L 0 { [ c (v ) ] ( ˆP +( x ˆϕ) qπ + U(x)ρ(x)+W cos c x dy ˆP (y) )} dimensionless parameters t = T Λ d d /c, = vλ d d /c, u = U Λ d 4 d /c,w= W d /cλ disorder fluctuations d = [ p4 u dl coth t B 0(p, ] t ) dt = t dl [ du = 3 p dl coth ] u t quantum phase slips [ d = π q w dl coth ( q 4 t B, ) ] t [ dt = π q w dl coth ( q 4 t B, ) ] t t dw = [ q dl 4 coth ] w t w u c=8/q c=6/p t t t t u thermal classical disordered quantum disordered p= quantum critical u 0 pinned qp>4 3? unpinned phase slips u 0 pinned qp<4 3 phase slips u * u * w * w * u * numerical calculated cross-over phase diagram for disordered D CDWs combined influence of disorder and quantum phase slips (T =0) [Glatz & Nattermann, cond-mat (00)] 8

9 Future Projects analysis of disorder effects in effective models for stripe phases, in particular in coupled Luttinger-Liquids systems as electronic models for stripe phases interplay between charge and spin order in consideration of quantum fluctuations and planar anisotropie; spin/charge separation effects of topological defects [rüger & Scheidl, cond-mat (00)] (A) (B) J J [0] non equilibrium properties methodically related projects: disordered Wigner Crystals, vortex lattices, Quantum- Hall Stripe Phase, scattering between Quantum-Hall edges 9

10 Cooperations in SFB 608 B: transport measurements electronic properties of stripe phases B3: temperature dependence of transport properties search of Non-Fermi-Liquid behavior; susceptibility measurements magnetic order C: neutron scattering structural aspects of stripe phases D: microscopic fundamentals of effective field theories D5: numerical studies of electronic properties D6: gauge theoretical models of strongly fluctuating magnets 0

11 Literature [] B eimer et al., Phys. Rev. B 46, 4034 (99). [] S Wakimoto et al., Phys. Rev. B 6, 3699 (000). [3] J. Zaanen und O. Gunnarson, 40, 739 (989). [4] V. J. Emery, S. A. ivelson, and J. M. Tranquada, Proc. Natl. Acad. Sci. USA 96, 884 (999). [5] J. M. Tranquada et al., Nature375, 56 (995). [6]. Yamada et al., Phys. Rev. B 57, 665 (998). [7] S. A. ivelson, E. Fradkin, V. J. ivelson, Nature 393, 550 (998). [8] A. Aharony et al., Phys. Rev. Lett. 60, 330 (988). [9] V. J. Emery and S. A. ivelson and T. C. Lubensky, Phys. Rev. Lett. 85, 60 (000); A. Vishwanath and D. Carpentier, Phys. Rev. Lett. 86, 676 (00). [0] J. Zaanen et al., cond-mat/0003