Sliding Luttinger Liquids

Transcription

1 Sliding Luttinger Liquids C. L. Kane T. C. Lubensky R. Mukhoadyay I. Introduction The D Luttinger Liquid II. The Sliding Phase A 2D Luttinger liquid University of Pennsylvania Penn / NEC Mukhoadyay, Kane, Lubensky, PRB 200; Ohern, Lubensky, Toner PRL 999; Emery, Fradkin, Kivelson, Lubensky PRL 200; Vishwanath, Carentier PRL 200; Sondhi, Yang PRB 200. III. Instabilities of the Sliding Phase The Fractional Quantum Hall Effect from D Bosonization Kane, Mukhoadyay, Lubensky PRL 2002.

2 Weakly Couled D Electron Systems Stri Phases of Curate Suerconductors Quantum Hall Smectic Phases Weakly Couled Wires e.g. Nanotube roes Theoretical Motivation: Can the owerful techniques from D be used to understand strongly correlated states in higher dimensions?

3 Three Views of the D Electron Gas. Non Interacting Fermi Liquid 2. Reulsive Interactions Almost a crystal ψ L ψ R 3. Attractive Interactions Almost a suerconductor Luttinger Liquid ( ) 2 L = µθ 8π g g 2 = ( µϕ) 8π ψψ i L R e ϕ Power Law Correlations with exonent deending on interactions. Analogous to classical 2D XY model Bosonization: ψ LR e ϕ θ i( ± )/2, Non Interacting g = < Reulsive > Attractive

4 Couled Luttinger Liquids Exect instabilities due to couling between wires. Charge Density Wave OCDW = cos( θi θ j) < ij> 2. Suerconductor OSC = cos 2( ϕi ϕ j) < ij> 3. 2D Fermi Liquid Renormalization Grou Analysis: dλ d O = ψ ψ + ψ ψ α FL < ij> Li Lj Ri Rj = (2 ) λ α α H Relevant if < 2. α int = λ Oˆ α α = 2g CDW = 2/ g SC = ( g+ / g)/2 FL Is the Luttinger Liquid always unstable?

5 Two Kinds of Interactions:. Forward Scattering H = V θ θ + V ϕ ϕ θ ϕ FS ij x i x j ij x i x j ij 2. Interchannel Scattering Oˆ, Oˆ, Oˆ,..., many more CDW SC FL Resonsible for Instabilities Dimensions deend on H FS Sliding Luttinger Liquid Choose H FS to make all O α irrelevant Smectic Metal Anisotroic Electrical Conductivity Power Law correlations (like D L.L.) Collective Modes roagate in 2D An anisotroic 2D Luttinger Liquid Analogous to Sliding Phase of couled classical 2D XY models Ohern, Lubensky, Toner PRL 99

6 Model Interaction Ohern, Lubensky, Toner PRL 99 ϕ θ κ ( q ) V ( q )/ V ( q ) = K(+ λ cos q + λ cos 2 q ) 2 Nearest neighbor model λ 2 = 0, nearest neighbor CDW, SC and FL terms only. λ Emery,Fradkin,Kivelson, Lubensky PRL 0 More general model K Vishwanath, Carentier PRL 0 Mukhoadyay, Kane, Lubensky PRB 0 Sondhi, Yang PRB 0 Higher order oerators reduce region of stability - Further neighbor CDW, SC, FL - Correlated hoing, etc. Sliding hase stable close to boundary of instability to transverse CDW with wavevector q 0

7 Nearest neighbor model Emery,Fradkin,Kivelson, Lubensky PRL 0 Nearest neighbor CDW, SC and FL terms only. Model Interaction: ϕ θ κ ( q ) V ( q )/ V ( q ) = K( + λ cos q ) λ - Further neighbor CDW, SC, FL - Correlated hoing, etc. K (~ /g) Higher order oerators lead to instability

8 Stabilizing the Sliding hase Model Interaction: κ( q ) = K(+ λ cosq + λ cos2 q ) 2 Ohern, Lubensky, Toner PRL 99 Vishwanath, Carentier PRL 0 Mukhoadyay, Kane, Lubensky PRB 0 Large density fluctuations at wavevector q 0 for small δ frustrate CDW formation. Values of λ, λ 2 for which SLL hase is stable to a large class of oerators for some K λ Vishwanath, Carentier PRL 0 λ 2 Perendicular Magnetic field increases region of stability by eliminating suerconducting instability. Sondhi, Yang, PRB 0

9 Crossed Sliding Luttinger Liquid Interactions between erendicular wires are marginal but do not affect dimensions of oerators. Tunneling between erendicular wires is irrelevant in sliding hase. Electrical conductivity isotroic at low (but finite) temerature. An isotroic 2D Luttinger Liquid

10 Instabilities of the Sliding Phase Integer Quantum Hall Effect From Bosonization: H = λ cos[ ϕ ϕ + θ + θ ]/ 2 int i i+ i i+ i= Switch Partners H λ cos θ λ ( θ ) = int i+ / 2 i+ / 2 i= i= Ga in bulk Edge Mode φ = ϕ -θ remains galess 2

11 Laughlin State: ν = /3 3 article correlated tunneling rocess H = λ cos[ ϕ ϕ + 3( θ + θ )]/ 2 int i i+ i i+ i= Rescale: θ = 3 θ H = λ cos[ ϕ ϕ + θ + θ ]/ 2 int i i+ i i+ i= Switch Partners H λ cos θ λ ( θ ) = int i+ / 2 i+ / 2 i= i= 2 Edge Mode: g=/3 Chiral Luttinger Liquid ~ 2π soliton in θ : Charge e/3 quasiarticle

12 Three Categories I. Quantum Hall States Allowed at secial magnetic fields Generalized Hierarchy cos Θ Θ 2 : Bulk Ga + Edge states II. Crystals ν = 2/3 ν = 2/3 0 K = K = 0 3 Allowed at any magnetic field: cos Θ Θ 2 : 2D Phonon mode H = λ cos[ m θ + n ϕ ] int i= Crystal of Electrons i+ i+ Crystal of Laughlin Quasiarticles v = 2 m n m (Haldane-Halerin Hierarchy) ( Bilayer state ) = n = 0 III. Degenerate Oerators Difficult to analyze : cos Θ Θ 2 Fermi Liquid (B=0) ν = /2: Comosite Fermi Liquid?

13 Conclusion I. Sliding Luttinger Liquid Anisotroic 2D hase with ower law correlations characteristic of D Luttinger Liquid. Residual Coulings irrelevant II. Instabilities of Sliding Phase D bosonization offers a new, concrete framework for describing the fractional quantum Hall effect. III. Can this be used to describe other strongly correlated states? Non Abelian Quantum Hall States Sin Liquid states, sin/charge searation.