4 December 2003NYU Colloquium1 Electronic Liquid Crystals Novel Phases of Electrons in Two Dimensions Alan Dorsey University of Florida Collaborators: Leo Radzihovsky (U Colorado) Carlos Wexler (U Missouri) Mouneim Ettouhami (UF) Support from the NSF

4 December 2003NYU Colloquium2 Competing interactions Long range repulsive force: uniform phase Short range attractive force: compact structures Competition between forces  inhomogeneous phase. Ferromagnetic films, ferrofluids, type-I superconductors, block copolymers

4 December 2003NYU Colloquium3 Ferrofluid in a Hele-Shaw cell Ferrofluid: colloid of 1 micron spheres. Fluid becomes magnetized in an applied field. Hele-Shaw cell: ferrofluid between two glass plates Surface tension competes with dipole-dipole interaction…

4 December 2003NYU Colloquium4 Results courtesy of Ken Cooper

4 December 2003NYU Colloquium5 Modulated phases Langmuir monolayer (phospholipid and cholesterol) Ferromagnetic film (magnetic garnet)

4 December 2003NYU Colloquium6 Liquid crystals T smectic-Csmectic-Anematicisotropic

4 December 2003NYU Colloquium7 Outline Overview of the two dimensional electron gas and the quantum Hall effect Theoretical and experimental evidence for a charge density wave? Liquid crystal physics in quantum Hall systems—smectics and nematics Quantum theory of the nematic phase

4 December 2003NYU Colloquium8 Two-dimensional electron gas (2DEG) Created in GaAs/AlGaAs heterostructures Magnetic field quantizes electron motion into highly degenerate Landau levels B AlGaAs E F N= Magnetic length Experiments at

4 December 2003NYU Colloquium9 The quantum Hall effect Filling fraction (per spin): State of the art mobility reveals interaction effects No Hall effect at half filling

4 December 2003NYU Colloquium10 Charge density wave in 2D? Hartree-Fock [Fogler et al. (1996)] predicts a CDW in higher LLs. Shown to be exact by Moessner and Chalker (1996). CDWs proposed by Fukuyama et al. (1979) as the ground state of a partially filled LL, but the Laughlin liquid has a lower energy. What happens in higher LLs (lower magnetic fields)?

4 December 2003NYU Colloquium11 Hartree-Fock treatment of CDW direct or “Hartree” termexchange or “Fock” term Direct vs. exchange balance leads to stripes or bubbles Direct: repulsive long range Coulomb interaction Exchange: attractive short range interaction

4 December 2003NYU Colloquium12 Experimental evidence dc transport: Lilly et al. (1999) Microwave conductivity: R. Lewis & L. Engel (NHMFL)

4 December 2003NYU Colloquium13 Experimental details Anisotropy can be reoriented with an in- plane field (new features at 5/2, 7/2) Transition at 100 mK “Easy” direction [110] “Native” anisotropy energy about 1 mK No QHE: “compressible” state

4 December 2003NYU Colloquium14 A charge density wave? Transport anisotropy consistent with CDW state BUT: Transport in static CDW would be too anisotropic Formation energy of several K, not mK Data also consistent with an anisotropic liquid Fluctuations must be important [Fradkin&Kivelson (1999), MacDonald&Fisher (2000)]!

4 December 2003NYU Colloquium15 The quantum Hall smectic Classical smectic is a “layered liquid” Stripe fluctuations lead to a “quantum Hall smectic” Wexler&ATD (2001): find elastic properties from HFA

4 December 2003NYU Colloquium16 Order in two dimensions Problem: in 2D phonons destroy the positional order but preserve the orientational order. However, this ignores dislocations (=half a layer inserted into crystal). Topological character. Dislocation energy in a smectic is finite, there will be a nonzero density. Dislocations further reduce the orientational order.

4 December 2003NYU Colloquium17 The quantum Hall nematic Dislocations “melt” the smectic [Toner&Nelson (1982)]. Algebraic orientational order:

4 December 2003NYU Colloquium18 Nematic to isotropic transition Low temperature phase is better described as a nematic [Cooper et al (2001)]. Local stripe order persists at high temperatures. Nematic to isotropic transition occurs via a disclination unbinding (Kosterlitz-Thouless) transition. Wexler&ATD: start from HFA and find transition at 200 mK, vs mK in experiments.

4 December 2003NYU Colloquium19 Quantum theory of the QHN Classical theory overestimates anisotropy below 20 mK. Are quantum fluctuations the culprit? Quantum fluctuations can unbind dislocations at T=0. Radzihovsky&ATD (PRL, 2002): use dynamics of local smectic layers as a guide. Make contact with hydrodynamics.

4 December 2003NYU Colloquium20 Theoretical digression… The collective degrees of freedom are the rotations of the dislocation-free domains (nematogens). Their angular momenta and directors are conjugate. Commutation relations are derived in the high field limit, and lead to an unusual quantum rotor model. Broken rotational symmetry leads to a Goldstone mode with anisotropic dispersion: Note that

4 December 2003NYU Colloquium21 Predictions QHN exhibits true long range order at zero temperature; quantum fluctuations important below 20 mK. QHN unstable to weak disorder. Glass phase? Tunneling probes low energy excitations. See a pseudogap at low bias. Damping of Goldstone mode due to coupling to quasiparticles. Resistivity anisotropy proportional to nematic order parameter [conjectured by Fradkin et al. (2000)].

4 December 2003NYU Colloquium22 New directions Start from half-filled fermi liquid state. Can interactions cause the FS to spontaneously deform? Variational wavefunctions? Experimental probes: tunneling, magnetic focusing, surface acoustic waves. Relation to nanoscale phase separation in other systems (e.g., cuprate superconductors)?

4 December 2003NYU Colloquium23 Summary Fascinating problem of orientationally ordered point particles!