Gravity models & condensed matter: improving interconnections

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1 Gravity models & condensed matter: improving interconnections Annalisa Marzuoli Dipartimento di Matematica PAFT 2012 Vietri sul Mare, April 2,

2 Prehistory: Witten s triplet W dimensional gravity as an exactly soluble system Nucl. Phys. B311 ( ) 46 W2. Quantum field theory and the Jones polynomial Commun. Math.Phys. 121 (1989) 351 W3. Gauge theories and integrable lattice models Nucl. Phys. B322 (1989) 629 had a deep influence on theoretical physics and mathematics in the 1990 s, and contributed to breaking the borders between the relativistic-geometric and the statisticalcondensed matter communities. 2

3 Quantized 3D Chern-Simons topological field theory: common background of Witten s triplet and more... Chern-Simons Gauge Theory As A String Theory ( Witten 1995 ) The world-sheet model (time evolution of a string) in this case involves a topological sigma model. Instanton contributions to the sigma model give rise to Wilson line insertions in the space-time Chern-Simons theory. Low-dimensional topology & geometry: topological invariants of knots and 3-manifolds, Vassiliev invariants,, state-sum models (Reshetikhin,Turaev, Viro, 1989 ): applications in (discretized ) quantum gravity models. Chern-Simons effective actions in planar condensed matter phenomena, based e.g. on the deep analogy between a canonically quantized CS theory and quantum mechanical Landau levels of fermionic systems. Topological Quantum Computing (M. Freedman 1998 ) 3

4 Topological phases & (non-abelian) anyonic dynamics are supposed to provide decoherence-free & fault-tolerant quantum computing Theoretical background models 3D quantum Chern-Simons 2D Boundary Conformal Field Theories 2D Lattice gauge theories Experimental settings 2D electron systems in B field: Fractional Q Hall Effect Cold atoms in optical lattices Bose-Einstein condensates Topological insulators???? time > Das Sarma, Freedman, Nayak, Simon, Stern Non-Abelian anyons and topological quantum computation, arxiv: (Rev. Mod. Phys. 2008) 4

5 Basic restriction on the class of models Computational universality can be achieved by resorting to background settings supporting non-chiral quasi-particle excitations [M Freedman at al. A class of PT-invariant topological phases of interacting electrons, ArXiv: cond-mat/ ] 2D Lattice models with PT-invariant topological phases (e.g. Levin & Wen string nets) (a) Doubled Chern-Simons theory for SU(2)k with k 3 (Das Sarma et al.) (b) Turaev-Viro type state sum models (SU(2)k - colored) (Kadar, Marzuoli & Rasetti ) 5

6 Universality for Q-computing motivation (a) Double CS (b) Turaev -Viro Suitabl e reformu 6 3D BFmercoledì 11 aprile 2012

7 BF-theory with (or without) a cosmological constant Topological Quantum Field Theory defined in any (spacetime) dimension Has been used and is used (possibly with constraints) as the reference model in 3 and 4D quantum gravity models * Colored triangulations *Loop QG Group Field Theory Extended to manifolds with boundary (WZW induced on 2D boundary) Quantum observables in 3D BF : v.e.v. of Wilson loop operators topological invariants of closed knotted curves are the same encountered in Chern-Simons (SU(2)k-colored Jones polynomials and HOMFLY polynomials for SU(N)) [A Cattaneo, P Cotta-Ramusino, J Fröhlich, M Martellini Topological BF theories in 3 and 4 dimensions, J. Math. Phys. 36 (1995) 6137] 7

8 Effective actions of the BF-type (& Q-computing) From Witten s triplet to present achievements (theor. & exper.) the story would last too long then focus on the issue of modelling the (monolayer) graphene effective action with a 3D BF-type TQFT supporting degenerate, topologically protected ground states non-abelian anyons A M, G Palumbo, BF-theory in graphene: a route to topological quantum computing? ArXiv:

9 CS-BF: classical actions (compact gauge group) (I) 9

10 BF classical action (II) 10

11 (Graphene in brief) At room temperature, near the Fermi points, electrons exhibit a relativistic behavior (2+1) D massless Dirac equation 11

12 Two triangular sub-lattices A,B 12

13 Low-energy action for fermions in graphene Topological defects in the honeycomb lattice generate vortices For this reason: coupling with U(1) x U(1) gauge fields aν and bν identified as the ordinary em field and the chiral gauge field [ Ryu et al '09] 13

14 (Sketch of further steps) The replica method takes care of disorder-averaged physical quantities, so that the gauge fields are 1- forms in (the Lie algebra of) U(N) x U(N) Wick rotation on the action S Fermionic degrees of freedom integrated out Regularization (Pauli-Villars) Level-rank duality in gauge th.: N k (CS-coupling constant) Effective action 14

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16 IDENTIFICATION: BF with cosmological constant 16

17 Quantum observables associated with (perturbative expansions of) holonomies along closed knotted curves embedded in an ambient 3-manifold [Cattaneo et al., 1995] 17

18 Summarizing The BF effective action for monolayer graphene provides a natural identification of the relevant gauge fields explicit expressions for dealing with quasi-particles excitations associated with closed paths surrounding vortices a natural playground for topologically- protected, anyonic type computing (at room temperature) 18

19 Editorial of Physical Review X 2, (2012): Current Graphene Research: Going beyond the Pure Monolayer the perfect atomic lattice ( ) both for new fundamental physics and for innovative technological applications, in particular, electronics. Interface between a monolayer graphene and a semiconductor New electron properties of trilayer graphene Doping of pure monolayer graphene 19

20 Graphene allotropes (curved graphene) Tools: QFT in curved spacetime [A Iorio, G Lambiase The Hawking-Unruh phenomenon on graphene, arxiv: ] Improved BF approach (curved background manifolds, boundaries, edge effects) Phenomenological modelling Helfrich s form for the surface energy of membranes and foams 20

21 Ideal graphene monolayer (hexagonal tiling of the plane) Deformations: curvature effects? (realistic deposition processes) 21

22 GRAPHENE TOPOLOGY (from G. Benedek, Pavia 2009) p 5 pentagons, p 6 ( 1) hexagons, p 7 heptagons GRAPHENE [Novoselov, Geim et al. (2004)] FULLERENE [Curl, Kroto, Smalley (1985)] NANOTUBE [S. Iijima (1991)] SCHWARZITE [P. Milani, G.Benedek et al.(2002)]

23 GRAPHENE TOPOLOGY (from G. Benedek, Pavia 2009) p 5 pentagons, p 6 ( 1) hexagons, p 7 heptagons GRAPHENE [Novoselov, Geim et al. (2004)] FULLERENE [Curl, Kroto, Smalley (1985)] NANOTUBE [S. Iijima (1991)] SCHWARZITE [P. Milani, G.Benedek et al.(2002)]

24 GRAPHENE TOPOLOGY (from G. Benedek, Pavia 2009) p 5 pentagons, p 6 ( 1) hexagons, p 7 heptagons GRAPHENE [Novoselov, Geim et al. (2004)] FULLERENE [Curl, Kroto, Smalley (1985)] NANOTUBE [S. Iijima (1991)] SCHWARZITE [P. Milani, G.Benedek et al.(2002)]

25 GRAPHENE TOPOLOGY (from G. Benedek, Pavia 2009) p 5 pentagons, p 6 ( 1) hexagons, p 7 heptagons GRAPHENE [Novoselov, Geim et al. (2004)] FULLERENE [Curl, Kroto, Smalley (1985)] NANOTUBE [S. Iijima (1991)] SCHWARZITE [P. Milani, G.Benedek et al.(2002)]

26 Sample of schwarzite (from G. Benedek, Pavia 2009) (Minimal 3-periodic hyperbolic surfaces) 1000 nm 23

27 Simulations of random carbon deposition * Flat regions (open, tubular) * Fullerens C60 * Schwarzites provide intriguing 3D visualization of foam-like structures of increasing topological complexity D. Donadio, L. Colombo and G. Benedek, (2004) Phys. Rev. B 70,

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29 Helfrich s form for the surface energy of membranes and foams POLIMI - 6 where Gauss curvature mean curvature elastic constant for cylindrical deformations elastic constant for elliptic/hyperbolic deformations Valid for all graphenes! from DFT calculations: ev/å 2 ev ev

31 NA 8

32 (Topology is quite fashionable) [Angular dependence of the growth velocity which changes from multi-lobes to a circle (isotropic growth) as the shape parameter r approaches unity. (a) Graphene dominated six-fold symmetry growth. (b) A four-fold symmetry growth front when the diffusion on Cu limits graphene growth. (c) A two-fold symmetry coupled with the four-fold symmetry of Cu lattices and hexagonal symmetry of graphene.] Lili Fan, Jie Zou, Zhen Li, Xiao Li, Kunlin Wang, Jinquan Wei, Minlin Zhong, Dehai Wu, Zhiping Xu and Hongwei Zhu Topology evolution of graphene in chemical vapor deposition, a combined theoretical/experimental approach toward shape control of graphene domains Nanotechnology 23 (2012) (8pp) 29

GRAPHENE: A NEW STAR IN MATERIAL SCIENCE

GRAPHENE: A NEW STAR IN MATERIAL SCIENCE S. Sahoo 1 & A. K. Dutta 2 Department of Physics, National Institute of Technology Durgapur-713209, West Bengal, India. 1 E-mail: sukadevsahoo@yahoo.com 2 E-mail: