lattice that you cannot do with graphene! or... Antonio H. Castro Neto
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1 Theoretical Aspects What you can do with cold atomsof on agraphene honeycomb lattice that you cannot do with graphene! or... Antonio H. Castro Neto
2 2
3 Outline 1. Graphene for beginners 2. Fermion-Fermion interactions 3. Fermi-Bose mixtures 4. Gauge fields and pseudo-magnetism 5. Bilayers 6. Non-equilibrium many-body physics 7. Conclusions
4 Origin of Dirac fermions in graphene Benzene molecule p z states π state A t ~ 0.1 ev A t ~ 2.7 ev Unit cell B t ~ 2.7 ev Next Nearest Nearest neighbors neighbors
5 In momentum space van Hove singularities Dirac Cone Vanishing density of states
6 Mathematically } 2 X 2 - Sub-latices Px + i Py - Expansion around K point } Chirality (helicity) + = right handed; - = left handed K+ K Γ +
7 Fermion-Fermion Interactions: µ = 0 U( - ) 2 2 In Graphene: U(r) = e /(ε r) K/U = e /(ε v ) RG: marginally irrelevant 2 In AMO lattices: U(r) = g δ(r) K/U = v F /(g E) RG: irrelevant F Phase transitions can occur at strong coupling requiring exact methods 7
8 Fermion-Fermion Interactions: µ = ± t Condition hard to obtain in clean Graphene van Hove singularities = New Many-Body States! 8
9 Fermion-Boson Mixtures Competing Orders in two-dimensional Bose-Fermi mixtures, by L. Mathey, S-W. Tsai, and A.H. Castro Neto, Phys. Rev. Lett. 97, (2006) Bose condensation Retarded Interactions 9
10 Functional Renormalization Group Method: 2D Superfluidity Renormalization-group approach to superconductivity: from weak to strong electron-phonon coupling, by S.-W. Tsai, A. H. Castro Neto, R. Shankar, and D. K. Campbell, Philosophical Magazine 86, 2631 (2006). µ V ~ 2 Unreachable in CM: Lattice instabilities Allen-Dynes _T* is the temperature below which a superconducting gap appears. _ Kosterlitz-Thouless transition: quasi-long range order driving by vortex unbinding. 10 McMillan regime
11 Nature or symmetry of the order parameter _ The vortices of a p + i p order parameter are Majorana fermions _ Have been proposed for quantum computation - D. A. Ivanov, Phys. Rev. Lett. 86, 268 (2001). p+ i p 11
12 Gauge Fields in the honeycomb lattice At long wavelengths the effect of local changes in the hopping parameters is equivalent to the addition to vector and scalar potentials to the Dirac equation
13 Changing the hopping parameters by changing the lattice Strain Tensor u( r ) Compression and Dilation Strain and Shear Uniform changes do not produce gauge fields but... 13
14 V. Pereira, AHCN, N. Peres, Phys. Rev. B 80, (2009) Spatial Anisotropy: 1D system Armchair direction Zigzag direction Opening of gaps (hard in CM)
15 Dirac fermions in a uniform magnetic field 0
16 E ω C 0 - ω C Finite geometry E F L x Odd IQHE
17 Inverse engineering: from the magnetic field obtain the distortion _ Quantum Hall Effect (Integer and Fractional) without a magnetic field _ Mimic the effect of enormous magnetic fields (neutron stars, etc) Uniform Magnetic Field: Guinea, F., et al., Nature Physics (2009) 17 6, 30.
18 Bilayers 18
19 Minimal Model 2 Sets of Massive Dirac Particles 19
20 2 van Hove Enhanced Density of States -> Enhanced Interactions Electron-electron interactions and the phase diagram of a graphene bilayer by Johan Nilsson, A. H. Castro Neto, N. M. R. Peres, and F. Guinea, Phys. Rev. B 73, (2006). 20
21 A bilayer experiment not possible in CM: Many-Body Non-Equilibrium Problem t=0 t>0 ɣ = 0 1 ɣ = 0 1 T*>T > T KT Decoupled incoherent superfluids Vortex-anti-vortex pairs destroy order? T T* Disordered Ordered T KT J A system that is fast driven by a phase transition can generate topological defects at a density with is related to the rate at which the transition is crossed. This Kibble-Zurek (KZ) effect is believed to have occurred just after the Big Bang, when the emerging phase of the vacuum became disordered, giving rise to a large number of topological defects such as cosmic strings, which may be the primal ingredient for the formation of 21 galaxies.
22 Vortex Anti-Vortex Phase locking transition of coupled low dimensional superfluids by Ludwig Mathey, A. Polkovnikov, and A. H. Castro Neto, Europhysics Letters 81, (2007). 22
23 Conclusions: _ In cold atom systems it is possible to reach situations which are hard, or even impossible in real graphene. _ This unconventional situations can lead to new many-body states that are rather unique and unusual in condensed matter. _ The control over the lattice allows for study pseudo-magnetic field phenomena without the presence of a real magnetic field. _ Permits to put the system into extreme situations which can only be obtained in certain astronomical environments. _ Non-equilibrium many-body physics can be studied in a controlled environment. 23
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