Simulation of Quantum Many-Body Systems
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1 Numerical Quantum Simulation of Matteo Rizzi - KOMET JGU Mainz Vorstellung der Arbeitsgruppen WS 14-15
2 QMBS: An interdisciplinary topic entanglement structure of relevant states anyons for q-memory & q-computation (Theoretical) Condensed Matter & Material Science emergent gauge theories, topological phenomena, exotic (quasi-)particles Quantum Information High-Energy Theory quantum gates & memories based on collective effects Quantum Optics (Cold Atoms, Trapped Ions, NV-Centers,...) artificial lattices & gauges (e.g. finite fermion density & real-time dynamics)
3 Q-Info driven numerics: DMRG & Tensor Networks Generic description of a many-body Hilbert space is exponentially expensive numbers Area-law for entanglement entropy generic state Physically accessible states Eisert, Cramer, Plenio RMP 82, 277 ( 10) Product states
4 generic state Q-Info driven numerics: DMRG & Tensor Networks Generic description of a many-body Hilbert space is exponentially expensive numbers Area-law for entanglement entropy Physically accessible states Eisert, Cramer, Plenio RMP 82, 277 ( 10) Product states
5 generic state Q-Info driven numerics: DMRG & Tensor Networks Generic description of a many-body Hilbert space is exponentially expensive Economic description by Tensor Networks : (variational RG schemes, DMRG) Schollwock, Ann. Phys. 326, 96 (2011) Area-law for entanglement entropy numbers numbers Physically accessible states Eisert, Cramer, Plenio RMP 82, 277 ( 10) Product states
6 Q-Info driven numerics: DMRG & Tensor Networks Generic description of a many-body Hilbert space is exponentially expensive Economic description by Tensor Networks : (variational RG schemes, DMRG) Schollwock, Ann. Phys. 326, 96 (2011) numbers numbers plenty of different decompositions in tensor products: MPS PEPS TTN see also the AG Orús MERA
7 Quantum Simulations & Engineering Computation on classical platforms Quantum Many-Body Systems Physicists Toy Models Quantum Simulator & Engineering pose new questions? imagination into real-world see also the AG Windpassinger / Schmidt-Kaler / Gerritsma
8 Interplay of geometry, gauges and interactions (1D) Wright et al., PRL 110, (2013) Iê - - Optimal regime for persistent current L a a W a 1.0 l= l= l= l= l=38.2 l= g M.Cominotti, D. Rossini, M. Rizzi, F. Hekking, A. Minguzzi, PRL 113, (2014)
9 Interplay of geometry, gauges and interactions (1D) strong transverse confinement 2D 1D 60 nm x 15 µm reduces dimensionality Greiner et al., PRL 87, (2001) Moritz et al., PRL 91, (2003) 1D peculiarity: interactions more relevant at lower density!! in 1D quantum fluctuations are crucially important: no true long range order! mean-field Luttinger Liquid Ideal Bosons need for simulations :) Ideal Fermions Gross-Pitaevski description M.Cominotti, D. Rossini, M. Rizzi, F. Hekking, A. Minguzzi, PRL 113, (2014)
10 Other recent works a. b. c. d. e. f. Tunable cold-atom platform for relativistic fermions & topological insulators PRA (2010) / PRL (2010) NJP (2012) / PoS 193, 036 (2014) Stability of quantum memories based on Kitaev-Majorana anyons PRB 88, (2013) + arxiv: (t) (t) tr / 2 t 0 (J 1 ) (a) time (J 1 ) (c) (b) time (J 1 ) N N = 8 N = 12 N = 16 N = 20 N = 24 Trapped ultracold fermions in non-abelian gauge potentials Sci. Rep. 1, 43 (2011) + arxiv:1411.xxxx Adaptive gauge approach to Tree Tensor Networks Phys. Rev. B, 90, (2014)
11 Possible B.Sc. / M.Sc. Projects 1. contribute to the design of flat bands: * learn the basics of optical trapping of atoms & artificial creation of magnetic fields (gauge) * compute Bloch & Wannier functions of a non-square lattice & use 2nd quantization to derive the Hubbard model * help to decide the proper approximations 2. investigate particles in a magnetic field: 2a) on a lattice: * learn about Peierls phase & Harper Hamiltonian * compute your own fractal Hofstadter butterfly 2a) in the continuum: * learn about Landau Levels and dimensional reduction * play with polynomials & co. OR with (existent) numerics * determine pseudopotentials for long-range interactions
12 Possible B.Sc. / M.Sc. Projects 3. get acquainted with anyons * solve exactly (via Gaussians) some fermionic problems (related to superconductors, spin-chains, topo. systems) * learn about anyons and their funny exchange rules * perform some own calculation on prototypical models (Fortran/Matlab/Mathematica) + learn about proposed physical implementations & speculate on new ones (via atoms, ions, etc.)
13 Possible B.Sc. / M.Sc. Projects 4. explore ground state manifolds: * learn about Berry curvature * learn about quantum phase transitions * play with exactly solvable models * reproduce recent results with s=1/2 + tackle new s=1 5. compute entanglement spectrum * learn about basics of DMRG & many-body entanglement * study & reproduce recent results on phase transitions + get in contact with Luttinger liquids + tackle some unexplored example, e.g. ES of 2LL-->1LL
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