Simulation of Quantum Many-Body Systems
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1 Numerical Quantum Simulation of Matteo Rizzi - KOMET 7 - JGU Mainz Vorstellung der Arbeitsgruppen WS 15-16
2 recent developments in control of quantum objects (e.g., cold atoms, trapped ions) General Framework quantum simulation of difficult problems for classical computers quantum engineering of synthetic states of matter
3 General Framework recent developments in control of quantum objects (e.g., cold atoms, trapped ions) quantum information look on many-body systems quantum simulation of difficult problems for classical computers relevant Hilbert corner & efficient numerics (e.g., tensor networks) quantum memories & processors quantum engineering of synthetic states of matter
4 General Framework recent developments in control of quantum objects (e.g., cold atoms, trapped ions) quantum information look on many-body systems quantum simulation of difficult problems for classical computers quantum engineering of synthetic states of matter relevant Hilbert corner & efficient numerics (e.g., tensor networks) quantum memories & processors MY FOCUS geometry + gauge fields + interactions topological states spin & orbital persistent currents anyons frustrated systems entanglement spectrum new, complementary approaches to cond-mat problems
5 Quantum Simulations & Engineering Computation on classical platforms Physicists Toy Models Quantum Many-Body Systems see also the AG Windpassinger / Schmidt-Kaler / Gerritsma
6 Quantum Simulations & Engineering Computation on classical platforms Quantum Many-Body Systems Physicists Toy Models Quantum Simulator & Engineering see also the AG Windpassinger / Schmidt-Kaler / Gerritsma
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 Quantum Simulations & Engineering Computation on classical platforms Physicists Toy Models pose new questions? Quantum Many-Body Systems Quantum Simulator & Engineering imagination into real-world k d = k 1 k 2 = k d e x see also the AG Windpassinger / Schmidt-Kaler / Gerritsma
9 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)
10 Q-Info driven numerics: DMRG & Tensor Networks Generic description of a many-body Hilbert space is exponentially expensive numbers
11 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
12 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
13 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
14 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
15 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)
16 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) Trapped ultracold fermions in non-abelian gauge potentials Sci. Rep. 1, 43 (2011) + PRB 91, (2015)
17 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: PRA 91, (2015) Trapped ultracold fermions in non-abelian gauge potentials Sci. Rep. 1, 43 (2011) + PRB 91, (2015) + (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
18 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: PRA 91, (2015) Trapped ultracold fermions in non-abelian gauge potentials Sci. Rep. 1, 43 (2011) + PRB 91, (2015) Adaptive gauge approach to Tree Tensor Networks PRB, 90, (2014) arxiv: (NJP) + (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
19 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 of non-square lattices & use 2nd quantization to derive Hubbard model * help to decide the proper approximations
20 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 of non-square lattices & use 2nd quantization to derive 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
21 Possible B.Sc. / M.Sc. Projects 3. get acquainted with anyons * learn about anyons and their funny exchange rules * solve exactly (via Gaussians) some fermionic problems (related to superconductors, spin-chains, topo. systems) * perform some own calculation on prototypical models (Fortran/Matlab/Mathematica) + learn about proposed physical implementations & speculate on new ones (via atoms, ions, etc.)
22 Possible B.Sc. / M.Sc. Projects 4. implement time-evol. in Tree-Tensor Networks: * learn about different solutions of the time-dep. Schrödinger Equation via Tensor Network Ansatz * apply them for periodic boundary conditions (new!) building on flexible existing libraries & codes * (master) exploit these tools to tackle non-equilibrium! [e.g., quenches in disordered systems]
23 Our group YOU! J. Jünemann M. Bischoff A. Haller Thanks for your attention!
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