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Computational approaches for quantum many-body systems

Computational approaches for quantum many-body systems. HGSFP Graduate Days SS2019 Martin Gärttner. Course overview. Lecture 1: Introduction to many-body spin systems Quantum Ising model, Bloch sphere, tensor structure, exact diagonalization Lecture 2: Collective spin models

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Computational approaches for quantum many-body systems

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  1. Computational approaches for quantum many-body systems HGSFP Graduate Days SS2019 Martin Gärttner

  2. Course overview Lecture 1: Introduction to many-body spin systems Quantum Isingmodel,Bloch sphere, tensor structure, exact diagonalization Lecture 2: Collective spin models LMG model, symmetry, semi-classical methods,Monte Carlo Lecture 3: Entanglement Mixed states, partial trace, Schmidt decomposition Lecture 4: Tensor network states Area laws, matrix product states,tensor contraction, AKLT model Lecture 5: DMRG and other variational approaches Energy minimization, PEPS and MERA, neural quantum states

  3. Learning goals After today you will be able to … • … explain how to find ground states using the MPS ansatz (DMRG). • … dig deeper into tensor network states (PEPS and MERA) • … explain alternative variationalansätze inspired by neural networks.

  4. Tensor network states beyond MPS:Extensions and applications • Projected entangled pair states (PEPS): → extension to 2D • Problem: No efficient contraction:

  5. Tensor network states beyond MPS:Extensions and applications • Multiscale entanglement renormalization ansatz (MERA) • Treat quantum critical ground states ADS (2+1) CFT (1+1)

  6. Libraries for Tensor network states • iTensor: C++ library for tensor network state calculations. http://itensor.org/ • ALPS (Algorithms and Libraries for Physics Simulations). Contains many different numerical methods for quantum many-body systems, not only spins. Especially also quantum Monte Carlo methods. MPS: http://alps.comp-phys.org/static/mps_doc/ • Open Source MPS (OSMPS), Python frontend! https://openmps.sourceforge.io/

  7. Other variational approaches Variational Monte Carlo (VMC): Local energies: Sample states according to

  8. See also: Deng , Li, Das Sarma, PRX 2017, PRB 2017 Gao, Duan, Nat. Commun. 2017 Cirac et al., PRX 2018 Clark, J. Phys. A 2018 Moore, arXiv2017 Carleo, Nomura, Imada, arxiv 2018 Freitas, Morigi, Dunjko, arXiv2018 …… Neural-network quantum states [Carleo and Troyer, Science 2017] • Restricted Boltzmann machine Classical networks: probability

  9. Neural-network quantum states • Efficient evaluation . . . visible hidden

  10. Neural-network quantum states • Finding ground states: Stochastic reconfiguration method Minimize energy functional Learning rate Determine gradients by Monte Carlo sampling • Imaginary time evolution

  11. Neural-network quantum states • Time evolution: Time dependent variational Monte Carlo Minimize deviation from SE solution in each step Time-dependent variational principle time step Determine gradients by Monte Carlo sampling • Real time evolution

  12. References • Ulrich Schollwoeck: The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96 (2011) • Time dependent variational principle: Phys. Rev. Lett. 107, 070601 (2011) • MERA and AdS/CFT: e.g. Phys. Rev. D 86, 065007 (2012) • Neural network quantum state ansatz: Giuseppe Carleo, Matthias Troyer: Solving the Quantum Many-Body Problem with Artificial Neural Networks, Science 355, 602 (2017)

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