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Tensor networks and the numerical study of quantum and classical systems on infinite lattices PowerPoint Presentation
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Tensor networks and the numerical study of quantum and classical systems on infinite lattices

Tensor networks and the numerical study of quantum and classical systems on infinite lattices

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Tensor networks and the numerical study of quantum and classical systems on infinite lattices

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  1. Tensor networks and the numerical study of quantum and classical systems on infinite lattices RománOrús School of Physical Sciences, The University of Queensland, Brisbane (Australia) in collaboration with Guifré Vidal and Jacob Jordan Trobada de Nadal 2006 ECM, December 21st 2006

  2. Outline 0.- Introduction Matrix Product States (MPS) Matrix Product Density Operators (MPDO) Projected Entangled Pair States (PEPS) 1.- Entanglement renormalization of environment degrees of freedom Infinite 1-dimensional thermal states and disentanglers Infinite 1-dimensional master equations 2.- Contraction of infinite 2-dimensional tensor networks Critical correlators of the classical Ising model 3.- Outlook

  3. Outline 0.-Introduction Matrix Product States (MPS) Matrix Product Density Operators (MPDO) Projected Entangled Pair States (PEPS) 1.- Entanglement renormalization of environment degrees of freedom Infinite 1-dimensional thermal states and disentanglers Infinite 1-dimensional master equations 2.- Contraction of infinite 2-dimensional tensor networks Critical correlators of the classical Ising model 3.- Outlook

  4. Introduction State of a quantum system of n spins 1/2: coefficients (very inneficient to handle classically) A natural ansatz for relevant states of quantum mechanical systems is given in terms of the contraction of an appropriate tensor network: Inspires classical techniques to compute properties of quantum systems which are free from the sign problem, and which can be implemented in the thermodynamic limit

  5. Any quantum state can be represented as an MPS, with large enough . For finite systems, the state is represented with parameters, instead of . Physical observables (e.g. correlators) can be computed in time. DMRG Dynamics Imaginary-time evolution Thermal states Master equations Great in 1 spatial dimension because of the logarithmic scaling of the entaglement entropy[Vidal et al., 2003] Matrix Product States (MPS) … … Bonds of dimension Physical local system of dimension [Afflek et al., 1987] [Fannes et al., 1992] [White, 1992][Ostlund and Rommer, 1995] [Vidal, 2003]

  6. Purification of local dimension … … Bonds of dimension Physical local system of dimension Any density operator can be represented as an MPDO, with large enough and For finite systems, the state is represented with parameters, instead of . Physical observables (e.g. correlators) can be computed in time. Useful in the computation of 1-dimensional thermal states. Matrix Product Density Operators (MPDO) [Verstraete, García-Ripoll, Cirac, 2004]

  7. For finite systems, the state is represented with parameters, instead of . Physical observables (e.g. correlators) can be computed in time. Exact contraction of an arbitrary PEPS for a finite system is an #P-Complete problem [N. Schuch et al., 2006]. Successfully applied to variationally compute the ground state of finite quantum systems in 2 spatial dimensions (up to 11 x 11 sites, [Murg, Verstraete and Cirac, 2006]). Projected Entangled Pair States (PEPS) [Verstraete and Cirac, 2004] … Physical local system of dimension … … Bonds of dimension …

  8. Outline 0.-Introduction Matrix Product States (MPS) Matrix Product Density Operators (MPDO) Projected Entangled Pair States (PEPS) 1.- Entanglement renormalization of environment degrees of freedom Infinite 1-dimensional thermal states and disentanglers Infinite 1-dimensional master equations 2.- Contraction of infinite 2-dimensional tensor networks Critical correlators of the classical Ising model 3.- Outlook

  9. Outline 0.- Introduction Matrix Product States (MPS) Matrix Product Density Operators (MPDO) Projected Entangled Pair States (PEPS) 1.-Entanglement renormalization of environment degrees of freedom Infinite 1-dimensional thermal states and disentanglers Infinite 1-dimensional master equations 2.- Contraction of infinite 2-dimensional tensor networks Critical correlators of the classical Ising model 3.- Outlook

  10. MPDO MPS-like … … … … [Zwolak and Vidal, 2004] Both ansatzs can be applied to compute thermal states. However, MPDOs can introduce unphysical correlations between the environment degrees of freedom environment “Unnecessary” entanglement! swap On thermal states in 1 spatial dimension… OR

  11. Less expensive representation U Disentangler (renormalization of correlations flowing across the environment) Disentanglers on the environment of MPDOs swap This effect is not negligible in the computation of thermal states with MPDOs

  12. Quantum Ising spin chain, Schmidt coefficients of the MPS-like representation BIG!!!

  13. … It is possible to introduce “disentangling isometries” acting in the environment subspace that truncate the proliferation of indices at each step M M M M W M M M BUT… M M M M M M M M Proliferation of indices makes “naive” simulation not feasible Simulating master equations with MPDOs Kraus operators

  14. Quantum Ising spin chain with amplitude damping,

  15. Quantum Ising spin chain with amplitude damping, with and without partial disentanglement

  16. Outline 0.- Introduction Matrix Product States (MPS) Matrix Product Density Operators (MPDO) Projected Entangled Pair States (PEPS) 1.-Entanglement renormalization of environment degrees of freedom Infinite 1-dimensional thermal states and disentanglers Infinite 1-dimensional master equations 2.- Contraction of infinite 2-dimensional tensor networks Critical correlators of the classical Ising model 3.- Outlook

  17. Outline 0.- Introduction Matrix Product States (MPS) Matrix Product Density Operators (MPDO) Projected Entangled Pair States (PEPS) 1.- Entanglement renormalization of environment degrees of freedom Infinite 1-dimensional thermal states and disentanglers Infinite 1-dimensional master equations 2.-Contraction of infinite 2-dimensional tensor networks Critical correlators of the classical Ising model 3.- Outlook

  18. … … … The difficult problem of a PEPS… In order to compute expected values of observables, one must necessarily contract the PEPS tensor network, and this is an #P-Complete problem in general. For finite systems, there is a variational technique to efficiently approximate such a contraction (up to lattices of 11 x 11 spins) due to Verstraete and Cirac (2004). We have developed a technique to contract the whole PEPS tensor network in the thermodynamic limit for translationally-invariant systems.

  19. The difficult problem of a PEPS… In order to compute expected values of observables, one must necessarily contract the PEPS tensor network, and this is an NP-hard problem in general. For finite systems, there is a variational technique to efficiently approximate such a contraction (up to lattices of 11 x 11 spins) due to Verstraete and Cirac. We have developed a technique to contract the whole PEPS tensor network in the thermodynamic limit for translationally-invariant systems. … … … …

  20. … … … The difficult problem of a PEPS… In order to compute expected values of observables, one must necessarily contract the PEPS tensor network, and this is an NP-hard problem in general. For finite systems, there is a variational technique to efficiently approximate such a contraction (up to lattices of 11 x 11 spins) due to Verstraete and Cirac. We have developed a technique to contract the whole PEPS tensor network in the thermodynamic limit for translationally-invariant systems.

  21. Action of non-unitary gates on an infinite MPS … … Can be efficiently computed, taking care of orthonormalization issues … The difficult problem of a PEPS… In order to compute expected values of observables, one must necessarily contract the PEPS tensor network, and this is an NP-hard problem in general. For finite systems, there is a variational technique to efficiently approximate such a contraction (up to lattices of 11 x 11 spins) due to Verstraete and Cirac. We have developed a technique to contract the whole PEPS tensor network in the thermodynamic limit for translationally-invariant systems. Boundary MPS with bond dimension

  22. … … … The difficult problem of a PEPS… In order to compute expected values of observables, one must necessarily contract the PEPS tensor network, and this is an NP-hard problem in general. For finite systems, there is a variational technique to efficiently approximate such a contraction (up to lattices of 11 x 11 spins) due to Verstraete and Cirac. We have developed a technique to contract the whole PEPS tensor network in the thermodynamic limit for translationally-invariant systems. Iterate until a fixed point is found

  23. … … … The difficult problem of a PEPS… In order to compute expected values of observables, one must necessarily contract the PEPS tensor network, and this is an NP-hard problem in general. For finite systems, there is a variational technique to efficiently approximate such a contraction (up to lattices of 11 x 11 spins) due to Verstraete and Cirac. We have developed a technique to contract the whole PEPS tensor network in the thermodynamic limit for translationally-invariant systems. Iterate until a fixed point is found

  24. … … … The difficult problem of a PEPS… In order to compute expected values of observables, one must necessarily contract the PEPS tensor network, and this is an NP-hard problem in general. For finite systems, there is a variational technique to efficiently approximate such a contraction (up to lattices of 11 x 11 spins) due to Verstraete and Cirac. We have developed a technique to contract the whole PEPS tensor network in the thermodynamic limit for translationally-invariant systems. Iterate until a fixed point is found

  25. The difficult problem of a PEPS… In order to compute expected values of observables, one must necessarily contract the PEPS tensor network, and this is an NP-hard problem in general. For finite systems, there is a variational technique to efficiently approximate such a contraction (up to lattices of 11 x 11 spins) due to Verstraete and Cirac. We have developed a technique to contract the whole PEPS tensor network in the thermodynamic limit for translationally-invariant systems. Once there is convergence, contract it from the other side and compute e.g. correlators on the diagonal with the obtained MPS … … … … … …

  26. The difficult problem of a PEPS… In order to compute expected values of observables, one must necessarily contract the PEPS tensor network, and this is an NP-hard problem in general. For finite systems, there is a variational technique to efficiently approximate such a contraction (up to lattices of 11 x 11 spins) due to Verstraete and Cirac. We have developed a technique to contract the whole PEPS tensor network in the thermodynamic limit for translationally-invariant systems. Once there is convergence, contract it from the other side and compute e.g. correlators on the diagonal with the obtained MPS … … r … … r

  27. An example: classical Ising model at criticality It is possible to build a quantum PEPS such that the expected values correspond to those of the classical ensemble exact Very good agreement up to ~100 sites with modest computational effort!

  28. Outline 0.- Introduction Matrix Product States (MPS) Matrix Product Density Operators (MPDO) Projected Entangled Pair States (PEPS) 1.- Entanglement renormalization of environment degrees of freedom Infinite 1-dimensional thermal states and disentanglers Infinite 1-dimensional master equations 2.-Contraction of infinite 2-dimensional tensor networks Critical correlators of the classical Ising model 3.- Outlook

  29. Outline 0.- Introduction Matrix Product States (MPS) Matrix Product Density Operators (MPDO) Projected Entangled Pair States (PEPS) 1.- Entanglement renormalization of environment degrees of freedom Infinite 1-dimensional thermal states and disentanglers Infinite 1-dimensional master equations 2.- Contraction of infinite 2-dimensional tensor networks Critical correlators of the classical Ising model 3.- Outlook

  30. Soon application to compute the ground state properties and dynamics of infinite quantum many-body systems in 2 spatial dimensions in collaboration with G. Vidal, J. Jordan, F. Verstraete and I. Cirac Outlook Question: why tensor networks are good for you? Answer: because, potentially, you can apply them to study… strongly-correlated quantum many-body systems in 1, 2, and more spatial dimensions, in the finite case and in the thermodynamic limit, Hubbard models, high-Tc superconductivity, frustrated lattices, topological effects, finite-temperature systems, systems away from equilibrium, master equations and dissipative systems, classical statistical models, quantum field theories on infinite lattices, at finite temperature and away from equilibrium, effects of boundary conditions, RG transformations, computational complexity of physical systems, etc