1. Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús 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 1.- Entanglement renormalization of environment degrees of freedom 2.- Contraction of infinite 2-dimensional tensor networks 3.- Outlook Matrix Product States (MPS) Matrix Product Density Operators (MPDO) Projected Entangled Pair States (PEPS) Infinite 1-dimensional thermal states and disentanglers Infinite 1-dimensional master equations Critical correlators of the classical Ising model

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

4. State of a quantum system of n spins 1/2: coefficients (very inneficient to handle classically) Introduction 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. Matrix Product States (MPS) [Afflek et al., 1987] [Fannes et al., 1992] [White, 1992] [Ostlund and Rommer, 1995] [Vidal, 2003] Physical local system of dimension Bonds of dimension For finite systems, the state is represented with parameters, instead of. Any quantum state can be represented as an MPS, with large enough. Physical observables (e.g. correlators) can be computed in time. Great in 1 spatial dimension because of the logarithmic scaling of the entaglement entropy [Vidal et al., 2003] DMRG Dynamics Imaginary-time evolution Thermal states Master equations ……

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

7. Projected Entangled Pair States (PEPS) Physical local system of dimension Bonds of dimension 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]). …… … … [Verstraete and Cirac, 2004]

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

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

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

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

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

13. Simulating master equations with MPDOs Kraus operators W M It is possible to introduce “disentangling isometries” acting in the environment subspace that truncate the proliferation of indices at each step BUT…

14. Quantum Ising spin chain with amplitude damping,

15. with and without partial disentanglement

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

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

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. 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 Action of non-unitary gates on an infinite MPS Can be efficiently computed, taking care of orthonormalization issues

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

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

30. 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 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