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Tensor networks and Symmetries

Sukhbinder Singh, Macquarie University, Sydney

(Session 9b : Friday from 4:00pm - 4:30pm)

Tensors networks methods, which are based on ideas of entanglement and renormalization group, have significantly progressed our understanding of strongly correlated quantum lattice systems in recent years. Examples of popular tensor network states include the matrix product state (MPS)[1], which results naturally from both Wilson’s numerical renormalization group[2] and White’s density matrix renormalization group(DMRG)[3], and the multi-scale entanglement renormalization ansatz (MERA) [4] which is based a specific RG scheme known as entanglement renormalization [5]. These tensor networks have been applied to the exploration of frustrated antiferromagnets, interacting fermions, quantum criticality, topological order and symmetry protected order, and more recently, the MERA has been used to explore[6] the holographic correspondence[7] of string theory. The careful incorporation[8] of lattice symmetries (both spacetime and internal symmetries) in tensor networks is playing an increasingly important role in these applications. In this talk I will outline some aspects of this role in the context of the MPS and the MERA where symmetries have been exploited to (i) target specific quantum number sectors of the Hilbert space, which subsequently also allows for the efficient simulation of lattice systems of anyons[9], (ii) identification of the quantum order of a ground state from its tensor network description[10], and (iii) to realize certain symmetry features of the holographic correspondence in the MERA. References [1] S. Ostlund and S. Rommer, Phys. Rev. Lett. 75, 3537 (1995). [2] K.G. Wilson, Rev. Mod. Phys. 47, 4, 773 (1975). [3] S.R. White, Phys. Rev. Lett. 69, 2863 (1992). [4] G. Vidal, Phys. Rev. Lett. 101, 110501 (2008). [5] G. Vidal, Phys. Rev. Lett. 99, 220405 (2007). [6] B. Swingle, Phys. Rev. D 86, 065007 (2012). [7] J. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998). [8] S. Singh, R. N. C. Pfeifer, and G. Vidal, Phys. Rev. A 82, 050301 (2010). [9] S. Singh, R. N. C. Pfeifer, G. Vidal, and G. K. Brennen. arXiv:1311.0967 (2013). [10] S. Singh and G. Vidal. Phys. Rev. B 88, 121108(R) (2013).