Tensor Network Renormalization.

We introduce a coarse-graining transformation for tensor networks that can be applied to study both the partition function of a classical statistical system and the Euclidean path integral of a quantum many-body system. The scheme is based upon the insertion of optimized unitary and isometric tensors (disentanglers and isometries) into the tensor network and has, as its key feature, the ability to remove short-range entanglement or correlations at each coarse-graining step. Removal of short-range entanglement results in scale invariance being explicitly recovered at criticality. In this way we obtain a proper renormalization group flow (in the space of tensors), one that in particular (i) is computationally sustainable, even for critical systems, and (ii) has the correct structure of fixed points, both at criticality and away from it. We demonstrate the proposed approach in the context of the 2D classical Ising model.

[1]  J. Preskill,et al.  Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence , 2015, 1503.06237.

[2]  R. Pfeifer,et al.  Boundary quantum critical phenomena with entanglement renormalization , 2009, 0912.1642.

[3]  L. Kadanoff.  Scaling laws for Ising models near T(c) , 1966 .

[4]  T. Takayanagi,et al.  Surface/State Correspondence as a Generalized Holography , 2015, 1503.03542.

[5]  Xiao-Gang Wen,et al.  Tensor-entanglement renormalization group approach as a unified method for symmetry breaking and topological phase transitions , 2008 .

[6]  Tadashi Takayanagi,et al.  Holographic geometry of cMERA for quantum quenches and finite temperature , 2013, 1311.6095.

[7]  G. Vidal.  Entanglement renormalization. , 2005, Physical review letters.

[8]  Z. Y. Xie,et al.  Second renormalization of tensor-network states. , 2008, Physical review letters.

[9]  Michael Levin,et al.  Tensor renormalization group approach to two-dimensional classical lattice models. , 2006, Physical review letters.

[10]  Ning Bao,et al.  Consistency conditions for an AdS multiscale entanglement renormalization ansatz correspondence , 2015, 1504.06632.

[11]  B. Normand,et al.  Phase Transitions of Ferromagnetic Potts Models on the Simple Cubic Lattice , 2014, 1405.1179.

[12]  G. Evenbly,et al.  Tensor Network States and Geometry , 2011, 1106.1082.

[13]  G. Evenbly,et al.  Tensor Network Renormalization Yields the Multiscale Entanglement Renormalization Ansatz. , 2015, Physical review letters.

[14]  B. Swingle,et al.  Entanglement Renormalization and Holography , 2009, 0905.1317.

[15]  Matthew J. Rosseinsky,et al.  Physical Review B , 2011 .

[16]  White,et al.  Density-matrix algorithms for quantum renormalization groups. , 1993, Physical review. B, Condensed matter.

[17]  J. Cardy,et al.  Conformal invariance, the central charge, and universal finite-size amplitudes at criticality. , 1986, Physical review letters.

[18]  J. Maldacena,et al.  Time evolution of entanglement entropy from black hole interiors , 2013, 1303.1080.

[19]  Glen Evenbly,et al.  Algorithms for tensor network renormalization , 2015, 1509.07484.

[20]  J. Molina-Vilaplana,et al.  Entanglement, tensor networks and black hole horizons , 2014, 1403.5395.

[21]  Z. Y. Xie,et al.  Renormalization of tensor-network states , 2010, 1002.1405.

[22]  G. Vidal.  Class of quantum many-body states that can be efficiently simulated. , 2006, Physical review letters.

[23]  Xiao-Gang Wen,et al.  Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order , 2009, 0903.1069.

[24]  K. Wilson.  The renormalization group: Critical phenomena and the Kondo problem , 1975 .

[25]  G. Evenbly,et al.  Algorithms for Entanglement Renormalization: Boundaries, Impurities and Interfaces , 2013, 1312.0303.

[26]  Leo P. Kadanoff,et al.  Real Space Renormalization in Statistical Mechanics , 2013, 1301.6323.

[27]  H. Matsueda,et al.  Tensor network and a black hole , 2012, 1208.0206.

[28]  F. Verstraete,et al.  Entanglement renormalization for quantum fields in real space. , 2011, Physical review letters.

[29]  T. Xiang,et al.  Accurate determination of tensor network state of quantum lattice models in two dimensions. , 2008, Physical review letters.

[30]  K. Wilson.  Renormalization Group Methods , 1975 .

[31]  F. Verstraete,et al.  Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions , 2004, cond-mat/0407066.

[32]  C. Itzykson,et al.  Conformal Invariance , 1987 .

[33]  Z. Y. Xie,et al.  Coarse-graining renormalization by higher-order singular value decomposition , 2012, 1201.1144.

[34]  F. Verstraete,et al.  Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems , 2008, 0907.2796.

[35]  F. Eckert,et al.  Coarse graining methods for spin net and spin foam models , 2011, 1109.4927.

[36]  G. Evenbly,et al.  Algorithms for entanglement renormalization , 2007, 0707.1454.

[37]  Ning Bao,et al.  Consistency Conditions for an AdS/MERA Correspondence , 2015, 1504.06632.

[38]  White,et al.  Density matrix formulation for quantum renormalization groups. , 1992, Physical review letters.

[39]  R. Pfeifer,et al.  Entanglement renormalization, scale invariance, and quantum criticality , 2008, 0810.0580.