Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order Xie Chen, 1 Zheng-Cheng Gu, 2 and Xiao-Gang Wen 1 1 Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2 Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA (Dated: Jul., 2010) Two gapped quantum ground states in the same phase are connected by an adiabatic evolution which gives rise to a local unitary transformation that maps between the states. On the other hand, gapped ground states remain within the same phase under local unitary transformations. Therefore, local unitary transformations define an equivalence relation and the equivalence classes are the universality classes that define the different phases for gapped quantum systems. Since local unitary transformations can remove local entanglement, the above equivalence/universality classes correspond to pattern of long range entanglement, which is the essence of topological order. The local unitary transformation also allows us to define a wave function renormalization scheme, under which a wave function can flow to a simpler one within the same equivalence/universality class. Using such a setup, we find conditions on the possible fixed-point wave functions where the local unitary transformations have finite dimensions. The solutions of the conditions allow us to classify this type of topological orders, which generalize the string-net classification of topological orders. We also describe an algorithm of wave function renormalization induced by local unitary transformations. The algorithm allows us to calculate the flow of tensor-product wave functions which are not at the fixed points. This will allow us to calculate topological orders as well as symmetry breaking orders in a generic tensor-product state. Contents I. Introduction – new states beyond symmetry breaking 1 II. Short-range and long-range quantum entanglement 3 III. Quantum phases and local unitary evolutions 3 IV. Topological order is a pattern of long-range entanglement 5 V. The LU evolutions and quantum circuits 5 VI. Symmetry breaking orders and symmetry protected topological orders 6 VII. Local unitary transformation and wave function renormalization 7 VIII. Wave function renormalization and a classification of topological order 8 A. Quantum states on a graph 8 B. The structure of entanglement in a fixed-point wave function 9 C. The first type of wave function renormalization 10 D. The second type of wave function renormalization 12 E. The fixed-point wave functions from the fixed-point gLU transformations 13 IX. Simple solutions of the fixed-point conditions 15 A. Unimportant phase factors in the solutions 15 B. N = 1 loop state 15 C. N = 1 string-net state 16 D. An N = 2 string-net state – the Z3 state 17 X. A classification of time reversal invariant topological orders 17 XI. Wave function renormalization for tensor product states 18 A. Motivation 18 B. Tensor product states 19 C. Renormalization algorithm 19 1. Step 1: F-move 19 2. Step 2: P-move 20 3. Complications: corner double line 21 XII. Applications of the renormalization for tensor product states 21 A. Renormalization on square lattice 22 B. Ising symmetry breaking phase 23 C. Z2 topological ordered phase 24 XIII. Summary 25 Appendix: Equivalence relation between quantum states in the same phase 26 References 28 I. INTRODUCTION – NEW STATES BEYOND SYMMETRY BREAKING According to the principle of emergence, the rich prop- erties and the many different forms of materials originate from the different ways in which the atoms are ordered in the materials. Landau symmetry-breaking theory pro- vides a general understanding of those different orders and resulting rich states of matter. 1,2 It points out that different orders really correspond to different symmetries arXiv:1004.3835v2 [cond-mat.str-el] 28 Jul 2010