Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order (1004.3835v2)

Abstract: 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 \emph{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.

• The paper establishes that local unitary transformations remove short-range entanglement to distinguish topologically ordered and gapped quantum states.
• It develops a wave function renormalization framework using tensor networks, verified on models like the Ising model and Z₂ gauge theory.
• The findings deepen our understanding of long-range entanglement in quantum phase transitions and pave the way for advances in quantum materials and computation.

Local Unitary Transformation, Long-Range Quantum Entanglement, Wave Function Renormalization, and Topological Order

The paper under discussion explores several interconnected subjects central to the understanding of quantum phases of matter. It provides a framework for understanding quantum states through the lens of local unitary (LU) transformations, long-range entanglement, wave function renormalization, and topological order. At its core, the paper establishes an equivalence relation and explores the universality classes of gapped quantum systems via these transformative concepts.

The authors begin by discussing how two gapped quantum ground states can be adiabatically connected through LU transformations, thereby defining a phase equivalence where these transformations become indicators of phase unity. Such phases, characterized by topological order, are determined by patterns of long-range quantum entanglement—a key deviation from the conventional, symmetry-breaking paradigm defined by Landau's theory.

The paper posits that local unitary transformations can eliminate short-range entanglement, hence differentiating short-range entangled (SRE) states that all belong to a single equivalence class from states exhibiting topological order, which are inherently long-range entangled (LRE). The implication here is profound: topological order and quantum phases are elegantly described by classes of long-range entangled states, which cannot be reduced to product states even by LU transformations.

Moving forward, the authors construct a framework for wave function renormalization leveraging these LU transformations, which simplify wave functions to their essential, entanglement-free forms, identified as fixed points. The fixed-point wave function renormalization uncovers conditions and patterns of entanglement necessary for characterizing different topological orders. By introducing a series of conditions involving the permutations of local indices—described intricately through tensors—the authors formalize an approach to classify topological orders without symmetry constraints.

The practical mechanism by which this renormalization is applied is through tensor networks and simulations on tensor product states (TPS), providing a robust tool in identifying phases of matter computationally. The renormalization algorithm developed in the paper is applied in models like the Ising model and the $\mathbb{Z}_2$ gauge theory to verify its capability in distinguishing between trivial and topologically ordered phases, demonstrating the stability of topological phases against local perturbations.

The implications of this research lie in its potential to rationalize observed and theoretical quantum phenomena that evade simple symmetry-breaking explanations, such as those found in high-$T_c$ superconductors and quantum Hall systems. Furthermore, the approach suggests pathways to examining stability properties and identifying new order types beyond characteristic boundaries dictated by symmetries. As insights into wave function renormalization deepen, this framework opens the door to advancing our understanding of quantum phase transitions, symmetry-protected topological phases, and potentially, the nature of exotic states of matter that include fractons and non-Abelian anyons.

Looking ahead, the established conditions on topological classification could influence computational methodologies in simulating quantum materials, phases, or even informing quantum error correction in topological quantum computing schemes. Additionally, this paper sets the stage for exploring how these theoretical constructs and numerical strategies could be extended or modified to incorporate time-reversal symmetry and non-trivial Chern-Simons theories, adapting our current understanding to more complex systems and potentially undiscovered materials or models.

In summary, the discussed paper provides a comprehensive foundation for utilizing LU transformations and wave function renormalization in classifying quantum states by topological order, presenting significant advancements both in theory and computational simulation. This work holds the promise of broadening our structural understanding of quantum matter—moving beyond traditional paradigms and accommodation for richer phenomenologies observed in condensed matter physics.