Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order

TL;DR

This work formalizes local unitary (LU) transformations as the defining equivalence relation for gapped quantum phases, identifying topological order with patterns of long-range entanglement that survive LU flow. It introduces generalized LU (gLU) moves and a wave-function renormalization scheme, yielding two core moves (F-move and P-move) that drive fixed-point classifications of 2D topological orders via data $(N_{ijk}, F^{ijm,oldsymbol{eta}}_{kln,oldsymbol{ u}}, P_i^{kj,oldsymbol{eta}})$ and amplitudes $A^i$. Concrete fixed-point solutions (e.g., $ ext{Z}_2$, double-semion, $ ext{Z}_3$ string-nets) illustrate how these data encode known topological orders and their TR-invariant variants; the framework extends to tensor-product states (TPS) allowing practical phase identification through a TPS renormalization algorithm. The paper also connects LU-based classifications to quantum circuits, symmetry protections, and tensor-category structures, offering a scalable approach to classify complex 2D quantum phases and analyze stability under local perturbations.

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.

Figures (22)

• Figure 1: (Color online) Energy spectrum of a gapped system as a function of a parameter $s$ in the Hamiltonian. (a,b) For gapped system, a quantum phase transition can happen only when energy gap closes. (a) describes a first order quantum phase transition (caused by level crossing). (b) describes a continuous quantum phase transition which has a continuum of gapless excitations at the transition point. (c) and (d) cannot happen for generic states. A gapped system may have ground state degeneracy, where the energy splitting between the ground states vanishes when system size $L\to \infty$: $\lim_{L\to \infty} \eps=0$. The energy gap $\Del$ between ground and excited states on the other hand remains finite as $L\to \infty$.
• Figure 2: (Color online) (a) A graphic representation of a quantum circuit, which is formed by (b) unitary operations on patches of finite size $l$. The green shading represents a causal structure.
• Figure 3: (Color online) (a) The possible phases for a Hamiltonian $H(g_1,g_2)$ without any symmetry. (b) The possible phases for a Hamiltonian $H_\text{symm}(g_1,g_2)$ with some symmetries. The shaded regions in (a) and (b) represent the phases with short range entanglement ( those ground states can be transformed into a direct product state via a generic LU transformations that do not have any symmetry.)
• Figure 4: (Color online) A finite depth quantum circuit can transform a state $|\Phi\>$ into a direct-product state, if and only if the state $|\Phi\>$ has no long-range quantum entanglement. Here, a dot represents a site with physical degrees of freedom. A vertical line carries an index that label the different physical states on a site. The presence of horizontal lines between dots represents quantum entanglement.
• Figure 5: (Color online) A piece-wise local unitary transformation can transform some degrees of freedom in a state $\Phi\>$ into a direct product. Removing/adding the degrees of freedom in the form of direct product defines an additional equivalence relation that defines the topological order (or classes of long-range entanglement).

Theorems & Definitions (1)

• Theorem 1