Gapped quantum liquids and topological order, stochastic local transformations and emergence of unitarity

TL;DR

The paper reframes topological order as a property of gapped quantum liquids (gapped states that remain well-behaved under system-size growth) and develops a hierarchy of transformations to classify phases. It introduces generalized local unitaries (gLU) to connect ground-state subspaces across sizes and generalized stochastic local (gSL) transformations to probe long-range entanglement, revealing an emergent unitarity in topological phases. The work distinguishes stable topological orders from unstable symmetry-breaking ones using new invariants, notably the tri-topological entropy $S^ ext{tri}_ ext{topo}$ alongside $S^ ext{qua}_ ext{topo}$, and shows that symmetry-breaking states can be probabilistically driven to product states while topological orders resist such conversion. Together, these insights redefine long-range entanglement, provide practical diagnostics for finite systems, and illuminate foundational aspects of unitarity and quantum phases in many-body systems.

Abstract

In this work we present some new understanding of topological order, including three main aspects: (1) It was believed that classifying topological orders corresponds to classifying gapped quantum states. We show that such a statement is not precise. We introduce the concept of \emph{gapped quantum liquid} as a special kind of gapped quantum states that can "dissolve" any product states on additional sites. Topologically ordered states actually correspond to gapped quantum liquids with stable ground-state degeneracy. Symmetry-breaking states for on-site symmetry are also gapped quantum liquids, but with unstable ground-state degeneracy. (2) We point out that the universality classes of generalized local unitary (gLU) transformations (without any symmetry) contain both topologically ordered states and symmetry-breaking states. This allows us to use a gLU invariant -- topological entanglement entropy -- to probe the symmetry-breaking properties hidden in the exact ground state of a finite system, which does not break any symmetry. This method can probe symmetry- breaking orders even without knowing the symmetry and the associated order parameters. (3) The universality classes of topological orders and symmetry-breaking orders can be distinguished by \emph{stochastic local (SL) transformations} (i.e.\ \emph{local invertible transformations}): small SL transformations can convert the symmetry-breaking classes to the trivial class of product states with finite probability of success, while the topological-order classes are stable against any small SL transformations, demonstrating a phenomenon of emergence of unitarity. This allows us to give a new definition of long-range entanglement based on SL transformations, under which only topologically ordered states are long-range entangled.

Figures (11)

• Figure 1: (Color online) (a) A graphic representation of a quantum circuit, which is form by (b) unitary operations on blocks of finite size $l$. The green shading represents a causal structure.
• Figure 2: The two rows of Hamiltonians describe two gapped quantum systems. The two rows connected by LU transformations represent the equivalence relation between the two gapped quantum systems, whose equivalence classes are gapped quantum phases. We may view $H_N$ as a projector that defines its ground-state subspace. The ground-state subspaces of two equivalent systems are connected by the LU transformations.
• Figure 3: The two rows define two gapped quantum liquid systems via gLU transformations. The two rows connected by LU transformations represent the equivalence relation between two gapped quantum liquid systems, whose equivalence classes are gapped quantum liquid phases.
• Figure 4: (Color online) Two systems (a) and (c), with size $N_k$ and $N_{k+1}$, are described by $H_{N_k}$ and $H_{N_{k+1}}$ respectively. (a) $\to$ (b) is an LA transformation where we add $N_{k+1}-N_k$ qubits to the system $H_{N_k}$ to obtain the Hamiltonian $H_{N_k} + \sum_i Z_i$ for the combined system (b). Under the LA transformation, the ground states of $H_{N_k}$ is tensored with a product state to obtain the ground states of $H_{N_k}+ \sum_i Z_i$. In (b) $\to$ (c), we transform the ground-state subspace of $H_{N_k} + \sum_i Z_i$ to the ground-state subspace of $H_{N_{k+1}}$ via an LU transformation.
• Figure 5: Toric code as a gapped quantum liquid: toric code of $N_k$ qubits on an arbitrary 2D lattice, where the green dots represent qubits sitting on the link of the lattice (given by solid lines). By adding $N_{k+1}-N_{k}$ qubits (red dots), the gLU transformation $H_{N_k}\to H_{N_{k+1}}$ 'dissolves' the red qubits in the new lattice (with both the solid lines and dashed lines).

Theorems & Definitions (23)

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