Topological Phase Condition: Tensor Network Study

• Topological Phase Condition is defined as the precise, symmetry-enforced criterion that dictates when a quantum system transitions between distinct, globally ordered phases.
• It employs tensor network representations to encode many-body entanglement by enforcing a Z2 symmetry on virtual indices, ensuring only physically admissible local perturbations occur.
• The approach uses topological entanglement entropy as a quantitative metric to diagnose the robustness of topological order and distinguish it from trivial phases.

A topological phase condition defines the precise circumstances under which a system undergoes a transition between phases distinguished by robust global topological properties, rather than local order parameters. This concept, central to understanding quantum matter and strongly correlated systems, dictates the mathematical and physical constraints required to preserve or alter topological order, particularly in the presence of perturbations or deformations. In the framework of tensor product state representations for quantum phases, the topological phase condition is encoded as a necessary symmetry constraint on the virtual indices of the tensor network ansatz, ensuring that only variations compatible with the symmetry genuinely correspond to local, physically admissible perturbations of the system’s Hamiltonian.

1. Tensor Product State Representation and Topological Order

The tensor product state (TPS) or tensor network approach encodes many-body quantum states as a contraction of local tensors associated with lattice sites. Each tensor contains “physical” indices (corresponding to the local Hilbert space degrees of freedom) and “virtual” or “inner” indices, which are contracted between neighboring sites to reproduce correlations and entanglement structure.

For $\mathbb{Z}_2$ topologically ordered systems, the relevant TPS can be constructed by grouping three qubits (for example) at each vertex and defining a tensor $T$ with components such that only even parity configurations of the inner indices (i.e., closed strings or closed loops) are allowed nonzero amplitude,

$$T_{[000]}(000) = 1, \quad T_{[011]}(011) = 1,\quad T_{[101]}(101) = 1,\quad T_{[110]}(110) = 1$$

(all other components zero). This structure is essential for encoding the long-range entanglement properties characteristic of topological order.

2. Symmetry Constraints: The Necessary Condition

A principal result is that arbitrary local deformations of the tensor components do not, in general, correspond to physical, local perturbations of the Hamiltonian. The necessary condition for a TPS variation to preserve topological phase is invariance under a specific $\mathbb{Z}_2$ symmetry operation acting on the virtual indices,

$T \mapsto (Z \otimes Z \otimes Z) T = T,$

where $Z|0\rangle = |0\rangle$ and $Z|1\rangle = -|1\rangle$ is the Pauli $Z$ operator. This symmetry enforces that only even strings (closed loops) exist in the virtual configuration, prohibiting “open” (defect) string excitations. Local tensor variations that preserve this symmetry, such as tuning the “string tension” by introducing a weight factor $g$ to closed strings, correspond to allowable physical perturbations (e.g., Hamiltonian deformations that remain local). In contrast, symmetry-breaking tensor deformations (e.g., assigning nonzero weight to open strings) introduce global, nonlocal defects incompatible with local Hamiltonian perturbations and result in the collapse of topological order.

3. Robustness Diagnosis via Topological Entanglement Entropy

The topological entanglement entropy (TEE), denoted $S_\text{tp}$, is a global invariant that provides a quantitative measure of topological order. It is computed from combinations of entanglement entropies for multiple regions: $$S_\text{tp} = S_A + S_B + S_C - S_{AB} - S_{AC} - S_{BC} + S_{ABC}.$$ For the ideal $\mathbb{Z}_2$ TPS, $S_\text{tp}=1$ signals nontrivial topological order. When small, symmetry-preserving variations are introduced (e.g., modifying the closed loop weights), $S_\text{tp}$ remains robust as long as the $\mathbb{Z}_2$ symmetry is intact. However, upon introducing symmetry-breaking perturbations, $S_\text{tp}$ drops to zero in the thermodynamic limit, marking a transition out of the topological phase.

4. Implications for Numerical Simulation and Phase Classification

This symmetry-based topological phase condition imposes strict requirements for variational and numerical searches of topological quantum phases using TPS or projected entangled pair states (PEPS). To faithfully capture and distinguish topological phases, the variational ansatz must be constrained to the subspace preserving the relevant invariant gauge group (here, $\mathbb{Z}_2$ symmetry). Failure to enforce the symmetry may yield numerically reasonable energy minima, yet these states can be topologically trivial as diagnosed by vanishing topological entanglement entropy. The topological phase condition thus guides the construction and refinement of variational algorithms and is essential for reliable, universal characterization of quantum phases.

5. Universal Features: Invariant Gauge Group and Quantum Phase Classification

The invariant gauge group (IGG), such as $\mathbb{Z}_2$ for the discussed model, encapsulates the symmetry structure of the tensor network’s virtual indices and serves as a fingerprint for the universal topological properties of the phase. The IGG framework generalizes to other groups for more complex topological orders, underlying a broader program of quantum phase classification via tensor network symmetries. Only tensor networks with virtual symmetry transformations corresponding to the IGG of the target topological phase can faithfully represent its universal properties and correctly encode transitions to trivial phases or to other topological orders.

6. Mathematical Formulation and Summary Table

The following table summarizes the mathematical structure central to the TPS-based topological phase condition:

Concept	Mathematical Expression	Physical Meaning
TPS representation	$$|\psi\rangle = \sum C(T^{i_1}[k_1] T^{i_2}[k_2] \ldots) |k_1 k_2 \ldots\rangle$$	Wavefunction via contracted local tensors
Symmetry constraint	$T \rightarrow (Z \otimes Z \otimes Z) T = T$	$\mathbb{Z}_2$ invariance of virtual indices
TEE (Kitaev-Preskill)	$$S_\text{tp} = S_A + S_B + S_C - S_{AB} - S_{AC} - S_{BC} + S_{ABC}$$	Measure of nonlocal topological entanglement
Robust phase	$S_\text{tp}=1$ (TPS variation preserves symmetry)	Phase is topologically ordered
Trivial phase	$S_\text{tp}=0$ (symmetry broken by TPS variation)	Topological order destroyed

7. Theoretical Implications and Future Directions

The establishment of symmetry-based topological phase conditions in tensor network representations bridges numerical simulation with theoretical classification. It substantiates the necessity of enforcing gauge symmetry constraints to distinguish physically allowed (local) tensor deformations from unphysical, globally trivializing perturbations. This principle generalizes to non-Abelian and higher-dimensional topological orders by characterizing suitable IGGs and their manifestation in TPS/PEPS structures. Furthermore, the explicit link between the virtual symmetry and topological entanglement entropy provides a rigorous diagnostic for tracking phase transitions and universal properties across quantum Hamiltonians.