Emergence of long-range entanglement and odd-even effect in periodic generalized quantum cluster models

1. Problem Statement

The paper addresses a significant gap in the understanding of one-dimensional (1D) topological phases and quantum entanglement:

• Limitation of Standard Measures: In 1D systems, standard bipartite entanglement measures (like entanglement entropy) often fail to distinguish topological phases from trivial ones because they are dominated by non-universal boundary contributions and lack informative sub-leading corrections.
• Open vs. Periodic Boundary Conditions (OBC vs. PBC): While generalized quantum cluster models (generalizations of the cluster state) are well-studied under OBC, where they exhibit Symmetry-Protected Topological (SPT) order, the entanglement structure under Periodic Boundary Conditions (PBC) has been largely overlooked. It is generally assumed that PBC systems in these models do not support long-range entanglement.
• The Core Question: Can long-range entanglement persist in generalized quantum cluster models under PBC, and if so, what specific conditions govern its emergence?

2. Methodology

The authors employ a combination of analytical exact solutions and numerical simulations to investigate the entanglement properties of a generalized quantum cluster model.

• Model Definition:
  The system is defined by the Hamiltonian:
  $$H = J \sum_{j=1}^{N} \sigma^x_j \tau^z_{j,m} \sigma^x_{j+m} + H_z$$
  where $\tau^z_{j,m} = \sigma^z_{j+1} \cdots \sigma^z_{j+m-1}$ represents an interaction range of $m$ spins, $N$ is the system size, and $H_z = -h \sum \sigma^z_j$ is a transverse field introducing quantum fluctuations. Periodic boundary conditions ($\sigma_{N+1} = \sigma_1$) are applied.

• Analytical Approach (Zero-Field Limit):

  • The authors use the Jordan-Wigner transformation to map the spin system to fermions.
  • They decompose the Hamiltonian using projectors based on fermion parity ($P_z$), resulting in two distinct fermionic Hamiltonians ($H_+$ and $H_-$) corresponding to Periodic (PBC) and Anti-Periodic (APBC) boundary conditions for the fermions.
  • They perform a Bogoliubov transformation to diagonalize the Hamiltonians and analyze the ground-state degeneracy structure as a function of $N$ (system size) and $m$ (interaction range).
  • They utilize the Kramers-Wannier transformation (Appendix A) to provide an alternative perspective on the ground-state degeneracy and geometric frustration.

• Entanglement Diagnostics:
  To detect long-range entanglement, the authors move beyond standard entanglement entropy ($S_l$) and focus on Quantum Conditional Mutual Information (QCMI):

  • Tripartite QCMI ($S^{t}_{cmi}$): $S_{AB} + S_{BC} - S_B - S_{ABC}$.
  • Four-part QCMI / Disconnected Entanglement Entropy ($S^{q}_{cmi}$ or $S_D$): Defined as $S_{AB} + S_{BC} - S_B - S_{ABC}$ (equivalent to the disconnected entropy for specific partitions).
  • Rationale: The four-part QCMI is constructed to systematically cancel local and boundary-driven contributions. A non-zero value is a robust indicator of genuine long-range entanglement, unlike bipartite measures which can be non-zero due to short-range correlations.

• Numerical Analysis:
  The authors compute entanglement entropy and QCMI for various system sizes ($N$) and interaction ranges ($m$) under infinitesimal but finite transverse fields ($h \to 0^+$) and large fields ($h > J$).

3. Key Contributions and Results

A. The "Odd-Even" Effect on Ground State Degeneracy

The analytical solution reveals a striking dependence of ground-state degeneracy on the parity of $N$ and $m$:

• Case 1: $N$ is odd, $m$ is odd.
  • The system exhibits a $2^N$-fold degenerate ground state manifold.
  • This degeneracy arises from geometric frustration: the system cannot satisfy the antiferromagnetic alignment globally due to the odd number of sites and the specific interaction range.
  • The ground states include specific fermionic excitations (e.g., $c^\dagger_0|\phi_+\rangle$, $\eta^\dagger_q|\phi_+\rangle$, etc.).
• Case 2: All other combinations ($N$ even, or $m$ even).
  • The system possesses a unique ground state (or a simple 2-fold degeneracy in specific even cases) with a gapped spectrum.

B. Emergence of Long-Range Entanglement

The central finding is that long-range entanglement exists under PBC if and only if both $N$ and $m$ are odd.

• Four-part QCMI ($S^{q}_{cmi}$):
  • When $N, m \in \text{odd}$: $S^{q}_{cmi}$ remains non-zero even in the presence of an infinitesimal transverse field. This non-vanishing value directly signals genuine long-range entanglement.
  • When $N$ or $m$ is even: $S^{q}_{cmi}$ vanishes, indicating the absence of long-range entanglement (only short-range correlations exist).
• Entanglement Entropy ($S_l$):
  • For the $N, m \in \text{odd}$ case, $S_l$ does not saturate to a constant as the block size $l$ increases; instead, it grows logarithmically or linearly depending on the regime, showing dependence on the global system size (non-local factor $\alpha = l/N$).
  • For other cases, $S_l$ follows the standard area law, saturating quickly based on the interaction range $m$.

C. Robustness Against Quantum Fluctuations

The authors investigate the stability of this entanglement under a large transverse field ($h$):

• Phase Transition: The system undergoes a quantum phase transition at $h = J$.
• Gapless vs. Gapped:
  • For $N, m \in \text{odd}$, the low-energy spectrum remains gapless for $h < J$ due to the massive ground-state degeneracy. The long-range entanglement (non-zero $S^{q}_{cmi}$) persists in this gapless phase.
  • For other cases, the system is gapped for $h < J$ and transitions to a trivial paramagnetic phase for $h > J$, where all long-range correlations vanish.
• Critical Scaling: At the critical point ($h=J$), the entanglement entropy exhibits logarithmic scaling $S \sim \frac{m}{6} \log_2 N$, consistent with Conformal Field Theory (CFT) predictions for the Ising universality class, where the central charge scales with the interaction range $m$.

D. Physical Origin

The paper attributes the emergence of long-range entanglement to geometric frustration.

• In the limit $m=1$, the model reduces to the transverse-field Ising chain on an odd-sized ring, which is frustrated and cannot satisfy all antiferromagnetic bonds, leading to degeneracy.
• For $m > 1$ (odd), the generalized model inherits this frustration, creating a highly degenerate manifold that supports non-local correlations and long-range entanglement, protected by the specific parity of the system size and interaction range.

4. Significance

• Redefining 1D Topology: The work challenges the conventional wisdom that 1D PBC systems in cluster models cannot support long-range entanglement. It demonstrates that system size parity is a critical control parameter.
• Diagnostic Tool: It validates the four-part conditional mutual information (disconnected entropy) as a superior diagnostic tool for 1D topological phases, capable of distinguishing genuine long-range entanglement from short-range correlations where bipartite measures fail.
• Robustness: The findings show that topological features (long-range entanglement) can be robust against quantum fluctuations (transverse fields) in specific parity sectors, suggesting new avenues for designing robust quantum memory or resources for measurement-based quantum computation in 1D systems.
• Universality: The connection between the interaction range $m$ and the central charge of the critical theory provides a deeper understanding of how interaction range influences universality classes in quantum phase transitions.

In summary, Zheng and Chen establish that the interplay between system size ($N$) and interaction range ($m$) dictates the topological nature of generalized quantum cluster models under PBC, with the odd-odd condition being the unique gateway to a gapless, long-range entangled phase.