Dynamical splitting and a nodal Bose liquid in 2d chiral XYZ model

Problem and Motivation

The paper investigates a class of quantum many-body Hamiltonians characterized by a structure termed "dynamical splitting." In these systems, the Hamiltonian terms can be partitioned into two sets, $H_A$ and $H_B$, acting on the same physical degrees of freedom, such that every term in $H_A$ commutes with every term in $H_B$, despite terms within each set failing to commute. This structure differs from standard symmetry-based simplifications (which decompose the Hilbert space into a direct sum) by enabling a decomposition into a tensor product of effective degrees of freedom within fixed central sectors.

The primary focus is the chiral XYZ model, a geometrically frustrated spin-1/2 model on a triangular lattice. This model was previously introduced as a special limit of a Majorana–Hubbard model and discussed in the context of subsystem codes. The author aims to determine the nature of its ground state, specifically investigating whether it realizes a gapped topological phase (analogous to the Kalmeyer–Laughlin chiral spin liquid) or a gapless phase, given its possession of anticommuting non-contractible line symmetries that enforce topology-dependent degeneracy.

Methodology

The study employs a combination of analytical large-$N$ techniques and exact diagonalization (ED), facilitated by the dynamical splitting structure.

1. Dynamical Splitting and Tensor-Product Decomposition:
   The author demonstrates that for the chiral XYZ model, the operator algebras generated by the two families of triangle interactions ($A$ and $B$) commute. By fixing the eigenvalues of the commuting center of the joint algebra (generated by subsystem line symmetries), the total Hilbert space decomposes as:
   $$H_\lambda \simeq \mathbb{C}^2_P \otimes \mathbb{C}^{2q}_A \otimes \mathbb{C}^{2q}_B$$
   where $\mathbb{C}^2_P$ corresponds to a logical qubit enforcing exact degeneracy, and $q = (L-1)(L-2)/2$ for an $L \times L$ torus. The Hamiltonian in a fixed sector $\lambda$ becomes:
   $$H|_{H_\lambda} = \mathbb{1}_P \otimes [H_A(\lambda) \otimes \mathbb{1}_B + \mathbb{1}_A \otimes H_B(\lambda)]$$
   This reduces the computational cost of diagonalization. Instead of diagonalizing a matrix of dimension $2^{2q+1}$, one diagonalizes two matrices of dimension $2^q$, effectively allowing the study of systems with approximately twice the number of spins accessible to standard ED.

2. Large-$N$ Generalization:
   The model is generalized to $Z_N$ clock variables. In the $N \to \infty$ limit, the system is treated semiclassically. The author derives the dispersion relation of the soft modes and analyzes the stability of the nodal structure against perturbations using Renormalization Group (RG) arguments.

3. Exact Diagonalization (ED):
   Using the dynamical splitting, the author performs ED on the $N=2$ (spin-1/2) model on lattices up to $9 \times 9$. They analyze:

   • The finite-size scaling of the many-body gap.
   • Bipartite entanglement entropy scaling ($S_1$) on tori and cylinders.
   • The response to various perturbations that preserve or break subsystem symmetries and dynamical splitting.

Key Contributions and Results

1. Algebraic Structure and Symmetries:
The paper establishes that the chiral XYZ model possesses $O(L)$ subsystem symmetries (row, column, and diagonal string operators). Crucially, a pair of anticommuting row and column operators enforces an exact, topology-dependent degeneracy (at least $2^g$ on a genus-$g$ surface). The states within this multiplet are locally indistinguishable, a feature typically associated with gapped topological order, yet the author shows this coexists with gaplessness.

2. Large-$N$ Solution:
In the $N \to \infty$ limit, the model is found to be gapless. The excitation spectrum exhibits three nodal lines in the Brillouin zone defined by $k_x = 0$, $k_y = 0$, and $k_x + k_y = 0$. The dispersion relation is given by:
$$\omega(k) \sim \left| \sin \frac{k_x}{2} \sin \frac{k_y}{2} \sin \frac{k_x + k_y}{2} \right|$$
This identifies the ground state as a nodal Bose liquid. The author notes that the subsystem symmetries protect these nodal lines.

3. Numerical Evidence for $N=2$:

• Gaplessness: ED results for the spin-1/2 model show that the many-body gap decreases rapidly with system size ($L$), consistent with a gapless state. The scaling suggests a dynamical exponent $z > 1$.
• Entanglement Scaling: The bipartite entanglement entropy scales as $L \log L$ for half-torus cuts. On cylindrical geometries, the entropy follows a "chord" scaling form characteristic of 1+1-dimensional conformal field theories (CFT), with the number of effective channels increasing with the transverse size. This provides strong evidence for a nodal Bose liquid at $N=2$.

4. Stability and Proximate Phases:
The author studies three types of deformations:

• Split-preserving deformation: A deformation that maintains both dynamical splitting and subsystem symmetries. At large $N$, this induces a first-order transition between uniform and inhomogeneous nodal liquids. At $N=2$, ED shows a reconstruction of the center sector consistent with this transition, though the inhomogeneous state's nature remains less clear.
• Breaking dynamical splitting (preserving subsystem symmetries): Interpolating between the chiral XYZ model and two decoupled Wen-plaquette models. The results suggest a finite-coupling transition from the gapless nodal liquid to a gapped phase with $Z_2 \times Z_2$ topological order (two copies of the Wen-plaquette model).
• Explicitly breaking subsystem symmetries: Adding a uniform onsite field. Large-$N$ RG analysis indicates that subsystem-symmetry-breaking perturbations are relevant, suggesting the nodal liquid is unstable. ED on small lattices ($3 \times 3, 4 \times 4$) shows a reorganization of the low-energy spectrum suggestive of $C_3$ symmetry breaking, but the limited system sizes prevent a conclusive determination of whether an arbitrarily weak field immediately gaps the system.

Significance and Claims

The paper claims to provide the first exact diagonalization evidence for a gapless nodal Bose liquid in a two-dimensional geometrically frustrated spin model that simultaneously possesses exact, topology-dependent degeneracy usually associated with gapped topological order.

The significance of the work lies in:

1. Methodological Advancement: The demonstration that "dynamical splitting" allows for the study of unusually large 2D systems (up to $9 \times 9$ spins) via exact diagonalization, bypassing the exponential scaling of the Hilbert space.
2. Physical Insight: The identification of a phase where gaplessness and topological-like degeneracy coexist, challenging the intuition that such degeneracy implies a gapped topological phase.
3. Robustness: The finding that the nodal liquid phase is robust against deformations that preserve subsystem symmetries (even if dynamical splitting is broken) but is likely unstable to explicit symmetry-breaking perturbations.

The author concludes that while the microscopic $N=2$ Hamiltonian may be fine-tuned with respect to generic local perturbations, the nodal liquid phase itself is a distinct, highly entangled state accessible through the unique algebraic structure of the chiral XYZ model. They do not claim the model is a realization of a specific experimental system but rather a theoretical laboratory for exploring these phenomena.