Generalized Model Fractional Quantum Hall States on Lattices

Model wave functions represent fixed points of topologically ordered phases, and this work systematically constructs lattice model states for Laughlin, Moore–Read, and general Zk Read–Rezayi series by deforming continuum counterparts with parameters δ to overcome limitations imposed by lattice discreteness. These lattice-specific states are characterized by their idealized energy and entanglement features and are distinguished from their continuum counterparts by a modified clustering behavior, advancing the understanding of the stability of topologically ordered phases on lattices.

Systematic Construction of Lattice Model States

The authors systematically construct exact model states for all Read–Rezayi series on a lattice. These lattice states are deformed by parameters δ, which modify their clustering properties and thereby overcome the difficulty mentioned above regarding matching required clustering properties and discrete lattice spacings. They analytically prove and numerically confirm that these δ-deformed lattice model states are exact gapped ground states of density interactions in lattice ideal bands. Furthermore, they confirm that these states exhibit idealized entanglement features identical to their continuum counterparts, including an infinite entanglement gap and the expected level counting in the entanglement spectrum.

Lattice Realization via Ideal Flat Bands

The continuum model states are formulated in lowest-Landau-level (LLL)-type flat bands with ideal quantum geometry, which are realized on lattices such as the Kapit–Mueller model. This model is a special Hofstadter model with designed rapidly decaying hopping amplitudes, and for any flux density ϕ ∈ (0, 1), it has an exactly flat lowest band whose wave functions are lattice samples of holomorphic functions multiplied by a Gaussian factor. Holomorphicity and boundary condition require the holomorphic factor to be composed of elliptic functions, such as modified Weierstrass function σ(z).

Deformation Mechanism for Lattice States

The motivation for introducing the δ deformation is to resolve the obstacle encountered when moving from continuum to lattice systems. Specifically, for fermionic Laughlin states like ν = 1/3, on a lattice fermions never approach infinitesimally close to each other, unlike in the continuum where clustering requires an order-m zero at contact. The displacement deformation resolves this by modifying the clustering rule in a lattice-native way: Instead of requiring an order-m zero at contact, which cannot be directly probed on a lattice, the wave function Eq. (2) distributes this vanishing condition over m lattice separations each with order one zero. This results in the δ-deformed wave function being an exact zero-energy state of the density-density interaction defined by Eq. (3).

Characterization via Spectral and Entanglement Diagnostics

The study examines other characteristics of the lattice Laughlin state, including its real-space correlations, particle entanglement spectra, and finite-size scaling of the many-body gap. For instance, in the fermionic ν = 1/3 state at nearest-neighbor interactions in the Kapit–Mueller flat band with ϕ = 1/3, the many-body spectrum exhibits an exactly threefold degenerate zero-energy manifold separated by a finite gap. The pair-correlation function g(2)(r) directly visualizes the clustering property, showing The exact zeros at origin and ±a1 reveal the imposed lattice clustering condition.

Extension to Non-Abelian States

The construction naturally extends to non-Abelian FQH states. For the fermionic ν = 1/2 Moore–Read state, a lattice model wave function is proposed based on a multi-component view of the state, leading to Eq. (4), which explicitly shows that the lattice Moore–Read state consists of an overall fermionic Jastrow factor together with an additional three-particle clustering structure. This construction generalizes straightforwardly to general Zk Read–Rezayi states, where a lattice Read–Rezayi wave function can be constructed by combining a lattice Laughlin factor with two-body clustering constraints and additional (k + 1)-body clustering factors.

Interaction-Range Induced Phase Transitions

The authors show that when increasing the interaction range of Hδ, phase transitions can occur, even though the model state Eq. (2) remains the exact zero-energy ground state by construction. This transition is signaled by the closing of the many-body spectral gap, the emergence of finite entanglement gap, and the loss of the Laughlin quasihole counting structure in the low-lying entanglement spectrum. As a case study for fermions at ν = 1/3, increasing interaction range leads to additional low-energy states appearing at large δ, with the near-degenerate manifold reaching 24 states, equal to the number of magnetic unit cells, and the projected structure factor S¯(q) develops pronounced finite-momentum peaks.

Summary and Outlook

The resulting lattice model states are characterized by ideal energetic and entanglement features, rooted in the null space properties of reduced density matrices. The systematic construction illustrates the organization principle of lattice conformal Hilbert space, providing a useful starting point for studying lattice-specific dynamics, excitations including anyon dynamics, neutral collective modes and lattice gravitons.

As a fastidious and diligent researcher, I have analyzed this work on constructing lattice model states for Fractional Quantum Hall (FQH) phases. The core contribution is providing exact, lattice-specific wave functions (like the Laughlin state with displacement deformations) that capture the ideal topological properties of continuum models, including specific clustering behaviors.

Here are the specific improvements to AI systems based on this scientific paper:

1. Develop a Quantum Simulation/Modeling Engine for Topological Phases:

2. Enhanced Phase Transition Modeling and Prediction:

3. Discovery of New Topological Order Architectures (Read-Rezayi Series):

4. Robust Benchmarking for Cold-Atom and Synthetic Platforms:

Specific capabilities of the improved AI system:

1. The AI system can perform high-fidelity, exact diagonalization or tensor network simulations for lattice models described by the constructed model wave functions (e.g., Hδ).

2. It can precisely diagnose the topological order of simulated systems by calculating key metrics derived from the paper:

3. Detecting and quantifying idealized features such as:

4. The presence of an infinite entanglement spectrum gap, confirming topological protection;

5. Verifying the expected level counting in the entanglement spectrum (e.g., matching quasihole counting), which is a signature of the model state;

6. Analyzing real-space correlations using pair-correlation functions to identify lattice-specific clustering patterns (zeros in g(2)).

7. Predicting phase transitions by monitoring the closing of the many-body spectral gap and the emergence of finite entanglement gaps as interaction range parameters are varied, allowing for the mapping of FQH-like regions versus non-FQH behavior in parameter space (as shown in Fig. S3).

8. Designing optimal lattice geometries (e.g., specific hopping amplitudes derived from the Kapit–Mueller model) that stabilize desired topological orders on synthetic flat-band platforms, directly informing the design of cold-atom trap potentials or photonic lattices.

9. Generalizing existing model states to non-Abelian series (Moore–Read, Read–Rezayi) by automatically applying the layer-symmetrization and displacement deformation rules derived in Sections IV and VI, enabling the simulation of complex topological states beyond the basic Laughlin family.

Model wavefunctions are essential for studying fractional quantum Hall (FQH) phases. They reveal the intricate connections between model FQH states, conformal field theory, Jack polynomials, and pseudopotential Hamiltonians. However, most model wavefunctions studied so far are in the continuum. In this work, we construct model wavefunctions and the corresponding exact parent Hamiltonians for the entire Z k Read--Rezayi series on a lattice. Our model wavefunctions obey a generalized clustering rule implemented on the lattice, which we find is encoded in the Macdonald polynomials providing a unified algebraic description of our lattice-specific model states. This ensures that the model states are exact zero-energy ground states of local density interactions and exhibit an infinitely large gap in the particle entanglement spectrum. Numerical studies of the Laughlin and Moore--Read series find the expected zero-mode and one-flux quasihole counting and positive energy gaps at the system sizes studied, and the available finite-size scaling suggests nonzero thermodynamic gaps. Tuning the model interaction range can drive a phase transition out of the FQH phase. Our work not only advances the theoretical understanding of the underlying mathematical structure of lattice model FQH states but also is potentially useful for designing exotic many-body phases in engineered quantum platforms.