Flat-Top Ground State

• Flat-Top Ground State is a configuration characterized by uniform spatial or ensemble profiles, where particle density or wavefunction amplitudes form plateaus with sharp transitions at boundaries.
• Engineered interactions and boundary conditions, such as flat potentials and gauge phase control, enable practical realizations in lattice models, twisted graphene, and photonic crystal slabs.
• Robust energy minimization and flat-band degeneracies impose rigorous theoretical constraints and drive applications in quantum simulations, correlated electron systems, and device engineering.

A flat-top ground state refers to a configuration, eigenstate, or energy minimum whose spatial, occupation, or ensemble profile is uniform—often subject to constraints—over a region or lattice, with possible discontinuities or transitions in density or order parameters at boundaries or quantized jumps in conjugate variables. The precise manifestation depends on the physical system (classical particles, quantum matter, photonic fields), dimensionality, interaction type, and boundary conditions. The notion of “flat-top” thus encompasses uniform plateaus in particle density, wavefunction amplitudes, ensemble energies, or surface-state occupation, typically arising due to frustration, soft interactions, engineered band structures, or gauge-related phase boundary conditions.

1. Classical Particle Systems: Tower-Lattice Flat-Top Ground States

In one-dimensional continuous systems with nonnegative, bounded, range-1 pair potentials (such as the overlap potential $u(x) = (1 - |x|)\cdot 1_{[-1,1]}(x)$), classical ground state configurations are rigorously proven to form tower-lattice ground states for densities $p \geq 1$ (Suto, 2010). Each lattice site at integer coordinates hosts a “tower” of $m$ or $m+1$ particles, generating a profile that is flat except for differences at most of one particle between sites.

The total particle number for an interval $[0, n]$ is $N = m(n + 1) + r$ with $0 \leq r \leq n$; energy minimization yields: $E_n(N) = (n + 1) \frac{m(m-1)}{2} + r m$ where all pairwise interactions are local, given the range-1 cutoff.

The flat-top character in this context—ideally realized for $r=0$ (exact integer density)—arises from maximally uniform tower heights. This construction is independent of fine details (flat or cusp-like) of the potential at the origin, as long as the interaction remains bounded and the range is precisely matched to the lattice constant. Variants with flat vs. cusp potentials merely change the degeneracy (number of equivalent ground states), not the spatial flat-top structure.

2. Quantum Systems with Flat or Flat-Topped Potentials

For quantum particles confined in one-dimensional potentials of the form $V(x) = p(x/a)^N$, with $N \to \infty$, the flat-top potential limit is approached: for large $N$, the potential well becomes nearly constant over a region (infinite square well) (Weber, 2018, Weber, 2018).

In these systems, the ground state wavefunction transitions from a bell-shaped (harmonic oscillator, $N=2$) to a plateaued structure. The ground state energy is partitioned into kinetic and potential components: $$E = \left(\frac{\hbar^2}{2ma^2}\right) \left( \frac{\mu}{\hbar^2} \right)^{2/(N+2)}$$ This ensures a “flat-top” region in the spatial probability density for very large $N$.

Numerical and variational analyses show the wavefunction develops extended flat regions, and energy splitting between levels increases as the edge “stiffness” dominates. Standard Gaussian-like trial functions fail to capture the true flat-top spatial profile as $N$ increases, motivating refined optimal trial functions for variational calculations.

These results have direct implications in quantum nanostructures and zero-point energy models where potentials can be engineered to be nearly flat, leading to controllable uniform ground-state density.

3. Flat Bands and Topological/Correlated Quantum Ground States

Flat-top ground states appear as a corollary of flat-band physics, particularly in systems where the band structure features a dispersionless (macroscopically degenerate) manifold. This occurs, for example, in magic-angle twisted bilayer graphene (MATBG) and engineered Hubbard lattices (Müller et al., 2016, Stubbs et al., 25 Mar 2025, Deng et al., 2023, Pan et al., 2022). The degenerate ground-state manifold in such settings is composed of product states (e.g., ferromagnetic Slater determinants), and the “flat-top” attribute is manifest in uniform occupation across a subspace or in ensemble energies insensitive to fractional occupation or spin over a region.

In MATBG, the flat band interacting (FBI) Hamiltonian is frustration-free, its ground state manifold being the linear span of ferromagnetic Slater determinants (Stubbs et al., 25 Mar 2025). In correlated flat-band systems with quantum anomalous Hall ground states, the ensemble ground state possesses a robust gap, but compressibility and order parameters experience sharp thermodynamic transitions at much lower temperatures due to many-body excitonic effects (Pan et al., 2022). The ground-state ensemble energy is piecewise-linear and “flat” in electron number and fractional spin; derivative discontinuities in the Kohn-Sham potential arise at boundaries of the flat-top region (Goshen et al., 2023).

In quasi-2D lattices (Lieb lattice), entrance into a flat band causes superfluid pairing gap and superfluid density to switch from exponential to power-law dependence on interaction, and induces dramatic compressibility enhancement—consequences of the flat-top band density of states (Deng et al., 2023).

4. Photonics and Flat-Top Wavefunctions Engineered via Artificial Gauge Fields

Recent experiments demonstrate that wave systems—particularly photonic crystal slabs—can be engineered to exhibit flat-top ground states through artificial gauge fields imposed by locally controlled reflective phases (Li et al., 25 Sep 2025).

By applying the Byers–Yang theorem and continuously shifting boundary phase conditions, discrete oscillatory states morph into states with uniform, nonvanishing envelope over the confined region. In a ring-like geometry with gauge phase $\theta_{AB}$, the energy spectrum is shifted as: $$E_n \propto \left[ n - \frac{\theta_{AB}}{2\pi} \right]^2 / L^2$$ In the photonic crystal, varying the gap $g$ between a central bulk and bandgap-engineered boundary continuously tunes the so-called reflective phase, driving the system toward a flat-top ground state. Quantization conditions for confined states are modified accordingly: $2q_x L_a + \Theta_x(\theta_R+\theta_L) = 2m\pi$ At specific phase windings, the wavenumber $k \to 0$, yielding a homogeneous (flat-top) mode envelope.

Experimentally, the evolution toward the flat-top state is probed via lasing and direct near-field/far-field measurements, showing suppressed divergence angles and uniform modal emission spatially replicated across the structure. This state is robust to variations in geometry and material parameters, and its uniformity enables device-level replication of band-edge properties.

5. Flat Surface States and Multipole Texture in Orbital Systems

In multi-orbital tight-binding models on finite-thickness cubic lattices (e.g., $e_g$ orbital systems), flat-top ground states emerge as localized flat surface bands. For (111) and (110) surfaces, when the bulk band projected onto the surface Brillouin zone is gapped, the surface band is “flat”—displaying negligible dispersion—and hosts strong octupole moments (Kubo, 22 Aug 2024).

The orbital configuration (distinguished by Pauli-matrix representations in the Hamiltonian, parameterized by $t_1$, $t_2$, and $\alpha$) gives rise to fully polarized octupole moments at high-symmetry points, with surface states extending over the full Brillouin zone (except where topological winding numbers vanish). Such flat-top surface states are candidates for strong correlation, unconventional superconductivity, or higher-moment device applications.

In contrast, (001) surfaces lack nontrivial topology and do not support flat-top surface bands or octupole character.

6. Flat-Top States in Engineered Quantum Simulators

Recent quantum computation work uses Hamiltonian engineering and variational quantum eigensolver (VQE) approaches to resolve ground-state energies of large, frustrated flat-band systems—such as the Kagome antiferromagnet (Ahsan, 8 Jul 2025). Here, a hardware-efficient ansatz entangles over 100 qubits, with bond strengths locally engineered (e.g., defect triangles with $J'=2$) to mimic loop-flip dynamics and promote nearly uniform resonance (flat-top) of dimer states. The engineered Hamiltonian and ansatz reduce circuit complexity while preserving long-range correlations, yielding per-site ground-state energies closely matching theoretical thermodynamic limits.

7. Implications and Theoretical Constraints

The generalized flat-top (or “flat-plane”) condition in DFT constrains approximate functionals to exactly reproduce energy profiles that are linear in electron number and constant (“flat”) in fractional spin over specified ensemble regions. Transition points marked by derivative discontinuities in the Kohn–Sham potential are required for correct description of excitations and charge/spin transfer physics (Goshen et al., 2023).

In photonics and wave engineering, the emergence of flat-top ground states through artificial gauge fields offers new routes for robust quantum emitters, surface lasers, and antennas with tailored uniformity and modal control (Li et al., 25 Sep 2025). In quantum magnetism and strongly correlated electron systems, flat-top and flat-band ground states serve as organizing principles for identifying the structure and degeneracy of the ground state manifold and for designing phase diagrams of exotic matter.

Summary Table: Flat-Top Ground State Manifestations

Physical System	Flat-Top Manifestation	Key References
1D classical particles (range-1)	Uniform tower-lattice configurations, constant occupation per site	(Suto, 2010)
Quantum flat potentials	Plateau in ground-state wavefunction, energy partitioned kinetic/potential	(Weber, 2018, Weber, 2018)
Flat-band lattice models	Uniform occupation/ensemble energy, degenerate ground states	(Müller et al., 2016, Stubbs et al., 25 Mar 2025, Pan et al., 2022, Deng et al., 2023)
Photonic crystal slabs (gauge field)	Uniform mode envelope over region via artificial gauge phase	(Li et al., 25 Sep 2025)
Multiorbital surface states	Flat surface bands with octupole polarization	(Kubo, 22 Aug 2024)
Quantum simulators (Kagome lattice)	Nearly uniform RVB resonance via Hamiltonian engineering	(Ahsan, 8 Jul 2025)

The concept of flat-top ground states thus unifies distinct physical settings—quantum, classical, and wave systems—under the rubric of uniformity in ground-state profile arising from frustration, interaction-induced degeneracy, engineered boundary phase conditions, or topological constraints. These insights impose exact conditions on functional formalisms and guide the design of materials and devices for next-generation quantum, photonic, and correlated-electron technologies.