Quantum Hard Disk Model in Constrained Lattice Systems

• The quantum hard disk model is a constrained lattice framework where hard-core bosons obey extended exclusion rules, bridging quantum coherence with classical combinatorics.
• It exhibits stark contrasts between quantum and classical dynamics, featuring ballistic crystal melting in one dimension and persistent defect patterns in two dimensions.
• The model serves as a versatile platform for probing Hilbert-space fragmentation and many-body scars, with direct relevance to Rydberg atom experiments and advanced quantum simulations.

Searching arXiv for recent and foundational papers on the quantum hard disk model and closely related work. The quantum hard disk model is a lattice model of hard-core bosons with an extended exclusion constraint: particles occupy lattice sites, the total particle number is conserved, and no two particles may occupy sites within a prescribed exclusion range. In the form developed for one-dimensional chains and two-dimensional square lattices, the model is a quantum counterpart of the classical hard-disk problem on a lattice and admits a natural realization in systems of Rydberg atoms. Its defining feature is that the static excluded-volume structure coincides with the corresponding classical constrained lattice gas at infinite temperature, whereas the real-time dynamics are qualitatively different because stochastic motion is replaced by coherent Hamiltonian evolution. In this setting, one finds ballistic crystal melting in one dimension and defect- and interface-stabilized crystal structures in two dimensions, with the latter linked to Hilbert-space fragmentation and quantum many-body scars (Naik et al., 2023, Trigueros et al., 17 Mar 2025).

1. Microscopic formulation

The model is defined on a $d$-dimensional lattice with $L^d$ sites. At each site $i$ there is a hard-core boson with operators

$$a_i^\dagger,\; a_i,\qquad n_i=a_i^\dagger a_i\in\{0,1\},$$

and conserved total particle number

$M=\sum_i n_i.$

The density is

$\eta=\frac{M}{L^d}.$

The hard-disk constraint is an excluded-volume rule. In the principal case studied on a hypercubic lattice with lattice spacing set to $1$, nearest-neighbor exclusion is imposed: $$|\mathbf r_i-\mathbf r_j|=1\;\Rightarrow\; n_i n_j=0.$$ Geometrically, each particle occupies one site and excludes its nearest neighbors from being occupied, giving a lattice version of a hard disk of radius $1$. The constrained Hilbert space is therefore

$$\mathcal H=\mathrm{span}\left\{\,|\{n_i\}\rangle\;\middle|\; n_i=0,1,\; n_i n_j=0\ \text{if}\ |\mathbf r_i-\mathbf r_j|=1\,\right\}.$$

The Hamiltonian is purely kinetic but dressed by projectors that enforce the hard-disk constraint during motion: $L^d$0 with

$L^d$1

Here $L^d$2 projects onto states in which all nearest neighbors of site $L^d$3 are empty, so $L^d$4 allows a hop only when both the initial and final local environments satisfy the exclusion rule. There is no explicit interaction term; the interaction is purely kinematic through the projectors (Naik et al., 2023).

The model also admits a spin-$L^d$5 representation,

$L^d$6

so occupied sites correspond to spin up, or equivalently to Rydberg excitations. The construction generalizes to larger exclusion radius $L^d$7 through a finite-range exclusion graph,

$L^d$8

with

$L^d$9

and the same projected hopping structure. The paper focuses on $i$0 on open 1D chains and open 2D square lattices, but the structure extends straightforwardly to other lattice geometries and ranges (Naik et al., 2023).

2. Classical correspondence and static order

The classical hard-disk problem is an excluded-volume system with non-overlapping particles and no energetic interactions beyond overlap forbiddance. In the lattice version relevant here, the allowed configurations are exactly the occupation-number configurations satisfying the exclusion constraint. Because the quantum Hamiltonian is off-diagonal in the occupation basis and conserves particle number, the infinite-temperature static properties are purely combinatorial: the quantum infinite-temperature structure factor equals the classical one. Consequently, static phase diagrams, including the existence and location of crystalline order as a function of density, match those of the classical hard-core lattice gas (Naik et al., 2023).

For the two-dimensional square lattice with nearest-neighbor exclusion, the crystalline order parameter is the staggered structure factor at wavevector $i$1,

$i$2

where the trace is over all configurations in the density sector $i$3, and $i$4 is the number of states in that sector. In this regime the model exhibits a checkerboard crystalline phase for

$i$5

An important structural point is that this crystalline phase lies in the weak fragmentation regime of Hilbert space rather than at the extreme high-density limit. This matters because the persistence of order in the quantum dynamics cannot then be reduced to trivial freezing alone (Naik et al., 2023).

The static classical-quantum equivalence is therefore limited. It holds at $i$6 for static observables, but it does not extend to dynamical behavior. This distinction is central to the model: identical configuration spaces and identical infinite-temperature combinatorics coexist with sharply different transport, melting, and defect dynamics.

3. One-dimensional crystal melting

In one dimension, the model is defined on an open chain with nearest-neighbor exclusion $i$7. The dynamical problem studied most directly is the melting of a finite crystal droplet,

$i$8

namely a contiguous block of maximally packed particles at density $i$9 embedded in empty space. Melting is monitored through the density profile

$$a_i^\dagger,\; a_i,\qquad n_i=a_i^\dagger a_i\in\{0,1\},$$0

and operationally means the spreading of the initially localized high-density region toward the uniform density $$a_i^\dagger,\; a_i,\qquad n_i=a_i^\dagger a_i\in\{0,1\},$$1 within the relevant sector (Naik et al., 2023).

The principal result is a sharp quantum-classical dynamical contrast. In the quantum model, crystal melting is ballistic. In the corresponding classical model, defined as a random walk of hard disks in configuration space subject to the same exclusion constraint, melting is sub-diffusive. The quantum result follows from an exact mapping specific to the 1D nearest-neighbor exclusion problem: by removing one site to the right of each occupied site in every basis configuration, the excluded volume is gauged away, and the constrained many-body dynamics reduce to free hopping of fictitious particles. The density fronts then propagate with light-cone-like behavior,

$$a_i^\dagger,\; a_i,\qquad n_i=a_i^\dagger a_i\in\{0,1\},$$2

By contrast, the classical dynamics must preserve particle ordering, so the same geometric reduction yields single-file diffusion rather than free propagation. The spreading is then sub-diffusive and falls into the KPZ universality class: $$a_i^\dagger,\; a_i,\qquad n_i=a_i^\dagger a_i\in\{0,1\},$$3 The same microscopic excluded-volume constraint therefore produces parametrically faster quantum melting than classical melting. In one dimension this anomalous quantum transport is tied primarily to integrability and to the exact mapping to free-particle-like propagation rather than to scar physics (Naik et al., 2023).

4. Two-dimensional defects, interfaces, and crystal memory

On the open $$a_i^\dagger,\; a_i,\qquad n_i=a_i^\dagger a_i\in\{0,1\},$$4 square lattice, nearest-neighbor exclusion supports checkerboard crystal order at sufficiently high density. The central dynamical question in two dimensions is not global melting from a droplet, but the fate of defects and interfaces inside a crystalline background. The original study considered initial configurations in the crystalline regime containing defects and found that the long-time quantum occupations retain a detailed memory of the initial crystal pattern, whereas the corresponding classical random-walk dynamics wash that pattern out almost completely (Naik et al., 2023).

This memory is quantified by the autocorrelation function

$$a_i^\dagger,\; a_i,\qquad n_i=a_i^\dagger a_i\in\{0,1\},$$5

with

$$a_i^\dagger,\; a_i,\qquad n_i=a_i^\dagger a_i\in\{0,1\},$$6

The subtraction ensures that $$a_i^\dagger,\; a_i,\qquad n_i=a_i^\dagger a_i\in\{0,1\},$$7 if the system completely forgets the initial microscopic pattern and approaches a homogeneous density $$a_i^\dagger,\; a_i,\qquad n_i=a_i^\dagger a_i\in\{0,1\},$$8. In the quantum model, $$a_i^\dagger,\; a_i,\qquad n_i=a_i^\dagger a_i\in\{0,1\},$$9 decays initially but saturates to a nonzero constant. In the classical model, it decays toward zero up to small finite-size corrections (Naik et al., 2023).

A later extension introduced short-range soft-core interactions,

$M=\sum_i n_i.$0

where $M=\sum_i n_i.$1 denotes next-nearest-neighbor pairs. This perturbation models the dominant residual interaction beyond a strict nearest-neighbor blockade in Rydberg realizations. The resulting dynamics exhibit several distinct classes of behavior for interfaces: fast relaxation, slow relaxation with long plateaus, and persistent memory retention. Three prototypical initial states were studied in detail: removing particles in the second row, in the first row, or along half of a diagonal. The diagonal interface is the most robust case: $M=\sum_i n_i.$2 decays somewhat but saturates to a substantial nonzero value that is weakly dependent on $M=\sum_i n_i.$3 and on system size. By contrast, the corresponding classical constrained random walk yields $M=\sum_i n_i.$4 in all three representative cases, including the diagonal one (Trigueros et al., 17 Mar 2025).

The same work distinguishes point-like defects from interfaces. Point-like defects, whose size does not scale with $M=\sum_i n_i.$5, typically live in strongly fragmented high-density sectors and remain frozen for arbitrarily long times. Interfaces, whose vacancy number scales linearly with $M=\sum_i n_i.$6, can reside either in small fragments, where they are trivially static, or in the largest fragment, where their stability is a genuinely quantum effect. This suggests a heterogeneous dynamical landscape in which fully frozen, metastable, and permanently non-ergodic patterns coexist in two dimensions (Trigueros et al., 17 Mar 2025).

5. Hilbert-space fragmentation and quantum many-body scars

The model’s non-ergodic structure is organized by Hilbert-space fragmentation. In one dimension fragmentation is trivial except at maximal density $M=\sum_i n_i.$7. In two dimensions it becomes nontrivial as a function of density. The key geometric objects are “snakes”: diagonal chains of occupied sites on the square lattice. Because nearest-neighbor exclusion forbids particles from crossing such diagonals, a snake partitions the lattice into dynamically disconnected regions. Configurations containing snakes therefore belong to disconnected fragments of the many-body configuration graph (Naik et al., 2023).

Two fragmentation regimes are identified. In the weak fragmentation regime, for densities $M=\sum_i n_i.$8 but below a high-density threshold, the Hilbert space splits into one large ergodic fragment plus many small fragments. In the strong fragmentation regime,

$M=\sum_i n_i.$9

every configuration contains snakes or fully filled diagonals, and the Hilbert space breaks down into only small fragments, with no large ergodic component (Naik et al., 2023).

To diagnose quantum many-body scars, the eigenstates $\eta=\frac{M}{L^d}.$0 are characterized by their bipartite entanglement entropy,

$\eta=\frac{M}{L^d}.$1

and by a normalized Edwards-Anderson parameter. The unnormalized quantity is

$\eta=\frac{M}{L^d}.$2

and the normalized form is

$\eta=\frac{M}{L^d}.$3

Here $\eta=\frac{M}{L^d}.$4 for a single configuration and $\eta=\frac{M}{L^d}.$5 for a fully delocalized eigenstate over all basis states (Naik et al., 2023).

The spectral structure differs across regimes. In the non-fragmented regime, $\eta=\frac{M}{L^d}.$6 forms a single ETH-like entanglement dome and $\eta=\frac{M}{L^d}.$7 is small and featureless. In the strong fragmentation regime, almost all eigenstates have low entanglement and large $\eta=\frac{M}{L^d}.$8, reflecting trivial freezing. In the weak fragmentation regime, including densities near the crystalline phase, there is a bulk of ETH eigenstates but also towers of low-entanglement, high-$\eta=\frac{M}{L^d}.$9 eigenstates superimposed on the entanglement dome. These appear both in small fragments and, crucially, inside the large ergodic fragment. They are identified as quantum many-body scars. The persistence of two-dimensional crystal patterns and the stability of certain interfaces are attributed to large overlaps between nearly crystalline initial states and these scarred eigenstates (Naik et al., 2023).

The soft-core extension preserves this logic. Because the $1$0-term is diagonal in the occupation basis, it does not change the connectivity of the configuration graph and hence does not alter the fragmentation structure. Numerically, the number of scar states decreases with $1$1, but a finite population remains even up to relatively large $1$2. This suggests that both fragmentation and scar physics survive a substantial range of perturbations (Trigueros et al., 17 Mar 2025).

6. Realizations, generalizations, and terminological scope

Rydberg atom arrays provide the natural microscopic realization emphasized for this model. Each atom is treated as a two-level system with ground state $1$3 and Rydberg state $1$4, identified with $1$5 and $1$6, so that $1$7. A generic driven Rydberg Hamiltonian takes the form

$1$8

with van der Waals interactions $1$9. When the blockade radius covers nearest neighbors and the nearest-neighbor interaction is much larger than the other relevant energy scales,

$$|\mathbf r_i-\mathbf r_j|=1\;\Rightarrow\; n_i n_j=0.$$0

states with adjacent excitations are energetically forbidden, and the system can be projected onto the same constrained Hilbert space as the lattice hard-disk model. With an engineered exchange term, one obtains the effective particle-number-conserving Hamiltonian

$$|\mathbf r_i-\mathbf r_j|=1\;\Rightarrow\; n_i n_j=0.$$1

which is equivalent to the projected hopping Hamiltonian under the spin-boson mapping (Naik et al., 2023).

The experimental observables follow directly from this mapping: site-resolved $$|\mathbf r_i-\mathbf r_j|=1\;\Rightarrow\; n_i n_j=0.$$2, two-point correlations $$|\mathbf r_i-\mathbf r_j|=1\;\Rightarrow\; n_i n_j=0.$$3, the $$|\mathbf r_i-\mathbf r_j|=1\;\Rightarrow\; n_i n_j=0.$$4 structure factor, and autocorrelation observables of the $$|\mathbf r_i-\mathbf r_j|=1\;\Rightarrow\; n_i n_j=0.$$5 type. The model is therefore positioned as a platform for observing ballistic melting in 1D, persistent crystal patterns and interfaces in 2D, and scar-induced non-thermal behavior in constrained quantum matter (Naik et al., 2023, Trigueros et al., 17 Mar 2025).

The same literature also points to several generalizations: larger disk radius $$|\mathbf r_i-\mathbf r_j|=1\;\Rightarrow\; n_i n_j=0.$$6, different lattice geometries such as triangular, honeycomb, Kagome, ladders, or quasi-1D strips, anisotropic exclusions such as rods, soft-core interactions, disorder, and driven or Floquet variants. These directions are presented as ways to investigate how scars and fragmentation evolve as the constraint structure is modified (Naik et al., 2023).

A persistent source of confusion is nomenclature. Within the literature considered here, “quantum hard disks” denotes the constrained lattice model of hard-core bosons with projected hopping and excluded volume (Naik et al., 2023, Trigueros et al., 17 Mar 2025). Other papers use similar language for distinct constructions. A continuum two-dimensional analog based on affine quantization is obtained by replacing hard spheres with hard disks in a bosonic fluid with

$$|\mathbf r_i-\mathbf r_j|=1\;\Rightarrow\; n_i n_j=0.$$7

which is a continuum thermodynamic model rather than a constrained lattice dynamics problem (Fantoni, 26 Nov 2025). Separately, “quantum hard disk” has been used metaphorically for passively protected quantum storage in Heisenberg ferromagnets (Ouyang, 2019) and for quantum-enabled single-atom magnetic storage based on quantum tunneling of magnetization (Forrester et al., 2019), while “quantum disk” in holography denotes a non-commutative space generated by $$|\mathbf r_i-\mathbf r_j|=1\;\Rightarrow\; n_i n_j=0.$$8 rather than an excluded-volume many-body model (Almheiri et al., 2024). The lattice model discussed above is therefore a specific many-body usage, distinguished by geometric exclusion, coherent hopping, fragmentation, and scar-mediated two-dimensional dynamics.