--- title: Quantum Hard Disk Model in Constrained Lattice Systems url: https://www.emergentmind.com/topics/quantum-hard-disk-model type: topic --- # Quantum Hard Disk Model in Constrained Lattice Systems 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 [2311.16240, 2503.13664]. ## 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: \[ H=J\sum_{\langle i,j\rangle} P_i\big(a_i^\dagger a_j+a_j^\dagger a_i\big)P_j, \] with \[ P_i=\prod_{j:\,|\mathbf r_i-\mathbf r_j|=1}\frac{1-n_j}{2}. \] Here \(P_i\) projects onto states in which all nearest neighbors of site \(i\) are empty, so \(P_i a_i^\dagger a_j P_j\) 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 [2311.16240]. The model also admits a spin-\(1/2\) representation, \[ a_i\mapsto S_i^- ,\qquad a_i^\dagger\mapsto S_i^+,\qquad n_i\mapsto S_i^z+\tfrac{1}{2}, \] so occupied sites correspond to spin up, or equivalently to Rydberg excitations. The construction generalizes to larger exclusion radius \(R\) through a finite-range exclusion graph, \[ P_i^{(R)}=\prod_{j\in\mathcal N_R(i)}\frac{1-n_j}{2}, \] with \[ \mathcal N_R(i)=\{j\mid 0<|\mathbf r_i-\mathbf r_j|\le R\}, \] and the same projected hopping structure. The paper focuses on \(R=1\) on open 1D chains and open 2D square lattices, but the structure extends straightforwardly to other lattice geometries and ranges [2311.16240]. ## 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 [2311.16240]. For the two-dimensional square lattice with nearest-neighbor exclusion, the crystalline order parameter is the staggered structure factor at wavevector \(\vec\pi=(\pi,\pi)\), \[ C_{\infty}(\eta)=\frac{1}{N}\left(\frac{2\pi}{L^d}\right)^2 \sum_{i,j} e^{i\vec\pi\cdot(\mathbf r_i-\mathbf r_j)}\mathrm{Tr}(n_i n_j), \] where the trace is over all configurations in the density sector \(\eta\), and \(N\) is the number of states in that sector. In this regime the model exhibits a checkerboard crystalline phase for \[ \eta>\eta_c\simeq 0.37. \] 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 [2311.16240]. The static classical-quantum equivalence is therefore limited. It holds at \(T=\infty\) 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 \(n_i n_{i+1}=0\). The dynamical problem studied most directly is the melting of a finite crystal droplet, \[ |\psi(0)\rangle = |0\cdots 0\,\underbrace{1\,0\,1\,0\,\dots}_{M\ \text{particles}}\,0\cdots 0\rangle, \] namely a contiguous block of maximally packed particles at density \(1/2\) embedded in empty space. Melting is monitored through the density profile \[ \langle n_i(t)\rangle=\langle\psi(t)|n_i|\psi(t)\rangle, \] and operationally means the spreading of the initially localized high-density region toward the uniform density \(\eta\) within the relevant sector [2311.16240]. 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, \[ \text{Width}(t)\sim v t. \] 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: \[ \text{Width}(t)\propto t^\alpha,\qquad \alpha<\tfrac{1}{2}. \] 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 [2311.16240]. ## 4. Two-dimensional defects, interfaces, and crystal memory On the open \(L\times L\) 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 [2311.16240]. This memory is quantified by the autocorrelation function \[ G(t)=\frac{1}{L^2}\sum_{i=1}^{L^2} \big\langle \big(2n_i(t)-1\big)\big(2n_i(0)-1\big)\big\rangle-G^*, \] with \[ G^*=(2\eta-1)^2. \] The subtraction ensures that \(G(t)\to 0\) if the system completely forgets the initial microscopic pattern and approaches a homogeneous density \(\eta\). In the quantum model, \(G(t)\) decays initially but saturates to a nonzero constant. In the classical model, it decays toward zero up to small finite-size corrections [2311.16240]. A later extension introduced short-range soft-core interactions, \[ H = J\sum_{\langle i,j\rangle}P_i\big(a_i^\dagger a_j+a_j^\dagger a_i\big)P_j + \lambda\sum_{\langle\langle i,j\rangle\rangle} n_i n_j, \] where \(\langle\langle i,j\rangle\rangle\) 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: \(G(t)\) decays somewhat but saturates to a substantial nonzero value that is weakly dependent on \(\lambda\) and on system size. By contrast, the corresponding classical constrained random walk yields \(G(t)\to 0\) in all three representative cases, including the diagonal one [2503.13664]. The same work distinguishes point-like defects from interfaces. Point-like defects, whose size does not scale with \(L\), typically live in strongly fragmented high-density sectors and remain frozen for arbitrarily long times. Interfaces, whose vacancy number scales linearly with \(L\), 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 [2503.13664]. ## 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 \(\eta=1/2\). 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 [2311.16240]. Two fragmentation regimes are identified. In the weak fragmentation regime, for densities \(\eta\gtrsim 1/L\) but below a high-density threshold, the Hilbert space splits into one large ergodic fragment plus many small fragments. In the strong fragmentation regime, \[ \eta \ge \frac{1}{2} - \frac{\lceil L/2\rceil}{L^2}, \] every configuration contains snakes or fully filled diagonals, and the Hilbert space breaks down into only small fragments, with no large ergodic component [2311.16240]. To diagnose quantum many-body scars, the eigenstates \(|\Psi\rangle\) are characterized by their bipartite entanglement entropy, \[ S_A=-\mathrm{Tr}\big[\rho_A\log\rho_A\big],\qquad \rho_A=\mathrm{Tr}_{\bar A}(|\Psi\rangle\langle\Psi|), \] and by a normalized Edwards-Anderson parameter. The unnormalized quantity is \[ Q_{\mathrm{EA}} = \frac{1}{L^4}\sum_{i,j} \big|\langle\Psi|(2n_i-1)(2n_j-1)|\Psi\rangle\big|^2, \] and the normalized form is \[ Q= \frac{Q_{\mathrm{EA}}-(2\eta-1)^4}{1-(2\eta-1)^4}. \] Here \(Q=1\) for a single configuration and \(Q\to 0\) for a fully delocalized eigenstate over all basis states [2311.16240]. The spectral structure differs across regimes. In the non-fragmented regime, \(S_A\) forms a single ETH-like entanglement dome and \(Q\) is small and featureless. In the strong fragmentation regime, almost all eigenstates have low entanglement and large \(Q\), 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-\(Q\) 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 [2311.16240]. The soft-core extension preserves this logic. Because the \(\lambda\)-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 \(\lambda\), but a finite population remains even up to relatively large \(\lambda=0.8\). This suggests that both fragmentation and scar physics survive a substantial range of perturbations [2503.13664]. ## 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 \(|g\rangle\) and Rydberg state \(|r\rangle\), identified with \(|0\rangle\) and \(|1\rangle\), so that \(n_i=|r_i\rangle\langle r_i|\). A generic driven Rydberg Hamiltonian takes the form \[ H_{\mathrm{Ryd}} = \sum_i\left[\frac{\Omega}{2}(\sigma_i^+ + \sigma_i^-)-\Delta n_i\right] + \sum_{i