Quantum Kinetically Constrained Models

• Quantum KCMs are many-body quantum systems where allowed transitions depend on local facilitation rules, enabling constrained hopping and spin-flip dynamics.
• They employ mechanisms like vacancy-assisted moves and chiral symmetry to produce Hilbert-space fragmentation, exponential zero-mode manifolds, and state-dependent mobility edges.
• These models exhibit rich transport phenomena, slow relaxation, and even fractional quantum Hall-like topological order without explicit density interactions.

Searching arXiv for recent and foundational papers on quantum kinetically constrained models. Quantum kinetically constrained models are many-body quantum systems in which local coherent or dissipative moves are permitted only when nearby occupations satisfy a facilitation rule. In one common formulation, all configurations in Fock space are allowed in principle, but transitions between configurations are constrained by local occupation patterns; in another, the Hamiltonian contains local facilitation constraints so that a degree of freedom is allowed to flip only if some neighboring configuration is satisfied (Kourtis et al., 2015, Marić et al., 3 Oct 2025). This class of models connects classical glassy KCMs, constrained Hamiltonian dynamics, and open-system Lindbladian engineering, and has become a setting for slow heterogeneous relaxation, Hilbert-space fragmentation, zero modes, state-dependent mobility edges, anomalous transport, and even fractional-quantum-Hall-like topological order without explicit density-density interactions (Olmos et al., 2014, Badbaria et al., 2024, Jonay et al., 29 Apr 2025).

1. Formal structure and defining mechanisms

A general constrained hopping formulation is

$$\hat{\mathcal H} = \sum_{\mathbf i\neq \mathbf j} \left( \hat c^\dagger_{\mathbf i}\,\hat F_{\mathbf{ij}}\,\hat c_{\mathbf j} +\text{H.c.} \right),$$

where $\hat F_{\mathbf{ij}}$ is an operator-valued function of local densities (Kourtis et al., 2015). If $\hat F_{\mathbf{ij}}=t_{\mathbf{ij}}$, the model reduces to an ordinary tight-binding Hamiltonian. If $\hat F_{\mathbf{ij}}$ is built from products of hole-density projectors $(1-\hat n_{\mathbf l})$, the hopping amplitude becomes conditional on the local environment, and interactions are encoded kinetically rather than as explicit density-density energy terms (Kourtis et al., 2015).

This distinction is sharp in the contrast between static exclusion and genuine kinetic constraints. In the quantum hard-hexagon limit, nearest-neighbor occupation is forbidden outright through projected operators such as

$$\tilde c^\dagger_{\mathbf i} = \hat c^\dagger_{\mathbf i} \prod_{\{\mathbf j:|\mathbf j-\mathbf i|=1\}} (1-\hat n_{\mathbf j}),$$

so some Fock-space configurations are removed entirely. In vacancy-assisted hopping, by contrast,

$$\hat F^{\mathrm{KC}}_{\mathbf{ij}} = t_{\mathbf{ij}} \prod_{\{\mathbf l:|\mathbf l-\mathbf i|=1 \wedge |\mathbf l-\mathbf j|=1\}} (1-\hat n_{\mathbf l}),$$

and all configurations remain allowed, while only the transitions between them are constrained (Kourtis et al., 2015). This is the defining quantum-KCM move from static repulsion to facilitated dynamics.

Spin-chain realizations express the same principle in a different language. The XPX model,

$H_{\rm XPX} = \sum_{j=1}^L \left[ \sigma^x_{j-1}(\mathds{1}-\sigma^z_j)\sigma^x_{j+1} + \Delta \sigma^z_j \right],$

implements a constraint in which a down spin facilitates simultaneous flips of its two neighbors (Marić et al., 3 Oct 2025). The quantum East model,

$$\hat H = -\frac12\sum_{j=0}^{N} \hat n_j\bigl(e^{-s}\hat\sigma^x_{j+1}-1\bigr), \qquad \hat n_j=\frac{1-\hat\sigma^z_j}{2},$$

is the unidirectional version: site $j+1$ can flip only if site $\hat F_{\mathbf{ij}}$0 is excited (Badbaria et al., 2024). Particle-conserving East models replace spin flips by facilitated hopping,

$\hat F_{\mathbf{ij}}$1

with $\hat F_{\mathbf{ij}}$2 built from projectors onto facilitating particles to the left (Brighi et al., 2022).

A frequent misconception is that constrained dynamics is equivalent to adding large static interactions. The literature distinguishes them carefully: strong interactions may motivate constraints, but in quantum KCMs the constraint acts directly on tunneling matrix elements or jump operators, and coherent many-body effects depend on that distinction (Kourtis et al., 2015, Maity et al., 2024).

2. Fragmentation, zero modes, and many-body localization in Fock space

In several quantum KCMs, chiral symmetry organizes the constrained Hilbert space into a bipartite graph. For $\hat F_{\mathbf{ij}}$3-conserving East and East-West chains, the Hamiltonian anticommutes with a chiral operator $\hat F_{\mathbf{ij}}$4, so in a suitable basis

$\hat F_{\mathbf{ij}}$5

and the zero-mode count is bounded below by the sublattice mismatch,

$\hat F_{\mathbf{ij}}$6

When the many-body Hilbert space further fragments into disconnected sectors $\hat F_{\mathbf{ij}}$7, the stronger bound

$\hat F_{\mathbf{ij}}$8

applies, and the number of zero modes can increase parametrically because the constraints split the graph into components with different local imbalances (Nicolau et al., 24 Apr 2025).

This mechanism is explicit in particle-conserving quantum East models. Their chiral, facilitated hopping yields exponentially many disconnected sectors, including a largest dynamically active sector and many smaller ones labeled by frozen patterns. The simultaneous presence of constraints and chiral symmetry then produces an exponentially large zero-mode manifold, while the inversion-symmetric East-West model, which fragments much less strongly, typically saturates the simpler mismatch bound without the same fragmentation enhancement (Brighi et al., 2022, Nicolau et al., 24 Apr 2025).

The resulting non-ergodicity is not exhausted by fragmentation. “Fock space cages” provide a distinct mechanism: exact many-body eigenstates localized on finite subgraphs of the Fock-space connectivity graph through destructive interference, even when the relevant Krylov sector is connected (Jonay et al., 29 Apr 2025). In that construction, Fock-space vertices are computational-basis states and edges are nonzero Hamiltonian matrix elements; zero-energy cage states are compactly supported superpositions whose outgoing amplitudes cancel exactly. This is the many-body analogue of flat-band localization and Aharonov-Bohm cages, but realized in configuration space rather than real space (Jonay et al., 29 Apr 2025).

A related development is the notion of collective bound states. In $\hat F_{\mathbf{ij}}$9 East and East-West chains, compact localized states of single-particle physics are generalized to many-body bound states that remain eigenstates when padded by sufficiently long strings of empty sites. These can be combined into factorizable eigenstates,

$\hat F_{\mathbf{ij}}=t_{\mathbf{ij}}$0

with $\hat F_{\mathbf{ij}}=t_{\mathbf{ij}}$1 acting as decoupling regions (Nicolau et al., 24 Apr 2025). In the East model, a large fraction of the zero-mode subspace is of this factorizable type, so area-law or strictly zero-entanglement eigenstates proliferate exponentially.

These results distinguish several non-ergodic mechanisms that are often conflated: exact Hilbert-space fragmentation, interference-driven localization on connected graphs, and symmetry-protected zero-mode manifolds. Quantum KCMs exhibit all three, but they need not coincide (Jonay et al., 29 Apr 2025, Nicolau et al., 24 Apr 2025).

3. Slow relaxation, metastability, and state-dependent mobility edges

Large-coupling expansions reveal a hierarchical form of constrained quantum dynamics. In the XPX model, a MacDonald-type strong-coupling expansion in $\hat F_{\mathbf{ij}}=t_{\mathbf{ij}}$2 defines truncated effective Hamiltonians $\hat F_{\mathbf{ij}}=t_{\mathbf{ij}}$3 and a nested sequence of frozen product-state sets,

$\hat F_{\mathbf{ij}}=t_{\mathbf{ij}}$4

For range-$\hat F_{\mathbf{ij}}=t_{\mathbf{ij}}$5 freezing, the defining geometric rule is that any two $\hat F_{\mathbf{ij}}=t_{\mathbf{ij}}$6-spins are separated by at least $\hat F_{\mathbf{ij}}=t_{\mathbf{ij}}$7 $\hat F_{\mathbf{ij}}=t_{\mathbf{ij}}$8-spins (Marić et al., 3 Oct 2025). A level-$\hat F_{\mathbf{ij}}=t_{\mathbf{ij}}$9 state remains frozen up to times $\hat F_{\mathbf{ij}}$0, so the minimal distance between facilitators directly controls the relaxation time scale.

This hierarchy appears in both observables and state complexity. Time-averaged Krylov complexity for level-$\hat F_{\mathbf{ij}}$1 states collapses when plotted against $\hat F_{\mathbf{ij}}$2, identifying $\hat F_{\mathbf{ij}}$3 as the onset of substantial Hilbert-space spreading (Marić et al., 3 Oct 2025). Local autocorrelations develop metastable plateaus whose height can be computed perturbatively; for frozen states,

$\hat F_{\mathbf{ij}}$4

so the plateau height is essentially the initial density of up spins (Marić et al., 3 Oct 2025).

The quantum East model displays a different, but related, non-ergodic structure. Rather than a sharp many-body mobility edge in energy, it exhibits a state-dependent mobility edge defined operationally by classical simulability: for many initial product states below an energy density $\hat F_{\mathbf{ij}}$5, the maximal MPS bond dimension grows polynomially in time, while above $\hat F_{\mathbf{ij}}$6 it grows exponentially (Badbaria et al., 2024). In the intermediate regime $\hat F_{\mathbf{ij}}$7, the crossover in the fraction of “easy” states becomes size-independent as system size increases (Badbaria et al., 2024).

The same work correlates this dynamical crossover with many non-thermal eigenstates at finite energy density. DMRG-X finds localized kink eigenstates with density profiles

$\hat F_{\mathbf{ij}}$8

small energy variance, and area-law entanglement. For product states made of clusters separated by $\hat F_{\mathbf{ij}}$9 zeros, the inter-cluster entangling time obeys $(1-\hat n_{\mathbf l})$0 for $(1-\hat n_{\mathbf l})$1, producing exponentially long separability windows (Badbaria et al., 2024). The model is therefore neither a standard MBL system nor a conventional scarred model: there is no quenched disorder and no sparse scar tower, but there are many non-thermal, low-entanglement eigenstates in a clean constrained Hamiltonian (Badbaria et al., 2024).

A second misconception follows. Slow relaxation in quantum KCMs does not require disorder, and it need not signal many-body localization. The mechanisms isolated in current work are constraints, facilitator geometry, fragmentation, and interference in Fock space (Marić et al., 3 Oct 2025, Badbaria et al., 2024).

4. Transport, hydrodynamics, and trajectory-space structure

Particle-conserving constrained chains provide a direct arena for transport. The quantum East-West model of hard-core bosons,

$(1-\hat n_{\mathbf l})$2

interpolates between East and West facilitation through $(1-\hat n_{\mathbf l})$3 (Brighi et al., 2024). For a left-domain-wall initial state, the model exhibits localization, ballistic transport, diffusion, and superdiffusion depending on $(1-\hat n_{\mathbf l})$4: $(1-\hat n_{\mathbf l})$5 gives ballistic spreading, $(1-\hat n_{\mathbf l})$6 gives diffusive spreading, and $(1-\hat n_{\mathbf l})$7 gives superdiffusive behavior with $(1-\hat n_{\mathbf l})$8 (Brighi et al., 2024). Yet near infinite temperature the same model appears diffusively scaling overall, while its density-gradient profile remains intrinsically skewed for any $(1-\hat n_{\mathbf l})$9, implying an asymmetric dynamical structure factor inconsistent with ordinary isotropic diffusion (Brighi et al., 2024).

Energy transport in the constrained PXP chain provides a parallel example. At infinite temperature, the energy-energy correlator first displays oscillatory dynamics tied to families of eigenstates forming different $$\tilde c^\dagger_{\mathbf i} = \hat c^\dagger_{\mathbf i} \prod_{\{\mathbf j:|\mathbf j-\mathbf i|=1\}} (1-\hat n_{\mathbf j}),$$0 representations hidden within the spectrum, then crosses over into a broad superdiffusive regime (Ljubotina et al., 2022). The late-time exponent drifts slowly in the bare model, consistent with proximity to a nearby integrable point, but strong chemical-potential deformations do not restore diffusion; instead they produce a stable superdiffusive exponent $$\tilde c^\dagger_{\mathbf i} = \hat c^\dagger_{\mathbf i} \prod_{\{\mathbf j:|\mathbf j-\mathbf i|=1\}} (1-\hat n_{\mathbf j}),$$1 and scaling collapse of the spatial energy profile (Ljubotina et al., 2022).

These unitary transport results fit into a wider trajectory-space picture inherited from KCM theory. Driven classical KCMs exhibit first-order space-time transitions between dynamical phases of finite and vanishing entropy production, diagnosed by large-deviation functions $$\tilde c^\dagger_{\mathbf i} = \hat c^\dagger_{\mathbf i} \prod_{\{\mathbf j:|\mathbf j-\mathbf i|=1\}} (1-\hat n_{\mathbf j}),$$2 of trajectory ensembles (Speck et al., 2010). Tensor-network treatments of tilted generators in FA- and East-type models show that the corresponding leading eigenstates are low-entangled and that finite-size scaling of spectral gaps and phase coexistence can be resolved well beyond exact diagonalization (Bañuls et al., 2019). The literature treats these constructions as a methodological bridge to constrained quantum jump problems, where tilted Liouvillians replace tilted classical generators (Bañuls et al., 2019).

The common lesson is that constrained many-body systems may look chaotic in level statistics yet retain highly nontrivial transport and trajectory-space organization. Quantum KCMs therefore require both operator dynamics and large-deviation diagnostics; neither by itself gives a complete characterization (Brighi et al., 2024, Ljubotina et al., 2022).

5. Dissipative constructions and emergent constraints

Quantum KCMs also arise in open systems. A purely dissipative construction uses site-resolved jump operators

$$\tilde c^\dagger_{\mathbf i} = \hat c^\dagger_{\mathbf i} \prod_{\{\mathbf j:|\mathbf j-\mathbf i|=1\}} (1-\hat n_{\mathbf j}),$$3

with classical KCM constraints $$\tilde c^\dagger_{\mathbf i} = \hat c^\dagger_{\mathbf i} \prod_{\{\mathbf j:|\mathbf j-\mathbf i|=1\}} (1-\hat n_{\mathbf j}),$$4, such as $$\tilde c^\dagger_{\mathbf i} = \hat c^\dagger_{\mathbf i} \prod_{\{\mathbf j:|\mathbf j-\mathbf i|=1\}} (1-\hat n_{\mathbf j}),$$5 and $$\tilde c^\dagger_{\mathbf i} = \hat c^\dagger_{\mathbf i} \prod_{\{\mathbf j:|\mathbf j-\mathbf i|=1\}} (1-\hat n_{\mathbf j}),$$6 (Olmos et al., 2014). These Lindbladians were built so that diagonal observables have the same stationary values as in the corresponding classical KCMs, but the quantum jumps generate coherences and richer relaxation. In the East case, the excitation density relaxes much faster than the coherences: $$\tilde c^\dagger_{\mathbf i} = \hat c^\dagger_{\mathbf i} \prod_{\{\mathbf j:|\mathbf j-\mathbf i|=1\}} (1-\hat n_{\mathbf j}),$$7 decays on a time scale roughly three orders of magnitude longer than the density relaxation time, while in the FA case diagonal and off-diagonal observables relax on comparable scales (Olmos et al., 2014).

A complementary route derives constraints emergently from strong dephasing. For open quantum spin systems with a diagonal Hamiltonian $$\tilde c^\dagger_{\mathbf i} = \hat c^\dagger_{\mathbf i} \prod_{\{\mathbf j:|\mathbf j-\mathbf i|=1\}} (1-\hat n_{\mathbf j}),$$8, coherent perturbation $$\tilde c^\dagger_{\mathbf i} = \hat c^\dagger_{\mathbf i} \prod_{\{\mathbf j:|\mathbf j-\mathbf i|=1\}} (1-\hat n_{\mathbf j}),$$9, and dephasing rate $$\hat F^{\mathrm{KC}}_{\mathbf{ij}} = t_{\mathbf{ij}} \prod_{\{\mathbf l:|\mathbf l-\mathbf i|=1 \wedge |\mathbf l-\mathbf j|=1\}} (1-\hat n_{\mathbf l}),$$0, adiabatic elimination yields an effective classical master equation with configuration-dependent rates

$$\hat F^{\mathrm{KC}}_{\mathbf{ij}} = t_{\mathbf{ij}} \prod_{\{\mathbf l:|\mathbf l-\mathbf i|=1 \wedge |\mathbf l-\mathbf j|=1\}} (1-\hat n_{\mathbf l}),$$1

where $$\hat F^{\mathrm{KC}}_{\mathbf{ij}} = t_{\mathbf{ij}} \prod_{\{\mathbf l:|\mathbf l-\mathbf i|=1 \wedge |\mathbf l-\mathbf j|=1\}} (1-\hat n_{\mathbf l}),$$2 is the energy cost of the local move (Everest et al., 2016). Resonant moves remain fast, off-resonant moves are suppressed, and the resulting dynamics acquires emergent kinetic constraints. The paper constructs a reaction-diffusion model with approximately conserved bond number, mobile polymers, immobile plaquettes, timescale separation, and dynamical reducibility, showing that collective constrained dynamics can be engineered from microscopic open-system ingredients (Everest et al., 2016).

A more explicitly quantum construction uses strong GKSL dissipation to generate decoherence-free subspaces in which constrained unitary dynamics survives. With Hermitian commuting jump operators $$\hat F^{\mathrm{KC}}_{\mathbf{ij}} = t_{\mathbf{ij}} \prod_{\{\mathbf l:|\mathbf l-\mathbf i|=1 \wedge |\mathbf l-\mathbf j|=1\}} (1-\hat n_{\mathbf l}),$$3, the leading-order effective Hamiltonian in a sector labeled by the full eigenvalue pattern $$\hat F^{\mathrm{KC}}_{\mathbf{ij}} = t_{\mathbf{ij}} \prod_{\{\mathbf l:|\mathbf l-\mathbf i|=1 \wedge |\mathbf l-\mathbf j|=1\}} (1-\hat n_{\mathbf l}),$$4 is

$$\hat F^{\mathrm{KC}}_{\mathbf{ij}} = t_{\mathbf{ij}} \prod_{\{\mathbf l:|\mathbf l-\mathbf i|=1 \wedge |\mathbf l-\mathbf j|=1\}} (1-\hat n_{\mathbf l}),$$5

where $$\hat F^{\mathrm{KC}}_{\mathbf{ij}} = t_{\mathbf{ij}} \prod_{\{\mathbf l:|\mathbf l-\mathbf i|=1 \wedge |\mathbf l-\mathbf j|=1\}} (1-\hat n_{\mathbf l}),$$6 projects onto the corresponding decoherence-free subspace (Maity et al., 2024). Because the sector is fixed by the entire vector $$\hat F^{\mathrm{KC}}_{\mathbf{ij}} = t_{\mathbf{ij}} \prod_{\{\mathbf l:|\mathbf l-\mathbf i|=1 \wedge |\mathbf l-\mathbf j|=1\}} (1-\hat n_{\mathbf l}),$$7, not only by $$\hat F^{\mathrm{KC}}_{\mathbf{ij}} = t_{\mathbf{ij}} \prod_{\{\mathbf l:|\mathbf l-\mathbf i|=1 \wedge |\mathbf l-\mathbf j|=1\}} (1-\hat n_{\mathbf l}),$$8, the resulting unitary dynamics is more tightly constrained than in the analogous strong-interaction Hamiltonian construction (Maity et al., 2024).

The concrete example is the PXQ chain generated by two-site dissipation $$\hat F^{\mathrm{KC}}_{\mathbf{ij}} = t_{\mathbf{ij}} \prod_{\{\mathbf l:|\mathbf l-\mathbf i|=1 \wedge |\mathbf l-\mathbf j|=1\}} (1-\hat n_{\mathbf l}),$$9, with effective Hamiltonian

$H_{\rm XPX} = \sum_{j=1}^L \left[ \sigma^x_{j-1}(\mathds{1}-\sigma^z_j)\sigma^x_{j+1} + \Delta \sigma^z_j \right],$0

Its decoherence-free sectors contain frozen “10” blocks, and the unfrozen sector maps to free domain-wall motion. Under a uniform magnetic field, the effective domain-wall problem becomes a tilted tight-binding chain and exhibits Wannier-Stark-like localization; when two such PXQ chains are coupled by an inter-chain interaction, localization persists generically but partial delocalization appears along the special line $H_{\rm XPX} = \sum_{j=1}^L \left[ \sigma^x_{j-1}(\mathds{1}-\sigma^z_j)\sigma^x_{j+1} + \Delta \sigma^z_j \right],$1 (Maity et al., 2024).

These dissipative results establish that quantum KCMs are not restricted to closed Hamiltonian settings. They can be built directly from jump operators, or they can emerge as slow coherent dynamics inside noise-protected subspaces (Olmos et al., 2014, Everest et al., 2016, Maity et al., 2024).

6. Two-dimensional topological order and the scope of quantum KCMs

Quantum KCMs are not confined to glassiness or localization-like phenomena. A two-dimensional constrained hopping construction on topological flat bands shows that kinetic constraints alone can stabilize fractional-quantum-Hall-like order in lattices (Kourtis et al., 2015). The models use spinless fermions in triangular- and kagome-lattice Chern bands with lowest-band Chern number $H_{\rm XPX} = \sum_{j=1}^L \left[ \sigma^x_{j-1}(\mathds{1}-\sigma^z_j)\sigma^x_{j+1} + \Delta \sigma^z_j \right],$2, and replace unconstrained hopping by vacancy-assisted hopping of the form

$H_{\rm XPX} = \sum_{j=1}^L \left[ \sigma^x_{j-1}(\mathds{1}-\sigma^z_j)\sigma^x_{j+1} + \Delta \sigma^z_j \right],$3

so that a hop is allowed only when all common neighbors of the initial and final sites are empty (Kourtis et al., 2015).

At densities $H_{\rm XPX} = \sum_{j=1}^L \left[ \sigma^x_{j-1}(\mathds{1}-\sigma^z_j)\sigma^x_{j+1} + \Delta \sigma^z_j \right],$4 on the triangular lattice and $H_{\rm XPX} = \sum_{j=1}^L \left[ \sigma^x_{j-1}(\mathds{1}-\sigma^z_j)\sigma^x_{j+1} + \Delta \sigma^z_j \right],$5 on the kagome lattice, corresponding to $H_{\rm XPX} = \sum_{j=1}^L \left[ \sigma^x_{j-1}(\mathds{1}-\sigma^z_j)\sigma^x_{j+1} + \Delta \sigma^z_j \right],$6 filling of the lowest Chern band, exact diagonalization on finite tori finds the standard signatures of a Laughlin-like phase: three low-lying states, spectral flow under twisted boundary conditions, smooth many-body Berry curvature on the flux torus, and quantized Hall conductivity

$H_{\rm XPX} = \sum_{j=1}^L \left[ \sigma^x_{j-1}(\mathds{1}-\sigma^z_j)\sigma^x_{j+1} + \Delta \sigma^z_j \right],$7

The vacancy-assisted models show the same qualitative spectral flow as their hard-core counterparts, although with somewhat smaller gaps (Kourtis et al., 2015).

This result is conceptually important because the Hamiltonians contain no explicit density-density interaction term of the form $H_{\rm XPX} = \sum_{j=1}^L \left[ \sigma^x_{j-1}(\mathds{1}-\sigma^z_j)\sigma^x_{j+1} + \Delta \sigma^z_j \right],$8. The correlations needed for a fractional Chern insulator are generated by the combination of nontrivial band topology and kinetic constraints (Kourtis et al., 2015). A plausible implication is that the design space of quantum KCMs extends well beyond slow dynamics and non-ergodicity, into interaction-free routes to topological order.

The present literature therefore supports a broad but internally differentiated picture. Quantum KCMs include constrained hopping models, facilitated spin chains, dissipatively generated effective Hamiltonians, and open-system jump processes. Their characteristic phenomena include hierarchical freezing, fragmentation, collective bound states, Fock-space cages, state-dependent mobility edges, anomalous transport, and topological order (Jonay et al., 29 Apr 2025, Nicolau et al., 24 Apr 2025, Kourtis et al., 2015). At the same time, several issues remain open: thermodynamic confirmation of some topological phases, full classification of zero-mode subspaces and bound states, and the stability of constrained non-ergodic structures under generic perturbations (Kourtis et al., 2015, Jonay et al., 29 Apr 2025).