Disorder-Free Localization in Quantum Systems

• Disorder-free localization is a phenomenon in translationally invariant quantum systems where internal constraints generate effective disorder and hinder transport.
• It arises via emergent static fields, superselection sectors, and constrained dynamics in models including lattice gauge theories and coupled spin chains.
• The concept challenges traditional localization paradigms, influencing ergodicity breaking, entanglement growth, and experimental designs in quantum simulations.

Searching arXiv for recent and foundational papers on disorder-free localization. Disorder-free localization denotes a class of nonergodic phenomena in translationally invariant quantum systems whose Hamiltonians contain no externally imposed quenched randomness, yet whose dynamics exhibits localization-like suppression of transport and persistent memory of initial conditions. Across the modern literature, the term covers several related but distinct mechanisms: localization generated by emergent static backgrounds associated with exact local conserved quantities in gauge and gauge-like models (Smith et al., 2017); nonergodicity in lattice gauge theories arising from superpositions over many gauge superselection sectors (Karpov et al., 2020, Osborne et al., 2023, Chakraborty et al., 2022); finite-temperature sector averaging that makes temperature itself act as an effective disorder strength (Halimeh et al., 2022); and interaction-generated quasiperiodic textures in clean coupled subsystems that can produce robust many-body localization (MBL)-like behavior without external disorder (Gunawardana et al., 2024). The unifying feature is that the inhomogeneity relevant for localization is generated internally—through conserved charges, sector structure, constrained dynamics, or incommensurate coupling geometry—rather than inserted by hand as a random field.

1. Concept and scope

In the conventional Anderson and disorder-induced MBL paradigms, localization is tied to quenched disorder in the Hamiltonian. Disorder-free localization replaces this ingredient by internally generated static or quasi-static structure. In the earliest canonical construction, a translationally invariant spinless-fermion model coupled to bond spins admits an extensive family of conserved $\mathbb Z_2$ charges; after duality, each charge sector becomes free fermions in a static binary onsite potential (Smith et al., 2017). This established that localization can arise in a clean microscopic Hamiltonian through dynamically generated disorder encoded in constants of motion.

Subsequent work broadened the notion. In $\mathbb Z_2$ and $U(1)$ lattice gauge theories, local Gauss-law generators partition Hilbert space into superselection sectors labeled by background charges. If a homogeneous initial state spans exponentially many such sectors, observables become sector averages over different static charge configurations, producing disorder-free localization in the quench dynamics (Karpov et al., 2020, Osborne et al., 2023). In other settings, the effective inhomogeneity is not charge-sector disorder but emergent kinetic structure. The two-leg compass ladder and related plaquette Ising model map to free fermions coupled to localized $\mathbb Z_2$ variables, yielding finite-temperature disorder-free localization with logarithmic entanglement growth from sector superposition (Hart et al., 2020). Clean dipolar lattice gases can exhibit disorder-free quasi-localization through Hilbert-space shattering induced by long-range interactions (Li et al., 2021). A recent spin-chain realization replaces conserved charges by deterministic nonperiodic coupling textures: two incommensurate Heisenberg chains coupled quasiperiodically can undergo a disorder-free MBL transition because the interchain exchange profile itself becomes quasiperiodic and self-generated (Gunawardana et al., 2024).

This broader usage has made the term partly umbrella-like. In some models, the localized degrees of freedom are effectively noninteracting within each sector and the phenomenon is best viewed as emergent Anderson localization (Smith et al., 2017, Yang et al., 2019). In others, especially gauge theories and interacting spin systems, the emphasis is on ergodicity breaking, slow entanglement growth, and MBL-like phenomenology without quenched disorder (Osborne et al., 2023, Gunawardana et al., 2024). A recurring controversy is therefore whether a given instance realizes true asymptotic localization or only long-lived prethermal or quasi-localized behavior.

2. Sector-generated effective disorder

A central mechanism of disorder-free localization is the emergence of static disorder from exact local conservation laws. In the translationally invariant spin-fermion chain introduced in "Disorder-Free Localization" (Smith et al., 2017), the Hamiltonian is

$$\hat{H} = -J\sum_{\langle ij\rangle} \hat{\sigma}^z_{i,j} \hat{f}^\dag_{i} \hat{f}_{j} - h \sum_{i} \hat{\sigma}^x_{i-1, i} \hat{\sigma}^x_{i,i+1}.$$

A duality transformation yields conserved charges

$$\hat q_j \equiv \hat \tau_j^z (-1)^{\hat n_j}, \qquad [\hat q_i,\hat q_j]=0, \qquad [\hat q_j,\hat H]=0,$$

and dressed fermions $\hat c_j=\hat \tau_j^x \hat f_j$. In each fixed sector $\{q_j\}$, the Hamiltonian reduces to

$$\hat{H}_{\{q_j\}} = -J\sum_{\langle ij\rangle} \hat{c}^\dag_i \hat{c}_{j} + 2h\sum_{j} q_j \left(\hat{c}^\dag_j \hat{c}_j-\frac12\right),$$

namely free fermions in a static binary onsite potential (Smith et al., 2017). The clean initial spin-polarized state becomes an equal-weight superposition over all charge sectors, so physical observables are averages over all binary-disorder realizations. Localization is therefore exact in each sector, while the full microscopic model remains translation invariant.

Closely related mechanisms appear in other models with exact local moments or gauge charges. In the half-filled Ising-Kondo lattice on the square lattice,

$$\hat{H}=-t\sum_{\langle i,j\rangle,\sigma}\hat{c}_{i\sigma}^{\dag}\hat{c}_{j\sigma} +\frac{J}{2}\sum_{j\sigma}\hat{S}_{j}^{z} \hat{c}_{j\sigma}^{\dag}\hat{\sigma^z}\hat{c}_{j\sigma},$$

the conserved moments $\mathbb Z_2$0 render each sector a free-electron problem in a static binary field,

$\mathbb Z_2$1

producing what is termed “hidden Anderson localization” (Yang et al., 2019). The paper identifies an intermediate-coupling, high-temperature Anderson-localized regime between Fermi-liquid and Mott-insulating behavior, diagnosed through density of states, inverse participation ratio, and dc resistance (Yang et al., 2019).

The same structural reduction underlies later $\mathbb Z_2$2 gauge-theory work. A one-dimensional $\mathbb Z_2$3 lattice gauge model with fermions and link spins has conserved local charges

$\mathbb Z_2$4

and after duality becomes

$\mathbb Z_2$5

again a free-fermion problem in binary onsite disorder (Yang et al., 8 Sep 2025). In that setting, the localization mechanism is standard sector-generated emergent disorder, later used as a basis for studying open-system stabilization.

A related Floquet realization occurs in a discrete-time quantum walk coupled to immobile on-site spins. Because $\mathbb Z_2$6 for each site, the spin $\mathbb Z_2$7-components are exact local constants of motion. In each fixed $\mathbb Z_2$8 sector, the walker undergoes a quantum walk in a static binary phase landscape,

$\mathbb Z_2$9

and a homogeneous $U(1)$0-polarized spin product state induces equal-weight averaging over all such sectors (Danacı et al., 2020). Strong coupling $U(1)$1 yields exponential localization of the walker without any quenched disorder (Danacı et al., 2020).

3. Gauge-theory realizations

Lattice gauge theories have become a principal arena for disorder-free localization because Gauss-law generators furnish exact local conserved quantities and naturally define superselection sectors. In one dimension, disorder-free localization was established in $U(1)$2 and $U(1)$3 models with quenches from homogeneous initial states spanning many sectors (Smith et al., 2017, Papaefstathiou et al., 2020). In the modified $U(1)$4D $U(1)$5 lattice gauge theory of (Papaefstathiou et al., 2020), the staggered-fermion discretization reduces, sector by sector, to free fermions in an emergent static potential, producing persistent density imbalance and arrested correlation spreading after a global quench.

In two dimensions, the situation is more subtle because standard disorder-induced MBL is widely expected to be unstable. A major result of "Disorder-free localization in an interacting two-dimensional lattice gauge theory" (Karpov et al., 2020) is that a homogeneous, genuinely interacting 2D $U(1)$6 quantum link model can nevertheless exhibit nonergodic dynamics due to gauge constraints. The model Hamiltonian,

$U(1)$7

with Gauss-law generators

$U(1)$8

decomposes into sectors labeled by static charges $U(1)$9 (Karpov et al., 2020). Starting from a homogeneous superposition over sectors, expectation values become effective averages over charge patterns, while the charges constrain the set of flippable plaquettes. The authors derive a rigorous lower bound on the localization–delocalization transition from a correlated percolation problem and diagnose the localized regime through strongly bent light cones and finite-range spreading of line defects (Karpov et al., 2020).

A more comprehensive characterization of the transition in the same 2D $\mathbb Z_2$0 quantum link model is provided in (Chakraborty et al., 2022). There, the authors distinguish two mechanisms. First, the static charge pattern fragments real space into connected clusters of flippable plaquettes, governed by a classical percolation transition at

$\mathbb Z_2$1

Second, even on finite connected clusters, large $\mathbb Z_2$2 can induce effective nonergodicity through interference in configuration space. In the regime where large clusters are ergodic, the quantum localization transition coincides with the classical percolation transition and inherits the standard 2D site-percolation exponents

$\mathbb Z_2$3

(Chakraborty et al., 2022). This is one of the clearest cases in which a disorder-free localization transition in a 2D interacting model is linked to a controlled universality class.

The inclusion of dynamical matter in $\mathbb Z_2$4D was studied in the spin-$\mathbb Z_2$5 $\mathbb Z_2$6 quantum link model on a cylinder (Osborne et al., 2023). There, homogeneous product states with gauge links prepared in local $\mathbb Z_2$7-ground states span an extensive number of gauge sectors and display a finite late-time imbalance, whereas corresponding states in a single gauge sector thermalize (Osborne et al., 2023). The strength of localization depends strongly on the “propagation directionality” $\mathbb Z_2$8 of the initial matter pattern: smaller $\mathbb Z_2$9 yields stronger disorder-free localization, and configuration III with $$\hat{H} = -J\sum_{\langle ij\rangle} \hat{\sigma}^z_{i,j} \hat{f}^\dag_{i} \hat{f}_{j} - h \sum_{i} \hat{\sigma}^x_{i-1, i} \hat{\sigma}^x_{i,i+1}.$$0 exhibits comparable localization strength in $$\hat{H} = -J\sum_{\langle ij\rangle} \hat{\sigma}^z_{i,j} \hat{f}^\dag_{i} \hat{f}_{j} - h \sum_{i} \hat{\sigma}^x_{i-1, i} \hat{\sigma}^x_{i,i+1}.$$1D and $$\hat{H} = -J\sum_{\langle ij\rangle} \hat{\sigma}^z_{i,j} \hat{f}^\dag_{i} \hat{f}_{j} - h \sum_{i} \hat{\sigma}^x_{i-1, i} \hat{\sigma}^x_{i,i+1}.$$2D (Osborne et al., 2023). This suggests that the weakening of disorder-free localization in higher dimensions is not purely dimensional, but mediated by how many propagation channels the initial state activates.

The lattice Schwinger model offers a different perspective. In (Jeyaretnam et al., 2024), the strong-coupling regime of the one-dimensional Schwinger model is analyzed using exact diagonalization, matrix product states, and degenerate perturbation theory. The authors argue that the observed disorder-free localization phenomenology is best understood not as conventional MBL, nor primarily as confinement, but as a single ergodicity-breaking regime generated by approximate Hilbert-space fragmentation. In fixed background-charge sectors, an emergent dynamical constraint on hopping causes fragmented Krylov subspaces and sharp jumps in configurational entropy. Although the model exhibits Poisson-like spectral statistics, slow entanglement growth, and area-law-like eigenstates on accessible sizes, the paper explicitly avoids claiming a stable asymptotic MBL phase at finite $$\hat{H} = -J\sum_{\langle ij\rangle} \hat{\sigma}^z_{i,j} \hat{f}^\dag_{i} \hat{f}_{j} - h \sum_{i} \hat{\sigma}^x_{i-1, i} \hat{\sigma}^x_{i,i+1}.$$3 (Jeyaretnam et al., 2024).

4. Fragmentation, shattering, and classicality

Not all disorder-free localization mechanisms are reducible to static charge-sector disorder. Several works emphasize constrained dynamics and Hilbert-space fragmentation. The cleanest example is the dipolar hard-core boson chain of (Li et al., 2021), where the Hamiltonian

$$\hat{H} = -J\sum_{\langle ij\rangle} \hat{\sigma}^z_{i,j} \hat{f}^\dag_{i} \hat{f}_{j} - h \sum_{i} \hat{\sigma}^x_{i-1, i} \hat{\sigma}^x_{i,i+1}.$$4

is studied mainly at $$\hat{H} = -J\sum_{\langle ij\rangle} \hat{\sigma}^z_{i,j} \hat{f}^\dag_{i} \hat{f}_{j} - h \sum_{i} \hat{\sigma}^x_{i-1, i} \hat{\sigma}^x_{i,i+1}.$$5. Strong nearest-neighbor interactions alone conserve the number of nearest-neighbor bonds $$\hat{H} = -J\sum_{\langle ij\rangle} \hat{\sigma}^z_{i,j} \hat{f}^\dag_{i} \hat{f}_{j} - h \sum_{i} \hat{\sigma}^x_{i-1, i} \hat{\sigma}^x_{i,i+1}.$$6 and fragment Hilbert space, but typical states still delocalize within fragments. The $$\hat{H} = -J\sum_{\langle ij\rangle} \hat{\sigma}^z_{i,j} \hat{f}^\dag_{i} \hat{f}_{j} - h \sum_{i} \hat{\sigma}^x_{i-1, i} \hat{\sigma}^x_{i,i+1}.$$7 dipolar tail generates additional emergent conservation laws, beginning with $$\hat{H} = -J\sum_{\langle ij\rangle} \hat{\sigma}^z_{i,j} \hat{f}^\dag_{i} \hat{f}_{j} - h \sum_{i} \hat{\sigma}^x_{i-1, i} \hat{\sigma}^x_{i,i+1}.$$8, which further constrain resonant motion and “shatter” the fragmented Hilbert-space blocks (Li et al., 2021). The result is dramatically slowed dynamics, finite density-wave memory, and effective disorder-free localization on experimentally relevant timescales, though the paper stops short of asserting strict asymptotic clean-system MBL (Li et al., 2021).

A more radical claim is advanced in "Disorder-Free Localization as a Purely Classical Effect" (Sala et al., 2023). For the finite-$$\hat{H} = -J\sum_{\langle ij\rangle} \hat{\sigma}^z_{i,j} \hat{f}^\dag_{i} \hat{f}_{j} - h \sum_{i} \hat{\sigma}^x_{i-1, i} \hat{\sigma}^x_{i,i+1}.$$9 $$\hat q_j \equiv \hat \tau_j^z (-1)^{\hat n_j}, \qquad [\hat q_i,\hat q_j]=0, \qquad [\hat q_j,\hat H]=0,$$0D $$\hat q_j \equiv \hat \tau_j^z (-1)^{\hat n_j}, \qquad [\hat q_i,\hat q_j]=0, \qquad [\hat q_j,\hat H]=0,$$1 quantum link model,

$$\hat q_j \equiv \hat \tau_j^z (-1)^{\hat n_j}, \qquad [\hat q_i,\hat q_j]=0, \qquad [\hat q_j,\hat H]=0,$$2

the authors construct a classical cellular automaton preserving the same Gauss-law structure,

$$\hat q_j \equiv \hat \tau_j^z (-1)^{\hat n_j}, \qquad [\hat q_i,\hat q_j]=0, \qquad [\hat q_j,\hat H]=0,$$3

and show that it reproduces the persistent memory and exponentially localized correlation profiles of the quantum model at finite $$\hat q_j \equiv \hat \tau_j^z (-1)^{\hat n_j}, \qquad [\hat q_i,\hat q_j]=0, \qquad [\hat q_j,\hat H]=0,$$4 (Sala et al., 2023). Their conclusion is that, in this regularized QLM, disorder-free localization can persist in the thermodynamic limit as a purely classical effect of the finite-dimensional gauge-field truncation. Quantum interference is not necessary for the phenomenon, though finite $$\hat q_j \equiv \hat \tau_j^z (-1)^{\hat n_j}, \qquad [\hat q_i,\hat q_j]=0, \qquad [\hat q_j,\hat H]=0,$$5 can enhance it (Sala et al., 2023). This claim narrows the scope of “disorder-free localization” in such models and complicates attempts to interpret all gauge-theory nonergodicity as intrinsically quantum.

The compass-ladder realization (Hart et al., 2020) occupies an intermediate position. The two-leg orbital compass ladder maps onto free fermions moving in bond-disordered transverse-field Ising chains, where the disorder is generated by conserved rung charges

$$\hat q_j \equiv \hat \tau_j^z (-1)^{\hat n_j}, \qquad [\hat q_i,\hat q_j]=0, \qquad [\hat q_j,\hat H]=0,$$6

At finite temperature or for generic product states spanning many sectors, the model shows logarithmic entanglement growth

$$\hat q_j \equiv \hat \tau_j^z (-1)^{\hat n_j}, \qquad [\hat q_i,\hat q_j]=0, \qquad [\hat q_j,\hat H]=0,$$7

and power-law dynamical autocorrelations despite being free-fermionic within each sector (Hart et al., 2020). The work therefore illustrates how MBL-like diagnostics can emerge from sector superposition and dephasing without fully interacting disorder-induced MBL.

5. Temperature, dissipation, and stabilization

Because disorder-free localization often relies on sector structure rather than literal randomness, its dependence on temperature and dissipation can differ sharply from standard intuition. In "Temperature-Induced Disorder-Free Localization" (Halimeh et al., 2022), the initial state is not a coherent superposition of sectors but a thermal Gibbs ensemble,

$$\hat q_j \equiv \hat \tau_j^z (-1)^{\hat n_j}, \qquad [\hat q_i,\hat q_j]=0, \qquad [\hat q_j,\hat H]=0,$$8

block diagonal in the gauge-sector basis. Since the post-quench Hamiltonian preserves the generators, the time-evolved state remains a weighted sum of independently evolving sectors (Halimeh et al., 2022). At low temperature, the ensemble is concentrated in a single homogeneous sector and the system thermalizes. At higher temperature, many sectors acquire appreciable weight, broadening the effective background-charge distribution. Temperature thereby acts as an effective disorder strength. Exact diagonalization in both $$\hat q_j \equiv \hat \tau_j^z (-1)^{\hat n_j}, \qquad [\hat q_i,\hat q_j]=0, \qquad [\hat q_j,\hat H]=0,$$9 and $\hat c_j=\hat \tau_j^x \hat f_j$0 lattice gauge theories shows greater long-time imbalance with increasing temperature, and the effect scales with the ratio $\hat c_j=\hat \tau_j^x \hat f_j$1, where $\hat c_j=\hat \tau_j^x \hat f_j$2 is the preparation gap (Halimeh et al., 2022). This explicitly demonstrates that finite-temperature mixed states can induce, rather than suppress, disorder-free localization.

The fragility of gauge-theory disorder-free localization to gauge-breaking errors has motivated stabilization schemes. In (Halimeh et al., 2021), a translation-invariant protection term linear in the gauge generators,

$\hat c_j=\hat \tau_j^x \hat f_j$3

induces a quantum Zeno effect that suppresses sector mixing and restores disorder-free localization for times at least polynomial in $\hat c_j=\hat \tau_j^x \hat f_j$4, with numerically observed stabilization up to

$\hat c_j=\hat \tau_j^x \hat f_j$5

(Halimeh et al., 2021). The effective Zeno Hamiltonian is

$\hat c_j=\hat \tau_j^x \hat f_j$6

an emergent renormalized gauge theory that again preserves the sector structure (Halimeh et al., 2021).

A more elaborate variant is Stark gauge protection (Lang et al., 2022), where a linear gradient is applied not to matter density but to the local generators or pseudogenerators: $\hat c_j=\hat \tau_j^x \hat f_j$7 in the $\hat c_j=\hat \tau_j^x \hat f_j$8 case. A Magnus expansion shows that gauge-breaking terms are then locally suppressed by denominators of the form $\hat c_j=\hat \tau_j^x \hat f_j$9, and exact diagonalization plus Krylov evolution indicate stabilization or even enhancement of disorder-free localization up to all accessible times (Lang et al., 2022). Because this scheme does not localize gauge-invariant single-sector initial states, the resulting nonergodicity is interpreted as protected disorder-free localization rather than bona fide Stark MBL (Lang et al., 2022).

The $\{q_j\}$0 analog goes further. "Enhancing disorder-free localization through dynamically emergent local symmetries" (Halimeh et al., 2021) uses a local pseudogenerator $\{q_j\}$1, simpler than the exact $\{q_j\}$2, together with a translation-invariant penalty

$\{q_j\}$3

In the large-$\{q_j\}$4 limit, the quantum Zeno effect yields a renormalized gauge theory with an enlarged local symmetry associated with $\{q_j\}$5, generating more superselection sectors than the original model and thus stronger effective disorder (Halimeh et al., 2021). Disorder-free localization is therefore not merely protected but enhanced.

Open-system effects can also reinforce localization. In the 2025 work (Yang et al., 8 Sep 2025), a one-dimensional $\{q_j\}$6 lattice gauge model coupled to a Markovian bath via pair-jump operators

$\{q_j\}$7

retains higher fidelity with its initial $\{q_j\}$8-type density-wave states under dissipative evolution than under unitary dynamics (Yang et al., 8 Sep 2025). The steady-state density matrix develops sharp peaks on the initial pattern and its translated copies, indicating that carefully structured dissipation can stabilize a nonthermal manifold rather than wash it out.

6. Coupled clean subsystems and disorder-free MBL

A qualitatively different route to disorder-free localization appears in coupled clean many-body systems whose mutual interaction creates deterministic but nonperiodic textures. The most explicit recent example is "Disorder free many-body localization transition in two quasiperiodically coupled Heisenberg spin chains" (Gunawardana et al., 2024). The model consists of two spin-$\{q_j\}$9 isotropic Heisenberg chains,

$$\hat{H}_{\{q_j\}} = -J\sum_{\langle ij\rangle} \hat{c}^\dag_i \hat{c}_{j} + 2h\sum_{j} q_j \left(\hat{c}^\dag_j \hat{c}_j-\frac12\right),$$0

coupled by an $$\hat{H}_{\{q_j\}} = -J\sum_{\langle ij\rangle} \hat{c}^\dag_i \hat{c}_{j} + 2h\sum_{j} q_j \left(\hat{c}^\dag_j \hat{c}_j-\frac12\right),$$1-type exchange

$$\hat{H}_{\{q_j\}} = -J\sum_{\langle ij\rangle} \hat{c}^\dag_i \hat{c}_{j} + 2h\sum_{j} q_j \left(\hat{c}^\dag_j \hat{c}_j-\frac12\right),$$2

with exponentially decaying interchain couplings

$$\hat{H}_{\{q_j\}} = -J\sum_{\langle ij\rangle} \hat{c}^\dag_i \hat{c}_{j} + 2h\sum_{j} q_j \left(\hat{c}^\dag_j \hat{c}_j-\frac12\right),$$3

Because the two chains have incommensurate lattice constants,

$$\hat{H}_{\{q_j\}} = -J\sum_{\langle ij\rangle} \hat{c}^\dag_i \hat{c}_{j} + 2h\sum_{j} q_j \left(\hat{c}^\dag_j \hat{c}_j-\frac12\right),$$4

the interchain distances and hence the exchange profile vary quasiperiodically along the ladder (Gunawardana et al., 2024). Each clean chain therefore sees the other through a deterministic but nonperiodic interaction texture, which acts as self-generated disorder. Unlike Aubry–André-type quasiperiodic MBL, no external quasiperiodic onsite potential is imposed; the modulation arises from geometry and mutual coupling.

The authors combine exact diagonalization, TEBD, and DMRG to study entanglement growth and the inverse participation ratio (IPR). The entanglement entropy is

$$\hat{H}_{\{q_j\}} = -J\sum_{\langle ij\rangle} \hat{c}^\dag_i \hat{c}_{j} + 2h\sum_{j} q_j \left(\hat{c}^\dag_j \hat{c}_j-\frac12\right),$$5

while the infinite-temperature average IPR is

$$\hat{H}_{\{q_j\}} = -J\sum_{\langle ij\rangle} \hat{c}^\dag_i \hat{c}_{j} + 2h\sum_{j} q_j \left(\hat{c}^\dag_j \hat{c}_j-\frac12\right),$$6

In the thermal phase, $$\hat{H}_{\{q_j\}} = -J\sum_{\langle ij\rangle} \hat{c}^\dag_i \hat{c}_{j} + 2h\sum_{j} q_j \left(\hat{c}^\dag_j \hat{c}_j-\frac12\right),$$7 grows quickly and saturates to a volume-law value. In one example at $$\hat{H}_{\{q_j\}} = -J\sum_{\langle ij\rangle} \hat{c}^\dag_i \hat{c}_{j} + 2h\sum_{j} q_j \left(\hat{c}^\dag_j \hat{c}_j-\frac12\right),$$8 and $$\hat{H}_{\{q_j\}} = -J\sum_{\langle ij\rangle} \hat{c}^\dag_i \hat{c}_{j} + 2h\sum_{j} q_j \left(\hat{c}^\dag_j \hat{c}_j-\frac12\right),$$9, the saturated entropy scales approximately linearly with slope $$\hat{H}=-t\sum_{\langle i,j\rangle,\sigma}\hat{c}_{i\sigma}^{\dag}\hat{c}_{j\sigma} +\frac{J}{2}\sum_{j\sigma}\hat{S}_{j}^{z} \hat{c}_{j\sigma}^{\dag}\hat{\sigma^z}\hat{c}_{j\sigma},$$0, close to the maximal $$\hat{H}=-t\sum_{\langle i,j\rangle,\sigma}\hat{c}_{i\sigma}^{\dag}\hat{c}_{j\sigma} +\frac{J}{2}\sum_{j\sigma}\hat{S}_{j}^{z} \hat{c}_{j\sigma}^{\dag}\hat{\sigma^z}\hat{c}_{j\sigma},$$1 (Gunawardana et al., 2024). In the localized regime, e.g. $$\hat{H}=-t\sum_{\langle i,j\rangle,\sigma}\hat{c}_{i\sigma}^{\dag}\hat{c}_{j\sigma} +\frac{J}{2}\sum_{j\sigma}\hat{S}_{j}^{z} \hat{c}_{j\sigma}^{\dag}\hat{\sigma^z}\hat{c}_{j\sigma},$$2 and $$\hat{H}=-t\sum_{\langle i,j\rangle,\sigma}\hat{c}_{i\sigma}^{\dag}\hat{c}_{j\sigma} +\frac{J}{2}\sum_{j\sigma}\hat{S}_{j}^{z} \hat{c}_{j\sigma}^{\dag}\hat{\sigma^z}\hat{c}_{j\sigma},$$3, the entanglement curves collapse across system sizes and remain area-law-like up to $$\hat{H}=-t\sum_{\langle i,j\rangle,\sigma}\hat{c}_{i\sigma}^{\dag}\hat{c}_{j\sigma} +\frac{J}{2}\sum_{j\sigma}\hat{S}_{j}^{z} \hat{c}_{j\sigma}^{\dag}\hat{\sigma^z}\hat{c}_{j\sigma},$$4 (Gunawardana et al., 2024).

The resulting phase diagram in the Ising limit $$\hat{H}=-t\sum_{\langle i,j\rangle,\sigma}\hat{c}_{i\sigma}^{\dag}\hat{c}_{j\sigma} +\frac{J}{2}\sum_{j\sigma}\hat{S}_{j}^{z} \hat{c}_{j\sigma}^{\dag}\hat{\sigma^z}\hat{c}_{j\sigma},$$5 displays a thermal regime for approximately

$$\hat{H}=-t\sum_{\langle i,j\rangle,\sigma}\hat{c}_{i\sigma}^{\dag}\hat{c}_{j\sigma} +\frac{J}{2}\sum_{j\sigma}\hat{S}_{j}^{z} \hat{c}_{j\sigma}^{\dag}\hat{\sigma^z}\hat{c}_{j\sigma},$$6

and a robust MBL window for

$$\hat{H}=-t\sum_{\langle i,j\rangle,\sigma}\hat{c}_{i\sigma}^{\dag}\hat{c}_{j\sigma} +\frac{J}{2}\sum_{j\sigma}\hat{S}_{j}^{z} \hat{c}_{j\sigma}^{\dag}\hat{\sigma^z}\hat{c}_{j\sigma},$$7

(Gunawardana et al., 2024). A marginal or prethermal regime occurs for

$$\hat{H}=-t\sum_{\langle i,j\rangle,\sigma}\hat{c}_{i\sigma}^{\dag}\hat{c}_{j\sigma} +\frac{J}{2}\sum_{j\sigma}\hat{S}_{j}^{z} \hat{c}_{j\sigma}^{\dag}\hat{\sigma^z}\hat{c}_{j\sigma},$$8

Including interchain spin flips $$\hat{H}=-t\sum_{\langle i,j\rangle,\sigma}\hat{c}_{i\sigma}^{\dag}\hat{c}_{j\sigma} +\frac{J}{2}\sum_{j\sigma}\hat{S}_{j}^{z} \hat{c}_{j\sigma}^{\dag}\hat{\sigma^z}\hat{c}_{j\sigma},$$9 strongly enhances entanglement growth and delocalization. In the isotropic limit

$\mathbb Z_2$00

the system is always thermal according to the reported data (Gunawardana et al., 2024). For finite but strong anisotropy, the entanglement can grow logarithmically,

$\mathbb Z_2$01

signaling a prethermal quasi-localized regime rather than immediately stable localization (Gunawardana et al., 2024). The Ising limit is therefore singled out as the regime where the self-generated longitudinal fields are most static-like and localization is most robust.

This work is significant because it presents a concrete, fully interacting disorder-free MBL transition not based on exact local conserved charges of a free-fermion sector. The caveat, emphasized by the authors, is that the phase diagram relies partly on small-system exact diagonalization, especially at low incommensuration, and finite-time simulations cannot rule out eventual delocalization in marginal regimes (Gunawardana et al., 2024).

7. Diagnostics, experiments, and open questions

Disorder-free localization is diagnosed by a narrower and more model-dependent set of observables than standard MBL, because different mechanisms localize different subsystems or operate only after sector averaging. Common diagnostics include density imbalance or matter imbalance after a domain-wall quench (Smith et al., 2017, Halimeh et al., 2022, Osborne et al., 2023), connected density or spin correlators and their spreading fronts (Smith et al., 2017, Karpov et al., 2020, Chakraborty et al., 2022), inverse participation ratios in real or Fock space (Yang et al., 2019, Gunawardana et al., 2024, Li et al., 2021), level-spacing ratios and eigenstate entanglement (Chakraborty et al., 2022, Jeyaretnam et al., 2024), and entanglement growth laws distinguishing area-law saturation, logarithmic growth, and volume-law crossover (Hart et al., 2020, Gunawardana et al., 2024, Halimeh et al., 2024).

Two experimental directions have recently become prominent. One is analog or digital quantum simulation of gauge-theory disorder-free localization. "Observation of disorder-free localization and efficient disorder averaging on a quantum processor" (Gyawali et al., 2024) implements a translationally invariant $\mathbb Z_2$02 lattice gauge theory / dual disordered-spin model on superconducting hardware in both 1D and 2D. Exact local conserved quantities

$\mathbb Z_2$03

define symmetry sectors, and translationally invariant superposition states over these sectors exhibit localization of local energy perturbations whereas single-sector initial states thermalize (Gyawali et al., 2024). A methodological contribution is the use of quantum parallelism to sample exponentially many effective disorder realizations in a single coherent evolution for linear charge-commuting observables.

The other is disorder-free localization in continuum heavy–light mixtures. In the 2026 experiment (Finelli et al., 19 Jan 2026), a 3D $\mathbb Z_2$04–$\mathbb Z_2$05 mixture exhibits a crossover from normal diffusion to subdiffusion as interspecies interactions are increased and temperature lowered. The Li cloud’s mean-square width is fit to

$\mathbb Z_2$06

with $\mathbb Z_2$07 dropping below $\mathbb Z_2$08 in the anomalous regime, and a localized fraction up to about $\mathbb Z_2$09 emerges with no discernible dynamics over $\mathbb Z_2$10 ms (Finelli et al., 19 Jan 2026). The interpretation is that the heavy Cr atoms form a quasi-static random scattering landscape for the light Li atoms. The strongest claim is therefore effective partial localization and subdiffusion in a disorder-free, translationally invariant mixture, consistent with an Anderson-like model of resonant point scatterers rather than with standard Fermi-liquid transport (Finelli et al., 19 Jan 2026).

Several major questions remain open. One is asymptotic stability: many gauge-theory and constrained models show compelling localization-like behavior on accessible sizes and times, but it is often unclear whether this persists in the thermodynamic and infinite-time limits (Jeyaretnam et al., 2024, Li et al., 2021). Another concerns dimensionality. Although 2D gauge theories can display disorder-free localization through percolation or directional constraints (Karpov et al., 2020, Chakraborty et al., 2022, Osborne et al., 2023), the relation between these mechanisms and the avalanche instability arguments for conventional MBL remains unsettled. A third question is classification: some works establish exact sector-wise Anderson localization, others prethermal quasi-localization, others fragmentation-induced nonergodicity, and others robust interacting MBL-like phases. The shared label “disorder-free localization” therefore identifies a family resemblance rather than a single sharply delimited phase.

A recurring theme across the literature is that quenched disorder is not the only route to static inhomogeneity. Exact local charges, gauge superselection sectors, emergent constraints, deterministic incommensurate coupling graphs, and heavy slow subsystems can all generate effective disorder landscapes. Disorder-free localization is the collective name for the resulting failure of ergodicity in clean Hamiltonians, but its precise nature depends on which of these structures supplies the inhomogeneity and whether the resulting nonergodic behavior is exact, asymptotic, or only long-lived.