---
title: XX Models with Non-Local Constraints
url: https://www.emergentmind.com/topics/xx-models-with-non-local-constraints
type: topic
---

# XX Models with Non-Local Constraints

XX models with non-local constraints are XX-type spin systems in which otherwise standard nearest-neighbor exchange dynamics is modified by constraints, disorder terms, or dissipative structures whose effective action is non-local in the natural quasiparticle variables. A central example is a facilitated Rydberg spin chain that, after projection onto the constrained subspace and Kramers–Wannier duality, becomes an XX chain with non-local, correlated disorder generated by thermal positional fluctuations of the atoms [1811.01667]. A related open-system realization is the XX chain with non-local dephasing, where coherent combinations of dephasing terms alter the transport universality class and produce superdiffusive magnetization transport in the thermodynamic limit [2311.07375]. Together, these constructions show that XX dynamics can remain analytically recognizable while acquiring non-local structures that qualitatively change relaxation, transport, level statistics, and entanglement growth.

## 1. Constrained XX dynamics from a facilitated Rydberg chain

The Rydberg-chain construction begins from the standard Rydberg-atom Hamiltonian in the rotating-wave approximation,
\[
H \;=\; \Omega\sum_{j=1}^N\sigma_j^x \;+\;\Delta\sum_{j=1}^N n_j
\;+\;\frac{C_6}{2}\sum_{j\neq k}\frac{n_j\,n_k}{|r_j-r_k|^6}\,,
\qquad
n_j=\tfrac12(1+\sigma_j^z)\,.
\]
Here \(\ket{\downarrow}_j\) and \(\ket{\uparrow}_j\) label ground and Rydberg states on site \(j\). Facilitation is imposed by choosing
\[
\Delta=-V_0\equiv -\frac{C_6}{r_0^6},
\qquad |\Delta|\gg\Omega,
\]
so that an isolated spin flip is far off-resonant, while a flip adjacent to an already excited atom costs zero net energy [1811.01667].

After truncating all van-der-Waals tails beyond nearest neighbors and projecting out the energetically forbidden processes, one obtains the effective constrained Hamiltonian
\[
H_{\mathrm{eff}}
\;=\;
\Omega\sum_{j=1}^N P_j\,\sigma_j^x,
\qquad
P_j\;=\;\tfrac12\bigl(1-\sigma_{j-1}^z\,\sigma_{j+1}^z\bigr)\,.
\]
The projector \(P_j\) enforces that site \(j\) may flip only if at least one neighbor is already in \(\ket{\uparrow}\). In this form the model is not yet an XX chain; rather, the XX structure emerges after a duality transformation.

Using Kramers–Wannier, or domain-wall, duality, new Pauli operators \(\mu_j^\alpha\) are defined on the bonds \(j=1,\dots,N+1\) by
\[
\sigma_j^x=\mu_j^x\,\mu_{j+1}^x,\qquad
\sigma_j^z=\;(-1)^{j+1}\prod_{\ell=1}^j\mu_\ell^z.
\]
Under this mapping, the constrained flip term becomes precisely a nearest-neighbor XX coupling in the \(\mu\)-basis,
\[
H_{\mathrm{XX}}
\;=\;
\frac{\Omega}{2}\sum_{j=1}^{N}
\bigl(\,\mu_j^x\,\mu_{j+1}^x \;+\;\mu_j^y\,\mu_{j+1}^y\bigr).
\]
In the absence of any further interaction this is a free-fermion model [1811.01667].

This mapping is conceptually important because the facilitation constraint does not disappear; it is encoded in the dual description as freely hopping domain-wall quasiparticles in the clean limit. This suggests that the phrase “constraint” is best understood here as a restructuring of the effective degrees of freedom rather than as a purely kinematic prohibition.

## 2. How non-locality enters: correlated disorder in the dual XX chain

The non-local structure arises when thermal motion of the atoms is included. At finite temperature, the atomic positions fluctuate as
\[
r_j=j\,r_0 + \delta r_j,\qquad
\delta r_j\sim\mathcal{N}(0,\sigma^2).
\]
Restricting again to nearest-neighbor van-der-Waals interactions yields a site-dependent energy shift
\[
\delta V_j
= \frac{C_6}{|r_{j+1}-r_j|^6}-V_0
= \frac{C_6}{(r_0+\delta r_{j+1}-\delta r_j)^6}-\frac{C_6}{r_0^6},
\]
which enters the spin Hamiltonian as
\[
V_{\mathrm{dis}}
= \sum_{j=1}^{N-1}\delta V_j\;n_j\,n_{j+1}.
\]
Because \(\delta r_j-\delta r_{j+1}\) is Gaussian of variance \(2\sigma^2\), each \(\delta V_j\) has a sharply peaked, skewed distribution around zero, and different \(\delta V_j\)s are correlated because they share common \(\delta r\) variables [1811.01667].

In the original spin language this is a fluctuating nearest-neighbor two-body term. In the domain-wall basis, however, the same disorder becomes explicitly non-local:
\[
V_{\mathrm{dis}}
= \frac14\sum_{j=1}^{N-1}\delta V_j
\Bigl[\bigl((-1)^{j+1}\!\prod_{\ell=1}^j\mu_\ell^z\bigr)+1\Bigr]
\Bigl[\bigl((-1)^{j+2}\!\prod_{k=1}^{j+1}\mu_k^z\bigr)+1\Bigr],
\]
that is, strings of \(\mu^z\) of up to length \(N\) [1811.01667].

The resulting XX model is therefore not an XX chain with conventional onsite random fields or local bond randomness. Disorder and interactions become inextricably intertwined in the dual variables. A common misconception is that an XX-chain mapping necessarily implies purely free-fermion delocalized behavior. In this setting that conclusion fails because the dualized disorder is neither local nor independent; it is a correlated string operator. The clean XX limit remains free-fermionic, but the disordered model does not reduce to a standard Anderson-type problem.

## 3. Non-equilibrium probes of localization in the constrained model

The localization analysis is performed by evolving from the staggered initial state
\[
\ket{\cdots\uparrow\uparrow\downarrow\downarrow\uparrow\uparrow\downarrow\downarrow\cdots},
\]
which has no overlap with zero-energy “isolated excitation” eigenstates [1811.01667]. Three diagnostics are used: the imbalance, the half-chain entanglement entropy, and the level-statistics ratio.

The imbalance is defined by
\[
\mathcal I(t)
=\frac1{N-1}\sum_{j=1}^{N-1}(-1)^j
\bigl[n_j(1-n_{j+1}) +(1-n_j)n_{j+1}\bigr].
\]
For \(\sigma\lesssim10^{-3}\), the long-time imbalance satisfies \(\mathcal I(\infty)\to0\), corresponding to delocalized behavior. When \(\sigma\gtrsim10^{-2}\), \(\mathcal I(\infty)\approx\mathcal I(0)\sim1/2\), signaling retention of the initial structure [1811.01667].

The half-chain entanglement entropy is
\[
S(t)
= -\operatorname{Tr}\{\rho_{1/2}(t)\,\ln\rho_{1/2}(t)\},
\qquad
\rho_{1/2}=\operatorname{Tr}_{N/2+1\ldots N}\ket{\Psi(t)}\bra{\Psi(t)}.
\]
At weak disorder, \(S(t)\) rapidly saturates to an \(O(N)\) value with oscillations. At strong disorder, \(\sigma\gtrsim10^{-2}\), \(S(t)\) grows slowly, logarithmically in \(t\), before saturating at a much smaller value, which is identified as a hallmark of many-body localization [1811.01667].

The level-statistics ratio is based on ordered eigenvalues \(E_n\), gaps \(\Delta_n=E_{n+1}-E_n\), and
\[
r_n=\frac{\min(\Delta_n,\Delta_{n+1})}
{\max(\Delta_n,\Delta_{n+1})},
\qquad
\langle r\rangle=\frac1{\#}\sum_n r_n
\quad(\text{disorder-averaged}).
\]
Its behavior is non-monotone in \(\sigma\): \(\langle r\rangle\approx0.39\) for \(\sigma\lesssim10^{-3}\), again \(\langle r\rangle\approx0.39\) for \(\sigma\gtrsim10^{-1}\), and \(\langle r\rangle\approx0.53\) in the intermediate window \(\sigma\approx10^{-2}\) [1811.01667]. The first Poisson-like regime is integrable-like, the second is MBL-like, and the intermediate GOE regime is indicative of ergodic behavior and thermalization.

| Diagnostic | Weak disorder | Strong disorder |
|---|---|---|
| Imbalance \(\mathcal I(\infty)\) | \(\to 0\) for \(\sigma\lesssim10^{-3}\) | \(\approx \mathcal I(0)\sim1/2\) for \(\sigma\gtrsim10^{-2}\) |
| HCEE \(S(t)\) | Rapid saturation to an \(O(N)\) value | Logarithmic growth, then smaller saturation value |
| LSR \(\langle r\rangle\) | \(\approx 0.39\) for \(\sigma\lesssim10^{-3}\) | \(\approx 0.39\) for \(\sigma\gtrsim10^{-1}\) |

The coexistence of Poisson statistics in both the weak- and strong-disorder limits is sometimes misread as evidence against a disorder-driven crossover. The numerical interpretation given in the source is more specific: the weak-disorder limit is integrable-like because the clean XX chain is free-fermionic, while the strong-disorder limit is MBL-like because random long strings pin the domain walls [1811.01667].

## 4. Ergodic, non-ergodic, and MBL-like regimes

The physical intuition underlying the constrained XX description has four parts. First, the facilitation constraint turns the bare Rydberg chain into an effectively assisted flip model that supports domain-wall quasiparticles hopping freely in the clean limit. Second, thermal motion induces randomness in the nearest-neighbor interaction energy, producing a spatially fluctuating two-body term in the spin language. Third, in the dual domain-wall picture this becomes a non-local string structure in \(\mu^z\). Fourth, at strong disorder these random long strings pin the domain walls and severely inhibit transport, giving rise to MBL-like logarithmic entanglement growth and memory retention [1811.01667].

Numerically, the crossover scale is identified around
\[
\sigma_{\mathrm c}\sim10^{-2},
\]
that is, fluctuations of order \(1\%\) of \(r_0\). Below this scale the system is described as ergodic or delocalized; above it, as non-ergodic and MBL-like [1811.01667]. The paper formulates the result as a clear ergodic-to-non-ergodic crossover as one tunes the trap width \(\sigma\).

This regime structure is experimentally significant because the model is realized in a natural fashion in Rydberg quantum simulators. The localization mechanism does not rely on externally imposed, independent quenched onsite randomness. Instead, the disorder originates from spin-spin interactions and positional fluctuations. A plausible implication is that XX-type localization mechanisms in constrained platforms can differ substantially from the textbook route based on local random potentials.

The distinction between “MBL” and “MBL-like” matters. The reported evidence consists of imbalance retention, logarithmic entanglement growth, and Poissonian level statistics at strong disorder, together with the non-local string structure of the disorder term [1811.01667]. The source uses “MBL-like” language for the strong-disorder regime, which is therefore the precise encyclopedic description.

## 5. Boundary-driven XX chains with non-local dephasing

A distinct but closely related XX construction arises in open quantum systems. The boundary-driven XX chain with non-local dephasing is governed in the bulk by the standard XX Hamiltonian
\[
H \;=\;\sum_{j=1}^{L-1}\Bigl(\sigma^x_j\sigma^x_{j+1}+\sigma^y_j\sigma^y_{j+1}\Bigr),
\]
supplemented by three-site dephasing channels and boundary driving [2311.07375]. In the first superdiffusive case, the jump amplitude \(l_j\) is a coherent sum of \(\sigma^z_{j-1}\) and \(\sigma^z_{j-1}\sigma^z_j\sigma^z_{j+1}\); individually, each term would generate ordinary diffusion, while their coherent sum produces anomalous transport. A second choice,
\[
l_j =\frac{1}{\sqrt{6}}
\Bigl(\sigma^z_{j-1} -2\,\sigma^z_j +\sigma^z_{j+1}\Bigr),
\]
yields a different superdiffusive exponent [2311.07375].

The full Lindblad evolution is
\[
\frac{d\rho}{dt} \;=\;-i\,[H,\rho] \;+\;\sum_{j=2}^{L-1}\gamma\, \mathcal{L}^{(\rm deph)}_j(\rho) \;+\;\mathcal{L}^{(\rm bath)}(\rho),
\]
with \(\gamma=1\). Boundary baths impose a small magnetization bias \(\pm\mu\), and the nonequilibrium steady state carries a nonzero magnetization current. In the linear-bias regime the current scales as
\[
J\;\propto\;\frac{1}{L^\alpha},
\qquad
J\sim L^{-(z-1)},
\qquad
z=\alpha+1.
\]
For diffusion, \(z=2\) and \(J\sim1/L\). For the first non-local dephasing one finds \(\alpha=1/2\), hence \(z=3/2\). For the second one finds \(\alpha=2/3\), hence \(z=5/3\) [2311.07375].

| Non-local dephasing choice | Scaling of \(J\) | Dynamical exponent |
|---|---|---|
| Coherent sum yielding first superdiffusive case | \(\alpha=1/2\) | \(z=3/2\) |
| \(l_j \propto \sigma^z_{j-1}-2\sigma^z_j+\sigma^z_{j+1}\) | \(\alpha=2/3\) | \(z=5/3\) |

These exponents were confirmed by an exact hierarchy-of-correlations solution up to \(L=6000\) and matrix-product-operator simulations up to \(L\sim1000\) [2311.07375]. Further evidence comes from the mean-square spreading of a localized magnetization packet,
\[
{\rm MSD}(t) \sim t^{2/z},
\]
with predicted \(t^{4/3}\) for \(z=3/2\) and \(t^{6/5}\) for \(z=5/3\) [2311.07375].

This open-system example is not a facilitation constraint in the strict projector sense, but it belongs to the same broader family of XX models whose effective dynamics is shaped by non-local structure. The key mechanism is explicitly quantum: a coherent sum of two diffusive terms results in superdiffusion [2311.07375].

## 6. Stability of the non-local mechanism and broader significance

The non-local dephasing construction is delicate. Two perturbations restore diffusion. First, breaking the exact form of the dissipator by omitting the Jordan–Wigner phase factor in the three-site dephasing leads numerically to \(J\sim1/L\) for large \(L\), that is, normal diffusion with \(z=2\). Second, replacing the Hamiltonian by the XXZ form
\[
H \;=\;\sum_{j=1}^{L-1} \bigl(\sigma^x_j\sigma^x_{j+1} +\sigma^y_j\sigma^y_{j+1} +\Delta\,\sigma^z_j\sigma^z_{j+1}\bigr)
\]
and keeping the superdiffusive dissipator also restores diffusion, even for small anisotropy \(\Delta\approx0.2\) [2311.07375]. The conclusion drawn in the source is that superdiffusion is a delicate quantum-coherent effect requiring exactly the non-local jump operators that preserve integrability of the two-point hierarchy.

Taken together with the facilitated Rydberg construction, these results identify two distinct roles for non-local structure in XX systems. In the closed-chain case, non-locality arises through a duality map and converts correlated positional disorder into disorder strings that suppress transport and produce a non-ergodic, MBL-like regime [1811.01667]. In the driven-dissipative case, non-locality is engineered directly in the Lindblad operators and can enhance transport from diffusion to superdiffusion, but only in a finely tuned form [2311.07375].

A persistent misconception is that “XX” is synonymous with simple transport or trivial integrability. The evidence from both settings is narrower and more precise. The clean constrained model maps to a free-fermion XX chain, yet non-local, correlated disorder generated by the underlying platform leads to an ergodic-to-non-ergodic crossover [1811.01667]. Conversely, a boundary-driven XX chain with dephasing is usually expected to be diffusive, yet a coherent non-local modification of the dissipator produces \(z=3/2\) or \(z=5/3\) scaling before reverting to diffusion under generic perturbations [2311.07375]. This suggests that the salient distinction is not whether the Hamiltonian is of XX type, but how constraints, disorder, or dissipation are represented in the effective variables.

Source: https://www.emergentmind.com/topics/xx-models-with-non-local-constraints