Interwoven long-range order induced by random fields

Abstract: We propose a distinct type of long-range ordered phase that can occur in classical and quantum many-particle systems. It is induced by impurities and defects that locally break a subset of the order-parameter symmetries, i.e., by random-field disorder that couples to a composite vestigial order parameter. The proposed ``implectic'' phase is characterized by spontaneous symmetry breaking on the background of the spatially interwoven domain structure created by the random fields. We explicitly demonstrate the existence of this phase in a layered $J_1-J_2$ Ising magnet by means of large-scale Monte Carlo simulations. We then discuss numerous potential applications in systems featuring charge and spin density wave order including frustrated magnets, cuprate and iron-based superconductors, and ultracold atoms.

• The paper introduces the implectic phase, where composite random-field disorder induces coexistence of interwoven x- and y-stripe orders.
• It employs a layered J1-J2 Ising model with Monte Carlo simulations to reveal sharp thermal transitions and percolating domain networks.
• Results clarify how domain-induced symmetry breaking reconciles local disorder with global order, offering insights for high-Tc cuprates and iron pnictides.

Interwoven Long-Range Order Induced by Random Fields

Introduction and Theoretical Framework

"Interwoven long-range order induced by random fields" (2607.10337) presents a theoretical and computational study of a novel long-range ordered phase—termed the implectic phase—arising in the presence of specific random-field disorder in many-body systems. The disorder considered couples not to the primary order parameter but to a composite vestigial order parameter. The work leverages and extends symmetry-based classifications, as well as Imry-Ma/Aizenman-Wehr disorder arguments, focusing on three-dimensional systems where standard wisdom posits the destruction of long-range order at strong random field via domain formation.

Central to the work is the identification that for multi-symmetry-breaking order parameters, disorder can selectively break real-space symmetries locally, leading to composite domains. In three dimensions, these domains can percolate and themselves undergo sharp symmetry-breaking transitions for the remaining parameters, resulting in coexisting long-range order that is globally disorder-configurationally interwoven.

Prototypical Model: $J_1$-$J_2$ Layered Ising Magnet

The main platform is the layered $J_1$-$J_2$ Ising model on a cubic lattice, with Hamiltonian

$$H_0 = - J_1 \sum_{\langle ij \rangle} S_i S_j - J_2 \sum_{\langle\langle ij \rangle\rangle} S_i S_j - J_\perp \sum_{\langle ij \rangle_\perp} S_i S_j,$$

where $S_i = \pm 1$ and $J_1>0$ (ferromagnetic nearest-neighbor), $J_2<0$ (antiferromagnetic next-nearest-neighbor), and $J_\perp > 0$ (interlayer). For $|J_2|/J_1 > 1/2$, the ground state spontaneously breaks both Ising and $J_2$0 lattice rotation symmetry, resulting in stripe order—stripes can be oriented along $J_2$1 or $J_2$2.

Figure 1: (a) Couplings in the layered $J_2$3-$J_2$4 model. (b) Schematic of the stripe order in the $J_2$5 plane with orientation degeneracy for $J_2$6.

Stripe order is described by the two-component vector order parameter $J_2$7, which distinguishes the two degenerate stripe orientations. The vestigial nematic order parameter $J_2$8 captures breaking of the $J_2$9 symmetry independent of spin order.

Upon the introduction of a random field $J_1$0 coupling locally to the nematic order parameter, the system's Hamiltonian becomes

$J_1$1

This type of quenched disorder arises naturally in real systems via strain, site dilution, or defects creating local lattice anisotropy, distinguishing it fundamentally from random-field Ising models that couple directly to the magnetic order.

Monte Carlo Results and Characterization of the Implectic Phase

Behavior Without Random Field

Simulations at $J_1$2 demonstrate a conventional phase transition into uniform stripe order at $J_1$3, signaled by the nematic order parameter and the Binder cumulant.

Figure 2: (a) Nematic order parameter and (b) stripe Binder cumulant versus temperature for various system sizes. (c) $J_1$4 vs. $J_1$5 for individual samples at $J_1$6, with each sample aligned along $J_1$7 or $J_1$8, but not both. (d) Local magnetization profile.

The histogram in Figure 2(c) collapses along pure $J_1$9 or $J_2$0 axes, with domain formation absent.

Interwoven Phase at Strong Random Field

At strong random field $J_2$1, simulations uncover a sharp thermal transition at $J_2$2 indicated by the crossing of the average stripe Binder cumulant and divergent stripe susceptibility:

Figure 3: (a) Stripe Binder cumulant $J_2$3 and (b) stripe susceptibility $J_2$4 versus temperature for $J_2$5 and $J_2$6.

However, the nematic order parameter is suppressed, and direct spatial maps (Figure 4) show the existence of finite-size nematic domains of characteristic $J_2$710 lattice constants, with the global average vanishing for large $J_2$8:

Figure 4: (a) Absence of nematic long-range order: $J_2$9 versus $$H_0 = - J_1 \sum_{\langle ij \rangle} S_i S_j - J_2 \sum_{\langle\langle ij \rangle\rangle} S_i S_j - J_\perp \sum_{\langle ij \rangle_\perp} S_i S_j,$$0, decaying with increasing $$H_0 = - J_1 \sum_{\langle ij \rangle} S_i S_j - J_2 \sum_{\langle\langle ij \rangle\rangle} S_i S_j - J_\perp \sum_{\langle ij \rangle_\perp} S_i S_j,$$1; (b) real-space nematic domain structure in a typical layer.

Strikingly, the joint distribution of $$H_0 = - J_1 \sum_{\langle ij \rangle} S_i S_j - J_2 \sum_{\langle\langle ij \rangle\rangle} S_i S_j - J_\perp \sum_{\langle ij \rangle_\perp} S_i S_j,$$2 and $$H_0 = - J_1 \sum_{\langle ij \rangle} S_i S_j - J_2 \sum_{\langle\langle ij \rangle\rangle} S_i S_j - J_\perp \sum_{\langle ij \rangle_\perp} S_i S_j,$$3 at low $$H_0 = - J_1 \sum_{\langle ij \rangle} S_i S_j - J_2 \sum_{\langle\langle ij \rangle\rangle} S_i S_j - J_\perp \sum_{\langle ij \rangle_\perp} S_i S_j,$$4 (Figure 5) reveals that every disorder realization exhibits coexisting $$H_0 = - J_1 \sum_{\langle ij \rangle} S_i S_j - J_2 \sum_{\langle\langle ij \rangle\rangle} S_i S_j - J_\perp \sum_{\langle ij \rangle_\perp} S_i S_j,$$5- and $$H_0 = - J_1 \sum_{\langle ij \rangle} S_i S_j - J_2 \sum_{\langle\langle ij \rangle\rangle} S_i S_j - J_\perp \sum_{\langle ij \rangle_\perp} S_i S_j,$$6-stripe order. As $$H_0 = - J_1 \sum_{\langle ij \rangle} S_i S_j - J_2 \sum_{\langle\langle ij \rangle\rangle} S_i S_j - J_\perp \sum_{\langle ij \rangle_\perp} S_i S_j,$$7 increases, the points cluster around the bisector $$H_0 = - J_1 \sum_{\langle ij \rangle} S_i S_j - J_2 \sum_{\langle\langle ij \rangle\rangle} S_i S_j - J_\perp \sum_{\langle ij \rangle_\perp} S_i S_j,$$8, indicating percolation and coexistence. This is the diagnostic hallmark of the implectic phase.

Figure 5: Distribution of $$H_0 = - J_1 \sum_{\langle ij \rangle} S_i S_j - J_2 \sum_{\langle\langle ij \rangle\rangle} S_i S_j - J_\perp \sum_{\langle ij \rangle_\perp} S_i S_j,$$9 vs.\ $S_i = \pm 1$0 for $S_i = \pm 1$1 random-field samples at $S_i = \pm 1$2, $S_i = \pm 1$3, $S_i = \pm 1$4. All samples exhibit simultaneous order in both variables.

Order parameter products and coexistence metrics (Figure 6) quantitatively confirm the emergence of this interwoven order below $S_i = \pm 1$5. The coexistence parameter $S_i = \pm 1$6 approaches unity for large $S_i = \pm 1$7, consistent with a $S_i = \pm 1$8 equal mixing of $S_i = \pm 1$9 and $J_1>0$0 stripe orders.

Figure 6: (a) Product $J_1>0$1 vs.\ temperature, and (b) coexistence parameter $J_1>0$2, both indicating a sharp onset of simultaneous order below $J_1>0$3.

Domain Structure and Percolation

The picture emerging is that strong vestigial random field breaks nematic long-range order by forming finite domains, but in three dimensions, domains favoring $J_1>0$4 and $J_1>0$5 stripes percolate—each forms an interpenetrating network supporting long-range order of the corresponding stripe orientation. The implementation is fundamentally distinct from macroscopic phase coexistence: the symmetry is broken only within percolating subnetworks, not globally, and the system as a whole remains isotropic on average.

Weak random field ($J_1>0$6) does not support this architecture: in that case, the system reverts to conventional uniform stripe order, affirming the role of a disorder threshold predicted via Imry-Ma scaling and observed numerically.

Sufficient Conditions and Distinction from Conventional Phases

For the realization of implectic order, three conditions must be fulfilled:

• Quenched disorder must couple to a composite vestigial order: In practice, this often means random fields that break spatial but not time-reversal or other non-geometric symmetries.
• Dimensionality $J_1>0$7: Only then can independent percolating domain networks coexist; two dimensions cannot support such mutual spanning.
• Random-field strength must exceed $J_1>0$8: Sufficient to nucleate robust domain formation.

The phase diagram thereby involves three possibilities: a conventional stripe state (uniform, both spin and lattice symmetry broken), a paramagnet, and the implectic phase, where Ising symmetry is broken within percolating subnetworks but lattice symmetry remains unbroken globally. No vestigial nematic-only phase is realized in this model.

Experimental and Broader Implications

The physical mechanism uncovered has direct implications for charge and magnetic order in high-$J_1>0$9 cuprates, iron pnictides, and other systems characterized by intertwined orders. Many such materials display sharp macroscopic thermodynamic transitions (e.g., signatures of magnetic or charge order) while local probes report disrupted nematicity and nanoscale inhomogeneity—an apparent contradiction naturally resolved by the implectic order scenario.

Specifically, in systems like GdRhIn$J_2<0$0, where X-ray data report no global orthorhombic distortion yet magnetic stripes are observed, the phase inferred here provides a symmetry-consistent explanation: domain-level lattice symmetry breaking and interwoven order coexisting with global $J_2<0$1 invariance.

The results also open a route to engineering and observing such phases in ultracold atomic gases and provide guidance for interpreting domain-dominated local probe results in correlated electron materials. Detection of implectic order requires both observation of percolating domain structures and measurements establishing sharp transitions in observables tied to the residual symmetry breaking.

Numerical Techniques

The computational advances enabling this analysis include highly parallelized replica-exchange Monte Carlo (parallel tempering) permitting reliable equilibration on large systems with complex energy landscapes. Internal diagnostics (acceptance rates, observable convergence from distinct initial states) confirm equilibrium and accurate sampling of the physical properties in the thermodynamic limit.

Figure 7: (a) A slice of the constructed correlated random field $J_2<0$2 for $J_2<0$3, $J_2<0$4, exhibits domain structure. (b) The measured radial correlation function agrees with targeted exponential decay.

Figure 8: Parallel-tempering diagnostics for $J_2<0$5—mean replica swap acceptance (a), and visited temperature-span fractions for different initialization protocols (b) ensure correct sampling.

Conclusion

The work establishes the existence and robust characterization of a disorder-induced interwoven long-range ordered phase, the implectic phase, in the layered $J_2<0$6-$J_2<0$7 Ising model and, by extension, multi-component order systems under composite random-field disorder (2607.10337).

Key conclusions:

• Strong, locally symmetry-breaking (vestigial) random field in $J_2<0$8 dimensions leads to spatially interwoven, percolating domain networks supporting coexistence of several long-range orders.
• The phase transition is sharp and spontaneous, despite the absence of long-range vestigial order—contradicting conventional unwritten assumptions about disorder-induced destruction of order.
• The interplay between local domain structure and macroscopic global invariance provides a prescription for interpreting apparently paradoxical experimental results in quantum materials.
• Future work may extend these ideas to clock models, random anisotropy systems, and other multi-symmetry-broken phases, and develop experimental probes capable of unambiguously distinguishing implectic order from macroscopic phase separation.

The conceptual framework established prompts further investigation into the nature of domain-spanning transitions, real-space topology of interwoven phases, and the effect of anisotropies and correlated disorder—matters of both fundamental and applied significance in modern strongly correlated systems.