Scarred Ferromagnetic Phase

• Scarred ferromagnetic phase is a nonequilibrium, symmetry-broken regime in the long-range transverse-field Ising model where selective initial states evolve toward ferromagnetic order.
• It emerges through the dynamical selection of parity-paired scarred eigenstates with nonzero magnetization, defying standard thermal predictions.
• Finite-size scaling and domain-wall dynamics are key to its detection and experimental realization in platforms like trapped-ion and Rydberg atom simulators.

Searching arXiv for the specified paper and closely related references mentioned in the provided data. The scarred ferromagnetic phase is a dynamical symmetry-broken regime of the one-dimensional long-range transverse-field Ising model in which simple product states containing a few small magnetic domains evolve toward ferromagnetic equilibrium states, despite lying in an energy-density regime where finite-temperature thermodynamics predicts only paramagnetism. In "Scarred ferromagnetic phase in the long-range transverse-field Ising model" (Corps et al., 22 Jul 2025), the phase is identified through a large set of ferromagnetic scarred states distributed across different spectral regions and surrounded by paramagnetic states. The central result is not the recovery of an ordinary finite-temperature ordered phase, but the existence of a nonequilibrium route to long-time ferromagnetic order through selective population of symmetry-breaking scarred subspaces.

1. Model, symmetries, and observables

The underlying system is a one-dimensional transverse-field Ising model with power-law decaying Ising couplings in the $x$-direction and a transverse field along $z$. Its Hamiltonian is

$$\hat{H}_{\mathrm{TFIM}}= -\frac{1}{\mathcal{K}(\alpha)}\sum_{i<j}\frac{J}{|i-j|^\alpha}\,\hat{\sigma}_i^x \hat{\sigma}_j^x +h\sum_i \hat{\sigma}_i^z,$$

where $J$ is the interaction strength, $h$ the transverse field, and $\alpha$ the power-law exponent controlling the interaction range. For $\alpha<1$, the Kac normalization factor $\mathcal{K}(\alpha)$ is introduced to ensure extensivity. With periodic boundary conditions, distances are defined on a ring, and the interaction is implemented through $D_{ij}$ together with

$$\mathcal{K}(\alpha)=\frac{1}{N-1}\sum_{i<j} D_{ij}^{-\alpha}.$$

The model is invariant under the $z$0 parity operator

$z$1

Ferromagnetism is characterized by the magnetization along $z$2,

$z$3

Because $z$4 flips parity, any fixed-parity eigenstate satisfies $z$5. This point is essential: the relevant ferromagnetic information is not contained in the expectation value of $z$6 within a single parity eigenstate, but in appropriate parity-mixed or parity-paired structures.

The numerical analysis uses periodic boundary conditions and exploits translational and inversion symmetries, specifically the zero-momentum, positive-inversion sector, to optimize exact diagonalization. The principal parameter sets are $z$7, $z$8, $z$9, and $$\hat{H}_{\mathrm{TFIM}}= -\frac{1}{\mathcal{K}(\alpha)}\sum_{i<j}\frac{J}{|i-j|^\alpha}\,\hat{\sigma}_i^x \hat{\sigma}_j^x +h\sum_i \hat{\sigma}_i^z,$$0 for static spectral analysis; $$\hat{H}_{\mathrm{TFIM}}= -\frac{1}{\mathcal{K}(\alpha)}\sum_{i<j}\frac{J}{|i-j|^\alpha}\,\hat{\sigma}_i^x \hat{\sigma}_j^x +h\sum_i \hat{\sigma}_i^z,$$1, $$\hat{H}_{\mathrm{TFIM}}= -\frac{1}{\mathcal{K}(\alpha)}\sum_{i<j}\frac{J}{|i-j|^\alpha}\,\hat{\sigma}_i^x \hat{\sigma}_j^x +h\sum_i \hat{\sigma}_i^z,$$2, $$\hat{H}_{\mathrm{TFIM}}= -\frac{1}{\mathcal{K}(\alpha)}\sum_{i<j}\frac{J}{|i-j|^\alpha}\,\hat{\sigma}_i^x \hat{\sigma}_j^x +h\sum_i \hat{\sigma}_i^z,$$3, $$\hat{H}_{\mathrm{TFIM}}= -\frac{1}{\mathcal{K}(\alpha)}\sum_{i<j}\frac{J}{|i-j|^\alpha}\,\hat{\sigma}_i^x \hat{\sigma}_j^x +h\sum_i \hat{\sigma}_i^z,$$4 for nonequilibrium dynamics; and $$\hat{H}_{\mathrm{TFIM}}= -\frac{1}{\mathcal{K}(\alpha)}\sum_{i<j}\frac{J}{|i-j|^\alpha}\,\hat{\sigma}_i^x \hat{\sigma}_j^x +h\sum_i \hat{\sigma}_i^z,$$5, $$\hat{H}_{\mathrm{TFIM}}= -\frac{1}{\mathcal{K}(\alpha)}\sum_{i<j}\frac{J}{|i-j|^\alpha}\,\hat{\sigma}_i^x \hat{\sigma}_j^x +h\sum_i \hat{\sigma}_i^z,$$6, $$\hat{H}_{\mathrm{TFIM}}= -\frac{1}{\mathcal{K}(\alpha)}\sum_{i<j}\frac{J}{|i-j|^\alpha}\,\hat{\sigma}_i^x \hat{\sigma}_j^x +h\sum_i \hat{\sigma}_i^z,$$7 for the dynamical phase boundary (Corps et al., 22 Jul 2025).

2. Finite-temperature context and ETH expectations

The phase is defined relative to a regime in which standard thermodynamics does not support ferromagnetic order. In one-dimensional quantum long-range Ising chains, finite-temperature ferromagnetism occurs only for $$\hat{H}_{\mathrm{TFIM}}= -\frac{1}{\mathcal{K}(\alpha)}\sum_{i<j}\frac{J}{|i-j|^\alpha}\,\hat{\sigma}_i^x \hat{\sigma}_j^x +h\sum_i \hat{\sigma}_i^z,$$8; for $$\hat{H}_{\mathrm{TFIM}}= -\frac{1}{\mathcal{K}(\alpha)}\sum_{i<j}\frac{J}{|i-j|^\alpha}\,\hat{\sigma}_i^x \hat{\sigma}_j^x +h\sum_i \hat{\sigma}_i^z,$$9, the thermal phase is paramagnetic. The paper concentrates especially on $J$0, where no finite-temperature ferromagnetic phase is expected, so canonical and microcanonical descriptions predict

$J$1

at all finite temperatures (Corps et al., 22 Jul 2025).

The dynamical benchmark is the usual ETH framework. For an initial state

$J$2

its time evolution is

$J$3

The long-time average of a few-body observable $J$4 is then

$J$5

with $J$6. Under ETH, this agrees with the microcanonical prediction

$J$7

For $J$8, parity symmetry enforces $J$9 in every fixed-parity eigenstate. The conventional ETH conclusion is therefore paramagnetic equilibration. The scarred ferromagnetic phase is precisely the regime in which this expectation fails dynamically, even though the underlying model remains clean and translationally invariant. The generalized ETH framework invoked in the paper follows Gómez–Corps and Relaño (Gómez et al., 16 Jun 2025).

3. Ferromagnetic scarred eigenstates

The spectral diagnostic is based on near-degenerate opposite-parity pairs. Because the mean level spacing shrinks exponentially with $h$0, opposite-parity eigenstates $h$1 and $h$2 can maintain coherence over extremely long times and define an effective two-dimensional subspace

$h$3

Projecting $h$4 onto $h$5 yields two eigenvalues

$h$6

A symmetry-broken equilibrium is possible in the thermodynamic limit if and only if there exists a sequence of such subspaces for which

$h$7

In the paper, substantial nonzero $h$8 is used as the ferromagnetic scar diagnostic.

For $h$9, $\alpha$0, and $\alpha$1, the spectral organization depends strongly on $\alpha$2. At $\alpha$3 and $\alpha$4, the spectrum exhibits a clear low-energy ferromagnetic region with nonzero $\alpha$5, a wide mixed region, and a small high-energy paramagnetic region. At $\alpha$6, where no finite-temperature ferromagnetic phase exists, a banded structure appears across the spectrum: some $\alpha$7 have $\alpha$8 significantly different from zero, while nearby bands satisfy $\alpha$9. These nonzero-$\alpha<1$0 bands are the ferromagnetic scarred states. At $\alpha<1$1, scars persist but are less prevalent; at $\alpha<1$2 and $\alpha<1$3, the $\alpha<1$4 values cluster around zero and ferromagnetic scars become sparse or absent (Corps et al., 22 Jul 2025).

The counting analysis uses the threshold $\alpha<1$5 to define a scarred subspace. For $\alpha<1$6, $\alpha<1$7, and $\alpha<1$8, the number of such subspaces grows exponentially with $\alpha<1$9. For $\mathcal{K}(\alpha)$0, the exponential growth is slower. For $\mathcal{K}(\alpha)$1 and $\mathcal{K}(\alpha)$2, exponential growth is not supported. The significance of this result is that, even at $\mathcal{K}(\alpha)$3, where finite-temperature ferromagnetism is absent, the model still contains an extensive set of eigen-subspaces with large magnetization eigenvalues.

A common misconception is to identify these states with standard low-entanglement or low-participation-ratio scars characterized through multiple diagnostics. Here the primary indicator is narrower and more specific: the paper does not report entanglement entropy, participation ratios, inverse participation ratios, ETH scatter plots, Loschmidt echo diagnostics, or two-point correlators as scar criteria. It instead isolates generalized-ETH-violating symmetry-breaking subspaces through the $\mathcal{K}(\alpha)$4 spectrum. The model has no exact spectral degeneracies for finite $\mathcal{K}(\alpha)$5, and the generalized ETH violation is demonstrated directly by the nonthermal distribution of $\mathcal{K}(\alpha)$6.

4. Dynamical definition of the scarred ferromagnetic phase

The scarred ferromagnetic phase is defined dynamically. It occurs when simple product-state initial conditions with a few small magnetic domains selectively populate scarred subspaces with large $\mathcal{K}(\alpha)$7, producing

$\mathcal{K}(\alpha)$8

even though the thermal prediction at the same energy is $\mathcal{K}(\alpha)$9 (Corps et al., 22 Jul 2025).

The paper contrasts two $D_{ij}$0-basis product states at $D_{ij}$1, $D_{ij}$2, $D_{ij}$3, $D_{ij}$4: $D_{ij}$5

$D_{ij}$6

These states have similar mean energies and initial magnetizations along $D_{ij}$7, but different domain structures: $D_{ij}$8 has three very small domains, while $D_{ij}$9 has two relatively large domains.

Their spectral content is quantified by the local density of states $$\mathcal{K}(\alpha)=\frac{1}{N-1}\sum_{i<j} D_{ij}^{-\alpha}.$$0, displayed through the normalized energy variable

$$\mathcal{K}(\alpha)=\frac{1}{N-1}\sum_{i<j} D_{ij}^{-\alpha}.$$1

For both initial states, most spectral weight lies in the central $$\mathcal{K}(\alpha)=\frac{1}{N-1}\sum_{i<j} D_{ij}^{-\alpha}.$$2 of the spectrum, corresponding to an effective inverse temperature $$\mathcal{K}(\alpha)=\frac{1}{N-1}\sum_{i<j} D_{ij}^{-\alpha}.$$3. Nevertheless, the projection onto the parity-paired subspaces $$\mathcal{K}(\alpha)=\frac{1}{N-1}\sum_{i<j} D_{ij}^{-\alpha}.$$4 yields sharply different magnetization distributions: for $$\mathcal{K}(\alpha)=\frac{1}{N-1}\sum_{i<j} D_{ij}^{-\alpha}.$$5, the distribution is peaked at nonzero $$\mathcal{K}(\alpha)=\frac{1}{N-1}\sum_{i<j} D_{ij}^{-\alpha}.$$6; for $$\mathcal{K}(\alpha)=\frac{1}{N-1}\sum_{i<j} D_{ij}^{-\alpha}.$$7, it is centered near $$\mathcal{K}(\alpha)=\frac{1}{N-1}\sum_{i<j} D_{ij}^{-\alpha}.$$8. The resulting dynamics satisfy

$$\mathcal{K}(\alpha)=\frac{1}{N-1}\sum_{i<j} D_{ij}^{-\alpha}.$$9

Thus, states with a few small magnetic domains evolve toward ferromagnetic equilibrium states, whereas states with larger domains relax to the expected thermal paramagnetic equilibrium state.

The phase boundary is formulated in terms of domain fraction. For initial states composed of one or two domains, the relevant control parameter is $z$00. At $z$01, with $z$02 and $z$03, the long-time averaged magnetization $z$04 displays a threshold structure. For $z$05, there is a clear threshold at $z$06: few or small domains lead to ferromagnetic equilibrium, whereas larger $z$07 or no magnetic structure yield paramagnetic equilibrium. For $z$08, the same pattern survives with a smaller critical fraction. For $z$09, the boundary is less sharp, but the $z$10 data still suggest a ferromagnetic scarred phase at low $z$11.

5. Finite-size scaling, parameter dependence, and limits of the phase

The finite-size analysis indicates that the phenomenon is not a negligible spectral rarity in the long-range regime where it is observed. The number of scarred subspaces with $z$12 grows exponentially with $z$13 for $z$14, $z$15, and $z$16, remains slower but still increasing at $z$17, and ceases to show exponential support at $z$18 and $z$19 (Corps et al., 22 Jul 2025). This establishes a distinction between the long-range regime around and below $z$20 and the effectively short-range regime at larger $z$21, where the ferromagnetic scars become sparse or absent.

The transverse field $z$22 acts as a suppressing parameter. Appendix results show that at $z$23, increasing $z$24 progressively destroys the nonzero-$z$25 bands and leads to a conventional paramagnetic spectrum. At $z$26, increasing $z$27 shrinks the ferromagnetic region and enlarges the paramagnetic one. Operationally, the scarred ferromagnetic phase is strongest at small $z$28 and weakens as the field increases.

All reported results are obtained with periodic boundary conditions in the momentum-zero, positive-inversion sector. The authors state that other symmetry sectors yield qualitatively similar results, although at higher computational cost. Explicit robustness studies against disorder or a longitudinal field are not presented. The paper therefore establishes the phase most directly as a property of the clean periodic model, rather than as a fully characterized perturbatively stable phase in the broad sense.

6. Mechanistic interpretation, relation to other nonergodic phenomena, and experimental relevance

The mechanistic picture ties the phase to domain-wall dynamics, confinement, and generalized ETH violation. The selective population of large-$z$29 subspaces by initial states with a few small domains suggests that domain-wall configurations are not uniformly ergodic at small $z$30. The paper relates the observed long-time ferromagnetic order to bands of symmetry-breaking subspaces that remain dynamically accessible to particular initial conditions. This suggests that the scarred ferromagnetic phase is underpinned by restricted domain-wall propagation and by effective constraints analogous to emergent constants of motion, although the decisive diagnostic reported in the work remains the subspace magnetization eigenvalue $z$31.

In relation to broader nonergodic phenomena, these scars differ from many-body localization and from Hilbert-space fragmentation. They arise in a clean, translationally invariant model and do not rely on strong quenched disorder or strict kinetic constraints. The comparison drawn in the paper is closer to quantum many-body scarring in constrained models such as PXP, but the operative structure here is a long-range Ising system with parity-paired symmetry-breaking subspaces rather than local blockade constraints.

The work also points to plausible experimental implementations. Trapped-ion simulators, with tunable $z$32–$z$33, $z$34 on the order of kHz, controllable $z$35, and arrays of $z$36–$z$37, are identified as compatible with the relevant parameter regime. Rydberg atom chains and arrays provide another plausible platform. The preparation protocol is conceptually simple: initialize product states in the $z$38-basis with a few small domains, then monitor $z$39 and its long-time average. The principal signatures are a nonzero long-time magnetization where thermal ensembles predict $z$40, a threshold in the domain fraction $z$41, and the suppression of the effect as $z$42 increases (Corps et al., 22 Jul 2025).

Taken together, these results define the scarred ferromagnetic phase as a nonequilibrium ordered regime embedded in a thermally paramagnetic part of the spectrum. Its distinctive feature is not merely slow relaxation, but state-selective equilibration into symmetry-broken ferromagnetic long-time states in a model that, at the same energy density and for $z$43, should thermodynamically remain paramagnetic.