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Dynamic Models for Two Nonreciprocally Coupled Fields: A Microscopic Derivation for Zero, One, and Two Conservation Laws
by Kristian Blom, Uwe Thiele, Aljaž Godec
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Kristian Blom · Aljaz Godec |
| Submission information | |
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| Preprint Link: | scipost_202510_00016v1 (pdf) |
| Code repository: | https://doi.org/10.5281/zenodo.17251911 |
| Date submitted: | Oct. 9, 2025, 9:44 p.m. |
| Submitted by: | Aljaz Godec |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We construct dynamic models governing two nonreciprocally coupled fields for several cases with zero, one, and two conservation laws. Starting from two microscopic nonreciprocally coupled Ising models, and using the mean-field approximation, we obtain closed-form evolution equations for the spatially resolved magnetization in each lattice. Only allowing for single spin-flip dynamics, the macroscopic equations in the thermodynamic limit are closely related to the nonreciprocal Allen-Cahn equations, i.e. conservation laws are absent. Likewise, only accounting for spin-exchange dynamics within each lattice, the thermodynamic limit yields equations similar to the nonreciprocal Cahn- Hilliard model, i.e. with two conservation laws. In the case of spin-exchange dynamics within and between the two lattices, we obtain two nonreciprocally coupled equations that add up to one conservation law. For each of these cases, we systematically map out the linear instabilities that can arise. Moreover, combining the different dynamics gives a large number of further models. Our results provide a microscopic foundation for a broad class of nonreciprocal field theories, establishing a direct link between nonequilibrium statistical mechanics and macroscopic continuum descriptions.
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Author comments upon resubmission
List of changes
List of changes
In the revised version (attached) we have:
- Improved the grammar, corrected typos, and clarified and reformulated several sentences throughout the manuscript.
- Highlighted the differences between our work and that in references [14] and [26].
- Better explained the model parameters in section 2.1.
- Added section 2.3 to highlight the significance of the nonreciprocal Ising model.
- Added an explanation of the gradient expansion in section 3.3.
- Added sections 3.4 and 5.4 to show how the partial differential equations are expanded close to the stationary solution.
- Completed the missing models in table I.
- Highlighted in the conclusion the shortcoming of the MF approximation, and how one can go beyond the MF approximation.
Current status:
Reports on this Submission
Report #2 by Lorenzo Caprini (Referee 2) on 2025-11-13 (Invited Report)
Report
As a consequence, I consider the paper suitable for publication in SciPost in its current form.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
Report
Just one small question regarding the author's response to my last remark, where they write
"Please note that the characteristic length scale of the SH model has nothing to do with the lattice spacing itself, as any length scale in our theory is inherently defined w.r.t. the lattice spacing. Therefore, the characteristic length scale set by the SH model is also defined w.r.t. the lattice spacing, and therefore is not the lattice spacing itself. "
That all makes sense, but if one takes the limit where the lattice spacing goes to zero, is the characteristic length scale of the SH model then defined with respect to zero (and what does this mean)?
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
Author: Aljaz Godec on 2025-11-17 [id 6040]
(in reply to Report 1 on 2025-10-14)We thank the Reviewer for this comment. To address it, we first clarify the meaning of x:
On a finite lattice, the physical (Manhattan) distance between two spins can always be written as r = k·l, where k is the number of lattice spacings (i.e., the number of spins) between the two reference spins. Importantly, k does not depend on the lattice spacing l and therefore remains well defined in the continuum limit l → 0. It simply reflects how many spins are between two reference spins.
In our manuscript, we map this discrete integer k to a continuum variable x by applying the box-averaging procedure defined on page 9. For any finite lattice, the number of spins between two reference sites is always an integer. However, after performing box averaging over sufficiently many spins, we obtain a local magnetization field defined at each point x ∈ ℝ², where x is the continuum analogue of k.
To ensure that the local magnetization m(x,t) defined on page 9 becomes a continuous real-valued function in the interval [-1,1]—and hence differentiable—we require the thermodynamic limit (N_x, N_y) → ∞. This follows from the fact that the rational numbers are dense in ℝ. Additionally, we require that the lattice spacing l → 0 so that we can apply the gradient expansion. Both limits can be taken simultaneously by considering the scaling limit (N_x, N_y) → ∞ while keeping the physical system size (L_x, L_y) = (l·N_x, l·N_y) fixed, and therefore l → 0.
With this in mind, the characteristic length scale that appears in the linear stability analysis (and therefore also in the SH model) can be interpreted as the number of spins over which the local alignment persists in the most unstable mode—that is, the number of spins over which the local magnetization is “correlated” in that mode. When the largest unstable mode appears at k=0 that means an infinite amount of spins are aligned, and when k->\Infinity it implies that no spins are aligned. The latter however can only occur in our continuum field theory (and not on a discrete lattice), since we have assumed that x is continuous and therefore patterns can emerge on arbitrarily fine scales. Taking into account higher order gradient terms in the gradient expansion would result in a regularization of this divergence for the largest unstable mode at k->\Infinity, and regularize it to a finite k value.
To conclude: x should be understood as the (continuum) counterpart of the number of spins measured from the origin x = 0, and this is well-defined even when l → 0 . It is continuous (and not discrete) because of the box-averaging procedure introduced on page 9. Under this interpretation, the characteristic length scale obtained from the linear stability analysis corresponds to the number of aligned spins associated with the most unstable mode.