Dilated Coordinate Method in Nuclear Lattice EFT

• Dilated coordinate method is an adaptive mesh refinement strategy in nuclear lattice EFT that decouples the resolution requirements of short-range interactions from the infrared volume for extended states.
• It employs an analytic dilation map to transform a uniform auxiliary grid into a nonuniform physical grid, efficiently capturing both fine short-distance details and coarse long-distance tails.
• The approach accelerates convergence in few-body quantum problems through effective volume reduction and precise symmetrization, making it suitable for weakly bound and near-threshold systems.

Searching arXiv for the cited paper and closely related uses of the term. arxiv_search.query({"^^^^^^^^3search_query3 OR ti:\3^^^^^^^^"Dilated coordinate method\"^^^^^^^^3 OR all:\3^^^^^^^^"dilated coordinate method\"","start":^^^^3search_query3^^^^,"max_results":^^^^3id:(He et al., 17 Sep 2025) OR ti:\3search_query3^^^^}) arxiv_search.query returned: ^^^^3id:(He et al., 17 Sep 2025) OR ti:\3^^^^. (&&&^^^^3search_query3^^^^&&&) — "Dilated coordinate method for solving nuclear lattice effective field theory" ^^^^3 OR all:\3^^^^. (&&&^^^^3 OR all:\3^^^^&&&) — "Distributed Optimization via Energy Conservation Laws in Dilated Coordinates"

1. (Suh et al., 2022) — "Continuous-Time Analysis of Accelerated Gradient Methods via Conservation Laws in Dilated Coordinate Systems" Searching for adjacent nuclear-lattice and adaptive-coordinate context. arxiv_search.query({"^^^^^^^^3search_query3 lattice effective field theory\" adaptive mesh coordinate transformation weakly bound few-body systems","start":^^^^3search_query3^^^^,"max_results":^^^^3id:(He et al., 17 Sep 2025) OR ti:\3search_query3^^^^}) arxiv_search.query returned: ^^^^3id:(He et al., 17 Sep 2025) OR ti:\3^^^^. (&&&^^^^3search_query3^^^^&&&) — "Dilated coordinate method for solving nuclear lattice effective field theory" The dilated coordinate method is an adaptive-mesh refinement strategy for nuclear lattice effective field theory (NLEFT) that addresses the multiscale structure of weakly bound few-body systems by replacing the uniform physical lattice with an analytically dilated coordinate system. In its defining formulation, a uniform lattice in an auxiliary coordinate PRESERVED_PLACEHOLDER_^^^^3search_query3^^^^ is mapped into a non-uniform lattice in the physical coordinate PRESERVED_PLACEHOLDER_^^^^3id:(He et al., 17 Sep 2025) OR ti:\3^^^^, so that short-distance regions retain fine resolution while long-distance wave-function tails are represented on a coarser effective mesh. The method is introduced as a practical way to decouple the ultraviolet resolution requirement set by the lattice spacing from the infrared volume requirement imposed by shallow binding and near-threshold behavior (&&&^^^^3search_query3^^^^&&&).

^^^^3id:(He et al., 17 Sep 2025) OR ti:\3^^^^. Multiscale motivation in nuclear lattice effective field theory

In standard NLEFT, the Schrödinger equation is solved on a uniform cubic lattice with spacing PRESERVED_PLACEHOLDER_^^^^3 OR all:\3^^^^, and that same spacing sets the ultraviolet cutoff $\Lambda \sim \pi/a$. This is efficient for compact nuclear structure, but the cost becomes severe when the system has a small binding momentum $Q$ and therefore a large spatial extent. Shallow bound states, halo-like configurations, and near-threshold scattering states require both fine resolution at short distances and a very large box to accommodate the long tail of the wave function. On a uniform grid, these demands conflict directly: small $a$ is needed to resolve short-range interactions, while large volume is needed to suppress finite-volume artifacts (&&&^^^^3search_query3^^^^&&&).

The dilated coordinate method is designed to decouple those requirements. Its central idea is to use a coordinate transformation that gives a fine effective mesh where particles are close and a coarser mesh far away. This extends the useful physical volume without paying the full uniform-lattice cost. The method is therefore a coordinate-level analogue of adaptive mesh refinement, but it is implemented analytically rather than by local remeshing.

A plausible implication is that the method is particularly well matched to finite-volume-dominated problems in which the asymptotic region is smooth but spatially extended. The paper explicitly identifies weakly bound few-body systems as the target setting and presents the method as a foundation for broader $\textit{ab initio}$ applications in exotic nuclei and light hypernuclei (&&&^^^^3search_query3^^^^&&&).

^^^^3 OR all:\3^^^^. Analytic dilation map and transformed Schrödinger equation

The method introduces an auxiliary coordinate $\bm{s}$ and defines the physical coordinate by

$\bm{r}=\bm{s}\,f(s), \qquad s=|\bm{s}|,$

with scaling function

$$f(s)=1+(\lambda-1)\left[1-\exp\!\left(-s^6/R^6\right)\right],$$

where PRESERVED_PLACEHOLDER_^^^^3id:(He et al., 17 Sep 2025) OR ti:\3search_query3^^^^ is the radius up to which the lattice is essentially unchanged and PRESERVED_PLACEHOLDER_^^^^3id:(He et al., 17 Sep 2025) OR ti:\3id:(He et al., 17 Sep 2025) OR ti:\3^^^^ is the asymptotic dilation factor (&&&^^^^3search_query3^^^^&&&).

This mapping has two asymptotic regimes. For PRESERVED_PLACEHOLDER_^^^^3id:(He et al., 17 Sep 2025) OR ti:\3 OR all:\3^^^^, one has PRESERVED_PLACEHOLDER_^^^^3id:(He et al., 17 Sep 2025) OR ti:\33^^^^, so the grid is nearly uniform and preserves the original short-distance resolution. For PRESERVED_PLACEHOLDER_^^^^3id:(He et al., 17 Sep 2025) OR ti:\3^^^^4, the map approaches

PRESERVED_PLACEHOLDER_^^^^3id:(He et al., 17 Sep 2025) OR ti:\3^^^^5

so the physical spacing is enlarged by approximately PRESERVED_PLACEHOLDER_^^^^3id:(He et al., 17 Sep 2025) OR ti:\3^^^^6. A uniform lattice in PRESERVED_PLACEHOLDER_^^^^3id:(He et al., 17 Sep 2025) OR ti:\3^^^^7-space is thus converted into a non-uniform lattice in PRESERVED_PLACEHOLDER_^^^^3id:(He et al., 17 Sep 2025) OR ti:\3^^^^8-space, with the Jacobian controlling the local mesh stretching.

For the two-body problem, the coordinate Jacobian is given explicitly by

PRESERVED_PLACEHOLDER_^^^^3id:(He et al., 17 Sep 2025) OR ti:\3^^^^9

and matrix inversion yields

PRESERVED_PLACEHOLDER_^^^^3 OR all:\3search_query3^^^^

Starting from

PRESERVED_PLACEHOLDER_^^^^3 OR all:\3id:(He et al., 17 Sep 2025) OR ti:\3^^^^

the transformed equation in PRESERVED_PLACEHOLDER_^^^^3 OR all:\3 OR all:\3^^^^-space becomes

PRESERVED_PLACEHOLDER_^^^^3 OR all:\33^^^^

with

PRESERVED_PLACEHOLDER_^^^^3 OR all:\3^^^^4

The transformed problem is still solved on a uniform PRESERVED_PLACEHOLDER_^^^^3 OR all:\3^^^^5-grid, so derivatives can be evaluated using standard finite differences or FFTs, while the physics is encoded in the transformed operators. This is the operative computational mechanism of the method: the coordinate system is dilated, not the underlying solver (&&&^^^^3search_query3^^^^&&&).

3. Few-body formulations and the role of symmetrization

For three-body systems, the construction is formulated in relative coordinates PRESERVED_PLACEHOLDER_^^^^3 OR all:\3^^^^6 and PRESERVED_PLACEHOLDER_^^^^3 OR all:\3^^^^7, with the same dilation applied separately: PRESERVED_PLACEHOLDER_^^^^3 OR all:\3^^^^8 In the chosen coordinates, the three-body kinetic operator is

PRESERVED_PLACEHOLDER_^^^^3 OR all:\3^^^^9

and it is transformed using the same Jacobian machinery as in the two-body case (&&&^^^^3search_query3^^^^&&&).

Because the coordinate mapping depends on particle labeling, the method requires an explicit symmetrization step for identical bosons. The wave function is symmetrized in $\Lambda \sim \pi/a$^^^^3search_query3^^^^-space as

$\Lambda \sim \pi/a$^^^^3id:(He et al., 17 Sep 2025) OR ti:\3^^^^

The paper emphasizes that this step is not merely cosmetic: without it, some dilated-lattice calculations can converge to the wrong infinite-volume limit (&&&^^^^3search_query3^^^^&&&).

This feature distinguishes the method from a purely geometric mesh-stretching prescription. In few-body calculations, the transformed coordinates alter the representation of exchange symmetry, so symmetry restoration becomes part of the numerical formulation itself. This suggests that the coordinate transformation and the many-body Hilbert-space structure cannot be treated independently.

4. Convergence mechanism and effective-volume reduction

The principal computational advantage comes from the way dilation compresses asymptotic tails. A wave function with asymptotic decay $\Lambda \sim \pi/a$^^^^3 OR all:\3^^^^ is compressed in $\Lambda \sim \pi/a$3-space to something like $\Lambda \sim \pi/a$4 in the far region, so the tail is easier to capture on a finite grid. If a uniform lattice needs a radius $\Lambda \sim \pi/a$5, then after dilation the effective size becomes

$\Lambda \sim \pi/a$6

so the required box size is reduced from $\Lambda \sim \pi/a$7 to

$\Lambda \sim \pi/a$8

This is the mechanism by which the method accelerates convergence toward the infinite-volume limit: the interaction region retains the original resolution, while the long-distance region is represented with fewer lattice points (&&&^^^^3search_query3^^^^&&&).

The paper repeatedly frames the benefit in finite-volume terms. Weakly bound states are dominated by large-distance tails, and those tails are the source of slow uniform-lattice convergence. By enlarging the physical spacing only in the asymptotic region, the method preserves ultraviolet resolution while reducing the infrared burden. This suggests that the gain should be strongest precisely when the state is shallow, extended, or close to continuum threshold.

The method is not introduced as a generic replacement for standard NLEFT on all systems. Rather, it is presented as a demonstration on few-body systems accessible via direct diagonalization. The authors also stress that operator transformation is required for observables beyond energies, and that the finite-volume analysis is more subtle for singular or truly long-range interactions such as Coulomb (&&&^^^^3search_query3^^^^&&&).

5. Numerical demonstrations and observed regimes

The reported benchmarks are designed to test exactly the finite-volume and near-threshold regimes that motivate the construction. In a one-dimensional two-boson Gaussian model with a shallow bound state, the uniform lattice shows strong finite-volume dependence, whereas the dilated lattice with $\Lambda \sim \pi/a$9 fm and $Q$^^^^3search_query3^^^^ reaches the infinite-volume limit much faster. The paper states that, for the same number of lattice points, the dilated grid describes a much larger physical domain and captures the exponentially decaying tail more efficiently; in one benchmark, the uniform lattice needed roughly $Q$^^^^3id:(He et al., 17 Sep 2025) OR ti:\3^^^^ to converge, while the dilated lattice achieved comparable accuracy at a significantly smaller number of points (&&&^^^^3search_query3^^^^&&&).

In three dimensions, the same two-boson Gaussian test is repeated with parameters chosen to mimic typical nucleon-nucleon scales. With $Q$^^^^3 OR all:\3^^^^ fm and dilation parameters $Q$3 fm, $Q$4, the dilated lattice converges faster to the benchmark infinite-volume energy $Q$5 MeV. The method is then applied to the deuteron using leading-order pionless EFT,

$Q$6

with $Q$7 MeV and $Q$8 MeV. For $Q$9 fm, $a$^^^^3search_query3^^^^, and $a$^^^^3id:(He et al., 17 Sep 2025) OR ti:\3^^^^ fm, the dilated lattice converges faster to the experimental deuteron binding energy, $a$^^^^3 OR all:\3^^^^ MeV, than the uniform lattice. The remaining $a$3 MeV difference between the two lattice treatments is attributed to residual lattice artifacts associated with the singular LO pionless potential and, according to the paper, could be reduced by soft regularization.

A distinct test concerns the attractive Coulomb problem, where the continuum spectrum satisfies

$a$4

With $a$5 fm and $a$6, the dilated lattice is reported to recover the correct $a$7-fold degeneracy up to at least $a$8, while the uniform lattice begins to fail for higher principal quantum numbers because of finite-volume effects. This example is used to show that the method is valuable not only for bound-state energies but also for near-threshold physics.

For three-body systems, the paper reports that shallow bound states benefit the most. In the three-dimensional three-boson case, where computational scaling is roughly $a$9, the uniform lattice requires $\textit{ab initio}$^^^^3search_query3^^^^ for convergence, while the dilated lattice converges already around $\textit{ab initio}$^^^^3id:(He et al., 17 Sep 2025) OR ti:\3^^^^, corresponding to an estimated speedup of about $\textit{ab initio}$^^^^3 OR all:\3^^^^. The remaining difference between the converged energies is stated to be about $\textit{ab initio}$3 MeV or $\textit{ab initio}$4, and is attributed to tunable residual artifacts (&&&^^^^3search_query3^^^^&&&).

System	Parameters	Reported observation
^^^^3id:(He et al., 17 Sep 2025) OR ti:\3^^^^D two-boson Gaussian	$\textit{ab initio}$5 fm, $\textit{ab initio}$6	Reaches the infinite-volume limit much faster
3D two-boson Gaussian	$\textit{ab initio}$7 fm, $\textit{ab initio}$8 fm, $\textit{ab initio}$9	Faster convergence to $\bm{s}$^^^^3search_query3^^^^ MeV
Deuteron, LO pionless EFT	$\bm{s}$^^^^3id:(He et al., 17 Sep 2025) OR ti:\3^^^^ MeV, $\bm{s}$^^^^3 OR all:\3^^^^ MeV, $\bm{s}$3 fm, $\bm{s}$4, $\bm{s}$5 fm	Faster convergence to $\bm{s}$6 MeV
Attractive Coulomb	$\bm{s}$7 fm, $\bm{s}$8	Correct $\bm{s}$9-fold degeneracy recovered up to at least $\bm{r}=\bm{s}\,f(s), \qquad s=|\bm{s}|,$^^^^3search_query3^^^^
3D three-boson	uniform $\bm{r}=\bm{s}\,f(s), \qquad s=|\bm{s}|,$^^^^3id:(He et al., 17 Sep 2025) OR ti:\3^^^^, dilated $\bm{r}=\bm{s}\,f(s), \qquad s=|\bm{s}|,$^^^^3 OR all:\3^^^^	Estimated speedup about $\bm{r}=\bm{s}\,f(s), \qquad s=|\bm{s}|,$3

6. Scope, limitations, and terminological distinctions

The method is presented as a foundation for extensions to scattering and reaction processes, especially when combined with adiabatic projection or multichannel Monte Carlo methods. The paper also points to dripline nuclei and halo systems, where valence nucleons have spatially extended wave functions, and to light hypernuclei such as $\bm{r}=\bm{s}\,f(s), \qquad s=|\bm{s}|,$4, whose tiny separation energies make large effective volumes essential (&&&^^^^3search_query3^^^^&&&).

Its limitations are stated just as explicitly. The demonstration is restricted to few-body systems accessible via direct diagonalization rather than a full-scale production NLEFT algorithm for large nuclei. For observables beyond energies, operator transformation is required. For singular or truly long-range potentials, such as Coulomb, the finite-volume analysis is more subtle because there is no strict radius $\bm{r}=\bm{s}\,f(s), \qquad s=|\bm{s}|,$5 beyond which the interaction vanishes. In addition, the choice of dilation parameters $\bm{r}=\bm{s}\,f(s), \qquad s=|\bm{s}|,$6 must be optimized to balance resolution and volume, and residual lattice artifacts can remain if the mapping is not properly symmetrized or if the potential is too singular.

A common misconception is to treat “dilated coordinate method” as a generic label for unrelated techniques that also use dilation or dilated coordinates. In the optimization literature, “dilated coordinate systems” denote time-dependent rescalings such as $\bm{r}=\bm{s}\,f(s), \qquad s=|\bm{s}|,$7 used to derive conservation laws for accelerated gradient methods and distributed optimization, rather than adaptive spatial meshes in lattice EFT (Suh et al., 2022, &&&^^^^3 OR all:\3^^^^&&&). Likewise, the phrase should not be conflated with the morphological “dilated symmetric difference” for comparing registered binary images (&&&^^^^3id:(He et al., 17 Sep 2025) OR ti:\3^^^^9&&&) or with learnable-spacing dilated convolutions in CNNs (&&&^^^^3 OR all:\3search_query3^^^^&&&). In the NLEFT context, the term refers specifically to analytic spatial coordinate dilation as an adaptive-mesh mechanism for few-body quantum problems (&&&^^^^3search_query3^^^^&&&).