Direction-Selective Criticality

• Direction-selective criticality is a phenomenon where only specific directions or modes become unstable rather than exhibiting global softening across the system.
• It is characterized by geometric and compatibility selection principles that govern phase transitions in models like rotor Hamiltonians, nemato-elastic systems, and non-reciprocal networks.
• This concept enables precise predictions of phase behavior by linking collective mode softening to restricted eigenspaces and asymmetric propagation channels.

Direction-selective criticality denotes a class of critical phenomena in which instability, softening, or divergent response is not global across all relevant degrees of freedom, but is selected along particular directions, modes, symmetry sectors, or propagation channels. In recent arXiv literature, the term appears in several technically distinct settings: mean-field rotor Hamiltonians, where criticality is identified with the vanishing of curvature coefficients in a finite-dimensional collective sector of the microcanonical energy shell; spin-orbit-coupled magnets, where only one symmetry-inequivalent excitation branch becomes gapless; nemato-elastic systems, where Saint Venant compatibility suppresses incompatible nematic fluctuations and leaves only a compatible critical subspace; and directed networks and hypergraphs, where non-reciprocity, anchors, or feed-forward propagation reshape thresholds, scaling, and selected steady states (Cairano, 30 Mar 2026, Lee et al., 22 Apr 2026, Meese et al., 31 Jul 2025, Sun et al., 28 Jan 2026).

1. Core meaning and recurring definitions

Across these works, the “direction” relevant to criticality is not a single universal object. In the geometric rotor formulation, it is a unit vector in order-parameter space, probing perturbations $\delta m = u v$ of collective amplitudes $m = (m_1,\ldots,m_r)^{\mathsf T}$ (Cairano, 30 Mar 2026). In BNZS, it is a symmetry-distinct excitation sector associated with one of two Kramers-doublet-derived gaps, $\Delta_L$ and $\Delta_H$, under a field applied along a crystallographic direction (Lee et al., 22 Apr 2026). In nemato-elasticity, it is a momentum-space direction $\hat q$ that determines whether an orbital nematic fluctuation projects onto a compatible or incompatible helical subspace (Meese et al., 31 Jul 2025). In directed hypergraphs and non-reciprocal networks, it is embedded in asymmetric propagation itself: forward versus backward percolative channels, or the degree of reciprocity $\gamma$ in a non-Hermitian interaction matrix (Sun et al., 28 Jan 2026, Martorell et al., 2023).

A common contrast is with global criticality. BNZS explicitly distinguishes mode-selective or partial quantum criticality from “conventional, global quantum criticality,” where the entire low-energy spectrum softens collectively (Lee et al., 22 Apr 2026). The rotor framework similarly replaces a thermodynamic-only characterization by a spectral geometric criterion that selects which collective mode loses quadratic rigidity first (Cairano, 30 Mar 2026). Nemato-elastic theory makes the same point in another language: only fluctuations lying in a compatible doublet become critical, while other symmetry components remain massive even at the transition (Meese et al., 31 Jul 2025). These formulations differ in microscopic content, but they converge on a restricted-softening picture.

This suggests a broad conceptual distinction between two kinds of critical organization. In one, criticality is extensive across the low-energy sector. In the other, the system approaches instability through a constrained eigenspace, with the noncritical sector remaining stiff, gapped, ordered, or only weakly affected.

2. Geometric selection in mean-field rotor Hamiltonians

In finite-dimensional trigonometric mean-field rotor Hamiltonians, direction-selective criticality is formulated as a property of the extrinsic geometry of the microcanonical constant-energy shell

$\Sigma_E = \{x \in \Lambda \mid H(x)=E\},$

with $\Lambda \simeq \mathbb{R}^{2N}$. The key local observable is the trace of the Weingarten operator,

$W = \Pi [\nabla^2 H(x)] \Pi / \|\nabla H(x)\|,$

whose eigenvalues are the principal curvatures. In Euclidean gauge,

$$\operatorname{Tr} W = \operatorname{div} n = \frac{\Delta H}{\|\nabla H\|} - \frac{\nabla H^{\mathsf T} (\nabla^2 H)\nabla H}{\|\nabla H\|^3},$$

and in the unit-normal gauge it directly controls microcanonical entropy derivatives through

$m = (m_1,\ldots,m_r)^{\mathsf T}$0

with $m = (m_1,\ldots,m_r)^{\mathsf T}$1 (Cairano, 30 Mar 2026).

For Hamiltonians of the form

$m = (m_1,\ldots,m_r)^{\mathsf T}$2

with

$m = (m_1,\ldots,m_r)^{\mathsf T}$3

the mean curvature per particle admits the universal collective expansion

$m = (m_1,\ldots,m_r)^{\mathsf T}$4

where

$m = (m_1,\ldots,m_r)^{\mathsf T}$5

$m = (m_1,\ldots,m_r)^{\mathsf T}$6

The matrices $m = (m_1,\ldots,m_r)^{\mathsf T}$7 and $m = (m_1,\ldots,m_r)^{\mathsf T}$8 encode closure of the trigonometric family and branch covariance on the reference branch, while $m = (m_1,\ldots,m_r)^{\mathsf T}$9 contains the finite set of collective couplings (Cairano, 30 Mar 2026).

The selection principle is spectral. If

$\Delta_L$0

then criticality occurs when the smallest curvature eigenvalue vanishes,

$\Delta_L$1

equivalently $\Delta_L$2. The associated eigenvector $\Delta_L$3 is the critical channel. In this formulation, the energy shell loses quadratic geometric rigidity first along a distinguished collective direction, and phase transition onset is reinterpreted as a geometric instability intrinsic to $\Delta_L$4 (Cairano, 30 Mar 2026).

The worked examples make the criterion explicit. In the Hamiltonian Mean-Field model,

$\Delta_L$5

with $\Delta_L$6, $\Delta_L$7, and $\Delta_L$8, the collective curvature form is

$\Delta_L$9

so the critical condition $\Delta_H$0 yields $\Delta_H$1. In multimode diagonal sectors one has

$\Delta_H$2

hence $\Delta_H$3 and, on the disordered branch, $\Delta_H$4 (Cairano, 30 Mar 2026).

A central limitation is explicit in the same framework: the geometric criterion identifies the mechanism and the critical energy, but not the order of the transition. First- versus second-order behavior still requires additional analysis of $\Delta_H$5 and its derivatives, including microcanonical inflection-point analysis.

3. Sector-selective quantum and elastic criticality in condensed matter

In BaNd$\Delta_H$6ZnS$\Delta_H$7, direction-selective criticality takes the form of a field-induced, mode-selective quantum phase transition. Below the Néel temperature $\Delta_H$8, the Nd$\Delta_H$9 Kramers doublets produce two symmetry-inequivalent low-energy excitation sectors with gaps $\hat q$0 and $\hat q$1. For $\hat q$2, the lower gap $\hat q$3 softens continuously and collapses at $\hat q$4, while the higher gap $\hat q$5 remains finite; neutron diffraction and thermodynamics indicate that the $\hat q$6 component collapses near $\hat q$7, whereas $\hat q$8 long-range order persists until much higher fields, near $\hat q$9–$\gamma$0. The intermediate partially critical phase appears only for $\gamma$1; it is absent for $\gamma$2 and $\gamma$3, where $\gamma$4 and $\gamma$5 remain symmetry-equivalent under the field (Lee et al., 22 Apr 2026).

The thermodynamic signatures are correspondingly partial rather than global. At criticality, the ac susceptibility follows $\gamma$6 and collapses according to

$\gamma$7

with $\gamma$8 and $\gamma$9. The residual Sommerfeld coefficient $\Sigma_E = \{x \in \Lambda \mid H(x)=E\},$0 increases markedly and shows an apparent divergence near $\Sigma_E = \{x \in \Lambda \mid H(x)=E\},$1, indicating a dense set of gapless excitations confined to the critical symmetry sector. The transition is described as continuous within experimental resolution: Ehrenfest consistency holds for the heat-capacity anomalies, whereas Clausius-Clapeyron fails (Lee et al., 22 Apr 2026).

In nemato-elastic systems, the selection mechanism is instead enforced by compatibility. The strain tensor $\Sigma_E = \{x \in \Lambda \mid H(x)=E\},$2 must satisfy the Saint Venant relations,

$\Sigma_E = \{x \in \Lambda \mid H(x)=E\},$3

or equivalently

$\Sigma_E = \{x \in \Lambda \mid H(x)=E\},$4

In a co-rotating helical basis, these constraints split the five-component traceless nematic fluctuation space into a compatible doublet and noncritical amplitudes. The effective Gaussian kernel can be written as

$\Sigma_E = \{x \in \Lambda \mid H(x)=E\},$5

with masses

$\Sigma_E = \{x \in \Lambda \mid H(x)=E\},$6

Thus only $\Sigma_E = \{x \in \Lambda \mid H(x)=E\},$7 and $\Sigma_E = \{x \in \Lambda \mid H(x)=E\},$8 become critical at $\Sigma_E = \{x \in \Lambda \mid H(x)=E\},$9, while incompatible fluctuations remain gapped (Meese et al., 31 Jul 2025).

The directional selectivity follows from projection back to orbital channels. For an Ising nematic $\Lambda \simeq \mathbb{R}^{2N}$0, the critical directions are $\Lambda \simeq \mathbb{R}^{2N}$1 and $\Lambda \simeq \mathbb{R}^{2N}$2; for $\Lambda \simeq \mathbb{R}^{2N}$3, they are $\Lambda \simeq \mathbb{R}^{2N}$4 and $\Lambda \simeq \mathbb{R}^{2N}$5. A companion formulation describes the same bifurcation as “compatible instability,” emphasizing that the critical modes are protected from pinning by defect strains, while defects generate long-ranged random longitudinal and transverse conjugate fields only in noncritical helical channels (Meese et al., 31 Jul 2025). One consequence is that mean-field thermodynamics and widespread domain formation are not contradictory within this framework.

4. Directed propagation, non-reciprocity, and higher-order network criticality

A distinct use of direction-selective criticality appears in nonequilibrium directed systems. In a chain of adaptive excitable integrators, directionality is literal: coupling is strictly feed-forward, $\Lambda \simeq \mathbb{R}^{2N}$6, the drive enters only at $\Lambda \simeq \mathbb{R}^{2N}$7, and the boundary at $\Lambda \simeq \mathbb{R}^{2N}$8 is open. The adaptive thresholds obey

$\Lambda \simeq \mathbb{R}^{2N}$9

with $W = \Pi [\nabla^2 H(x)] \Pi / \|\nabla H(x)\|,$0, and the per-level gain

$W = \Pi [\nabla^2 H(x)] \Pi / \|\nabla H(x)\|,$1

self-organizes near marginal propagation, numerically $W = \Pi [\nabla^2 H(x)] \Pi / \|\nabla H(x)\|,$2–$W = \Pi [\nabla^2 H(x)] \Pi / \|\nabla H(x)\|,$3 and $W = \Pi [\nabla^2 H(x)] \Pi / \|\nabla H(x)\|,$4 in the fixed-threshold approximation. The model exhibits discrete scale invariance,

$W = \Pi [\nabla^2 H(x)] \Pi / \|\nabla H(x)\|,$5

with mixture and sum exponents near $W = \Pi [\nabla^2 H(x)] \Pi / \|\nabla H(x)\|,$6 and $W = \Pi [\nabla^2 H(x)] \Pi / \|\nabla H(x)\|,$7, respectively. Threshold and subthreshold spectra are Lorentzian with position-dependent corner frequencies, decreasing along the chain from $W = \Pi [\nabla^2 H(x)] \Pi / \|\nabla H(x)\|,$8 to $W = \Pi [\nabla^2 H(x)] \Pi / \|\nabla H(x)\|,$9 and from $$\operatorname{Tr} W = \operatorname{div} n = \frac{\Delta H}{\|\nabla H\|} - \frac{\nabla H^{\mathsf T} (\nabla^2 H)\nabla H}{\|\nabla H\|^3},$$0 to $$\operatorname{Tr} W = \operatorname{div} n = \frac{\Delta H}{\|\nabla H\|} - \frac{\nabla H^{\mathsf T} (\nabla^2 H)\nabla H}{\|\nabla H\|^3},$$1 (Martinez-Saito, 2022).

In non-reciprocal neural networks, directionality is encoded by the reciprocity parameter $$\operatorname{Tr} W = \operatorname{div} n = \frac{\Delta H}{\|\nabla H\|} - \frac{\nabla H^{\mathsf T} (\nabla^2 H)\nabla H}{\|\nabla H\|^3},$$2 in the coupling statistics,

$$\operatorname{Tr} W = \operatorname{div} n = \frac{\Delta H}{\|\nabla H\|} - \frac{\nabla H^{\mathsf T} (\nabla^2 H)\nabla H}{\|\nabla H\|^3},$$3

Linear stability of the quiescent state is governed by

$$\operatorname{Tr} W = \operatorname{div} n = \frac{\Delta H}{\|\nabla H\|} - \frac{\nabla H^{\mathsf T} (\nabla^2 H)\nabla H}{\|\nabla H\|^3},$$4

The system supports paramagnetic, ferromagnetic, and spin-glass-like regions. In the spin-glass region, reciprocal couplings produce marginal behavior, whereas decreasing reciprocity drives a smooth transition to chaos; the spin-glass region shrinks and disappears at $$\operatorname{Tr} W = \operatorname{div} n = \frac{\Delta H}{\|\nabla H\|} - \frac{\nabla H^{\mathsf T} (\nabla^2 H)\nabla H}{\|\nabla H\|^3},$$5. Dynamic mean-field theory identifies a selected separatrix state through

$$\operatorname{Tr} W = \operatorname{div} n = \frac{\Delta H}{\|\nabla H\|} - \frac{\nabla H^{\mathsf T} (\nabla^2 H)\nabla H}{\|\nabla H\|^3},$$6

in the noiseless case and

$$\operatorname{Tr} W = \operatorname{div} n = \frac{\Delta H}{\|\nabla H\|} - \frac{\nabla H^{\mathsf T} (\nabla^2 H)\nabla H}{\|\nabla H\|^3},$$7

with noise. In the ferromagnetic region, only fixed points are dynamically realizable; in the spin-glass region, the selected state is marginal in the ensemble description but single realizations generically display chaos, with $$\operatorname{Tr} W = \operatorname{div} n = \frac{\Delta H}{\|\nabla H\|} - \frac{\nabla H^{\mathsf T} (\nabla^2 H)\nabla H}{\|\nabla H\|^3},$$8 for sufficiently large $$\operatorname{Tr} W = \operatorname{div} n = \frac{\Delta H}{\|\nabla H\|} - \frac{\nabla H^{\mathsf T} (\nabla^2 H)\nabla H}{\|\nabla H\|^3},$$9 (Martorell et al., 2023).

Directed hypergraph percolation generalizes the same logic to higher-order interactions with asymmetric functional dependencies. A directed hyperedge maps an input set $m = (m_1,\ldots,m_r)^{\mathsf T}$00 to an output set $m = (m_1,\ldots,m_r)^{\mathsf T}$01, while anchor nodes encode indispensable participants. The renormalized availability is

$m = (m_1,\ldots,m_r)^{\mathsf T}$02

and the percolation threshold is controlled by

$m = (m_1,\ldots,m_r)^{\mathsf T}$03

The Hypergraph Giant In Component, Hypergraph Giant Out Component, and Hypergraph Giant Strongly Connected Component emerge simultaneously, but post-critical scaling differs. In finite-moment regimes,

$m = (m_1,\ldots,m_r)^{\mathsf T}$04

In maximally correlated heavy-tailed regimes, anomalous exponents depend on whether node or hyperedge percolation is considered, and anchor-free systems can exhibit modified composition rules such as $m = (m_1,\ldots,m_r)^{\mathsf T}$05 in the reported $m = (m_1,\ldots,m_r)^{\mathsf T}$06 regimes (Sun et al., 28 Jan 2026).

Taken together, these systems show that “direction-selective” need not refer only to spatial anisotropy. It can refer to asymmetric causal structure, forward/backward reachability, or non-reciprocal spectral selection of attractors and exponents.

5. Strain direction and optimization trade-offs as selectors of instability channels

In hole-doped manganite La$m = (m_1,\ldots,m_r)^{\mathsf T}$07Ca$m = (m_1,\ldots,m_r)^{\mathsf T}$08MnO$m = (m_1,\ldots,m_r)^{\mathsf T}$09, direction-selective criticality is implemented by uniaxial strain as a crystallographically resolved tuning field. The structural response is decomposed into breathing and Jahn-Teller modes,

$m = (m_1,\ldots,m_r)^{\mathsf T}$10

with $m = (m_1,\ldots,m_r)^{\mathsf T}$11, alongside site-average and site-selective combinations such as $m = (m_1,\ldots,m_r)^{\mathsf T}$12, $m = (m_1,\ldots,m_r)^{\mathsf T}$13, $m = (m_1,\ldots,m_r)^{\mathsf T}$14, and $m = (m_1,\ldots,m_r)^{\mathsf T}$15. Extreme uniaxial strain up to nearly $m = (m_1,\ldots,m_r)^{\mathsf T}$16 along $m = (m_1,\ldots,m_r)^{\mathsf T}$17, $m = (m_1,\ldots,m_r)^{\mathsf T}$18, $m = (m_1,\ldots,m_r)^{\mathsf T}$19, and $m = (m_1,\ldots,m_r)^{\mathsf T}$20 stabilizes qualitatively distinct responses rather than different strengths of one phase (Lee et al., 27 May 2026).

The selected channel depends on direction. Along $m = (m_1,\ldots,m_r)^{\mathsf T}$21, the response is predominantly cooperative Jahn-Teller: $m = (m_1,\ldots,m_r)^{\mathsf T}$22 grows, $m = (m_1,\ldots,m_r)^{\mathsf T}$23 remains vanishingly small, and the orbital pattern tends toward staggered $m = (m_1,\ldots,m_r)^{\mathsf T}$24 above about $m = (m_1,\ldots,m_r)^{\mathsf T}$25. Along $m = (m_1,\ldots,m_r)^{\mathsf T}$26, both Jahn-Teller and breathing amplitudes increase, but $m = (m_1,\ldots,m_r)^{\mathsf T}$27 and $m = (m_1,\ldots,m_r)^{\mathsf T}$28 become much larger above about $m = (m_1,\ldots,m_r)^{\mathsf T}$29, and both Mn sites evolve toward $m = (m_1,\ldots,m_r)^{\mathsf T}$30. Along the diagonals, strain is applied along Mn–O bonds, suppresses the in-plane rotation $m = (m_1,\ldots,m_r)^{\mathsf T}$31, and produces strong site selectivity in both $m = (m_1,\ldots,m_r)^{\mathsf T}$32 and $m = (m_1,\ldots,m_r)^{\mathsf T}$33, stabilizing the CF phase with ferro-orbital order within one Mn sublattice and C-type charge disproportionation between sublattices. Under biaxial strain the FM$m = (m_1,\ldots,m_r)^{\mathsf T}$34A-type AFM boundary appears near $m = (m_1,\ldots,m_r)^{\mathsf T}$35, whereas under diagonal uniaxial strain the FM metallic ground state remains stable to at least $m = (m_1,\ldots,m_r)^{\mathsf T}$36 (Lee et al., 27 May 2026).

A different but related selector appears in the spatiotemporal TDANN model of primate MT. There the control parameter is the balance between a contrastive objective and a spatial regularizer,

$m = (m_1,\ldots,m_r)^{\mathsf T}$37

with

$m = (m_1,\ldots,m_r)^{\mathsf T}$38

Varying $m = (m_1,\ldots,m_r)^{\mathsf T}$39 induces distinct map regimes: fragmented maps with many defects at $m = (m_1,\ldots,m_r)^{\mathsf T}$40; an intermediate regime near $m = (m_1,\ldots,m_r)^{\mathsf T}$41 with smooth direction slabs, pinwheels, fraction of direction-selective units $m = (m_1,\ldots,m_r)^{\mathsf T}$42 of about $m = (m_1,\ldots,m_r)^{\mathsf T}$43, median DSI about $m = (m_1,\ldots,m_r)^{\mathsf T}$44, median circular variance about $m = (m_1,\ldots,m_r)^{\mathsf T}$45, primary FWHM of $m = (m_1,\ldots,m_r)^{\mathsf T}$46, and pinwheel density near $m = (m_1,\ldots,m_r)^{\mathsf T}$47; then defect proliferation near $m = (m_1,\ldots,m_r)^{\mathsf T}$48; and oversmoothing near $m = (m_1,\ldots,m_r)^{\mathsf T}$49 (Gu et al., 12 May 2026).

The paper explicitly stops short of claiming established criticality. It reports bifurcation-like behavior, re-entrant defect density, and “qualitative hallmarks” suggestive of competing-energy landscapes, while noting that no finite-size scaling, critical exponents, or precise $m = (m_1,\ldots,m_r)^{\mathsf T}$50 are provided. This is therefore a near-critical or phase-like use of direction-selective organization rather than a demonstrated universality class (Gu et al., 12 May 2026).

6. Universality, distinctions, and open problems

The literature assigns different meanings to universality. In the rotor case, universality refers to the collective geometric expansion of $m = (m_1,\ldots,m_r)^{\mathsf T}$51 within a broad class of finite-dimensional trigonometric mean-field interactions, with model dependence reduced to a finite set of collective couplings and closure data (Cairano, 30 Mar 2026). In nemato-elasticity, universality refers to compatibility itself: the gauge constraints of elasticity suppress incompatible fluctuations independently of crystalline anisotropy in the ideal medium (Meese et al., 31 Jul 2025). In directed hypergraph percolation, by contrast, universality can break down: anomalous exponents depend on heavy tails, maximal correlations, anchors, and whether node or hyperedge percolation is performed (Sun et al., 28 Jan 2026). In BNZS, the extracted exponents are reported to be smaller than in conventional BEC- or Ising-like transitions, consistent with criticality confined to a restricted sector rather than global softening (Lee et al., 22 Apr 2026).

Several misconceptions are explicitly corrected by these works. Direction-selective criticality is not synonymous with ordinary anisotropy in real space. It may be selection in order-parameter space, as in the eigenvectors of $m = (m_1,\ldots,m_r)^{\mathsf T}$52 in rotor Hamiltonians; symmetry-sector selectivity, as in $m = (m_1,\ldots,m_r)^{\mathsf T}$53 while $m = (m_1,\ldots,m_r)^{\mathsf T}$54 stays finite in BNZS; compatibility-selected momentum directions in nematicity; forward/backward branching channels in directed hypergraphs; or crystallographic strain selection of Jahn-Teller versus breathing instabilities in manganites (Cairano, 30 Mar 2026, Lee et al., 22 Apr 2026, Lee et al., 27 May 2026, Sun et al., 28 Jan 2026). It also does not, by itself, fix transition order. The rotor framework requires additional thermodynamic analysis to distinguish first- from second-order transitions, and the MT study does not establish a critical point despite reporting phase-like behavior (Cairano, 30 Mar 2026, Gu et al., 12 May 2026).

The outstanding problems are correspondingly heterogeneous. Rotor models require branch-dependent analysis when multiple competing phases are present, including expansions around symmetry-broken branches (Cairano, 30 Mar 2026). Nemato-elastic theory leaves open dynamic criticality, explicit lattice-anisotropic implementations, and nonlinear fluctuation effects beyond the Gaussian sector (Meese et al., 31 Jul 2025). BNZS motivates angle-dependent field studies, neutron spectroscopy of branch-selective softening, and further tests of partially critical phases in spin-orbit-coupled rare-earth magnets (Lee et al., 22 Apr 2026). The directed-chain model leaves open continuous-time extensions, within-level spatial structure, and the behavior under strongly autocorrelated inputs (Martinez-Saito, 2022). The MT model identifies dense $m = (m_1,\ldots,m_r)^{\mathsf T}$55 sweeps, finite-area scaling, and defect statistics as the next step if a true critical point is to be claimed (Gu et al., 12 May 2026).

A plausible synthesis is that direction-selective criticality names a constrained route to instability: a system may approach criticality not by uniformly softening all relevant fluctuations, but by reorganizing its accessible fluctuation space so that only a selected subset becomes soft. The selector can be geometric curvature, compatibility, anisotropic exchange, non-reciprocal propagation, higher-order functional asymmetry, uniaxial strain, or an optimization trade-off. What unifies these otherwise disparate cases is the replacement of global softening by restricted critical channels.