Extraordinary-Log Phase in Critical Boundaries

• Extraordinary-log phase is a boundary universality class in O(N) systems where the critical bulk induces logarithmic rather than power-law decay at the boundary.
• The phase is characterized by a two-distance finite-size scaling structure, where short-distance correlations and long-distance plateaus are governed by distinct logarithmic exponents.
• This framework applies across classical and quantum systems, providing actionable insights into unconventional scaling and emergent symmetries in boundary critical phenomena.

Extraordinary-log phase is a boundary or defect universality class of critical $O(N)$ systems in which the bulk is tuned to criticality and the boundary lies on the extraordinary side of the phase diagram, yet boundary correlations are governed by logarithms rather than ordinary power laws. In the three-dimensional $XY$ model, Hu, Deng, and Lv reported that the surface two-point function along the boundary behaves as $g(r)\sim [\ln(r/r_0)]^{-\hat{\eta}}$, while finite systems display a large-distance plateau $g(L/2)\sim [\ln(L/l_0)]^{-\hat{\eta}'}$ rather than conventional algebraic scaling (Hu et al., 2021). Subsequent work established the same general logarithmic universality in open-surface Villain and Bose–Hubbard realizations, in plane defects embedded into 3D and 4D critical systems, and in several discrete models with emergent $O(2)$ symmetry (Sun et al., 2022).

1. Definition within boundary criticality

For a bulk critical system with a boundary, the standard surface universality classes are ordinary, special, and extraordinary. In the ordinary transition the bulk is critical while the surface is not ordered; in the special transition the surface is tuned to the verge between ordered and disordered behavior at the bulk critical point; in the extraordinary transition the bulk is critical and the surface is already ordered because of enhanced surface couplings. In conventional critical phenomena, the two-point function follows $g(r)\sim r^{2-d-\eta}$, and surface correlations in ordinary or special regimes follow $g_{\rm surf}(r)\sim r^{-(d_s-2+\eta_\parallel)}$, with $d_s=d-1$ (Hu et al., 2021).

The extraordinary-log scenario changes only the boundary sector, not the bulk universality class. For three-dimensional $O(N)$ systems with a two-dimensional boundary, the boundary is at its lower critical dimension for continuous symmetry breaking, and recent work describes the extraordinary regime by logarithmic universality rather than by a standard power law. In this regime the boundary remains on the extraordinary side of the phase diagram, but its long-distance correlations decay as powers of $\ln r$, and finite-size observables acquire logarithmic rather than algebraic violations of standard finite-size scaling (Toldin et al., 2024).

2. Correlation structure and two-distance finite-size scaling

The defining phenomenology of the extraordinary-log phase is the replacement of algebraic boundary decay by logarithmic decay. In the 3D $XY$0 model, the surface two-point function along the surface obeys

$XY$1

with a nonuniversal microscopic scale $XY$2 and a universal logarithmic exponent for fixed $XY$3 (Hu et al., 2021).

Hu, Deng, and Lv also proposed that finite systems exhibit a two-distance structure rather than a single $XY$4 scaling function. Their ansatz is

$XY$5

so that short-to-intermediate distances are controlled by an $XY$6-dependent logarithmic term, whereas large distances form an almost $XY$7-independent plateau whose height decays with $XY$8. In the 3D $XY$9 simulations they obtained $g(r)\sim [\ln(r/r_0)]^{-\hat{\eta}}$0, $g(r)\sim [\ln(r/r_0)]^{-\hat{\eta}}$1, and observed $g(r)\sim [\ln(r/r_0)]^{-\hat{\eta}}$2 (Hu et al., 2021).

This two-distance structure also explains why different observables probe different logarithmic exponents. The zero-momentum surface susceptibility is dominated by the plateau and scales as

$g(r)\sim [\ln(r/r_0)]^{-\hat{\eta}}$3

whereas nonzero-momentum fluctuations scale as

$g(r)\sim [\ln(r/r_0)]^{-\hat{\eta}}$4

A central point is therefore that extraordinary-log criticality is not a single-log correction to standard power-law finite-size scaling; it is a distinct boundary scaling structure with separate large-distance and finite-size logarithmic sectors (Hu et al., 2021).

3. Renormalization-group formulation and universal relations

A compact field-theoretic description is provided by a two-dimensional nonlinear $g(r)\sim [\ln(r/r_0)]^{-\hat{\eta}}$5-model for a unit $g(r)\sim [\ln(r/r_0)]^{-\hat{\eta}}$6-vector $g(r)\sim [\ln(r/r_0)]^{-\hat{\eta}}$7 on the boundary or defect,

$g(r)\sim [\ln(r/r_0)]^{-\hat{\eta}}$8

universally coupled to the three-dimensional critical bulk. The running boundary coupling obeys

$g(r)\sim [\ln(r/r_0)]^{-\hat{\eta}}$9

and the extraordinary-log phase corresponds to $g(L/2)\sim [\ln(L/l_0)]^{-\hat{\eta}'}$0, so that $g(L/2)\sim [\ln(L/l_0)]^{-\hat{\eta}'}$1 flows to zero only logarithmically with scale (Toldin et al., 2024).

This RG structure organizes the finite-size behavior of renormalization-group invariant observables. For open boundaries, the rescaled helicity modulus obeys

$g(L/2)\sim [\ln(L/l_0)]^{-\hat{\eta}'}$2

whereas for a single plane defect the coefficient is halved,

$g(L/2)\sim [\ln(L/l_0)]^{-\hat{\eta}'}$3

The boundary correlation-length ratio satisfies

$g(L/2)\sim [\ln(L/l_0)]^{-\hat{\eta}'}$4

and the Binder ratio approaches unity only logarithmically,

$g(L/2)\sim [\ln(L/l_0)]^{-\hat{\eta}'}$5

(Toldin et al., 2024).

The same framework predicts universal asymptotic combinations: $g(L/2)\sim [\ln(L/l_0)]^{-\hat{\eta}'}$6 and

$g(L/2)\sim [\ln(L/l_0)]^{-\hat{\eta}'}$7

for open boundaries, or

$g(L/2)\sim [\ln(L/l_0)]^{-\hat{\eta}'}$8

for a plane defect. Monte Carlo step-scaling analyses further resolved the subleading coefficient $g(L/2)\sim [\ln(L/l_0)]^{-\hat{\eta}'}$9: $O(2)$0 for $O(2)$1, and $O(2)$2 for $O(2)$3 (Toldin et al., 2024).

A second universal relation links logarithmic correlations to stiffness. In the 3D $O(2)$4 surface case, the plateau exponent satisfies

$O(2)$5

and with $O(2)$6 one obtains $O(2)$7, matching the direct estimate $O(2)$8 (Hu et al., 2021). In the Villain realization this same relation appears as

$O(2)$9

with $g(r)\sim r^{2-d-\eta}$0 and $g(r)\sim r^{2-d-\eta}$1 for $g(r)\sim r^{2-d-\eta}$2 (Sun et al., 2022).

4. Realizations in classical and quantum systems

The extraordinary-log phase now appears across several classical and quantum realizations. The examples below emphasize the breadth of the universality class rather than a single microscopic mechanism.

System	Boundary or defect regime	Distinguishing result
3D $g(r)\sim r^{2-d-\eta}$3 model with enhanced surface coupling	Extraordinary-log surface phase	$g(r)\sim r^{2-d-\eta}$4, $g(r)\sim r^{2-d-\eta}$5, $g(r)\sim r^{2-d-\eta}$6 (Hu et al., 2021)
Open-surface Villain representation	Classical–quantum bridge	$g(r)\sim r^{2-d-\eta}$7, $g(r)\sim r^{2-d-\eta}$8, and $g(r)\sim r^{2-d-\eta}$9 for $g_{\rm surf}(r)\sim r^{-(d_s-2+\eta_\parallel)}$0 (Sun et al., 2022)
2D Bose–Hubbard edge at the bulk O(2) quantum critical point	Quantum extraordinary-log phase	$g_{\rm surf}(r)\sim r^{-(d_s-2+\eta_\parallel)}$1 and $g_{\rm surf}(r)\sim r^{-(d_s-2+\eta_\parallel)}$2, with $g_{\rm surf}(r)\sim r^{-(d_s-2+\eta_\parallel)}$3 at the special point (Sun et al., 2022)
Plane defects in 3D and 4D $g_{\rm surf}(r)\sim r^{-(d_s-2+\eta_\parallel)}$4 systems	Defect extraordinary-log criticality	In 3D, $g_{\rm surf}(r)\sim r^{-(d_s-2+\eta_\parallel)}$5: $g_{\rm surf}(r)\sim r^{-(d_s-2+\eta_\parallel)}$6, $g_{\rm surf}(r)\sim r^{-(d_s-2+\eta_\parallel)}$7; in 4D $g_{\rm surf}(r)\sim r^{-(d_s-2+\eta_\parallel)}$8: $g_{\rm surf}(r)\sim r^{-(d_s-2+\eta_\parallel)}$9, $d_s=d-1$0 (Sun et al., 2023)
3D 6-state clock model with open surfaces	Intermediate extraordinary-log phase	Emergent $d_s=d-1$1 symmetry appears on the surface before the surface enters the $d_s=d-1$2 symmetry-breaking region (Zou et al., 2022)
3D antiferromagnetic 3-state Potts model with surface NNN couplings	Sublattice extraordinary-log phase	$d_s=d-1$3 for the dominant sublattice order parameter, with a new special point at $d_s=d-1$4 (Zhang et al., 2023)

The Villain study is especially important because it directly bridges classical 3D O(2) surface criticality and the open-edge quantum Bose–Hubbard model. In that framework the special transition has $d_s=d-1$5 and $d_s=d-1$6, while the extraordinary-log phase is characterized by $d_s=d-1$7 and $d_s=d-1$8, thereby reproducing the logarithmic finite-size scaling of the quantum edge stiffness (Sun et al., 2022).

Discrete-spin realizations show that extraordinary-log behavior does not require microscopic $d_s=d-1$9 symmetry. In the 3D 6-state clock model, the extraordinary-log phase occurs because an $O(N)$0 symmetry emerges on the surface before the surface enters the $O(N)$1 symmetry-breaking region as $O(N)$2 is increased at the bulk critical point (Zou et al., 2022). In the antiferromagnetic Potts case, frustration and sublattice structure produce a distinct sublattice extraordinary-log phase with a new special universality class (Zhang et al., 2023).

5. Defects, topology, and deconfined generalizations

Plane defects show that extraordinary-log criticality is not confined to open surfaces. In 3D plane defects embedded in periodic $O(N)$3 systems, the logarithmic relation $O(N)$4 is satisfied for $O(N)$5, but the same relation is violated for $O(N)$6: the 3D $O(N)$7 defect yields $O(N)$8, $O(N)$9, while the 3D $\ln r$0 defect yields $\ln r$1, $\ln r$2 (Sun et al., 2023). This indicates that the logarithmic universality class persists beyond the $\ln r$3 case, but the simplest stiffness–correlator relation is not universal across all $\ln r$4.

The same boundary phenomenon now appears in topological and deconfined settings. In the Kane–Mele–Hubbard model on the honeycomb lattice with zig-zag edges, tuned to the 3D $\ln r$5 bulk critical point, varying the edge Hubbard interaction produces an ordinary phase with a helical Luttinger liquid edge and an extraordinary-log phase with logarithmically diverging spin stiffness. The boundary transition occurs at $\ln r$6, and in the extraordinary-log phase the edge spectral function changes from a sharp helical mode to a broad continuum with pseudo-gap features and enhanced low-energy spin spectral weight (Toldin et al., 1 Aug 2025).

A closely related generalization arises at deconfined quantum critical points described by the non-compact $\ln r$7 theory. There the boundary Cooper-pair operator has extraordinary-log correlations

$\ln r$8

and a large-$\ln r$9 analysis yields

$XY$00

defining a new family of boundary universality classes parameterized by $XY$01 (Cui et al., 12 Jan 2026). This extends extraordinary-log behavior beyond Landau $XY$02 order parameters to deconfined gauge-theoretic criticality.

6. Counterexamples, competing extraordinary states, and current structure

Extraordinary-log behavior is not synonymous with every extraordinary boundary regime. A clean counterexample is the 3D tricritical $XY$03 model, where the boundary extraordinary transition is not extraordinary-log but truly ordered for any $XY$04. In that case the effective boundary nonlinear $XY$05-model coupling decays faster than $XY$06, so the renormalized boundary order parameter remains finite; the paper identifies this as the first nontrivial example of continuous symmetry breaking in 2D in the context of boundary criticality (Sun et al., 10 Jan 2025).

A second counterexample is the surface of a two-dimensional quantum Heisenberg antiferromagnet at its bulk $XY$07 quantum critical point. There the surface exhibits a direct special transition at $XY$08 with $XY$09 and $XY$10, followed by an extraordinary phase with long-range antiferromagnetic surface order. Fits to an extraordinary-log form produce inconsistent $XY$11 values across observables, whereas extrapolations assuming a finite surface order parameter are mutually consistent (Ding et al., 2021).

These counterexamples show that the extraordinary-log phase is one boundary fixed point among several. This suggests that lower critical dimensionality, emergent continuous symmetry, marginal boundary flows, and the microscopic structure of boundary degrees of freedom jointly determine whether the extraordinary regime is logarithmic or truly ordered. For open boundaries in the 3D $XY$12 model, recent finite-size scaling analyses treat $XY$13 and find negative subleading $XY$14-coefficients, a result that controls how the special and extraordinary-log structures evolve with $XY$15 (Toldin et al., 2024).

Extraordinary-log phase therefore occupies a sharply defined place in modern boundary critical theory. It preserves universality, scaling relations, and RG control, but replaces ordinary algebraic scaling by logarithmic decay, logarithmic stiffness growth, and logarithmic finite-size violations. In current usage, it denotes not merely a correction to scaling, but a distinct boundary or defect universality class of critical systems with continuous or emergent continuous symmetry.