Surface Criticality in Classical & Quantum Systems

• Surface criticality is the phenomenon where boundary regions display unique critical exponents and scaling laws distinct from the bulk, driven by reduced dimensionality and altered symmetry.
• It encompasses regimes like ordinary, extraordinary, and special transitions, each marked by specific order parameters and crossover behaviors in classical and quantum systems.
• Field-theoretical and experimental studies show that disorder, topological effects, and finite-size constraints further modify surface scaling, opening new research directions.

Surface criticality is the phenomenon in which critical behavior develops at the boundary of a system—real or effective—when the bulk is tuned to, or near, a critical point. This leads to distinct scaling laws, critical exponents, and universality classes for the surface, typically governed by reduced dimensionality, altered symmetry, or changed coupling parameters at the boundary. Unlike bulk criticality, which involves long-range fluctuations throughout the volume, surface criticality encodes how boundaries, interfaces, or edges exhibit universal scaling, crossover, or novel order both in classical and quantum systems. The subject spans statistical physics, condensed matter theory, field theory, and topological phases.

1. Definitions, Theoretical Structure, and Distinction from Bulk Criticality

Surface criticality first arises when boundary degrees of freedom acquire anomalous scaling distinct from the "bulk" (interior) system. In classical lattice models, semi-infinite or finite-geometry counterparts often reveal new "surface" exponents, such as the surface magnetization exponent $\beta_1$, surface-surface susceptibility exponent $\gamma_{11}$, and independent surface correlation exponents $\eta_\parallel$ (parallel to the surface) and $\eta_\perp$ (perpendicular) (Diep, 2024, Zhang et al., 2022). In percolation, one defines a surface order parameter $P_s$ for the fraction of surface sites in the infinite cluster, which exhibits its own exponent $\beta_s$ (Baek et al., 2010). In statistical mechanics, criticality of the surface can occur at, above, or below the bulk critical temperature ($T_c$), yielding ordinary, extraordinary, or special transitions; see below for precise classification.

Quantum versions occur both in conventional symmetry-breaking transitions and in topological systems. For example, surface quantum criticality can occur at the boundary of gapped topological phases tuned by symmetry breaking or proximity perturbations, leading to new universality classes (Karcher et al., 2021, Yang et al., 2020). In such contexts, surface criticality is protected by global topology or symmetry and can be robust against disorder, showing multifractal spectra or distinct CFT descriptions.

2. Universality Classes and Scaling: Ordinary, Extraordinary, and Special Transitions

The classification of surface criticality is rooted in boundary RG and mean-field analysis (Diep, 2024, Zhu et al., 10 Aug 2025, Zhang et al., 2022). Three primary universality classes exist when the bulk is tuned to criticality:

• Ordinary transition: The surface is locally disordered at $T=T_c$ (the bulk transition point) and only fluctuates critically due to bulk correlations; all singularities in surface observables are induced by the bulk.
• Extraordinary transition: The surface has already ordered (or developed long-range correlations) above $T_c$, and the bulk subsequently orders in the presence of an already-ordered surface.
• Special transition: Tuning the surface enhancement parameter (e.g., the ratio $J_s/J_b$ in magnets), the surface and bulk critical points coincide at a multicritical point with new scaling laws.

These regimes are characterized by distinct critical exponents. For example, in 3D Ising models, the ordinary transition has $\gamma_{11}$0, $\gamma_{11}$1, and $\gamma_{11}$2, while at the special transition $\gamma_{11}$3 and $\gamma_{11}$4 (Diep, 2024, Zhang et al., 2022). In the extraordinary regime for $\gamma_{11}$5-like or $\gamma_{11}$6 boundary symmetry, true long-range order is absent but parallel correlations decay logarithmically rather than algebraically ("extraordinary-log" phase) (Zhang et al., 2022).

Surface critical exponents obey generalized scaling and hyperscaling relations (such as $\gamma_{11}$7), but their numerical values, and in some cases even their qualitative behavior (e.g. logarithmic vs algebraic decay), differ radically from bulk exponents and strongly depend on boundary conditions or symmetry.

3. Field-Theoretical Approaches and Topological Surface Criticality

In quantum systems, field-theory developments reveal that surface criticality can be protected or even enforced by topology or symmetry. In particular, the surface of a 3D topological superconductor in class AIII, CI, or DIII supports massless Dirac or Majorana fermions whose low-energy disordered theory is described by perturbed Wess–Zumino–Novikov–Witten (WZNW) $\gamma_{11}$8-models with topological $\gamma_{11}$9 (Karcher et al., 2021). Remarkably, numerical and analytic studies show that in these classes, the entire energy spectrum of surface states remains at quantum criticality ("spectrum-wide surface quantum criticality" or SWQC): all finite-energy surface eigenstates exhibit universal multifractal statistics and critical conductance, analogous to the 2D quantum Hall plateau transition, but protected throughout the spectrum (see Table below; (Karcher et al., 2021, Niwazuki et al., 24 Nov 2025)).

TSC Class	Surface Dirac Theory	Quantum Hall Plateau	Field Theory (low-$\eta_\parallel$0)
AIII	2D Dirac, chiral U(1)	Class A (integer)	WZNW $\eta_\parallel$1, $\eta_\parallel$2
CI	2D Dirac, chiral SU(2)	Class C (spin)	WZNW $\eta_\parallel$3
DIII	2D Majorana, no spin	Class D (thermal)	WZNW $\eta_\parallel$4

SWQC is signaled by universal multifractal exponents (e.g., parabolic $\eta_\parallel$5), length-independent critical conductance, and the absence of both Anderson localization and weak antilocalization across the spectrum. The link to Pruisken models with topological angle $\eta_\parallel$6 is established at the field-theory level (Karcher et al., 2021).

4. Surface Criticality under Disorder, Topological Proximity, and Quantum Phase Transitions

Disorder and proximity effects yield rich surface criticality scenarios. For Dirac materials (e.g., topological insulator surfaces, Weyl/Dirac semimetals), boundaries can induce phase transitions distinct from the bulk, with critical disorder strength for surface localization ($\eta_\parallel$7) generically lower than that for the bulk ($\eta_\parallel$8). Dirac semimetals display sharp surface transitions at $\eta_\parallel$9 dependent on boundary conditions, and bifurcation points where the surface metallic phase precedes the bulk transition. Fermi arcs in Weyl semimetals, by contrast, preclude sharp surface criticality by acting as a metallic bath (Brillaux et al., 2020).

In topological superconductors, hybridization with trivial bands (trivializing proximity effect, TPE) can destroy spectrum-wide surface criticality, leading to the Anderson localization of all surface states except for symmetry-protected zero-modes, provided the surface-bulk band crossing is detached by a gap. However, in effective continuum Dirac models, even if trivial bands are present, increasing disorder "heals" the surface and reestablishes criticality, indicating the necessity of a microscopic lattice description to capture full surface fragility (Niwazuki et al., 24 Nov 2025).

For symmetry-breaking quantum transitions (e.g., quantum Ising models with special interactions), a boundary can induce surface criticality even at a first-order bulk transition: the healing length of the order parameter diverges logarithmically near the boundary as the transition is approached, a quantum analog of classical surface criticality (Nakamura et al., 2024).

Surface quantum criticality also arises as a precursor in topological phases: in 3D TRI topological superfluids/superconductors, time-reversal-symmetry-breaking (TRB) fields generically gap surface Majorana cones at infinitesimal strength, producing a $\eta_\perp$0D Majorana Dirac CFT exactly at zero TRB field—defining a surface QCP distinct from, and generally preceding, the bulk topological critical point (Yang et al., 2020).

5. Surface Criticality in Experiments, Applications, and Beyond Equilibrium

Surface criticality plays a key mechanistic role in a broad spectrum of systems. In ultra-thin magnetic films or 2D magnets, surface and bulk ordering temperatures, and fluctuation scaling, can be tuned by surface reconstruction, competing interactions, and Dzyaloshinskii–Moriya (DM) couplings, leading to unique dynamical relaxation and local mode softening at the surface (Diep, 2024).

In finite-size scaling and complex geometries (e.g., additive manufacturing), criticality is also manifest at surfaces: machine learning models robustly identify near-boundary pores as disproportionally critical failure origins, with "surface distance" surpassing all other geometric descriptors in predictive power (Mishra et al., 3 Feb 2026). This explicitly links surface criticality to boundary-driven failure mechanisms in materials science.

Nonequilibrium scenarios—such as periodically driven (kinetic) Ising models—produce genuinely new surface universality classes, distinct from equilibrium behavior, with surface critical exponents and phase diagrams that sharply deviate from their static counterparts (Park et al., 2012). The breaking of spatial symmetries at boundaries allows for this departure, while bulk exponents remain unchanged.

Surface criticality is also pivotal at the interface between entanglement physics and quantum phase transitions. The entanglement spectrum of a bipartitioned many-body state can be dominated either by bulk or surface criticalities, depending on the geometry of the "cut." Only bipartitions corresponding to ordinary boundaries reveal genuine bulk critical behavior in scaling of entanglement entropy, while extraordinary or special edge cuts inject non-universal surface terms that could mask or mimic bulk scaling (Zhu et al., 10 Aug 2025).

6. Universality, Field Theory, and Extensions to Topological and Rotating Systems

Surface criticality exhibits robust universality—scaling exponents and singularities are insensitive to microscopic details, depending only on symmetry and dimensionality. Field-theoretic advances demonstrate the existence of interacting conformal manifolds of fixed points (not just isolated fixed points as in the Wilson–Fisher paradigm) in boundary/topological quantum criticality; the number of marginal operators determines the dimension of the fixed-point manifold (Vijayan et al., 2024). Higher-loop corrections can break conformal manifolds into discrete multicritical points, leading to a rich structure of infrared-stable or unstable boundary phases.

The concept extends to nontrivial gravitational backgrounds: in black hole thermodynamics, a "surface" first law with horizon surface tension $\eta_\perp$1 conjugate to area $\eta_\perp$2 captures horizon criticality. Rotating black holes present surface phase transitions with mean-field exponents, building an explicit analogy between the membrane paradigm and surface statistical criticality (Hansen et al., 2016).

7. Perspectives and Open Problems

Surface criticality unifies boundary scaling, topological protection, disorder, and geometry at the interface of bulk phase transitions. Across percolation (Baek et al., 2010), quantum magnets (Diep, 2024, Zhang et al., 2022), topological phases (Karcher et al., 2021, Yang et al., 2020, Niwazuki et al., 24 Nov 2025), polymers (Foster, 2010), and engineered platforms (Nakamura et al., 2024), it delineates distinct universality classes and critical phenomena emergent only at boundaries.

Modern research continues to extend this paradigm to non-equilibrium, multi-field, and high-dimensional systems, and to explore the ramifications in entanglement theory, machine learning-based materials assessment, and even black-hole physics. Open problems include the full resolution of spectrum-wide quantum criticality at topological boundaries, the classification of boundary RG fixed-point manifolds, and the design of experimental probes of surface exponents and dynamic criticality in artificial systems.

Surface criticality remains a central theme for understanding how global phase transitions modulate, and are modulated by, the presence of boundaries across classical, quantum, and topological matter (Karcher et al., 2021, Diep, 2024, Yang et al., 2020, Niwazuki et al., 24 Nov 2025, Park et al., 2012).