Surface Criticality & Entanglement Scaling

• Surface criticality is the emergent universal scaling behavior at quantum critical points influenced by boundary conditions and impurities.
• Entanglement scaling quantifies how quantum correlations adjust with system size and subsystem geometry, separating bulk and surface contributions.
• Refined diagnostics, including entanglement entropy derivatives and spectrum analysis, reveal distinct edge modes and crossover regimes in critical systems.

Surface criticality refers to the emergent universal scaling behavior governed by boundaries or impurities at quantum critical points, influencing both local observables and entanglement properties. Entanglement scaling describes how quantum correlations, especially as quantified by measures such as the entanglement entropy or negativity, scale with system size or subsystem geometry at such critical points. The interplay between bulk universality and surface/boundary conditions leads to a diverse hierarchy of scaling regimes, with the entanglement structure strongly modified by the presence and type of boundary excitations. This article systematically reviews the theoretical foundations, scaling forms, and physical implications of surface criticality and entanglement scaling, with an emphasis on (2+1)D quantum systems, one-dimensional critical points with boundary impurities, and recent advances using quantum Monte Carlo and tensor network methods.

1. Boundary Universality Classes and Surface Criticality

At a conformally invariant quantum critical point (QCP) in $d$ dimensions, the presence of a physical boundary or an effective one induced by an entanglement cut selects a specific surface universality class. These classes—ordinary, special, and extraordinary—are determined by the low-energy spectrum realized at the surface and their coupling to bulk critical fluctuations:

• Ordinary class: The boundary is gapped in the adjacent phase, yielding no intrinsic low-energy edge states. For example, a cut through weak bonds in a dimerized system leaves a gapped 1D chain on the edge at criticality (Wang et al., 18 Jan 2026).
• Special class: The boundary is critical due to a gapless 1D mode such as an antiferromagnetic Heisenberg chain, e.g., a cut along strong bonds (Wang et al., 18 Jan 2026).
• Extraordinary class: The boundary develops spontaneous order at criticality, typically realized by a staggered cut producing a ferrimagnetic chain in the corresponding gapped phase (Wang et al., 18 Jan 2026, Zhu et al., 10 Aug 2025).

The assignment of boundary universality class controls which critical operators dominate surface scaling, as established in both classical and quantum systems.

2. Scaling Forms for Entanglement Entropy: Bulk and Surface Contributions

The entanglement entropy (EE) for a subregion $A$ of size $L$ in $d$ dimensions at a QCP traditionally obeys the area law,

$S(L) = a\,L^{d-1} + \gamma + \cdots,$

where $a$ is a non-universal coefficient and $\gamma$ a universal constant. However, the presence of surface criticality mandates a refined scaling ansatz (Wang et al., 18 Jan 2026, Zhu et al., 10 Aug 2025):

$$S(L, g) = a(g) L + \tilde{S}_b(x) - \tilde{S}_s(x),$$

for scaling variable $x = (g-g_c)L^{1/\nu}$, where $\tilde{S}_b(x)$ encodes the bulk critical contribution and $\tilde{S}_s(x)$ the surface-induced correction.

At criticality,

$S(L, g_c) = a(g_c)L + \gamma_b - \gamma_s,$

with $\gamma_b$ a bulk constant and $\gamma_s$ a universal surface constant, nonzero only if the entanglement boundary hosts gapless modes (Wang et al., 18 Jan 2026). This geometry dependence of $\gamma$ resolves longstanding puzzles about cut-sensitive constants in critical EE.

In the presence of corners with opening angle $\theta$, further logarithmic corrections appear:

$$S_2(A) = a\,|\partial A| - n_c\,c(\theta)\,\ln L + O(1),$$

with $c(\theta)$ a universal function controlled by the bulk conformal data. For standard unitary CFTs, $c(\theta) \ge 0$; negative values imply nonunitary criticality (Zhao et al., 2021).

3. Universal and Non-Universal Entanglement Features from Surface Modes

The scaling of the subleading constant $\gamma$ in the entanglement entropy is dictated by whether the entanglement cut supports gapless boundary modes:

• Gapped edge (ordinary class): $\gamma = +\gamma_b > 0$; the full constant arises from bulk critical correlations (Wang et al., 18 Jan 2026, Zhu et al., 10 Aug 2025).
• Gapless edge (“special” or “extraordinary” classes): $\gamma = \gamma_b - \gamma_s < 0$, with $\gamma_s > 0$ capturing the entanglement generated by surface-critical modes. The sign reversal of $\gamma$ is a sharp signature of the transition from gapped to gapless boundary universality classes (Wang et al., 18 Jan 2026).

Explicit numerical fits demonstrate these behaviors. For example, in the (2+1)D O(3) QCP, $a = 0.168(1)$, $\gamma = +0.069(4)$ for an ordinary/gapped edge, versus $a = 0.781(1)$, $\gamma = -0.764(9)$ for a special/gapless edge (Wang et al., 18 Jan 2026).

Moreover, the entanglement Hamiltonian faithfully retains the edge universality class, i.e., the entanglement spectrum coincides (up to rescaling) with the spectrum of a physical edge (Zhu et al., 10 Aug 2025):

Cut Type	Edge Mode Type	$\gamma$	Subleading Term Origin
Ordinary	Gapped	$+\gamma_b$	Bulk critical correlations
Special/Extraordinary	Gapless	$-\gamma_s$	Surface-induced criticality

Thus, analysis of the entanglement spectrum and subleading EE corrections jointly diagnoses the interplay of bulk and surface criticality.

4. Surface Criticality in Disordered, Metallic, and Impurity Systems

Surface criticality and entanglement scaling are central for a variety of quantum phases beyond translationally invariant QCPs:

• Disordered Fermion Systems: In random-dimer and Aubry–André models, the surface roughness and EE exhibit Family–Vicsek (FV) scaling, with scaling exponents $\alpha, \beta, z$ uniquely determined by the disorder class. The coexistence of localized and delocalized eigenstates in certain models produces anomalous FV exponents not accessible in classical or clean quantum systems (Fujimoto et al., 2021).
• Metals and Fermi Surfaces: For two-dimensional metals, the optimal tensor-network approximation enforces a finite correlation length $\xi_D$ and yields the finite-entanglement scaling law $S(L; D) \sim L\log[\xi_D f(L/\xi_D)]$, where $f(x)$ encodes crossover from metallic to gapped behavior and is sensitive to the geometry of both the Fermi surface and the cut (Mortier et al., 2023).
• Impurity Quantum Phase Transitions: At boundary QCPs such as in the two-impurity Kondo model, tripartite entanglement diverges with system size as a power law $E_j \sim L^{\lambda_j}$, with exponents $\lambda_1 \approx 0.19$, $\lambda_2 \approx 0.46$–$0.50$ reflecting the scaling dimensions of associated boundary operators. These divergences are stronger than the logarithmic scaling of bulk von Neumann entropy and serve as unambiguous indicators of surface or impurity criticality (Bayat, 2016).

5. Measurement-Induced Criticality and Boundary Operators

In quantum circuits or stabilizer tensor-network models exhibiting measurement-induced criticality, entanglement negativity and mutual information probe distinct boundary operators in the associated conformal field theory:

• In measurement-only (percolation) circuits, mutual information decays with boundary scaling dimension $\Delta_I=1/3$ (spin bcc field), while mutual negativity decays as $x^{-4}$, i.e., with $\Delta_N=2$ (boundary stress-tensor operator) (Sang et al., 2020).
• In “hybrid” circuit models with additional unitaries, $\Delta_N$ remains robustly around 3, even as $\Delta_I$ varies, marking a form of super-universality for negativity. This establishes negativity as a fine-grained probe sharply sensitive to the boundary universality class (Sang et al., 2020).

Model/Class	$\Delta_I$	$\Delta_N$	Key Operator
Percolation	$1/3$	$2.0$	Spin / Stress
Hybrid Clifford/Haar	$2.0$	$3.0$–$3.1$	Distinct Bdy Op

Thus, surface criticality in measurement-induced transitions is directly encoded in the scaling dimensions controlling boundary entanglement correlations.

6. Distinguishing Bulk and Surface Criticality: Diagnostics and Resolution

The total entanglement entropy in systems where the entanglement cut can induce intrinsic low-energy modes may no longer serve as a pure probe of bulk quantum criticality. To isolate bulk properties, two robust diagnostics have emerged:

• Derivative of Entanglement Entropy: The scaling of $\partial S/\partial g$ with system size collapses for all bipartition geometries, reliably extracting the bulk critical point and correlation-length exponent $\nu$, unaffected by cut geometry (Wang et al., 18 Jan 2026).
• Choice of Cut/Bipartition: Only cuts consistent with the ordinary boundary class—i.e., which do not produce extra gapless edge modes—yield an uncontaminated view of bulk entanglement scaling. Tilted or symmetry-restoring cuts may systematically mask or distort the bulk signature (Zhu et al., 10 Aug 2025).

The recognition of cut-dependent surface contributions underlines the necessity of careful subsystem selection and combined entanglement spectrum and correlation function analysis for unbiased diagnoses of universality at quantum critical points.

7. Nonunitary Criticality: Negative Corner Logarithms

At deconfined quantum critical points (DQC), such as the SO(5)-symmetric J–Q₃ model, the scaling of the second Rényi entropy reveals negative logarithmic corner terms:

$S_2(R) = a\,R + s\,\ln R + b,$

with $s < 0$ (Zhao et al., 2021). This result fundamentally violates the reflection-positivity bound $c(\theta) \ge 0$ for unitary CFTs and signals nonunitary behavior at the DQC, reinforcing conclusions obtained by analyzing nonuniversal and geometry-dependent constants in smooth bipartition scaling (Zhao et al., 2021, Zhu et al., 10 Aug 2025).

References Table

Topic/Phenomenon	arXiv	Main Content
Surface criticality and EE cuts	(Wang et al., 18 Jan 2026)	Cut-dependent constant $\gamma$ in EE, bulk vs. surface
DQCP entanglement, nonunitarity	(Zhao et al., 2021, Zhu et al., 10 Aug 2025)	Negative corner function, surface-exponent contamination
Disordered 1D models: SR/EE scaling	(Fujimoto et al., 2021)	FV scaling, anomalous exponents, partial localization
Tripartite/impurity entanglement	(Bayat, 2016)	Power-law divergence at boundaries, scaling exponents
2D metals: finite-entanglement scaling	(Mortier et al., 2023)	$L\log L$ scaling, Fermi surface geometry, tensor networks
Measurement-induced criticality	(Sang et al., 2020)	Negativity, boundary operators, super-universality

In conclusion, the precise interplay of surface criticality and entanglement scaling establishes entanglement as a fundamentally nonlocal yet boundary-sensitive probe of quantum criticality. The geometry of the bipartition, the presence of edge modes, and impurity/boundary-induced scaling dimensions collectively control the universal and non-universal features in the scaling structure of entanglement observables.