Holographic Interfaces in AdS/CFT

Updated 3 December 2025

Holographic interfaces are codimension-one defects in AdS/CFT that enable the study of field theories with spatially varying couplings, boundary conditions, and phase transitions.
They are constructed using diverse models such as thin-brane, Janus, and top-down supergravity approaches, which reveal non-perturbative insights into RG flows and operator dynamics.
Applications include quantifying energy transport, entanglement entropy modifications, and interface thermodynamics, providing a rigorous framework for understanding defect phenomena.

Holographic interfaces are codimension-one defects in the context of the AdS/CFT correspondence, enabling the paper of field theories with spatially varying couplings, distinct phases, or boundary conditions across an embedded interface. They provide a non-perturbative, gravitational dual arena for exploring interface conformal field theories (ICFT), renormalization group flows, symmetry breaking, energy transport, entanglement, and non-trivial topological or confining behaviors. The construction and analysis of holographic interfaces span bottom-up effective models, top-down string/supergravity reductions, and mathematically explicit realizations, capturing a range of interface phenomena from simple geometric deformations to intricate RG flows and non-local operator insertions.

1. Geometric Constructions and Model Classes

Holographic interfaces are implemented in bulk gravity as solutions exhibiting multiple asymptotically AdS boundaries, a codimension-one region such as a probe brane, or non-trivial warping interpolating between different vacua or coupling profiles. The principal classes are:

Thin-brane (Randall–Sundrum-type) models: Two (or more) AdS bulk regions of radii $\ell_L, \ell_R$ are glued along a codimension-one brane of tension $\sigma$ , whose embedding satisfies Israel junction conditions. These models encode reflection/transmission of energy, boundary entropy, and interface thermodynamics in terms of geometric invariants and brane parameters (Bachas et al., 2020, Baig et al., 2022, Bachas et al., 2021, Bachas et al., 2021).
Janus and RG flow solutions: Solutions with an explicit AdS $_d$ slicing (e.g., AdS $_{d+1}$ in “domain wall” form) realize smooth RG interfaces, often involving scalar fields dual to relevant operators. These allow the paper of interfaces across which couplings, vevs, or physical quantities interpolate, and support continuous deformations or jumping behavior (e.g., coupling, $\theta$ -angle, or symmetry breaking profiles) (Estes et al., 2012, Melby-Thompson et al., 2017, Ghodsi et al., 2022, Ghodsi et al., 4 Sep 2024).
Top-down supergravity and orbifold interfaces: Fully back-reacted string-theory reductions (e.g., half-BPS Janus, D1/D5 interfaces, F(4) supergravity constructions) enable explicit computation of spectra and correlators, and realize interfaces dual to defect or boundary CFTs with extended symmetry and global consistency (Gutperle et al., 2015, Chiodaroli et al., 2016, Karndumri, 30 Sep 2024, Harris et al., 31 Mar 2025, Bobev et al., 2020).
Non-conformal and non-supersymmetric interfaces: Interfaces involving localized perturbations or scalar operators can break conformal invariance in a spatially controlled manner, with frequency-dependent and possibly complex energy transport coefficients, as well as phase sensitivity (Liu et al., 26 Mar 2025).

2. Boundary Conditions, Saddles, and Solution Space

The boundary value problem for holographic interfaces consists of specifying asymptotic behavior and regularity conditions:

Two-boundary ("interface") solutions are constructed by solving metric and field equations with an AdS $_d$ slicing, imposing standard Dirichlet or mixed (double-trace) boundary conditions on both ends $u\to\pm\infty$ :

$\phi(u) \simeq \phi_- e^{-\Delta_- u/\ell} + \cdots + \phi_+ e^{-\Delta_+ u/\ell}$

where the sources $\phi_-$ may differ on each boundary, and $\phi_+$ encodes the vev. Interior regularity requires the scalar either runs off to a "good" Gubser singularity (confining flows) or bounces between extrema of the potential (Ghodsi et al., 4 Sep 2024, Ghodsi et al., 2022).

Saddles:
- Connected ("interface") saddles are single-bulk geometries connecting both boundaries.
- Disconnected ("product") saddles are two independent one-boundary solutions "glued" side by side.
- The dominance criterion at large $N_c$ is that the saddle with the lowest free energy $F$ dominates the semiclassical expansion.
- In confining or nontrivial RG-flow contexts, the product saddle typically dominates, implying factorization of physical observables in the planar limit (Ghodsi et al., 4 Sep 2024).
- Solution space admits more intricate branches characterized by the number and type of bounces and fragmentation, leading to phenomena like "walking" (prolonged near-conformal behavior) and flow fragmentation (extra boundaries) (Ghodsi et al., 2022).

3. Physical Observables: Free Energy and Correlators

The on-shell action (free energy) and observable correlators are computed by holographic renormalization:

Free energy: For two-boundary, AdS-sliced solutions in Einstein–dilaton gravity,

$F_{\text{conn}} = M_P^{d-1} \int d^d x \sqrt{\zeta} \left[ \cdots \right]_{u=+\infty} - \left[ \cdots \right]_{u=-\infty}$

with leading dependence on source parameters $\phi_-^\pm$ and boundary curvatures $R_\pm$ . Product saddles decompose into the sum of independent one-sided free energies (Ghodsi et al., 4 Sep 2024, Ghodsi et al., 2022).

Interface dominance and suppression: In confining holographic QFTs (with negative $V(\phi)$ , exponential run-off), numerics confirm $F_{\text{prod}} < F_{\text{conn}}$ for all parameter values; hence, normalized two-boundary correlators

$\langle O_L O_R \rangle \sim \exp(-\Delta F) \sim \exp(-c N_c^2)$

are exponentially suppressed and vanish in the planar limit, reflecting absence of pertinent interface-mediated interactions (Ghodsi et al., 4 Sep 2024). This is generically found to hold for holographic QFTs with standard confining potentials.

Two-point functions: For mixed-quantization (double-trace) interfaces (RG-flow defects), the two-point function between operators of dimensions $\Delta_-$ (UV side) and $\Delta_+$ (IR side) involves hypergeometric conformal blocks. The interface spectrum contains defect-localized operators with discrete scaling dimensions determined by boundary conditions. The 1-loop determinant yields the interface g-factor (entropy), matching CFT predictions at large $N$ (Melby-Thompson et al., 2017).

4. Energy Transport, Reflection, and Transmission

Interfaces act as local scatterers for energy and momentum:

Reflection and transmission coefficients: For 2d holographic ICFTs, an explicit thin-brane calculation establishes

$\mathcal{T}_L = \frac{2}{\ell_L (1/\ell_L + 1/\ell_R + 8\pi G \sigma)}$

with $\sigma$ the brane tension, and $c_{L,R}$ the central charges. The sum rule $\mathcal{T}_L + \mathcal{R}_L = 1$ holds; weighted averages match CFT null energy bounds (Bachas et al., 2020, Liu et al., 26 Mar 2025).

Frequency-dependence: Non-conformal interfaces with localized operators can exhibit complex, frequency-dependent transmission with oscillatory structure at intermediate $\omega$ , while approaching conformal interface values in the deep UV/IR. Special brane embeddings can enforce perfect transmission, mimicking topological defects (Liu et al., 26 Mar 2025).
Steady-state transport and entropy production: Holographic steady-state interface solutions connect thermal baths and display non-Killing event horizons, maximal entropy production, and a sharp phase transition in thermal conductivity, reflecting the interplay between classical scatterers and quantum coherence (Bachas et al., 2021).

5. Entanglement Entropy, Boundary Entropy, and Universal Relations

Holographic interfaces modify entanglement entropy (EE) in ways that both capture universal interface data and encode geometric structure:

Symmetric intervals: For intervals straddling the interface,

$S_A = \frac{c}{3} \ln \frac{l}{\epsilon} + \log g$

where $\log g$ is the boundary entropy (interface g-factor), determined by the minimal warp near the interface or brane parameters (Gutperle et al., 2015, Karch et al., 2021, Afxonidis et al., 12 Jul 2025).

Asymmetric intervals and effective central charge: For intervals with one endpoint at the interface, the coefficient of $\ln(l/\epsilon)$ is renormalized to an effective central charge $c_{\text{eff}} = c\, e^{A_*}$ , interpolating between the full and zero transmission limits. The limiting procedure connects $\log g$ and $c_{\text{eff}}$ precisely (Afxonidis et al., 12 Jul 2025).
Non-crossing intervals and strong subadditivity: Even intervals that do not cross the interface receive non-vanishing finite contributions to EE, required by strong subadditivity and the global geometric structure (Afxonidis et al., 12 Jul 2025, Karch et al., 2021).
Universal relations: All interface-dependent EE coefficients obey identities such as $\sigma_1 = c/6 + \sigma_2/2$ , expressing the additivity of endpoint contributions and linking geometric and information-theoretic quantities (Karch et al., 2021).

6. Extensions: RG Flows, Defect Operator Spectrum, and Topological Interfaces

Double-trace and RG flow interfaces: Mixed boundary conditions interpolate between different CFT quantizations, yielding exact expressions for two-point functions, boundary g-factors, and the spectrum of interface-localized (defect) operators. For instance, a scalar with mass in the unitarity window in AdS $_{d+1}$ supports a sharp interface between quantizations, with defect operator dimensions $\Delta_a = d/2 + a$ and full agreement with large- $N$ minimal model data (Melby-Thompson et al., 2017).
Topological and symmetric orbifold interfaces: In symmetric product orbifolds, interfaces realize analogues of permutation branes and admit a precise mapping to open-string sectors of AdS $_2$ -branes in string theory on AdS $_3 \times S^3 \times \mathbb{T}^4$ , with explicit transmission and reflection numbers fixed by combinatorial data (Harris et al., 31 Mar 2025).
Confining and walking phases: AdS-sliced confining QFTs on fixed AdS backgrounds support a taxonomy of solution branches, RG flow fragmentation, and walking phenomena, realized as transitions and degenerations in the bulk moduli space. The general interface solution space is understood as a phase diagram of wormhole-like flows, with interface observables computable in principle via holographic renormalization (Ghodsi et al., 4 Sep 2024, Ghodsi et al., 2022).

7. Interface Phase Structure, Thermodynamics, and Outlook

The bottom-up holographic interface model admits a full phase diagram based on thermodynamic criteria and geometric transitions:

Phase structure: The interface geometry can realize cold (no horizon), warm (one-sided horizon), hot (both-sided horizon), and intermediate "bubble" or negative specific heat phases. Transitions (e.g., Hawking–Page, wall capture by horizon) display first-order or sweeping behavior, affecting the interface entropy and operator spectrum (Bachas et al., 2021).
Dominance and factorization: In confining top-down and bottom-up constructions, the product (disconnected) saddle always dominates in the planar limit, enforcing exponential suppression of normalized cross-correlators and manifesting a strong notion of interface factorization (Ghodsi et al., 4 Sep 2024).
Non-equilibrium and topological aspects: Far-from-equilibrium steady state solutions, coherent/quantum conduction regimes, and the emergence of topological or perfectly transmitting interfaces are accessible, supporting a broad extension of interface holography to out-of-equilibrium, non-conformal, or topological contexts (Bachas et al., 2021, Liu et al., 26 Mar 2025, Harris et al., 31 Mar 2025).

These results collectively provide a rigorous, physically transparent, and mathematically controlled holographic dictionary for interfaces, encompassing free energy, correlator structure, transport, entanglement, and RG data within and beyond AdS/CFT. The robustness of product saddle dominance for confining holographic QFTs, the sharply defined universal relations for EE, and the flexibility to tune transmission/entropy via brane configurations form a foundation for ongoing advances in the interface, defect, and boundary holography program.

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