Double Trace Deformation in Holography

• Double trace deformation is a modification of CFTs by a quadratic term in a primary operator, driving controlled RG flows and altering holographic dual boundary conditions.
• It employs techniques such as the Hubbard–Stratonovich transformation to non-perturbatively analyze operator scaling dimensions and fixed point transitions.
• Its applications span modeling interface phenomena, constructing traversable wormholes, and exploring quantum information effects like entanglement and dissipation.

A double-trace deformation is a perturbation of a conformal field theory (CFT) by adding a term quadratic in a single-trace operator, typically of the form $(f/2) \int d^d x\, \mathcal{O}^2(x)$ where $f$ is a coupling. This deformation is distinguished by its non-linear structure, its controlled effects on renormalization group flows, and its well-defined gravitational dual in the AdS/CFT correspondence, where it is implemented by a modification of boundary conditions for the dual bulk field. Double-trace deformations serve as a paradigm for studying nontrivial flows between distinct CFTs, the emergence of interfaces and defects, entanglement and transport phenomena, and the construction of traversable wormholes via holographic duality.

1. Mathematical Structure and RG Flows

A double-trace deformation modifies the CFT action as

$$S_\text{CFT} \to S_\text{CFT} + \frac{f}{2} \int d^d x\, \mathcal{O}^2(x)$$

where $\mathcal{O}$ is a primary operator of the undeformed CFT. In the regime where the scaling dimension $\Delta$ of $\mathcal{O}$ satisfies $\Delta < d/2$, the deformation is relevant and triggers an RG flow from a UV fixed point (with dimension $\Delta_{-} = d/2 - \nu$ for $\nu = \sqrt{m^2 + d^2/4}$ in AdS/CFT) to an IR fixed point (where $\mathcal{O}$ acquires dimension $\Delta_{+} = d/2 + \nu$). This flow is encoded in the large-$N$ expansion and can be analyzed non-perturbatively using auxiliary fields via a Hubbard–Stratonovich transformation.

The deformed two-point function can be resummed exactly at large $N$: $\widetilde G_f(k) = \frac{1}{f + k^{d-2\Delta}}$ This deformation generalizes to spinorial operators and to conserved currents, with each case requiring an appropriate treatment of variational principles and boundary terms in the dual gravitational theory (Allais, 2010).

2. Holographic Dual and Boundary Conditions

In AdS/CFT, double-trace deformations are realized by altering the boundary conditions of the bulk field dual to $\mathcal{O}$: $$\phi(z,x) \sim \alpha(x) z^{d-\Delta} + \beta(x) z^\Delta$$ The standard and alternative quantizations correspond to Dirichlet ($\alpha=0$) and Neumann ($\beta=0$) conditions, respectively. A double-trace deformation leads to mixed (Robin-type) boundary conditions: $\alpha(x) = f\, \beta(x)$ This boundary modification encodes the RG flow between different quantizations (i.e., between conformal dimensions $\Delta_-$ and $\Delta_+$), and more generally, allows the construction of interfaces (see Sec. 4), spatially inhomogeneous flows, and even wormholes by imposing boundary conditions that relate fields on disconnected asymptotic regions.

For vector and higher-spin fields, analogous deformations can be implemented, but the allowed transformations form discrete (SL(2,ℤ), e.g., via Chern-Simons terms) or continuous (SL(2,ℝ), for scalar or vector double-trace) families, depending on gauge invariance and quantization conditions (Cottrell et al., 2017, Casper et al., 2017).

3. Physical Implications: RG Flows, Central Charge, and Pathologies

Double-trace deformations generate controlled RG flows whose properties can be extracted via both holographic and field-theoretic techniques. For scalar operators,

$$\Delta_\text{IR} = d - \Delta_\text{UV} + \mathcal{O}(1/N)$$

The change in the central charge along these flows obeys the c-theorem (or its higher-dimensional analogs). Holographically, the change in the central charge is

$$\frac{\Delta c}{c} = \frac{1}{N^2} \frac{L^{d+1} \Delta V}{2d}$$

where $\Delta V$ is the shift in the bulk effective potential due to the deformation. Computations confirm that $\Delta V < 0$, so $c$ decreases monotonically along the RG flow (Allais, 2010).

Not all double-trace deformations yield consistent flows: for certain "extremal" deformations ($\Delta = d/2 \pm 1$), the Källén-Lehmann representation reveals unphysical tachyonic or ghost-like excitations in the deformed spectrum, demonstrating that not all formal RG flows correspond to unitary or conformal UV fixed points (Porrati et al., 2016). The consequences of such pathologies are crucial for the global structure of theory space and the swampland of AdS/CFT.

4. Defects, Interfaces, and Mixed Boundary Conditions

Localizing double-trace deformations creates RG interfaces: domain walls separating CFTs with different operator content and scaling dimensions. In practice, this is implemented by turning on the deformation on half-space,

$$S_\text{def} = S_\text{CFT} + \lambda \int_{z>0} d^d x\, \mathcal{O}^2(x)$$

The resulting interface supports an array of defect operators whose spectra and OPE coefficients can be computed systematically at large $N$ by reformulating the theory with a Hubbard–Stratonovich field confined to half-space. The interface CFT data—for example, one-point functions of local operators and the interface central charge (or $g$-function)—match precisely with holographic calculations using "inhomogeneous" mixed boundary conditions for the dual bulk field (Giombi et al., 10 Jul 2024, Melby-Thompson et al., 2017). In Janus-type geometries, different quantizations are imposed on complementary spatial regions, naturally realizing RG domain walls.

Furthermore, the formalism has been generalized to celestial and wedge-like holography, where similar "double deformation" effects are induced by mixed boundary conditions even in asymptotically flat spacetimes (Fukada et al., 2023).

5. Traversable Wormholes and Quantum Information

Non-local double-trace deformations enable the construction of traversable AdS wormholes by coupling two decoupled CFTs: $$S_\text{tot} = S_{\mathrm{CFT}_1} + S_{\mathrm{CFT}_2} + \frac{1}{2}\int d^d x d^d y\, \lambda(x,y)\, O_1(x) O_2(y)$$ where $\lambda(x,y)$ decays in momentum space ensuring that the deformation is IR-localized. This induces mixed boundary conditions of the form

$$J^{(1)} = \alpha^{(1)} - \lambda \beta^{(2)},\quad J^{(2)} = \alpha^{(2)} - \lambda \beta^{(1)}$$

in the dual AdS spacetime, "gluing" the two asymptotically AdS regions in the infrared.

The modification of boundary conditions translates into the violation of the averaged null energy condition (ANEC) along the horizon, making classically non-traversable wormholes traversable (Gao et al., 2016, Ahn et al., 2022). The traversability is encoded in the bulk two-point correlation functions (which decay exponentially for high momenta and diverge on the boundary of the wormhole light cone) and in nonvanishing "pseudo"-entanglement entropy, since the global state becomes mixed due to the explicit coupling of the boundaries (Kawamoto et al., 5 Feb 2025). When the double-trace deformation is constructed from conserved current operators, the wormhole opening and quantum information transfer are limited by the charge diffusion constant, directly linking transport properties to the efficiency of teleportation protocols across the wormhole (Ahn et al., 2022).

Notably, there is a distinction in the global state structure between double-trace (model B) and Janus (interface, model A) deformations. For Janus, global time slices yield pure states, whereas for double-trace, they correspond to non-Hermitian (mixed) transition matrices, as evidenced by nonzero Renyi pseudo-entropy (Kawamoto et al., 5 Feb 2025).

6. Quantum Information, Entanglement, and Dissipation

Double-trace deformations in the Keldysh framework manifest as nonunitary Lindblad-type terms: $$L(\phi_+,\phi_-) = L_0(\phi_+) - L_0(\phi_-) + i\gamma (O_q)^2$$ with $O_q = (O_+ - O_-)/\sqrt{2}$. This structure captures the effective dynamics of open quantum systems, with scaling properties distinct from those found in equilibrium settings. Depending on the dimension $\Delta$ of the operator and spacetime dimension $d$, these deformations can lead to either new IR or UV fixed points, possibly nonthermal and nonunitary, thus broadening the set of critical phenomena accessible via double-trace methods (Meng, 2020).

In the context of entanglement entropy and its evolution under double-trace deformations, rigorous calculations reveal both positive and negative contributions to entanglement, nontrivial dependencies on geometry (e.g., black hole size), and phenomena such as shockwave-like time evolution and nonmonotonicity in quantum corrections, especially near RG interfaces or in the presence of strong gravitational back-reaction (Miyagawa et al., 2015, Song, 2016, Miyaji, 2018).

7. Applications and Extensions

Double-trace deformations underpin a range of applications:

• Pseudogap and Quantum Phase Transitions: In holographic superconductors, these deformations tune the pairing interaction and drive transitions between superconducting and pseudogap phases, with the plasma oscillations of the associated bulk bosonic fluid accounting for conductivity suppression (DeWolfe et al., 2016).
• CFT-to-Bath Coupling and Open Holography: When a CFT is coupled to a bath via a double-trace term, boundary conditions glue the bulk AdS region to external degrees of freedom, thus modeling measurement, decoherence, and subregion physics. These constructions yield insights into unitarity, edge mode dynamics, and gravitational anomalies (e.g., emergent graviton masses) in the presence of external observers (Geng, 2023, Karch et al., 2023).
• TT̄ Deformation: Double-trace deformations constructed from composite operators such as $T\overline{T}$ play a central role in integrable irrelevant deformations of 2d CFTs, with detailed rules for the computation of correlator corrections to all orders in perturbation theory (He et al., 2023).

The versatility of double-trace deformations, their tight link to boundary conditions and RG interfaces, and their centrality to the dynamical structure of holography, quantum gravity, and condensed matter dualities underscore their value as a fundamental tool in modern theoretical physics.