Defect Entropy in CFT and Material Science

• Defect entropy is an entropy-like measure that quantifies contributions from defects, boundaries, and interfaces in quantum field theories and material systems.
• Renormalized perturbative approaches and entanglement entropy methods reveal universal corrections, such as boundary g-factors and anomaly coefficients, in both 2D CFTs and higher-dimensional DCFTs/BCFTs.
• Applications span thermodynamic defect calculations in solids, micromagnetics, topological information measures, and holographic analyses of defect-modified quantum states.

“Defect entropy” denotes a family of entropy-like quantities attached to defects, boundaries, interfaces, defect-induced singular sets, or defect-modified quantum states. In two-dimensional CFT it includes the Affleck–Ludwig boundary entropy $s_B=\ln g_B$ and its defect counterpart obtained by folding; in higher-dimensional DCFT and BCFT it includes universal terms extracted from entanglement entropy, sphere partition functions, or defect free energies; in crystalline defect physics it denotes formation, migration, and configurational entropies of point defects; in micromagnetics it appears as an entropy defect measure $\mu$ associated with weak solutions of the eikonal system; and in quantum-information settings it includes Shannon, Rényi, relative, and stabilizer-based quantities that diagnose topological or geometric defects (Konechny et al., 2014, Katsika-Tsigourakou et al., 2017, Marconi, 2020, Ghasemi, 29 Jan 2026).

1. Conformal perturbation defects and boundary $g$-factors

A conformal perturbation defect arises when a 2D CFT$_{\rm UV}$ is perturbed on the half-plane $x_1>0$ by a relevant operator $\Phi(x)$ of dimension $\Delta=2-\delta$, with $\delta>0$ small. If the perturbed bulk theory flows to an infrared fixed point described by another CFT, the defect flows to a conformal defect between the ultraviolet and infrared fixed point CFTs. Folding the unperturbed half-plane $x_1<0$ converts the construction into a coupled bulk-plus-boundary flow on a single half-plane whose endpoint is a conformal boundary condition in $\mathrm{CFT}_{\rm UV}\otimes \mathrm{CFT}_{\rm IR}$; unfolding restores the defect interpretation. The boundary, or defect, entropy is defined from the cylinder partition function by isolating the finite prefactor $\mu$0, so that $\mu$1 (Konechny et al., 2014).

For the perturbative computation, one may work on a torus of periods $\mu$2, perturb $\mu$3 on half of it, and expand

$\mu$4

In the long-torus limit $\mu$5, the terms proportional to $\mu$6 reproduce the bulk vacuum-energy shift, while the finite residual pieces $\mu$7 sum to $\mu$8. Up to cubic order,

$\mu$9

The bulk coupling is renormalized by

$g$0

with infrared fixed point

$g$1

Using renormalized integrals $g$2 and $g$3, with

$g$4

one obtains the universal perturbative formula

$g$5

The $g$6 term is entirely fixed by the OPE coefficient $g$7 and is positive in a unitary theory, whereas the $g$8 term involves the three-loop coefficient $g$9 of the bulk $_{\rm UV}$0-function (Konechny et al., 2014).

For the short $_{\rm UV}$1 flow between neighboring A-series unitary minimal models $_{\rm UV}$2, with $_{\rm UV}$3, the specialization gives

$_{\rm UV}$4

Gaiotto’s candidate RG-domain-wall defect has exactly known

$_{\rm UV}$5

so the first two terms of the perturbative result agree exactly with the algebraic construction. This provides strong evidence that Gaiotto’s RG domain wall is the conformal perturbation defect associated with the short $_{\rm UV}$6 flow (Konechny et al., 2014).

2. Entanglement, localized entropy, and anomaly coefficients in DCFT and BCFT

In higher-dimensional defect and boundary conformal field theories, defect entropy is often defined by background subtraction from entanglement entropy. For planar codimension-one defects or boundaries, one chooses a spherical or hemispherical entangling surface centered on the defect or boundary, computes the Ryu–Takayanagi minimal surface, and then subtracts the ambient CFT contribution regulated with the same cutoff: $_{\rm UV}$7 After subtraction, the result takes the same form as a $_{\rm UV}$8-dimensional spherical entanglement entropy. In $_{\rm UV}$9,

$x_1>0$0

and the universal constants $x_1>0$1 and $x_1>0$2 are identified as higher-dimensional analogs of a boundary $x_1>0$3-function. In the D3/D5 DCFT, $x_1>0$4 decreases under defect-localized RG flows that reduce the number of defect hypermultiplets, while bulk Higgs-branch flows do not have a definite sign, consistent with the expectation that ambient flows need not obey a defect $x_1>0$5-theorem (Estes et al., 2014).

A different but related quantity is defect localized entropy, defined for the internal degrees of freedom $x_1>0$6 living on a $x_1>0$7-dimensional defect inside a $x_1>0$8-dimensional CFT. After a Casini–Huerta–Myers map, the von Neumann entropy of a defect subregion can be computed by replicating only the defect operator on the sphere: $x_1>0$9 At a defect RG fixed point, the universal part obeys

$\Phi(x)$0

Away from fixed points, refined defect $\Phi(x)$1-functions can be defined. For surface defects,

$\Phi(x)$2

and for volume defects,

$\Phi(x)$3

These monotonicity statements are derived from strong subadditivity applied within the defect worldvolume (Yuan et al., 2022).

For planar two-dimensional defects in $\Phi(x)$4 CFTs, the universal logarithmic term in the defect entanglement entropy is fixed by defect Weyl-anomaly central charges $\Phi(x)$5 and $\Phi(x)$6. If

$\Phi(x)$7

then for a sphere of radius $\Phi(x)$8 centered on the defect,

$\Phi(x)$9

The Averaged Null Energy Condition implies $\Delta=2-\delta$0. In 4D $\Delta=2-\delta$1 SYM with half-BPS surface operators,

$\Delta=2-\delta$2

while in the 6D $\Delta=2-\delta$3 theory,

$\Delta=2-\delta$4

The same analysis also shows that no universal Cardy-like formula relates the defect central charges to defect thermal entropy density across all DCFTs and BCFTs (Jensen et al., 2018).

For codimension-two superconformal monodromy defects, the defect entanglement entropy is defined by subtracting $\Delta=2-\delta$5 times the vacuum entanglement entropy from the defect result. The universal term $\Delta=2-\delta$6 is cutoff-independent and, in $\Delta=2-\delta$7, satisfies

$\Delta=2-\delta$8

$\Delta=2-\delta$9

$\delta>0$0

Explicit holographic examples show that this universal defect entanglement need not decrease along RG flows driven by non-localized bulk deformations: in both $\delta>0$1 and $\delta>0$2, the ratio of IR to UV universal defect entropy can be either larger or smaller than one, depending on parameters (Conti et al., 27 Nov 2025).

3. Relative, Shannon, and stabilizer formulations for topological defects

For a 2D CFT on a circle with a topological defect operator $\delta>0$3, one may define the reduced density matrix $\delta>0$4 on an interval $\delta>0$5 and compare it to the reduced density matrix associated with another defect $\delta>0$6. The defect relative entropy is

$\delta>0$7

and in the limit $\delta>0$8 it reduces to a classical Kullback–Leibler divergence,

$\delta>0$9

In diagonal RCFTs, topological defects are labeled by primaries $x_1<0$0, with

$x_1<0$1

so that

$x_1<0$2

The same framework yields a defect sandwiched Rényi relative entropy and a defect fidelity,

$x_1<0$3

A “defect relative sector” is defined as the set of defects with zero defect relative entropy. In the Ising model the sectors are $x_1<0$4 and $x_1<0$5; in tricritical Ising they include $x_1<0$6 and $x_1<0$7; and in $x_1<0$8 they are the pairs $x_1<0$9 (Ghasemi, 29 Jan 2026).

In symmetric product orbifold CFTs $\mathrm{CFT}_{\rm UV}\otimes \mathrm{CFT}_{\rm IR}$0, defect relative entropy again assumes a KL form. For universal defects labeled by irreducible representations $\mathrm{CFT}_{\rm UV}\otimes \mathrm{CFT}_{\rm IR}$1 of $\mathrm{CFT}_{\rm UV}\otimes \mathrm{CFT}_{\rm IR}$2, the relevant probability distribution is

$\mathrm{CFT}_{\rm UV}\otimes \mathrm{CFT}_{\rm IR}$3

and

$\mathrm{CFT}_{\rm UV}\otimes \mathrm{CFT}_{\rm IR}$4

For maximally fractional defects $\mathrm{CFT}_{\rm UV}\otimes \mathrm{CFT}_{\rm IR}$5, a second distribution enters,

$\mathrm{CFT}_{\rm UV}\otimes \mathrm{CFT}_{\rm IR}$6

so that

$\mathrm{CFT}_{\rm UV}\otimes \mathrm{CFT}_{\rm IR}$7

The first term is purely permutation-group data, while the second is controlled by the seed RCFT modular $\mathrm{CFT}_{\rm UV}\otimes \mathrm{CFT}_{\rm IR}$8-matrix (Ghasemi, 14 Feb 2026).

The same symmetric-orbifold defects contribute characteristic subleading terms to entanglement entropy. For a topological defect inserted in the middle of an interval, the leading logarithm is unchanged and

$\mathrm{CFT}_{\rm UV}\otimes \mathrm{CFT}_{\rm IR}$9

For universal defects,

$\mu$00

If the defect is placed at an endpoint of the interval, then for universal defects

$\mu$01

so the defect contribution becomes a Shannon-type entropy of squared characters. For non-universal defects $\mu$02, the boundary result acquires an additional seed-theory term,

$\mu$03

These formulas rely on the large-interval limit, vacuum dominance after modular transformation, and a diagonal seed RCFT for the non-universal lines (Gutperle et al., 2024).

A further information-theoretic diagnostic is the stabilizer Rényi entropy (SRE) of a 1D critical chain with boundaries or topological defects,

$\mu$04

In the replica CFT treatment, open boundaries produce

$\mu$05

for the Ising CFT, independent of $\mu$06 and of the Cardy boundary pair. A single topological defect gives a universal size-independent term

$\mu$07

and multiple defects encode fusion rules: in the Ising case,

$\mu$08

with the SRE selecting the lowest-energy fusion channel. Numerically at $\mu$09,

$\mu$10

and two duality defects satisfy $\mu$11 up to the effective chain-length shift $\mu$12 (Hoshino et al., 14 Jul 2025).

4. Entropy defect measures and rectifiable singular sets

In the micromagnetics model associated with Riviére–Serfaty energies, entropy enters in a measure-theoretic sense. The full micromagnetic energy for a unit-length magnetization $\mu$13 is

$\mu$14

where the stray field satisfies

$\mu$15

As $\mu$16, thin domain walls form, and bounded-energy sequences converge to limits satisfying

$\mu$17

that is, the eikonal system (Marconi, 2020).

Admissible weak limits are described by the class

$\mu$18

with kinetic function $\mu$19. The entropy defect measure $\mu$20 is defined by the kinetic identity

$\mu$21

Projecting $\mu$22 onto the $\mu$23-plane gives the total entropy defect

$\mu$24

a finite Radon measure on $\mu$25 (Marconi, 2020).

The main theorem states that the entropy dissipation concentrates on a countably $\mu$26-rectifiable set

$\mu$27

More precisely: $\mu$28 is countably $\mu$29-rectifiable, and $\mu$30 is concentrated on $\mu$31, so $\mu$32. The proof uses the kinetic formulation, a Lagrangian representation measure $\mu$33 on characteristic trajectories, a decomposition into shock and rarefaction trajectories, an optimal transport or coupling argument forcing $\mu$34 to sit on a union of Lipschitz graphs, and density estimates together with rectifiability criteria of Federer and Besicovitch (Marconi, 2020).

In this setting, the “entropy defect” is not a scalar entropy of a state but a measure encoding where the weak solution fails to behave as a smooth entropy solution. Physically, the rectifiable set $\mu$35 is the sharp-interface domain-wall network. The concentration of $\mu$36 and $\mu$37 on $\mu$38 rigorously identifies the one-dimensional structures supporting entropy dissipation and the limiting wall energy (Marconi, 2020).

5. Thermodynamic and configurational defect entropies in solids

In ionic-crystal defect thermodynamics, defect entropy appears as a formation or migration entropy attached to a specific defect process. For BaF$\mu$39, the relevant intrinsic processes are anion Frenkel-pair formation, fluorine vacancy migration, and fluorine interstitial migration, together with dielectric relaxation of a $\mu$40-F dipole. The relaxation time obeys

$\mu$41

and within transition-state theory the migration entropy can be deduced from the prefactor through

$\mu$42

or equivalently

$\mu$43

The reported numerical values are

$\mu$44

$\mu$45

$\mu$46

and for the uranium-related dielectric relaxation,

$\mu$47

Thus the uranium-dipole migration entropy is nearly two orders of magnitude smaller than the anion Frenkel formation entropy (Katsika-Tsigourakou et al., 2017).

These entropies are connected by the $\mu$48 model, which relates any defect Gibbs energy to bulk elastic and volumetric properties,

$\mu$49

From

$\mu$50

one obtains the universal ratio

$\mu$51

For BaF$\mu$52, using

$\mu$53

one finds

$\mu$54

and the measured entropies and enthalpies lie approximately on a straight line of that slope. The large Frenkel formation entropy is attributed to vacancy-plus-interstitial creation and substantial vibrational-mode softening, whereas the small uranium-dipole value reflects reorientation of a single F interstitial about a fixed $\mu$55 with fewer equivalent sites and smaller local vibrational changes (Katsika-Tsigourakou et al., 2017).

In a nonequilibrium materials framework, defect entropy becomes the configurational entropy of an explicit defect energy landscape. In the steepest-entropy-ascent quantum thermodynamic treatment of bilayer PtSe$\mu$56, the system is described by discrete energy eigenlevels $\mu$57, degeneracies $\mu$58, and occupation probabilities $\mu$59, with entropy

$\mu$60

The dynamics are governed by a SEAQT equation of motion that implements maximum instantaneous entropy production subject to conservation laws, driving $\mu$61 toward the canonical form

$\mu$62

The associated level free energy is

$\mu$63

A large degeneracy therefore lowers the free energy and produces entropic stabilization at finite temperature (Younis et al., 2022).

For PtSe$\mu$64, scanning tunneling microscopy identified six unique point defects and twenty combinations of multiple point defects in close proximity. The energy landscape was built from DFT defect energies and a Replica-Exchange Wang-Landau density of states for a $\mu$65 bilayer film containing $\mu$66 atoms and approximately $\mu$67 discrete eigenlevels. The model uses a $\mu$68-state Potts representation with $\mu$69, where $\mu$70 is a perfect cell and $\mu$71 labels one of the six defect types. During annealing, the system entropy rises from the initial nonequilibrium value to its equilibrium maximum, and the time-dependent weighted defect concentrations

$\mu$72

evolve toward equilibrium values. In this formulation, “defect entropy” is exactly the configurational entropy $\mu$73 of the many-defect landscape, and it simultaneously shifts free energies and drives the kinetics of defect rearrangement (Younis et al., 2022).

6. Holographic surface defects, double holography, and defect-modified quantum states

For half-BPS disorder-type surface defects in $\mu$74 SYM, holographic entanglement entropy can be computed in ten-dimensional bubbling geometries dual to the defect. For a ball-shaped region bisected by the surface defect, the minimal-surface calculation yields

$\mu$75

A second derivation based on adapting the Lewkowycz–Maldacena argument combines the expectation value of the defect operator and the one-point function of the stress tensor, but the two results agree only up to an additional term

$\mu$76

The final expression is therefore

$\mu$77

The origin of $\mu$78 is discussed in terms of possible conformal-anomaly, localized-stress-tensor, cutoff, or replica-trick subtleties (Gentle et al., 2015).

In double holography, a defect subregion can be realized as a finite region $\mu$79 of a defect CFT$\mu$80 living on a brane inside AdS$\mu$81, coupled to a bath CFT$\mu$82. The entanglement entropy is computed by a generalized area functional containing a bulk term and a DGP brane term, and the numerical solutions exhibit a phase transition. In the shrinking phase, relevant for subcritical parameters, the entanglement wedge and entropy collapse to zero as the bath subregion is removed; in the stable phase, relevant for supercritical parameters, they remain finite. In the semiclassical limit the entropy decomposes as

$\mu$83

with the bath CFT producing the dominant linear divergence and the induced brane gravity producing a universal logarithmic divergence. For reflected entropy of left-right halves of the defect subsystem, the classical geometry contributes a leading logarithmic divergence controlled by $\mu$84, whereas the quantum matter inside the entanglement wedge contributes only a finite term (Liu et al., 2023).

The defect-extremal-surface proposal for reflected entropy makes the brane contribution explicit. For an AdS bulk with a codimension-one defect carrying a lower-dimensional CFT, the generalized functional is

$\mu$85

In AdS$\mu$86/BCFT$\mu$87, the defect quantum term for an interval is

$\mu$88

and the resulting defect-extremal-surface formula for reflected entropy agrees exactly with the island formula after the bulk is decomposed into a brane-gravity region and a bath-CFT region. Time-dependent calculations in an evaporating black-hole model reproduce Page-curve behavior for reflected entropy (Li et al., 2021).

In six dimensions, surface-defect Rényi entropy admits a heat-kernel formulation on $\mu$89. The defect contribution to Rényi entropy is

$\mu$90

For a free scalar surface defect in $\mu$91,

$\mu$92

while for the free 2-form,

$\mu$93

For a free $\mu$94 tensor multiplet,

$\mu$95

At large $\mu$96, probe M2-brane embeddings produce defect indices

$\mu$97

$\mu$98

The free-field result is $\mu$99, whereas the large-$g$00 result is $g$01, reflecting the different scaling of defect degrees of freedom (Yuan et al., 2023).

A non-holographic but still information-theoretic example appears in heavy-meson spectroscopy on a point-like global-monopole background. The metric

$g$02

modifies the Schrödinger equation with the Cornell potential $g$03. The position- and momentum-space Shannon entropies,

$g$04

obey the BBM relation

$g$05

As the defect strengthens, meaning $g$06 decreases, $g$07 grows and $g$08 falls; as $g$09, the total entropy approaches the BBM lower bound. In this usage, defect entropy measures redistribution of localization information between position and momentum in a defect-deformed quantum bound state (Almeida et al., 2023).