Effective Central Charge

• Effective central charge is a measure of active conformal degrees of freedom in two-dimensional systems, defined via spectral shifts or finite-size scaling.
• It is computed using techniques such as thermal partition functions, entanglement entropy in interface CFTs, and replica derivative methods in random and monitored systems.
• Variants including time-dependent, holographically ensemble-selected, and interface-specific definitions illustrate its role in quantifying critical behavior and information transmission.

Searching arXiv for papers on “effective central charge” to anchor the article in current literature. Effective central charge is a context-dependent quantity that, in its standard two-dimensional conformal-field-theory usage, is defined by

$c_{\mathrm{eff}}=c-24h_{\min},$

or equivalently through the finite-size vacuum energy

$$E_0(R)=\frac{2\pi}{R}\left(\Delta_{\min}+\bar\Delta_{\min}-\frac{c}{12}\right)\equiv -\frac{\pi c_{\rm eff}}{6R}.$$

In unitary CFTs, where $\Delta_{\min}=\bar\Delta_{\min}=0$, one has $c_{\mathrm{eff}}=c$ (Cubero, 2017). In current literature, however, the same term is also used for several non-identical but structurally related objects: an interface-sensitive entanglement coefficient in interface CFTs, a replica-response coefficient $\left.\frac{dc_{IR}(R)}{dR}\right|_{R=1}$ in monitored or random systems, a time-dependent quantity after a quantum quench, and a running or ensemble-selected degree-of-freedom count in holographic and conformal-thermodynamic constructions (Karch et al., 2023, Patil et al., 10 Jul 2025, Promsiri et al., 22 Jun 2026).

1. Standard spectral and thermal definition

The equilibrium notion is tied to finite-size scaling and thermal partition functions. For a massive deformation of a two-dimensional CFT, the partition function on a cylinder of circumference $R$ and length $L$ defines a temperature-dependent effective central charge through

$c_{\rm thermal}(R)=\frac{6R}{\pi L}\log Z(R,L),$

which compares the massive theory to the CFT Casimir form (Cubero, 2017). In integrable models this quantity is computable by TBA, and it interpolates between the infrared and ultraviolet fixed-point values: $c_{\rm thermal}(R)\to c_{\rm IR}$ as $R\to\infty$ and $$E_0(R)=\frac{2\pi}{R}\left(\Delta_{\min}+\bar\Delta_{\min}-\frac{c}{12}\right)\equiv -\frac{\pi c_{\rm eff}}{6R}.$$0 as $$E_0(R)=\frac{2\pi}{R}\left(\Delta_{\min}+\bar\Delta_{\min}-\frac{c}{12}\right)\equiv -\frac{\pi c_{\rm eff}}{6R}.$$1 (Cubero, 2017).

For the free massive boson, the paper on quenches gives the explicit equilibrium result

$$E_0(R)=\frac{2\pi}{R}\left(\Delta_{\min}+\bar\Delta_{\min}-\frac{c}{12}\right)\equiv -\frac{\pi c_{\rm eff}}{6R}.$$2

with $$E_0(R)=\frac{2\pi}{R}\left(\Delta_{\min}+\bar\Delta_{\min}-\frac{c}{12}\right)\equiv -\frac{\pi c_{\rm eff}}{6R}.$$3 and $$E_0(R)=\frac{2\pi}{R}\left(\Delta_{\min}+\bar\Delta_{\min}-\frac{c}{12}\right)\equiv -\frac{\pi c_{\rm eff}}{6R}.$$4 (Cubero, 2017). This example illustrates the standard interpretation: $$E_0(R)=\frac{2\pi}{R}\left(\Delta_{\min}+\bar\Delta_{\min}-\frac{c}{12}\right)\equiv -\frac{\pi c_{\rm eff}}{6R}.$$5 is the coefficient that continues to count the effective short-distance conformal degrees of freedom away from criticality, even when the theory is massive.

A crucial distinction already appears here. The standard $$E_0(R)=\frac{2\pi}{R}\left(\Delta_{\min}+\bar\Delta_{\min}-\frac{c}{12}\right)\equiv -\frac{\pi c_{\rm eff}}{6R}.$$6 is a spectral quantity extracted from the dominant state in finite-size or modular asymptotics. It is not, by itself, an interface coefficient, a replica derivative, or a time-dependent observable. Those later uses are analogies or generalizations, not identical definitions.

2. Interface CFT and entanglement transmission

In interface CFTs, the effective central charge is defined from entanglement entropy rather than from the vacuum Casimir energy. When two CFTs are joined by a conformal interface, the entropy of an interval with an endpoint at the interface takes the universal form

$$E_0(R)=\frac{2\pi}{R}\left(\Delta_{\min}+\bar\Delta_{\min}-\frac{c}{12}\right)\equiv -\frac{\pi c_{\rm eff}}{6R}.$$7

and $$E_0(R)=\frac{2\pi}{R}\left(\Delta_{\min}+\bar\Delta_{\min}-\frac{c}{12}\right)\equiv -\frac{\pi c_{\rm eff}}{6R}.$$8 is determined by the lowest conformal dimension $$E_0(R)=\frac{2\pi}{R}\left(\Delta_{\min}+\bar\Delta_{\min}-\frac{c}{12}\right)\equiv -\frac{\pi c_{\rm eff}}{6R}.$$9 in the interface Hilbert space,

$\Delta_{\min}=\bar\Delta_{\min}=0$0

This quantity is not a simple function of the bulk central charges $\Delta_{\min}=\bar\Delta_{\min}=0$1 and $\Delta_{\min}=\bar\Delta_{\min}=0$2; it depends on the interface Hilbert space and therefore on the transmissive properties of the interface (Karch et al., 2023).

The same work shows that $\Delta_{\min}=\bar\Delta_{\min}=0$3 controls a broader family of entangling geometries. For intervals that probe only the bulk channels, the logarithmic coefficient is $\Delta_{\min}=\bar\Delta_{\min}=0$4, whereas for mixed geometries it is $\Delta_{\min}=\bar\Delta_{\min}=0$5 (Karch et al., 2023). Holographically and entropically, $\Delta_{\min}=\bar\Delta_{\min}=0$6 obeys the upper bound

$\Delta_{\min}=\bar\Delta_{\min}=0$7

with the totally transmissive interface saturating the upper endpoint and the totally reflective interface giving $\Delta_{\min}=\bar\Delta_{\min}=0$8 (Karch et al., 2023).

A stronger inequality was proposed later by separating information transmission from energy transmission. If $\Delta_{\min}=\bar\Delta_{\min}=0$9 is defined by the cross-interface stress-tensor two-point function,

$c_{\mathrm{eff}}=c$0

then the proposed hierarchy is

$c_{\mathrm{eff}}=c$1

Its interpretation is that the amount of energy transmission can never exceed the amount of information transmission (Karch et al., 2024). In holographic ICFTs this lower bound follows from the fact that $c_{\mathrm{eff}}=c$2 depends on an integrated property of the interface geometry, while $c_{\mathrm{eff}}=c$3 depends only on the minimal warp factor (Karch et al., 2024).

A further holographic refinement relates interface entropy and effective central charge directly. For an interval with one endpoint on the interface,

$c_{\mathrm{eff}}=c$4

and in the holographic class studied the effective central charge is

$c_{\mathrm{eff}}=c$5

where $c_{\mathrm{eff}}=c$6 is the minimum of the warp factor (Afxonidis et al., 12 Jul 2025). That paper shows that the endpoint-on-interface logarithm is the limiting form of the more general finite interface-entropy contribution, so $c_{\mathrm{eff}}=c$7 emerges as the coefficient governing the singular limit of interface entropy rather than as an unrelated parameter (Afxonidis et al., 12 Jul 2025).

3. Replica limits, randomness, and monitored systems

A distinct use of the term appears in replicated nonunitary field theories describing monitored systems or randomness. There one defines, not $c_{\mathrm{eff}}=c$8, but

$c_{\mathrm{eff}}=c$9

where $\left.\frac{dc_{IR}(R)}{dR}\right|_{R=1}$0 is the infrared central charge of the $\left.\frac{dc_{IR}(R)}{dR}\right|_{R=1}$1-copy replicated theory (Patil et al., 10 Jul 2025). At $\left.\frac{dc_{IR}(R)}{dR}\right|_{R=1}$2, the theory reduces to the original clean CFT, so $\left.\frac{dc_{IR}(R)}{dR}\right|_{R=1}$3; the nontrivial information is therefore in the derivative with respect to replica number, not in the value itself (Patil et al., 10 Jul 2025).

For replica actions of the form

$\left.\frac{dc_{IR}(R)}{dR}\right|_{R=1}$4

with $\left.\frac{dc_{IR}(R)}{dR}\right|_{R=1}$5 relevant, the paper proves nonperturbatively that

$\left.\frac{dc_{IR}(R)}{dR}\right|_{R=1}$6

under three assumptions: no additive operator renormalization for the perturbing operator, no replica symmetry breaking, and analyticity in $\left.\frac{dc_{IR}(R)}{dR}\right|_{R=1}$7 on $\left.\frac{dc_{IR}(R)}{dR}\right|_{R=1}$8 (Patil et al., 10 Jul 2025). The proof adapts the Zamolodchikov sum rule to the $\left.\frac{dc_{IR}(R)}{dR}\right|_{R=1}$9 limit and rewrites the relevant replica correlators as averages over a single-copy theory coupled to a real randomness field $R$0, yielding a negative-definite expression for $R$1 (Patil et al., 10 Jul 2025).

This replica-response effective central charge is explicitly not the ordinary Virasoro central charge of the physical $R$2 theory. Its role is instead to control universal finite-size effects in monitored or random ensembles, especially the Shannon entropy of the measurement record (Patil et al., 10 Jul 2025). The same paper compares it with the conventional disorder quantity

$R$3

which is relevant for uncorrelated quenched disorder, and argues—under an extra assumption on zeros of $R$4—that the $R$5 slope may satisfy the opposite inequality relative to the clean central charge (Patil et al., 10 Jul 2025).

4. Time-dependent effective central charge after a quantum quench

The equilibrium construction has also been extended to non-equilibrium dynamics. The proposed time-dependent effective central charge $R$6 is defined by comparing the return amplitude

$R$7

to the corresponding quantity in a CFT quench (Cubero, 2017). The strip free energy is decomposed as

$R$8

and the CFT expression motivates a non-equilibrium $R$9, whose physical version is taken to be the time average

$L$0

The averaging is essential because $L$1 generically oscillates at finite times (Cubero, 2017).

For a large mass quench of a free boson, the paper finds

$L$2

matching the expectation that the quench drives the system from an infrared massive state to a late-time regime probing the ultraviolet free-boson CFT (Cubero, 2017). The same framework constructs an “Ising to Tricritical Ising” quench in which the effective central charge evolves from

$L$3

to

$L$4

corresponding to the central charges of the first two unitary minimal models (Cubero, 2017).

The inverse process is argued to be impossible within the same methods. More generally, for pure initial states the paper finds

$L$5

while $L$6 itself need not be monotone and can oscillate or overshoot at intermediate times (Cubero, 2017). In the staircase model, the short-time evolution displays a discrete staircase structure whose plateaus are determined by minimal-model central charges, providing a non-equilibrium analogue of the equilibrium staircase TBA picture (Cubero, 2017).

5. Holographic and conformal-thermodynamic extensions

Recent holographic work has extended the idea still further by allowing the central charge itself to vary across an ensemble of theories. In conformal thermodynamics, the boundary conformal factor is promoted to a thermodynamic parameter so that the field-theory volume $L$7 and the central charge $L$8 vary independently, with

$L$9

In the fixed $c_{\rm thermal}(R)=\frac{6R}{\pi L}\log Z(R,L),$0 ensemble, the paper defines a central-charge Rényi entropy by

$c_{\rm thermal}(R)=\frac{6R}{\pi L}\log Z(R,L),$1

and shows that it can be written thermally as

$c_{\rm thermal}(R)=\frac{6R}{\pi L}\log Z(R,L),$2

This construction is interpreted as a Rényi measure over a statistical ensemble of holographic CFTs with fluctuating degrees of freedom, rather than over states of a single fixed theory (Promsiri et al., 22 Jun 2026).

In that framework the central charge becomes $c_{\rm thermal}(R)=\frac{6R}{\pi L}\log Z(R,L),$3-dependent through the thermodynamic solution,

$c_{\rm thermal}(R)=\frac{6R}{\pi L}\log Z(R,L),$4

with $c_{\rm thermal}(R)=\frac{6R}{\pi L}\log Z(R,L),$5 determined by the Rényi index (Promsiri et al., 22 Jun 2026). The paper explicitly discusses this $c_{\rm thermal}(R)=\frac{6R}{\pi L}\log Z(R,L),$6 as an effective central charge selected by the generalized Rényi ensemble: at small $c_{\rm thermal}(R)=\frac{6R}{\pi L}\log Z(R,L),$7, larger-$c_{\rm thermal}(R)=\frac{6R}{\pi L}\log Z(R,L),$8 theories contribute more broadly, whereas at large $c_{\rm thermal}(R)=\frac{6R}{\pi L}\log Z(R,L),$9 the entropy is dominated by the most probable theories (Promsiri et al., 22 Jun 2026). A characteristic temperature $c_{\rm thermal}(R)\to c_{\rm IR}$0 and the associated characteristic Rényi index $c_{\rm thermal}(R)\to c_{\rm IR}$1 separate a dominant-theory regime $c_{\rm thermal}(R)\to c_{\rm IR}$2 from a multi-theory regime $c_{\rm thermal}(R)\to c_{\rm IR}$3 (Promsiri et al., 22 Jun 2026).

A different holographic generalization appears in sine dilaton gravity, where a monotone flow central charge $c_{\rm thermal}(R)\to c_{\rm IR}$4 is constructed in domain-wall gauge,

$c_{\rm thermal}(R)\to c_{\rm IR}$5

In that model the ultraviolet limit reproduces the Liouville central charge $c_{\rm thermal}(R)\to c_{\rm IR}$6, while the infrared limit flows to pure JT gravity with $c_{\rm thermal}(R)\to c_{\rm IR}$7 (Mahapatra et al., 25 Jan 2026). This is a running $c_{\rm thermal}(R)\to c_{\rm IR}$8-function rather than the standard $c_{\rm thermal}(R)\to c_{\rm IR}$9, but it belongs to the same broader family of “effective” degree-of-freedom measures.

6. Distinctions, neighboring notions, and recurrent misconceptions

A recurring misconception is to treat every quantity labeled “effective central charge” as the same observable. The literature surveyed here shows that this is not the case. The standard finite-size quantity $R\to\infty$0, the interface entanglement coefficient, the replica derivative $R\to\infty$1, the quench-dependent $R\to\infty$2, and the $R\to\infty$3-dependent ensemble-selected $R\to\infty$4 are all different constructions, even when they play analogous roles.

A second misconception is to identify any appearance of “central charge” with an effective central charge. Several papers in adjacent areas explicitly do not do this. The warped $R\to\infty$5 black-hole analysis computes ordinary left- and right-moving central charges $R\to\infty$6 and states that it does not define $R\to\infty$7 (Gupta et al., 2010). The string-theoretic $R\to\infty$8 computation evaluates the vacuum expectation value of the spacetime central charge operator and likewise does not discuss the standard effective central charge (Troost, 2011). Periodically driven Kitaev chains exhibit a universal entanglement central charge for the Floquet ground state, but the paper explicitly does not introduce a distinct “Floquet effective central charge” (Yates et al., 2018).

The same distinction applies to experimental and numerical extraction of central charge. Central spin decoherence, Wang–Landau sampling, and measurements on a universal quantum processor all target the ordinary CFT central charge $R\to\infty$9, not an effective central charge in the nonunitary or vacuum-shifted sense (Wei, 2017, Belov et al., 2016, Köylüoğlu et al., 2024). Likewise, the chiral central charge $$E_0(R)=\frac{2\pi}{R}\left(\Delta_{\min}+\bar\Delta_{\min}-\frac{c}{12}\right)\equiv -\frac{\pi c_{\rm eff}}{6R}.$$00 extracted from a single bulk wave function in $$E_0(R)=\frac{2\pi}{R}\left(\Delta_{\min}+\bar\Delta_{\min}-\frac{c}{12}\right)\equiv -\frac{\pi c_{\rm eff}}{6R}.$$01-dimensional topological phases is a thermal-transport quantity and not the standard $$E_0(R)=\frac{2\pi}{R}\left(\Delta_{\min}+\bar\Delta_{\min}-\frac{c}{12}\right)\equiv -\frac{\pi c_{\rm eff}}{6R}.$$02 of $$E_0(R)=\frac{2\pi}{R}\left(\Delta_{\min}+\bar\Delta_{\min}-\frac{c}{12}\right)\equiv -\frac{\pi c_{\rm eff}}{6R}.$$03-dimensional CFT (Kim et al., 2021).

The term also has close relatives that are best regarded as analogues rather than synonyms. The defect and boundary charges $$E_0(R)=\frac{2\pi}{R}\left(\Delta_{\min}+\bar\Delta_{\min}-\frac{c}{12}\right)\equiv -\frac{\pi c_{\rm eff}}{6R}.$$04 and $$E_0(R)=\frac{2\pi}{R}\left(\Delta_{\min}+\bar\Delta_{\min}-\frac{c}{12}\right)\equiv -\frac{\pi c_{\rm eff}}{6R}.$$05, defined from logarithmic defect contributions to entanglement entropy, are central-charge analogues for higher-dimensional defects, but they are not ordinary Virasoro central charges and are not presented as $$E_0(R)=\frac{2\pi}{R}\left(\Delta_{\min}+\bar\Delta_{\min}-\frac{c}{12}\right)\equiv -\frac{\pi c_{\rm eff}}{6R}.$$06 (Estes et al., 2018). Interface CFT provides the closest modern example of a distinct but established alternative usage, because there $$E_0(R)=\frac{2\pi}{R}\left(\Delta_{\min}+\bar\Delta_{\min}-\frac{c}{12}\right)\equiv -\frac{\pi c_{\rm eff}}{6R}.$$07 is rigorously the coefficient of a logarithmic entanglement term and is bounded by the smaller bulk central charge (Karch et al., 2023, Karch et al., 2024).

Taken together, these developments establish effective central charge as a family of universal coefficients that quantify how many conformal degrees of freedom are effectively active after one introduces a vacuum shift, an interface, a replica deformation, a quench, or a theory-space ensemble. The precise definition is therefore inseparable from the physical question being asked.