Flow Central Charge in QFTs

• Flow Central Charge is a measure that characterizes the universal RG flow by connecting UV and IR fixed points through anomaly matching and irreversible evolution.
• It is extracted using techniques such as modular flow, return amplitude matching, and non-perturbative RG methods, as illustrated in sine-Gordon and CFT models.
• The concept extends to non-equilibrium dynamics where an effective central charge tracks quantum quenches, revealing energy injection and emergent topological orders.

The flow central charge characterizes the emergent universal content and anomaly structure of quantum field theories (QFTs) under renormalization group (RG) evolution and dynamical protocols such as quenches or entanglement-generated modular flow. Its most prominent instantiation is the “$c$-function”, which interpolates between the ultraviolet (UV) and infrared (IR) fixed-point central charges, capturing the irreversibility of RG flow, anomaly matching, and topological order. Modern developments have extended its definition beyond equilibrium, allowing extraction from dynamical entanglement, return amplitudes, or non-perturbative functional RG flows.

1. Modular Hamiltonian, Modular Flow, and Central Charge

For a pure ground state $|\psi\rangle$ on a Hilbert space $\mathcal{H}_A\otimes\mathcal{H}_{\bar{A}}$, the reduced density matrix on region $A$ is $$\rho_A=\mathrm{Tr}_{\bar{A}}|\psi\rangle\langle\psi|$$. The modular Hamiltonian is $K_A := -\ln\rho_A$, and the modular flow defines a family of unitaries $U_A(s):=\exp[iK_A s]$, with $s\in\mathbb{R}$ (Fan, 2022).

The modular flow induces entanglement dynamics: perturbing $|\psi\rangle$ to $|\psi(s)\rangle=U_A(s)|\psi\rangle$, the linear response of the entanglement entropy yields

$$\partial_s S_A(s)\big|_{s=0} = i\,\mathrm{Tr}\,[\rho_A\,[K_A, \ln\rho_A]].$$

In the case $A=A'$, this vanishes, but considering entropies on overlapping regions recovers non-trivial expressions.

For (1+1)D conformal field theories (CFTs) on the circle, explicit evaluation exposes its dependence on the chiral central charge $c_-=c-c̄$, connecting modular flow to the gravitational anomaly structure:

$\partial_s S_{BC}|_{0} = (\pi c_- / 6)\,F(x,y).$

For pure global states, modular flow reveals that genuine (1+1)D theories cannot carry perturbative gravitational anomaly ($c_-=0$).

2. Generalization to Non-Equilibrium: Effective Central Charge Flow

Under a quantum quench, time-dependent “effective central charge” $c_{\rm eff}(t)$ tracks RG-like evolution out of equilibrium (Cubero, 2017). Defined by matching the partition function or return amplitude to that of a CFT with central charge $c_{\rm eff}(t)$,

$$c_{\rm eff}(t) = \frac{1}{t} \int_0^t dt' \,\Delta c_{\rm eff}'(t'),$$

where $\Delta c_{\rm eff}'(t)$ is formulated by comparing the strip free energy of the massive theory to an auxiliary CFT. At $t=0$, $c_{\rm eff}(0)$ recovers the pre-quench IR central charge; at $t\to\infty$ the observable approaches the UV value associated with high-energy degrees of freedom.

Protocols studied include:

• Free boson mass quench: $c_{\rm eff}(0)=0 \rightarrow c_{\rm eff}(\infty)=1$.
• Ising $\rightarrow$ Tricritical Ising quench: $$c_{\rm eff}(0)=1/2 \rightarrow c_{\rm eff}(\infty)=7/10$$.
• Staircase model quench: displays “staircase” increases in $c_{\rm eff}(t)$ corresponding to minimal CFT central charges (Cubero, 2017).

3. Non-Perturbative RG and the Zamolodchikov $c$-Function

In two-dimensional models, the scale-dependent $c$-function $c_k$ can be formulated within non-perturbative functional RG (FRG), which tracks RG flows from UV to IR (Bacsó et al., 2015). For the sine-Gordon model, the Wetterich equation governs the flow of the effective action $\Gamma_k[\varphi]$, and the $c$-function is extracted by projecting the trace anomaly onto the Polyakov action:

$$k\,\partial_k\,c_k = \frac{[k\,\partial_k\,\tilde{V}_k''(\varphi_0)]^2}{[1+\tilde{V}_k''(\varphi_0)]^3}$$

in the Local Potential Approximation (LPA). For sine-Gordon,

$$\tilde{V}_k(\varphi) = -\tilde{u}_k \cos(\beta \varphi),$$

and the flow equations are solved with respect to the regulator choice; universality is seen at low frequency ($\beta^2\to 0$), matching the Ising result $\Delta c=1$.

The phase diagram separates into regions based on $\beta^2$:

• $\beta^2 < 8\pi$: $c_{UV}=1 \rightarrow c_{IR}=0$, i.e., $\Delta c = 1$.
• $\beta^2 > 8\pi$: flows from infinite coupling in the UV to IR Gaussian ($c_{UV}=c_{IR}=1$).

Going beyond LPA, incorporation of running wavefunction renormalization is necessary for quantitative accuracy in $c_k$ across the entire RG trajectory.

4. Modular Flow and Extraction of Chiral Central Charge in Topological Phases

For gapped (2+1)D systems with chiral edge modes, the modular commutator in entanglement-generated flow permits computation of the edge chiral central charge, encoding the bulk-boundary anomaly consequences (Fan, 2022):

$$c_- := \frac{3i}{\pi}\mathrm{Tr}\,[\rho\,[K_{AB},K_{BC}]],$$

and the geometric form (Kim et al.):

$$\Delta S \equiv \partial_s S_{BC}|_0 = 2\pi i\,c_-\,I[\partial A],$$

with $I[\partial A]$ the normalized total geodesic curvature of the region boundary.

Numerical checks: | Model | Topological Phase | $c_-$ Value | |----------------------------------------|--------------------------|-----------------| | Quantum Hall (disk, $\nu=1$) | Radius $\gg$ mag. length | $c_-\to 1$ | | Quantum Hall (annulus partition) | | $c_-\approx 0$ | | Lattice $p+ip$ superconductor ($20^2$) | $0<\mu<4$, $4<\mu<8$ | Plateau at $\pm 1/2$ |

These computational results verify the modular flow definition of $c_-$ and its bulk-edge correspondence.

5. RG Irreversibility and Dynamical Flow Constraints

Time-dependent $c_{\rm eff}(t)$ and scale-dependent $c_k$ share an irreversible character: under dynamical quench or RG flow, the effective central charge exhibits

$c_{\rm eff}(\infty) \geq c_{\rm eff}(0),$

reflecting energy injection and the sampling of higher central charge degrees of freedom. Finite-time oscillations may occur but the time-averaged value is physically meaningful and parallels the thermal $c$-function monotonicity in equilibrium (Cubero, 2017).

This non-equilibrium $c_{\rm eff}(t)$ generalizes the c-theorem: while exact monotonicity is absent out-of-equilibrium, the correspondence between initial (IR) and final (UV) central charges, and the impossibility of returning to lower values under unitary evolution, encode the fundamental irreversibility of RG flow.

6. Significance and Limitations

The flow central charge provides a powerful unifying framework for quantifying RG trajectories, anomaly matching, and bulk-edge correspondence in both equilibrium and non-equilibrium quantum field theories. Its precise extraction depends sensitively on the chosen definition—modular flow, return amplitude matching, or non-perturbative RG projection. While numerical and analytic techniques establish quantitative agreement for simple models, care must be exercised (e.g., necessity of wavefunction renormalization in FRG, correct modular cut topology in entanglement protocols). Higher harmonics and naive LPA truncations may yield unreliable results except in limiting cases.

A plausible implication is that future extensions to higher dimensions, interacting topological phases, or arbitrary quenched protocols demand careful specification of modular Hamiltonians, entanglement structures, and scale-dependent anomaly content. The flow central charge remains foundational for the study of universal RG properties, quantum anomalies, and topological order.