Driven Inhomogeneous CFT

• Driven inhomogeneous CFT is a framework combining spatially modulated Hamiltonians with structured driving to analytically explore quantum and cosmological phenomena.
• Structured drive protocols, including quasiperiodic and random sequences, reveal ergodicity breaking, prethermal regimes, and distinct heating phases.
• Analytical techniques extend to transport, entanglement, and holographic duality, linking operator evolution with observable measures in quantum systems.

Driven inhomogeneous conformal field theory (CFT) encompasses a broad class of quantum and classical systems where spatial inhomogeneities, structured driving, and temporal modulation are integrated into the framework of CFT. These models combine the versatility of conformal symmetry with arbitrary spatial backgrounds and nontrivial driving protocols, yielding analytical control, rich dynamical phase structure, and significant implications both for quantum matter and for cosmology. This entry presents a systematic account of the principles, mathematical structures, and physical phenomena characterizing driven inhomogeneous CFTs, incorporating recent advances in operator formalism, quantum circuits, ergodicity breaking, non-unitary dynamics, and holography.

1. Mathematical Framework and Operator Formulation

Driven inhomogeneous CFTs generalize the standard conformal Hamiltonian to include spatially dependent velocity profiles and arbitrary background geometries. The canonical form of the Hamiltonian is

$H = \int_{0}^{L} dx\, v(x) T_{00}(x)$

where $v(x)$ is a smooth function encoding spatial inhomogeneity and $T_{00}(x)$ is the energy density. In two-dimensional systems, such a Hamiltonian is a smearing of the stress tensor over the spatial slice, often realized as a spatially modulated combination of Virasoro generators: $$H = \frac{1}{2\pi} \int_{0}^{L}\left[ v(x) T(x) + \bar{v}(x) \bar{T}(x) \right] dx$$ Key operator evolution is then described via Mӧbius transformations ($SL(2,\mathbb{R})$ or $SL(2,\mathbb{C})$ matrices), with the drive protocol determined by a prescribed sequence or structure of inhomogeneous Hamiltonians. In recent works (Erdmenger et al., 25 Aug 2025), the implementation relies on a chirally split renormalization scheme, where the curved-space Weyl anomaly—essential for consistency in backgrounds that evolve with time—is realized as the difference between Schrödinger and Heisenberg picture Hamiltonians. This approach connects directly to quantum circuit models based on Virasoro symmetry, with unitary evolution generated by the background metric coinciding with that of a Virasoro quantum circuit.

2. Structured Drives and Ergodicity Breaking

Many driven inhomogeneous CFTs employ temporally structured drive sequences, far from simple periodicity. Examples include aperiodic Thue–Morse driving, random multipolar driving, and quasiperiodic protocols. Each drive step is represented by a Mӧbius matrix, and the evolution trajectory in operator space becomes a product of these matrices. The cumulative evolution is mapped onto a nonlinear classical dynamical system, $\mathcal{K}$, whose invariant sets and fixed points dictate ergodicity breaking: $|\psi_j\rangle = U_j \ldots U_1 |\psi_0\rangle$ where $U_i$ are Mӧbius operators reflecting the spatial modulation and driving parameters. Invariant regions (Region I, etc.) in the dynamical system result in non-heating phases with vanishing Lyapunov exponent, while generic choices lead to heating (ergodic) behavior with positive Lyapunov exponent (Mo et al., 27 May 2025). Prethermal phases emerge when the drive parameters approach, but do not coincide with, these fine-tuned regions; the prethermal lifetime is triply tunable and controlled by the deviation $K$, multipolar order $\eta$, and preimage order $\xi$: $t^{*} \sim K^{-2(\eta-\xi)}$

3. Phase Structure and Non-Unitary Dynamics

Floquet driven inhomogeneous CFTs display a nuanced phase structure, typically classified as heating (hyperbolic), non-heating (elliptic), and phase transition (parabolic) regimes (Das et al., 2021, Fan et al., 2020). Stroboscopic trajectories of operator evolution under spatially modulated Hamiltonians reveal heating when attractive fixed points exist (energy and entanglement entropy grow unboundedly), and non-heating when operator trajectories cycle without accumulation.

The inclusion of non-unitary (e.g. imaginary time) steps, realized as postselected weak measurements, leads to the emergence of loxodromic $SL(2,\mathbb{C})$ transformations with interior fixed points on the unit disk. In such protocols, any initial state generically relaxes to a unique steady state with finite energy and entanglement entropy (Lapierre et al., 2 May 2025). Systematic alternation between different types of non-unitary operations realizes purification phase transitions where, as a function of drive parameters, the density matrix either purifies to a single coherent state or remains mixed indefinitely, with the universal critical exponent $\nu=1/2$.

Further, sacrificing Hermiticity by adopting $SU(2)$ deformed Hamiltonians introduces a robust non-heating phase of nonzero measure. These compact group structures impede heating even under binary-disordered driving, due to emergent compactness in the effective transfer matrix dynamics.

4. Transport, Entanglement, and Thermodynamic Observables

Driven inhomogeneous CFTs are analytically tractable in the calculation of transport and entanglement measures. The evolution of the energy density, charge density, currents, and their multi-point correlation functions can be obtained exactly via mapping to homogeneous correlators with “dressed” operators (Moosavi, 2019). The treatment incorporates quantum anomalies—most notably through Schwarzian derivatives in transformation properties—resulting in corrections such as a generalized Wiedemann–Franz law at finite frequencies: $$\operatorname{Re}\,\kappa_{th}(\omega) = D_{th}\pi \delta(\omega) + \operatorname{Re}\,\kappa_{th}^{reg}(\omega)$$ Renyi and von Neumann entropies can be calculated via twist operator conformal blocks, including exact entanglement entropy formulas for spatially inhomogeneous setups: $$S_n(x) = \frac{n+1}{12n} \ln \left[ L^2 (1 - (x/L)^2)^{3/2} \right]$$ Non-equilibrium and quench protocols induce emergent spatial structure—localized energy peaks, inhomogeneous growth of entanglement, and spatial singularities in correlators—determined by the interplay of the geometric conformal mapping and the drive parameters (Das et al., 2021, Jiang et al., 11 Apr 2024).

5. Cosmological and Holographic Perspectives

In cosmology, CFT-driven models employ inhomogeneous ensembles—via the microcanonical density matrix over all compatible initial geometries—leading to thermal inflation and a primordial power spectrum with a distinctive thermal red tilt (Barvinsky, 2013). The path integral over metrics and fields, with compact Euclidean time, replaces the vacuum initial state with a thermal ensemble, modifying the observable statistics of the cosmic microwave background.

The holographic dual construction for driven inhomogeneous CFTs utilizes generalized Ba~nados solutions, Roberts mappings, and curved-space backgrounds (Li et al., 16 Feb 2025, Erdmenger et al., 25 Aug 2025). The central technical advance is the classification of Hamiltonians via Virasoro coadjoint orbits, connecting boundary deformations to specific bulk geometries. In the AdS$_3$/CFT$_2$ correspondence, non-heating and heating phases map to AdS$_2$ global and black-hole slices, respectively (Das et al., 2022). The appearance of “new horizons” in the bulk geometry—distinct from thermodynamic event horizons—can be directly traced to fixed point structures in the underlying $SL(2,\mathbb{R})$ or $SL(2,\mathbb{C})$ dynamical maps. Modular flow establishes operator reconstruction behind these horizons, extending entanglement wedge reconstruction methods from high-energy theory to condensed matter settings.

6. Physical Realizations and Experimental Implications

The theory of driven inhomogeneous CFTs has direct relevance to one-dimensional quantum systems, non-interacting Fermi gases in optical traps, integrable spin chains, and strongly correlated lattice models. Structured driving protocols—quasiperiodic, random, or aperiodic—enable robust suppression or enhancement of heating, which can be precisely controlled via the parameters of the dynamical system. Ultracold atomic gases in experiments display spatial inhomogeneities naturally suited to the geometric formulation (Dubail et al., 2016). Further, the analytic machinery developed for operator evolution, entanglement measures, and transport coefficients matches closely (to leading order) with numerical simulations of critical lattice models, providing essential benchmarks for quantum simulation platforms.

7. Outlook, Classification, and Geometric Insights

Recent work classifies Hamiltonians by their Virasoro coadjoint orbit, enabling systematic paper of new classes of inhomogeneous deformations—including those associated with KdV charges, not just linear combinations of Virasoro modes (Li et al., 16 Feb 2025). Entanglement Berry phases acquired along these orbits reflect underlying geometric properties of the infinite-dimensional symmetry group. The chirally split renormalization scheme, geometric group theory (cf. Lie–Cartan), and the interplay of anomaly structure and operator dynamics establish a unified story: driven inhomogeneous CFTs encapsulate the deep connection between infinite-dimensional symmetries, renormalization schemes, and quantum circuit evolution.

This comprehensive framework underpins the analytical tractability, dynamical control, and geometric elegance of driven inhomogeneous CFTs, positioning them as central objects in the paper of out-of-equilibrium quantum systems, quantum critical phenomena, and quantum cosmology.