Quantum information geometry of driven CFTs

TL;DR

The paper develops a dual viewpoint on driven two-dimensional conformal field theories by combining spacetime-diffeomorphism dynamics with quantum information geometry. It shows that driving a 2d CFT can be reformulated as evolving the theory on a time-dependent spacetime metric and, simultaneously, as trajectories on Virasoro state spaces equipped with the Bogoliubov–Kubo–Mori (BKM) metric. This framework yields concrete tools to quantify dissipation, complexity, and ergodicity via geodesics in the BKM geometry, and it uncovers a rich structure for Möbius (SL(2,R)) driving with heating/non-heating phases, as well as a holographic dual description in terms of evolving BTZ black holes. The work also connects to Euler–Arnold theory, optimal transport, and classical information geometry in appropriate limits, offering a comprehensive toolkit for analyzing non-equilibrium dynamics in driven CFTs. Overall, it provides a unified geometric lens for non-equilibrium CFT dynamics and their holographic interpretations, with explicit formulas for stress tensors, relative entropy, and information-geometric quantities across driving protocols.

Abstract

Driven quantum systems exhibit a large variety of interesting and sometimes exotic phenomena. Of particular interest are driven conformal field theories (CFTs) which describe quantum many-body systems at criticality. In this paper, we develop both a spacetime and a quantum information geometry perspective on driven 2d CFTs. We show that for a large class of driving protocols the theories admit an alternative but equivalent formulation in terms of a CFT defined on a spacetime with a time-dependent metric. We prove this equivalence both in the operator formulation as well as in the path integral description of the theory. A complementary quantum information geometric perspective for driven 2d CFTs employs the so-called Bogoliubov-Kubo-Mori (BKM) metric, which is the counterpart of the Fisher metric of classical information theory, and which is obtained from a perturbative expansion of relative entropy. We compute the BKM metric for the universal sector of Virasoro excitations of a thermal state, which captures a large class of driving protocols, and find it to be a useful tool to classify and characterize different types of driving. For Möbius driving by the SL(2,R) subgroup, the BKM metric becomes the hyperbolic metric on the unit disk. We show how the non-trivial dynamics of Floquet driven CFTs is encoded in the BKM geometry via Möbius transformations. This allows us to identify ergodic and non-ergodic regimes in the driving. We also explain how holographic driven CFTs are dual to driven BTZ black holes with evolving horizons. The deformation of the black hole horizon towards and away from the asymptotic boundary provides a holographic understanding of heating and cooling in Floquet CFTs.

Figures (14)

• Figure 1: Two-step driving. We alternate between the Hamiltonian $H_1$ for a time $T_1$ and the Hamiltonian $H_2$ for a time $T_2$.
• Figure 2: Phase diagram of a two-step process. The shaded region is the heating region where $|\mathrm{Tr}{M}|>2$.
• Figure 3: Representation of the two-step process in the BKM geometry after 10 steps. The trajectory on the hyperbolic disk is represented with alternating blue and red coloring corresponding to each step. In the non-heating phase, the process remains in a finite region, while in the heating phase, it diverges to the asymptotic boundary. The parameters chosen here are $\lambda=0.7, T_1=1.6$, $T_2=3$ for the non-heating phase and $T_2=4$ for the heating phase.
• Figure 4: Two-step process in the non-heating phase. The green points correspond to the positions of the process after each period. They lie on the process circle. Ergodicity of the dynamics on this circle, characterized by an angle $\theta$, leads to ergodicity of the process in a region $\mathcal{R}$ of a disk. Each step is a rotation of angle $T_k$ around a center the point $u_k$ representing the Hamiltonian $H_k$. The red and blue points are $u_1$ and $u_2$ corresponding to $H_1$ and $H_2$. This picture corresponds to 1000 steps and $\lambda=1$.
• Figure 5: Heating phase after 50 steps. The parameters are $\lambda=0.7, T_1=1.6, T_2=4$.