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arXiv:2404.07884 · Journal of High Energy Physics

New horizons for inhomogeneous quenches and Floquet CFT

Hanzhi Jiang, Márk Mezei

Published Apr 2, 2025• 11 citations
arXivHTMLPDF

Abstract

A bstract A fruitful avenue in investigating out-of-equilibrium quantum many-body systems is to abruptly change their Hamiltonian and study the subsequent evolution of their quantum state. If this is done once, the setup is called a quench, while if it is done periodically, it is called Floquet driving. We consider the solvable setup of a two-dimensional CFT driven by Hamiltonians built out of conformal symmetry generators: in this case, the quantum dynamics can be understood using two-dimensional geometry. We investigate how the dynamics is reflected in the holographic dual three-dimensional spacetime and find new horizons. We argue that bulk operators behind the new horizons are reconstructable by virtue of modular flow.

DOI
10.1007/jhep04(2025)025

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A bstract A fruitful avenue in investigating out-of-equilibrium quantum many-body systems is to abruptly change their Hamiltonian and study the subsequent evolution of their quantum state. If this is done once, the setup is called a quench, while if it is done periodically, it is called Floquet driving. We consider the solvable setup of a two-dimensional CFT driven by Hamiltonians built out of conformal symmetry generators: in this case, the quantum dynamics can be understood using two-dimensional geometry. We investigate how the dynamics is reflected in the holographic dual three-dimensional spacetime and find new horizons. We argue that bulk operators behind the new horizons are reconstructable by virtue of modular flow.
In this article, we explore the connection between the heating phase of periodically driven CFTs and the Modular Hamiltonian of a subregion in the vacuum state. We show that the heating phase Hamiltonian corresponds to the Modular Hamiltonian, with the fixed points mapping to the endpoints of the subregion. In the bulk dual, we find that these fixed points correspond to the Ryu-Takayanagi surface of the AdS-Rindler wedge. Consequently, the entanglement entropy associated to the boundary interval within two fixed points exactly matches with the Rindler entropy of AdS-Rindler. We observe the emergent Virasoro algebra in the boundary quantization of the Modular Hamiltonian has a striking similarity with the emergent near Horizon Virasoro algebra. This is a consequence of the fact that while obtaining the boundary Virasoro algebra, a cut-off with conformal boundary condition around the fixed point is introduced, which in the bulk is related to a stretched horizon, with an emergent two-dimensional conformal symmetry. We also argue that as one tunes the parameter space of Floquet Hamiltonians to transition from the non-heating to the heating phase the operator algebra type changes from Von Neumann type III to III1III_1III1​ factor, providing a non-equilibrium analogue of the Hawking-Page transition.
This paper investigates the dynamical phases of Floquet Conformal Field Theories (CFTs) in space-time dimensions greater than two. Building upon our previous work which introduced quaternionic representations for studying Floquet dynamics in higher-dimensional CFTs, we now explore more general square pulse drive protocols that go beyond a single SU(1,1) S U ( 1 , 1 ) subgroup. We demonstrate that, for multi-step drive protocols, the system exhibits distinct dynamical phases characterized by the nature of the eigenvalues of the quaternionic matrix representing time evolution in a single cycle, leading to different stroboscopic responses. Our analysis establishes a fundamental geometric interpretation where these dynamical phases directly correspond to the presence or absence of Killing horizons in the base space of the CFT and in a higher dimensional AdS space on which a putative dual lives. The heating phase is associated with a non-extremal horizon, the critical phase with an extremal horizon which disappears in the non-heating phase. We develop perturbative approaches to compute the Floquet Hamiltonians in different regimes and show, how tuning drive parameters can lead to horizons, providing a geometric framework for understanding heating phenomena in driven conformal systems.
This paper investigates the dynamical phases of Floquet Conformal Field Theories (CFTs) in space-time dimensions greater than two. Building upon our previous work [1] which introduced quaternionic representations for studying Floquet dynamics in higher dimensional CFTs, we now explore more general square pulse drive protocols that go beyond a single SU(1,1) subgroup. We demonstrate that, for multi-step drive protocols, the system exhibits distinct dynamical phases characterized by the nature of the eigenvalues of the quaternionic matrix representing time evolution in a single cycle, leading to different stroboscopic responses. Our analysis establishes a fundamental geometric interpretation where these dynamical phases directly correspond to the presence or absence of Killing horizons in the base space of the CFT and in a higher dimensional AdS space on which a putative dual lives. The heating phase is associated with a non-extremal horizon, the critical phase with an extremal horizon which disappears in the non-heating phase. We develop perturbative approaches to compute the Floquet Hamiltonians in different regimes and show, how tuning drive parameters can lead to horizons, providing a geometric framework for understanding heating phenomena in driven conformal systems.
A bstract We find classes of driven conformal field theories (CFT) in d + 1 dimensions with d > 1, whose quench and floquet dynamics can be computed exactly. The setup is suitable for studying periodic drives, consisting of square pulse protocols for which Hamiltonian evolution takes place with different deformations of the original CFT Hamiltonian in successive time intervals. These deformations are realized by specific combinations of conformal generators with a deformation parameter β ; the β < 1 ( β > 1) Hamiltonians can be unitarily related to the standard (Lüscher-Mack) CFT Hamiltonians. The resulting time evolution can be then calculated by performing appropriate conformal transformations. For d ≤ 3 we show that the transformations can be easily obtained in a quaternion formalism. Evolution with such a single Hamiltonian yields qualitatively different time dependences of observables depending on the value of β , with exponential decays characteristic of heating for β > 1, oscillations for β < 1 and power law decays for β = 1. This manifests itself in the behavior of the fidelity, unequal-time correlator, and the energy density at the end of a single cycle of a square pulse protocol with different hamiltonians in successive time intervals. When the Hamiltonians in a cycle involve generators of a single SU(1, 1) subalgebra we calculate the Floquet Hamiltonian. We show that one can get dynamical phase transitions for any β by varying the time period of a cycle, where the system can go from a non-heating phase which is oscillatory as a function of the time period to a heating phase with an exponentially damped behavior. Our methods can be generalized to other discrete and continuous protocols. We also point out that our results are expected to hold for a broader class of QFTs that possesses an SL(2, C ) symmetry with fields that transform as quasi-primaries under this. As an example, we briefly comment on celestial CFTs in this context.
A bstract 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 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.
A bstract We explore non-equilibrium processes in two-dimensional conformal field theories (2d CFTs) due to the growth of operators induced by inhomogeneous and homogeneous Hamiltonians by investigating the time dependence of the partition function, energy density, and entanglement entropy. The non-equilibrium processes considered in this paper are constructed out of the Lorentzian and Euclidean time evolution governed by different Hamiltonians. We explore the effect of the time ordering on entanglement dynamics so that we find that in a free boson CFT and RCFTs, this time ordering does not affect the entanglement entropy, while in the holographic CFTs, it does. Our main finding is that in the holographic CFTs, the non-unitary time evolution induced by the inhomogeneous Hamiltonian can retain the initial state information longer than in the unitary time evolution.