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arXiv:2309.04540 · Physical Review Letters

Inhomogeneous Quantum Quenches of Conformal Field Theory with Boundaries

Xinyu Liu, Alexander McDonald, Tokiro Numasawa, Biao Lian, Shinsei Ryu

Published Jun 5, 2025• 5 citations
arXivHTMLPDF

Abstract

We develop a method to calculate generic time-dependent correlation functions for inhomogeneous quantum quenches in (1+1)-dimensional conformal field theory (CFT) induced by sudden Hamiltonian deformations that modulate the energy density inhomogeneously. Our Letter particularly focuses on the effects of spatial boundaries, which have remained unresolved by previous analytical methods. For generic postquench Hamiltonian, we develop a generic method to calculate the correlations by mirroring the system, which otherwise are Euclidean path integrals in complicated spacetime geometries difficult to calculate. On the other hand, for a special class of inhomogeneous postquench Hamiltonians, including the Möbius and sine-square-deformation Hamiltonians, we show that the quantum quenches exhibit simple boundary effects calculable from Euclidean path integrals in a straightforward strip spacetime geometry. Applying our method to the time evolution of entanglement entropy, we find that, for generic cases, the entanglement entropy shows discontinuities (shockwave fronts) propagating from the boundaries. In contrast, such discontinuities are absent in cases with simple boundary effects. We verify that our generic CFT formula matches well with numerical calculations from free-fermion tight-binding models for various quench scenarios.

DOI
10.1103/physrevlett.134.220404

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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.
A bstract We study quantum quench processes in (1+1)-dimensional conformal field theory (CFT) in which the initial thermal equilibrium (Gibbs) state is time-evolved by spatially inhomogeneous Hamiltonians, the so-called Möbius and sine-square-deformed (SSD) Hamiltonians. We found that, when the quench is induced by the SSD Hamiltonian, almost all the degrees of freedom are asymptotically gathered at a single point, resulting in a point-like excitation. This excitation, which we dub black hole-like excitation, carries as much information as the total thermal entropy. In contrast, other parts of the system approach the low-entropy (low-temperature) state at late times. For the quench by the Möbius Hamiltonian, we instead found an eternal periodic oscillation of physical quantities such as von Neumann entropy for subsystems. When the CFT admits a holographic dual description, the SSD quench induces a time-dependent, inhomogeneous deformation of the bulk black hole horizon, which, at late enough times, “touches” the boundary. Our quench setups can be used as a way to create low-temperature states, and, also, simulate the formation and evaporation processes of black holes.
Abstract In this work, we study the time evolution of the entanglement Hamiltonian during the process of thermalization in a (1+1)-dimensional conformal field theory (CFT) after a quantum quench from a special class of initial states. In particular, we focus on a subsystem which is a finite interval at the end of a semi-infinite line. Based on conformal mappings, the exact forms of both entanglement Hamiltonian and entanglement spectrum of the subsystem can be obtained. Aside from various interesting features, it is found that in the infinite time limit the entanglement Hamiltonian and entanglement spectrum are exactly the same as those in the thermal ensemble. The entanglement spectrum approaches the steady state spectrum exponentially in time. We also study the modular flows generated by the entanglement Hamiltonian in Minkowski spacetime, which provides us with an intuitive picture of how the entanglement propagates and how the subsystem is thermalized. Furthermore, the effect of a generic initial state is also discussed.
Abstract For two-dimensional conformal field theories (CFTs) in the ground state, it is known that a conformal interface along the entanglement cut can suppress the entanglement entropy from to , where L is the length of the subsystem A , and is the effective central charge which depends on the transmission property of the conformal interface. In this work, by making use of conformal mappings, we show that a conformal interface has the same effect on entanglement evolution in non-equilibrium cases, including global, local and certain inhomogeneous quantum quenches. I.e. a conformal interface suppresses the time evolution of entanglement entropy by effectively replacing the central charge c with , where is exactly the same as that in the ground state case. We confirm this conclusion by a numerical study on a critical fermion chain. Furthermore, based on the quasi-particle picture, we conjecture that this conclusion holds for an arbitrary quantum quench in CFTs, as long as the initial state can be described by a regularized conformal boundary state.
We study the problem of a quantum quench in which the initial state is the ground state of an inhomogeneous Hamiltonian, in two different models, conformal field theory and ordinary free field theory, which are known to exhibit thermalization of finite regions in the homogeneous case. We derive general expressions for the evolution of the energy flow and correlation functions, as well as the entanglement entropy in the conformal case. Comparison of the results from the two approaches in the regime of their common validity shows agreement up to a point further discussed. Unlike for the thermal analogue, the evolution in our problem is non-diffusive and can be physically interpreted using an intuitive picture of quasiparticles emitted from the initial time hypersurface and propagating semiclassically.
We investigate a class of local quantum quenches in which the conformal boundary condition of a (1+1)-dimensional conformal field theory is abruptly changed. We derive a remarkably simple and universal expression for the time evolution of one-point functions on the half-line. This result provides a direct description of the propagation of the disturbance generated by the quench and, in turn, allows us to determine the dynamics of bipartite entanglement for subsystems adjacent to the boundary. We show that, once the subsystem becomes fully causally connected to the quench event, the entanglement entropy undergoes a sharp finite jump whose magnitude is universally given by the logarithm of the ratio of the boundary g-factors associated with the initial and final boundary conditions. We benchmark these analytical predictions against Matrix Product State simulations of the critical Ising spin chain, finding excellent agreement. The numerical analysis further allows us to investigate the time evolution of the spin-flip entanglement asymmetry, revealing how the symmetry-breaking perturbation emitted from the boundary propagates through the system. Our results uncover universal dynamical signatures of boundary quenches and establish a direct connection between nonequilibrium entanglement dynamics and boundary critical phenomena.