Local Quenches from a Krylov Perspective

TL;DR

This work develops and applies a Krylov-space framework to two-dimensional CFT local quenches, deriving Lanczos coefficients, spread complexity, and Krylov entropies for joining, splitting, and operator quenches across finite, infinite, and finite-temperature geometries. A central result is that both spread complexity and Krylov entropy scale with the CFT central charge, with Krylov entropies exhibiting a universal logarithmic time growth, while spread complexity carries a nonuniversal UV prefactor. The analysis reveals an emergent SL(2,$ ext{R}$) symmetry governing Krylov dynamics in these quench setups and establishes a holographic correspondence: the rate of spread complexity maps to the proper bulk momentum of a bulk probe in AdS/BCFT, linking boundary complexity to bulk motion of the end-of-the-world brane. These findings position Krylov-space diagnostics as powerful, universal probes of non-equilibrium dynamics in interacting QFTs and their holographic duals. Potential impacts include informing non-equilibrium analyses in 2D CFTs, guiding holographic interpretations of complexity, and motivating further links between Krylov data and bulk geometric quantities.

Abstract

In this work, we investigate local quench dynamics in two-dimensional conformal field theories using Krylov space methods. We derive Lanczos coefficients, spread complexity, and Krylov entropies for local joining and splitting quenches in theories on an infinite line, a circle, a finite interval, and at finite temperature. We examine how these quantities depend on the central charge of the underlying conformal field theory and find that both spread complexity and Krylov entropy are proportional to it. Interestingly, Krylov entropies evolve logarithmically with time, mirroring standard entanglement entropies, making them useful for extracting the central charge. In the large central charge limit, using holography, we establish a connection between the rate of spread complexity and the proper momentum of the tip of the end-of-the world brane, which probes the bulk analogously to a point particle. Our results further demonstrate that spread complexity and Krylov entropies are powerful tools for probing non-equilibrium dynamics of interacting quantum systems.

Figures (12)

• Figure 1: Pictorial representation of the three local quenches discussed in this work. a) Joining quench: The initial state is the product of two states $|A\rangle$ and $|B\rangle$ on two complementary intervals, and we let the system evolve via a Hamiltonian, which couples them. b) Splitting quench: The initial state is homogeneous, and the evolution Hamiltonian $H_A+H_B$ splits the system into two complementary parts. c) Operator quench: The initial state is the ground state of a Hamiltonian $H$ locally excited by an operator $O(x_0)$, and the system is evolved through $H$.
• Figure 2: In the left panel, we show the two-dimensional geometry whose CFT partition function gives the return amplitude after a joining quench of two intervals of size $L/2$. This geometry can be mapped to the half-plane shown in the right panel. The expression of $a$ in terms of $\tau$ and $L$ is given in \ref{['eq:def_parameter_a']}. The curves $\Gamma$ and $\Gamma'$ used to compute the free energy \ref{['eq:variationF_Schwarz']} are represented as blue dashed lines.
• Figure 3: We show the absolute value of the return amplitude after a joining quench of free quantum chains. In both cases, the initial state is a product of identical ground states of the model with $L/2$ sites. The system is let evolve for $t>0$ with the same Hamiltonian, now defined on $L$ sites. The data are reported as a function of time for various choices of $L$. The solid curves correspond to the CFT predictions (\ref{['eq:Snormalized']}) with $c=1$ and the additive constant obtained through a fit of $\varepsilon$, resulting in different values for the two models. The points in the left panel are obtained for the tight-binding model with Hamiltonian \ref{['eq:FFHamiltonian']} and correspond to a fitted value $\varepsilon=0.215$. In the right panel, the curves are obtained considering a harmonic chain with Hamiltonian \ref{['eq:HC Hamiltonian']}, Dirichlet boundary conditions, and a vanishing frequency. In this case, the fit gives $\varepsilon=0.620$.
• Figure 4: In the left panel, we show the two-dimensional geometry whose CFT partition function gives the return amplitude after a splitting quench of an infinite system into two semi-infinite systems. This geometry can be mapped to the half-plane shown in the right panel. The curves $\Gamma$ and $\Gamma'$ used to compute the free energy \ref{['eq:variationF_splitting_v1']} are represented as blue dashed lines.
• Figure 5: The absolute value of the return amplitude after a splitting quench in the tight-binding model with Hamiltonian \ref{['eq:FFHamiltonian']}. The initial state is the ground state of the chain $L$ sites. The system is let evolve for $t>0$ with the sum of the Hamiltonian of the model defined on the first $L/2$ sites and the Hamiltonian of the chain on the last $L/2$ sites. The data are reported as a function of time for various choices of $L$. The solid curve corresponds to the CFT predictions (\ref{['eq:Snormalized_infinite']}) with $c=1$ and the additive constant obtained through a fit of $\varepsilon=0.225$. We focus on the initial times after the splitting quench in the chain to compare the return amplitude with the CFT predictions obtained for the splitting of a system on an infinite line.