Krylov Complexity Meets Confinement

TL;DR

The paper investigates confinement in the one-dimensional Ising chain with transverse and longitudinal fields by diagnosing Krylov state complexity, $C_k(t)$, following quenches in various phases. It shows strong suppression and oscillatory behavior of $C_k(t)$ within the confining ferromagnetic regime, while quenches in the non-confining paramagnetic phase exhibit larger, less restricted growth; crossing the critical point to the ferromagnetic phase yields dramatically larger complexity, signaling weak confinement. A semiclassical two-body bound-state model yields meson masses, which are imprinted as peaks in the power spectrum of Krylov complexity, $S_k(\omega)$, matching Bohr–Sommerfeld predictions for the bound-state spectrum. The authors further implement a Full Orthogonalization Lanczos algorithm to construct the Krylov basis and map dynamics to a semi-infinite chain, enabling operator-independent Krylov spectroscopy of confinement. Overall, the work links quantum-information measures with nonperturbative confinement phenomena and suggests Krylov complexity as a robust diagnostic tool with potential extensions to lattice gauge theories and non-equilibrium dynamics.

Abstract

In high-energy physics, confinement denotes the tendency of fundamental particles to remain bound together, preventing their observation as free, isolated entities. Interestingly, analogous confinement behavior emerges in certain condensed matter systems, for instance, in the Ising model with both transverse and longitudinal fields, where domain walls become confined into meson-like bound states as a result of a longitudinal field-induced linear potential. In this work, we employ the Ising model to demonstrate that Krylov state complexity--a measure quantifying the spread of quantum information under the repeated action of the Hamiltonian on a quantum state--serves as a sensitive and quantitative probe of confinement. We show that confinement manifests as a pronounced suppression of Krylov complexity growth following quenches within the ferromagnetic phase in the presence of a longitudinal field, reflecting slow correlation dynamics. In contrast, while quenches within the paramagnetic phase exhibit enhanced complexity with increasing longitudinal field, reflecting the absence of confinement, those crossing the critical point to the ferromagnetic phase reveal a distinct regime characterized by orders-of-magnitude larger complexity and display trends of weak confinement. Notably, in the confining regime, the complexity oscillates at frequencies corresponding to the meson masses, with its power-spectrum peaks closely matching the semiclassical predictions.

Figures (6)

• Figure 1: Krylov complexity $C_k(t)$ following a quench from a fully ferromagnetic initial state to $h_x = 0.25$ within the ferromagnetic phase, shown for various values of the longitudinal field $h_z$. A pronounced suppression in the complexity is observed upon introducing the longitudinal field. Inset (a) highlights marked finite-size effects in $C_k(t)$ at $h_z = 0$, while inset (b) illustrates the suppression of these effects for $h_z = 0.4$ as a consequence of confinement. For the main plot, we have considered a system of size $L = 14$.
• Figure 2: Krylov complexity $C_k(t)$ following a quench from a paramagnetic initial state at $h_x=2$ to $h_x = 1.75$ within the paramagnetic phase, shown for different values of the longitudinal field $h_z$. The observed increase in the amplitude of complexity with increasing $h_z$ indicates the absence of confinement. The system size used is $L = 14$.
• Figure 3: Krylov complexity $C_k(t)$ following a quench from a paramagnetic initial state at $h_x=2$ to $h_x = 0.25$, crossing the quantum critical point ($h_x = 1$), shown for various values of the longitudinal field $h_z$. The complexity initially increases with $h_z$ before eventually decreasing at larger field strengths which might indicate weak confinement. Notably, its magnitude is several orders higher than that observed for quenches performed entirely within the paramagnetic or ferromagnetic phases. The system size used is $L = 14$.
• Figure 4: Power spectrum $S_k(\omega)$ of the Krylov complexity for a quench within the ferromagnetic phase to $h_z = 0.2$ and $h_x=0.25$, starting from a fully polarized ferromagnetic state. The high-frequency peaks correspond to the two meson masses $m_1 = 4.025$ and $m_2 = 4.702$ obtained from the semiclassical analysis. The spectrum also captures the relative spacing between meson masses $m_{12}$. We consider a system of size $L = 14$ with Krylov complexity evolved up to time $t = 100$ with a time step of $\Delta t = 0.1$ for computing the power spectrum using DFT.
• Figure S1: Semiclassical bound-state energy levels in $\omega(p,P)$ obtained from the solutions of Eq. \ref{['BS1']}. Dashed vertical lines indicate the turning points $p_{a,b}$. Horizontal lines denote the meson mass values. (a) Bound states for $h_x = 0.25, h_z = 0.2, P = 0$. (b) Bound states for $h_x = 0.25, h_z = 0.1, P = 0$.