Berry connection and quantum geometry in time-dependent systems with instantaneous quantum integrable field theory

TL;DR

This work addresses quantum geometric effects in time-dependent many-body systems whose low-energy dynamics are governed instantaneously by a quantum integrable field theory. It proves a theorem that Berry connection matrix elements between instantaneous eigenstates are determined by at most two-particle excitations in the thermodynamic limit, enabling analytic access via two-particle form factors. The authors illustrate the framework with a driven Ising chain whose scaling limit is the quantum $E_8$ field theory, introducing a quantum geometric potential that renormalizes instantaneous gaps and drives many-body Landau-Zener tunneling, as quantified by Loschmidt-echo spectral entropy. Using truncated conformal-space and truncated lattice free-fermion approaches, they reveal a threshold at $\kappa_c=1$ with hyperscaling near the threshold and a cascade of entropy peaks for $\kappa<1$, suggesting universal dynamics and potential experimental realization in platforms like Rydberg arrays.

Abstract

We study many-body quantum geometric effects in time-dependent system with emergent quantum integrable field theory instantaneously. We establish a theorem stating that the Berry connection matrix thus all associated geometric quantities of the system can be precisely characterized by excitations up to two particles from the initial quantum integrable system. To illustrate the many-body geometric influence, we analyze an Ising chain subjected to both a small longitudinal field and a slowly rotating transverse field, whose low-energy physics in the scaling limit is instantaneously governed by the quantum $E_8$ integrable field theory. Focusing on the quantum geometric potential (QGP), we show the QGP continuously suppresses the instantaneous energy gaps with decreasing longitudinal field, thereby enhancing many-body Landau-Zener tunneling as evidenced by the Loschmidt echo and its associated spectral entropy. The critical threshold for the longitudinal field strength is determined,where the spectral entropy linearly increases with system size and exhibits hyperscaling behavior when approaching to the threshold. As the longitudinal field passes the threshold and decreases toward zero, the QGP continuously leads to vanishing instantaneous energy gaps involving more low-energy excitations, resulting in increasing spectral entropy indicative of many-body Landau-Zener tunneling. Our results unveil telltale quantum geometric signatures in time-dependent many-body systems, elucidating the intricate interplay between quantum geometry and dynamics.

Figures (4)

• Figure 1: An illustration of the setup for the theorem. (a) The initial wavefunction of an eigenstate in an IFT contains $N$ quasiparticles with periodic boundary conditions, represented as a cylinder with circumference $L$. The blue circles represent the quasiparticles. (b) The time evolution of the initial wavefunction under the unitary transformation $\mathscr{U}_a(t)$, with the effective adiabatic Hamiltonian given by Eq. (\ref{['Eq:Hadia']}). The blue line corresponds to the initial configuration in (a), and the red space-time network illustrates $\mathscr{U}_a(t)$.
• Figure 2: (a) Comparison of the QGP between the ground state (vacuum of the $\mathcal{H}_{E_8}$) and the three lightest single-particle states of the $\mathcal{H}_{E_8}$, calculated with both the $\mathcal{H}_{E_8}$ form factor theory and the boosted TCSA algorithm TCSA. Here, $\Lambda$ is the truncation parameter satisfying $E^2-P^2\leq \Lambda^2$. (b) Analytical results of the EEG between the ground state and the 8 single $E_8$ particle state with zero momentum, with $0<\kappa \le 2$. The zeros of EEG appear at $\kappa=8/15$ for all the single $E_8$ particles. (c) The zeros of EEG as a function of $A_1$'s rapidity $\vartheta_1$, for the two-particle states $A_1A_1$ to $A_1A_4$, with a zero momentum constraint $m_{i}\sinh\vartheta_1+m_{j}\sinh\vartheta_2=0$. (d) Zeros of EEG between two four $E_8$ particle states. While both the two groups of particles are of particle type $A_{1\sim4}$, the rapidities for one of the states takes $(\vartheta_1,\vartheta_2,\vartheta_3,\vartheta_4)=(0.2,0.2,0.1,0.1)$, and for the other one takes $\vartheta_1=\vartheta_2=\vartheta_4=0.1$ and $\vartheta_3 = 0.2421,0.2422,0.2425$, corresponding to the zeros being from large to small, respectively.
• Figure 3: $S_{LE}$$vs.$$\kappa$ for different system sizes $L$'s. The $S_{LE}$ at $\kappa=1$ linearly increases with $L$ [inset (a)], along with more peaks emerging for $\kappa \lesssim 8/15$. When $\kappa$ approaches 1, the $S_{LE}$ collapses on the same curve for different $L$ [inset (b)], implying hyperscaling behavior of $S_{\text{LE}}$ w.r.t scaling parameter $(\kappa-1)^{8/15}L$.
• Figure S1: $-p \ln p$$vs.$$\omega$ for different $\kappa$ from $0.2$ to $2$ with $\Delta\kappa=0.2$, corresponding to the curves from the bottom to the top, after a Lorentzian broadening with $\alpha=0.001$.