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Evolution of quantum geometric tensor of 1D periodic systems after a quench

Jia-Chen Tang, Xu-Yang Hou, Yu-Huan Huang, Hao Guo. Chih-Chun Chien

TL;DR

The paper analyzes post-quench dynamics of the quantum geometric tensor (QGT) in 1D periodic systems, using the SSH model as a concrete testbed. It shows that post-quench QGT components map to physical variances and covariances: $Q_{kk}=g_{kk}$ with $g_{kk}(t)= ext{Var}( ext{x})+2t ext{Cov}( ext{x}, ext{v}_k)+t^2 ext{Var}( ext{v}_k)$, $Q_{tt}=g_{tt}= ext{Var}(H_f)$, and $Q_{kt}= ext{Cov}( ext{x},H_f)$, rendering a linear-$t$ and an oscillatory contribution from inter-band coherence. The SSH analysis provides explicit expressions involving Berry connections, group velocities, and energy variance, and numerical results show how these quantities shape the momentum-resolved QGT and its ballistic $t^2$ growth. The work highlights the QGT as a comprehensive probe of nonequilibrium quantum geometry, with curvature and dispersion encoded in local observables rather than global topological invariants. Overall, the QGT links geometric language to concrete nonequilibrium observables, enabling momentum- and time-resolved insights into quantum dynamics after quenches.

Abstract

We investigate the post-quench dynamics of the quantum geometric tensor (QGT) of 1D periodic systems with a suddenly changed Hamiltonian. The diagonal component with respect to the crystal momentum gives a metric corresponding to the variance of the time-evolved position, and its coefficient of the quadratic term in time is the group-velocity variance, signaling ballistic wavepacket dispersion. The other diagonal QGT component with respect to time reveals the energy variance. The off-diagonal QGT component features a real part as a covariance and an imaginary part representing a quench-induced curvature. Using the Su-Schrieffer-Heeger (SSH) model as an example, our numerical results of different quenches confirm that the post-quench QGT is governed by physical quantities and local geometric objects from the initial state and post-quench bands, such as the Berry connection, group velocities, and energy variance. Furthermore, the connections between the QGT and physical observables suggest the QGT as a comprehensive probe for nonequilibrium phenomena.

Evolution of quantum geometric tensor of 1D periodic systems after a quench

TL;DR

The paper analyzes post-quench dynamics of the quantum geometric tensor (QGT) in 1D periodic systems, using the SSH model as a concrete testbed. It shows that post-quench QGT components map to physical variances and covariances: with , , and , rendering a linear- and an oscillatory contribution from inter-band coherence. The SSH analysis provides explicit expressions involving Berry connections, group velocities, and energy variance, and numerical results show how these quantities shape the momentum-resolved QGT and its ballistic growth. The work highlights the QGT as a comprehensive probe of nonequilibrium quantum geometry, with curvature and dispersion encoded in local observables rather than global topological invariants. Overall, the QGT links geometric language to concrete nonequilibrium observables, enabling momentum- and time-resolved insights into quantum dynamics after quenches.

Abstract

We investigate the post-quench dynamics of the quantum geometric tensor (QGT) of 1D periodic systems with a suddenly changed Hamiltonian. The diagonal component with respect to the crystal momentum gives a metric corresponding to the variance of the time-evolved position, and its coefficient of the quadratic term in time is the group-velocity variance, signaling ballistic wavepacket dispersion. The other diagonal QGT component with respect to time reveals the energy variance. The off-diagonal QGT component features a real part as a covariance and an imaginary part representing a quench-induced curvature. Using the Su-Schrieffer-Heeger (SSH) model as an example, our numerical results of different quenches confirm that the post-quench QGT is governed by physical quantities and local geometric objects from the initial state and post-quench bands, such as the Berry connection, group velocities, and energy variance. Furthermore, the connections between the QGT and physical observables suggest the QGT as a comprehensive probe for nonequilibrium phenomena.
Paper Structure (19 sections, 27 equations, 3 figures)

This paper contains 19 sections, 27 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Theoretical Formalism of the Post-quench QGT
  3. Quantum Geometric Tensor
  4. Post-quench QGT as Operator Variance and Covariance
  5. Diagonal components $Q_{kk}$ and $Q_{tt}$
  6. Off-diagonal component $Q_{kt}$
  7. QGT dynamics of the SSH Model
  8. The SSH Model
  9. Diagonal component $Q_{kk}$
  10. Diagonal component $Q_{tt}$
  11. Off-diagonal component $Q_{kt}$
  12. Numerical examples
  13. Diagonal component $Q_{kk}$
  14. Diagonal component $Q_{tt}$
  15. Off-diagonal component $Q_{kt}$
  16. ...and 4 more sections

Figures (3)

  • Figure 1: Quantum-metric dynamics $g_{kk}(k,t)$ for four quench protocols. For each case, the left (right) panel shows the 3D surface (contour map). The top (bottom) row shows quenches staying on the same side of (crossing) the gap closing point $m=1$. The values of $m_i$ and $m_f$ are labeled above each case.
  • Figure 2: Post-quench temporal metric $g_{tt}(k)$. Blue dot-dash line: $m_i=0.5, m_f=0.1$; Green dotted line: $m_i=1.1, m_f=2.0$; Black dash line: $m_i=1.5, m_f=0.1$; Red solid line: $m_i=0.9, m_f=2.0$.
  • Figure 3: Off-diagonal component $Q_{kt}(k,t)$ for the same four quench protocols in Fig. \ref{['Fig_g_kk']}. For each set, the left (right) plot shows the imaginary (real) part. Here $k\in[-\pi,\pi]$ and $t\in[0,20]$. The values of $m_i$ and $m_f$ are labeled above each panel.