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

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.

• The paper develops a general post-quench quantum geometric tensor (QGT) framework for 1D periodic systems-four key particles are variance of the evolved group-velocity operator positions and covariances for $Q_{kk}$ and $Q_{kt}$, and the energy variance $\Delta E^{2}$.Ecotontatalgy, which is directly related to $Q_{tt}$.
• This formalism, tested on a SSH model, demonstrates how $m=$ 2 shell regimes can alter wavepacket dispersion and quasiparticle energy variance.
• Practical insights into quench-induced geometric spreading and energy fluctuations.

Overview

This paper by Tang, Hou, Huang, Guo, and Chien develops a general framework for the post-quench dynamics of the quantum geometric tensor (QGT) of one-dimensional periodic systems, taking $\boldsymbol{\lambda}=(k,t)$ as the parameters and using the Su-Schrieffer-Heeger (SSH) model as the concrete realization (2601.19152). The central observation is that while unitary evolution generated by the same Hamiltonian leaves the QGT invariant (a $U(1)$ gauge transformation along $k$), a sudden quench breaks this symmetry and drives the state along a non-symmetric trajectory on the quantum-state manifold. The resulting QGT acquires secular terms whose coefficients are identifiable with familiar physical quantities: group-velocity variance, energy variance, Berry connections, and position–velocity covariances.

The paper's main structural result is that each component of the post-quench QGT admits an operator interpretation:

Component	Physical meaning	Time dependence
$Q_{kk} = g_{kk}$	Variance of the time-evolved position operator	$g_{kk}^{(0)} + g_{kk}^{(1)}t + g_{kk}^{(2)}t^2$
$Q_{tt} = g_{tt}$	Energy variance $\Delta E^2$ of $H_f$	Time-independent
$Q_{kt}$	Covariance $\mathrm{Cov}(\hat{x}, H_f)$; imaginary part is a quench-induced curvature on the $U(1)$0 plane	Linear growth plus oscillations

These identifications follow from recognizing that $U(1)$1 acts as the position operator for Bloch states and that $U(1)$2 brings down $U(1)$3 via the Schrödinger equation. The three components are further constrained by the positive-semidefiniteness of the variance-covariance matrix, giving $U(1)$4, which bounds geometric spreading against energy fluctuations throughout the post-quench evolution.

Operator formalism for the post-quench QGT

For a Bloch state, the quantum metric equals the variance of the maximally localized Wannier center: $U(1)$5. Under post-quench Heisenberg evolution with static $U(1)$6, the position operator integrates to $U(1)$7 with $U(1)$8, so the metric expands as

$U(1)$9

The dominant $k$0 term has coefficient equal to the group-velocity variance, signaling ballistic wavepacket dispersion; this parallels the known $k$1 scaling of Fisher information in quantum metrology [PhysRevLett.124.060402]. The temporal component reduces exactly to $k$2, independent of time since the energy variance computed in the evolved state equals that of the initial state expanded in post-quench eigenstates. The off-diagonal component satisfies $k$3, decomposable into a symmetrized covariance plus an antisymmetric part proportional to $k$4.

A key conceptual point is why secular terms appear only after a quench. Without a quench, $k$5 merely phases each momentum eigenstate — a $k$6 transformation preserving $k$7 (proved explicitly in the appendix). After a quench, the initial state decomposes into two eigenstates of $k$8, making the effective evolution a $k$9 rotation; since $Q_{kk} = g_{kk}$0, the trajectory leaves the symmetry orbit of $Q_{kk} = g_{kk}$1, and the linear and quadratic coefficients quantify this deviation.

Analytical results for the SSH model

Applying the formalism to the SSH chain with dimerization ratio $Q_{kk} = g_{kk}$2, quenched from $Q_{kk} = g_{kk}$3 to $Q_{kk} = g_{kk}$4, the authors obtain closed-form expressions for all components. Writing the initial ground state as $Q_{kk} = g_{kk}$5, the metric expansion reads

$Q_{kk} = g_{kk}$6

with

$Q_{kk} = g_{kk}$7

$Q_{kk} = g_{kk}$8

Here $Q_{kk} = g_{kk}$9 are the pre-/post-quench Berry connections, $g_{kk}^{(0)} + g_{kk}^{(1)}t + g_{kk}^{(2)}t^2$0 mixes them with the overlap factor $g_{kk}^{(0)} + g_{kk}^{(1)}t + g_{kk}^{(2)}t^2$1, and $g_{kk}^{(0)} + g_{kk}^{(1)}t + g_{kk}^{(2)}t^2$2 encodes the group velocity times the transition probability. The quadratic coefficient is manifestly the group-velocity variance, containing the quantum uncertainty factor $g_{kk}^{(0)} + g_{kk}^{(1)}t + g_{kk}^{(2)}t^2$3. In the no-quench limit ($g_{kk}^{(0)} + g_{kk}^{(1)}t + g_{kk}^{(2)}t^2$4), $g_{kk}^{(0)} + g_{kk}^{(1)}t + g_{kk}^{(2)}t^2$5 and $g_{kk}^{(0)} + g_{kk}^{(1)}t + g_{kk}^{(2)}t^2$6, so both secular terms vanish and the static metric is recovered — a consistency check confirmed by a general appendix proof.

The temporal component takes the compact form

$g_{kk}^{(0)} + g_{kk}^{(1)}t + g_{kk}^{(2)}t^2$7

which vanishes without a quench and diverges at the gap-closing point $g_{kk}^{(0)} + g_{kk}^{(1)}t + g_{kk}^{(2)}t^2$8, reflecting enhanced energy fluctuations when bands touch. Notably, $g_{kk}^{(0)} + g_{kk}^{(1)}t + g_{kk}^{(2)}t^2$9 vanishes at $Q_{tt} = g_{tt}$0 because the SSH Hamiltonian there is proportional to $Q_{tt} = g_{tt}$1 with $Q_{tt} = g_{tt}$2-independent eigenstates — a symmetry-locking mechanism suppressing quench-induced energy variance at high-symmetry points.

The off-diagonal component obeys

$Q_{tt} = g_{tt}$3

with quench-activated prefactor $Q_{tt} = g_{tt}$4. Because $Q_{tt} = g_{tt}$5 is odd in $Q_{tt} = g_{tt}$6 while all other factors are even, $Q_{tt} = g_{tt}$7 is odd under $Q_{tt} = g_{tt}$8, so its Brillouin-zone integral vanishes. The authors emphasize an important distinction: the quench-induced curvature $Q_{tt} = g_{tt}$9 is a local object on the extended $\Delta E^2$0 plane and carries no additional topological information, unlike the adiabatic Berry phase ($\Delta E^2$1 or $\Delta E^2$2 depending on the winding number). Even for trivial-phase parameters, nonzero local curvature fluctuations appear but integrate to zero.

Numerical results across quench protocols

Four protocols are examined: $\Delta E^2$3 and $\Delta E^2$4 (same side of the gap closing), and $\Delta E^2$5 and $\Delta E^2$6 (crossing $\Delta E^2$7). The momentum-resolved maps of $\Delta E^2$8 show quadratically growing ridges near $\Delta E^2$9 and oscillatory peaks at $H_f$0 where $H_f$1 extinguishes the secular terms. A clear dichotomy emerges: when the group-velocity variance is large (protocols starting near criticality with large jumps, i.e., panels (a) and (d)), the smooth ballistic $H_f$2 ridge overwhelms oscillations; when it is modest ((b) and (c)), coherent inter-band oscillations persist as visible ripples.

Two conclusions follow directly from these comparisons. First, the decisive factors governing the response are the proximity of the initial state to the gap-closing point and the magnitude of the parameter change — not whether the quench crosses the topological phase boundary. Quenches (a) and (d), which differ in whether they cross $H_f$3, produce qualitatively similar ballistic behavior, while (b) and (c) resemble each other despite also differing in topology. Second, the linear-in-time growth of $H_f$4, governed by the same coefficient $H_f$5 satisfying $H_f$6, serves as an early-time indicator of the subsequent ballistic spreading in $H_f$7.

For $H_f$8, the peak height follows the ordering $H_f$9, revealing a competition between quench strength $Q_{kt}$0 and initial closeness to criticality; the protocol $Q_{kt}$1 maximizes both factors. Since $Q_{kt}$2 sets a Mandelstam–Tamm timescale $Q_{kt}$3, the momentum-resolved temporal metric identifies which modes respond most rapidly after the quench — though the actual oscillation frequency remains set by the spectrum $Q_{kt}$4, so $Q_{kt}$5 bounds rather than determines the dynamics.

For $Q_{kt}$6, the sign of $Q_{kt}$7 near $Q_{kt}$8 acts as a direct marker of the quench direction ($Q_{kt}$9 versus $\mathrm{Cov}(\hat{x}, H_f)$0): the sign flips between protocols (a) and (d) because $\mathrm{Cov}(\hat{x}, H_f)$1 changes sign across $\mathrm{Cov}(\hat{x}, H_f)$2 (analyzed in detail in the appendix, including the divergence $\mathrm{Cov}(\hat{x}, H_f)$3 near criticality).

Limitations and open questions

The analysis is restricted to single-particle, noninteracting two-band systems in one dimension with periodic boundary conditions; interactions, disorder, higher dimensions, and mixed states lie outside its scope. The identification of $\mathrm{Cov}(\hat{x}, H_f)$4 with position variance relies on the crystal-momentum parametrization and the Wannier-center construction, so its extension to other parameter choices is not established here. The claim that the quench-induced curvature $\mathrm{Cov}(\hat{x}, H_f)$5 carries no topological content is demonstrated only for the SSH model, where oddness in $\mathrm{Cov}(\hat{x}, H_f)$6 forces a vanishing integral; whether analogous curvatures can yield nonzero integrals, or connect to dynamical topological invariants, in other models remains open. Finally, the paper establishes the QGT as a diagnostic of nonequilibrium dynamics but does not address experimental extraction of the full $\mathrm{Cov}(\hat{x}, H_f)$7-dependent tensor, leaving open how the predicted secular and oscillatory structures could be resolved in time- or momentum-resolved spectroscopies such as those already used to measure equilibrium QGTs.

Conclusion

The paper shows that the full QGT, evaluated with respect to $\mathrm{Cov}(\hat{x}, H_f)$8 after a quantum quench, decomposes into physically transparent pieces: a metric growing ballistically with coefficient equal to the group-velocity variance, a time-independent energy variance, and a covariance whose real part grows linearly and whose imaginary part constitutes a quench-induced local curvature. For the SSH model, closed-form expressions demonstrate that the post-quench geometry is controlled by the Berry connections, group velocities, energy variance, and band overlaps of the initial and final Hamiltonians — with enhancement near the gap-closing point — rather than by the equilibrium topological invariant itself. These results position the QGT as a comprehensive, momentum-resolved probe of nonequilibrium phenomena, complementing existing spectroscopic and tomographic measurements of quantum geometry.