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arxiv: 2601.19152 · v1 · submitted 2026-01-27 · 🪐 quant-ph · cond-mat.other

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 This is my paper

Pith reviewed 2026-05-16 11:25 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.other
keywords quantum geometric tensorquench dynamics1D periodic systemsBerry connectiongroup velocity varianceenergy varianceSSH modelnonequilibrium geometry
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The pith

After a sudden quench in 1D periodic systems the quantum geometric tensor components map directly onto position variance, energy variance, and a quench-induced curvature using initial Berry connections and final band velocities.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies how the quantum geometric tensor evolves immediately after a sudden Hamiltonian change in one-dimensional periodic quantum systems. Different tensor components turn out to equal concrete observables: the crystal-momentum diagonal supplies a metric whose quadratic time growth is fixed by the group-velocity variance, the time diagonal equals the energy variance, and the off-diagonal real part is a covariance while its imaginary part is a curvature created by the quench. These relations follow because the time-evolved state is assembled from the initial state's geometric data together with the post-quench dispersion. The findings are checked numerically on the Su-Schrieffer-Heeger chain for several quench protocols, suggesting the tensor can serve as a compact geometric diagnostic for nonequilibrium dynamics.

Core claim

In one-dimensional periodic systems the post-quench quantum geometric tensor is governed by physical quantities and local geometric objects drawn from the initial state and the post-quench bands, including the Berry connection, group velocities, and energy variance. Its momentum-diagonal component supplies the metric for the variance of the time-evolved position operator, with the coefficient of the quadratic time term equal to the group-velocity variance that signals ballistic wave-packet dispersion. The time-diagonal component equals the energy variance, while the off-diagonal component has a real part that is a covariance and an imaginary part that represents the curvature induced by the

What carries the argument

The quantum geometric tensor of the time-evolved Bloch states, whose diagonal and off-diagonal elements are shown to equal position and energy variances plus a quench-generated curvature constructed from the initial Berry connection and post-quench group velocities.

If this is right

  • The quadratic coefficient in the position metric directly measures the group-velocity variance and therefore the rate of ballistic spreading.
  • The time-diagonal component of the tensor equals the energy variance of the evolved state.
  • The imaginary part of the off-diagonal component isolates the curvature generated solely by the quench protocol.
  • The full set of relations supplies a geometric probe that links initial-state Berry geometry to post-quench observables without requiring full wave-function reconstruction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the relations generalize, cold-atom experiments could extract QGT components indirectly by measuring wave-packet widths after a lattice quench.
  • The same geometric construction might be applied to Floquet or periodically driven 1D chains to identify new curvature terms.
  • Extension to two-dimensional lattices would test whether the post-quench tensor still reduces to local band quantities or acquires topological corrections.
  • The covariance term could serve as a diagnostic for correlations between position and energy fluctuations in nonequilibrium steady states.

Load-bearing premise

The numerical mappings found for the SSH model are assumed to hold exactly, without extra corrections, for arbitrary one-dimensional periodic systems.

What would settle it

An exact analytic or high-precision numerical calculation of the post-quench QGT for a different 1D model such as the Rice-Mele chain, checking whether every component matches the predicted expressions involving only the initial Berry connection, final velocities, and energy variance.

Figures

Figures reproduced from arXiv: 2601.19152 by Hao Guo. Chih-Chun Chien, Jia-Chen Tang, Xu-Yang Hou, Yu-Huan Huang.

Figure 1
Figure 1. Figure 1: Quantum-metric dynamics gkk(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 mi and mf are labeled above each case. Near the Brillouin zone boundaries k = ±π, sharp, nar￾row peaks oscillate periodically in time, stemming solely from … view at source ↗
Figure 2
Figure 2. Figure 2: Post-quench temporal metric gtt(k). Blue dot-dash line: mi = 0.5, mf = 0.1; Green dotted line: mi = 1.1, mf = 2.0; Black dash line: mi = 1.5, mf = 0.1; Red solid line: mi = 0.9, mf = 2.0. lution. The smooth or rippled ridge in the momentum-resolved evolution of gkk mirrors the magnitude of the time￾independent coefficient g (2) kk = Var(ˆvk). Larger group￾velocity variance in [PITH_FULL_IMAGE:figures/full… view at source ↗
Figure 3
Figure 3. Figure 3: Off-diagonal component Qkt(k, t) for the same four quench protocols in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper claims that in 1D periodic systems after a sudden quench, the quantum geometric tensor (QGT) evolves such that its k-diagonal component corresponds to the variance of the time-evolved position, with the quadratic time coefficient being the group-velocity variance indicating ballistic dispersion; the time-diagonal component gives the energy variance; and the off-diagonal has real part as covariance and imaginary part as quench-induced curvature. Numerical results in the SSH model confirm that the post-quench QGT is governed by quantities like Berry connection, group velocities, and energy variance from initial and post-quench states, suggesting QGT as a probe for nonequilibrium phenomena.

Significance. If the claimed mappings hold for general 1D systems, this work would provide a geometric framework for understanding post-quench dynamics, connecting the QGT directly to physical observables and local geometric properties. This could be significant for nonequilibrium quantum many-body physics, offering new ways to probe ballistic spreading and energy fluctuations. The numerical confirmation in the SSH model supports the idea, but the lack of a general derivation reduces the immediate impact.

major comments (1)
  1. [Abstract] The central claim that the post-quench QGT components map exactly to the Berry connection, group-velocity variance, energy variance, and quench-induced curvature for arbitrary 1D periodic systems rests only on numerical confirmation in the SSH model for specific quenches. No model-independent analytic derivation is provided from the time-evolution operator U(t) = exp(-i H_f t) acting on initial Bloch states, which is necessary to rule out potential corrections from non-adiabatic overlaps or protocol-specific details.
minor comments (1)
  1. The abstract states that numerical results confirm the relations but provides no details on the specific quenches, error bars, or how the data selection was performed, which would help assess the robustness of the findings.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful review and for highlighting the need for a clearer demonstration of generality. We address the major comment below and will revise the manuscript accordingly to strengthen the analytic foundation of our claims.

read point-by-point responses

  1. Referee: [Abstract] The central claim that the post-quench QGT components map exactly to the Berry connection, group-velocity variance, energy variance, and quench-induced curvature for arbitrary 1D periodic systems rests only on numerical confirmation in the SSH model for specific quenches. No model-independent analytic derivation is provided from the time-evolution operator U(t) = exp(-i H_f t) acting on initial Bloch states, which is necessary to rule out potential corrections from non-adiabatic overlaps or protocol-specific details.

    Authors: We agree that an explicit model-independent derivation is essential to support the generality of the mappings. In the revised manuscript we will add a dedicated analytic section deriving the post-quench QGT directly from the action of U(t) = exp(-i H_f t) on the initial Bloch states |u_k(0)>. Starting from the definition of the QGT in the time-evolved basis and using only the periodicity of the lattice and the completeness of the Bloch basis (without assuming any specific form of H_f beyond being periodic), we show that the k-diagonal component reduces to the position variance whose quadratic time coefficient is the group-velocity variance, the time-diagonal component is the energy variance, and the off-diagonal components yield the covariance and the quench-induced Berry curvature term. The derivation explicitly demonstrates the absence of non-adiabatic overlap corrections in one dimension. We will also update the abstract to reflect that the mappings are now supported by both the general derivation and the SSH numerics. revision: yes

Circularity Check

0 steps flagged

No circularity detected; relations derived from standard QM definitions and time evolution

full rationale

The paper defines the post-quench QGT directly from the time-evolved Bloch wavefunctions under the sudden quench protocol using the standard expression for the quantum geometric tensor in terms of overlaps. The mappings to Berry connection, group-velocity variance, energy variance, and quench-induced curvature are obtained by expanding the time-dependent overlaps and extracting coefficients of the resulting quadratic forms in t; these steps follow from the definition of the time-evolution operator and do not reduce to the target quantities by construction. Numerical results in the SSH model serve only as confirmation for specific cases, not as fitting or redefinition of the general expressions. No load-bearing self-citations, ansatz smuggling, or uniqueness theorems from prior author work are invoked to close the derivation. The chain remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard quantum mechanics for Bloch states in periodic systems and the sudden quench approximation, with no free parameters or invented entities introduced.

axioms (2)
  • standard math Bloch theorem and geometric phases apply to 1D periodic systems
    Invoked for defining the quantum geometric tensor in terms of crystal momentum and band structures.
  • domain assumption Sudden quench approximation for instantaneous Hamiltonian change
    Assumed to model the time evolution after the quench without intermediate dynamics.

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Reference graph

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