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Quantum adiabaticity in many-body systems and almost-orthogonality in complementary subspace

Jyong-Hao Chen, Vadim Cheianov

TL;DR

This paper explains why adiabatic fidelity $\\mathcal{F}(\\lambda)$ and ground-state overlap $\\mathcal{C}(\\lambda)$ often align in driven many-body quantum systems by identifying two complementary mechanisms: perturbative behavior at small evolution parameters and almost-orthogonality in the complementary subspace at large system size. It develops reverse triangle inequalities connecting $\\mathcal{F}$, $\\mathcal{C}$, and an auxiliary overlap $\\mathcal{D}_{\\mathrm{un}}$, and shows that $\\mathcal{D}_{\\mathrm{un}}$ becomes small in both regimes, explaining the observed closeness. The authors illustrate the framework with the driven Rice–Meile model (non-interacting) and a driven interacting Kitaev chain, demonstrating robustness of almost-orthogonality and providing refined bounds on $\\mathcal{F}$ through a scaling form $\\mathcal{D}(\\lambda)\\approx\\mathcal{C}(\\lambda)^s$. They also discuss implications for adiabatic breakdown via a finite-size scaling of the driving rate $\\Gamma_N$, and offer practical diagnostics for finite many-body adiabaticity. Overall, the results provide a conceptually clear and quantitatively useful picture of adiabaticity in driven many-body dynamics with potential applicability to a range of quantum technologies.

Abstract

We investigate why, in quantum many-body systems, the adiabatic fidelity and the overlap between the initial state and instantaneous ground states often yield nearly identical values. Our analysis suggests that this phenomenon results from an interplay between two intrinsic limits of many-body systems: the limit of small evolution parameters and the limit of large system sizes. In the former case, conventional perturbation theory provides a straightforward explanation. In the latter case, a key insight is that pairs of vectors in the Hilbert space orthogonal to the initial state tend to become nearly orthogonal as the system size increases. We illustrate these general findings with two representative models of driven many-body systems: the driven Rice-Mele model and the driven interacting Kitaev chain model.

Quantum adiabaticity in many-body systems and almost-orthogonality in complementary subspace

TL;DR

This paper explains why adiabatic fidelity and ground-state overlap often align in driven many-body quantum systems by identifying two complementary mechanisms: perturbative behavior at small evolution parameters and almost-orthogonality in the complementary subspace at large system size. It develops reverse triangle inequalities connecting , , and an auxiliary overlap , and shows that becomes small in both regimes, explaining the observed closeness. The authors illustrate the framework with the driven Rice–Meile model (non-interacting) and a driven interacting Kitaev chain, demonstrating robustness of almost-orthogonality and providing refined bounds on through a scaling form . They also discuss implications for adiabatic breakdown via a finite-size scaling of the driving rate , and offer practical diagnostics for finite many-body adiabaticity. Overall, the results provide a conceptually clear and quantitatively useful picture of adiabaticity in driven many-body dynamics with potential applicability to a range of quantum technologies.

Abstract

We investigate why, in quantum many-body systems, the adiabatic fidelity and the overlap between the initial state and instantaneous ground states often yield nearly identical values. Our analysis suggests that this phenomenon results from an interplay between two intrinsic limits of many-body systems: the limit of small evolution parameters and the limit of large system sizes. In the former case, conventional perturbation theory provides a straightforward explanation. In the latter case, a key insight is that pairs of vectors in the Hilbert space orthogonal to the initial state tend to become nearly orthogonal as the system size increases. We illustrate these general findings with two representative models of driven many-body systems: the driven Rice-Mele model and the driven interacting Kitaev chain model.
Paper Structure (20 sections, 1 theorem, 86 equations, 9 figures)

This paper contains 20 sections, 1 theorem, 86 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Setup
  4. Orthogonal decomposition
  5. A motivating limit and interpretations
  6. Insights from perturbative expansion in $\lambda$
  7. Insights from almost-orthogonality in the complementary subspace under large system size
  8. Reverse triangle inequalities
  9. Illustrative example I: non-interacting Hamiltonians
  10. Asymptotic form of the overlap $\mathcal{D}(\lambda)$ and implications
  11. Illustrative example II: interacting fermions
  12. Summary and Outlook
  13. Funding information
  14. Alternative derivation of Eq. (\ref{['eq: inequality perp basis']})
  15. Perturbative expansion in $\lambda$
  16. ...and 5 more sections

Key Result

Lemma 1

The three real-valued quantities, $\sqrt{\mathcal{F}(\lambda)}$ (eq: define adiabatic fidelity), $\cos\theta(\lambda)\,\sqrt{\mathcal{C}(\lambda)}$ (eq: define GOC), and $\sqrt{\mathcal{D}^{\,}_{\mathrm{un}}(\lambda)}$ (eq: define un normalized D overlap), obey a set of (reverse) triangle inequaliti

Figures (9)

  • Figure 1: Depict of the triangle relationship (\ref{['eq: reverse triangle inequalities']}) between the three real-valued quantities: $\sqrt{\mathcal{F}(\lambda)}$ (\ref{['eq: define adiabatic fidelity']}), $\cos\theta(\lambda)\,\sqrt{\mathcal{C}(\lambda)}$ (\ref{['eq: define GOC']}), and $\sqrt{\mathcal{D}^{\,}_{\mathrm{un}}(\lambda)}$ (\ref{['eq: define un normalized D overlap']}), where $\theta(\lambda)$ is the Bures angle (\ref{['eq: bures angle']}).
  • Figure 2: (Color online) Various quantities are calculated numerically for the driven Rice-Mele model (\ref{['eq: driven RM model']}) with the value of parameters shown in Eq. (\ref{['eq: numerical parameter']}) for system size $N=10,200$ and $1000.$ The fourth row overlays the curves from the second and third rows, with the vertical axis on a logarithmic scale. Further explanation is provided in the main text.
  • Figure 3: (Color online) Various quantities, $\sqrt{\mathcal{C}(\lambda)}$ (\ref{['eq: define GOC']}), $\sqrt{\mathcal{D}(\lambda)}$ (\ref{['eq: define overlap of orthogonal components']}), $\sqrt{\mathcal{D}^{\,}_{\mathrm{un}}(\lambda)}$ (\ref{['eq: define un normalized D overlap']}), $\sqrt{\mathcal{C}(\lambda)^{1/2}}$, and $s(\lambda)$ (\ref{['eq: sD postulate form']}) for the driven Rice-Mele model (\ref{['eq: driven RM model']}) are plotted as a function of $\lambda$ or $N$.
  • Figure 4: (Color online) Compare the behavior of the function $g(\lambda)=g^{\,}_{1}(\lambda)+g^{\,}_{2}(\lambda)$ (\ref{['eq: new improved bound with alpha']}) with the function $f(\lambda)=f^{\,}_{1}(\lambda)+f^{\,}_{2}(\lambda)$ (\ref{['eq: summarized inequalities one side']}) for $N=200.$
  • Figure 5: (Color online) Bounds on the adiabatic fidelity $\mathcal{F}(\lambda)$ for $N=200$ and $N=1000$ using Eq. (\ref{['eq: summarized inequalities one side']}) [blue-shaded region], Eq. (\ref{['eq: new improved bound with alpha']}) [red-shaded region], and Eq. (\ref{['eq: a new improved bound']}) with $s=1/2$ [green-shaded region] and $s=1$ [yellow-shaded region]. For both figures, the actual adiabatic fidelity $\mathcal{F}(\lambda)$ is the black curve, while the red curve is for the ground state overlap $\mathcal{C}(\lambda)$, which is, however, not distinct from $\mathcal{F}(\lambda)$.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Lemma
  • proof