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Breaking of clustering and macroscopic coherence under the lens of asymmetry measures

Florent Ferro

Abstract

In one-dimensional systems, spontaneous symmetry breaking (SSB) states are fragile by nature, as the injection of a non-zero energy density above the ground state is expected to restore the symmetry. This instability implies that local perturbations can lead to macroscopic correlation profiles, a breaking of clustering properties and even macroscopic quantum superpositions. In this work, we investigate the effect of interaction on this phenomenology by considering an interacting model with conserved domain wall number, that possesses a ferromagnetic ground state breaking the Z2 symmetry of the Hamiltonian. We first show that a local quench in this system amplifies quantum interferences, producing a macroscopic magnetisation profile that directly reflects the scattering phase of the model. Then, we use two asymmetry measures, namely the Entanglement Asymmetry (EA) and Quantum Fisher Information (QFI), to characterise the quantum coherence associated with the fluctuations of the magnetisation. By focusing on subsystems comparable in size to the light-cone of the perturbation, we confirm the emergence of macroscopic quantum coherence throughout the whole perturbed region. Finally, we discuss the link between EA and QFI and show that the variance/EA inequality for pure state can be generalised to a QFI/EA inequality for mixed states.

Breaking of clustering and macroscopic coherence under the lens of asymmetry measures

Abstract

In one-dimensional systems, spontaneous symmetry breaking (SSB) states are fragile by nature, as the injection of a non-zero energy density above the ground state is expected to restore the symmetry. This instability implies that local perturbations can lead to macroscopic correlation profiles, a breaking of clustering properties and even macroscopic quantum superpositions. In this work, we investigate the effect of interaction on this phenomenology by considering an interacting model with conserved domain wall number, that possesses a ferromagnetic ground state breaking the Z2 symmetry of the Hamiltonian. We first show that a local quench in this system amplifies quantum interferences, producing a macroscopic magnetisation profile that directly reflects the scattering phase of the model. Then, we use two asymmetry measures, namely the Entanglement Asymmetry (EA) and Quantum Fisher Information (QFI), to characterise the quantum coherence associated with the fluctuations of the magnetisation. By focusing on subsystems comparable in size to the light-cone of the perturbation, we confirm the emergence of macroscopic quantum coherence throughout the whole perturbed region. Finally, we discuss the link between EA and QFI and show that the variance/EA inequality for pure state can be generalised to a QFI/EA inequality for mixed states.
Paper Structure (29 sections, 131 equations, 5 figures)

This paper contains 29 sections, 131 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Setup
  3. CBA description
  4. Scaling limit of the transverse magnetisation
  5. Scaling limit of the Reduced Density matrix
  6. Exact RDM
  7. Scaling limit
  8. Pure and trivial sectors
  9. Diagonalisation of the $1-$ particle sector
  10. Entanglement asymmetry
  11. Quantum Fisher Information
  12. Link with variance:
  13. Sensitivity to pure coherence:
  14. Normalisation:
  15. Discussion: Link between EA and QFI
  16. ...and 14 more sections

Figures (5)

  • Figure 1: Magnetisation profile and connected correlations after a spin flip at site $l=0$ on the state $\ket{\Uparrow}$ for different values of $\Delta$. Points correspond to exact values of the magnetisation while the solid curve represents the asymptotic limit of Eq.\ref{['eq:magnetisation_limit']}. In the two rightmost plots the dashed line represent the maximal bound state velocity $v^B_{max}=\min(1, 1/(2\Delta))$.
  • Figure 2: Numerical diagonalisation of the scattering phase using its truncated Fourier representation $S_{nm}$ with $|n|, |m|\le 250$. For $\Delta = 0.8$, $\sum_\jmath \lambda_\jmath \approx 0.996611$ and $\sum_\jmath \lambda_\jmath^2 \approx 0.999991$, while for $\Delta = 1.2$$\sum_\jmath \lambda_\jmath \approx 1$ and $\sum_\jmath \lambda_\jmath^2 \approx 0.1$ up to at least $10^{-8}$ precision. Left and center: representation of the first modes of the scattering phase. The dashed lines represent the momenta $\pm \cos^{-1}\Delta$ at which the scattering phase becomes non-analytic. Right: eigenvalues of the truncated Fourier representation of the scattering phase.
  • Figure 3: Probability $p(\delta)$ to find two domain walls separated by a distance $\delta$. Left: time-independent contribution to the probability distribution (bound states). Right: scaling limit of the time-dependent part of the distribution (scattering states). The numerical data is compared with the (coarsed grained) underlying continuous distribution.
  • Figure 4: Left: Time evolution of asymmetry after a spin flip. Right: Comparison between the asymptotic prediction and the numerical values of $\delta \mathcal{S}(\rho_A, Z_A) = (\Delta S(\rho_A, Z_A)-\Delta S_0)/\log t$ for different subsystem sizes.
  • Figure 5: Left: time evolution of the rQFI for two subsystems of lengths $|A| = 31$ and $|A=49|$ centered around the initial spin flip, compared with the analytical bounds for the rQFI in the scaling limit, see Eq. \ref{['eq:qfi_pure']}. Right: higher-order corrections to the rQFI, see Eq. \ref{['eq:qfi_cut']}, \ref{['eq:moments']} and \ref{['eq:qfi_bounds_cut']}.