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Robustness of quantum symmetries against perturbations

Paolo Facchi, Marilena Ligabò, Vito Viesti

TL;DR

This work develops an algebraic framework to understand how quantum symmetries behave under perturbations of the Hamiltonian. It proves that a symmetry $S$ is robust against a perturbation $V$ if and only if $S$ commutes with the perturbed spectral projections, i.e. $[S,P_n(0)]=0$ for all $n$ when $H$ has a pure-point spectrum, and extends this to arbitrary sets of perturbations via intersections of single-perturbation robust algebras, yielding a von Neumann-algebraic structure. Completely robust symmetries are shown to be exactly the elements of the bicommutant $\{H\}''$, i.e. bounded functions of $H$, with the infinite-dimensional case revealing a wandering range that can scale sublinearly with the perturbation strength and the emergence of quantum adiabatic invariants $S_\varepsilon=U(\varepsilon) S U(\varepsilon)^{\dagger}$. The results illuminate the long-time stability of conserved quantities under perturbations and have potential implications for quantum thermodynamics, quenches, and equilibrium-state stability.

Abstract

We investigate quantum symmetries in terms of their large-time stability with respect to perturbations of the Hamiltonian. We find a complete algebraic characterization of the set of symmetries robust against a single perturbation and we use such result to characterize their stability with respect to arbitrary sets of perturbations.

Robustness of quantum symmetries against perturbations

TL;DR

This work develops an algebraic framework to understand how quantum symmetries behave under perturbations of the Hamiltonian. It proves that a symmetry is robust against a perturbation if and only if commutes with the perturbed spectral projections, i.e. for all when has a pure-point spectrum, and extends this to arbitrary sets of perturbations via intersections of single-perturbation robust algebras, yielding a von Neumann-algebraic structure. Completely robust symmetries are shown to be exactly the elements of the bicommutant , i.e. bounded functions of , with the infinite-dimensional case revealing a wandering range that can scale sublinearly with the perturbation strength and the emergence of quantum adiabatic invariants . The results illuminate the long-time stability of conserved quantities under perturbations and have potential implications for quantum thermodynamics, quenches, and equilibrium-state stability.

Abstract

We investigate quantum symmetries in terms of their large-time stability with respect to perturbations of the Hamiltonian. We find a complete algebraic characterization of the set of symmetries robust against a single perturbation and we use such result to characterize their stability with respect to arbitrary sets of perturbations.

Paper Structure

This paper contains 13 sections, 11 theorems, 98 equations.

Table of Contents

  1. Notation and preliminaries
  2. $\mathcal{P}$-robustness
  3. Robustness against a single perturbation
  4. Eternal Block-Diagonal Approximation
  5. Fragile and robust components of a symmetry
  6. Proof of Theorems
  7. $\mathcal{P}$-robustness and complete robustness
  8. Symmetry-restricted perturbations
  9. Completely robust symmetries
  10. Wandering range and adiabatic invariants
  11. Wandering range of a robust symmetry
  12. Quantum adiabatic invariants
  13. Conclusions

Key Result

Theorem 1.1

Let $V$ be a symmetric and $H$-bounded operator with $H$-bound $a_V$. Then for all $\varepsilon \in \mathbb{R}$, with $\left|\varepsilon\right| a_V <1$, the operator $H+\varepsilon{V}$ is self-adjoint on $D(H)$.

Theorems & Definitions (30)

  • Definition 1.1
  • Remark 1.1
  • Definition 1.2
  • Theorem 1.1: Kato-Rellich
  • Definition 2.1
  • Theorem 3.1: Kato
  • Theorem 3.2
  • Remark 3.1
  • Theorem 3.3
  • Remark 3.2
  • ...and 20 more