Towards Verifications of Krylov Complexity
Krylov complexity is considered to provide a measure of the growth of operators evolving under Hamiltonian dynamics. The main strategy is the analysis of the structure of Krylov subspace $\mathcal {K}_M(\mathcal {H},\eta )$ spanned by the multiple applications of the Liouville operator $\mathcal {L}...
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| Vydáno v: | Progress of theoretical and experimental physics Ročník 2024; číslo 6 |
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| Hlavní autor: | |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Oxford
Oxford University Press
01.06.2024
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| Témata: | |
| ISSN: | 2050-3911, 2050-3911 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | Krylov complexity is considered to provide a measure of the growth of operators evolving under Hamiltonian dynamics. The main strategy is the analysis of the structure of Krylov subspace $\mathcal {K}_M(\mathcal {H},\eta )$ spanned by the multiple applications of the Liouville operator $\mathcal {L}$ defined by the commutator in terms of a Hamiltonian $\mathcal {H}$, $\mathcal {L}:=[\mathcal {H},\cdot ]$ acting on an operator η, $\mathcal {K}_M(\mathcal {H},\eta )=\text{span}\lbrace \eta ,\mathcal {L}\eta ,\ldots ,\mathcal {L}^{M-1}\eta \rbrace$. For a given inner product (·, ·) of the operators, the orthonormal basis $\lbrace \mathcal {O}_n\rbrace$ is constructed from $\mathcal {O}_0=\eta /\sqrt{(\eta ,\eta )}$ by the Lanczos algorithm. The moments $\mu _m=(\mathcal {O}_0,\mathcal {L}^m\mathcal {O}_0)$ are closely related to the important data {bn}, called Lanczos coefficients. I present exact and explicit expressions of the moments {μm} for 16 quantum mechanical systems that are exactly solvable in both the Schrödinger and Heisenberg pictures. The operator η is the variable of the eigenpolynomials. Among them, six systems show a clear sign of “noncomplexity” with vanishing higher Lanczos coefficients bm = 0, m ≥ 3. |
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| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 2050-3911 2050-3911 |
| DOI: | 10.1093/ptep/ptae073 |