Operator K-complexity in DSSYK: Krylov complexity equals bulk length Article Swipe
Marco Ambrosini
,
Eliezer Rabinovici
,
A. Sánchez-Garrido
,
Ruth Shir
,
Julian Sonner
·
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.1007/jhep08(2025)059
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.1007/jhep08(2025)059
A bstract In this paper we study the notion of complexity under time evolution in chaotic quantum systems with holographic duals. Continuing on from our previous work, we turn our attention to the issue of Krylov complexity upon the insertion of a class of single-particle operators in the double-scaled SYK model. Such an operator is described by a matter-chord insertion, which splits the theory into left/right sectors, allowing us, via chord-diagram technology, to compute two different notions of complexity associated to the operator insertion: first a Krylov operator complexity, and second the Krylov complexity of a state obtained by an operator acting on the thermofield double state. We will provide both an analytic proof and detailed numerical evidence, that both Krylov complexities arise from a recursively defined basis of states characterized by a constant total chord number. As a consequence, in all cases we are able to establish that Krylov complexity is given by the expectation value of a length operator acting on the Hilbert space of the theory, expressed in terms of basis states, organized by left and right chord number. We find analytic expressions for the semiclassical limit of K-complexity, and study how the size of the operator encodes the scrambling dynamics upon the matter insertion in Krylov language. We furthermore determine the effective Hamiltonian governing the evolution of K-complexity, showing that evolution on the Krylov chain can equivalently be understood as a particle moving in a Morse potential. A particular type of triple scaling limit allows to access the gravitational sector of the theory, in which the geometrical nature of K-complexity is assured by virtue of being a total chord length, in an analogous fashion to what was found in [1] for the K-complexity of the thermofield double state.
Related Topics
Concepts
Physics
Operator (biology)
Mathematical physics
Theoretical physics
Chemistry
Biochemistry
Gene
Transcription factor
Repressor
Metadata
- Type
- article
- Language
- en
- Landing Page
- https://doi.org/10.1007/jhep08(2025)059
- https://link.springer.com/content/pdf/10.1007/JHEP08(2025)059.pdf
- OA Status
- diamond
- Cited By
- 5
- References
- 55
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4413134687
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
-
https://openalex.org/W4413134687Canonical identifier for this work in OpenAlex
- DOI
-
https://doi.org/10.1007/jhep08(2025)059Digital Object Identifier
- Title
-
Operator K-complexity in DSSYK: Krylov complexity equals bulk lengthWork title
- Type
-
articleOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2025Year of publication
- Publication date
-
2025-08-07Full publication date if available
- Authors
-
Marco Ambrosini, Eliezer Rabinovici, A. Sánchez-Garrido, Ruth Shir, Julian SonnerList of authors in order
- Landing page
-
https://doi.org/10.1007/jhep08(2025)059Publisher landing page
- PDF URL
-
https://link.springer.com/content/pdf/10.1007/JHEP08(2025)059.pdfDirect link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
-
diamondOpen access status per OpenAlex
- OA URL
-
https://link.springer.com/content/pdf/10.1007/JHEP08(2025)059.pdfDirect OA link when available
- Concepts
-
Physics, Operator (biology), Mathematical physics, Theoretical physics, Chemistry, Biochemistry, Gene, Transcription factor, RepressorTop concepts (fields/topics) attached by OpenAlex
- Cited by
-
5Total citation count in OpenAlex
- Citations by year (recent)
-
2025: 5Per-year citation counts (last 5 years)
- References (count)
-
55Number of works referenced by this work
- Related works (count)
-
10Other works algorithmically related by OpenAlex
Full payload
| id | https://openalex.org/W4413134687 |
|---|---|
| doi | https://doi.org/10.1007/jhep08(2025)059 |
| ids.doi | https://doi.org/10.1007/jhep08(2025)059 |
| ids.openalex | https://openalex.org/W4413134687 |
| fwci | 54.15818414 |
| type | article |
| title | Operator K-complexity in DSSYK: Krylov complexity equals bulk length |
| biblio.issue | 8 |
| biblio.volume | 2025 |
| biblio.last_page | |
| biblio.first_page | |
| topics[0].id | https://openalex.org/T10287 |
| topics[0].field.id | https://openalex.org/fields/26 |
| topics[0].field.display_name | Mathematics |
| topics[0].score | 0.9995999932289124 |
| topics[0].domain.id | https://openalex.org/domains/3 |
| topics[0].domain.display_name | Physical Sciences |
| topics[0].subfield.id | https://openalex.org/subfields/2608 |
| topics[0].subfield.display_name | Geometry and Topology |
| topics[0].display_name | Algebraic structures and combinatorial models |
| topics[1].id | https://openalex.org/T10849 |
| topics[1].field.id | https://openalex.org/fields/26 |
| topics[1].field.display_name | Mathematics |
| topics[1].score | 0.9986000061035156 |
| topics[1].domain.id | https://openalex.org/domains/3 |
| topics[1].domain.display_name | Physical Sciences |
| topics[1].subfield.id | https://openalex.org/subfields/2607 |
| topics[1].subfield.display_name | Discrete Mathematics and Combinatorics |
| topics[1].display_name | Finite Group Theory Research |
| topics[2].id | https://openalex.org/T10304 |
| topics[2].field.id | https://openalex.org/fields/26 |
| topics[2].field.display_name | Mathematics |
| topics[2].score | 0.996999979019165 |
| topics[2].domain.id | https://openalex.org/domains/3 |
| topics[2].domain.display_name | Physical Sciences |
| topics[2].subfield.id | https://openalex.org/subfields/2608 |
| topics[2].subfield.display_name | Geometry and Topology |
| topics[2].display_name | Geometric and Algebraic Topology |
| is_xpac | False |
| apc_list | |
| apc_paid | |
| concepts[0].id | https://openalex.org/C121332964 |
| concepts[0].level | 0 |
| concepts[0].score | 0.8619120121002197 |
| concepts[0].wikidata | https://www.wikidata.org/wiki/Q413 |
| concepts[0].display_name | Physics |
| concepts[1].id | https://openalex.org/C17020691 |
| concepts[1].level | 5 |
| concepts[1].score | 0.6378324031829834 |
| concepts[1].wikidata | https://www.wikidata.org/wiki/Q139677 |
| concepts[1].display_name | Operator (biology) |
| concepts[2].id | https://openalex.org/C37914503 |
| concepts[2].level | 1 |
| concepts[2].score | 0.3442436456680298 |
| concepts[2].wikidata | https://www.wikidata.org/wiki/Q156495 |
| concepts[2].display_name | Mathematical physics |
| concepts[3].id | https://openalex.org/C33332235 |
| concepts[3].level | 1 |
| concepts[3].score | 0.32999134063720703 |
| concepts[3].wikidata | https://www.wikidata.org/wiki/Q18362 |
| concepts[3].display_name | Theoretical physics |
| concepts[4].id | https://openalex.org/C185592680 |
| concepts[4].level | 0 |
| concepts[4].score | 0.0 |
| concepts[4].wikidata | https://www.wikidata.org/wiki/Q2329 |
| concepts[4].display_name | Chemistry |
| concepts[5].id | https://openalex.org/C55493867 |
| concepts[5].level | 1 |
| concepts[5].score | 0.0 |
| concepts[5].wikidata | https://www.wikidata.org/wiki/Q7094 |
| concepts[5].display_name | Biochemistry |
| concepts[6].id | https://openalex.org/C104317684 |
| concepts[6].level | 2 |
| concepts[6].score | 0.0 |
| concepts[6].wikidata | https://www.wikidata.org/wiki/Q7187 |
| concepts[6].display_name | Gene |
| concepts[7].id | https://openalex.org/C86339819 |
| concepts[7].level | 3 |
| concepts[7].score | 0.0 |
| concepts[7].wikidata | https://www.wikidata.org/wiki/Q407384 |
| concepts[7].display_name | Transcription factor |
| concepts[8].id | https://openalex.org/C158448853 |
| concepts[8].level | 4 |
| concepts[8].score | 0.0 |
| concepts[8].wikidata | https://www.wikidata.org/wiki/Q425218 |
| concepts[8].display_name | Repressor |
| keywords[0].id | https://openalex.org/keywords/physics |
| keywords[0].score | 0.8619120121002197 |
| keywords[0].display_name | Physics |
| keywords[1].id | https://openalex.org/keywords/operator |
| keywords[1].score | 0.6378324031829834 |
| keywords[1].display_name | Operator (biology) |
| keywords[2].id | https://openalex.org/keywords/mathematical-physics |
| keywords[2].score | 0.3442436456680298 |
| keywords[2].display_name | Mathematical physics |
| keywords[3].id | https://openalex.org/keywords/theoretical-physics |
| keywords[3].score | 0.32999134063720703 |
| keywords[3].display_name | Theoretical physics |
| language | en |
| locations[0].id | doi:10.1007/jhep08(2025)059 |
| locations[0].is_oa | True |
| locations[0].source.id | https://openalex.org/S187585107 |
| locations[0].source.issn | 1029-8479, 1126-6708, 1127-2236 |
| locations[0].source.type | journal |
| locations[0].source.is_oa | True |
| locations[0].source.issn_l | 1029-8479 |
| locations[0].source.is_core | True |
| locations[0].source.is_in_doaj | True |
| locations[0].source.display_name | Journal of High Energy Physics |
| locations[0].source.host_organization | https://openalex.org/P4310319965 |
| locations[0].source.host_organization_name | Springer Nature |
| locations[0].source.host_organization_lineage | https://openalex.org/P4310319965 |
| locations[0].source.host_organization_lineage_names | Springer Nature |
| locations[0].license | cc-by |
| locations[0].pdf_url | https://link.springer.com/content/pdf/10.1007/JHEP08(2025)059.pdf |
| locations[0].version | publishedVersion |
| locations[0].raw_type | journal-article |
| locations[0].license_id | https://openalex.org/licenses/cc-by |
| locations[0].is_accepted | True |
| locations[0].is_published | True |
| locations[0].raw_source_name | Journal of High Energy Physics |
| locations[0].landing_page_url | https://doi.org/10.1007/jhep08(2025)059 |
| locations[1].id | pmh:oai:cds.cern.ch:2928032 |
| locations[1].is_oa | False |
| locations[1].source.id | https://openalex.org/S4306402194 |
| locations[1].source.issn | |
| locations[1].source.type | repository |
| locations[1].source.is_oa | False |
| locations[1].source.issn_l | |
| locations[1].source.is_core | False |
| locations[1].source.is_in_doaj | False |
| locations[1].source.display_name | CERN Document Server (European Organization for Nuclear Research) |
| locations[1].source.host_organization | https://openalex.org/I67311998 |
| locations[1].source.host_organization_name | European Organization for Nuclear Research |
| locations[1].source.host_organization_lineage | https://openalex.org/I67311998 |
| locations[1].license | |
| locations[1].pdf_url | |
| locations[1].version | submittedVersion |
| locations[1].raw_type | |
| locations[1].license_id | |
| locations[1].is_accepted | False |
| locations[1].is_published | False |
| locations[1].raw_source_name | |
| locations[1].landing_page_url | http://cds.cern.ch/record/2928032 |
| locations[2].id | pmh:oai:doaj.org/article:18169044f6064498a229c9fe25c217e2 |
| locations[2].is_oa | False |
| locations[2].source.id | https://openalex.org/S4306401280 |
| locations[2].source.issn | |
| locations[2].source.type | repository |
| locations[2].source.is_oa | False |
| locations[2].source.issn_l | |
| locations[2].source.is_core | False |
| locations[2].source.is_in_doaj | False |
| locations[2].source.display_name | DOAJ (DOAJ: Directory of Open Access Journals) |
| locations[2].source.host_organization | |
| locations[2].source.host_organization_name | |
| locations[2].license | |
| locations[2].pdf_url | |
| locations[2].version | submittedVersion |
| locations[2].raw_type | article |
| locations[2].license_id | |
| locations[2].is_accepted | False |
| locations[2].is_published | False |
| locations[2].raw_source_name | Journal of High Energy Physics, Vol 2025, Iss 8, Pp 1-92 (2025) |
| locations[2].landing_page_url | https://doaj.org/article/18169044f6064498a229c9fe25c217e2 |
| indexed_in | crossref, doaj |
| authorships[0].author.id | https://openalex.org/A5115615985 |
| authorships[0].author.orcid | |
| authorships[0].author.display_name | Marco Ambrosini |
| authorships[0].countries | CH |
| authorships[0].affiliations[0].institution_ids | https://openalex.org/I114457229 |
| authorships[0].affiliations[0].raw_affiliation_string | University of Geneva |
| authorships[0].institutions[0].id | https://openalex.org/I114457229 |
| authorships[0].institutions[0].ror | https://ror.org/01swzsf04 |
| authorships[0].institutions[0].type | education |
| authorships[0].institutions[0].lineage | https://openalex.org/I114457229 |
| authorships[0].institutions[0].country_code | CH |
| authorships[0].institutions[0].display_name | University of Geneva |
| authorships[0].author_position | first |
| authorships[0].raw_author_name | M. Ambrosini |
| authorships[0].is_corresponding | False |
| authorships[0].raw_affiliation_strings | University of Geneva |
| authorships[1].author.id | https://openalex.org/A5090476157 |
| authorships[1].author.orcid | |
| authorships[1].author.display_name | Eliezer Rabinovici |
| authorships[1].countries | CH, IL |
| authorships[1].affiliations[0].institution_ids | https://openalex.org/I67311998 |
| authorships[1].affiliations[0].raw_affiliation_string | CERN, Theoretical Physics Department, CH-1211, Geneva 23, Switzerland |
| authorships[1].affiliations[1].institution_ids | https://openalex.org/I197251160 |
| authorships[1].affiliations[1].raw_affiliation_string | Racah Institute of Physics, The Hebrew University, 9190401, Jerusalem, Israel |
| authorships[1].institutions[0].id | https://openalex.org/I67311998 |
| authorships[1].institutions[0].ror | https://ror.org/01ggx4157 |
| authorships[1].institutions[0].type | facility |
| authorships[1].institutions[0].lineage | https://openalex.org/I67311998 |
| authorships[1].institutions[0].country_code | CH |
| authorships[1].institutions[0].display_name | European Organization for Nuclear Research |
| authorships[1].institutions[1].id | https://openalex.org/I197251160 |
| authorships[1].institutions[1].ror | https://ror.org/03qxff017 |
| authorships[1].institutions[1].type | education |
| authorships[1].institutions[1].lineage | https://openalex.org/I197251160 |
| authorships[1].institutions[1].country_code | IL |
| authorships[1].institutions[1].display_name | Hebrew University of Jerusalem |
| authorships[1].author_position | middle |
| authorships[1].raw_author_name | E. Rabinovici |
| authorships[1].is_corresponding | False |
| authorships[1].raw_affiliation_strings | CERN, Theoretical Physics Department, CH-1211, Geneva 23, Switzerland, Racah Institute of Physics, The Hebrew University, 9190401, Jerusalem, Israel |
| authorships[2].author.id | https://openalex.org/A5007023872 |
| authorships[2].author.orcid | https://orcid.org/0000-0003-2313-5859 |
| authorships[2].author.display_name | A. Sánchez-Garrido |
| authorships[2].countries | GB |
| authorships[2].affiliations[0].institution_ids | https://openalex.org/I43439940 |
| authorships[2].affiliations[0].raw_affiliation_string | School of Mathematical Sciences, University of Southampton, Southampton, SO17 1BJ, UK |
| authorships[2].institutions[0].id | https://openalex.org/I43439940 |
| authorships[2].institutions[0].ror | https://ror.org/01ryk1543 |
| authorships[2].institutions[0].type | education |
| authorships[2].institutions[0].lineage | https://openalex.org/I43439940 |
| authorships[2].institutions[0].country_code | GB |
| authorships[2].institutions[0].display_name | University of Southampton |
| authorships[2].author_position | middle |
| authorships[2].raw_author_name | A. Sánchez-Garrido |
| authorships[2].is_corresponding | False |
| authorships[2].raw_affiliation_strings | School of Mathematical Sciences, University of Southampton, Southampton, SO17 1BJ, UK |
| authorships[3].author.id | https://openalex.org/A5088050489 |
| authorships[3].author.orcid | https://orcid.org/0000-0003-2248-6913 |
| authorships[3].author.display_name | Ruth Shir |
| authorships[3].countries | LU |
| authorships[3].affiliations[0].institution_ids | https://openalex.org/I186903577 |
| authorships[3].affiliations[0].raw_affiliation_string | Department of Physics and Materials Science, University of Luxembourg, L-1511, Luxembourg, Luxembourg |
| authorships[3].institutions[0].id | https://openalex.org/I186903577 |
| authorships[3].institutions[0].ror | https://ror.org/036x5ad56 |
| authorships[3].institutions[0].type | education |
| authorships[3].institutions[0].lineage | https://openalex.org/I186903577 |
| authorships[3].institutions[0].country_code | LU |
| authorships[3].institutions[0].display_name | University of Luxembourg |
| authorships[3].author_position | middle |
| authorships[3].raw_author_name | R. Shir |
| authorships[3].is_corresponding | True |
| authorships[3].raw_affiliation_strings | Department of Physics and Materials Science, University of Luxembourg, L-1511, Luxembourg, Luxembourg |
| authorships[4].author.id | https://openalex.org/A5082868104 |
| authorships[4].author.orcid | https://orcid.org/0000-0002-6179-1729 |
| authorships[4].author.display_name | Julian Sonner |
| authorships[4].countries | CH |
| authorships[4].affiliations[0].institution_ids | https://openalex.org/I114457229 |
| authorships[4].affiliations[0].raw_affiliation_string | University of Geneva |
| authorships[4].institutions[0].id | https://openalex.org/I114457229 |
| authorships[4].institutions[0].ror | https://ror.org/01swzsf04 |
| authorships[4].institutions[0].type | education |
| authorships[4].institutions[0].lineage | https://openalex.org/I114457229 |
| authorships[4].institutions[0].country_code | CH |
| authorships[4].institutions[0].display_name | University of Geneva |
| authorships[4].author_position | last |
| authorships[4].raw_author_name | J. Sonner |
| authorships[4].is_corresponding | False |
| authorships[4].raw_affiliation_strings | University of Geneva |
| has_content.pdf | True |
| has_content.grobid_xml | True |
| is_paratext | False |
| open_access.is_oa | True |
| open_access.oa_url | https://link.springer.com/content/pdf/10.1007/JHEP08(2025)059.pdf |
| open_access.oa_status | diamond |
| open_access.any_repository_has_fulltext | False |
| created_date | 2025-10-10T00:00:00 |
| display_name | Operator K-complexity in DSSYK: Krylov complexity equals bulk length |
| has_fulltext | False |
| is_retracted | False |
| updated_date | 2025-11-06T03:46:38.306776 |
| primary_topic.id | https://openalex.org/T10287 |
| primary_topic.field.id | https://openalex.org/fields/26 |
| primary_topic.field.display_name | Mathematics |
| primary_topic.score | 0.9995999932289124 |
| primary_topic.domain.id | https://openalex.org/domains/3 |
| primary_topic.domain.display_name | Physical Sciences |
| primary_topic.subfield.id | https://openalex.org/subfields/2608 |
| primary_topic.subfield.display_name | Geometry and Topology |
| primary_topic.display_name | Algebraic structures and combinatorial models |
| related_works | https://openalex.org/W4391375266, https://openalex.org/W2935759653, https://openalex.org/W3105167352, https://openalex.org/W54078636, https://openalex.org/W2954470139, https://openalex.org/W1501425562, https://openalex.org/W2902782467, https://openalex.org/W3084825885, https://openalex.org/W2298861036, https://openalex.org/W2271181815 |
| cited_by_count | 5 |
| counts_by_year[0].year | 2025 |
| counts_by_year[0].cited_by_count | 5 |
| locations_count | 3 |
| best_oa_location.id | doi:10.1007/jhep08(2025)059 |
| best_oa_location.is_oa | True |
| best_oa_location.source.id | https://openalex.org/S187585107 |
| best_oa_location.source.issn | 1029-8479, 1126-6708, 1127-2236 |
| best_oa_location.source.type | journal |
| best_oa_location.source.is_oa | True |
| best_oa_location.source.issn_l | 1029-8479 |
| best_oa_location.source.is_core | True |
| best_oa_location.source.is_in_doaj | True |
| best_oa_location.source.display_name | Journal of High Energy Physics |
| best_oa_location.source.host_organization | https://openalex.org/P4310319965 |
| best_oa_location.source.host_organization_name | Springer Nature |
| best_oa_location.source.host_organization_lineage | https://openalex.org/P4310319965 |
| best_oa_location.source.host_organization_lineage_names | Springer Nature |
| best_oa_location.license | cc-by |
| best_oa_location.pdf_url | https://link.springer.com/content/pdf/10.1007/JHEP08(2025)059.pdf |
| best_oa_location.version | publishedVersion |
| best_oa_location.raw_type | journal-article |
| best_oa_location.license_id | https://openalex.org/licenses/cc-by |
| best_oa_location.is_accepted | True |
| best_oa_location.is_published | True |
| best_oa_location.raw_source_name | Journal of High Energy Physics |
| best_oa_location.landing_page_url | https://doi.org/10.1007/jhep08(2025)059 |
| primary_location.id | doi:10.1007/jhep08(2025)059 |
| primary_location.is_oa | True |
| primary_location.source.id | https://openalex.org/S187585107 |
| primary_location.source.issn | 1029-8479, 1126-6708, 1127-2236 |
| primary_location.source.type | journal |
| primary_location.source.is_oa | True |
| primary_location.source.issn_l | 1029-8479 |
| primary_location.source.is_core | True |
| primary_location.source.is_in_doaj | True |
| primary_location.source.display_name | Journal of High Energy Physics |
| primary_location.source.host_organization | https://openalex.org/P4310319965 |
| primary_location.source.host_organization_name | Springer Nature |
| primary_location.source.host_organization_lineage | https://openalex.org/P4310319965 |
| primary_location.source.host_organization_lineage_names | Springer Nature |
| primary_location.license | cc-by |
| primary_location.pdf_url | https://link.springer.com/content/pdf/10.1007/JHEP08(2025)059.pdf |
| primary_location.version | publishedVersion |
| primary_location.raw_type | journal-article |
| primary_location.license_id | https://openalex.org/licenses/cc-by |
| primary_location.is_accepted | True |
| primary_location.is_published | True |
| primary_location.raw_source_name | Journal of High Energy Physics |
| primary_location.landing_page_url | https://doi.org/10.1007/jhep08(2025)059 |
| publication_date | 2025-08-07 |
| publication_year | 2025 |
| referenced_works | https://openalex.org/W4386314537, https://openalex.org/W2117187503, https://openalex.org/W2014489580, https://openalex.org/W2203723895, https://openalex.org/W2904845127, https://openalex.org/W2960588259, https://openalex.org/W3081039832, https://openalex.org/W3083018415, https://openalex.org/W4288060965, https://openalex.org/W4224215230, https://openalex.org/W2997918644, https://openalex.org/W3203968734, https://openalex.org/W4402500802, https://openalex.org/W4386292029, https://openalex.org/W4410382270, https://openalex.org/W4213161738, https://openalex.org/W2900051186, https://openalex.org/W2808052156, https://openalex.org/W2581653392, https://openalex.org/W4309082140, https://openalex.org/W4385894947, https://openalex.org/W4406247669, https://openalex.org/W4389478385, https://openalex.org/W3001297680, https://openalex.org/W3007873191, https://openalex.org/W2047749712, https://openalex.org/W2117686912, https://openalex.org/W3037042650, https://openalex.org/W2786750559, https://openalex.org/W4206175892, https://openalex.org/W2035864055, https://openalex.org/W2059586807, https://openalex.org/W3037850448, https://openalex.org/W4386955777, https://openalex.org/W3209620443, https://openalex.org/W2090518157, https://openalex.org/W4206036716, https://openalex.org/W2089191553, https://openalex.org/W4385429136, https://openalex.org/W2405851928, https://openalex.org/W3178119464, https://openalex.org/W4403923088, https://openalex.org/W4380681741, https://openalex.org/W4411716704, https://openalex.org/W2344612405, https://openalex.org/W3145066249, https://openalex.org/W2045978557, https://openalex.org/W3102564242, https://openalex.org/W3134594783, https://openalex.org/W4406690211, https://openalex.org/W4221131106, https://openalex.org/W3102226719, https://openalex.org/W3124875895, https://openalex.org/W3101529462, https://openalex.org/W3099526883 |
| referenced_works_count | 55 |
| abstract_inverted_index.A | 0, 241 |
| abstract_inverted_index.a | 41, 57, 85, 95, 124, 132, 138, 158, 234, 238, 270 |
| abstract_inverted_index.As | 137 |
| abstract_inverted_index.In | 2 |
| abstract_inverted_index.We | 107, 182, 211 |
| abstract_inverted_index.an | 52, 99, 111, 275 |
| abstract_inverted_index.as | 233 |
| abstract_inverted_index.be | 231 |
| abstract_inverted_index.by | 56, 98, 131, 153, 176, 266 |
| abstract_inverted_index.in | 14, 46, 140, 170, 208, 237, 257, 274, 282 |
| abstract_inverted_index.is | 54, 151, 264 |
| abstract_inverted_index.of | 9, 34, 40, 43, 77, 94, 128, 157, 166, 172, 190, 197, 220, 244, 254, 262, 268, 287 |
| abstract_inverted_index.on | 22, 102, 162, 225 |
| abstract_inverted_index.to | 31, 72, 80, 146, 249, 278 |
| abstract_inverted_index.we | 5, 27, 143 |
| abstract_inverted_index.SYK | 49 |
| abstract_inverted_index.[1] | 283 |
| abstract_inverted_index.all | 141 |
| abstract_inverted_index.and | 89, 114, 178, 192 |
| abstract_inverted_index.are | 144 |
| abstract_inverted_index.can | 229 |
| abstract_inverted_index.for | 186, 284 |
| abstract_inverted_index.how | 194 |
| abstract_inverted_index.our | 24, 29 |
| abstract_inverted_index.the | 7, 32, 38, 47, 62, 81, 91, 103, 154, 163, 167, 187, 195, 198, 201, 205, 214, 218, 226, 251, 255, 259, 285, 288 |
| abstract_inverted_index.two | 74 |
| abstract_inverted_index.us, | 68 |
| abstract_inverted_index.via | 69 |
| abstract_inverted_index.was | 280 |
| abstract_inverted_index.Such | 51 |
| abstract_inverted_index.able | 145 |
| abstract_inverted_index.both | 110, 119 |
| abstract_inverted_index.find | 183 |
| abstract_inverted_index.from | 23, 123 |
| abstract_inverted_index.into | 64 |
| abstract_inverted_index.left | 177 |
| abstract_inverted_index.size | 196 |
| abstract_inverted_index.that | 118, 148, 223 |
| abstract_inverted_index.this | 3 |
| abstract_inverted_index.time | 12 |
| abstract_inverted_index.turn | 28 |
| abstract_inverted_index.type | 243 |
| abstract_inverted_index.upon | 37, 204 |
| abstract_inverted_index.what | 279 |
| abstract_inverted_index.will | 108 |
| abstract_inverted_index.with | 18 |
| abstract_inverted_index.Morse | 239 |
| abstract_inverted_index.arise | 122 |
| abstract_inverted_index.basis | 127, 173 |
| abstract_inverted_index.being | 269 |
| abstract_inverted_index.cases | 142 |
| abstract_inverted_index.chain | 228 |
| abstract_inverted_index.chord | 135, 180, 272 |
| abstract_inverted_index.class | 42 |
| abstract_inverted_index.first | 84 |
| abstract_inverted_index.found | 281 |
| abstract_inverted_index.given | 152 |
| abstract_inverted_index.issue | 33 |
| abstract_inverted_index.limit | 189, 247 |
| abstract_inverted_index.paper | 4 |
| abstract_inverted_index.proof | 113 |
| abstract_inverted_index.right | 179 |
| abstract_inverted_index.space | 165 |
| abstract_inverted_index.state | 96 |
| abstract_inverted_index.study | 6, 193 |
| abstract_inverted_index.terms | 171 |
| abstract_inverted_index.total | 134, 271 |
| abstract_inverted_index.under | 11 |
| abstract_inverted_index.value | 156 |
| abstract_inverted_index.which | 60, 258 |
| abstract_inverted_index.work, | 26 |
| abstract_inverted_index.Krylov | 35, 86, 92, 120, 149, 209, 227 |
| abstract_inverted_index.access | 250 |
| abstract_inverted_index.acting | 101, 161 |
| abstract_inverted_index.allows | 248 |
| abstract_inverted_index.double | 105, 290 |
| abstract_inverted_index.duals. | 20 |
| abstract_inverted_index.length | 159 |
| abstract_inverted_index.matter | 206 |
| abstract_inverted_index.model. | 50 |
| abstract_inverted_index.moving | 236 |
| abstract_inverted_index.nature | 261 |
| abstract_inverted_index.notion | 8 |
| abstract_inverted_index.second | 90 |
| abstract_inverted_index.sector | 253 |
| abstract_inverted_index.splits | 61 |
| abstract_inverted_index.state. | 106, 291 |
| abstract_inverted_index.states | 129 |
| abstract_inverted_index.theory | 63 |
| abstract_inverted_index.triple | 245 |
| abstract_inverted_index.virtue | 267 |
| abstract_inverted_index.Hilbert | 164 |
| abstract_inverted_index.assured | 265 |
| abstract_inverted_index.bstract | 1 |
| abstract_inverted_index.chaotic | 15 |
| abstract_inverted_index.compute | 73 |
| abstract_inverted_index.defined | 126 |
| abstract_inverted_index.encodes | 200 |
| abstract_inverted_index.fashion | 277 |
| abstract_inverted_index.length, | 273 |
| abstract_inverted_index.notions | 76 |
| abstract_inverted_index.number. | 136, 181 |
| abstract_inverted_index.provide | 109 |
| abstract_inverted_index.quantum | 16 |
| abstract_inverted_index.scaling | 246 |
| abstract_inverted_index.showing | 222 |
| abstract_inverted_index.states, | 174 |
| abstract_inverted_index.systems | 17 |
| abstract_inverted_index.theory, | 168, 256 |
| abstract_inverted_index.allowing | 67 |
| abstract_inverted_index.analytic | 112, 184 |
| abstract_inverted_index.constant | 133 |
| abstract_inverted_index.detailed | 115 |
| abstract_inverted_index.dynamics | 203 |
| abstract_inverted_index.obtained | 97 |
| abstract_inverted_index.operator | 53, 82, 87, 100, 160, 199 |
| abstract_inverted_index.particle | 235 |
| abstract_inverted_index.previous | 25 |
| abstract_inverted_index.sectors, | 66 |
| abstract_inverted_index.analogous | 276 |
| abstract_inverted_index.attention | 30 |
| abstract_inverted_index.described | 55 |
| abstract_inverted_index.determine | 213 |
| abstract_inverted_index.different | 75 |
| abstract_inverted_index.effective | 215 |
| abstract_inverted_index.establish | 147 |
| abstract_inverted_index.evidence, | 117 |
| abstract_inverted_index.evolution | 13, 219, 224 |
| abstract_inverted_index.expressed | 169 |
| abstract_inverted_index.governing | 217 |
| abstract_inverted_index.insertion | 39, 207 |
| abstract_inverted_index.language. | 210 |
| abstract_inverted_index.numerical | 116 |
| abstract_inverted_index.operators | 45 |
| abstract_inverted_index.organized | 175 |
| abstract_inverted_index.Continuing | 21 |
| abstract_inverted_index.associated | 79 |
| abstract_inverted_index.complexity | 10, 36, 78, 93, 150 |
| abstract_inverted_index.insertion, | 59 |
| abstract_inverted_index.insertion: | 83 |
| abstract_inverted_index.left/right | 65 |
| abstract_inverted_index.particular | 242 |
| abstract_inverted_index.potential. | 240 |
| abstract_inverted_index.scrambling | 202 |
| abstract_inverted_index.understood | 232 |
| abstract_inverted_index.Hamiltonian | 216 |
| abstract_inverted_index.complexity, | 88 |
| abstract_inverted_index.expectation | 155 |
| abstract_inverted_index.expressions | 185 |
| abstract_inverted_index.furthermore | 212 |
| abstract_inverted_index.geometrical | 260 |
| abstract_inverted_index.holographic | 19 |
| abstract_inverted_index.recursively | 125 |
| abstract_inverted_index.technology, | 71 |
| abstract_inverted_index.thermofield | 104, 289 |
| abstract_inverted_index.K-complexity | 263, 286 |
| abstract_inverted_index.complexities | 121 |
| abstract_inverted_index.consequence, | 139 |
| abstract_inverted_index.equivalently | 230 |
| abstract_inverted_index.matter-chord | 58 |
| abstract_inverted_index.K-complexity, | 191, 221 |
| abstract_inverted_index.characterized | 130 |
| abstract_inverted_index.chord-diagram | 70 |
| abstract_inverted_index.double-scaled | 48 |
| abstract_inverted_index.gravitational | 252 |
| abstract_inverted_index.semiclassical | 188 |
| abstract_inverted_index.single-particle | 44 |
| cited_by_percentile_year.max | 98 |
| cited_by_percentile_year.min | 97 |
| corresponding_author_ids | https://openalex.org/A5088050489 |
| countries_distinct_count | 4 |
| institutions_distinct_count | 5 |
| corresponding_institution_ids | https://openalex.org/I186903577 |
| citation_normalized_percentile.value | 0.99768588 |
| citation_normalized_percentile.is_in_top_1_percent | True |
| citation_normalized_percentile.is_in_top_10_percent | True |