Undecidability of the spectral gap View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2015-12

AUTHORS

Toby S. Cubitt, David Perez-Garcia, Michael M. Wolf

ABSTRACT

The spectral gap--the energy difference between the ground state and first excited state of a system--is central to quantum many-body physics. Many challenging open problems, such as the Haldane conjecture, the question of the existence of gapped topological spin liquid phases, and the Yang-Mills gap conjecture, concern spectral gaps. These and other problems are particular cases of the general spectral gap problem: given the Hamiltonian of a quantum many-body system, is it gapped or gapless? Here we prove that this is an undecidable problem. Specifically, we construct families of quantum spin systems on a two-dimensional lattice with translationally invariant, nearest-neighbour interactions, for which the spectral gap problem is undecidable. This result extends to undecidability of other low-energy properties, such as the existence of algebraically decaying ground-state correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phase-estimation algorithm followed by a universal Turing machine. The spectral gap depends on the outcome of the corresponding 'halting problem'. Our result implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless, and that there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics. More... »

PAGES

207

References to SciGraph publications

  • 2013-09. Stability of Frustration-Free Hamiltonians in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 2009-04. The Power of Quantum Systems on a Line in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1971-09. Undecidability and nonperiodicity for tilings of the plane in INVENTIONES MATHEMATICAE
  • 1988-09. Valence bond ground states in isotropic quantum antiferromagnets in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 2012-12. Fractionalized excitations in the spin-liquid state of a kagome-lattice antiferromagnet in NATURE
  • 1931-12. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I in MONATSHEFTE FÜR MATHEMATIK
  • 2010-03. Spin liquids in frustrated magnets in NATURE
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1038/nature16059

    DOI

    http://dx.doi.org/10.1038/nature16059

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1046900490

    PUBMED

    https://www.ncbi.nlm.nih.gov/pubmed/26659181


    Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
    Incoming Citations Browse incoming citations for this publication using opencitations.net

    JSON-LD is the canonical representation for SciGraph data.

    TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

    [
      {
        "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
        "about": [
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0202", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Atomic, Molecular, Nuclear, Particle and Plasma Physics", 
            "type": "DefinedTerm"
          }, 
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/02", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Physical Sciences", 
            "type": "DefinedTerm"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "University of Cambridge", 
              "id": "https://www.grid.ac/institutes/grid.5335.0", 
              "name": [
                "Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK", 
                "DAMTP, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Cubitt", 
            "givenName": "Toby S.", 
            "id": "sg:person.01220303047.01", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.01220303047.01"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "Autonomous University of Madrid", 
              "id": "https://www.grid.ac/institutes/grid.5515.4", 
              "name": [
                "Departamento de An\u00e1lisis Matem\u00e1tico and IMI, Facultad de CC Matem\u00e1ticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain", 
                "ICMAT, C/Nicol\u00e1s Cabrera, Campus de Cantoblanco, 28049 Madrid, Spain"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Perez-Garcia", 
            "givenName": "David", 
            "id": "sg:person.01246774316.49", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.01246774316.49"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "Technical University Munich", 
              "id": "https://www.grid.ac/institutes/grid.6936.a", 
              "name": [
                "Department of Mathematics, Technische Universit\u00e4t M\u00fcnchen, 85748 Garching, Germany"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Wolf", 
            "givenName": "Michael M.", 
            "id": "sg:person.01014223314.43", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.01014223314.43"
            ], 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "sg:pub.10.1007/bf01418780", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1009316877", 
              "https://doi.org/10.1007/bf01418780"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01418780", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1009316877", 
              "https://doi.org/10.1007/bf01418780"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1016/0025-5408(73)90167-0", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1012450990"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1016/0025-5408(73)90167-0", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1012450990"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00220-013-1762-6", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1013552523", 
              "https://doi.org/10.1007/s00220-013-1762-6"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1038/nature11659", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1014270440", 
              "https://doi.org/10.1038/nature11659"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01700692", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1016008840", 
              "https://doi.org/10.1007/bf01700692"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1002/j.1538-7305.1961.tb03975.x", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1017716034"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01218021", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1018417561", 
              "https://doi.org/10.1007/bf01218021"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01218021", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1018417561", 
              "https://doi.org/10.1007/bf01218021"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1103/physrevb.50.3037", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1029990619"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1103/physrevb.50.3037", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1029990619"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1103/physrevb.69.104431", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1032053995"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1103/physrevb.69.104431", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1032053995"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1063/1.3490195", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1032476255"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1112/plms/s2-42.1.230", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1032985688"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1038/nature08917", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1042830450", 
              "https://doi.org/10.1038/nature08917"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1038/nature08917", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1042830450", 
              "https://doi.org/10.1038/nature08917"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00220-008-0710-3", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1046386965", 
              "https://doi.org/10.1007/s00220-008-0710-3"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00220-008-0710-3", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1046386965", 
              "https://doi.org/10.1007/s00220-008-0710-3"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1103/physrevlett.50.1153", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1060788368"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1103/physrevlett.50.1153", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1060788368"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1103/revmodphys.36.856", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1060838380"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1103/revmodphys.36.856", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1060838380"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1126/science.1201080", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1062464379"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1137/s0097539704445226", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1062879550"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1137/s0097539796300921", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1062880100"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1364/on.11.2.000011", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1065242236"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1109/focs.2009.22", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1095180902"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "2015-12", 
        "datePublishedReg": "2015-12-01", 
        "description": "The spectral gap--the energy difference between the ground state and first excited state of a system--is central to quantum many-body physics. Many challenging open problems, such as the Haldane conjecture, the question of the existence of gapped topological spin liquid phases, and the Yang-Mills gap conjecture, concern spectral gaps. These and other problems are particular cases of the general spectral gap problem: given the Hamiltonian of a quantum many-body system, is it gapped or gapless? Here we prove that this is an undecidable problem. Specifically, we construct families of quantum spin systems on a two-dimensional lattice with translationally invariant, nearest-neighbour interactions, for which the spectral gap problem is undecidable. This result extends to undecidability of other low-energy properties, such as the existence of algebraically decaying ground-state correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phase-estimation algorithm followed by a universal Turing machine. The spectral gap depends on the outcome of the corresponding 'halting problem'. Our result implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless, and that there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics.", 
        "genre": "research_article", 
        "id": "sg:pub.10.1038/nature16059", 
        "inLanguage": [
          "en"
        ], 
        "isAccessibleForFree": true, 
        "isFundedItemOf": [
          {
            "id": "sg:grant.6540663", 
            "type": "MonetaryGrant"
          }, 
          {
            "id": "sg:grant.3940235", 
            "type": "MonetaryGrant"
          }
        ], 
        "isPartOf": [
          {
            "id": "sg:journal.1018957", 
            "issn": [
              "0090-0028", 
              "1476-4687"
            ], 
            "name": "Nature", 
            "type": "Periodical"
          }, 
          {
            "issueNumber": "7581", 
            "type": "PublicationIssue"
          }, 
          {
            "type": "PublicationVolume", 
            "volumeNumber": "528"
          }
        ], 
        "name": "Undecidability of the spectral gap", 
        "pagination": "207", 
        "productId": [
          {
            "name": "readcube_id", 
            "type": "PropertyValue", 
            "value": [
              "df6a67a1b84a2875db5d82ed1751adabcbe8bb17f9dfe9d40c27c21dd7967343"
            ]
          }, 
          {
            "name": "pubmed_id", 
            "type": "PropertyValue", 
            "value": [
              "26659181"
            ]
          }, 
          {
            "name": "nlm_unique_id", 
            "type": "PropertyValue", 
            "value": [
              "0410462"
            ]
          }, 
          {
            "name": "doi", 
            "type": "PropertyValue", 
            "value": [
              "10.1038/nature16059"
            ]
          }, 
          {
            "name": "dimensions_id", 
            "type": "PropertyValue", 
            "value": [
              "pub.1046900490"
            ]
          }
        ], 
        "sameAs": [
          "https://doi.org/10.1038/nature16059", 
          "https://app.dimensions.ai/details/publication/pub.1046900490"
        ], 
        "sdDataset": "articles", 
        "sdDatePublished": "2019-04-10T15:39", 
        "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
        "sdPublisher": {
          "name": "Springer Nature - SN SciGraph project", 
          "type": "Organization"
        }, 
        "sdSource": "s3://com-uberresearch-data-dimensions-target-20181106-alternative/cleanup/v134/2549eaecd7973599484d7c17b260dba0a4ecb94b/merge/v9/a6c9fde33151104705d4d7ff012ea9563521a3ce/jats-lookup/v90/0000000001_0000000264/records_8664_00000426.jsonl", 
        "type": "ScholarlyArticle", 
        "url": "https://www.nature.com/articles/nature16059"
      }
    ]
     

    Download the RDF metadata as:  json-ld nt turtle xml License info

    HOW TO GET THIS DATA PROGRAMMATICALLY:

    JSON-LD is a popular format for linked data which is fully compatible with JSON.

    curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1038/nature16059'

    N-Triples is a line-based linked data format ideal for batch operations.

    curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1038/nature16059'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1038/nature16059'

    RDF/XML is a standard XML format for linked data.

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1038/nature16059'


     

    This table displays all metadata directly associated to this object as RDF triples.

    162 TRIPLES      21 PREDICATES      49 URIs      21 LITERALS      9 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1038/nature16059 schema:about anzsrc-for:02
    2 anzsrc-for:0202
    3 schema:author Nffb88204c39e4e7eb03c7623c92fb5a2
    4 schema:citation sg:pub.10.1007/bf01218021
    5 sg:pub.10.1007/bf01418780
    6 sg:pub.10.1007/bf01700692
    7 sg:pub.10.1007/s00220-008-0710-3
    8 sg:pub.10.1007/s00220-013-1762-6
    9 sg:pub.10.1038/nature08917
    10 sg:pub.10.1038/nature11659
    11 https://doi.org/10.1002/j.1538-7305.1961.tb03975.x
    12 https://doi.org/10.1016/0025-5408(73)90167-0
    13 https://doi.org/10.1063/1.3490195
    14 https://doi.org/10.1103/physrevb.50.3037
    15 https://doi.org/10.1103/physrevb.69.104431
    16 https://doi.org/10.1103/physrevlett.50.1153
    17 https://doi.org/10.1103/revmodphys.36.856
    18 https://doi.org/10.1109/focs.2009.22
    19 https://doi.org/10.1112/plms/s2-42.1.230
    20 https://doi.org/10.1126/science.1201080
    21 https://doi.org/10.1137/s0097539704445226
    22 https://doi.org/10.1137/s0097539796300921
    23 https://doi.org/10.1364/on.11.2.000011
    24 schema:datePublished 2015-12
    25 schema:datePublishedReg 2015-12-01
    26 schema:description The spectral gap--the energy difference between the ground state and first excited state of a system--is central to quantum many-body physics. Many challenging open problems, such as the Haldane conjecture, the question of the existence of gapped topological spin liquid phases, and the Yang-Mills gap conjecture, concern spectral gaps. These and other problems are particular cases of the general spectral gap problem: given the Hamiltonian of a quantum many-body system, is it gapped or gapless? Here we prove that this is an undecidable problem. Specifically, we construct families of quantum spin systems on a two-dimensional lattice with translationally invariant, nearest-neighbour interactions, for which the spectral gap problem is undecidable. This result extends to undecidability of other low-energy properties, such as the existence of algebraically decaying ground-state correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phase-estimation algorithm followed by a universal Turing machine. The spectral gap depends on the outcome of the corresponding 'halting problem'. Our result implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless, and that there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics.
    27 schema:genre research_article
    28 schema:inLanguage en
    29 schema:isAccessibleForFree true
    30 schema:isPartOf N9c3665a1844244dcb699174a767da242
    31 Neb549f9e6b0346b790347d091b252a32
    32 sg:journal.1018957
    33 schema:name Undecidability of the spectral gap
    34 schema:pagination 207
    35 schema:productId N7314952efd7d4441b8f69993c001cc8c
    36 N7335f4ab9ebb4207b6fd339cdad885db
    37 N81e8c05009314608b08be77109df601a
    38 Nbc55e49213a54771aec1700546a614dc
    39 Ne0082bca869644ecbce693d077f48a3f
    40 schema:sameAs https://app.dimensions.ai/details/publication/pub.1046900490
    41 https://doi.org/10.1038/nature16059
    42 schema:sdDatePublished 2019-04-10T15:39
    43 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    44 schema:sdPublisher Nd4beab8377d244238a8e02b6cb885f95
    45 schema:url https://www.nature.com/articles/nature16059
    46 sgo:license sg:explorer/license/
    47 sgo:sdDataset articles
    48 rdf:type schema:ScholarlyArticle
    49 N7314952efd7d4441b8f69993c001cc8c schema:name nlm_unique_id
    50 schema:value 0410462
    51 rdf:type schema:PropertyValue
    52 N7335f4ab9ebb4207b6fd339cdad885db schema:name pubmed_id
    53 schema:value 26659181
    54 rdf:type schema:PropertyValue
    55 N81e8c05009314608b08be77109df601a schema:name doi
    56 schema:value 10.1038/nature16059
    57 rdf:type schema:PropertyValue
    58 N974c0ecf37bf426b9eac525361beae91 rdf:first sg:person.01246774316.49
    59 rdf:rest Nb29f7f5c46a14e43a331c277665e1b62
    60 N9c3665a1844244dcb699174a767da242 schema:volumeNumber 528
    61 rdf:type schema:PublicationVolume
    62 Nb29f7f5c46a14e43a331c277665e1b62 rdf:first sg:person.01014223314.43
    63 rdf:rest rdf:nil
    64 Nbc55e49213a54771aec1700546a614dc schema:name dimensions_id
    65 schema:value pub.1046900490
    66 rdf:type schema:PropertyValue
    67 Nd4beab8377d244238a8e02b6cb885f95 schema:name Springer Nature - SN SciGraph project
    68 rdf:type schema:Organization
    69 Ne0082bca869644ecbce693d077f48a3f schema:name readcube_id
    70 schema:value df6a67a1b84a2875db5d82ed1751adabcbe8bb17f9dfe9d40c27c21dd7967343
    71 rdf:type schema:PropertyValue
    72 Neb549f9e6b0346b790347d091b252a32 schema:issueNumber 7581
    73 rdf:type schema:PublicationIssue
    74 Nffb88204c39e4e7eb03c7623c92fb5a2 rdf:first sg:person.01220303047.01
    75 rdf:rest N974c0ecf37bf426b9eac525361beae91
    76 anzsrc-for:02 schema:inDefinedTermSet anzsrc-for:
    77 schema:name Physical Sciences
    78 rdf:type schema:DefinedTerm
    79 anzsrc-for:0202 schema:inDefinedTermSet anzsrc-for:
    80 schema:name Atomic, Molecular, Nuclear, Particle and Plasma Physics
    81 rdf:type schema:DefinedTerm
    82 sg:grant.3940235 http://pending.schema.org/fundedItem sg:pub.10.1038/nature16059
    83 rdf:type schema:MonetaryGrant
    84 sg:grant.6540663 http://pending.schema.org/fundedItem sg:pub.10.1038/nature16059
    85 rdf:type schema:MonetaryGrant
    86 sg:journal.1018957 schema:issn 0090-0028
    87 1476-4687
    88 schema:name Nature
    89 rdf:type schema:Periodical
    90 sg:person.01014223314.43 schema:affiliation https://www.grid.ac/institutes/grid.6936.a
    91 schema:familyName Wolf
    92 schema:givenName Michael M.
    93 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.01014223314.43
    94 rdf:type schema:Person
    95 sg:person.01220303047.01 schema:affiliation https://www.grid.ac/institutes/grid.5335.0
    96 schema:familyName Cubitt
    97 schema:givenName Toby S.
    98 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.01220303047.01
    99 rdf:type schema:Person
    100 sg:person.01246774316.49 schema:affiliation https://www.grid.ac/institutes/grid.5515.4
    101 schema:familyName Perez-Garcia
    102 schema:givenName David
    103 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.01246774316.49
    104 rdf:type schema:Person
    105 sg:pub.10.1007/bf01218021 schema:sameAs https://app.dimensions.ai/details/publication/pub.1018417561
    106 https://doi.org/10.1007/bf01218021
    107 rdf:type schema:CreativeWork
    108 sg:pub.10.1007/bf01418780 schema:sameAs https://app.dimensions.ai/details/publication/pub.1009316877
    109 https://doi.org/10.1007/bf01418780
    110 rdf:type schema:CreativeWork
    111 sg:pub.10.1007/bf01700692 schema:sameAs https://app.dimensions.ai/details/publication/pub.1016008840
    112 https://doi.org/10.1007/bf01700692
    113 rdf:type schema:CreativeWork
    114 sg:pub.10.1007/s00220-008-0710-3 schema:sameAs https://app.dimensions.ai/details/publication/pub.1046386965
    115 https://doi.org/10.1007/s00220-008-0710-3
    116 rdf:type schema:CreativeWork
    117 sg:pub.10.1007/s00220-013-1762-6 schema:sameAs https://app.dimensions.ai/details/publication/pub.1013552523
    118 https://doi.org/10.1007/s00220-013-1762-6
    119 rdf:type schema:CreativeWork
    120 sg:pub.10.1038/nature08917 schema:sameAs https://app.dimensions.ai/details/publication/pub.1042830450
    121 https://doi.org/10.1038/nature08917
    122 rdf:type schema:CreativeWork
    123 sg:pub.10.1038/nature11659 schema:sameAs https://app.dimensions.ai/details/publication/pub.1014270440
    124 https://doi.org/10.1038/nature11659
    125 rdf:type schema:CreativeWork
    126 https://doi.org/10.1002/j.1538-7305.1961.tb03975.x schema:sameAs https://app.dimensions.ai/details/publication/pub.1017716034
    127 rdf:type schema:CreativeWork
    128 https://doi.org/10.1016/0025-5408(73)90167-0 schema:sameAs https://app.dimensions.ai/details/publication/pub.1012450990
    129 rdf:type schema:CreativeWork
    130 https://doi.org/10.1063/1.3490195 schema:sameAs https://app.dimensions.ai/details/publication/pub.1032476255
    131 rdf:type schema:CreativeWork
    132 https://doi.org/10.1103/physrevb.50.3037 schema:sameAs https://app.dimensions.ai/details/publication/pub.1029990619
    133 rdf:type schema:CreativeWork
    134 https://doi.org/10.1103/physrevb.69.104431 schema:sameAs https://app.dimensions.ai/details/publication/pub.1032053995
    135 rdf:type schema:CreativeWork
    136 https://doi.org/10.1103/physrevlett.50.1153 schema:sameAs https://app.dimensions.ai/details/publication/pub.1060788368
    137 rdf:type schema:CreativeWork
    138 https://doi.org/10.1103/revmodphys.36.856 schema:sameAs https://app.dimensions.ai/details/publication/pub.1060838380
    139 rdf:type schema:CreativeWork
    140 https://doi.org/10.1109/focs.2009.22 schema:sameAs https://app.dimensions.ai/details/publication/pub.1095180902
    141 rdf:type schema:CreativeWork
    142 https://doi.org/10.1112/plms/s2-42.1.230 schema:sameAs https://app.dimensions.ai/details/publication/pub.1032985688
    143 rdf:type schema:CreativeWork
    144 https://doi.org/10.1126/science.1201080 schema:sameAs https://app.dimensions.ai/details/publication/pub.1062464379
    145 rdf:type schema:CreativeWork
    146 https://doi.org/10.1137/s0097539704445226 schema:sameAs https://app.dimensions.ai/details/publication/pub.1062879550
    147 rdf:type schema:CreativeWork
    148 https://doi.org/10.1137/s0097539796300921 schema:sameAs https://app.dimensions.ai/details/publication/pub.1062880100
    149 rdf:type schema:CreativeWork
    150 https://doi.org/10.1364/on.11.2.000011 schema:sameAs https://app.dimensions.ai/details/publication/pub.1065242236
    151 rdf:type schema:CreativeWork
    152 https://www.grid.ac/institutes/grid.5335.0 schema:alternateName University of Cambridge
    153 schema:name DAMTP, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
    154 Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK
    155 rdf:type schema:Organization
    156 https://www.grid.ac/institutes/grid.5515.4 schema:alternateName Autonomous University of Madrid
    157 schema:name Departamento de Análisis Matemático and IMI, Facultad de CC Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain
    158 ICMAT, C/Nicolás Cabrera, Campus de Cantoblanco, 28049 Madrid, Spain
    159 rdf:type schema:Organization
    160 https://www.grid.ac/institutes/grid.6936.a schema:alternateName Technical University Munich
    161 schema:name Department of Mathematics, Technische Universität München, 85748 Garching, Germany
    162 rdf:type schema:Organization
     




    Preview window. Press ESC to close (or click here)


    ...