{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:23:23Z","timestamp":1760239403067,"version":"build-2065373602"},"reference-count":51,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,11,12]],"date-time":"2020-11-12T00:00:00Z","timestamp":1605139200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003706","name":"Korea Research Institute of Standards and Science","doi-asserted-by":"publisher","award":["Grant No. CAP-15-08-KRISS"],"award-info":[{"award-number":["Grant No. CAP-15-08-KRISS"]}],"id":[{"id":"10.13039\/501100003706","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>For the identification of non-trivial quantum phase, we exploit a Bell-type correlation that is applied to the one-dimensional spin-1 XXZ chain. It is found that our generalization of bipartite Bell correlation can take a decomposed form of transverse spin correlation together with high-order terms. The formulation of the density-matrix renormalisation group is utilized to obtain the ground state of a given Hamiltonian with non-trivial phase. Subsequently Bell-type correlation is evaluated through the analysis of the matrix product state. Diverse classes of quantum phase transitions in the spin-1 model are identified precisely through the evaluation of the first and the second moments of the generalized Bell correlations. The role of high-order terms in the criticality has been identified and their physical implications for the quantum phase have been revealed.<\/jats:p>","DOI":"10.3390\/e22111282","type":"journal-article","created":{"date-parts":[[2020,11,12]],"date-time":"2020-11-12T10:00:32Z","timestamp":1605175232000},"page":"1282","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Bell-Type Correlation at Quantum Phase Transitions in Spin-1 Chain"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1184-9248","authenticated-orcid":false,"given":"Dongkeun","family":"Lee","sequence":"first","affiliation":[{"name":"Department of Physics, Sogang University, 35, Baekbeom-ro, Mapo-gu, Seoul 04107, Korea"},{"name":"Research Institute for Basic Science, Sogang University, 35, Baekbeom-ro, Mapo-gu, Seoul 04107, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4165-6172","authenticated-orcid":false,"given":"Wonmin","family":"Son","sequence":"additional","affiliation":[{"name":"Department of Physics, Sogang University, 35, Baekbeom-ro, Mapo-gu, Seoul 04107, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"074002","DOI":"10.1088\/1361-6633\/aabf61","article-title":"Genuine quantum correlations in quantum many-body systems: A review of recent progress","volume":"81","author":"Sanpera","year":"2018","journal-title":"Rep. Prog. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1103\/RevModPhys.80.517","article-title":"Entanglement in many-body systems","volume":"80","author":"Amico","year":"2008","journal-title":"Rev. Mod. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1038\/416608a","article-title":"Scaling of entanglement close to a quantum phase transition","volume":"416","author":"Osterloh","year":"2002","journal-title":"Nature"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"032110-14","DOI":"10.1103\/PhysRevA.66.032110","article-title":"Entanglement in a simple quantum phase transition","volume":"66","author":"Osborne","year":"2002","journal-title":"Phys. Rev. A"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1103\/RevModPhys.82.277","article-title":"Colloquium: Area laws for the entanglement entropy","volume":"82","author":"Eisert","year":"2010","journal-title":"Rev. Mod. Phys"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"227902","DOI":"10.1103\/PhysRevLett.90.227902","article-title":"Entanglement in Quantum Critical Phenomena","volume":"90","author":"Vidal","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"087201","DOI":"10.1103\/PhysRevLett.92.087201","article-title":"Diverging Entanglement Length in Gapped Quantum Spin Systems","volume":"92","author":"Verstraete","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"042306","DOI":"10.1103\/PhysRevA.71.042306","article-title":"Localizable entanglement","volume":"71","author":"Popp","year":"2005","journal-title":"Phys. Rev. A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"022302","DOI":"10.1103\/PhysRevA.79.022302","article-title":"Quantum instability and edge entanglement in the quasi-long-range order","volume":"79","author":"Son","year":"2009","journal-title":"Phys. Rev. A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1007\/BF00669912","article-title":"Sensitive observables of quantum mechanics","volume":"7","author":"Capasso","year":"1973","journal-title":"Int. J. Theor. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4277","DOI":"10.1103\/PhysRevA.40.4277","article-title":"Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model","volume":"40","author":"Werner","year":"1989","journal-title":"Phys. Rev. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"030404","DOI":"10.1103\/PhysRevLett.115.030404","article-title":"Entanglement and Nonlocality are Inequivalent for Any Number of Parties","volume":"115","author":"Augusiak","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"062101","DOI":"10.1103\/PhysRevA.82.062101","article-title":"Nonlocality and entanglement in the XYmodel","volume":"82","author":"Batle","year":"2010","journal-title":"Phys. Rev. A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"052128","DOI":"10.1103\/PhysRevA.85.052128","article-title":"Bell inequalities and entanglement at quantum phase transitions in the XXZmodel","volume":"85","author":"Justino","year":"2012","journal-title":"Phys. Rev. A"},{"key":"ref_15","first-page":"60004","article-title":"Nonviolation of Bell\u2019s inequality in translation invariant systems","volume":"100","author":"Saguia","year":"2013","journal-title":"Europhys. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"012311-4","DOI":"10.1103\/PhysRevA.77.012311","article-title":"Scaling properties of fidelity in the spin-1 anisotropic model","volume":"77","author":"Tzeng","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"125603","DOI":"10.1088\/0953-8984\/25\/12\/125603","article-title":"Quantum phases of dimerized and frustrated Heisenberg spin chains with s = 1\/2, 1 and 3\/2: An entanglement entropy and fidelity study","volume":"25","author":"Goli","year":"2013","journal-title":"J. Phys. Condens. Matter"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"184428","DOI":"10.1103\/PhysRevB.93.184428","article-title":"Quantum correlations and coherence in spin-1 Heisenberg chains","volume":"93","author":"Malvezzi","year":"2016","journal-title":"Phys. Rev. B"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"040404","DOI":"10.1103\/PhysRevLett.88.040404","article-title":"Bell Inequalities for Arbitrarily High-Dimensional Systems","volume":"88","author":"Collins","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"060406","DOI":"10.1103\/PhysRevLett.96.060406","article-title":"Generic Bell Inequalities for Multipartite Arbitrary Dimensional Systems","volume":"96","author":"Son","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"022116","DOI":"10.1103\/PhysRevA.98.022116","article-title":"Generalized nonlocality criteria under the correlation symmetry","volume":"98","author":"Bae","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"235106","DOI":"10.1103\/PhysRevB.87.235106","article-title":"Phase diagram of the anisotropic spin-2 XXZ model: Infinite-system density matrix renormalization group study","volume":"87","author":"Zaletel","year":"2013","journal-title":"Phys. Rev. B"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"6372","DOI":"10.1103\/PhysRevB.34.6372","article-title":"Phase diagrams and correlation exponents for quantum spin chains of arbitrary spin quantum number","volume":"34","author":"Schulz","year":"1986","journal-title":"Phys. Rev. B"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2896","DOI":"10.1103\/PhysRevB.46.2896","article-title":"Critical behavior of anisotropic spin-S Heisenberg chains","volume":"46","author":"Alcaraz","year":"1992","journal-title":"Phys. Rev. B"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"4038","DOI":"10.1103\/PhysRevLett.76.4038","article-title":"Phase Diagram of S=1 Bond-Alternating XXZ Chains","volume":"76","author":"Kitazawa","year":"1996","journal-title":"Phys. Rev. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1153","DOI":"10.1103\/PhysRevLett.50.1153","article-title":"Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the One-Dimensional Easy-Axis N\u00e9el State","volume":"50","author":"Haldane","year":"1983","journal-title":"Phys. Rev. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"464","DOI":"10.1016\/0375-9601(83)90631-X","article-title":"Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model","volume":"93","author":"Haldane","year":"1983","journal-title":"Phys. Lett. A"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1103\/RevModPhys.86.419","article-title":"Bell nonlocality","volume":"86","author":"Brunner","year":"2014","journal-title":"Rev. Mod. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1103\/PhysRevLett.59.799","article-title":"Rigorous results on valence-bond ground states in antiferromagnets","volume":"59","author":"Affleck","year":"1987","journal-title":"Phys. Rev. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"4709","DOI":"10.1103\/PhysRevB.40.4709","article-title":"Preroughening transitions in crystal surfaces and valence-bond phases in quantum spin chains","volume":"40","author":"Nijs","year":"1989","journal-title":"Phys. Rev. B"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1103\/PhysRevB.45.304","article-title":"Hidden Z2\u00d7Z2 symmetry breaking in Haldane-gap antiferromagnets","volume":"45","author":"Kennedy","year":"1992","journal-title":"Phys. Rev. B"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"155131","DOI":"10.1103\/PhysRevB.80.155131","article-title":"Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order","volume":"80","author":"Gu","year":"2009","journal-title":"Phys. Rev. B"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"035107","DOI":"10.1103\/PhysRevB.83.035107","article-title":"Classification of gapped symmetric phases in one-dimensional spin systems","volume":"83","author":"Chen","year":"2011","journal-title":"Phys. Rev. B"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"235128","DOI":"10.1103\/PhysRevB.84.235128","article-title":"Complete classification of one-dimensional gapped quantum phases in interacting spin systems","volume":"84","author":"Chen","year":"2011","journal-title":"Phys. Rev. B"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"075125","DOI":"10.1103\/PhysRevB.85.075125","article-title":"Symmetry protection of topological phases in one-dimensional quantum spin systems","volume":"85","author":"Pollmann","year":"2012","journal-title":"Phys. Rev. B"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"104401","DOI":"10.1103\/PhysRevB.67.104401","article-title":"Ground-state phase diagram of S=1XXZchains with uniaxial single-ion-type anisotropy","volume":"67","author":"Chen","year":"2003","journal-title":"Phys. Rev. B"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"880","DOI":"10.1103\/PhysRevLett.23.880","article-title":"Proposed Experiment to Test Local Hidden-Variable Theories","volume":"23","author":"Clauser","year":"1969","journal-title":"Phys. Rev. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/BF00417500","article-title":"Quantum generalizations of Bell\u2019s inequality","volume":"4","year":"1980","journal-title":"Lett. Math. Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"147902","DOI":"10.1103\/PhysRevLett.91.147902","article-title":"Efficient Classical Simulation of Slightly Entangled Quantum Computations","volume":"91","author":"Vidal","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_40","first-page":"401","article-title":"Matrix Product State Representations","volume":"7","author":"Verstraete","year":"2007","journal-title":"Quantum Inf. Comput."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.aop.2010.09.012","article-title":"The density-matrix renormalization group in the age of matrix product states","volume":"326","year":"2011","journal-title":"Ann. Phys."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/j.aop.2014.06.013","article-title":"A practical introduction to tensor networks: Matrix product states and projected entangled pair states","volume":"349","year":"2014","journal-title":"Ann. Phys."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Hauschild, J., and Pollmann, F. (2018). Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy). SciPost Phys. Lect. Notes, 5. Available online: https:\/\/github.com\/tenpy\/tenpy.","DOI":"10.21468\/SciPostPhysLectNotes.5"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"110403","DOI":"10.1103\/PhysRevLett.97.110403","article-title":"Quantum Phase Transitions in Matrix Product Systems","volume":"97","author":"Wolf","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_45","first-page":"074003-11","article-title":"Non-local Correlations in the Haldane Phase for an XXZ Spin-1 Chain: A Perspective from Infinite Matrix Product State Representation","volume":"81","author":"Li","year":"2012","journal-title":"J. Phys. Soc. Jpn"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"224202","DOI":"10.1103\/PhysRevB.78.224402","article-title":"Finite-size scaling of string order parameters characterizing the Haldane phase","volume":"78","author":"Ueda","year":"2008","journal-title":"Phys. Rev. B"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"2421","DOI":"10.1103\/PhysRevB.40.2421","article-title":"Spin correlation function of the S=1 antiferromagnetic Heisenberg chain by the large-cluster-decomposition Monte Carlo method","volume":"40","author":"Nomura","year":"1989","journal-title":"Phys. Rev. B"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"220402","DOI":"10.1103\/PhysRevB.84.220402","article-title":"Accurate determination of the Gaussian transition in spin-1 chains with single-ion anisotropy","volume":"84","author":"Hu","year":"2011","journal-title":"Phys. Rev. B"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1256","DOI":"10.1126\/science.1247715","article-title":"Detecting nonlocality in many-body quantum states","volume":"344","author":"Tura","year":"2014","journal-title":"Science"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1126\/science.aad8665","article-title":"Bell correlations in a Bose-Einstein condensate","volume":"352","author":"Schmied","year":"2016","journal-title":"Science"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"170604","DOI":"10.1103\/PhysRevLett.123.170604","article-title":"Bell Correlations at Ising Quantum Critical Points","volume":"123","author":"Piga","year":"2019","journal-title":"Phys. Rev. Lett."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/11\/1282\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:32:31Z","timestamp":1760178751000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/11\/1282"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,12]]},"references-count":51,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2020,11]]}},"alternative-id":["e22111282"],"URL":"https:\/\/doi.org\/10.3390\/e22111282","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2020,11,12]]}}}