{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:07:31Z","timestamp":1760148451870,"version":"build-2065373602"},"reference-count":196,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,5,9]],"date-time":"2023-05-09T00:00:00Z","timestamp":1683590400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Funda\u00e7\u00e3o para a Ci\u00eancia e Tecnologia (Portugal)","award":["UID\/CTM\/04540\/2019"],"award-info":[{"award-number":["UID\/CTM\/04540\/2019"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Herein, aspects of entanglement and fidelity and their use in condensed matter systems are briefly reviewed. Both static and time-dependent situations are considered. Different signatures of phases and phase transitions are discussed, including the dynamic aspects of the evolution across a critical point. Some emphasis is placed on the use of entanglement in phase transitions with no clear order parameters and no symmetry breaking.<\/jats:p>","DOI":"10.3390\/sym15051055","type":"journal-article","created":{"date-parts":[[2023,5,10]],"date-time":"2023-05-10T01:40:02Z","timestamp":1683682802000},"page":"1055","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Entanglement and Fidelity: Statics and Dynamics"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8276-6485","authenticated-orcid":false,"given":"Pedro D.","family":"Sacramento","sequence":"first","affiliation":[{"name":"CeFEMA and Departamento de F\u00edsica, Instituto Superior T\u00e9cnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisbon, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,9]]},"reference":[{"key":"ref_1","first-page":"19","article-title":"On the theory of phase transitions","volume":"7","author":"Landau","year":"1937","journal-title":"Nature"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1133","DOI":"10.1103\/PhysRevLett.17.1133","article-title":"Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models","volume":"17","author":"Mermin","year":"1966","journal-title":"Phys. Rev. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"L124","DOI":"10.1088\/0022-3719\/5\/11\/002","article-title":"Long range order and metastability in two dimensional solids and superfluids. (Application of dislocation theory)","volume":"5","author":"Kosterlitz","year":"1972","journal-title":"J. Phys. C"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1181","DOI":"10.1088\/0022-3719\/6\/7\/010","article-title":"Ordering, metastability and phase transitions in two-dimensional systems","volume":"6","author":"Kosterlitz","year":"1973","journal-title":"J. Phys. C"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"040502","DOI":"10.1103\/RevModPhys.89.040502","article-title":"Nobel Lecture: Topological quantum matter","volume":"89","author":"Haldane","year":"2017","journal-title":"Rev. Mod. Phys."},{"key":"ref_6","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":"93A","author":"Haldane","year":"1983","journal-title":"Phys. Lett."},{"key":"ref_7","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_8","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_9","unstructured":"Nielsen, M.A., and Chuang, I.L. (2000). Quantum Computation and Quantum Information, Cambridge University Press."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"010504","DOI":"10.1103\/PhysRevLett.101.010504","article-title":"Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States","volume":"101","author":"Li","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"245123","DOI":"10.1103\/PhysRevB.94.245123","article-title":"Reduced density matrix and order parameters of a topological insulator","volume":"94","author":"Yu","year":"2016","journal-title":"Phys. Rev. B"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"031123","DOI":"10.1103\/PhysRevE.74.031123","article-title":"Ground state overlap and quantum phase transitions","volume":"74","author":"Zanardi","year":"2006","journal-title":"Phys. Rev. E"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Sachdev, S. (1999). Quantum Phase Transitions, Cambridge University Press.","DOI":"10.1017\/CBO9780511622540"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"4371","DOI":"10.1142\/S0217979210056335","article-title":"Fidelity approach to quantum phase transitions","volume":"24","author":"Gu","year":"2010","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_15","first-page":"48","article-title":"Ground state entanglement in quantum spin chains","volume":"4","author":"Latorre","year":"2004","journal-title":"Quantum Inf. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"052322","DOI":"10.1103\/PhysRevA.71.052322","article-title":"Ground-state entanglement in the XXZ model","volume":"71","author":"Gu","year":"2005","journal-title":"Phys. Rev. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"042330","DOI":"10.1103\/PhysRevA.68.042330","article-title":"Entanglement, quantum phase transition, and scaling in the XXZ chain","volume":"68","author":"Gu","year":"2003","journal-title":"Phys. Rev. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"086402","DOI":"10.1103\/PhysRevLett.93.086402","article-title":"Entanglement and Quantum Phase Transition in the Extended Hubbard Model","volume":"93","author":"Gu","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"196406","DOI":"10.1103\/PhysRevLett.95.196406","article-title":"Entanglement Scaling in the One-Dimensional Hubbard Model at Criticality","volume":"95","author":"Larsson","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"062318","DOI":"10.1103\/PhysRevA.73.062318","article-title":"Concurrence in collective models","volume":"73","author":"Vidal","year":"2006","journal-title":"Phys. Rev. A"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Vidal, J., Dusuel, S., and Barthel, T. (2007). Entanglement entropy in collective models. J. Stat. Mech., P01015.","DOI":"10.1088\/1742-5468\/2007\/01\/P01015"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2245","DOI":"10.1103\/PhysRevLett.80.2245","article-title":"Entanglement of formation of an arbitrary state of two qubits","volume":"80","author":"Wootters","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1088\/1367-2630\/6\/1\/022","article-title":"Mean-field approximations and multipartite thermal correlations","volume":"6","author":"Vedral","year":"2004","journal-title":"New. J. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"025002","DOI":"10.1088\/1751-8113\/41\/2\/025002","article-title":"Universal role of correlation entropy in critical phenomena","volume":"41","author":"Gu","year":"2008","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"12143","DOI":"10.1088\/1751-8113\/40\/40\/009","article-title":"Effects of environmental parameters to total, quantum and classical correlations","volume":"40","author":"Chan","year":"2007","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"032314","DOI":"10.1103\/PhysRevA.65.032314","article-title":"Computable measure of entanglement","volume":"65","author":"Vidal","year":"2002","journal-title":"Phys. Rev. A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"4273","DOI":"10.1063\/1.1497700","article-title":"Global entanglement in multiparticle systems","volume":"43","author":"Meyer","year":"2002","journal-title":"J. Math. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"010305","DOI":"10.1103\/PhysRevA.73.010305","article-title":"Genuine multipartite entanglement in quantum phase transitions","volume":"73","author":"Rigolin","year":"2006","journal-title":"Phys. Rev. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"170401","DOI":"10.1103\/PhysRevLett.97.170401","article-title":"Multipartite Entanglement Signature of Quantum Phase Transitions","volume":"97","author":"Rigolin","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"030503","DOI":"10.1103\/PhysRevLett.95.030503","article-title":"Natural multiparticle entanglement in a Fermi gas","volume":"95","author":"Lunkes","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_31","unstructured":"Heaney, L., Anders, J., and Vedral, V. (2006). Spatial entanglement of a free Bosonic field. arXiv."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"144523","DOI":"10.1103\/PhysRevB.71.144523","article-title":"Entanglement of electron spins in superconductors","volume":"71","author":"Oh","year":"2005","journal-title":"Phys. Rev. B"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Zanardi, P., Cozzini, M., and Giorda, P. (2007). Ground state fidelity and quantum phase transitions in free Fermi systems. J. Stat. Mech., L02002.","DOI":"10.1088\/1742-5468\/2007\/02\/L02002"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"014439","DOI":"10.1103\/PhysRevB.75.014439","article-title":"Quantum phase transitions and quantum fidelity in free fermion graphs","volume":"75","author":"Cozzini","year":"2007","journal-title":"Phys. Rev. B"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"110601","DOI":"10.1103\/PhysRevLett.98.110601","article-title":"Ground-State Fidelity and Bipartite Entanglement in the Bose-Hubbard Model","volume":"98","author":"Buonsante","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"115119","DOI":"10.1103\/PhysRevB.75.115119","article-title":"Ground-state properties of the attractive one-dimensional Bose-Hubbard model","volume":"75","author":"Oelkers","year":"2007","journal-title":"Phys. Rev. B"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"061108","DOI":"10.1103\/PhysRevE.76.061108","article-title":"Fidelity and quantum phase transition for the Heisenberg chain with next-nearest-neighbor interaction","volume":"76","author":"Chen","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"180403","DOI":"10.1103\/PhysRevB.76.180403","article-title":"Ground-state fidelity in one-dimensional gapless models","volume":"76","author":"Yang","year":"2007","journal-title":"Phys. Rev. B"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"412001","DOI":"10.1088\/1751-8113\/41\/41\/412001","article-title":"Fidelity and quantum phase transitions","volume":"41","author":"Zhou","year":"2008","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"492002","DOI":"10.1088\/1751-8113\/41\/49\/492002","article-title":"Fidelity approach to quantum phase transitions: Finite size scaling for quantum Ising model in a transverse field","volume":"41","author":"Zhou","year":"2008","journal-title":"J. Phys. Math. Gen."},{"key":"ref_41","unstructured":"Zhou, H.Q. (2007). Renormalization group flows and quantum phase transitions: Fidelity versus entanglement. arXiv."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"100603","DOI":"10.1103\/PhysRevLett.99.100603","article-title":"Information-Theoretic Differential Geometry of Quantum Phase Transitions","volume":"99","author":"Zanardi","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"095701","DOI":"10.1103\/PhysRevLett.99.095701","article-title":"Quantum Critical Scaling of the Geometric Tensors","volume":"99","author":"Venuti","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"022101","DOI":"10.1103\/PhysRevE.76.022101","article-title":"Fidelity, dynamic structure factor, and susceptibility in critical phenomena","volume":"76","author":"You","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"032111","DOI":"10.1103\/PhysRevA.77.032111","article-title":"Intrinsic relation between ground-state fidelity and the characterization of a quantum phase transition","volume":"77","author":"Chen","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"245109","DOI":"10.1103\/PhysRevB.77.245109","article-title":"Fidelity susceptibility, scaling, and universality in quantum critical phenomena","volume":"77","author":"Gu","year":"2008","journal-title":"Phys. Rev. B"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"115410","DOI":"10.1103\/PhysRevB.78.115410","article-title":"Fidelity approach to the Hubbard model","volume":"78","author":"Cozzini","year":"2008","journal-title":"Phys. Rev. B"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"155111","DOI":"10.1103\/PhysRevB.77.155111","article-title":"Entanglement, fidelity, and topological entropy in a quantum phase transition to topological order","volume":"77","author":"Hamma","year":"2008","journal-title":"Phys. Rev. B"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"010301","DOI":"10.1103\/PhysRevA.78.010301","article-title":"Fidelity analysis of topological quantum phase transitions","volume":"78","author":"Abasto","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"012304","DOI":"10.1103\/PhysRevA.78.012304","article-title":"Fidelity susceptibility and long-range correlation in the Kitaev honeycomb model","volume":"78","author":"Yang","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"012321","DOI":"10.1103\/PhysRevA.79.012321","article-title":"Thermal states of the Kitaev honeycomb model: Bures metric analysis","volume":"79","author":"Abasto","year":"2009","journal-title":"Phys. Rev. A"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"070602","DOI":"10.1103\/PhysRevLett.98.070602","article-title":"Breakdown of a Topological Phase: Quantum Phase Transition in a Loop Gas Model with Tension","volume":"98","author":"Trebst","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"014403","DOI":"10.1103\/PhysRevB.80.014403","article-title":"Singularities in ground-state fidelity and quantum phase transitions for the Kitaev model","volume":"80","author":"Zhao","year":"2009","journal-title":"Phys. Rev. B"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"062350","DOI":"10.1103\/PhysRevA.81.062350","article-title":"Reduced fidelity in the Kitaev honeycomb model","volume":"81","author":"Wang","year":"2010","journal-title":"Phys. Rev. A"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"054433","DOI":"10.1103\/PhysRevB.77.054433","article-title":"Quantum topological phase transition at the microscopic level","volume":"77","author":"Castelnovo","year":"2008","journal-title":"Phys. Rev. B"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"060301","DOI":"10.1103\/PhysRevA.79.060301","article-title":"Reduced fidelity in topological quantum phase transitions","volume":"79","author":"Eriksson","year":"2009","journal-title":"Phys. Rev. A"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"032109","DOI":"10.1103\/PhysRevA.75.032109","article-title":"Mixed-state fidelity and quantum criticality at finite temperature","volume":"75","author":"Zanardi","year":"2007","journal-title":"Phys. Rev. A"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"062318","DOI":"10.1103\/PhysRevA.76.062318","article-title":"Bures metric over thermal state manifolds and quantum criticality","volume":"76","author":"Zanardi","year":"2007","journal-title":"Phys. Rev. A"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"011129","DOI":"10.1103\/PhysRevE.77.011129","article-title":"Macroscopic distinguishability between quantum states defining different phases of matter: Fidelity and the Uhlmann geometric phase","volume":"77","author":"Vieira","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"052302","DOI":"10.1103\/PhysRevA.77.052302","article-title":"Fidelity between partial states as a signature of quantum phase transitions","volume":"77","author":"Sacramento","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1103\/RevModPhys.74.197","article-title":"The role of relative entropy in quantum information theory","volume":"74","author":"Vedral","year":"2002","journal-title":"Rev. Mod. Phys."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"184517","DOI":"10.1103\/PhysRevB.76.184517","article-title":"Entanglement signatures of the quantum phase transition induced by a magnetic impurity in a superconductor","volume":"76","author":"Sacramento","year":"2007","journal-title":"Phys. Rev. B"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"032110","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_64","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_65","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_66","doi-asserted-by":"crossref","first-page":"P06002","DOI":"10.1088\/1742-5468\/2004\/06\/P06002","article-title":"Entanglement entropy and quantum field theory","volume":"2004","author":"Calabrese","year":"2004","journal-title":"J. Stat. Mech."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"1291","DOI":"10.1103\/PhysRevLett.71.1291","article-title":"Average entropy of a subsystem","volume":"71","author":"Page","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"200602","DOI":"10.1103\/PhysRevLett.124.200602","article-title":"Multifractality Meets Entanglement: Relation for Nonergodic Extended States","volume":"124","author":"Tomasi","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"180604","DOI":"10.1103\/PhysRevLett.125.180604","article-title":"Eigenstate Entanglement Entropy in Random Quadratic Hamiltonians","volume":"125","author":"Lydzba","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"L205","DOI":"10.1088\/0305-4470\/36\/14\/101","article-title":"Calculation of reduced density matrices from correlation functions","volume":"36","author":"Peschel","year":"2003","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"064412","DOI":"10.1103\/PhysRevB.64.064412","article-title":"Density-matrix spectra of solvable fermionic systems","volume":"64","author":"Chung","year":"2001","journal-title":"Phys. Rev. B"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"075111","DOI":"10.1103\/PhysRevB.69.075111","article-title":"Many-body density matrices for free fermions","volume":"69","author":"Cheong","year":"2004","journal-title":"Phys. Rev. B"},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"504003","DOI":"10.1088\/1751-8113\/42\/50\/504003","article-title":"Reduced density matrices and entanglement entropy in free lattice models","volume":"42","author":"Peschel","year":"2009","journal-title":"J. Phys. A"},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1007\/s13538-012-0074-1","article-title":"Entanglement in solvable many-particle models","volume":"42","author":"Peschel","year":"2012","journal-title":"Braz. J. Phys."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"016801","DOI":"10.1103\/PhysRevLett.103.016801","article-title":"Topological Entanglement and Clustering of Jain Hierarchy States","volume":"103","author":"Regnault","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"077202","DOI":"10.1103\/PhysRevLett.105.077202","article-title":"Entanglement Spectra of Quantum Heisenberg Ladders","volume":"105","author":"Poilblanc","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"100405","DOI":"10.1103\/PhysRevLett.106.100405","article-title":"Extracting Excitations from Model State Entanglement","volume":"106","author":"Sterdyniak","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"116805","DOI":"10.1103\/PhysRevLett.105.116805","article-title":"Nonlocal Order in Gapless Systems: Entanglement Spectrum in Spin Chains","volume":"105","author":"Thomale","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"130502","DOI":"10.1103\/PhysRevLett.104.130502","article-title":"Entanglement Spectrum of Topological Insulators and Superconductors","volume":"104","author":"Fidkowski","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"241102","DOI":"10.1103\/PhysRevB.82.241102","article-title":"Entanglement and inversion symmetry in topological insulators","volume":"82","author":"Turner","year":"2010","journal-title":"Phys. Rev. B"},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"245132","DOI":"10.1103\/PhysRevB.83.245132","article-title":"Inversion-symmetric topological insulators","volume":"83","author":"Hughes","year":"2011","journal-title":"Phys. Rev. B"},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"195103","DOI":"10.1103\/PhysRevB.84.195103","article-title":"Trace index and spectral flow in the entanglement spectrum of topological insulators","volume":"84","author":"Alexandradinata","year":"2011","journal-title":"Phys. Rev. B"},{"key":"ref_83","first-page":"041033","article-title":"Measurement Protocol for the Entanglement Spectrum of Cold Atoms","volume":"6","author":"Pichler","year":"2016","journal-title":"Phys. Rev. X"},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"064439","DOI":"10.1103\/PhysRevB.81.064439","article-title":"Entanglement spectrum of a topological phase in one dimension","volume":"81","author":"Pollmann","year":"2010","journal-title":"Phys. Rev. B"},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"134509","DOI":"10.1103\/PhysRevB.81.134509","article-title":"Effects of interactions on the topological classification of free fermion systems","volume":"81","author":"Fidkowski","year":"2010","journal-title":"Phys. Rev. B"},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"075102","DOI":"10.1103\/PhysRevB.83.075102","article-title":"Topological phases of one-dimensional fermions: An entanglement point of view","volume":"83","author":"Turner","year":"2011","journal-title":"Phys. Rev. B"},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1103\/RevModPhys.82.277","article-title":"Area laws for the entanglement entropy","volume":"82","author":"Eisert","year":"2010","journal-title":"Rev. Mod. Phys."},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"070502","DOI":"10.1103\/PhysRevLett.100.070502","article-title":"Area Laws in Quantum Systems: Mutual Information and Correlations","volume":"100","author":"Wolf","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"902","DOI":"10.1038\/nphys2465","article-title":"Identifying topological order by entanglement entropy","volume":"8","author":"Jiang","year":"2012","journal-title":"Nature Phys."},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"110404","DOI":"10.1103\/PhysRevLett.96.110404","article-title":"Topological Entanglement Entropy","volume":"96","author":"Kitaev","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"110405","DOI":"10.1103\/PhysRevLett.96.110405","article-title":"Detecting Topological Order in a Ground State Wave Function","volume":"96","author":"Levin","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_92","first-page":"180504","article-title":"Topological superconductivity induced by a triple-q magnetic structure","volume":"80","author":"Ali","year":"2010","journal-title":"Phys. Rev. B"},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"977","DOI":"10.1016\/j.aop.2008.11.002","article-title":"A symmetry principle for topological quantum order","volume":"324","author":"Nussinov","year":"2009","journal-title":"Ann. Phys."},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"245115","DOI":"10.1103\/PhysRevB.73.245115","article-title":"Entanglement entropy and the Berry phase in the solid state","volume":"73","author":"Ryu","year":"2006","journal-title":"Phys. Rev. B"},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"214407","DOI":"10.1103\/PhysRevB.75.214407","article-title":"Topological entanglement entropy in the quantum dimer model on the triangular lattice","volume":"75","author":"Furukawa","year":"2007","journal-title":"Phys. Rev. B"},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"067201","DOI":"10.1103\/PhysRevLett.109.067201","article-title":"Nature of the Spin-Liquid Ground State of the S=1\/2 Heisenberg Model on the Kagome Lattice","volume":"109","author":"Depenbrock","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"024424","DOI":"10.1103\/PhysRevB.86.024424","article-title":"Spin liquid ground state of the spin-1\/2 square J1-J2 Heisenberg model","volume":"86","author":"Jiang","year":"2012","journal-title":"Phys. Rev. B"},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"165134","DOI":"10.1103\/PhysRevB.84.165134","article-title":"Anomalies in the entanglement properties of the square-lattice Heisenberg model","volume":"84","author":"Kallin","year":"2011","journal-title":"Phys. Rev. B"},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"235141","DOI":"10.1103\/PhysRevB.84.235141","article-title":"Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations","volume":"84","author":"Chen","year":"2011","journal-title":"Phys. Rev. B"},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"125119","DOI":"10.1103\/PhysRevB.86.125119","article-title":"Theory and classification of interacting integer topological phases in two dimensions: A Chern-Simons approach","volume":"86","author":"Lu","year":"2012","journal-title":"Phys. Rev. B"},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1080\/00018739500101566","article-title":"Topological orders and edge excitations in fractional quantum Hall states","volume":"44","author":"Wen","year":"1995","journal-title":"Adv. Phys."},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"155138","DOI":"10.1103\/PhysRevB.82.155138","article-title":"Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order","volume":"82","author":"Chen","year":"2010","journal-title":"Phys. Rev. B"},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"10267","DOI":"10.1103\/PhysRevB.61.10267","article-title":"Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect","volume":"61","author":"Read","year":"2000","journal-title":"Phys. Rev. B"},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"080501","DOI":"10.1103\/PhysRevLett.105.080501","article-title":"Entanglement Entropy and Entanglement Spectrum of the Kitaev Model","volume":"105","author":"Yao","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"425702","DOI":"10.1088\/0953-8984\/26\/42\/425702","article-title":"Entanglement entropy and entanglement spectrum of triplet topological superconductors","volume":"26","author":"Oliveira","year":"2014","journal-title":"J. Phys. Cond. Matt."},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"094504","DOI":"10.1103\/PhysRevB.79.094504","article-title":"Topological phases of noncentrosymmetric superconductors: Edge states, Majorana fermions, and non-Abelian statistics","volume":"79","author":"Sato","year":"2009","journal-title":"Phys. Rev. B"},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"075128","DOI":"10.1103\/PhysRevB.84.075128","article-title":"Topological entanglement entropy of Z2 spin liquids and lattice Laughlin states","volume":"84","author":"Zhang","year":"2011","journal-title":"Phys. Rev. B"},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"37011","DOI":"10.1209\/0295-5075\/105\/37011","article-title":"Hall conductivity as bulk signature of topological transitions in superconductors","volume":"105","author":"Sacramento","year":"2014","journal-title":"Europhys. Lett."},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"10812","DOI":"10.1103\/PhysRevB.46.10812","article-title":"Mapping of the two-channel Kondo problem to a resonant-level model","volume":"46","author":"Emery","year":"1992","journal-title":"Phys. Rev. B"},{"key":"ref_110","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1088\/0022-3719\/18\/1\/020","article-title":"The thermodynamics of multichannel Kondo problem","volume":"18","author":"Tsvelik","year":"1985","journal-title":"J. Phys. C"},{"key":"ref_111","doi-asserted-by":"crossref","first-page":"5481","DOI":"10.1088\/0022-3719\/18\/28\/017","article-title":"Thermodynamics of the n-channel Kondo problem (numerical solution)","volume":"18","author":"Desgranges","year":"1985","journal-title":"J. Phys. C"},{"key":"ref_112","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1080\/00018739300101534","article-title":"Multichannel Kondo problem and some applications","volume":"42","author":"Schlottmann","year":"1993","journal-title":"Adv. Phys."},{"key":"ref_113","doi-asserted-by":"crossref","unstructured":"Carr, L. (2011). Understanding Quantum Phase Transitions, CRC Press.","DOI":"10.1201\/b10273"},{"key":"ref_114","unstructured":"Wen, X.G. (2004). Quantum Field Theory of Many-Body Systems, Oxford University."},{"key":"ref_115","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/j.aop.2013.05.014","article-title":"Construct order parameters from the reduced density matrix spectra","volume":"336","author":"Gu","year":"2013","journal-title":"Ann. Phys."},{"key":"ref_116","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1140\/epjb\/e2016-70361-6","article-title":"Density matrix spectra and order parameters in the 1D extended Hubbard model","volume":"89","author":"Yu","year":"2016","journal-title":"Eur. Phys. J."},{"key":"ref_117","doi-asserted-by":"crossref","first-page":"047211","DOI":"10.1103\/PhysRevLett.96.047211","article-title":"Systematic Derivation of Order Parameters through Reduced Density Matrices","volume":"96","author":"Furukawa","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_118","doi-asserted-by":"crossref","unstructured":"Henley, C.L., and Changlani, H.J. (2014). Density-matrix based numerical methods for discovering order and correlations in interacting systems. J. Stat. Mech., P11002.","DOI":"10.1088\/1742-5468\/2014\/11\/P11002"},{"key":"ref_119","doi-asserted-by":"crossref","first-page":"212402","DOI":"10.1103\/PhysRevB.79.212402","article-title":"Correlation density matrix: An unbiased analysis of exact diagonalizations","volume":"79","author":"Cheong","year":"2009","journal-title":"Phys. Rev. B"},{"key":"ref_120","doi-asserted-by":"crossref","first-page":"014505","DOI":"10.1103\/PhysRevB.90.014505","article-title":"Fermion fractionalization to Majorana fermions in a dimerized Kitaev superconductor","volume":"90","author":"Wakatsuki","year":"2014","journal-title":"Phys. Rev. B"},{"key":"ref_121","doi-asserted-by":"crossref","first-page":"1698","DOI":"10.1103\/PhysRevLett.42.1698","article-title":"Solitons in Polyacetylene","volume":"42","author":"Su","year":"1979","journal-title":"Phys. Rev. Lett."},{"key":"ref_122","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1070\/1063-7869\/44\/10S\/S29","article-title":"Unpaired Majorana fermions in quantum wires","volume":"44","author":"Kitaev","year":"2001","journal-title":"Phys. Usp."},{"key":"ref_123","doi-asserted-by":"crossref","first-page":"115113","DOI":"10.1103\/PhysRevB.99.115113","article-title":"Detection of topological phases by quasilocal operators","volume":"99","author":"Yu","year":"2019","journal-title":"Phys. Rev. B"},{"key":"ref_124","doi-asserted-by":"crossref","first-page":"245131","DOI":"10.1103\/PhysRevB.101.245131","article-title":"Energy bonds as correlators for long-range symmetry-protected topological models and models with long-range topological order","volume":"101","author":"Yu","year":"2020","journal-title":"Phys. Rev. B"},{"key":"ref_125","doi-asserted-by":"crossref","first-page":"014503","DOI":"10.1103\/PhysRevD.99.014503","article-title":"Symmetry-protected topological phases in lattice gauge theories: Topological QED2","volume":"99","author":"Magnifico","year":"2019","journal-title":"Phys. Rev. D"},{"key":"ref_126","doi-asserted-by":"crossref","first-page":"247202","DOI":"10.1103\/PhysRevLett.112.247202","article-title":"Wave Function and Strange Correlator of Short-Range Entangled States","volume":"112","author":"You","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_127","doi-asserted-by":"crossref","first-page":"165123","DOI":"10.1103\/PhysRevB.92.165123","article-title":"Quantum Monte Carlo study of strange correlator in interacting topological insulators","volume":"92","author":"Wu","year":"2015","journal-title":"Phys. Rev. B"},{"key":"ref_128","doi-asserted-by":"crossref","first-page":"177203","DOI":"10.1103\/PhysRevLett.121.177203","article-title":"Mapping Topological to Conformal Field Theories through strange Correlators","volume":"121","author":"Vanhove","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_129","unstructured":"Lepori, L., Burrello, M., Trombettoni, A., and Paganelli, S. (2022). Strange correlators for topological quantum systems from bulk-boundary correspondence. arXiv."},{"key":"ref_130","unstructured":"Zhang, J.-H., Qi, Y., and Bi, Z. (2022). Strange Correlation Function for Average Symmetry-Protected Topological Phases. arXiv."},{"key":"ref_131","doi-asserted-by":"crossref","first-page":"041105","DOI":"10.1103\/PhysRevB.87.041105","article-title":"Holographic symmetries and generalized order parameters for topological matter","volume":"87","author":"Cobanera","year":"2013","journal-title":"Phys. Rev. B"},{"key":"ref_132","doi-asserted-by":"crossref","first-page":"104420","DOI":"10.1103\/PhysRevB.76.104420","article-title":"Quantum fidelity and quantum phase transitions in matrix product states","volume":"76","author":"Cozzini","year":"2007","journal-title":"Phys. Rev. B"},{"key":"ref_133","doi-asserted-by":"crossref","first-page":"157203","DOI":"10.1103\/PhysRevLett.95.157203","article-title":"Geometric Phases and Criticality in Spin-Chain Systems","volume":"95","author":"Carollo","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_134","doi-asserted-by":"crossref","first-page":"077206","DOI":"10.1103\/PhysRevLett.96.077206","article-title":"Scaling of Geometric Phases Close to the Quantum Phase Transition in the XY Spin Chain","volume":"96","author":"Zhu","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_135","unstructured":"Hamma, A. (2006). Berry Phases and Quantum Phase Transitions. arXiv."},{"key":"ref_136","first-page":"1271","article-title":"Geometric Phases and Critical Phenomena in a Chain of Interacting Spins","volume":"463","author":"Reuter","year":"2007","journal-title":"Proc. Roy. Soc. Lond. A"},{"key":"ref_137","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1016\/0375-9601(92)90823-5","article-title":"Fluid-dimer critical point in S=1\/2 antiferromagnetic Heisenberg chain with next nearest neighbor interactions","volume":"169","author":"Okamoto","year":"1992","journal-title":"Phys. Lett. A"},{"key":"ref_138","doi-asserted-by":"crossref","first-page":"10003","DOI":"10.1209\/0295-5075\/87\/10003","article-title":"Scaling dimension of fidelity susceptibility in quantum phase transitions","volume":"87","author":"Gu","year":"2009","journal-title":"Europhys. Lett."},{"key":"ref_139","doi-asserted-by":"crossref","first-page":"P10032","DOI":"10.1088\/1742-5468\/2014\/10\/P10032","article-title":"Boundary fidelity and entanglement in the symmetry protected topological phase of the SSH model","volume":"2014","author":"Sirker","year":"2014","journal-title":"J. Stat. Mech."},{"key":"ref_140","unstructured":"Fuchs, C.A. (1995). Distinguishability and Accessible Information in Quantum Theory. [Ph.D. Thesis, University of New Mexico]."},{"key":"ref_141","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1016\/0034-4877(86)90055-8","article-title":"Parallel transport and \u201cquantum holonomy\u201d along density operators","volume":"24","author":"Uhlmann","year":"1986","journal-title":"Rep. Math. Phys."},{"key":"ref_142","doi-asserted-by":"crossref","first-page":"130401","DOI":"10.1103\/PhysRevLett.112.130401","article-title":"Uhlmann Phase as a Topological Measure for One-Dimensional Fermion Systems","volume":"112","author":"Viyuela","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_143","doi-asserted-by":"crossref","first-page":"076407","DOI":"10.1103\/PhysRevLett.113.076407","article-title":"Topological Indices for Open and Thermal Systems Via Uhlmann\u2019s Phase","volume":"113","author":"Huang","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_144","doi-asserted-by":"crossref","first-page":"076408","DOI":"10.1103\/PhysRevLett.113.076408","article-title":"Two-Dimensional Density-Matrix Topological Fermionic Phases: Topological Uhlmann Numbers","volume":"113","author":"Viyuela","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_145","doi-asserted-by":"crossref","first-page":"015702","DOI":"10.1103\/PhysRevLett.119.015702","article-title":"Uhlmann Connection in Fermionic Systems Undergoing Phase Transitions","volume":"119","author":"Mera","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_146","doi-asserted-by":"crossref","first-page":"245141","DOI":"10.1103\/PhysRevB.98.245141","article-title":"Fidelity and Uhlmann connection analysis of topological phase transitions in two dimensions","volume":"98","author":"Amin","year":"2018","journal-title":"Phys. Rev. B"},{"key":"ref_147","doi-asserted-by":"crossref","first-page":"085127","DOI":"10.1103\/PhysRevB.103.085127","article-title":"Interferometric geometry from symmetry-broken Uhlmann gauge group with applications to topological phase transitions","volume":"103","author":"Silva","year":"2021","journal-title":"Phys. Rev. B"},{"key":"ref_148","doi-asserted-by":"crossref","unstructured":"Hou, X.-Y., Wang, X., Zhou, Z., Guo, H., and Chien, C.-C. (2023). Geometric phases of mixed quantum states: A comparative study of interferometric and Uhlmann phases. arXiv.","DOI":"10.1103\/PhysRevB.107.165415"},{"key":"ref_149","doi-asserted-by":"crossref","first-page":"051126","DOI":"10.1103\/PhysRevE.78.051126","article-title":"Reduced fidelity susceptibility and its finite-size scaling behaviors in the Lipkin-Meshkov-Glick model","volume":"78","author":"Ma","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_150","doi-asserted-by":"crossref","first-page":"062302","DOI":"10.1103\/PhysRevA.78.062302","article-title":"Partial-state fidelity and quantum phase transitions induced by continuous level crossing","volume":"78","author":"Kwok","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_151","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_152","doi-asserted-by":"crossref","first-page":"174425","DOI":"10.1103\/PhysRevB.79.174425","article-title":"Reduced-fidelity approach for quantum phase transitions in spin-1\/2 dimerized Heisenberg chains","volume":"79","author":"Xiong","year":"2009","journal-title":"Phys. Rev. B"},{"key":"ref_153","doi-asserted-by":"crossref","first-page":"062318","DOI":"10.1103\/PhysRevA.84.062318","article-title":"Fidelity spectrum and phase transitions of quantum systems","volume":"84","author":"Sacramento","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_154","doi-asserted-by":"crossref","first-page":"1350106","DOI":"10.1142\/S0217979213501063","article-title":"A spin chain with spiral orders: Perspectives of quantum information and mechanical response","volume":"27","author":"Gu","year":"2013","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_155","doi-asserted-by":"crossref","first-page":"094512","DOI":"10.1103\/PhysRevB.89.094512","article-title":"Entanglement modes and topological phase transitions in superconductors","volume":"89","author":"Oliveira","year":"2014","journal-title":"Phys. Rev. B"},{"key":"ref_156","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1016\/j.aop.2018.11.021","article-title":"Vanishing k-space fidelity and phase diagram\u2019s bulk\u2013edge\u2013bulk correspondence","volume":"401","author":"Sacramento","year":"2019","journal-title":"Ann. Phys."},{"key":"ref_157","doi-asserted-by":"crossref","first-page":"P04010","DOI":"10.1088\/1742-5468\/2005\/04\/P04010","article-title":"Evolution of entanglement entropy in one-dimensional systems","volume":"2005","author":"Calabrese","year":"2005","journal-title":"J. Stat. Mech."},{"key":"ref_158","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1007\/BF01645779","article-title":"The finite group velocity of quantum spin systems","volume":"28","author":"Lieb","year":"1972","journal-title":"Commun. Math. Phys."},{"key":"ref_159","doi-asserted-by":"crossref","first-page":"050405","DOI":"10.1103\/PhysRevLett.98.050405","article-title":"Relaxation in a Completely Integrable Many-Body Quantum System: An Ab Initio Study of the Dynamics of the Highly Excited States of 1D Lattice Hard-Core Bosons","volume":"98","author":"Rigol","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_160","doi-asserted-by":"crossref","first-page":"140405","DOI":"10.1103\/PhysRevLett.106.140405","article-title":"Generalized Thermalization in an Integrable Lattice System","volume":"106","author":"Cassidy","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_161","doi-asserted-by":"crossref","first-page":"854","DOI":"10.1038\/nature06838","article-title":"Thermalization and its mechanism for generic isolated quantum systems","volume":"452","author":"Rigol","year":"2008","journal-title":"Nature"},{"key":"ref_162","doi-asserted-by":"crossref","first-page":"245107","DOI":"10.1103\/PhysRevB.87.245107","article-title":"Reduced density matrix after a quantum quench","volume":"87","author":"Fagotti","year":"2013","journal-title":"Phys. Rev. B"},{"key":"ref_163","doi-asserted-by":"crossref","first-page":"017","DOI":"10.21468\/SciPostPhys.4.3.017","article-title":"Entanglement dynamics after quantum quenches in generic integrable systems","volume":"4","author":"Alba","year":"2018","journal-title":"SciPost Phys."},{"key":"ref_164","doi-asserted-by":"crossref","first-page":"794","DOI":"10.1126\/science.aaf6725","article-title":"Quantum thermalization through entanglement in an isolated many-body system","volume":"353","author":"Kaufman","year":"2016","journal-title":"Science"},{"key":"ref_165","doi-asserted-by":"crossref","first-page":"020505","DOI":"10.1103\/PhysRevLett.109.020505","article-title":"Measuring Entanglement Growth in Quench Dynamics of Bosons in an Optical Lattice","volume":"109","author":"Daley","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_166","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1038\/nature15750","article-title":"Measuring entanglement entropy in a quantum many-body system","volume":"528","author":"Islam","year":"2015","journal-title":"Nature"},{"key":"ref_167","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1126\/science.aau0818","article-title":"Probing entanglement in a many-body-localized system","volume":"364","author":"Lukin","year":"2019","journal-title":"Science"},{"key":"ref_168","doi-asserted-by":"crossref","unstructured":"Torlai, G., Tagliacozzo, L., and de Chiara, G. (2014). Dynamics of the entanglement spectrum in spin chains. J. Stat. Mech., P06001.","DOI":"10.1088\/1742-5468\/2014\/06\/P06001"},{"key":"ref_169","doi-asserted-by":"crossref","first-page":"104303","DOI":"10.1103\/PhysRevB.89.104303","article-title":"Dynamics of entanglement entropy and entanglement spectrum crossing a quantum phase transition","volume":"89","author":"Canovi","year":"2014","journal-title":"Phys. Rev. B"},{"key":"ref_170","doi-asserted-by":"crossref","first-page":"250601","DOI":"10.1103\/PhysRevLett.121.250601","article-title":"Topological Entanglement-Spectrum Crossing in Quench Dynamics","volume":"121","author":"Gong","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_171","doi-asserted-by":"crossref","first-page":"104511","DOI":"10.1103\/PhysRevB.88.104511","article-title":"Quantum quench in a p+ip superfluid: Winding numbers and topological states far from equilibrium","volume":"88","author":"Foster","year":"2013","journal-title":"Phys. Rev. B"},{"key":"ref_172","doi-asserted-by":"crossref","first-page":"042125","DOI":"10.1103\/PhysRevE.89.042125","article-title":"Survival probability of an edge Majorana in a one-dimensional p-wave superconducting chain under sudden quenching of parameters","volume":"89","author":"Rajak","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_173","doi-asserted-by":"crossref","first-page":"032138","DOI":"10.1103\/PhysRevE.90.032138","article-title":"Fate of Majorana fermions and Chern numbers after a quantum quench","volume":"90","author":"Sacramento","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_174","doi-asserted-by":"crossref","first-page":"236403","DOI":"10.1103\/PhysRevLett.115.236403","article-title":"Quantum Quenches in Chern Insulators","volume":"115","author":"Caio","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_175","doi-asserted-by":"crossref","first-page":"062117","DOI":"10.1103\/PhysRevE.93.062117","article-title":"Edge mode dynamics of quenched topological wires","volume":"93","author":"Sacramento","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_176","doi-asserted-by":"crossref","first-page":"250602","DOI":"10.1103\/PhysRevLett.122.250602","article-title":"Sub-ballistic growth of r\u00e9nyi entropies due to diffusion","volume":"122","author":"Rakovszky","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_177","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1038\/s42005-020-0366-7","article-title":"Entanglement growth in diffusive systems","volume":"3","author":"Znidaric","year":"2020","journal-title":"Commun. Phys."},{"key":"ref_178","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1038\/s42005-021-00594-4","article-title":"Entanglement growth in diffusive systems with large spin","volume":"4","author":"Rakovszky","year":"2021","journal-title":"Commun. Phys."},{"key":"ref_179","doi-asserted-by":"crossref","first-page":"075142","DOI":"10.1103\/PhysRevB.104.075142","article-title":"Unveiling quantum phase transitions by fidelity mapping","volume":"104","author":"Tang","year":"2021","journal-title":"Phys. Rev. B"},{"key":"ref_180","doi-asserted-by":"crossref","first-page":"135704","DOI":"10.1103\/PhysRevLett.110.135704","article-title":"Dynamical Quantum Phase Transitions in the Transverse-Field Ising Model","volume":"110","author":"Heyl","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_181","doi-asserted-by":"crossref","first-page":"054001","DOI":"10.1088\/1361-6633\/aaaf9a","article-title":"Dynamical quantum phase transitions: A review","volume":"81","author":"Heyl","year":"2018","journal-title":"Rep. Prog. Phys."},{"key":"ref_182","doi-asserted-by":"crossref","first-page":"064304","DOI":"10.1103\/PhysRevB.97.064304","article-title":"Bulk-boundary correspondence for dynamical phase transitions in one-dimensional topological insulators and superconductors","volume":"97","author":"Sedlmayr","year":"2018","journal-title":"Phys. Rev. B"},{"key":"ref_183","doi-asserted-by":"crossref","first-page":"014301","DOI":"10.1103\/PhysRevB.101.014301","article-title":"Quasiperiodic dynamical quantum phase transitions in multiband topological insulators and connections with entanglement entropy and fidelity susceptibility","volume":"101","author":"Maslowski","year":"2020","journal-title":"Phys. Rev. B"},{"key":"ref_184","doi-asserted-by":"crossref","first-page":"085416","DOI":"10.1103\/PhysRevB.93.085416","article-title":"Dynamical topological order parameters far from equilibrium","volume":"93","author":"Budich","year":"2016","journal-title":"Phys. Rev. B"},{"key":"ref_185","doi-asserted-by":"crossref","first-page":"075130","DOI":"10.1103\/PhysRevB.104.075130","article-title":"Local measures of dynamical quantum phase transitions","volume":"104","author":"Halimeh","year":"2021","journal-title":"Phys. Rev. B"},{"key":"ref_186","doi-asserted-by":"crossref","first-page":"250601","DOI":"10.1103\/PhysRevLett.122.250601","article-title":"Dynamical Topological Quantum Phase Transitions in Nonintegrable Models","volume":"122","author":"Hubig","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_187","doi-asserted-by":"crossref","first-page":"200602","DOI":"10.1103\/PhysRevLett.126.200602","article-title":"Observing Dynamical Quantum Phase Transitions through Quasilocal String Operators","volume":"126","author":"Bandyopadhyay","year":"2021","journal-title":"Phys. Rev. Lett"},{"key":"ref_188","first-page":"041018","article-title":"Determination of Dynamical Quantum Phase Transitions in Strongly Correlated Many-Body Systems Using Loschmidt Cumulants","volume":"11","author":"Peotta","year":"2021","journal-title":"Phys. Rev. X"},{"key":"ref_189","doi-asserted-by":"crossref","first-page":"180303","DOI":"10.1103\/PhysRevB.96.180303","article-title":"Mixed state dynamical quantum phase transitions","volume":"96","author":"Bhattacharya","year":"2017","journal-title":"Phys. Rev. B"},{"key":"ref_190","doi-asserted-by":"crossref","first-page":"094110","DOI":"10.1103\/PhysRevB.97.094110","article-title":"Dynamical phase transitions at finite temperature from fidelity and interferometric Loschmidt echo induced metrics","volume":"97","author":"Mera","year":"2018","journal-title":"Phys. Rev. B"},{"key":"ref_191","doi-asserted-by":"crossref","first-page":"045147","DOI":"10.1103\/PhysRevB.97.045147","article-title":"Fate of dynamical phase transitions at finite temperatures and in open systems","volume":"97","author":"Sedlmayr","year":"2018","journal-title":"Phys. Rev. B"},{"key":"ref_192","doi-asserted-by":"crossref","first-page":"130603","DOI":"10.1103\/PhysRevLett.121.130603","article-title":"Dynamical Quantum Phase Transitions: A Geometric Picture","volume":"121","author":"Lang","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_193","doi-asserted-by":"crossref","first-page":"085104","DOI":"10.1103\/PhysRevB.104.085104","article-title":"Correlations and dynamical quantum phase transitions in an interacting topological insulator","volume":"104","author":"Yu","year":"2021","journal-title":"Phys. Rev. B"},{"key":"ref_194","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1080\/14789940801912366","article-title":"Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems","volume":"57","author":"Verstraete","year":"2008","journal-title":"Adv. Phys."},{"key":"ref_195","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1146\/annurev-conmatphys-031016-025507","article-title":"Diagonalizing transfer matrices and matrix product operators: A medley of exact and computational methods","volume":"8","author":"Haegeman","year":"2016","journal-title":"Annu. Rev. Cond. Matt. Phys."},{"key":"ref_196","doi-asserted-by":"crossref","first-page":"065007","DOI":"10.1103\/PhysRevD.86.065007","article-title":"Entanglement renormalization and holography","volume":"86","author":"Swingle","year":"2012","journal-title":"Phys. Rev. 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