{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T00:56:34Z","timestamp":1773104194858,"version":"3.50.1"},"reference-count":233,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,15]],"date-time":"2022-09-15T00:00:00Z","timestamp":1663200000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Chongqing University"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Fidelity mechanics is formalized as a framework for investigating critical phenomena in quantum many-body systems. Fidelity temperature is introduced for quantifying quantum fluctuations, which, together with fidelity entropy and fidelity internal energy, constitute three basic state functions in fidelity mechanics, thus enabling us to formulate analogues of the four thermodynamic laws and Landauer\u2019s principle at zero temperature. Fidelity flows, which are irreversible, are defined and may be interpreted as an alternative form of renormalization group flows. Thus, fidelity mechanics offers a means to characterize both stable and unstable fixed points: divergent fidelity temperature for unstable fixed points and zero-fidelity temperature and (locally) maximal fidelity entropy for stable fixed points. In addition, fidelity entropy behaves differently at an unstable fixed point for topological phase transitions and at a stable fixed point for topological quantum states of matter. A detailed analysis of fidelity mechanical-state functions is presented for six fundamental models\u2014the quantum spin-1\/2 XY model, the transverse-field quantum Ising model in a longitudinal field, the quantum spin-1\/2 XYZ model, the quantum spin-1\/2 XXZ model in a magnetic field, the quantum spin-1 XYZ model, and the spin-1\/2 Kitaev model on a honeycomb lattice for illustrative purposes. We also present an argument to justify why the thermodynamic, psychological\/computational, and cosmological arrows of time should align with each other, with the psychological\/computational arrow of time being singled out as a master arrow of time.<\/jats:p>","DOI":"10.3390\/e24091306","type":"journal-article","created":{"date-parts":[[2022,9,15]],"date-time":"2022-09-15T22:11:10Z","timestamp":1663279870000},"page":"1306","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Fidelity Mechanics: Analogues of the Four Thermodynamic Laws and Landauer\u2019s Principle"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4030-6886","authenticated-orcid":false,"given":"Huan-Qiang","family":"Zhou","sequence":"first","affiliation":[{"name":"Centre for Modern Physics, Chongqing University, Chongqing 400044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qian-Qian","family":"Shi","sequence":"additional","affiliation":[{"name":"Centre for Modern Physics, Chongqing University, Chongqing 400044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yan-Wei","family":"Dai","sequence":"additional","affiliation":[{"name":"Centre for Modern Physics, Chongqing University, Chongqing 400044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Sachdev, S. (1999). Quantum Phase Transitions, Cambridge University Press.","DOI":"10.1017\/CBO9780511622540"},{"key":"ref_2","unstructured":"Wen, X.-G. (2004). Quantum Field Theory of Many-Body Systems, Oxford University Press."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Nishimori, H., and Ortiz, G. (2011). Elements of Phase Transitions and Critical Phenomena, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780199577224.001.0001"},{"key":"ref_4","unstructured":"Landau, L.D., Lifshitz, E.M., and Pitaevskii, E.M. (1999). Statistical Physics, Butterworth-Heinemann."},{"key":"ref_5","unstructured":"Anderson, P.W. (1997). Basic Notions of Condensed Matter Physics, Addison-Wesley: The Advanced Book Program, Addison-Wesley."},{"key":"ref_6","unstructured":"Coleman, S. (1975). An Introduction to Spontaneous Symmetry Breakdown and Gauge Fields: Laws of Hadronic Matter, Academic."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1103\/PhysicsPhysiqueFizika.2.263","article-title":"Scaling laws for Ising model near Tc*","volume":"2","author":"Kadanoff","year":"1966","journal-title":"Physics"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"773","DOI":"10.1103\/RevModPhys.47.773","article-title":"The renormalization group: Critical phenomena and the Kondo problem","volume":"47","author":"Wilson","year":"1975","journal-title":"Rev. Mod. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"597","DOI":"10.1103\/RevModPhys.46.597","article-title":"The renormalization group in the theory of critical behavior","volume":"46","author":"Fisher","year":"1974","journal-title":"Rev. Mod. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Zinn-Justin, J. (2002). Quantum Field Theory and Critical Phenomena, Clarendon Press.","DOI":"10.1093\/acprof:oso\/9780198509233.001.0001"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1769","DOI":"10.1103\/PhysRevD.16.1769","article-title":"Quantum field theories on a lattice: Variational methods for arbitrary coupling strengths and the Ising model in a transverse magnetic field","volume":"16","author":"Drell","year":"1977","journal-title":"Phys. Rev. D"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1500","DOI":"10.1103\/PhysRevLett.38.1500","article-title":"Kondo Lattice: Real-Space Renormalization-Group Approach","volume":"38","author":"Jullien","year":"1977","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1142\/S0217979290000139","article-title":"Topological orders in regid states","volume":"4","author":"Wen","year":"1990","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3045","DOI":"10.1103\/RevModPhys.82.3045","article-title":"Colloquium: Topological insulators","volume":"82","author":"Hasan","year":"2010","journal-title":"Rev. Mod. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1057","DOI":"10.1103\/RevModPhys.83.1057","article-title":"Topological insulators and superconductors","volume":"83","author":"Qi","year":"2011","journal-title":"Rev. Mod. Phys."},{"key":"ref_16","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_17","doi-asserted-by":"crossref","first-page":"L02002","DOI":"10.1088\/1742-5468\/2007\/02\/L02002","article-title":"Ground state fidelity and quantum phase transitions in free Fermi systems","volume":"2007","author":"Zanardi","year":"2007","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_18","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_19","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_20","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_21","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_22","doi-asserted-by":"crossref","first-page":"180403(R)","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_23","doi-asserted-by":"crossref","first-page":"012311","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_24","doi-asserted-by":"crossref","first-page":"115119","DOI":"10.1103\/PhysRevB.75.115119","article-title":"Links, Ground-state properties of the attractive one-dimensional Bose-Hubbard model","volume":"75","author":"Oelkers","year":"2007","journal-title":"Phys. Rev. B"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"P07011","DOI":"10.1088\/1742-5468\/2008\/07\/P07011","article-title":"Ground state fidelity of Luttinger liquids: A wavefunctional approach","volume":"2008","author":"Fjaerestad","year":"2008","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_26","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_27","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 the quantum Ising model in a transverse field","volume":"41","author":"Zhou","year":"2008","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"080601","DOI":"10.1103\/PhysRevLett.100.080601","article-title":"Ground State Fidelity from Tensor Network Representations","volume":"100","author":"Zhou","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_29","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_30","doi-asserted-by":"crossref","first-page":"L10001","DOI":"10.1088\/1742-5468\/2011\/10\/L10001","article-title":"Kosterlitz-Thouless phase transition and ground state fidelity: A novel perspective from matrix product states","volume":"2011","author":"Wang","year":"2011","journal-title":"J. Stat. Mech."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1016\/j.aop.2005.10.005","article-title":"Anyons in an exactly solved model and beyond","volume":"321","author":"Kitaev","year":"2006","journal-title":"Ann. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"247201","DOI":"10.1103\/PhysRevLett.98.247201","article-title":"Exact results for spin dynamics and fractionalization in the Kitaev model","volume":"98","author":"Baskaran","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"075001","DOI":"10.1088\/1751-8113\/41\/7\/075001","article-title":"Exact results of the Kitaev model on a hexagonal lattice: Spin states, string and brane correlators, and anyonic excitations","volume":"41","author":"Chen","year":"2008","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_34","first-page":"610","article-title":"Destruction of long range order in one dimensional and two dimensional systems having a continuous symmetry group I. classical systems","volume":"34","author":"Berezinskii","year":"1972","journal-title":"Sov. Phys. JETP"},{"key":"ref_35","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 Solid State Phys."},{"key":"ref_36","unstructured":"Nielsen, M.A., and Chuang, I.L. (2000). Quantum Computation and Quantum Information, Cambridge University Press."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"2863","DOI":"10.1103\/PhysRevLett.69.2863","article-title":"Density matrix formulation for quantum renormalization groups","volume":"69","author":"White","year":"1992","journal-title":"Phys. Rev. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1103\/RevModPhys.77.259","article-title":"The density-matrix renormalization group","volume":"77","year":"2005","journal-title":"Rev. Mod. Phys."},{"key":"ref_39","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_40","doi-asserted-by":"crossref","first-page":"504004","DOI":"10.1088\/1751-8113\/42\/50\/504004","article-title":"Renormalization and tensor product states in spin chains and lattices","volume":"42","author":"Cirac","year":"2009","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"250602","DOI":"10.1103\/PhysRevLett.101.250602","article-title":"Classical simulation of infinite-size quantum lattice systems in two spatial dimensions","volume":"101","author":"Jordan","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_42","unstructured":"Shi, Q.-Q., Li, S.-H., Zhao, J.-H., and Zhou, H.-Q. (2009). Graded projected entangled-pair state representations and an algorithm for translationally invariant strongly correlated electronic systems on infinite-size lattices in two spatial dimensions. arXiv."},{"key":"ref_43","unstructured":"Li, S.-H., Shi, Q.-Q., and Zhou, H.-Q. (2010). Ground-state phase diagram of the two-dimensional t-J model. arXiv."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"052338","DOI":"10.1103\/PhysRevA.81.052338","article-title":"Fermionic projected entangled pair states","volume":"81","author":"Kraus","year":"2010","journal-title":"Phys. Rev. A"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"245110","DOI":"10.1103\/PhysRevB.81.245110","article-title":"Fermionic implementation of projected entangled pair states algorithm","volume":"81","author":"Verstraete","year":"2010","journal-title":"Phys. Rev. B"},{"key":"ref_46","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_47","doi-asserted-by":"crossref","first-page":"040502","DOI":"10.1103\/PhysRevLett.93.040502","article-title":"Efficient simulation of one-dimensional quantum many-body systems","volume":"93","author":"Vidal","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"070201","DOI":"10.1103\/PhysRevLett.98.070201","article-title":"Classical simulation of infinite-size quantum lattice systems in one spatial dimension","volume":"98","author":"Vidal","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Zhou, H.-Q. (2008). Deriving local order parameters from tensor network representations. arXiv.","DOI":"10.1103\/PhysRevLett.100.080601"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"064401","DOI":"10.1103\/PhysRevB.86.064401","article-title":"Tensor network states and ground-state fidelity for quantum spin ladders","volume":"86","author":"Li","year":"2012","journal-title":"Phys. Rev. B"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1073\/pnas.97.1.28","article-title":"The theory of everything","volume":"97","author":"Laughlin","year":"2000","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_52","first-page":"730","article-title":"\u201cIrreversibility\u201d of the flux of the renormalization group in a 2D field theory","volume":"43","author":"Zamolodchikov","year":"1986","journal-title":"JETP Lett."},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Cardy, J. (1996). Scaling and Renormalization in Statistical Physics, Cambridge University Press.","DOI":"10.1017\/CBO9781316036440"},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Francesco, P.D., Mathieu, P., and S\u00e9nxexchal, D. (1997). Conformal Field Theory, Springer.","DOI":"10.1007\/978-1-4612-2256-9"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"P10004","DOI":"10.1088\/1742-5468\/2010\/10\/P10004","article-title":"The ubiquitous \u2018c\u2019: From the Stefan-Boltzmann law to quantum information","volume":"2010","author":"Cardy","year":"2010","journal-title":"J. Stat. Mech."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1007\/JHEP12(2011)099","article-title":"On renormalization group flows in four dimensions","volume":"12","author":"Komargodski","year":"2011","journal-title":"J. High Energy Phys."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"5163","DOI":"10.1103\/PhysRevD.54.5163","article-title":"Field theory entropy, the H theorem, and the renormalization group","volume":"54","author":"Gaite","year":"1996","journal-title":"Phys. Rev. D"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1147\/rd.53.0183","article-title":"Irreversibility and heat generation in the computing process","volume":"5","author":"Landauer","year":"1961","journal-title":"IBM J. Res. Dev."},{"key":"ref_59","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_60","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_61","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_62","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_63","first-page":"334","article-title":"Magnetic-field phase transition in a one-dimensional system of electrons with attraction","volume":"27","author":"Dzhaparidze","year":"1978","journal-title":"JETP Lett."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1103\/PhysRevLett.42.65","article-title":"Ground State, spectrum, and phase diagram of two-dimensional incommensurate crystals","volume":"42","author":"Pokrovsky","year":"1979","journal-title":"Phys. Rev. Lett."},{"key":"ref_65","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_66","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_67","unstructured":"McCulloch, I.P. (2008). Infinite size density matrix renormalization group, revisited. arXiv."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"P10014","DOI":"10.1088\/1742-5468\/2007\/10\/P10014","article-title":"From density-matrix renormalization group to matrix product states","volume":"2007","author":"McCulloch","year":"2007","journal-title":"J. Stat. Mech."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"054423","DOI":"10.1103\/PhysRevB.85.054423","article-title":"Ground-state properties of antiferromagnetic anisotropic S = 1 Heisenberg spin chains","volume":"85","author":"Peters","year":"2012","journal-title":"Phys. Rev. B"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"032304","DOI":"10.1103\/PhysRevA.81.032304","article-title":"Quantum key distribution over probabilistic quantum repeaters","volume":"81","author":"Huang","year":"2010","journal-title":"Phys. Rev. A"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"062341","DOI":"10.1103\/PhysRevA.93.062341","article-title":"Geometric entanglement and quantum phase transitions in two-dimensional quantum lattice models","volume":"93","author":"Shi","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_72","unstructured":"Papadimitriou, C.H. (1994). Computational Complexity, Addison-Wesley."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"060101R","DOI":"10.1103\/PhysRevE.79.060101","article-title":"Quantum phase transitions in a two-dimensional quantum XYX model: Ground-state fidelity and entanglement","volume":"79","author":"Li","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"4045","DOI":"10.1016\/j.physleta.2011.09.014","article-title":"Bifurcation in ground-state fidelity for a one-dimensional spin model with competing two-spin and three-spin interactions","volume":"375","author":"Wang","year":"2011","journal-title":"Phys. Lett. A"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"015306","DOI":"10.1088\/1751-8113\/45\/1\/015306","article-title":"Ground-state fidelity and Kosterlitz-Thouless phase transition for the spin-1\/2 Heisenberg chain with next-to-the-nearest-neighbor interaction","volume":"45","author":"Wang","year":"2012","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1103\/PhysRev.60.252","article-title":"Statistics of the two-dimensional ferromagnet. Part I","volume":"60","author":"Kramers","year":"1941","journal-title":"Phys. Rev."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"197201","DOI":"10.1103\/PhysRevLett.100.197201","article-title":"Theory of ground state factorization in quantum cooperative systems","volume":"100","author":"Giampaolo","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"224434","DOI":"10.1103\/PhysRevB.79.224434","article-title":"Separability and ground-state factorization in quantum spin systems","volume":"79","author":"Giampaolo","year":"2009","journal-title":"Phys. Rev. B"},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"207202","DOI":"10.1103\/PhysRevLett.104.207202","article-title":"Probing quantum frustrated systems via factorization of the ground state","volume":"104","author":"Giampaolo","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/0378-4371(82)90217-5","article-title":"Antiferromagnetic long-range order in the anisotropic quantum spin chain","volume":"112","author":"Kurmann","year":"1982","journal-title":"Phys. A"},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"167203","DOI":"10.1103\/PhysRevLett.93.167203","article-title":"Studying quantum spin systems through entanglement estimators","volume":"93","author":"Roscilde","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"147208","DOI":"10.1103\/PhysRevLett.94.147208","article-title":"Entanglement and factorized ground states in two-dimensional quantum antiferromagnets","volume":"94","author":"Roscilde","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_83","unstructured":"Messiah, A. (1966). Quantum Mechanics, John Wiley and Sons."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"365303","DOI":"10.1088\/1751-8113\/42\/36\/365303","article-title":"Transitionless quantum driving","volume":"42","author":"Berry","year":"2009","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"115703","DOI":"10.1103\/PhysRevLett.109.115703","article-title":"Assisted finite-rate adiabatic passage across a quantum critical point: Exact solution for the quantum Ising model","volume":"109","author":"Campo","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_86","doi-asserted-by":"crossref","unstructured":"Mersini-Houghton, L., and Vaas, R. (2012). Time After Time-Big Bang Cosmology and the Arrows of Time. The Arrows of Time, Fundamental Theories of Physics, Springer.","DOI":"10.1007\/978-3-642-23259-6"},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1126\/science.177.4047.393","article-title":"More is different: Broken symmetry and the nature of the hierarchical structure of science","volume":"177","author":"Anderson","year":"1972","journal-title":"Science"},{"key":"ref_88","unstructured":"Kroemer, H., and Kittel, C. (1980). Thermal Physics, W. H. Freeman Company. [2nd ed.]."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"1387","DOI":"10.1088\/0305-4470\/9\/8\/029","article-title":"Topology of cosmic domains and strings","volume":"9","author":"Kibble","year":"1976","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/0370-1573(80)90091-5","article-title":"Some implications of a cosmological phase transition","volume":"67","author":"Kibble","year":"1980","journal-title":"Phys. Rep."},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1038\/317505a0","article-title":"Cosmological experiments in superfluid helium?","volume":"317","author":"Zurek","year":"1985","journal-title":"Nature"},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/S0370-1573(96)00009-9","article-title":"Cosmological experiments in condensed matter systems","volume":"276","author":"Zurek","year":"1996","journal-title":"Phys. Rep."},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"035701","DOI":"10.1103\/PhysRevLett.95.035701","article-title":"The simplest quantum model supporting the Kibble-Zurek mechanism of topological defect production: Landau-Zener transitions from a new perspective","volume":"95","author":"Damski","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"105701","DOI":"10.1103\/PhysRevLett.95.105701","article-title":"Dynamics of a quantum phase transition","volume":"95","author":"Zurek","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"245701","DOI":"10.1103\/PhysRevLett.95.245701","article-title":"Dynamics of a quantum phase transition: Exact solution of the quantum Ising model","volume":"95","author":"Dziarmaga","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_96","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_97","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_98","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1007\/BF02345020","article-title":"Particle creation by black holes","volume":"43","author":"Hawking","year":"1975","journal-title":"Commun. Math. Phys."},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1038\/248030a0","article-title":"Black Hole Explosions?","volume":"248","author":"Hawking","year":"1974","journal-title":"Nature"},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"245104","DOI":"10.1103\/PhysRevB.83.245104","article-title":"Possible proximity of the Mott insulating iridate Na2IrO3 to a topological phase: Phase diagram of the Heisenberg-Kitaev model in a magnetic field","volume":"83","author":"Jiang","year":"2011","journal-title":"Phys. Rev. B"},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"014418","DOI":"10.1103\/PhysRevB.98.014418","article-title":"Dynamical and topological properties of the Kitaev model in a [111] magnetic field","volume":"98","author":"Gohlke","year":"2018","journal-title":"Phys. Rev. B"},{"key":"ref_102","unstructured":"Shi, Q.-Q., Dai, Y.-W., Zhou, H.-Q., and McCulloch, I. (2022). Fractal dimension and the counting rule of the Goldstone modes. arXiv."},{"key":"ref_103","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_104","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/BF01646487","article-title":"There are no Goldstone bosons in two dimensions","volume":"31","author":"Coleman","year":"1973","journal-title":"Commun. Math. Phys."},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"251602","DOI":"10.1103\/PhysRevLett.108.251602","article-title":"Unified description of Nambu-Goldstone bosons without Lorentz invariance","volume":"108","author":"Watanabe","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"091601","DOI":"10.1103\/PhysRevLett.110.091601","article-title":"Counting rule for Nambu-Goldstone modes in nonrelativistic systems","volume":"110","author":"Hidaka","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1063\/1.1485555","article-title":"Singular limits","volume":"55","author":"Berry","year":"2002","journal-title":"Phys. Today"},{"key":"ref_108","first-page":"0355107","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_109","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_110","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_111","doi-asserted-by":"crossref","first-page":"165135","DOI":"10.1103\/PhysRevB.93.165135","article-title":"SU(3) quantum critical model emerging from a spin-1 topological phase","volume":"93","author":"Rao","year":"2016","journal-title":"Phys. Rev. B"},{"key":"ref_112","doi-asserted-by":"crossref","first-page":"125441","DOI":"10.1103\/PhysRevB.86.125441","article-title":"Detection of symmetry-protected topological phases in one dimension","volume":"86","author":"Pollmann","year":"2012","journal-title":"Phys. Rev. B"},{"key":"ref_113","doi-asserted-by":"crossref","first-page":"177204","DOI":"10.1103\/PhysRevLett.114.177204","article-title":"Distinct trivial phases protected by a point-group symmetry in quantum spin chains","volume":"114","author":"Fuji","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_114","doi-asserted-by":"crossref","first-page":"085146","DOI":"10.1103\/PhysRevB.102.085146","article-title":"Symmetry-protected trivial phases and quantum phase transitions in an anisotropic antiferromagnetic spin-1 biquadratic model","volume":"102","author":"Chen","year":"2020","journal-title":"Phys. Rev. B"},{"key":"ref_115","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_116","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1016\/0003-4916(61)90115-4","article-title":"Two soluble models of an antiferromagnetic chain","volume":"16","author":"Lieb","year":"1961","journal-title":"Ann. Phys."},{"key":"ref_117","unstructured":"Lieb, E., and Mattis, D. (1966). Mathematical Physics in One Dimension, Academic Press."},{"key":"ref_118","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/0003-4916(70)90270-8","article-title":"The one-dimensional Ising model with a transverse field","volume":"57","author":"Pfeuty","year":"1970","journal-title":"Ann. Phys."},{"key":"ref_119","doi-asserted-by":"crossref","first-page":"125142","DOI":"10.1103\/PhysRevB.88.125142","article-title":"Universal aspects in the behavior of the entanglement spectrum in one dimension: Scaling transition at the factorization point and ordered entangled structures","volume":"88","author":"Giampaolo","year":"2013","journal-title":"Phys. Rev. B"},{"key":"ref_120","doi-asserted-by":"crossref","first-page":"8467","DOI":"10.1088\/1751-8113\/40\/29\/019","article-title":"Ellipses of constant entropy in the XY spin chain","volume":"40","author":"Franchini","year":"2007","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_121","doi-asserted-by":"crossref","first-page":"641","DOI":"10.2969\/aspm\/01910641","article-title":"Integrable field theory from conformal field theory","volume":"19","author":"Zamolodchikov","year":"1989","journal-title":"Adv. Stud. Pure Math."},{"key":"ref_122","doi-asserted-by":"crossref","first-page":"834","DOI":"10.1103\/PhysRevLett.26.834","article-title":"One-Dimensional anisotropic Heisenberg chain","volume":"26","author":"Baxter","year":"1971","journal-title":"Phys. Rev. Lett."},{"key":"ref_123","doi-asserted-by":"crossref","first-page":"832","DOI":"10.1103\/PhysRevLett.26.832","article-title":"Eight-Vertex Model in Lattice Statistics","volume":"26","author":"Baxter","year":"1971","journal-title":"Phys. Rev. Lett."},{"key":"ref_124","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/0003-4916(72)90335-1","article-title":"Partition function of the Eight-Vertex lattice model","volume":"70","author":"Baxter","year":"1972","journal-title":"Ann. Phys."},{"key":"ref_125","unstructured":"Baxter, R.J. (1982). Exactly Solved Models in Statistical Mechanics, Academic Press."},{"key":"ref_126","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/BF01087245","article-title":"Spectrum and scattering of excitations in the one-dimensional isotropic Heisenberg model","volume":"24","author":"Faddeev","year":"1984","journal-title":"J. Math. Sci."},{"key":"ref_127","unstructured":"Korepin, V.E., Bogoliubov, N.M., and Izergin, A.G. (1996). Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press."},{"key":"ref_128","doi-asserted-by":"crossref","first-page":"1384","DOI":"10.1063\/1.1705048","article-title":"Anisotropic Linear Magnetic Chain","volume":"7","author":"Cloizeaux","year":"1966","journal-title":"J. Math. Phys."},{"key":"ref_129","doi-asserted-by":"crossref","first-page":"1115","DOI":"10.1063\/1.1664947","article-title":"Thermodynamics of a one-dimensional system of bosons with repulsive delta-function interaction","volume":"10","author":"Yang","year":"1969","journal-title":"J. Math. Phys."},{"key":"ref_130","doi-asserted-by":"crossref","first-page":"3908","DOI":"10.1103\/PhysRevB.12.3908","article-title":"Calculation of critical exponents in two dimensions from quantum field theory in one dimension","volume":"12","author":"Luther","year":"1975","journal-title":"Phys. Rev. B"},{"key":"ref_131","unstructured":"Essler, F.H.L., and Konik, R.M. (2005). From Fields to Strings: Circumnavigating Theoretical Physics, World Scientific."},{"key":"ref_132","doi-asserted-by":"crossref","first-page":"6241","DOI":"10.1103\/PhysRevB.58.6241","article-title":"Magnetization plateaux in N-leg spin ladders","volume":"58","author":"Cabra","year":"1998","journal-title":"Phys. Rev. B"},{"key":"ref_133","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_134","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1007\/BF01218021","article-title":"Valence bond ground states in isotropic quantum antiferromagnets","volume":"115","author":"Affleck","year":"1988","journal-title":"Commun. Math. Phys."},{"key":"ref_135","doi-asserted-by":"crossref","first-page":"3337","DOI":"10.1103\/PhysRevB.43.3337","article-title":"Spontaneous dimerization in quantum-spin chains","volume":"43","author":"Chubukov","year":"1991","journal-title":"Phys. Rev. B"},{"key":"ref_136","doi-asserted-by":"crossref","first-page":"3620","DOI":"10.1103\/PhysRevB.51.3620","article-title":"Search for the nondimerized quantum nematic phase in the spin-1 chain","volume":"51","year":"1995","journal-title":"Phys. Rev. B"},{"key":"ref_137","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1143\/PTPS.145.138","article-title":"Quantum monte carlo methods","volume":"145","author":"Kawashima","year":"2002","journal-title":"Prog. Theor. Phys. Suppl."},{"key":"ref_138","doi-asserted-by":"crossref","first-page":"052401","DOI":"10.1103\/PhysRevB.68.052401","article-title":"Effective field theory for the S = 1 quantum nematic","volume":"68","author":"Ivanov","year":"2003","journal-title":"Phys. Rev. B"},{"key":"ref_139","doi-asserted-by":"crossref","first-page":"054433","DOI":"10.1103\/PhysRevB.72.054433","article-title":"Probable absence of a quadrupolar spin-nematic phase in the bilinear-biquadratic spin-1 chain","volume":"72","author":"Buchta","year":"2005","journal-title":"Phys. Rev. B"},{"key":"ref_140","doi-asserted-by":"crossref","first-page":"240404","DOI":"10.1103\/PhysRevLett.95.240404","article-title":"Phase diagram of spin-1 bosons on one-dimensional lattices","volume":"95","author":"Rizzi","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_141","doi-asserted-by":"crossref","first-page":"144426","DOI":"10.1103\/PhysRevB.74.144426","article-title":"Spin nematics correlations in bilinear-biquadratic S = 1 spin chains","volume":"74","author":"Schmid","year":"2006","journal-title":"Phys. Rev. B"},{"key":"ref_142","doi-asserted-by":"crossref","first-page":"014410","DOI":"10.1103\/PhysRevB.73.014410","article-title":"Renormalization algorithm for the calculation of spectra of interacting quantum systems","volume":"73","author":"Porras","year":"2006","journal-title":"Phys. Rev. B"},{"key":"ref_143","doi-asserted-by":"crossref","first-page":"050303(R)","DOI":"10.1103\/PhysRevA.75.050303","article-title":"Quantum state transfer in spin-1 chains","volume":"75","author":"Eckert","year":"2007","journal-title":"Phys. Rev. A"},{"key":"ref_144","doi-asserted-by":"crossref","first-page":"015007","DOI":"10.1088\/2399-6528\/aa7470","article-title":"Bilinear-biquadratic spin-1 rings: An SU(2)-symmetric MPS algorithm for periodic boundary conditions","volume":"1","author":"Rakov","year":"2017","journal-title":"J. Phys. Commun."},{"key":"ref_145","doi-asserted-by":"crossref","first-page":"104431","DOI":"10.1103\/PhysRevB.81.104431","article-title":"Infinite matrix product states, conformal field theory, and the Haldane-Shastry model","volume":"81","author":"Cirac","year":"2010","journal-title":"Phys. Rev. B"},{"key":"ref_146","doi-asserted-by":"crossref","first-page":"P11014","DOI":"10.1088\/1742-5468\/2011\/11\/P11014","article-title":"Quantum spin Hamiltonians for the SU(2)k WZW model","volume":"2011","author":"Nielsen","year":"2011","journal-title":"J. Stat. Mech."},{"key":"ref_147","doi-asserted-by":"crossref","first-page":"11836","DOI":"10.1103\/PhysRevB.44.11836","article-title":"Period tripling in the bilinear-biquadratic antiferromagnetic S = 1 chain","volume":"44","year":"1991","journal-title":"Phys. Rev. B"},{"key":"ref_148","doi-asserted-by":"crossref","first-page":"3795","DOI":"10.1103\/PhysRevB.12.3795","article-title":"Model for a multicomponent quantum system","volume":"12","author":"Sutherland","year":"1975","journal-title":"Phys. Rev. B"},{"key":"ref_149","doi-asserted-by":"crossref","first-page":"4621","DOI":"10.1103\/PhysRevB.40.4621","article-title":"Spectrum of the biquadratic spin-1 antiferromagnetic chain","volume":"40","author":"Barber","year":"1989","journal-title":"Phys. Rev. B"},{"key":"ref_150","doi-asserted-by":"crossref","first-page":"180402(R)","DOI":"10.1103\/PhysRevB.65.180402","article-title":"Unveiling order behind complexity: Coexistence of ferromagnetism and Bose-Einstein condensation","volume":"65","author":"Batista","year":"2002","journal-title":"Phys. Rev. B"},{"key":"ref_151","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1088\/0953-8984\/2\/2\/016","article-title":"Exact results on the dimerisation transition in SU(n) antiferromagnetic chains","volume":"2","author":"Affleck","year":"1990","journal-title":"Phys. Condens. Matter"},{"key":"ref_152","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1016\/0375-9601(82)90764-2","article-title":"The picture of low-lying excitations in the isotropic Heisenberg chain of arbitrary spins","volume":"87","author":"Takhtajan","year":"1982","journal-title":"Phys. Lett. A"},{"key":"ref_153","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1016\/0550-3213(83)90668-5","article-title":"Exact solution of the isotropic Heisenberg chain with arbitrary spins: Thermodynamics of the model","volume":"215","author":"Babujian","year":"1983","journal-title":"Nucl. Phys. B"},{"key":"ref_154","doi-asserted-by":"crossref","first-page":"L15","DOI":"10.1088\/0305-4470\/23\/1\/004","article-title":"Spin-s quantum chains and Temperley-Lieb algebras","volume":"23","author":"Batchelor","year":"1990","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_155","doi-asserted-by":"crossref","first-page":"P05018","DOI":"10.1088\/1742-5468\/2010\/05\/P05018","article-title":"Quantum spin chains of Temperley-Lieb type: Periodic boundary conditions, spectral multiplicities and finite temperature","volume":"2010","author":"Aufgebauer","year":"2010","journal-title":"J. Stat. Mech."},{"key":"ref_156","doi-asserted-by":"crossref","first-page":"081112","DOI":"10.1103\/PhysRevB.94.081112","article-title":"Universal entanglement spectra in critical spin chains","volume":"94","author":"Lundgren","year":"2016","journal-title":"Phys. Rev. B"},{"key":"ref_157","doi-asserted-by":"crossref","first-page":"P07017","DOI":"10.1088\/1742-5468\/2015\/07\/P07017","article-title":"Entanglement analysis of isotropic spin-1 chains","volume":"2015","author":"Thomale","year":"2015","journal-title":"J. Stat. Mech."},{"key":"ref_158","unstructured":"Dai, Y.-W., Shi, Q.-Q., Zhou, H.-Q., and McCulloch, I. (2022). Absence of a critical nematic phase in the vicinity of the SU(3) ferromagnetic point for the one-dimensional spin-1 bilinear-biquadratic model. arXiv."},{"key":"ref_159","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":"Rommelse","year":"1989","journal-title":"Phys. Rev. B"},{"key":"ref_160","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1103\/PhysRevB.45.304","article-title":"Hidden Z2 \u00d7 Z2 symmetry breaking in Haldane-gap antiferromagnets","volume":"45","author":"Kennedy","year":"1992","journal-title":"Phys. Rev. B"},{"key":"ref_161","doi-asserted-by":"crossref","first-page":"7469","DOI":"10.1088\/0953-8984\/4\/36\/019","article-title":"Hidden Z2 \u00d7 Z2 symmetry in quantum spin chains with arbitrary integer spin","volume":"4","author":"Oshikawa","year":"1992","journal-title":"J. Phys. Condens. Matter"},{"key":"ref_162","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1140\/epjb\/e2003-00299-7","article-title":"On c = 1 critical phases in anisotropic spin-1 chains","volume":"35","author":"Boschi","year":"2003","journal-title":"Eur. Phys. J. B"},{"key":"ref_163","doi-asserted-by":"crossref","first-page":"224402","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_164","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1038\/nphys287","article-title":"A toolbox for lattice-spin models with polar molecules","volume":"2","author":"Micheli","year":"2006","journal-title":"Nat. Phys."},{"key":"ref_165","doi-asserted-by":"crossref","first-page":"087204","DOI":"10.1103\/PhysRevLett.98.087204","article-title":"Topological characterization of quantum phase transitions in a spin-1\/2 model","volume":"98","author":"Feng","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_166","doi-asserted-by":"crossref","first-page":"022319","DOI":"10.1103\/PhysRevA.82.022319","article-title":"Quantum-Information approach to the quantum phase transition in the Kitaev honeycomb model","volume":"82","author":"Cui","year":"2010","journal-title":"Phys. Rev. A"},{"key":"ref_167","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1007\/BF01645742","article-title":"The four laws of black hole mechanics","volume":"31","author":"Bardeen","year":"1973","journal-title":"Commun. Math. Phys."},{"key":"ref_168","first-page":"125","article-title":"Cosmology today","volume":"143","author":"Spergel","year":"2014","journal-title":"Daedalus Am. Acad. Arts Sci."},{"key":"ref_169","doi-asserted-by":"crossref","first-page":"150","DOI":"10.3390\/entropy-e10030150","article-title":"Information equation of state","volume":"10","author":"Gough","year":"2008","journal-title":"Entropy"},{"key":"ref_170","doi-asserted-by":"crossref","first-page":"924","DOI":"10.3390\/e13040924","article-title":"Holographic dark information energy","volume":"13","author":"Gough","year":"2011","journal-title":"Entropy"},{"key":"ref_171","doi-asserted-by":"crossref","first-page":"1135","DOI":"10.3390\/e15031135","article-title":"Holographic dark information energy: Predicted dark energy measurement","volume":"15","author":"Gough","year":"2013","journal-title":"Entropy"},{"key":"ref_172","doi-asserted-by":"crossref","first-page":"1902","DOI":"10.3390\/e16041902","article-title":"A dynamic dark information energy consistent with planck data","volume":"16","author":"Gough","year":"2014","journal-title":"Entropy"},{"key":"ref_173","unstructured":"Gogolin, A.O., Nersesyan, A.A., and Tsvelik, A.M. (1998). Bosonization and Strongly Correlated Systems, Cambridge University Press."},{"key":"ref_174","doi-asserted-by":"crossref","unstructured":"Rao, S. (2001). An introduction to bosonization and some of its applications. Field Theories in Condensed Matter Systems, Hindustan Book Agency.","DOI":"10.1007\/978-93-86279-07-1_6"},{"key":"ref_175","unstructured":"Shi, Q.-Q., Dai, Y.-W., Li, S.-H., and Zhou, H.-Q. (2022). Instability of the Luttinger liquids towards an exotic quantum state of matter with highly degenerate ground states: An anisotropic extension of the ferromagnetic spin-1 biquadratic model. arXiv."},{"key":"ref_176","doi-asserted-by":"crossref","first-page":"021601","DOI":"10.1103\/PhysRevLett.111.021601","article-title":"Massive Nambu-Goldstone bosons","volume":"111","author":"Watanabe","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_177","doi-asserted-by":"crossref","first-page":"056006","DOI":"10.1103\/PhysRevD.91.056006","article-title":"Dispersion relations of Nambu-Goldstone modes at finite temperature and density","volume":"91","author":"Hayata","year":"2015","journal-title":"Phys. Rev. D"},{"key":"ref_178","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/j.aop.2014.12.009","article-title":"Counting rule of Nambu-Goldstone modes for internal and spacetime symmetries: Bogoliubov theory approach","volume":"354","author":"Takahashi","year":"2015","journal-title":"Ann. Phys."},{"key":"ref_179","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1023\/B:JOSS.0000019827.74407.2d","article-title":"Spontaneous breaking of Lie and current algebras","volume":"115","author":"Nambu","year":"2004","journal-title":"J. Stat. Phys."},{"key":"ref_180","doi-asserted-by":"crossref","first-page":"125013","DOI":"10.1103\/PhysRevD.84.125013","article-title":"Number of Nambu-Goldstone bosons and its relation to charge densities","volume":"84","author":"Watanabe","year":"2011","journal-title":"Phys. Rev. D"},{"key":"ref_181","first-page":"031057","article-title":"Effective Lagrangian for Nonrelativistic Systems","volume":"4","author":"Watanabe","year":"2014","journal-title":"Phys. Rev. X"},{"key":"ref_182","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/S0370-2693(01)01265-5","article-title":"Kaon condensation and Goldstone\u2019s theorem","volume":"522","author":"Schafer","year":"2001","journal-title":"Phys. Lett. B"},{"key":"ref_183","doi-asserted-by":"crossref","first-page":"111601","DOI":"10.1103\/PhysRevLett.88.111601","article-title":"Spontaneous symmetry breaking with abnormal number of Nambu-Goldstone bosons and kaon condensate","volume":"88","author":"Miransky","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_184","doi-asserted-by":"crossref","first-page":"011602","DOI":"10.1103\/PhysRevLett.110.011602","article-title":"Implications of relativity on nonrelativistic Goldstone theorems: Gapped excitations at finite charge density","volume":"110","author":"Nicolis","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_185","first-page":"1","article-title":"An introduction to spontaneous symmetry breaking","volume":"11","author":"Beekman","year":"2019","journal-title":"SciPost Phys. Lect. Notes"},{"key":"ref_186","first-page":"P02001","article-title":"Permutation operators, entanglement entropy, and the XXZ spin chain in the limit \u0394 \u2192 \u22121+","volume":"1102","author":"Doyon","year":"2011","journal-title":"J. Stat. Mech."},{"key":"ref_187","doi-asserted-by":"crossref","first-page":"120401","DOI":"10.1103\/PhysRevLett.108.120401","article-title":"Entanglement entropy of highly degenerate states and fractal dimensions","volume":"108","author":"Doyon","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_188","doi-asserted-by":"crossref","first-page":"012301","DOI":"10.1103\/PhysRevA.71.012301","article-title":"Logarithmic divergence of the block entanglement entropy for the ferromagnetic Heisenberg model","volume":"71","author":"Popkov","year":"2005","journal-title":"Phys. Rev. A"},{"key":"ref_189","doi-asserted-by":"crossref","first-page":"032327","DOI":"10.1103\/PhysRevA.72.032327","article-title":"Entangling power of permutation-invariant quantum states","volume":"72","author":"Popkov","year":"2005","journal-title":"Phys. Rev. A"},{"key":"ref_190","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1098\/rspa.1971.0067","article-title":"Relations between \u2018percolation\u2019 and \u2018colouring\u2019 problems and other graph theoretical problems associated with regular planar lattices: Some exact results for the percolation problem","volume":"322","author":"Temperley","year":"1971","journal-title":"Proc. R. Soc. Lond. A"},{"key":"ref_191","doi-asserted-by":"crossref","unstructured":"Martin, P. (1991). Potts Models and Related Problems in Statistical Mechanics, World Scientific.","DOI":"10.1142\/0983"},{"key":"ref_192","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF01389127","article-title":"Index for subfactors","volume":"72","author":"Jones","year":"1983","journal-title":"Invent. Math."},{"key":"ref_193","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1016\/0550-3213(87)90332-4","article-title":"Two-dimensional critical systems labelled by Dynkin diagrams","volume":"285","author":"Pasquier","year":"1987","journal-title":"Nucl. Phys. B"},{"key":"ref_194","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1007\/BF01404457","article-title":"Hecke algebras of type An and subfactors","volume":"92","author":"Wenzl","year":"1988","journal-title":"Invent. Math."},{"key":"ref_195","doi-asserted-by":"crossref","first-page":"3285","DOI":"10.1143\/JPSJ.55.3285","article-title":"Virasoro Algebra, von Neumann algebra and critical eight-vertex SOS models","volume":"55","author":"Kuniba","year":"1986","journal-title":"J. Phys. Soc. Jpn."},{"key":"ref_196","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/BF01014383","article-title":"Eight-Vertex SOS model and generalized Rogers-Ramanujan-type identities","volume":"35","author":"Andrews","year":"1984","journal-title":"J. Stat. Phys."},{"key":"ref_197","doi-asserted-by":"crossref","first-page":"L489","DOI":"10.1088\/0305-4470\/36\/38\/101","article-title":"On spin systems related to the Temperley-Lieb algebra","volume":"36","author":"Kulish","year":"2003","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_198","doi-asserted-by":"crossref","first-page":"200401","DOI":"10.1103\/PhysRevLett.115.200401","article-title":"Tensor network renormalization yields the multiscale entanglement renormalization ansatz","volume":"115","author":"Evenbly","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_199","doi-asserted-by":"crossref","first-page":"180405","DOI":"10.1103\/PhysRevLett.115.180405","article-title":"Tensor network renormalization","volume":"115","author":"Evenbly","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_200","doi-asserted-by":"crossref","first-page":"931","DOI":"10.1007\/s10955-014-0983-1","article-title":"Algorithms for entanglement renormalization: Boundaries, impurities and interfaces","volume":"157","author":"Evenbly","year":"2014","journal-title":"J. Stat. Phys."},{"key":"ref_201","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1007\/BF01331938","article-title":"\u00dcber das Paulische \u00c4quivalenzverbot","volume":"47","author":"Jordan","year":"1928","journal-title":"Z. Phys."},{"key":"ref_202","first-page":"23","article-title":"On the theory of superfluidity","volume":"11","author":"Bogoliubov","year":"1947","journal-title":"J. Phys."},{"key":"ref_203","doi-asserted-by":"crossref","first-page":"193101","DOI":"10.1103\/PhysRevB.76.193101","article-title":"Exact mapping between classical and topological orders in two-dimensional spin systems","volume":"76","author":"Chen","year":"2007","journal-title":"Phys. Rev. B"},{"key":"ref_204","unstructured":"Feng, X.-Y., Zhang, G.-M., and Xiang, T. (2006). Topological characterization of quantum phase transitions in a S = 1\/2 spin model. arXiv."},{"key":"ref_205","doi-asserted-by":"crossref","first-page":"042307","DOI":"10.1103\/PhysRevA.68.042307","article-title":"Geometric measure of entanglement and applications to bipartite and multipartite quantum states","volume":"68","author":"Wei","year":"2003","journal-title":"Phys. Rev. A"},{"key":"ref_206","doi-asserted-by":"crossref","first-page":"155301","DOI":"10.1088\/1751-8121\/ab78cd","article-title":"Duality and ground-state phase diagram for the quantum XYZ model with arbitrary spin-s in one spatial dimension","volume":"53","author":"Shi","year":"2020","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_207","doi-asserted-by":"crossref","first-page":"2489","DOI":"10.1103\/PhysRevD.32.2489","article-title":"Arrow of time in cosmology","volume":"32","author":"Hawking","year":"1985","journal-title":"Phys. Rev. D"},{"key":"ref_208","doi-asserted-by":"crossref","first-page":"743","DOI":"10.1007\/BF00670829","article-title":"Memory systems, computation, and the second law of thermodynamics","volume":"31","author":"Wolpert","year":"1992","journal-title":"Int. J. Theor. Phys."},{"key":"ref_209","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1119\/1.1783900","article-title":"The physics of now","volume":"73","author":"Hartle","year":"2005","journal-title":"Am. J. Phys."},{"key":"ref_210","doi-asserted-by":"crossref","first-page":"052102","DOI":"10.1103\/PhysRevE.89.052102","article-title":"Relation between the psychological and thermodynamic arrows of time","volume":"89","author":"Mlodinow","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_211","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1007\/s10701-007-9202-0","article-title":"The arrow of time: From universe time-asymmetry to local irreversible processes","volume":"38","author":"Aiello","year":"2008","journal-title":"Found. Phys."},{"key":"ref_212","unstructured":"Vaas, R. (2004). Time before time\u2013classifications of universes in contemporary cosmology, and how to avoid the antinomy of the beginning and eternity of the world. arXiv."},{"key":"ref_213","doi-asserted-by":"crossref","first-page":"802","DOI":"10.1038\/299802a0","article-title":"A single quantum cannot be cloned","volume":"299","author":"Wootters","year":"1982","journal-title":"Nature"},{"key":"ref_214","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/0375-9601(82)90084-6","article-title":"Communication by EPR devices","volume":"92","author":"Dieks","year":"1982","journal-title":"Phys. Lett. A"},{"key":"ref_215","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1038\/404130b0","article-title":"Impossibility of deleting an unknown quantum state","volume":"404","author":"Pati","year":"2000","journal-title":"Nature"},{"key":"ref_216","doi-asserted-by":"crossref","first-page":"080502","DOI":"10.1103\/PhysRevLett.98.080502","article-title":"Quantum information cannot be completely hidden in correlations: Implications for the black-hole information paradox","volume":"98","author":"Braunstein","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_217","doi-asserted-by":"crossref","first-page":"080401","DOI":"10.1103\/PhysRevLett.106.080401","article-title":"Experimental test of the quantum no-hiding theorem","volume":"106","author":"Samal","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_218","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1063\/1.3086114","article-title":"The no-cloning theorem","volume":"2","author":"Wootters","year":"2009","journal-title":"Phys. Today"},{"key":"ref_219","doi-asserted-by":"crossref","first-page":"227205","DOI":"10.1103\/PhysRevLett.93.227205","article-title":"Density matrix renormalization group and periodic boundary conditions: A quantum information perspective","volume":"93","author":"Verstraete","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_220","doi-asserted-by":"crossref","first-page":"125104","DOI":"10.1103\/PhysRevB.83.125104","article-title":"Exploiting translational invariance in matrix product state simulations of spin chains with periodic boundary conditions","volume":"83","author":"Pirvu","year":"2011","journal-title":"Phys. Rev. B"},{"key":"ref_221","unstructured":"Perez-Garcia, D., Verstraete, F., Wolf, M.M., and Cirac, J.I. (2006). Matrix product state representations. arXiv."},{"key":"ref_222","doi-asserted-by":"crossref","first-page":"045003","DOI":"10.1103\/RevModPhys.93.045003","article-title":"Matrix product states and projected entangled pair states: Concepts, symmetries, theorems","volume":"93","author":"Cirac","year":"2021","journal-title":"Rev. Mod. Phys."},{"key":"ref_223","doi-asserted-by":"crossref","first-page":"245101","DOI":"10.1103\/PhysRevB.86.245101","article-title":"Projected entangled pair states at finite temperature: Imaginary time evolution with ancillas","volume":"86","author":"Czarnik","year":"2012","journal-title":"Phys. Rev. B"},{"key":"ref_224","doi-asserted-by":"crossref","first-page":"035120","DOI":"10.1103\/PhysRevB.92.035120","article-title":"Projected entangled pair states at finite temperature: Iterative self-consistent bond renormalization for exact imaginary time evolution","volume":"92","author":"Czarnik","year":"2015","journal-title":"Phys. Rev. B"},{"key":"ref_225","doi-asserted-by":"crossref","first-page":"214409","DOI":"10.1103\/PhysRevB.95.214409","article-title":"Finite-temperature fidelity and von Neumann entropy in the honeycomb spin lattice with quantum Ising interaction","volume":"95","author":"Dai","year":"2017","journal-title":"Phys. Rev. B"},{"key":"ref_226","doi-asserted-by":"crossref","first-page":"032324","DOI":"10.1103\/PhysRevA.84.032324","article-title":"Scaling of ground-state fidelity in the thermodynamic limit: XY model and beyond","volume":"84","author":"Rams","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_227","doi-asserted-by":"crossref","first-page":"214302","DOI":"10.1103\/PhysRevB.83.214302","article-title":"Fidelity susceptibility and general quench near an anisotropic quantum critical point","volume":"83","author":"Mukherjee","year":"2011","journal-title":"Phys. Rev. B"},{"key":"ref_228","first-page":"381","article-title":"Conformal symmetry of critical fluctuation","volume":"12","author":"Polyakov","year":"1970","journal-title":"JETP Lett."},{"key":"ref_229","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_230","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1103\/PhysRevB.27.613","article-title":"Ground-state properties of a spin-1 antiferromagnetic chain","volume":"27","author":"Botet","year":"1983","journal-title":"Phys. Rev. B"},{"key":"ref_231","doi-asserted-by":"crossref","first-page":"3304","DOI":"10.1103\/PhysRevB.53.3304","article-title":"Onset of incommensurability at the valence-bond-solid point in the S = 1 quantum spin chain","volume":"53","author":"Jolicoeur","year":"1996","journal-title":"Phys. Rev. B"},{"key":"ref_232","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1016\/0550-3213(86)90167-7","article-title":"Exact critical exponents for quantum spin chains, non-linear \u03c3-models at \u03b8 = \u03c0 and the quantum hall effect","volume":"265","author":"Affleck","year":"1986","journal-title":"Nucl. Phys. B"},{"key":"ref_233","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"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/9\/1306\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:32:11Z","timestamp":1760142731000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/9\/1306"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,15]]},"references-count":233,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2022,9]]}},"alternative-id":["e24091306"],"URL":"https:\/\/doi.org\/10.3390\/e24091306","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,15]]}}}