{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T05:09:52Z","timestamp":1778735392023,"version":"3.51.4"},"reference-count":79,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2026,4,26]],"date-time":"2026-04-26T00:00:00Z","timestamp":1777161600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The discovery of dynamical quantum phase transitions (DQPTs) has fundamentally challenged the traditional view that phase transitions only occur in thermal equilibrium. Experimental platforms and 1D numerical methods, like matrix product states (MPS), have made great progress. However, exploring true 2D DQPTs remains difficult due to finite-size limitations and the geometric biases of quasi-1D cylinder mappings. Here, we bypass these limitations by deploying a tree tensor network (TTN) approach. This allows us to directly compute the quench dynamics of the transverse-field Ising model (TFIM) on an open 2D square lattice. Because the TTN architecture naturally mirrors 2D lattice connectivity, we can extract the global Loschmidt echo. Our simulations reveal that while deep quenches yield standard DQPTs, quenching within the ferromagnetic phase produces an anomalous dynamical response. In this regime, the rate function exhibits sharp non-analytic peaks even as the macroscopic order parameter maintains its initial sign. This decoupled behavior strongly indicates that local spin excitations drive 2D DQPTs, rather than the macroscopic domain-wall motions seen in 1D chains. These results provide a quantitative numerical baseline for understanding non-equilibrium quantum matter in higher dimensions.<\/jats:p>","DOI":"10.3390\/e28050495","type":"journal-article","created":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:43:51Z","timestamp":1777448631000},"page":"495","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Tree Tensor Network Simulation of Dynamical Quantum Phase Transitions in the 2D Transverse-Field Ising Model"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-4508-2743","authenticated-orcid":false,"given":"Xiangyue","family":"Zhang","sequence":"first","affiliation":[{"name":"College of Science, National University of Defense Technology, Changsha 410073, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dizhou","family":"Xie","sequence":"additional","affiliation":[{"name":"College of Science, National University of Defense Technology, Changsha 410073, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1244-965X","authenticated-orcid":false,"given":"Yongqiang","family":"Li","sequence":"additional","affiliation":[{"name":"College of Science, National University of Defense Technology, Changsha 410073, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,4,26]]},"reference":[{"key":"ref_1","unstructured":"Karch, S., Bandyopadhyay, S., Sun, Z.H., Impertro, A., Huh, S., Rodr\u00edguez, I.P., Wienand, J.F., Ketterle, W., Heyl, M., and Polkovnikov, A. (2025). Probing quantum many-body dynamics using subsystem Loschmidt echos. arXiv."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"030501","DOI":"10.1103\/RevModPhys.95.030501","article-title":"Nobel Lecture: Multiple equilibria","volume":"95","author":"Parisi","year":"2023","journal-title":"Rev. Mod. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Sachdev, S. (2011). Quantum Phase Transitions, Cambridge University Press.","DOI":"10.1017\/CBO9780511973765"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Cardy, J. (1996). Scaling and Renormalization in Statistical Physics, Cambridge University Press.","DOI":"10.1017\/CBO9781316036440"},{"key":"ref_5","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_6","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_7","doi-asserted-by":"crossref","first-page":"130601","DOI":"10.1103\/PhysRevLett.120.130601","article-title":"Dynamical quantum phase transitions in spin chains with long-range interactions","volume":"120","author":"Heyl","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"014302","DOI":"10.1103\/PhysRevB.96.014302","article-title":"Emergent topology and dynamical quantum phase transitions in two-dimensional closed quantum systems","volume":"96","author":"Bhattacharya","year":"2017","journal-title":"Phys. Rev. B"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"134313","DOI":"10.1103\/PhysRevB.96.134313","article-title":"Dynamical quantum phase transitions in systems with continuous symmetry breaking","volume":"96","author":"Weidinger","year":"2017","journal-title":"Phys. Rev. B"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"134427","DOI":"10.1103\/PhysRevB.96.134427","article-title":"Dynamical phase diagram of quantum spin chains with long-range interactions","volume":"96","author":"Halimeh","year":"2017","journal-title":"Phys. Rev. B"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"033111","DOI":"10.1103\/PhysRevResearch.2.033111","article-title":"Quasiparticle origin of dynamical quantum phase transitions","volume":"2","author":"Halimeh","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"ref_12","unstructured":"Cao, K., Li, M., Jiang, X.P., Chen, S., and Wang, J. (2026, March 19). Maximal Entanglement and Frozen Information: A Unified Framework for Dynamical Quantum Phase Transitions. Preprint. Available online: https:\/\/arxiv.org\/abs\/2601.04535."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"104309","DOI":"10.1103\/cfs4-ccs6","article-title":"Dynamical quantum phase transitions through the lens of mode dynamics","volume":"113","author":"Mitra","year":"2026","journal-title":"Phys. Rev. B"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Zheng, Z.Y., Liu, X., Lin, S., Zhang, Y., and Chen, S. (2026, March 19). Loschmidt Echo Zeros in Finite-Size Quantum Systems with Linear Quench. Preprint. Available online: https:\/\/arxiv.org\/abs\/2504.00483.","DOI":"10.1103\/qc3w-s38g"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"200602","DOI":"10.1103\/PhysRevLett.126.200602","article-title":"Subsystem Loschmidt Echo as a Measure of Dynamical Quantum Phase Transitions","volume":"126","author":"Bandyopadhyay","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"075114","DOI":"10.1103\/PhysRevB.92.075114","article-title":"Dynamical quantum phase transitions in the Kitaev honeycomb model","volume":"92","author":"Schmitt","year":"2015","journal-title":"Phys. Rev. B"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"121107","DOI":"10.1103\/PhysRevB.99.121107","article-title":"Dynamical quantum phase transitions in collapse and revival oscillations of a quenched superfluid","volume":"99","author":"Heyl","year":"2019","journal-title":"Phys. Rev. B"},{"key":"ref_18","first-page":"043102","article-title":"Dynamical quantum phase transitions in the one-dimensional extended Fermi\u2013Hubbard model","volume":"2022","year":"2022","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"040602","DOI":"10.1103\/PhysRevLett.126.040602","article-title":"Entanglement View of Dynamical Quantum Phase Transitions","volume":"126","author":"Michailidis","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"033090","DOI":"10.1103\/PhysRevResearch.5.033090","article-title":"Anatomy of dynamical quantum phase transitions","volume":"5","author":"Desaules","year":"2023","journal-title":"Phys. Rev. Res."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"150601","DOI":"10.1103\/PhysRevLett.122.150601","article-title":"Confined Dynamics in Long-Range Interacting Quantum Spin Chains","volume":"122","author":"Liu","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"014434","DOI":"10.1103\/PhysRevB.100.014434","article-title":"Dynamical criticality and domain-wall coupling in long-range Hamiltonians","volume":"100","author":"Defenu","year":"2019","journal-title":"Phys. Rev. B"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1038\/nphys3934","article-title":"Real-time dynamics of confined topological excitations in a one-dimensional magnet","volume":"13","author":"Kormos","year":"2017","journal-title":"Nat. Phys."},{"key":"ref_24","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_25","unstructured":"Luo, L.Y., Li, W.L., Xu, B.M., and Li, Z. (2026, March 19). The Intrinsic Connection between Dynamical Phase Transitions and Magnetization in the 1D XY Model. Preprint. Available online: https:\/\/arxiv.org\/abs\/2602.01259."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"023194","DOI":"10.1103\/PhysRevResearch.7.023194","article-title":"Dynamical quantum phase transition and thermal equilibrium in the lattice Thirring model","volume":"7","author":"Cichy","year":"2025","journal-title":"Phys. Rev. Res."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1038\/nature13461","article-title":"Quasiparticle engineering and entanglement propagation in a quantum many-body system","volume":"511","author":"Jurcevic","year":"2014","journal-title":"Nature"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"080501","DOI":"10.1103\/PhysRevLett.119.080501","article-title":"Direct observation of dynamical quantum phase transitions in an interacting many-body system","volume":"119","author":"Jurcevic","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"885","DOI":"10.1103\/RevModPhys.80.885","article-title":"Many-body physics with ultracold gases","volume":"80","author":"Bloch","year":"2008","journal-title":"Rev. Mod. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1038\/s41567-017-0013-8","article-title":"Observation of dynamical vortices after quenches in a system with topology","volume":"14","author":"Vogel","year":"2018","journal-title":"Nat. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1038\/nature24654","article-title":"Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator","volume":"551","author":"Zhang","year":"2017","journal-title":"Nature"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"044080","DOI":"10.1103\/PhysRevApplied.11.044080","article-title":"Observation of a dynamical quantum phase transition by a superconducting qubit simulation","volume":"11","author":"Guo","year":"2019","journal-title":"Phys. Rev. Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"eaba4935","DOI":"10.1126\/sciadv.aba4935","article-title":"Probing dynamical phase transitions with a superconducting quantum simulator","volume":"6","author":"Xu","year":"2020","journal-title":"Sci. Adv."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"167998","DOI":"10.1016\/j.aop.2019.167998","article-title":"Time-evolution methods for tensor-network states","volume":"411","author":"Paeckel","year":"2019","journal-title":"Ann. Phys."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"100402","DOI":"10.1103\/PhysRevLett.130.100402","article-title":"Theory of dynamical phase transitions in quantum systems with symmetry-breaking eigenstates","volume":"130","author":"Corps","year":"2023","journal-title":"Phys. Rev. Lett."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"013250","DOI":"10.1103\/PhysRevResearch.4.013250","article-title":"Dynamical phase transitions in the two-dimensional transverse-field Ising model","volume":"4","author":"Hashizume","year":"2022","journal-title":"Phys. Rev. Res."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"016801","DOI":"10.1103\/PhysRevLett.121.016801","article-title":"Detecting equilibrium and dynamical quantum phase transitions in Ising chains via out-of-time-ordered correlators","volume":"121","author":"Heyl","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_38","first-page":"05LT02","article-title":"Stochastic approach to dynamical quantum phase transitions","volume":"52","author":"Doyon","year":"2019","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"022320","DOI":"10.1103\/PhysRevA.74.022320","article-title":"Classical simulation of quantum many-body systems with a tree tensor network","volume":"74","author":"Shi","year":"2006","journal-title":"Phys. Rev. A"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"024","DOI":"10.21468\/SciPostPhys.8.2.024","article-title":"Time dependent variational principle for tree tensor networks","volume":"8","author":"Bauernfeind","year":"2020","journal-title":"SciPost Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"070","DOI":"10.21468\/SciPostPhys.9.5.070","article-title":"Studying dynamics in two-dimensional quantum lattices using tree tensor network states","volume":"9","author":"Kloss","year":"2020","journal-title":"SciPost Phys."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Silvi, P., Tschirsich, F., Gerster, M., J\u00fcnemann, J., Jaschke, D., Rizzi, M., and Montangero, S. (2019). The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems. SciPost Phys. Lect. Notes, 8.","DOI":"10.21468\/SciPostPhysLectNotes.8"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Krinitsin, W., Tausendpfund, N., Heyl, M., Rizzi, M., and Schmitt, M. (2026, March 19). Time Evolution of the Quantum Ising Model in Two Dimensions Using Tree Tensor Networks. Preprint. Available online: https:\/\/arxiv.org\/abs\/2505.07612.","DOI":"10.1103\/kj16-sqcm"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"174310","DOI":"10.1103\/7jzt-xhn6","article-title":"Simulating quantum dynamics in two-dimensional lattices with tensor network influence functional belief propagation","volume":"112","author":"Park","year":"2025","journal-title":"Phys. Rev. B"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"165117","DOI":"10.1103\/56px-qsxx","article-title":"Diagonal isometric form for tensor network states in two dimensions","volume":"113","author":"Sappler","year":"2026","journal-title":"Phys. Rev. B"},{"key":"ref_46","unstructured":"Vovrosh, J., Juli\u00e0-Farr\u00e9, S., Krinitsin, W., Kaicher, M., Hayes, F., Gottlob, E., Kshetrimayum, A., Bidzhiev, K., J\u00e4ger, S.B., and Schmitt, M. (2026, March 19). Simulating Dynamics of the Two-Dimensional Transverse-Field Ising Model: A Comparative Study of Large-Scale Classical Numerics. Preprint. Available online: https:\/\/arxiv.org\/abs\/2511.19340."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"020302","DOI":"10.1103\/PRXQuantum.6.020302","article-title":"Real-time operator evolution in two and three dimensions via sparse Pauli dynamics","volume":"6","author":"Chan","year":"2025","journal-title":"PRX Quantum"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"140506","DOI":"10.1103\/PhysRevLett.98.140506","article-title":"Computational complexity of projected entangled pair states","volume":"98","author":"Schuch","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"8494","DOI":"10.1103\/PhysRevB.57.8494","article-title":"Critical behavior of the two-dimensional Ising model in a transverse field: A density-matrix renormalization calculation","volume":"57","year":"1998","journal-title":"Phys. Rev. B"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"066110","DOI":"10.1103\/PhysRevE.66.066110","article-title":"Cluster Monte Carlo simulation of the transverse Ising model","volume":"66","author":"Deng","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"094101","DOI":"10.1103\/PhysRevB.102.094101","article-title":"Worm-algorithm-type simulation of the quantum transverse-field Ising model","volume":"102","author":"Huang","year":"2020","journal-title":"Phys. Rev. B"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"033024","DOI":"10.1103\/PhysRevResearch.3.033024","article-title":"Higher-form symmetry breaking at Ising transitions","volume":"3","author":"Zhao","year":"2021","journal-title":"Phys. Rev. Res."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1146\/annurev-conmatphys-020911-125018","article-title":"Studying two-dimensional systems with the density matrix renormalization group","volume":"3","author":"Stoudenmire","year":"2012","journal-title":"Annu. Rev. Condens. Matter Phys."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"235127","DOI":"10.1103\/PhysRevB.80.235127","article-title":"Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law","volume":"80","author":"Tagliacozzo","year":"2009","journal-title":"Phys. Rev. B"},{"key":"ref_55","unstructured":"Tausendpfund, N., Rizzi, M., Krinitsin, W., and Schmitt, M. (2026, March 19). TTN.jl\u2014A Tree Tensor Network Library for Calculating Groundstates and Solving Time Evolution. Available online: https:\/\/zenodo.org\/records\/14421855."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"070601","DOI":"10.1103\/PhysRevLett.107.070601","article-title":"Time-dependent variational principle for quantum lattices","volume":"107","author":"Haegeman","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"165116","DOI":"10.1103\/PhysRevB.94.165116","article-title":"Unifying time evolution and optimization with matrix product states","volume":"94","author":"Haegeman","year":"2016","journal-title":"Phys. Rev. B"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"240402","DOI":"10.1103\/9bsk-x9rw","article-title":"Roughening Dynamics of Interfaces in the Two-Dimensional Quantum Ising Model","volume":"134","author":"Krinitsin","year":"2025","journal-title":"Phys. Rev. Lett."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"120601","DOI":"10.1103\/PhysRevLett.129.120601","article-title":"Localization and Melting of Interfaces in the Two-Dimensional Quantum Ising Model","volume":"129","author":"Balducci","year":"2022","journal-title":"Phys. Rev. Lett."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"024306","DOI":"10.1103\/PhysRevB.107.024306","article-title":"Interface dynamics in the two-dimensional quantum Ising model","volume":"107","author":"Balducci","year":"2023","journal-title":"Phys. Rev. B"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"5338","DOI":"10.1103\/PhysRevB.28.5338","article-title":"Roughening transition in quantum interfaces","volume":"28","author":"Fradkin","year":"1983","journal-title":"Phys. Rev. B"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1098\/rsta.1951.0006","article-title":"The growth of crystals and the equilibrium structure of their surfaces","volume":"243","author":"Burton","year":"1951","journal-title":"Philos. Trans. R. Soc. Lond. Ser. A"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1103\/PhysRevLett.31.549","article-title":"Structural transition in the Ising-model interface","volume":"31","author":"Weeks","year":"1973","journal-title":"Phys. Rev. Lett."},{"key":"ref_64","doi-asserted-by":"crossref","unstructured":"Pave\u0161i\u0107, L., Jaschke, D., and Montangero, S. (2024). Constrained dynamics and confinement in the two-dimensional quantum Ising model. arXiv.","DOI":"10.1103\/PhysRevB.111.L140305"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1103\/PhysRev.65.117","article-title":"Crystal statistics. I. A two-dimensional model with an order-disorder transition","volume":"65","author":"Onsager","year":"1944","journal-title":"Phys. Rev."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"205136","DOI":"10.1103\/PhysRevB.91.205136","article-title":"Exploring dynamical phase transitions and prethermalization with quantum noise of excitations","volume":"91","author":"Smacchia","year":"2015","journal-title":"Phys. Rev. B"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"054304","DOI":"10.1103\/mkll-nd46","article-title":"Scaling and universality at noisy quench dynamical quantum phase transitions","volume":"112","author":"Ansari","year":"2025","journal-title":"Phys. Rev. B"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1080\/00018732.2016.1198134","article-title":"From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics","volume":"65","author":"Kafri","year":"2016","journal-title":"Adv. Phys."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2009.05.002","article-title":"The large deviation approach to statistical mechanics","volume":"478","author":"Touchette","year":"2009","journal-title":"Phys. Rep."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"021001","DOI":"10.1103\/RevModPhys.91.021001","article-title":"Many-body localization, thermalization, and entanglement","volume":"91","author":"Abanin","year":"2019","journal-title":"Rev. Mod. Phys."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"086501","DOI":"10.1088\/1361-6633\/ac73a0","article-title":"Quantum many-body scars and Hilbert space fragmentation: A review of exact results","volume":"85","author":"Moudgalya","year":"2022","journal-title":"Rep. Prog. Phys."},{"key":"ref_72","unstructured":"Katschke, L., Farrell, R.C., Borla, U., Pollet, L., and Halimeh, J.C. (2026, March 19). Finite-Temperature Dynamical Phase Diagram of the 2 + 1D Quantum Ising Model. Preprint. Available online: https:\/\/arxiv.org\/abs\/2602.16772v1."},{"key":"ref_73","first-page":"013830","article-title":"Virtual photons in the ground state of a dissipative system","volume":"92","author":"Cirio","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_74","doi-asserted-by":"crossref","unstructured":"Carollo, A., Spagnolo, B., and Valenti, D. (2018). Symmetric Logarithmic Derivative of Fermionic Gaussian States. Entropy, 20.","DOI":"10.20944\/preprints201805.0288.v1"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"2341","DOI":"10.3390\/e17042341","article-title":"Multi-State Quantum Dissipative Dynamics in Sub-Ohmic Environment: The Strong Coupling Regime","volume":"17","author":"Valenti","year":"2015","journal-title":"Entropy"},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1140\/epjb\/e2009-00155-x","article-title":"Lifetime of the superconductive state in short and long Josephson junctions","volume":"70","author":"Augello","year":"2009","journal-title":"Eur. Phys. J. B"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"043332","DOI":"10.1103\/PhysRevResearch.2.043332","article-title":"Voltage drop across Josephson junctions for L\u00e9vy noise detection","volume":"2","author":"Guarcello","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"ref_78","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_79","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."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/5\/495\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T04:15:53Z","timestamp":1778732153000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/5\/495"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,26]]},"references-count":79,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2026,5]]}},"alternative-id":["e28050495"],"URL":"https:\/\/doi.org\/10.3390\/e28050495","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,26]]}}}