Exploring quantum coherence, spin squeezing and entanglement in an extended spin-1/2 XX chain. (English) Zbl 07898024
Summary: In this study, we explore the ground state phase diagram of the spin-1/2 XX chain model, which features \(XZY-YZX\)-type three-spin interactions (TSIs). This model, while seemingly simple, reveals a rich tapestry of quantum behaviors. Our analysis relies on several key metrics. The ‘\(l_1\)-norm of coherence’ helps us identify coherent states within the phase diagram, which represent states capable of superposition and interference. We employ the ‘spin squeezing parameter’ to pinpoint unique coherent states characterized by isotropic noise in all directions, making them invaluable for quantum metrology. Additionally, we utilize the ‘entanglement entropy’ to determine which of these coherent states exhibit entanglement, indicating states that cannot be fully described by local variables. Our research unveils diverse regions within the phase diagram, each characterized by coherent, squeezed, or entangled states, offering insights into the quantum phenomena underling these systems. We also study the critical scaling versus the system size for the mentioned quantities.
MSC:
| 81P68 | Quantum computation |
References:
| [1] | Zhang, W.M., Gilmore, R.: Rev. Mod. Phys. 62, 867 (1990) |
| [2] | Sanders, B.C.: J. Phys. A Math. Theor. 45, 244002 (2012) · Zbl 1248.81084 |
| [3] | Pezze, L.; Smerzi, A.; Oberthaler, MK; Schmied, R.; Treutlein, P., Quantum metrology with nonclassical states of atomic ensembles, Rev. Modern Phys., 90, 2018 · doi:10.1103/RevModPhys.90.035005 |
| [4] | Li, L.; Wang, QW; Shen, SQ; Li, M., Quantum coherence measures based on Fisher information with applications, Phys. Rev. A, 103, 2021 · doi:10.1103/PhysRevA.103.012401 |
| [5] | Baumgratz, T.; Crame, M.; Plenio, MB, Quantifying coherence, Phys. Rev. Lett., 113, 2014 · doi:10.1103/PhysRevLett.113.140401 |
| [6] | Baumgratz, T.; Cramer, M.; Plenio, MB, Quantum coherence as a resource, Rev. Mod. Phys., 89, 2017 · doi:10.1103/RevModPhys.89.041003 |
| [7] | Radcliffe, JM, Some properties of coherent spin states, J. Phys. A Gen. Phys., 4, 313, 1971 · doi:10.1088/0305-4470/4/3/009 |
| [8] | Kitagawa, M.; Ueda, M., Squeezed spin states, Phys. Rev. A, 47, 5138, 1993 · doi:10.1103/PhysRevA.47.5138 |
| [9] | Wineland, DJ; Bollinger, JJ; Itano, WM; Moore, FL; Heinzen, DJ, Spin squeezing and reduced quantum noise in spectroscopy, Phys. Rev. A, 46, R6797, 1992 · doi:10.1103/PhysRevA.46.R6797 |
| [10] | Messikh, A.; Ficek, Z.; Wahiddin, MRB, Spin squeezing as a measure of entanglement in a two-qubit system, Phys. Rev. A, 68, 2003 · doi:10.1103/PhysRevA.68.064301 |
| [11] | Carrasco, SC; Goerz, MH; Li, Z.; Colombo, S.; Vuletić, V.; Malinovsky, VS, Extreme spin squeezing via optimized one-axis twisting and rotations, Phys. Rev. Appl., 17, 6, 2022 · doi:10.1103/PhysRevApplied.17.064050 |
| [12] | Ma, J.; Wang, X.; Sun, CP; Nori, F., Quantum spin squeezing, Phys. Rep., 509, 89, 2011 · doi:10.1016/j.physrep.2011.08.003 |
| [13] | Giovannetti, V.; Lloyd, S.; Maccone, L., Quantum-enhanced measurements: beating the standard quantum limit, Science, 306, 1330, 2004 · doi:10.1126/science.1104149 |
| [14] | André, A.; ørensen, AS; Lukin, MD, Stability of atomic clocks based on entangled atoms, Phys. Rev. Lett., 92, 2004 · doi:10.1103/PhysRevLett.92.230801 |
| [15] | Gross, C.; Zibold, T.; Nicklas, E.; Esteve, J.; Oberthaler, MK, Nonlinear atom interferometer surpasses classical precision limit, Nature, 464, 1165, 2010 · doi:10.1038/nature08919 |
| [16] | Pezze, L.; Smerzi, A.; Oberthaler, MK; Schmied, R.; Treutlein, P., Quantum metrology with nonclassical states of atomic ensembles, Rev. Mod. Phys., 90, 2018 · doi:10.1103/RevModPhys.90.035005 |
| [17] | Pedrozo-Peñafiel, E.; Colombo, S.; Shu, C.; Adiyatullin, AF; Li, Z.; Mendez, E.; Braverman, B.; Kawasaki, A.; Akamatsu, D.; Xiao, Y.; Vuletić, V., Entanglement on an optical atomic-clock transition, Nature, 588, 414, 2020 · doi:10.1038/s41586-020-3006-1 |
| [18] | Sinatra, A., Spin-squeezed states for metrology, Appl. Phys. Lett., 120, 2022 · doi:10.1063/5.0084096 |
| [19] | Comparin, T.; Mezzacapo, F.; Roscilde, T., Robust spin squeezing from the tower of states of U (1)-symmetric spin Hamiltonians, Phys. Rev. A, 105, 2022 · doi:10.1103/PhysRevA.105.022625 |
| [20] | Kaubruegger, R.; Silvi, P.; Kokail, C.; van Bijnen, R.; Rey, AM; Ye, J.; Kaufman, AM; Zoller, P., Variational spin-squeezing algorithms on programmable quantum sensors, Phys. Rev. Lett., 123, 2019 · doi:10.1103/PhysRevLett.123.260505 |
| [21] | Marciniak, CD; Feldker, T.; Pogorelov, I.; Kaubruegger, R.; Vasilyev, DV; van Bijnen, R.; Schindler, P.; Zoller, P.; Blatt, R.; Monz, T., Optimal metrology with programmable quantum sensors, Nature, 603, 604, 2022 · doi:10.1038/s41586-022-04435-4 |
| [22] | Cheraghi, Hadi; Mahdavifar, Saeed; Johannesson, Henrik, Achieving spin-squeezed states by quench dynamics in a quantum chain, Phys. Rev. B, 105, 2022 · doi:10.1103/PhysRevB.105.024425 |
| [23] | Lyu, C.; Tang, X.; Li, J.; Xu, X.; Yung, MH; Bayat, A., Variational quantum simulation of long-range interacting systems, New J. Phys., 25, 2023 · Zbl 07866292 · doi:10.1088/1367-2630/acd571 |
| [24] | Vidal, J.; Palacios, G.; Mosseri, R., Entanglement in a second-order quantum phase transition, Phys. Rev. A, 69, 2004 · doi:10.1103/PhysRevA.69.022107 |
| [25] | Liu, WF; Ma, J.; Wang, X., Quantum Fisher information and spin squeezing in the ground state of the XY model, J. Phys. A Math. Theor., 46, 4, 2013 · Zbl 1267.81202 · doi:10.1088/1751-8113/46/4/045302 |
| [26] | Dusuel, S.; Vidal, J., Finite-size scaling exponents of the Lipkin-Meshkov-Glick model, Phys. Rev. Lett., 93, 2004 · doi:10.1103/PhysRevLett.93.237204 |
| [27] | Ma, Jian; Wang, Xiaoguang, Fisher information and spin squeezing in the Lipkin-Meshkov-Glick model, Phys. Rev. A, 80, 2009 · doi:10.1103/PhysRevA.80.012318 |
| [28] | Pezze, L.; Smerzi, A.; Oberthaler, MK; Schmied, R.; Treutlein, P., Quantum metrology with nonclassical states of atomic ensembles, Rev. Mod. Phys., 90, 2018 · doi:10.1103/RevModPhys.90.035005 |
| [29] | Frerot, I.; Roscilde, T., Quantum critical metrology, Phys. Rev. Lett., 121, 2018 · doi:10.1103/PhysRevLett.121.020402 |
| [30] | Makhalov, V.; Satoor, T.; Evrard, A.; Chalopin, T.; Lopes, R.; Nascimbene, S., Probing quantum criticality and symmetry breaking at the microscopic level, Phys. Rev. Lett., 123, 2019 · doi:10.1103/PhysRevLett.123.120601 |
| [31] | Bao, H.; Duan, J.; Jin, S.; Xingda, L.; Li, P.; Weizhi, Q.; Wang, M.; Novikova, I.; Mikhailov, EE; Zhao, K-F; Mølmer, K.; Shen, H.; Xiao, Y., Probing quantum criticality and symmetry breaking at the microscopic level, Nature, 581, 159, 2020 · doi:10.1038/s41586-020-2243-7 |
| [32] | Hayashida, K.; Makihara, T.; Marquez Peraca, N.; Fallas Padilla, D.; Pu, H.; Kono, J.; Bamba, M., Perfect intrinsic squeezing at the superradiant phase transition critical point, Sci. Rep., 13, 1, 2526, 2023 · doi:10.1038/s41598-023-29202-x |
| [33] | Frérot, I., Fadel, M., Lewenstein, M.: Probing quantum correlations in many-body systems: a review of scalable methods. https://arxiv.org/abs/2302.00640 |
| [34] | Xin, L., Barrios, M., Cohen, J.T., Chapman, M.S.: Squeezed ground states in a Spin-1 Bose-Einstein condensate https://arxiv.org/abs/2202.12338 |
| [35] | van Enk, SJ; Hirota, O., Entangled coherent states: teleportation and decoherence, Phys. Rev. A, 64, 2001 · doi:10.1103/PhysRevA.64.022313 |
| [36] | Sanders, BC, Review of entangled coherent states, J. Phys. A Math. Theor., 45, 2012 · Zbl 1248.81084 · doi:10.1088/1751-8113/45/24/244002 |
| [37] | Wang, C.; Gao, YY; Philip Reinhold, RW; Heeres, NO; Chou, K.; Axline, C.; Reagor, M.; Jacob Blumoff, KM; Sliwa, L.; Frunzio, SM; Girvin, LJ; Mirrahimi, M.; Devoret, MH; Schoelkopf, RJ, A Schrödinger cat living in two boxes, Science, 352, 1087, 2016 · Zbl 1355.81020 · doi:10.1126/science.aaf2941 |
| [38] | Wang, Z.; Bao, Z.; Yukai, W.; Li, Y.; Cai, W.; Wang, W.; Ma, Y.; Cai, T.; Han, X.; Wang, J.; Song, Y.; Sun, L.; Zhang, H.; Duan, L., An ultra-high gain single-photon transistor in the microwave regime, Sci. Adv., 8, 1778, 2022 · doi:10.1126/sciadv.abn1778 |
| [39] | Truong Minh Duc, Nonclassical Properties of the Superposition of Three-Mode Photon-Added Trio Coherent State, Int. J. Theor. Phys., 59, 3206, 2020 · Zbl 1450.81046 · doi:10.1007/s10773-020-04573-3 |
| [40] | Mansour, M.; Dahbi, Z.; Essakhi, M.; Salah, A., Quantum correlations through spin coherent states, Int. J. Theor. Phys., 60, 21568, 2021 · Zbl 1528.81041 · doi:10.1007/s10773-021-04831-y |
| [41] | Yan, PS; Zhou, L.; Zhong, W., Measurement-based entanglement purification for entangled coherent states, Front. Phys., 17, 21501, 2022 · doi:10.1007/s11467-021-1103-8 |
| [42] | Bennett, CH; Bernstein, HJ; Popescu, S.; Schumacher, B., Concentrating partial entanglement by local operations, Phys. Rev. A, 53, 2046, 1996 · doi:10.1103/PhysRevA.53.2046 |
| [43] | Kitaev, A.; Preskill, J., Topological entanglement entropy, Phys. Rev. Lett., 96, 0110404, 2006 · doi:10.1103/PhysRevLett.96.110404 |
| [44] | Laflorencie, N., Quantum entanglement in condensed matter systems, Phys. Rep., 646, 1, 2016 · doi:10.1016/j.physrep.2016.06.008 |
| [45] | Bergschneider, A.; Klinkhamer, VM; Becher, JH; Klemt, R.; Palm, L.; Zurn, G.; Jochim, S.; Preiss, PM, Experimental characterization of two-particle entanglement through position and momentum correlations, Nat. Phys., 15, 640, 2019 · doi:10.1038/s41567-019-0508-6 |
| [46] | Calabrese, P.; Cardy, J., Entanglement entropy and quantum field theory, J. Stat. Mech., 06, P06002, 2004 · Zbl 1082.82002 |
| [47] | Hastings, MB, An area law for one dimensional quantum systems, J. Stat. Mech., 08, P0802, 2007 · Zbl 1456.82046 |
| [48] | Gottlieb, D.; Rossler, J., Exact solution of a spin chain with binary and ternary interactions of Dzialoshinsky-Moriya type, Phys. Rev. B, 60, 9232, 1999 · doi:10.1103/PhysRevB.60.9232 |
| [49] | Lou, P.; Wen-Chin, W.; Ming-Che, C., Quantum phase transition in spin-12 XX Heisenberg chain with three-spin interaction, Phys. Rev. B, 70, 2004 · doi:10.1103/PhysRevB.70.064405 |
| [50] | Topilko, M.; Krokhmalskii, T.; Derzhko a, O.; Ohanyan, V., Magnetocaloric effect in spin-1/2 XX chains with three-spin interactions, Eur. Phys. J. B, 85, 278, 2012 · doi:10.1140/epjb/e2012-30359-8 |
| [51] | Liu, X., Chiral phase and quantum phase transitions of anisotropic XY chains with three-site interactions, J. Stat. Mech., 01, P01003, 2012 |
| [52] | Liu, X., Erratum: chiral phase and quantum phase transitions of anisotropic XY chains with three-site interactions, J. Stat. Mech., 11, E11001, 2012 · doi:10.1088/1742-5468/2012/11/E11001 |
| [53] | Lei, S.; Tong, P., Quantum discord in the transverse field XY chains with three-spin interaction, Phys. B, 463, 1-6, 2015 · doi:10.1016/j.physb.2015.01.031 |
| [54] | Ming-Liang, H.; Gao, Y-Y; Fan, H., Steered quantum coherence as a signature of quantum phase transitions in spin chains, Phys. Rev. A, 101, 2020 · doi:10.1103/PhysRevA.101.032305 |
| [55] | Thakur, P.; Durganandini, P., Orbital antiferroelectricity and higher dimensional magnetoelectric order in the spin-1/2 XX chain extended with three-spin interactions, Eur. Phys. J. B, 96, 51, 2023 · doi:10.1140/epjb/s10051-023-00522-1 |
| [56] | Bermudez, A.; Porras, D.; Martin-Delgado, MA, Competing many-body interactions in systems of trapped ions, Phys. Rev. A, 79, 060303(R), 2009 · doi:10.1103/PhysRevA.79.060303 |
| [57] | Katz, O.; Feng, L.; Risinger, A., Demonstration of three- and four-body interactions between trapped-ion spins, Nat. Phys., 19, 1452, 2023 · doi:10.1038/s41567-023-02102-7 |
| [58] | Andrade, B., Engineering an effective three-spin Hamiltonian in trapped-ion systems for applications in quantum simulation, Q. Sci. Technol., 7, 2022 |
| [59] | Peng, X.; Zhang, J.; Jiangfeng, D.; Dieter, S., Quantum simulation of a system with competing two- and three-body interactions, Phys. Rev. Lett., 103, 2009 · doi:10.1103/PhysRevLett.103.140501 |
| [60] | Cheong, S-A; Henley, CL, Many-body density matrices for free fermions, Phys. Rev. B, 69, 07511, 2004 · doi:10.1103/PhysRevB.69.075111 |
| [61] | Peschel, I., Calculation of reduced density matrices from correlation functions, J. Phys. A Math. Gen., 36, L205-L208, 2003 · Zbl 1049.82011 · doi:10.1088/0305-4470/36/14/101 |
| [62] | Vidal, G.; Latorre, JI; Rico, E.; Kitaev, A., Entanglement in quantum critical phenomena, Phys. Rev. Lett., 90, 2003 · doi:10.1103/PhysRevLett.90.227902 |
| [63] | Peschel, I.; Eisler, V., Reduced density matrices and entanglement entropy in free lattice models, J. Phys. A Math. Theor., 42, 2009 · Zbl 1179.81032 · doi:10.1088/1751-8113/42/50/504003 |
| [64] | Peschel, I., Special review: entanglement in solvable many-particle models, Braz. J. Phys., 42, 267, 2012 · doi:10.1007/s13538-012-0074-1 |
| [65] | Khastehdel F, F.; Mahdavifar, S.; Afrousheh, K., Entangled unique coherent line in the ground-state phase diagram of the spin-1/2 XX chain model with three-spin interaction, Phys. Rev. E, 109, 044142, 2024 · doi:10.1103/PhysRevE.109.044142 |
| [66] | Wick, GC, The evaluation of the collision matrix, Phys. Rev., 80, 268, 1950 · Zbl 0040.13006 · doi:10.1103/PhysRev.80.268 |
| [67] | Wootters, WK, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett., 80, 2245, 1998 · Zbl 1368.81047 · doi:10.1103/PhysRevLett.80.2245 |
| [68] | Amico, L.; Fazio, R.; Osterloh, A.; Vedral, V., Entanglement in many-body systems, Rev. Mod. Phys., 80, 517, 2008 · Zbl 1205.81009 · doi:10.1103/RevModPhys.80.517 |
| [69] | Horodecki, R.; Horodecki, P.; Horodecki, M.; Horodecki, K., Quantum entanglement, Rev. Mod. Phys., 81, 865, 2009 · Zbl 1205.81012 · doi:10.1103/RevModPhys.81.865 |
| [70] | Osborne, TJ; Nielsen, MA, Entanglement in a simple quantum phase transition, Phys. Rev. A, 66, 2002 · doi:10.1103/PhysRevA.66.032110 |
| [71] | Osterloh, A.; Amico, L.; Falci, G.; Fazio, R., Scaling of entanglement close to a quantum phase transition, Nature, 416, 608, 2002 · doi:10.1038/416608a |
| [72] | Vidal, G.; Latorre, JI; Rico, E.; Kitaev, A., Entanglement in quantum critical phenomena, Phys. Rev. A, 90, 2003 |
| [73] | Wu, L-A; Sarandy, MS; Lidar, DA, Quantum phase transitions and bipartite entanglement, Phys. Rev. Lett., 93, 2004 · doi:10.1103/PhysRevLett.93.250404 |
| [74] | Fumani, FK; Nemati, S.; Mahdavifar, S.; Darooneh, AH, Magnetic entanglement in spin-1/2 XY chains, Phys. A., 445, 256, 2016 · Zbl 1400.81026 · doi:10.1016/j.physa.2015.11.004 |
| [75] | Soltani, MR; Fumani, FK; Mahdavifar, S., Ising in a transverse field with added transverse dzyaloshinskii-moriya interaction, J. Magn. Magn. Mater., 476, 580, 2019 · doi:10.1016/j.jmmm.2018.12.019 |
| [76] | Fumani, FK; Beradze, B.; Nemati, S.; Mahdavifar, S.; Japaridze, GI, Quantum correlations in the spin-1/2 Heisenberg XXZ chain with modulated Dzyaloshinskii-Moriya interaction, J. Magn. Magn. Mater., 518, 2021 · doi:10.1016/j.jmmm.2020.167411 |
| [77] | Fumani, FK; Nemati, S.; Mahdavifar, S., Quantum critical lines in the ground state phase diagram of spin-1/2 frustrated transverse-field ising chains, Ann. Phys., 533, 2000384, 2020 · Zbl 07766316 · doi:10.1002/andp.202000384 |
| [78] | Satoori, S.; Mahdavifar, S.; Vahedi, J., Entanglement and quantum correlations in the XX spin-1/2 honeycomb lattice, Sci. Rep., 12, 17991, 2022 · doi:10.1038/s41598-022-19945-4 |
| [79] | Satoori, S.; Mahdavifar, S.; Vahedi, J., Quantum correlations in the frustrated XY model on the honeycomb lattice, Sci. Rep., 13, 16034, 2023 · doi:10.1038/s41598-023-43080-3 |
| [80] | Eisert, J.; Cramer, M.; Plenio, MB, Area laws for the entanglement entropy - a review, Rev. Mod. Phys., 82, 277, 2010 · Zbl 1205.81035 · doi:10.1103/RevModPhys.82.277 |
| [81] | Nishioka, T., Entanglement entropy: holography and renormalization group, Rev. Mod. Phys., 90, 2018 · doi:10.1103/RevModPhys.90.035007 |
| [82] | Zhao, J.; Wang, Y-C; Yan, Z.; Cheng, M.; Zi, YM, Scaling of entanglement entropy at deconfined quantum criticality, Phys. Rev. Lett., 128, 2022 · doi:10.1103/PhysRevLett.128.010601 |
| [83] | Latorre, JI; Riera, A., A short review on entanglement in quantum spin systems, J. Phys. A Math. Theor., 42, 2009 · Zbl 1179.81031 · doi:10.1088/1751-8113/42/50/504002 |
| [84] | Huelga, SF, Improvement of frequency standards with quantum entanglement, Phys. Rev. Lett., 79, 3865, 1997 · doi:10.1103/PhysRevLett.79.3865 |
| [85] | Escher, BM; de Matos Filho, RL; Davidovich, L., General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology, Nat. Phys., 7, 406, 2011 · doi:10.1038/nphys1958 |
| [86] | Giovannetti, V.; Lloyd, S.; Maccone, L., Quantum-enhanced measurements: beating the standard quantum limit, Science, 306, 1330, 2004 · doi:10.1126/science.1104149 |
| [87] | Giovannetti, V.; Lloyd, S.; Maccone, L., Advances in quantum metrology, Nat. Photonics, 5, 222, 2011 · doi:10.1038/nphoton.2011.35 |
| [88] | Davis, E.; Bentsen, G.; Schleier-Smith, M., Approaching the Heisenberg limit without single-particle detection, Phys. Rev. Lett., 116, 2016 · doi:10.1103/PhysRevLett.116.053601 |
| [89] | Zhiyao, H.; Li, Q.; Zhang, X.; Huang, L-G; Zhang, H.; Yong-Chun, L., Spin squeezing with arbitrary quadratic collective-spin interactions, Phys. Rev. A, 108, 2023 · doi:10.1103/PhysRevA.108.023722 |
| [90] | Balazadeh, L.; Najarbashi, G.; Tavana, A., Quantum renormalization of \(l_1\)-norm and relative entropy of coherence in quantum spin chains, Q. Inf. Process., 19, 181, 2020 · Zbl 1508.82016 · doi:10.1007/s11128-020-02677-7 |
| [91] | Vidal, G.; Latorre, JI; Rico, E.; Kitaev, A., Entanglement in quantum critical phenomena, Phys. Rev. Lett., 90, 2003 · doi:10.1103/PhysRevLett.90.227902 |
| [92] | Latorre, JI; Rico, E.; Vidal, G., Ground state entanglement in quantum spin chains, Quant. Info. Comp., 4, 48, 2004 · Zbl 1175.82017 |
| [93] | Jin, B-Q; Korepin, VE, Quantum spin chain, Toeplitz determinants and the fisher-Hartwig conjecture, J. Stat. Phys., 116, 79, 2004 · Zbl 1142.82314 · doi:10.1023/B:JOSS.0000037230.37166.42 |
| [94] | Gotta, L.; Mazza, L.; Simon, P.; Roux, G., Pairing in spinless fermions and spin chains with next-nearest neighbor interactions, Phys. Rev. A, 3, 2021 |
| [95] | Ruhman, J.; Altman, E., Topological degeneracy and pairing in a one-dimensional gas of spinless fermions, Phys. Rev. B, 96, 2017 · doi:10.1103/PhysRevB.96.085133 |
| [96] | Kane, CL; Stern, A.; Halperin, BI, Pairing in luttinger liquids and quantum hall states, Phys. Rev. X, 7, 2017 |
| [97] | Mattioli, M.; Dalmonte, M.; Lechner, W.; Pupillo, G., Cluster Luttinger liquids of Rydberg-dressed atoms in optical lattices, Phys. Rev. Lett., 111, 2013 · doi:10.1103/PhysRevLett.111.165302 |
| [98] | Dalmonte, M.; Lechner, W.; Cai, Z.; Mattioli, M.; Läuchli, AM; Pupillo, G., Cluster Luttinger liquids and emergent supersymmetric conformal critical points in the one-dimensional soft-shoulder Hubbard model, Phys. Rev. B, 92, 2015 · doi:10.1103/PhysRevB.92.045106 |
| [99] | He, Y.; Tian, B.; Pekker, D.; Mong, RSK, Emergent mode and bound states in single-component one-dimensional lattice fermionic systems, Phys. Rev. B, 100, 201101(R), 2019 · doi:10.1103/PhysRevB.100.201101 |
| [100] | D’Emidio, J.; Block, MS; Kaul, RK, R’enyi entanglement entropy of critical SU(N) spin chains, Phys. Rev. B, 92, 2015 · doi:10.1103/PhysRevB.92.054411 |
| [101] | D’Emidio, J., Entanglement entropy from nonequilibrium work, Phys. Rev. Lett., 124, 2020 · doi:10.1103/PhysRevLett.124.110602 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
