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These interactions can be accounted for by a long-wavelength relativistic theory, where the phonons are described by a coarse-grained Klein-Gordon field <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>&amp;#x03D5;<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> locally coupled to the spins that acts as a carrier, leading to an analogue of pion-mediated Yukawa interactions. In the vicinity of a structural transition of the ion crystal, one must go beyond the Klein-Gordon fields, and include additional <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>&amp;#x03BB;<\/mml:mi><mml:msup><mml:mi>&amp;#x03D5;<\/mml:mi><mml:mn>4<\/mml:mn><\/mml:msup><\/mml:math> terms responsible for phonon-phonon scattering. This leads to quantum effects that can be expressed by Feynman loop integrals that modify the range of the Yukawa-type spin interactions; an effect that could be used to probe the underlying fixed point of this quantum field theory (QFT). Unfortunately, the rigidity of the trapped-ion crystal makes it challenging to observe genuine quantum effects, such as the flow of the critical point with the quartic coupling <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>&amp;#x03BB;<\/mml:mi><\/mml:math>. We hereby show that thermal effects, which can be controlled by laser cooling, can unveil this flow through the appearance of thermal masses in interacting QFTs. We perform self-consistent calculations that resum certain Feynman diagrams and, additionally, go beyond mean-field theory to predict how measurements on the trapped-ion spin system can probe key properties of the <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>&amp;#x03BB;<\/mml:mi><mml:msup><mml:mi>&amp;#x03D5;<\/mml:mi><mml:mn>4<\/mml:mn><\/mml:msup><\/mml:math> QFT.<\/jats:p>","DOI":"10.22331\/q-2024-07-15-1411","type":"journal-article","created":{"date-parts":[[2024,7,15]],"date-time":"2024-07-15T14:39:36Z","timestamp":1721054376000},"page":"1411","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":["Thermal masses and trapped-ion quantum spin models: a self-consistent approach to Yukawa-type interactions in the <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>&amp;#x3BB;<\/mml:mo><\/mml:mrow><mml:mspace width=\"negativethinmathspace\"\/><mml:msup><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>&amp;#x3D5;<\/mml:mo><\/mml:mrow><mml:mn>4<\/mml:mn><\/mml:msup><\/mml:math> model"],"prefix":"10.22331","volume":"8","author":[{"given":"Pablo Vi\u00f1as","family":"Mart\u00ednez","sequence":"first","affiliation":[{"name":"Instituto de F\u00edsica Te\u00f3rica, UAM-CSIC, Universidad Aut\u00f3noma de Madrid, Cantoblanco, 28049 Madrid, Spain."}]},{"given":"Esperanza","family":"L\u00f3pez","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica Te\u00f3rica, UAM-CSIC, Universidad Aut\u00f3noma de Madrid, Cantoblanco, 28049 Madrid, Spain."}]},{"given":"Alejandro","family":"Bermudez","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica Te\u00f3rica, UAM-CSIC, Universidad Aut\u00f3noma de Madrid, Cantoblanco, 28049 Madrid, Spain."}]}],"member":"9598","published-online":{"date-parts":[[2024,7,15]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"A. Ac\u00edn, I. Bloch, H. Buhrman, T. Calarco, C. Eichler, J. Eisert, D. Esteve, N. Gisin, S. J. Glaser, F. Jelezko, S. Kuhr, M. Lewenstein, M. F. Riedel, P. O. Schmidt, R. Thew, A. Wallraff, I. Walmsley, and F. K. Wilhelm, New Journal of Physics 20, 080201 (2018).","DOI":"10.1088\/1367-2630\/aad1ea"},{"key":"1","doi-asserted-by":"publisher","unstructured":"M. Mohseni, P. Read, H. Neven, S. Boixo, V. Denchev, R. Babbush, A. Fowler, V. Smelyanskiy, and J. Martinis, Nature 543, 171 (2017).","DOI":"10.1038\/543171a"},{"key":"2","unstructured":"M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000)."},{"key":"3","doi-asserted-by":"publisher","unstructured":"L. Postler, S. Heu$\\beta$en, I. Pogorelov, M. Rispler, T. Feldker, M. Meth, C. D. Marciniak, R. Stricker, M. Ringbauer, R. Blatt, P. Schindler, M. M\u00fcller, and T. Monz, Nature 605, 675 (2022).","DOI":"10.1038\/s41586-022-04721-1"},{"key":"4","doi-asserted-by":"publisher","unstructured":"S. Krinner, N. Lacroix, A. Remm, A. Di Paolo, E. Genois, C. Leroux, C. Hellings, S. Lazar, F. Swiadek, J. Herrmann, G. J. Norris, C. K. Andersen, M. M\u00fcller, A. Blais, C. Eichler, and A. Wallraff, Nature 605, 669 (2022).","DOI":"10.1038\/s41586-022-04566-8"},{"key":"5","doi-asserted-by":"publisher","unstructured":"Y. Zhao, Y. Ye, H.-L. Huang, Y. Zhang, D. Wu, H. Guan, Q. Zhu, Z. Wei, T. He, S. Cao, F. Chen, T.-H. Chung, H. Deng, D. Fan, M. Gong, C. Guo, S. Guo, L. Han, N. Li, S. Li, Y. Li, F. Liang, J. Lin, H. Qian, H. Rong, H. Su, L. Sun, S. Wang, Y. Wu, Y. Xu, C. Ying, J. Yu, C. Zha, K. Zhang, Y.-H. Huo, C.-Y. Lu, C.-Z. Peng, X. Zhu, and J.-W. Pan, Phys. Rev. Lett. 129, 030501 (2022).","DOI":"10.1103\/PhysRevLett.129.030501"},{"key":"6","doi-asserted-by":"publisher","unstructured":"C. Ryan-Anderson, N. C. Brown, M. S. Allman, B. Arkin, G. Asa-Attuah, C. Baldwin, J. Berg, J. G. Bohnet, S. Braxton, N. Burdick, J. P. Campora, A. Chernoguzov, J. Esposito, B. Evans, D. Francois, J. P. Gaebler, T. M. Gatterman, J. Gerber, K. Gilmore, D. Gresh, A. Hall, A. Hankin, J. Hostetter, D. Lucchetti, K. Mayer, J. Myers, B. Neyenhuis, J. Santiago, J. Sedlacek, T. Skripka, A. Slattery, R. P. Stutz, J. Tait, R. Tobey, G. Vittorini, J. Walker, and D. Hayes, (2022), 10.48550\/ARXIV.2208.01863.","DOI":"10.48550\/ARXIV.2208.01863"},{"key":"7","doi-asserted-by":"publisher","unstructured":"R. Acharya, I. Aleiner, and R. e. al.. Allen, (2022), 10.1038\/s41586-022-05434-1.","DOI":"10.1038\/s41586-022-05434-1"},{"key":"8","doi-asserted-by":"publisher","unstructured":"D. Aharonov and M. Ben-Or, SIAM J. Comput. 38, 1207 (1998).","DOI":"10.1137\/S0097539799359385"},{"key":"9","doi-asserted-by":"publisher","unstructured":"A. Montanaro, npj Quantum Information 2, 15023 (2016).","DOI":"10.1038\/npjqi.2015.23"},{"key":"10","doi-asserted-by":"publisher","unstructured":"J. Preskill, Quantum 2, 79 (2018).","DOI":"10.22331\/q-2018-08-06-79"},{"key":"11","doi-asserted-by":"publisher","unstructured":"K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru-Guzik, Rev. Mod. Phys. 94, 015004 (2022).","DOI":"10.1103\/RevModPhys.94.015004"},{"key":"12","doi-asserted-by":"publisher","unstructured":"F. A. et al., Nature 574, 505\u2013510 (2019).","DOI":"10.1038\/s41586-019-1666-5"},{"key":"13","doi-asserted-by":"publisher","unstructured":"H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.-C. Peng, Y.-H. Luo, J. Qin, D. Wu, X. Ding, Y. Hu, P. Hu, X.-Y. Yang, W.-J. Zhang, H. Li, Y. Li, X. Jiang, L. Gan, G. Yang, L. You, Z. Wang, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, Science 370, 1460 (2020).","DOI":"10.1126\/science.abe8770"},{"key":"14","doi-asserted-by":"publisher","unstructured":"R. P. Feynman, International journal of theoretical physics 21, 467 (1982).","DOI":"10.1007\/BF02650179"},{"key":"15","doi-asserted-by":"publisher","unstructured":"J. I. Cirac and P. Zoller, Nature Physics 8, 264 (2012).","DOI":"10.1038\/nphys2275"},{"key":"16","doi-asserted-by":"publisher","unstructured":"E. Altman, K. R. Brown, G. Carleo, L. D. Carr, E. Demler, C. Chin, B. DeMarco, S. E. Economou, M. A. Eriksson, K.-M. C. Fu, M. Greiner, K. R. Hazzard, R. G. Hulet, A. J. Koll\u00e1r, B. L. Lev, M. D. Lukin, R. Ma, X. Mi, S. Misra, C. Monroe, K. Murch, Z. Nazario, K.-K. Ni, A. C. Potter, P. Roushan, M. Saffman, M. Schleier-Smith, I. Siddiqi, R. Simmonds, M. Singh, I. Spielman, K. Temme, D. S. Weiss, J. Vu\u010dkovi\u0107, V. Vuleti\u0107, J. Ye, and M. Zwierlein, PRX Quantum 2, 017003 (2021).","DOI":"10.1103\/PRXQuantum.2.017003"},{"key":"17","doi-asserted-by":"publisher","unstructured":"A. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller, Nature 607, 667 (2022).","DOI":"10.1038\/s41586-022-04940-6"},{"key":"18","doi-asserted-by":"publisher","unstructured":"S. McArdle, S. Endo, A. Aspuru-Guzik, S. C. Benjamin, and X. Yuan, Rev. Mod. Phys. 92, 015003 (2020).","DOI":"10.1103\/RevModPhys.92.015003"},{"key":"19","doi-asserted-by":"publisher","unstructured":"I. Bloch, J. Dalibard, and S. Nascimb\u00e8ne, Nature Physics 8, 267 (2012).","DOI":"10.1038\/nphys2259"},{"key":"20","doi-asserted-by":"publisher","unstructured":"R. Blatt and C. F. Roos, Nature Physics 8, 277 (2012).","DOI":"10.1038\/nphys2252"},{"key":"21","doi-asserted-by":"publisher","unstructured":"C. Monroe, W. C. Campbell, L.-M. Duan, Z.-X. Gong, A. V. Gorshkov, P. W. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano, P. Richerme, C. Senko, and N. Y. Yao, Rev. Mod. Phys. 93, 025001 (2021).","DOI":"10.1103\/RevModPhys.93.025001"},{"key":"22","doi-asserted-by":"publisher","unstructured":"M. C. Ba\u00f1uls, R. Blatt, J. Catani, A. Celi, J. I. Cirac, M. Dalmonte, L. Fallani, K. Jansen, M. Lewenstein, S. Montangero, C. A. Muschik, B. Reznik, E. Rico, L. Tagliacozzo, K. Van Acoleyen, F. Verstraete, U.-J. Wiese, M. Wingate, J. Zakrzewski, and P. Zoller, The European Physical Journal D 74, 165 (2020).","DOI":"10.1140\/epjd\/e2020-100571-8"},{"key":"23","doi-asserted-by":"publisher","unstructured":"M. Suzuki, Communications in Mathematical Physics 51, 183 (1976).","DOI":"10.1007\/BF01609348"},{"key":"24","doi-asserted-by":"publisher","unstructured":"S. Lloyd, Science 273, 1073 (1996).","DOI":"10.1126\/science.273.5278.1073"},{"key":"25","doi-asserted-by":"publisher","unstructured":"R. D. Somma, G. Ortiz, E. H. Knill, and J. Gubernatis, in SPIE Proceedings, edited by E. Donkor, A. R. Pirich, and H. E. Brandt (SPIE, 2003).","DOI":"10.1117\/12.487249"},{"key":"26","doi-asserted-by":"publisher","unstructured":"D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. 81, 3108 (1998).","DOI":"10.1103\/PhysRevLett.81.3108"},{"key":"27","doi-asserted-by":"publisher","unstructured":"M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. A 45, 6493 (1992).","DOI":"10.1103\/PhysRevA.45.6493"},{"key":"28","doi-asserted-by":"publisher","unstructured":"B. P. Lanyon, C. Hempel, D. Nigg, M. M\u00fcller, R. Gerritsma, F. Z\u00e4hringer, P. Schindler, J. T. Barreiro, M. Rambach, G. Kirchmair, M. Hennrich, P. Zoller, R. Blatt, and C. F. Roos, Science 334, 57 (2011).","DOI":"10.1126\/science.1208001"},{"key":"29","doi-asserted-by":"publisher","unstructured":"O. Katz, L. Feng, A. Risinger, C. Monroe, and M. Cetina, (2022), 10.1038\/s41567-023-02102-7.","DOI":"10.1038\/s41567-023-02102-7"},{"key":"30","doi-asserted-by":"publisher","unstructured":"J. T. Barreiro, M. M\u00fcller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller, and R. Blatt, Nature 470, 486 (2011).","DOI":"10.1038\/nature09801"},{"key":"31","doi-asserted-by":"publisher","unstructured":"P. Schindler, M. M\u00fcller, D. Nigg, J. T. Barreiro, E. A. Martinez, M. Hennrich, T. Monz, S. Diehl, P. Zoller, and R. Blatt, Nature Physics 9, 361 (2013).","DOI":"10.1038\/nphys2630"},{"key":"32","doi-asserted-by":"publisher","unstructured":"K. G. Wilson, Phys. Rev. D 10, 2445 (1974).","DOI":"10.1103\/PhysRevD.10.2445"},{"key":"33","doi-asserted-by":"publisher","unstructured":"J. Kogut and L. Susskind, Phys. Rev. D 11, 395 (1975).","DOI":"10.1103\/PhysRevD.11.395"},{"key":"34","doi-asserted-by":"publisher","unstructured":"C. Gattringer and C. B. Lang, Quantum chromodynamics on the lattice, Vol. 788 (Springer, Berlin, 2010).","DOI":"10.1007\/978-3-642-01850-3"},{"key":"35","doi-asserted-by":"publisher","unstructured":"E. A. Martinez, C. A. Muschik, P. Schindler, D. Nigg, A. Erhard, M. Heyl, P. Hauke, M. Dalmonte, T. Monz, P. Zoller, and R. Blatt, Nature 534, 516 (2016).","DOI":"10.1038\/nature18318"},{"key":"36","doi-asserted-by":"publisher","unstructured":"N. H. Nguyen, M. C. Tran, Y. Zhu, A. M. Green, C. H. Alderete, Z. Davoudi, and N. M. Linke, PRX Quantum 3, 020324 (2022).","DOI":"10.1103\/PRXQuantum.3.020324"},{"key":"37","doi-asserted-by":"publisher","unstructured":"A. S\u00f8rensen and K. M\u00f8lmer, Phys. Rev. Lett. 82, 1971 (1999).","DOI":"10.1103\/PhysRevLett.82.1971"},{"key":"38","doi-asserted-by":"publisher","unstructured":"A. S\u00f8rensen and K. M\u00f8lmer, Phys. Rev. A 62, 022311 (2000).","DOI":"10.1103\/PhysRevA.62.022311"},{"key":"39","doi-asserted-by":"publisher","unstructured":"D. Porras and J. I. Cirac, Phys. Rev. Lett. 92, 207901 (2004).","DOI":"10.1103\/PhysRevLett.92.207901"},{"key":"40","doi-asserted-by":"publisher","unstructured":"A. Friedenauer, H. Schmitz, J. T. Glueckert, D. Porras, and T. Schaetz, Nature Physics 10.1038\/nphys1032.","DOI":"10.1038\/nphys1032"},{"key":"41","doi-asserted-by":"publisher","unstructured":"K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, Nature 465, 590 (2010).","DOI":"10.1038\/nature09071"},{"key":"42","doi-asserted-by":"publisher","unstructured":"J. W. Britton, B. C. Sawyer, A. C. Keith, C.-C. J. Wang, J. K. Freericks, H. Uys, M. J. Biercuk, and J. J. Bollinger, Nature 484, 489 (2012).","DOI":"10.1038\/nature10981"},{"key":"43","doi-asserted-by":"publisher","unstructured":"R. Islam, C. Senko, W. C. Campbell, S. Korenblit, J. Smith, A. Lee, E. E. Edwards, C.-C. J. Wang, J. K. Freericks, and C. Monroe, Science 340, 583 (2013).","DOI":"10.1126\/science.1232296"},{"key":"44","doi-asserted-by":"publisher","unstructured":"C. Senko, J. Smith, P. Richerme, A. Lee, W. C. Campbell, and C. Monroe, Science 345, 430 (2014).","DOI":"10.1126\/science.1251422"},{"key":"45","doi-asserted-by":"publisher","unstructured":"P. Jurcevic, B. P. Lanyon, P. Hauke, C. Hempel, P. Zoller, R. Blatt, and C. F. Roos, Nature 511, 202 (2014).","DOI":"10.1038\/nature13461"},{"key":"46","doi-asserted-by":"publisher","unstructured":"P. Richerme, Z.-X. Gong, A. Lee, C. Senko, J. Smith, M. Foss-Feig, S. Michalakis, A. V. Gorshkov, and C. Monroe, Nature 511, 198 (2014).","DOI":"10.1038\/nature13450"},{"key":"47","doi-asserted-by":"publisher","unstructured":"J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W. Hess, P. Hauke, M. Heyl, D. A. Huse, and C. Monroe, Nature Physics 12, 907 (2016).","DOI":"10.1038\/nphys3783"},{"key":"48","doi-asserted-by":"publisher","unstructured":"J. Zhang, G. Pagano, P. W. Hess, A. Kyprianidis, P. Becker, H. Kaplan, A. V. Gorshkov, Z.-X. Gong, and C. Monroe, Nature 551, 601 (2017).","DOI":"10.1038\/nature24654"},{"key":"49","doi-asserted-by":"publisher","unstructured":"W. L. Tan, P. Becker, F. Liu, G. Pagano, K. S. Collins, A. De, L. Feng, H. B. Kaplan, A. Kyprianidis, R. Lundgren, W. Morong, S. Whitsitt, A. V. Gorshkov, and C. Monroe, Nature Physics 17, 742 (2021).","DOI":"10.1038\/s41567-021-01194-3"},{"key":"50","doi-asserted-by":"publisher","unstructured":"W. Morong, F. Liu, P. Becker, K. S. Collins, L. Feng, A. Kyprianidis, G. Pagano, T. You, A. V. Gorshkov, and C. Monroe, Nature 599, 393 (2021).","DOI":"10.1038\/s41586-021-03988-0"},{"key":"51","doi-asserted-by":"publisher","unstructured":"A. Bermudez, G. Aarts, and M. M\u00fcller, Physical Review X 7 (2017a), 10.1103\/physrevx.7.041012.","DOI":"10.1103\/physrevx.7.041012"},{"key":"52","doi-asserted-by":"publisher","unstructured":"G. Mart\u00edn-V\u00e1zquez, G. Aarts, M. M\u00fcller, and A. Bermudez, (2021), 10.1103\/PRXQuantum.3.020352.","DOI":"10.1103\/PRXQuantum.3.020352"},{"key":"53","doi-asserted-by":"publisher","unstructured":"P. Nevado and D. Porras, Phys. Rev. A 93, 013625 (2016).","DOI":"10.1103\/PhysRevA.93.013625"},{"key":"54","doi-asserted-by":"publisher","unstructured":"A. Bermudez, L. Tagliacozzo, G. Sierra, and P. Richerme, Phys. Rev. B 95, 024431 (2017b).","DOI":"10.1103\/PhysRevB.95.024431"},{"key":"55","doi-asserted-by":"publisher","unstructured":"O. Klein, Zeitschrift f\u00fcr Physik 37, 895 (1926).","DOI":"10.1007\/BF01397481"},{"key":"56","doi-asserted-by":"publisher","unstructured":"W. Gordon, Zeitschrift f\u00fcr Physik 40, 117 (1926).","DOI":"10.1007\/BF01390840"},{"key":"57","doi-asserted-by":"publisher","unstructured":"Y. Hidaka, S. Iso, and K. Shimada, (2022), 10.1103\/PhysRevD.107.085003.","DOI":"10.1103\/PhysRevD.107.085003"},{"key":"58","doi-asserted-by":"publisher","unstructured":"S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967).","DOI":"10.1103\/PhysRevLett.19.1264"},{"key":"59","doi-asserted-by":"publisher","unstructured":"K. G. Wilson and J. Kogut, Physics Reports 12, 75 (1974).","DOI":"10.1016\/0370-1573(74)90023-4"},{"key":"60","doi-asserted-by":"publisher","unstructured":"S. Fishman, G. De Chiara, T. Calarco, and G. Morigi, Phys. Rev. B 77, 064111 (2008).","DOI":"10.1103\/PhysRevB.77.064111"},{"key":"61","doi-asserted-by":"publisher","unstructured":"D. Podolsky, E. Shimshoni, P. Silvi, S. Montangero, T. Calarco, G. Morigi, and S. Fishman, Phys. Rev. B 89, 214408 (2014).","DOI":"10.1103\/PhysRevB.89.214408"},{"key":"62","doi-asserted-by":"publisher","unstructured":"L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 (1974).","DOI":"10.1103\/PhysRevD.9.3320"},{"key":"63","doi-asserted-by":"publisher","unstructured":"S. Weinberg, Phys. Rev. D 9, 3357 (1974).","DOI":"10.1103\/PhysRevD.9.3357"},{"key":"64","doi-asserted-by":"publisher","unstructured":"M. \u0160a\u0161ura and V. Buzek, Journal of Modern Optics 49, 1593 (2002).","DOI":"10.1080\/09500340110115497"},{"key":"65","doi-asserted-by":"publisher","unstructured":"C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, Applied Physics Reviews 6, 021314 (2019).","DOI":"10.1063\/1.5088164"},{"key":"66","doi-asserted-by":"crossref","unstructured":"P. Ghosh, ``Ion traps. clarendon,&apos;&apos; (1995).","DOI":"10.1093\/oso\/9780198539957.001.0001"},{"key":"67","doi-asserted-by":"publisher","unstructured":"D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev. Mod. Phys. 75, 281 (2003a).","DOI":"10.1103\/RevModPhys.75.281"},{"key":"68","doi-asserted-by":"publisher","unstructured":"D. F. V. James, Applied Physics B 66, 181 (1998).","DOI":"10.1007\/s003400050373"},{"key":"69","doi-asserted-by":"publisher","unstructured":"C. Marquet, F. Schmidt-Kaler, and D. F. V. James, Applied Physics B 76, 199 (2003).","DOI":"10.1007\/s00340-003-1097-7"},{"key":"70","doi-asserted-by":"publisher","unstructured":"A. Altland and B. D. Simons, Condensed Matter Field Theory, 2nd ed. (Cambridge University Press, 2010).","DOI":"10.1017\/CBO9780511789984"},{"key":"71","doi-asserted-by":"publisher","unstructured":"J. P. Schiffer, Phys. Rev. Lett. 70, 818 (1993).","DOI":"10.1103\/PhysRevLett.70.818"},{"key":"72","doi-asserted-by":"publisher","unstructured":"D. H. E. Dubin, Phys. Rev. Lett. 71, 2753 (1993).","DOI":"10.1103\/PhysRevLett.71.2753"},{"key":"73","doi-asserted-by":"publisher","unstructured":"D. G. Enzer, M. M. Schauer, J. J. Gomez, M. S. Gulley, M. H. Holzscheiter, P. G. Kwiat, S. K. Lamoreaux, C. G. Peterson, V. D. Sandberg, D. Tupa, A. G. White, R. J. Hughes, and D. F. V. James, Phys. Rev. Lett. 85, 2466 (2000).","DOI":"10.1103\/PhysRevLett.85.2466"},{"key":"74","doi-asserted-by":"publisher","unstructured":"W. Greiner and J. Reinhardt, Field quantization (1996).","DOI":"10.1007\/978-3-642-61485-9"},{"key":"75","doi-asserted-by":"publisher","unstructured":"H. J. Rothe, Lattice Gauge Theories : An Introduction (Fourth Edition), Vol. 43 (World Scientific Publishing Company, 2012) Chap. 3.","DOI":"10.1142\/8229"},{"key":"76","doi-asserted-by":"publisher","unstructured":"A. Bermudez and M. B. Plenio, Phys. Rev. Lett. 109, 010501 (2012).","DOI":"10.1103\/PhysRevLett.109.010501"},{"key":"77","doi-asserted-by":"publisher","unstructured":"E. Shimshoni, G. Morigi, and S. Fishman, Phys. Rev. Lett. 106, 010401 (2011).","DOI":"10.1103\/PhysRevLett.106.010401"},{"key":"78","unstructured":"I. Affleck, Field theory methods and quantum critical phenomena Course 10 (North-Holland, Netherlands, 1990)."},{"key":"79","doi-asserted-by":"publisher","unstructured":"V. J. Emery, Correlated Electron Systems (WORLD SCIENTIFIC, 1993).","DOI":"10.1142\/1882"},{"key":"80","doi-asserted-by":"publisher","unstructured":"``Correspondence from discrete to continuum models,&apos;&apos; in Quantum Electron Liquids and High-TcSuperconductivity (Springer Berlin Heidelberg, Berlin, Heidelberg, 1995) pp. 109\u2013124.","DOI":"10.1007\/978-3-540-47678-8_5"},{"key":"81","unstructured":"D. T. Son, (2002), arXiv:hep-ph\/0204199."},{"key":"82","doi-asserted-by":"publisher","unstructured":"in Theory of Elasticity (Third Edition), edited by E. LIFSHITZ, A. KOSEVICH, and L. PITAEVSKII (Butterworth-Heinemann, Oxford, 1986) third edition ed., p. iv.","DOI":"10.1016\/B978-0-08-057069-3.50003-6"},{"key":"83","unstructured":"K. Thorne and R. Blandford, Modern Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics (Princeton University Press, 2017)."},{"key":"84","doi-asserted-by":"publisher","unstructured":"S. Coleman and E. Weinberg, Phys. Rev. D 7, 1888 (1973).","DOI":"10.1103\/PhysRevD.7.1888"},{"key":"85","doi-asserted-by":"publisher","unstructured":"R. Jackiw, Phys. Rev. D 9, 1686 (1974).","DOI":"10.1103\/PhysRevD.9.1686"},{"key":"86","doi-asserted-by":"publisher","unstructured":"K. G. Wilson and M. E. Fisher, Phys. Rev. Lett. 28, 240 (1972).","DOI":"10.1103\/PhysRevLett.28.240"},{"key":"87","doi-asserted-by":"publisher","unstructured":"W. Loinaz and R. S. Willey, Phys. Rev. D 58, 076003 (1998).","DOI":"10.1103\/PhysRevD.58.076003"},{"key":"88","doi-asserted-by":"publisher","unstructured":"D. Schaich and W. Loinaz, Phys. Rev. D 79, 056008 (2009).","DOI":"10.1103\/PhysRevD.79.056008"},{"key":"89","doi-asserted-by":"publisher","unstructured":"T. Sugihara, Journal of High Energy Physics 2004, 007 (2004).","DOI":"10.1088\/1126-6708\/2004\/05\/007"},{"key":"90","doi-asserted-by":"publisher","unstructured":"A. H. Asit K. De, J. Maiti, and T. 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D 106, L071501 (2022).","DOI":"10.1103\/PhysRevD.106.L071501"},{"key":"96","doi-asserted-by":"publisher","unstructured":"S.-J. Chang, Phys. Rev. D 13, 2778 (1976).","DOI":"10.1103\/PhysRevD.13.2778"},{"key":"97","doi-asserted-by":"publisher","unstructured":"J. Chiaverini et al., Nature 432, 602 (2004).","DOI":"10.1038\/nature03074"},{"key":"98","doi-asserted-by":"publisher","unstructured":"P. Schindler, J. T. Barreiro, T. Monz, V. Nebendahl, D. Nigg, M. Chwalla, M. Hennrich, and R. Blatt, Science 332, 1059 (2011).","DOI":"10.1126\/science.1203329"},{"key":"99","doi-asserted-by":"publisher","unstructured":"D. Nigg, M. M\u00fcller, E. A. Martinez, P. Schindler, M. Hennrich, T. Monz, M. A. Martin-Delgado, and R. Blatt, Science 345, 302 (2014).","DOI":"10.1126\/science.1253742"},{"key":"100","doi-asserted-by":"publisher","unstructured":"N. M. Linke, M. Gutierrez, K. A. Landsman, C. Figgatt, S. Debnath, K. R. Brown, and C. Monroe, Science Advances 3 (2017), 10.1126\/sciadv.1701074.","DOI":"10.1126\/sciadv.1701074"},{"key":"101","doi-asserted-by":"publisher","unstructured":"V. Negnevitsky, M. Marinelli, K. K. Mehta, H.-Y. Lo, C. Fl\u00fchmann, and J. P. Home, Nature 563, 527 (2018).","DOI":"10.1038\/s41586-018-0668-z"},{"key":"102","doi-asserted-by":"publisher","unstructured":"C. Fluehmann, T. Nguyen, M. Marinelli, V. Negnevitsky, K. Mehta, and J. Home, Nature 566, 513 (2019).","DOI":"10.1038\/s41586-019-0960-6"},{"key":"103","doi-asserted-by":"publisher","unstructured":"R. Stricker et al., Nature 585, 207 (2020).","DOI":"10.1038\/s41586-020-2667-0"},{"key":"104","doi-asserted-by":"publisher","unstructured":"B. de Neeve, T. L. Nguyen, T. Behrle, and J. Home, (2020), 10.1038\/s41567-021-01487-7.","DOI":"10.1038\/s41567-021-01487-7"},{"key":"105","doi-asserted-by":"publisher","unstructured":"A. Erhard, H. Poulsen Nautrup, M. Meth, L. Postler, R. Stricker, M. Stadler, V. Negnevitsky, M. Ringbauer, P. Schindler, H. J. Briegel, R. Blatt, N. Friis, and T. Monz, Nature 589, 220 (2021).","DOI":"10.1038\/s41586-020-03079-6"},{"key":"106","doi-asserted-by":"publisher","unstructured":"L. Egan, D. M. Debroy, C. Noel, A. Risinger, D. Zhu, D. Biswas, M. Newman, M. Li, K. R. Brown, M. Cetina, and C. Monroe, Nature 598, 281 (2021).","DOI":"10.1038\/s41586-021-03928-y"},{"key":"107","doi-asserted-by":"publisher","unstructured":"D. M. Debroy, L. Egan, C. Noel, A. Risinger, D. Zhu, D. Biswas, M. Cetina, C. Monroe, and K. R. Brown, Phys. Rev. Lett. 127, 240501 (2021).","DOI":"10.1103\/PhysRevLett.127.240501"},{"key":"108","doi-asserted-by":"publisher","unstructured":"C. Ryan-Anderson, J. G. Bohnet, K. Lee, D. Gresh, A. Hankin, J. P. Gaebler, D. Francois, A. Chernoguzov, D. Lucchetti, N. C. Brown, T. M. Gatterman, S. K. Halit, K. Gilmore, J. A. Gerber, B. Neyenhuis, D. Hayes, and R. P. Stutz, Phys. Rev. X 11, 041058 (2021).","DOI":"10.1103\/PhysRevX.11.041058"},{"key":"109","doi-asserted-by":"publisher","unstructured":"J. Hilder, D. Pijn, O. Onishchenko, A. Stahl, M. Orth, B. Lekitsch, A. Rodriguez-Blanco, M. M\u00fcller, F. Schmidt-Kaler, and U. G. Poschinger, Phys. Rev. X 12, 011032 (2022).","DOI":"10.1103\/PhysRevX.12.011032"},{"key":"110","doi-asserted-by":"publisher","unstructured":"A. Bermudez, X. Xu, R. Nigmatullin, J. O&apos;Gorman, V. Negnevitsky, P. Schindler, T. Monz, U. G. Poschinger, C. Hempel, J. Home, F. Schmidt-Kaler, M. Biercuk, R. Blatt, S. Benjamin, and M. M\u00fcller, Phys. Rev. X 7, 041061 (2017c).","DOI":"10.1103\/PhysRevX.7.041061"},{"key":"111","doi-asserted-by":"publisher","unstructured":"S. Gulde, M. Riebe, G. P. T. Lancaster, C. Becher, J. Eschner, H. H\u00e4ffner, F. Schmidt-Kaler, I. L. Chuang, and R. Blatt, Nature 421, 48 (2003).","DOI":"10.1038\/nature01336"},{"key":"112","doi-asserted-by":"publisher","unstructured":"M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri, and D. J. Wineland, Nature 429, 737 (2004).","DOI":"10.1038\/nature02608"},{"key":"113","doi-asserted-by":"publisher","unstructured":"M. Riebe, H. H\u00e4ffner, C. F. Roos, W. H\u00e4nsel, J. Benhelm, G. P. T. Lancaster, T. W. K\u00f6rber, C. Becher, F. Schmidt-Kaler, D. F. V. James, and R. Blatt, Nature 429, 734 (2004).","DOI":"10.1038\/nature02570"},{"key":"114","doi-asserted-by":"publisher","unstructured":"J. Chiaverini, J. Britton, D. Leibfried, E. Knill, M. D. Barrett, R. B. Blakestad, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, T. Schaetz, and D. J. Wineland, Science 308, 997 (2005).","DOI":"10.1126\/science.1110335"},{"key":"115","doi-asserted-by":"publisher","unstructured":"T. Monz, D. Nigg, E. A. Martinez, M. F. Brandl, P. Schindler, R. Rines, S. X. Wang, I. L. Chuang, and R. Blatt, Science 351, 1068 (2016).","DOI":"10.1126\/science.aad9480"},{"key":"116","doi-asserted-by":"publisher","unstructured":"C. Figgatt, D. Maslov, K. A. Landsman, N. M. Linke, S. Debnath, and C. Monroe, Nature Communications 8, 1918 (2017).","DOI":"10.1038\/s41467-017-01904-7"},{"key":"117","doi-asserted-by":"publisher","unstructured":"Y. Wan, D. Kienzler, S. D. Erickson, K. H. Mayer, T. R. Tan, J. J. Wu, H. M. Vasconcelos, S. Glancy, E. Knill, D. J. Wineland, A. C. Wilson, and D. Leibfried, Science 364, 875 (2019).","DOI":"10.1126\/science.aaw9415"},{"key":"118","doi-asserted-by":"publisher","unstructured":"D. T. C. Harty, T. P. et al. Allcock, C. J. Ballance, L. Guidoni, H. A. Janacek, N. M. Linke, D. N. Stacey, and D. M. Lucas, Phys. Rev. Lett. 113, 220501 (2014).","DOI":"10.1103\/PhysRevLett.113.220501"},{"key":"119","doi-asserted-by":"publisher","unstructured":"C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol, and D. M. Lucas, Phys. Rev. Lett. 117, 060504 (2016).","DOI":"10.1103\/PhysRevLett.117.060504"},{"key":"120","doi-asserted-by":"publisher","unstructured":"J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, Phys. Rev. Lett. 117, 060505 (2016).","DOI":"10.1103\/PhysRevLett.117.060505"},{"key":"121","doi-asserted-by":"publisher","unstructured":"T. P. Harty, M. A. Sepiol, D. T. C. Allcock, C. J. Ballance, J. E. Tarlton, and D. M. Lucas, Phys. Rev. Lett. 117, 140501 (2016).","DOI":"10.1103\/PhysRevLett.117.140501"},{"key":"122","doi-asserted-by":"publisher","unstructured":"A. Erhard, J. J. Wallman, L. Postler, M. Meth, R. Stricker, E. A. Martinez, P. Schindler, T. Monz, J. Emerson, and R. Blatt, Nature Communications 10, 5347 (2019).","DOI":"10.1038\/s41467-019-13068-7"},{"key":"123","doi-asserted-by":"publisher","unstructured":"G. Zarantonello, H. Hahn, J. Morgner, M. Schulte, A. Bautista-Salvador, R. F. Werner, K. Hammerer, and C. Ospelkaus, Phys. Rev. Lett. 123, 260503 (2019).","DOI":"10.1103\/PhysRevLett.123.260503"},{"key":"124","doi-asserted-by":"publisher","unstructured":"D. Kielpinski, C. Monroe, and D. J. Wineland, Nature 417, 709 (2002).","DOI":"10.1038\/nature00784"},{"key":"125","doi-asserted-by":"publisher","unstructured":"V. Kaushal, B. Lekitsch, A. Stahl, J. Hilder, D. Pijn, C. Schmiegelow, A. Bermudez, M. M\u00fcller, F. Schmidt-Kaler, and U. Poschinger, AVS Quantum Science 2, 014101 (2020).","DOI":"10.1116\/1.5126186"},{"key":"126","doi-asserted-by":"publisher","unstructured":"J. P. Home, D. Hanneke, J. D. Jost, J. M. Amini, D. Leibfried, and D. J. Wineland, Science 325, 1227 (2009).","DOI":"10.1126\/science.1177077"},{"key":"127","doi-asserted-by":"publisher","unstructured":"S. Debnath, N. M. Linke, C. Figgatt, K. A. Landsman, K. Wright, and C. Monroe, Nature 536, 63 (2016).","DOI":"10.1038\/nature18648"},{"key":"128","doi-asserted-by":"publisher","unstructured":"C. Figgatt, A. Ostrander, N. M. Linke, K. A. Landsman, D. Zhu, D. Maslov, and C. Monroe, Nature 572, 368 (2019).","DOI":"10.1038\/s41586-019-1427-5"},{"key":"129","doi-asserted-by":"publisher","unstructured":"I. Pogorelov, T. Feldker, C. D. Marciniak, L. Postler, G. Jacob, O. Krieglsteiner, V. Podlesnic, M. Meth, V. Negnevitsky, M. Stadler, B. H\u00f6fer, C. W\u00e4chter, K. Lakhmanskiy, R. Blatt, P. Schindler, and T. Monz, PRX Quantum 2, 020343 (2021).","DOI":"10.1103\/PRXQuantum.2.020343"},{"key":"130","doi-asserted-by":"publisher","unstructured":"C. Hempel, C. Maier, J. Romero, J. McClean, T. Monz, H. Shen, P. Jurcevic, B. P. Lanyon, P. Love, R. Babbush, A. Aspuru-Guzik, R. Blatt, and C. F. Roos, Phys. Rev. X 8, 031022 (2018).","DOI":"10.1103\/PhysRevX.8.031022"},{"key":"131","doi-asserted-by":"publisher","unstructured":"Y. Nam, J.-S. Chen, N. C. Pisenti, K. Wright, C. Delaney, D. Maslov, K. R. Brown, S. Allen, J. M. Amini, J. Apisdorf, K. M. Beck, A. Blinov, V. Chaplin, M. Chmielewski, C. Collins, S. Debnath, K. M. Hudek, A. M. Ducore, M. Keesan, S. M. Kreikemeier, J. Mizrahi, P. Solomon, M. Williams, J. D. Wong-Campos, D. Moehring, C. Monroe, and J. Kim, npj Quantum Information 6, 33 (2020).","DOI":"10.1038\/s41534-020-0259-3"},{"key":"132","doi-asserted-by":"publisher","unstructured":"C. Kokail, C. Maier, R. van Bijnen, T. Brydges, M. K. Joshi, P. Jurcevic, C. A. Muschik, P. Silvi, R. Blatt, C. F. Roos, and P. Zoller, Nature 569, 355 (2019).","DOI":"10.1038\/s41586-019-1177-4"},{"key":"133","doi-asserted-by":"publisher","unstructured":"U.-J. Wiese, Annalen der Physik 525, 777 (2013).","DOI":"10.1002\/andp.201300104"},{"key":"134","doi-asserted-by":"publisher","unstructured":"E. Zohar, J. I. Cirac, and B. Reznik, Reports on Progress in Physics 79, 014401 (2015).","DOI":"10.1088\/0034-4885\/79\/1\/014401"},{"key":"135","doi-asserted-by":"publisher","unstructured":"M. Dalmonte and S. Montangero, Contemporary Physics 57, 388 (2016).","DOI":"10.1080\/00107514.2016.1151199"},{"key":"136","doi-asserted-by":"publisher","unstructured":"M.-C. Ba\u00f1uls, R. Blatt, J. Catani, A. Celi, J. Cirac, M. Dalmonte, L. Fallani, K. Jansen, M. Lewenstein, S. Montangero, C. Muschik, B. Reznik, E. Rico Ortega, L. Tagliacozzo, K. Acoleyen, F. Verstraete, U.-J. Wiese, M. Wingate, J. Zakrzewski, and P. Zoller, The European Physical Journal D 74 (2020), 10.1140\/epjd\/e2020-100571-8.","DOI":"10.1140\/epjd\/e2020-100571-8"},{"key":"137","doi-asserted-by":"publisher","unstructured":"M. C. Ba\u00f1uls and K. Cichy, Reports on Progress in Physics 83, 024401 (2020).","DOI":"10.1088\/1361-6633\/ab6311"},{"key":"138","doi-asserted-by":"publisher","unstructured":"M. Aidelsburger, L. Barbiero, A. Bermudez, T. Chanda, A. Dauphin, D. Gonz\u00e1lez-Cuadra, P. R. Grzybowski, S. Hands, F. Jendrzejewski, J. J\u00fcnemann, G. Juzeli\u016bnas, V. Kasper, A. Piga, S.-J. Ran, M. Rizzi, G. Sierra, L. Tagliacozzo, E. Tirrito, T. V. Zache, J. Zakrzewski, E. Zohar, and M. Lewenstein, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380, 20210064 (2022).","DOI":"10.1098\/rsta.2021.0064"},{"key":"139","doi-asserted-by":"publisher","unstructured":"N. Klco, A. Roggero, and M. J. Savage, Reports on Progress in Physics 85, 064301 (2022).","DOI":"10.1088\/1361-6633\/ac58a4"},{"key":"140","doi-asserted-by":"publisher","unstructured":"C. W. Bauer, Z. Davoudi, A. B. Balantekin, T. Bhattacharya, M. Carena, W. A. de Jong, P. Draper, A. El-Khadra, N. Gemelke, M. Hanada, D. Kharzeev, H. Lamm, Y.-Y. Li, J. Liu, M. Lukin, Y. Meurice, C. Monroe, B. Nachman, G. Pagano, J. Preskill, E. Rinaldi, A. Roggero, D. I. Santiago, M. J. Savage, I. Siddiqi, G. Siopsis, D. Van Zanten, N. Wiebe, Y. Yamauchi, K. Yeter-Aydeniz, and S. 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