Geodesic
Spectral gap of the antiferromagnetic Lipkin–Meshkov–Glick model
Teoretičeskaâ i matematičeskaâ fizika, Tome 195 (2018) no. 2, pp. 256-268 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study the spectral property of the supersymmetric (SUSY) antiferromagnetic Lipkin–Meshkov–Glick (LMG) model with an even number of spins and explicitly construct the supercharges of the model. Using the exact form of the SUSY ground state, we introduce simple trial variational states for the first excited states. We show numerically that they provide a relatively accurate upper bound for the spectral gap (the energy difference between the ground state and first excited states) in all parameter ranges, but because it is an upper bound, it does not allow rigorously determining whether the model is gapped or gapless. To answer this question, we obtain a nontrivial lower bound for the spectral gap and thus show that the antiferromagnetic SUSY LMG model is gapped for any even number of spins.
Keywords: integrable model, Lipkin–Meshkov–Glick model, many-body system, spectral gap. 
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     title = {Spectral gap of the~antiferromagnetic {Lipkin{\textendash}Meshkov{\textendash}Glick} model},
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R. Unanyan. Spectral gap of the antiferromagnetic Lipkin–Meshkov–Glick model. Teoretičeskaâ i matematičeskaâ fizika, Tome 195 (2018) no. 2, pp. 256-268. http://geodesic.mathdoc.fr/item/TMF_2018_195_2_a6/

[1] H. J. Lipkin, N. Meshkov, A. J. Glick, “Validity of many-body approximation methods for a solvable model: (I). Exact solutions and perturbation theory”, Nucl. Phys., 62:2 (1965), 188–198 | DOI | MR

[2] J. I. Cirac, M. Lewenstein, K. Mølmer, P. Zoller, “Quantum superposition states of Bose–Einstein condensates”, Phys. Rev. A, 57:2 (1998), 1208–1218 | DOI

[3] R. G. Unanyan, M. Fleischhauer, “Decoherence-free generation of many-particle entanglement by adiabatic ground-state transitions”, Phys. Rev. Lett., 90:13 (2003), 133601, 4 pp. ; R. G. Unanyan, C. Ionescu, M. Fleischhauer, “Many-particle entanglement in the gaped antiferromagnetic Lipkin model”, Phys. Rev. A, 72:2 (2005), 022326, 7 pp. | DOI | DOI

[4] J. Larson, “Circuit QED scheme for the realization of the Lipkin–Meshkov–Glick model”, Europhys. Lett., 90:5 (2010), 54001, 6 pp., arXiv: ; Y.-C. Zhang, X.-F. Zhou, X. Zhou, G.-C. Guo, Z.-W. Zhou, “Cavity-assisted single-mode and two-mode spin-squeezed states via phase-locked atom–photon coupling”, Phys. Rev. Lett., 118:8 (2017), 083604, 6 pp. 1004.5041 | DOI | DOI

[5] E. Witten, “Dynamical breaking of supersymmetry”, Nucl. Phys. B, 188:3 (1981), 513–554 ; Л. Э. Генденштейн, И. В. Криве, “Суперсимметрия в квантовой механике”, УФН, 146:4 (1985), 553–590 ; F. Cooper, A. Khare, U. Sukhatme, “Supersymmetry and quantum mechanics”, Phys. Rep., 267:5–6 (1995), 267–385 | DOI | Zbl | DOI | DOI | MR | DOI | MR

[6] A. Garg, “Quenched spin tunneling and diabolical points in magnetic molecules. I. Symmetric configurations”, Phys. Rev. B, 64:9 (2001), 094413, 15 pp. ; E. Keçecioǧlu, A. Garg, “$SU(2)$ instantons with boundary jumps and spin tunneling in magnetic molecules”, Phys. Rev. Lett., 88:23 (2002), 237205, 4 pp. | DOI | DOI

[7] C. Eckart, “The theory and calculation of screening constants”, Phys. Rev., 36:5 (1930), 878–892 | DOI

[8] L. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics. Theory and Application, Encyclopedia of Mathematics and its Applications, 8, Addison–Wesley, Reading, MA, 1981 ; Д. А. Варшалович, А. Н. Москалев, В. К. Херсонский, Квантовая теория углового момента, Наука, Л., 1975 | MR | MR

[9] G. Sege, Ortogonalnye mnogochleny, Fizmatgiz, M., 1962 | MR | MR | Zbl

[10] H. Weyl, “Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)”, Math. Ann., 71:4 (1912), 441–479 ; W. Fulton, “Eigenvalues, invariant factors, highest weights, and Schubert calculus”, Bull. Amer. Math. Soc. (N. S.), 37:3 (2000), 209–249 | DOI | MR | Zbl | DOI | MR | Zbl

[11] R. Blatt, C. F. Roos, “Quantum simulations with trapped ions”, Nature Phys., 8:4 (2012), 277–284 | DOI