Abstract
In recent years, various aspects of theoretical models with long range interactions have attracted attention, ranging from out-of-time-ordered correlators to entanglement. In the present paper, entanglement properties of a simple non-local model with long-range interactions in the form of a fractional Laplacian is investigated in both static and a quantum quench scenario. Logarithmic negativity, which is a measure for entanglement in mixed states is calculated numerically. In the static case, it is shown that the presence of long-range interaction ensures that logarithmic negativity decays much slower with distance compared to short-range models. For a sudden quantum quench, the temporal evolution of the logarithmic negativity reveals that, in contrast to short-range models, logarithmic negativity exhibits no revivals for long-range interactions for the time intervals considered. To further support this result, a simpler measure of entanglement, namely the entanglement entropy is also studied for this class of models.
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References
M.B. Hastings and T. Koma, Spectral gap and exponential decay of correlations, Commun. Math. Phys. 265 (2006) 781 [math-ph/0507008] [INSPIRE].
R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81 (2009) 865 [quant-ph/0702225] [INSPIRE].
M.B. Plenio, Logarithmic Negativity: A Full Entanglement Monotone That is not Convex, Phys. Rev. Lett. 95 (2005) 090503 [quant-ph/0505071] [INSPIRE].
K. Audenaert, J. Eisert, M.B. Plenio and R.F. Werner, Entanglement Properties of the Harmonic Chain, Phys. Rev. A 66 (2002) 042327 [quant-ph/0205025] [INSPIRE].
J. Angel-Ramelli, C. Berthiere, V.G.M. Puletti and L. Thorlacius, Logarithmic Negativity in Quantum Lifshitz Theories, JHEP 09 (2020) 011 [arXiv:2002.05713] [INSPIRE].
M.R. Mohammadi Mozaffar and A. Mollabashi, Logarithmic Negativity in Lifshitz Harmonic Models, J. Stat. Mech. 1805 (2018) 053113 [arXiv:1712.03731] [INSPIRE].
S. Marcovitch, A. Retzker, M.B. Plenio and B. Reznik, Critical and noncritical long-range entanglement in Klein-Gordon fields, Phys. Rev. A 80 (2009) 012325 [arXiv:0811.1288] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in quantum field theory, Phys. Rev. Lett. 109 (2012) 130502 [arXiv:1206.3092] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in extended systems: A field theoretical approach, J. Stat. Mech. 1302 (2013) P02008 [arXiv:1210.5359] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Finite temperature entanglement negativity in conformal field theory, J. Phys. A 48 (2015) 015006 [arXiv:1408.3043] [INSPIRE].
N. Shiba and T. Takayanagi, Volume Law for the Entanglement Entropy in Non-local QFTs, JHEP 02 (2014) 033 [arXiv:1311.1643] [INSPIRE].
B. Basa, G. La Nave and P.W. Phillips, Classification of nonlocal actions: Area versus volume entanglement entropy, Phys. Rev. D 101 (2020) 106006 [arXiv:1907.09494] [INSPIRE].
T.-C. Lu and T. Grover, Structure of Quantum Entanglement at a Finite Temperature Critical Point, Phys. Rev. Res. 2 (2020) 043345 [arXiv:1907.01569] [INSPIRE].
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Part. Diff. Eq. 32 (2007) 1245 [math/0608640].
M. Ghasemi Nezhadhaghighi and M.A. Rajabpour, Entanglement dynamics in short and long-range harmonic oscillators, Phys. Rev. B 90 (2014) 205438 [arXiv:1408.3744] [INSPIRE].
K. Kim, M.-S. Chang, R. Islam, S. Korenblit, L.-M. Duan and C. Monroe, Entanglement and tunable spin-spin couplings between trapped ions using multiple transverse modes, Phys. Rev. Lett. 103 (2009) 120502 [arXiv:0905.0225].
T. Koffel, M. Lewenstein and L. Tagliacozzo, Entanglement entropy for the long range Ising chain, Phys. Rev. Lett. 109 (2012) 267203 [arXiv:1207.3957] [INSPIRE].
R.G. Unanyan and M. Fleischhauer, Entanglement dynamics in harmonic-oscillator chains, Phys. Rev. A 89 (2014) 062330 [arXiv:1011.4838].
N. Nessi, A. Iucci and M.A. Cazalilla, Quantum Quench and Prethermalization Dynamics in a Two-Dimensional Fermi Gas with Long-range Interactions, Phys. Rev. Lett. 113 (2014) 210402 [arXiv:1401.1986] [INSPIRE].
M.R. Mohammadi Mozaffar and A. Mollabashi, Entanglement Evolution in Lifshitz-type Scalar Theories, JHEP 01 (2019) 137 [arXiv:1811.11470] [INSPIRE].
M.R.M. Mozaffar and A. Mollabashi, Time scaling of entanglement in integrable scale-invariant theories, Phys. Rev. Res. 4 (2022) L022010 [arXiv:2106.14700] [INSPIRE].
B. Nachtergaele, Y. Ogata and R. Sims, Propagation of correlations in quantum lattice systems, J. Stat. Phys. 124 (2006) 1.
M. Foss-Feig, Z.-X. Gong, C.W. Clark and A.V. Gorshkov, Nearly linear light cones in long-range interacting quantum systems, Phys. Rev. Lett. 114 (2015) 157201 [arXiv:1410.3466].
Z.-X. Gong, M. Foss-Feig, S. Michalakis and A.V. Gorshkov, Persistence of locality in systems with power-law interactions, Phys. Rev. Lett. 113 (2014) 030602 [arXiv:1401.6174].
A.Y. Guo, M.C. Tran, A.M. Childs, A.V. Gorshkov and Z.-X. Gong, Signaling and scrambling with strongly long-range interactions, Phys. Rev. A 102 (2020) 010401 [arXiv:1906.02662] [INSPIRE].
A.M. Frassino and O. Panella, Quantization of nonlocal fractional field theories via the extension problem, Phys. Rev. D 100 (2019) 116008 [arXiv:1907.00733] [INSPIRE].
A. Serafini, Quantum continuous variables: a primer of theoretical methods, CRC press, Boca Raton, U.S.A. (2017).
J.S. Cotler, M.P. Hertzberg, M. Mezei and M.T. Mueller, Entanglement Growth after a Global Quench in Free Scalar Field Theory, JHEP 11 (2016) 166 [arXiv:1609.00872] [INSPIRE].
A. Coser, E. Tonni and P. Calabrese, Entanglement negativity after a global quantum quench, J. Stat. Mech. 1412 (2014) P12017 [arXiv:1410.0900] [INSPIRE].
H. Wichterich, J. Molina-Vilaplana and S. Bose, Scaling of entanglement between separated blocks in spin chains at criticality, Phys. Rev. A 80 (2009) 010304 [arXiv:0811.1285] [INSPIRE].
P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. 0504 (2005) P04010 [cond-mat/0503393] [INSPIRE].
M. Fagotti and P. Calabrese, Entanglement entropy of two disjoint blocks in XY chains, J. Stat. Mech. 1004 (2010) P04016 [arXiv:1003.1110] [INSPIRE].
V. Alba and P. Calabrese, Entanglement and thermodynamics after a quantum quench in integrable systems, Proc. Nat. Acad. Sci. 114 (2017) 7947.
R. Modak, V. Alba and P. Calabrese, Entanglement revivals as a probe of scrambling in finite quantum systems, J. Stat. Mech. 2008 (2020) 083110 [arXiv:2004.08706] [INSPIRE].
M.A. Rajabpour and S. Sotiriadis, Quantum quench in long-range field theories, Phys. Rev. B 91 (2015) 045131 [arXiv:1409.6558] [INSPIRE].
V. Alba and P. Calabrese, Quantum information scrambling after a quantum quench, Phys. Rev. B 100 (2019) 115150 [arXiv:1903.09176] [INSPIRE].
S.R. Das, D.A. Galante and R.C. Myers, Universal scaling in fast quantum quenches in conformal field theories, Phys. Rev. Lett. 112 (2014) 171601 [arXiv:1401.0560] [INSPIRE].
S.R. Das, D.A. Galante and R.C. Myers, Universality in fast quantum quenches, JHEP 02 (2015) 167 [arXiv:1411.7710] [INSPIRE].
S.R. Das, D.A. Galante and R.C. Myers, Smooth and fast versus instantaneous quenches in quantum field theory, JHEP 08 (2015) 073 [arXiv:1505.05224] [INSPIRE].
A. Larkin and Y.N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, Sov. Phys. JETP 28 (1969) 1200.
A. Kitaev, A simple model of quantum holography, in KITP strings seminar and Entanglement. Vol. 12, Kavli Institute for Theoretical Physics, Santa Barbara, U.S.A. (2015), pg. 26.
S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
E.B. Rozenbaum, S. Ganeshan and V. Galitski, Lyapunov Exponent and Out-of-Time-Ordered Correlator’s Growth Rate in a Chaotic System, Phys. Rev. Lett. 118 (2017) 086801 [arXiv:1609.01707] [INSPIRE].
J. Chávez-Carlos et al., Quantum and Classical Lyapunov Exponents in Atom-Field Interaction Systems, Phys. Rev. Lett. 122 (2019) 024101 [arXiv:1807.10292] [INSPIRE].
C.-J. Lin and O.I. Motrunich, Out-of-time-ordered correlators in a quantum Ising chain, Phys. Rev. B 97 (2018) 144304 [arXiv:1801.01636] [INSPIRE].
E. Iyoda and T. Sagawa, Scrambling of Quantum Information in Quantum Many-Body Systems, Phys. Rev. A 97 (2018) 042330 [arXiv:1704.04850] [INSPIRE].
K. Hashimoto, K. Murata and R. Yoshii, Out-of-time-order correlators in quantum mechanics, JHEP 10 (2017) 138 [arXiv:1703.09435] [INSPIRE].
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Roy, P. Aspects of entanglement in non-local field theories with fractional Laplacian. J. High Energ. Phys. 2022, 101 (2022). https://doi.org/10.1007/JHEP06(2022)101
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DOI: https://doi.org/10.1007/JHEP06(2022)101