Monte Carlo simulation of distance dependent quantum entanglement in mixed XXZ Heisenberg spin-1/2 chains

Year 2024, Volume: 2 Issue: 2, 131 - 151, 27.09.2024

İzzet Paruğ Duru , Şahin Aktaş

Abstract

The quantum entanglement of mixed XXZ Heisenberg spin-1/2 chain is examined. We quantify localizable entanglement (LE) in terms of upper/lower bounds through Quantum Monte Carlo simulations. Loop algorithm is chosen to numerically calculate thermodynamic quantities including spin-spin correlations. The exchange coupling, Zeeman energy, and dipolar interaction are literally taken into account. Findings summarize that the strength of dipole-dipole interaction (D) and external magnetic field (Bz) are notable in entanglement formation driving creation and extinction of entanglement. The creation and extinction of entanglement depend on D and Bz. Furthermore, strong fields at the critical temperatures lead a non-monotonic/monotonic behavior introducing revival phenomena. Nevertheless, strong D provides the distance-dependent stability of LE values, preserving unity.

Keywords

mixed magnetic state, quantum entanglement, Monte Carlo simulation, sudden death, critical temperature

Supporting Institution

Scientific Research Projects Commission of Marmara University

Project Number

FEN-C-DRP-120613-0273

Thanks

Authors thank Scientific Research Projects Commission of Marmara University due to its valuable support with a project number of FEN-C-DRP-120613-0273.

References

• Modławska, J., & Grudka, A., (2008). Nonmaximally Entangled States Can Be Better for Multiple Linear Optical Teleportation. Physical Review Letters, 100, 110503.
• Cavalcanti, D., Skrzypczyk, P., & Šupić, I., (2017).All Entangled States can Demonstrate Nonclassical Teleportation. Physical Review Letters, 119, 110501.
• Z.A.Sabegh, R., & Mahmoudi, M., (2018). Spatially dependent atom-photon entanglement. Scientific Reports, 8, 13840.
• Loss, D., & DiVincenzo, D., (1998). Quantum computation with quantum dots. Physical Review A, 57, 120.
• Jürgen Audretsch (2007), The Quantum Computer. In Entangled Systems: New Directions in Quantum Physics (pp.219-245), Weinheim, Germany: John Wiley and Sons, Ltd.
• Belsley, M. (2014). Introduction to Quantum Information Science, by Vlatko Vedral. Contemporary Physics, 55, 124.
• DiVincenzo, D. (1997). Quantum computation and spin physics (invited). Journal of Applied Physics, 81, 4602-4607.
• Zheng, S., & Guo, G., (2000). Efficient Scheme for Two-Atom Entanglement and Quantum Information Processing in Cavity QED. Physical Review Letters, 85, 2392-2395.
• Bennett, C., & DiVincenzo, D., (2000). Quantum information and computation. Nature, 404, 1476-4687.
• Eggert, S., Affleck, I., & Takahashi, M., (1994). Susceptibility of the spin 1/2 Heisenberg antiferromagnetic chain. Physical Review Letters, 73, 332-335.
• Hammar, P., Stone, M., Reich, D., Broholm, C., Gibson, P., Turnbull, M., Landee, C., & Oshikawa, M., (1999). Characterization of a quasi-one-dimensional spin-1/2 magnet which is gapless and paramagnetic for g μ B H ≲ J and k_B T≪ J. Physical Review B, 59, 1008-1015.
• Androvitsaneas, P., Fytas, N., Paspalakis, E., & Terzis, A.F., (2012). Quantum Monte Carlo simulations revisited: The case of anisotropic Heisenberg chains. Philosophical Magazine, 92, 4649-4656.
• Barma, M., & Shastry, B., (1978). Classical equivalents of one-dimensional quantum-mechanical systems. Physical Review B, 18, 3351-3359.
• Handscomb, D. (1964). A Monte Carlo method applied to the Heisenberg ferromagnet. Mathematical Proceedings of the Cambridge Philosophical Society, 60, 115-122.
• Harada, K., & Kawashima, N., (2001). Loop Algorithm for Heisenberg Models with Biquadratic Interaction and Phase Transitions in Two Dimensions. Journal of the Physical Society of Japan, 70, 13-16.
• Huang, Y., & Su, G., (2017). Quantum Monte Carlo study of the spin-1/2 honeycomb Heisenberg model with mixed antiferromagnetic and ferromagnetic interactions in external magnetic fields. Physical Review E, 95, 052147.
• Sandvik, A., & Kurkijärvi, J., (1991). Quantum Monte Carlo simulation method for spin systems. Physical Review B, 43, 5950-5961.
• Deger, C., Aksu, P., & Yildiz, F., (2016). Effect of Interdot Distance on Magnetic Behavior of 2-D Ni Dot Arrays. IEEE Transactions on Magnetics, 52, 1-4.
• Duru, I., Değer, C., Kalaycı, T., & Arucu, M., (2015). A computational study on magnetic effects of Zn_(1-x) Cr_x O type diluted magnetic semiconductor. Journal of Magnetism and Magnetic Materials, 396 pp. 268-274.
• Arnesen, M., Bose, S., & Vedral, V., (2001). Natural Thermal and Magnetic Entanglement in the 1D Heisenberg Model. Physical Review Letters, 87, 017901.
• Marchukov, O.V., & Zinner, N., (2016). Quantum spin transistor with a Heisenberg spin chain. Nature Communications, 7, 13070.
• Renes, J., Miyake, A., Brennen, G., & Bartlett, S., (2013). Holonomic quantum computing in symmetry-protected ground states of spin chains. New Journal of Physics, 15, 025020.
• Apollaro, T., Lorenzo, S., Sindona, A., Paganelli, S., Giorgi, G., & Plastina, F., (2015). Many-qubit quantum state transfer via spin chains. Physica Scripta, T165 pp. 014036.
• Wang, X., (2002). Threshold temperature for pairwise and many-particle thermal entanglement in the isotropic Heisenberg model. Physical Review A, 66, 044305.
• Wang, X., (2001). Entanglement in the quantum Heisenberg XY model. Physical Review A, 64, 012313.
• Rigolin, G., (2004). Thermal entanglement in the two-qubit Heisenberg XYZ model. International Journal of Quantum Information, 2, 393-405.
• Androvitsaneas, P., Paspalakis, E., & Terzis, A., (2012). A quantum Monte Carlo study of the localizable entanglement in anisotropic ferromagnetic Heisenberg chains. Annals of Physics, 327, 212-223.
• Sinyagin, A., Belov, A., Tang, Z., & Kotov, N., (2006). Monte Carlo Computer Simulation of Chain Formation from Nanoparticles. Journal of Physical Chemistry B, 110, 7500-7507.
• Kim, I., (2013). Long-Range Entanglement is Necessary for a Topological Storage of Quantum Information. Physical Review Letters, 111, 080503.
• Elman, S., Bartlett, S., & Doherty, A., (2017). Long-range entanglement for spin qubits via quantum Hall edge modes. Physical Review B, 96, 115407.
• Bitko, D., Rosenbaum, T., & Aeppli, G., (1996). Quantum Critical Behavior for a Model Magnet. Physical Review Letters, 77, 940-943.
• Chakraborty, P., Henelius, P., Kjønsberg, H., Sandvik, A., & Girvin, S., (2004). Theory of the magnetic phase diagram of LiHoF_4. Physical Review B, 70, 144411.
• Bramwell, S., & Gingras, M., (2001). Spin Ice State in Frustrated Magnetic Pyrochlore Materials. Science, 294, 1495-1501.
• Castelnovo, C. R., & Sondhi, S., (2008). Magnetic Monopoles in Spin Ice. Nature, 451, 42-45.
• Mengotti, E., Heyderman, L., Bisig, A., Fraile Rodríguez, A., Le Guyader, L., Nolting, F., & Braun, H., (2009). Dipolar energy states in clusters of perpendicular magnetic nanoislands. Journal of Applied Physics, 105, 113113.
• Lahaye, T., Menotti, C., Santos, L., Lewenstein, M., & Pfau, T., (2009). The physics of dipolar bosonic quantum gases. Reports on Progress in Physics. 72, 126401.
• Peter, D., Müller, S., Wessel, S., & Büchler, H., (2012). Anomalous Behavior of Spin Systems with Dipolar Interactions. Physical Review Letters, 109, 025303.
• Islam, R., Senko, C., Campbell, W., Korenblit, S., Smith, J., Lee, A., Edwards, E., Wang, C., Freericks, J., & Monroe, C., (2013). Emergence and Frustration of Magnetism with Variable-Range Interactions in a Quantum Simulator. Science, 340, 583-587.
• Jurcevic, P., & Roos, C., (2014). Quasiparticle engineering and entanglement propagation in a quantum many-body system. Nature, 511, 202-205.
• Richerme, P., & Monroe, C., (2014). Non-local propagation of correlations in quantum systems with long-range interactions. Nature, j511 pp. 198-201.
• Mahmoudian S., Rademaker L., Ralko A., Fratini S., & Dobrosavljevic V., (2015). Glassy Dynamics in Geometrically Frustrated Coulomb Liquids without Disorder. Physical Review Letters, 115, 025701.
• Bohnet, J., Sawyer, B., Britton, J., Wall, M., Rey, A., Foss-Feig, M., & Bollinger, J., (2016). Quantum spin dynamics and entanglement generation with hundreds of trapped ions. Science, 352, 1297-1301.
• Sahling, S., & Lorenzo, E., (2015). Experimental realization of long-distance entanglement between spins in antiferromagnetic quantum spin chains. Nature Physics, 15, 255-260.
• Osborne, T., & Nielsen, M., (2002). Entanglement in a simple quantum phase transition. Physical Review A, 66, 032110.
• Vidal, G., Latorre, J., Rico, E., & Kitaev, A., (2003). Entanglement in Quantum Critical Phenomena. Physical Review Letters, 90, 227902.
• Bravo, B., Cabra, D., Gomez Albarracin, F., & Rossini, G., (2017). Long-range interactions in antiferromagnetic quantum spin chains. Physical Review B, 96, 054441.
• Duru, I.P., & Aktas, S., (2019). Localizable entanglement of isotropic antiferromagnetic spin-1/2 chain. Turkish Journal of Physics. 43 pp. 272 - 279.
• Bauer, B., Carr, L., Evertz, H., Feiguin, A., Freire, J., Fuchs, S., Gamper, L., Gukelberger, J., Gull, E., Guertler, S., Hehn, A., Igarashi, R., Isakov, S., Koop, D., Ma, P., Mates, P., Matsuo, H., Parcollet, O., Pawlowski, G., Picon, J., Pollet, L., Santos, E., Scarola, V., Schollwöck, U., Silva, C., Surer, B., Todo, S., Trebst, S., Troyer, M., Wall, M., Werner, P., & Wessel, S., (2011). The ALPS project release 2.0: open source software for strongly correlated systems. Journal of Statistical Mechanics: Theory and Experiment, 2011, P05001.
• DiVincenzo & Uhlmann, A., (1999). Entanglement of Assistance. Quantum Computing and Quantum Communications, pp. 247-257.
• Laustsen, T., Verstraete, F., & Van Enk, S., (2003). Local vs. Joint Measurements for the Entanglement of Assistance. Quantum Information and Computation, 3, 64-83.
• Popp, M., Verstraete, F., Martin-Delgado, M., & Cirac, J., (2005). Localizable entanglement. Physical Review A, 71, 042306.
• Todo, S., & Kato, K., (2001). Cluster Algorithms for General S Quantum Spin Systems. Physical Review Letters, 87, 047203.
• Vedral, V., & Plenio, M., (1998). Entanglement measures and purification procedures. Physical Review A, 57, 1619-1633.
• Qi, X., Gao, T., & Yan, F., (2017). Lower bounds of concurrence for N-qubit systems and the detection of k-nonseparability of multipartite quantum systems. Quantum Information Process, 16, 23.
• Xue-Na Zhu, M., & Fei, S., (2018). A lower bound of concurrence for multipartite quantum systems. Quantum Information Processing, 17, 30.
• Cornelio, M., (2013). Multipartite monogamy of the concurrence. Physical Review A, 87, 032330.
• Evertz, H. G., Lana, G., & Marcu, M. (1993). Cluster algorithm for vertex models. Physical Review Letters, 70(7), 875.
• Evertz, H. G., & Marcu, M. (1993). The loop-cluster algorithm for the case of the 6 vertex model. Nuclear Physics B-Proceedings Supplements, 30, 277–280.
• Kawashima, N., Gubernatis, J. E., & Evertz, H. G. (1994). Loop algorithms for quantum simulations of fermion models on lattices. Physical Review B, 50(1), 136.
• Sandvik, A. W. (1992). A generalization of Handscomb’s quantum Monte Carlo scheme-application to the 1D Hubbard model. Journal of Physics A: Mathematical and General, 25(13), 3667.
• Sandvik, A. W. (1997). Finite-size scaling of the ground-state parameters of the two-dimensional Heisenberg model. Physical Review B, 56(18), 11678.
• Sandvik, A. W., & Kurkijärvi, J. (1991). Quantum Monte Carlo simulation method for spin systems. Physical Review B, 43(7), 5950.
• Scalettar, R. T. (1998). World-line quantum Monte Carlo. Quantum Monte Carlo Methods in Physics and Chemistry, 525, 65.
• Swendsen, R. H., & Wang, J.-S. (1987). Nonuniversal critical dynamics in Monte Carlo simulations. Physical Review Letters, 58(2), 86.
• Fortuin, C. M. (1969). Physica (Utrecht) 57, 536 (1972); PW Kasteleyn and CM Fortuin. J. Phys. Soc. Jpn. Suppl, 26(11).
• Fortuin, C. M., & Kasteleyn, P. W. (1972). On the random-cluster model: I. Introduction and relation to other models. Physica, 57(4), 536–564.
• Duru, İ.P., & Aktaş, Ş. (2022). Dipole-Dipole Effect to Limits of Entanglement in Multipartite Spin Chain: A Monte Carlo Study. International Journal of Advances in Engineering and Pure Sciences, 34(2), 305-316. https://doi.org/10.7240/jeps.1032914