There are some concepts that are accepted in our daily life but are not trivial in physics. One of them is the cluster property that means there exist no relations between two events which are sufficiently separated. In the works recently published by the author, the extensive and quantitative examination has been made about the violation of cluster property in the correlation function of the spin operator for the quantum spin system. These works have shown that, when we include the symmetry breaking interaction, the effect by the violation is proportional to the inverse of the system size. Therefore this effect is tinny since the system size is quite large. In order to find the effect due to the violation even when the size is large, we propose a new system where additional spins couple with the spin system on the square lattice, where the coupling constant between these systems being assumed to be small. Applying the perturbation theory, we obtain the effective Hamiltonian for the additional system. This Hamiltonian includes Curie-Weiss model that is induced by the violation of the cluster property. Then we find that this effective Hamiltonian has the factor which is the inverse of the system size. Since Curie-Weiss model, which is known to be exactly soluble, has to contain this factor so that the thermodynamical properties are well-defined, the essential factor for the Hamiltonian is determined by the coupling and the strength of the symmetry breaking interaction. Our conclusion is, therefore, that it is possible to observe the effect by the violation of the cluster property at the inverse temperature whose order is given by these parameters.
References
[1]
Bell, J.S. (1964) On the Einstein-Podolsky-Rosen Paradox. Physics, 1, 195.
https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195
[2]
Kochen, S. and Specker, E.P. (1967) The Problem of Hidden Variables in Quantum Mechanics. Journal of Mathematics and Mechanics, 17, 59-87.
https://doi.org/10.1512/iumj.1968.17.17004
[3]
Hardy, L. (1993) Nonlocality of Two Particles without Inequalities for Almost All Entangled States. Physics Review Letters, 71, 1665-1668.
https://doi.org/10.1103/PhysRevLett.71.1665
[4]
’tHooft, G. (2006) The Mathematical Basis for Deterministic Quantum Mechanics. In: Beyond the Quantum, World Scientific, Singapore, 3-19.
https://doi.org/10.1142/9789812771186_0001
[5]
Horodecki, R., Horodecki, P., Horodecki, M. and Horodecki, K. (2009) Quantum Entanglement. Review of Modern Physics, 81, 865.
https://doi.org/10.1103/RevModPhys.81.865
[6]
Deutsch, D. (1985) Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer. Proceedings of the Royal Society of London. Series A, 400, 97. https://doi.org/10.1098/rspa.1985.0070
[7]
Nielsen, M.A. and Chuang, I.L. (2000) Quantum Computation and Quantum Information. Cambridge University, Cambridge.
[8]
Gühne, O. and Tóth, G. (2009) Entanglement Detection. Physics Reports, 474, 1-75.
https://doi.org/10.1016/j.physrep.2009.02.004
[9]
Ekert, A., Alves, C., Oi, D., Horodecki, M., Horodecki, P. and Kwek, L. (2002) Direct Estimations of Linear and Nonlinear Functionals of a Quantum State. Physical Review Letters, 88, Article ID: 217901.
https://doi.org/10.1103/PhysRevLett.88.217901
[10]
Napoli, C., Bromley, T.R., Cianciaruso, M., Piani, M., Johnston, N. and Adesso, G. (2016) Robustness of Coherence: An Operational and Observable Measure of Quantum Coherence. Physical Review Letters, 116, Article ID: 150502.
https://doi.org/10.1103/PhysRevLett.116.150502
[11]
Amico, L., Fazio, R., Osterloh, A. and Vedral, V. (2008) Entanglement in Many-Body Systems. Review of Modern Physics, 80, 517.
https://doi.org/10.1103/RevModPhys.80.517
[12]
Islam, R., Ma, R., Preiss, P.M., Tai, M.E., Lukin, A., Rispoli, M. and Greiner, M. (2015) Measuring Entanglement Entropy in a Quantum Many-Body System. Nature, 528, 77. https://doi.org/10.1038/nature15750
[13]
Cianciaruso, M., Ferro, L., Giampaolo, S.M., Zonzo, G. and Illuminati, F. (2014) Classical Nature of Ordered Phases: Origin of Spontaneous Symmetry Breaking.
[14]
Shi, Y. (2003) Quantum Disentanglement in Long-Range Orders and Spontaneous Symmetry Breaking. Physics Letters A, 309, 254-261.
https://doi.org/10.1016/S0375-9601(03)00128-2
[15]
Heaney, L., Anders, J., Kaszlikowski, D. and Vedral, V. (2007) Spatial Entanglement from Off-Diagonal Long-Range Order in a Bose-Einstein Condensate. Physics Review A, 76, Article ID: 053605. https://doi.org/10.1103/PhysRevA.76.053605
[16]
Hamma, A., Giampaolo, S. and Illuminati, F. (2016) Mutual Information and Spontaneous Symmetry Breaking. Physics Review A, 93, Article ID: 012303.
https://doi.org/10.1103/PhysRevA.93.012303
[17]
Weinberg, S. (1995) The Quantum Theory of Fields. Vol. 2, Cambridge Univ. Press, Cambridge. https://doi.org/10.1017/CBO9781139644167
Strocchi, F. (1978) Local and Covariant Gauge Quantum Field Theories. Cluster Property, Superselection Rules, and the Infrared Problem. Physics Review D, 17, 2010-2021. https://doi.org/10.1103/PhysRevD.17.2010
[20]
Lowdon, P. (2016) Conditions on the Violation of the Cluster Decomposition Property in QCD. Journal of Mathematical Physics, 57, Article ID: 102302.
https://doi.org/10.1063/1.4965715
[21]
Xu, S. and Fan, S. (2017) Generalized Cluster Decomposition Principle Illustrated in Waveguide Quantum Electrodynamics. Physics Review A, 95, Article ID: 063809.
https://doi.org/10.1103/PhysRevA.95.063809
[22]
Frohlich, J. and Rodríguez, P. (2017) On Cluster Properties of Classical Ferromagnets in an External Magnetic Field. Journal of Statistical Physics, 166, 828-840. https://doi.org/10.1007/s10955-016-1556-2
[23]
Shimizu, A. and Miyadera, T. (2002) Cluster Property and Robustness of Ground States of Interacting Many Bosons. Journal of the Physical Society of Japan, 71, 56-59. https://doi.org/10.1143/JPSJ.71.56
[24]
Shimizu, A. and Miyadera, T. (2002) Stability of Quantum States of Finite Macroscopic Systems against Classical Noises, Perturbations from Environments, and Local Measurements. Physical Review Letters, 89, Article ID: 270403.
https://doi.org/10.1103/PhysRevLett.89.270403
[25]
Dell’Anna, L., Salberger, O., Barbiero, L., Trombettoni, A. and Korepin, V. (2016) Violation of Cluster Decomposition and Absence of Light Cones in Local Integer and Half-Integer Spin Chains. Physics Review B, 94, Article ID: 155140.
https://doi.org/10.1103/PhysRevB.94.155140
[26]
Munehisa, T. (2018) Violation of Cluster Property in Quantum Antiferromagnet. World Journal of Condensed Matter Physics, 8, 1-22.
https://doi.org/10.4236/wjcmp.2018.81001
[27]
Richter, J., Schulenburg, J. and Honecker, A. (2004) Quantum Magnetism. In: Schollwock, U., Richter, J., Farnell, D.J.J. and Bishop, R.F., Eds., Lecture Note in Physics, Volume 645, Springer-Verlag, Berlin Heidelberg, 85-153.
Auerbach, A. (1994) Interacting Electrons and Quantum Magnetism. Springer-Verlag, Berlin Heidelberg. https://doi.org/10.1007/978-1-4612-0869-3
[30]
Hatano, N. and Suzuki, M. (1993) Quantum Monte Carlo Methods in Condensed Matter Physics. World Scientific, Singapore, 13-47.
https://doi.org/10.1142/9789814503815_0002
[31]
De Raedt, H. and von der Linden, W. (1995) The Monte Carlo Method in Condensed Matter Physics. Springer-Verlag, Berlin Heidelberg, 249-284.
[32]
Manousakis, E. (1991) The Spin 1/2 Heisenberg Antiferromagnet on a Square Lattice and Its Application to the Cuprous Oxides. Review Modern of Physics, 63, 1.
https://doi.org/10.1103/RevModPhys.63.1
[33]
Landee, C. and Turnbull, M. (2013) Recent Developments in Low-Dimensional Copper(II) Molecular Magnets. European Journal of Inorganic Chemistry, 2013, 2266-2285. https://doi.org/10.1002/ejic.201300133
[34]
Munehisa, T. (2018) Violation of Cluster Property in Heisenberg Antiferromagnet. World Journal of Condensed Matter Physics, 8, 203-229.
https://doi.org/10.4236/wjcmp.2018.84015
[35]
Kochmanski, M., Paszkiewicz, T. and Wolski, S. (2013) Curie-Weiss Magnet: A Simple Model of Phase Transition. European Journal of Physics, 34.
[36]
Chayes, L., Crawford, N., Ioffe, D. and Levit, A. (2008) The Phase Diagram of the Quantum Curie-Weiss Model. Journal of Statistical Physics, 133,131.
[37]
Carneiro, C. and Pellegrino, G. (2018) Analysis of Quantum Phase Transition in Some Different Curie-Weiss Models: A Unified Approach.
[38]
Predescu, C. (2002) Local Variational Principle. Review Physics Review E, 66, Article ID: 066133. https://doi.org/10.1103/PhysRevE.66.066133
[39]
Prato, D. and Barraco, D. (1996) Bogoliubov Inequality. Revista Mexicana de Físiro, 42, 145-150.
