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Degenerate States in Nonlinear Sigma Model with U(1) Symmetry

—For Study on Violation of Cluster Property

DOI: 10.4236/wjcmp.2021.113003, PP. 29-52

Keywords: Quantum Nonlinear Sigma Model, U(1) Symmetry, Cluster Property, Spontaneous Symmetry Breaking, Degenerate States

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Abstract:

Entanglement in quantum theory is a concept that has confused many scientists. This concept implies that the cluster property, which means no relations between sufficiently separated two events, is non-trivial. In the works for some quantum spin systems, which have been recently published by the author, extensive and quantitative examinations were made about the violation of cluster property in the correlation function of the spin operator. The previous study of these quantum antiferromagnets showed that this violation is induced by the degenerate states in the systems where the continuous symmetry spontaneously breaks. Since this breaking is found in many materials such as the high temperature superconductors and the superfluidity, it is an important question whether we can observe the violation of the cluster property in them. As a step to answer this question we study a quantum nonlinear sigma model with U(1) symmetry in this paper. It is well known that this model, which has been derived as an effective model of the quantum spin systems, can also be applied to investigations of many materials. Notifying that the existence of the degenerate states is essential for the violation, we made numerical calculations in addition to theoretical arguments to find these states in the nonlinear sigma model. Then, successfully finding the degenerate states in the model, we came to a conclusion that there is a chance to observe the violation of cluster property in many materials to which the nonlinear sigma model applies.

References

[1]  Bell, J.S. (1964) On the Einstein-Podolsky-Rosen Paradox. Physics Physique Fizika, 1, 195-200.
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]  Horodecki, R., Horodecki, P., Horodecki, M. and Horodecki, K. (2003) Quantum Entanglement. Review of Modern Physics, 81, 865-942.
https://doi.org/10.1103/RevModPhys.81.865
[5]  ’t Hooft, G. (2006) The Mathematical Basis for Deterministic Quantum Mechanics. In: Nieuwenhuizen, T.M., Mehmani, B., Špička, V., Aghdami, M.J. and Khrennikov, A.Y., Eds., Beyond the Quantum, World Scientific, Singapore, 3-19.
https://doi.org/10.1142/9789812771186_0001
[6]  Gühne, O. and Tóth, G. (2009) Entanglement Detection. Physics Reports, 474, 1-75.
https://doi.org/10.1016/j.physrep.2009.02.004
[7]  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
[8]  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
[9]  Nielsen, M.A. and Chuang, I.L. (2000) Quantum Computation and Quantum Information. Cambridge University, Cambridge.
[10]  Deutsch, D. (1985) Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer. Proceedings of the Royal Society of London. Series A, 400, 97-117.
https://doi.org/10.1098/rspa.1985.0070
[11]  Weinberg, S. (1995) The Quantum Theory of Fields. Vol. 2, Cambridge University, Cambridge.
[12]  Strocchi, F. (2008) Symmetry Breaking. Vol. 732, Springer, Berlin.
https://doi.org/10.1007/978-3-540-73593-9
[13]  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. Physical Review B, 94, Article ID: 155140.
https://doi.org/10.1103/PhysRevB.94.155140
[14]  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
[15]  Shimizu, A. and Miyadera, T. (2002) Stability of Quantum States of Finite Macroscopic Systems against Classical Noises. Perturbations from Environments, and Local Measurements, 89, Article ID: 270403.
https://doi.org/10.1103/PhysRevLett.89.270403
[16]  Xu, S. and Fan, S. (2017) Generalized C Luster Decomposition Principle Illustrated in Waveguide Quantum Electrodynamics. Physical Review A, 95, Article ID: 063809.
https://doi.org/10.1103/PhysRevA.95.063809
[17]  Fröhlich, 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
[18]  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
[19]  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
[20]  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
[21]  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
[22]  Munehisa, T. (2020) Quantum Curie-Weiss Magnet Induced by Violation of Cluster Property. World Journal of Condensed Matter Physics, 10, 27-52.
https://doi.org/10.4236/wjcmp.2020.102003
[23]  Anderson, P.W. (1984) Basic Notions of Condensed Matter Physics. Benjamin/ Cummings, Menlo Park.
[24]  Kochmanski, M., Paszkiewicz, T. and Wolski, S. (2013) Curie-Weiss Magnet?—A Simple Model of Phase Transition. European Journal of Physics, 34, Article No. 1555.
https://doi.org/10.1088/0143-0807/34/6/1555
[25]  Chayes, L., Crawford, N., Ioffe, D. and Levit. A. (2008) The Phase Diagram of the Quantum Curie-Weiss Model. Journal of Statistical Physics, 133, Article No. 131.
https://doi.org/10.1007/s10955-008-9608-x
[26]  Carneiro, C. and Pellegrino, G. (2018) Analysis of Quantum Phase Transition in Some Different Curie-Weiss Models: A Unified Approach.
[27]  Richter, J., Schulenburg, J. and Honecker, A. (2004) Quantum Magnetism in Two Dimensions: From Semi-Classical Néel Order to Magnetic Disorder. In: Schollwöck, U., Richter, J., Farnell, D.J.J. and Bishop, R.F., Eds., Quantum Magnetism, Vol. 645, Springer-Verlag, Berlin, 85-153.
https://doi.org/10.1007/BFb0119592
[28]  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, 1-62.
https://doi.org/10.1103/RevModPhys.63.1
[29]  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.201390053
[30]  Haldane, F. (1983) Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the One-Dimensional Easy-Axis Néel State. Physical Review Letters, 50, 1153-1156.
https://doi.org/10.1103/PhysRevLett.50.1153
[31]  Chakravarty, S., Halperin, B, and Nelson, S. (1989) Two-Dimensional Quantum Heisenberg Antiferromagnet at Low Temperatures. Physical Review B, 39, 2344-2371.
https://doi.org/10.1103/PhysRevB.39.2344
[32]  Chubukov, A., Sachdev, S. and Ye, J. (1994) Theory of Two-Dimensional Quantum Heisenberg Antiferromagnets with a Nearly Critical Ground State. Physical Review B, 49, 11919-11961.
https://doi.org/10.1103/PhysRevB.49.11919
[33]  Auerbach, A. (1994) Interacting Electrons and Quantum Magnetism. Springer-Verlag, New York.
https://doi.org/10.1007/978-1-4612-0869-3
[34]  Chen, Y. and Neto, A. (2000) Effective Field Theory for Layered Quantum Antiferromagnets with Nonmagnetic Impurities. Physical Review B, 61, Article No. R3772-R3775.
https://doi.org/10.1103/PhysRevB.61.R3772
[35]  Dupre, T. (1996) Localization Transition in Three Dimensions: Monte Carlo Simulation of a Nonlinear Sigma Model. Physical Review B, 54, 12763-12774.
https://doi.org/10.1103/PhysRevB.54.12763
[36]  Schaefer, L. and Wegner, F. (1980) Disordered System with N Orbitals Per Site: Lagrange Formulation, Hyperbolic Symmetry, and Goldstone Modes. Zeitschrift für Physik B Condensed Matter, 38, 113-126.
https://doi.org/10.1007/BF01598751
[37]  Bulgadaev, S. (2000) D-Dimensional Conformal Sigma-Models and Their Topological Excitation. arXiv:hep-th/0008017v1.
[38]  Alles, B., Borisenko, O. and Papa, A. (2018) Finite Density 2D O(3) Sigma Model: Dualization and Numerical Simulations. Physics Review D, 98, Article ID: 114508.
https://doi.org/10.1103/PhysRevD.98.114508
[39]  Abanov, A. and Wiegmann, P. (2000) Chiral Non-Linear Sigma-Models as Models for Topological Superconductivity.
[40]  Scherer, S. (2002) Introduction to Chiral Perturbation Theory.
[41]  Schmudgen, K. (1983) On the Heisenberg Commutation Relation. I. Journal of Functional Analysis, 50, 8-49.
https://doi.org/10.1016/0022-1236(83)90058-7
[42]  Albeverio, S. and Sengupta, A. (2016) Complex Phase Space and Weyl’s Commutation Relations. Expositiones Mathematicae, 34, 249-286.
https://doi.org/10.1016/j.exmath.2015.12.006
[43]  Arai, A. (2016) A Family of Inequivalent Weyl Representations of Canonical Commutation Relations with Applications to Quantum Field Theory. Reviews in Mathematical Physics, 28, Article ID: 1650007.
https://doi.org/10.1142/S0129055X16500070
[44]  Munehisa, T. and Munehisa, Y. (2003) A New Approach to Stochastic State Selections in Quantum Spin Systems. Journal of the Physical Society of Japan, 72, 2759-2765.
https://doi.org/10.1143/JPSJ.72.2759
[45]  Munehisa, T. and Munehisa, Y. (2004) The Stochastic State Selection Method for Energy Eigenvalues in the Shastry-Sutherland Model. Journal of the Physical Society of Japan, 73, 340-347.
https://doi.org/10.1143/JPSJ.73.340
[46]  Munehisa, T. and Munehisa, Y. (2004) Numerical Study for an Equilibrium in the Recursive Stochastic State Selection Method.
[47]  Munehisa, T. and Munehisa, Y. (2004) A Recursive Method of the Stochastic State Selection for Quantum Spin Systems. Journal of the Physical Society of Japan, 73, 2245-2251.
https://doi.org/10.1143/JPSJ.73.2245
[48]  Munehisa, T. and Munehisa, Y. (2006) The Stochastic State Selection Method Combined with the Lanczos Approach to Eigenvalues in Quantum Spin Systems. Journal of Physics: Condensed Matter, 18, 2327-2335.
https://doi.org/10.1088/0953-8984/18/7/018
[49]  Munehisa, T. and Munehisa, Y. (2007) An Equilibrium for Frustrated Quantum Spin Systems in the Stochastic State Selection Method. Journal of Physics: Condensed Matter, 19, 196202-196220.
https://doi.org/10.1088/0953-8984/19/19/196202
[50]  Munehisa, T. and Munehisa, Y. (2009) A Constrained Stochastic State Selection Method Applied to Frustrated Quantum Spin Systems. Journal of Physics: Condensed Matter, 21, 236008-236019.
https://doi.org/10.1088/0953-8984/21/23/236008
[51]  Munehisa, T. and Munehisa, Y. (2010) Numerical Study of the Spin-1/2 Heisenberg Antiferromagnet on a 48-Site Triangular Lattice Using the Stochastic State Selection Method.
[52]  Hatano, N. and Suzuki, M. (1993) Quantum Monte Carlo and Related Methods: Recent Developments. In: Suzuki, M., Ed., Quantum Monte Carlo Methods in Condensed Matter Physics, World Scientific, Singapore, 13-47.
https://doi.org/10.1142/9789814503815_0002
[53]  De Raedt, H. and von der Linden, W. (1995) The Monte Carlo Method in Condensed Matter Physics. Springer-Verlag, Berlin, Heidelberg, 249-284.
[54]  Kawashima, N. (2002) Quantum Monte Carlo Methods. Progress of Theoretical Physics Supplement, 145, 138-148.
https://doi.org/10.1143/PTPS.145.138

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