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Degenerate States in Nonlinear Sigma Model with SU(2) Symmetry

DOI: 10.4236/wjcmp.2023.131002, PP. 14-39

Keywords: Quantum Nonlinear Sigma Model, SU(2): Special Unitary Group in Two Dimensions, Cluster Property, Spontaneous Symmetry Breaking, Degenerate States, Spin-Weighted Harmonics

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Abstract:

Entanglement in quantum theory is a peculiar concept to scientists. With this concept we are forced to re-consider the cluster property which means that one event is irrelevant to another event when they are fully far away. In the recent works we showed that the quasi-degenerate states induce the violation of cluster property in antiferromagnets when the continuous symmetry breaks spontaneously. We expect that the violation of cluster property will be observed in other materials too, because the spontaneous symmetry breaking is found in many systems such as the high temperature superconductors and the superfluidity. In order to examine the cluster property for these materials, we studied a quantum nonlinear sigma model with U(1) symmetry in the previous work. There we showed that the model does have quasi-degenerate states. In this paper we study the quantum nonlinear sigma model with SU(2) symmetry. In our approach we first define the quantum system on the lattice and then adopt the representation where the kinetic term is diagonalized. Since we have no definition on the conjugate variable to the angle variable, we use the angular momentum operators instead for the kinetic term. In this representation we introduce the states with the fixed quantum numbers and carry out numerical calculations using quantum Monte Carlo methods and other methods. Through analytical and numerical studies, we conclude that the energy of the quasi-degenerate state is proportional to the squared total angular momentum as well as to the inverse of the lattice size.

References

[1]  Auerbach, A. (1994) Interacting Electrons and Quantum Magnetism. Springer-Verlag, Berlin, Heidelberg.
https://doi.org/10.1007/978-1-4612-0869-3
[2]  Chen, Y. and Castro Neto, A.H. (2000) Effective Field Theory for Layered Quantum Antiferromagnets with Nonmagnetic Impurities. Physics Review B, 61, R3772-R3775.
https://doi.org/10.1103/PhysRevB.61.R3772
[3]  Dupré, T. (1996) Localization Transition in Three Dimensions: Monte Carlo Simulation of Nonlinear σ Model. Physics Review B, 54, 12763-12774.
https://doi.org/10.1103/PhysRevB.54.12763
[4]  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
[5]  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
[6]  Chakravarty, S., Halperin, B. and Nelson, D.R. (1989) Two-Dimensional Quantum Heisenberg Antiferromagnet at Low Temperatures. Physics Review B, 39, 2344-2371.
https://doi.org/10.1103/PhysRevB.39.2344
[7]  Chubukov, A., Sachdev, S. and Ye, J. (1994) Theory of Two-Dimensional Quantum Heisenberg Antiferromagnets with a Nearly Critical Ground State. Physics Review B, 49, 11919-11961.
https://doi.org/10.1103/PhysRevB.49.11919
[8]  Bulgadaev, S. (2000) D-Dimensional Confromal Sigma-Models and Their Topological Excitation. arXiv:hep-th/0008017
[9]  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
[10]  Abanov, A. and Wiegmann, P. (2000) Chiral Non-Linear Sigma-Models as Models for Topological Superconductivity. Physical Review Letters, 86, 1319.
[11]  Scherer, S. (2002) Introduction to Chiral Perturbation Theory.
arXiv:hep-ph/0210398.
[12]  da Costa, R. (1981) Quantum Mechanics of a Constrained Particle. Physical Review A, 23, 1982-1987.
https://doi.org/10.1103/PhysRevA.23.1982
[13]  da Silva, L., Bastos, C. and Ribeiro, F. (2017) Quantum Mechanics of a Constrained Particle and the Problem of Prescribed Geometry-Induced Potential. Annals of Physics, 379, 13-33.
https://doi.org/10.1016/j.aop.2017.02.012
[14]  Dandoloff, R., Jensen, B. and Saxena, A. (2014) Generalized Anti-Centrifugal Potential. Physics Letters A, 378, 510-513.
https://doi.org/10.1016/j.physleta.2013.12.016
[15]  Dandoloff, R. (2019) Topologically Stable States of the Anti-Centrifugal Potential. Journal of Modern Physics, 10, 1002-1005.
https://doi.org/10.4236/jmp.2019.108066
[16]  Weinberg, S. (1995) The Quantum Theory of Fields. Vol. 2, Cambridge University Press, Cambridge.
[17]  Strocchi, F. (2008) Symmetry Breaking. In: Lecture Note Physics, Vol. 732, Springer, Berlin.
https://doi.org/10.1007/978-3-540-73593-9
[18]  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
[19]  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
[20]  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
[21]  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, Vol. 645, Springer-Verlag, Berlin Heidelberg, 85-153.
https://doi.org/10.1007/BFb0119592
[22]  Manousakis, E. (1991) The Spin-1/2Heisenberg 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
[23]  Landee, C. and Turnbull, M. (2013) Recent Developments in Low-Dimensional Copper (II) Molecular Magnets. European Journal of Inorganic Chemistry, 2013, 2250.
https://doi.org/10.1002/ejic.201300268
[24]  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
[25]  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
[26]  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
[27]  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
[28]  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
[29]  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
[30]  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
[31]  Munehisa, T. (2021) Degenerate States in Nonlinear Sigma Model with U(1) Symmetry—For Study on Violation of Cluster Property. World Journal of Condensed Matter Physics, 11, 29-52.
https://doi.org/10.4236/wjcmp.2021.113003
[32]  Schmüdgen, 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
[33]  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
[34]  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 1650007.
https://doi.org/10.1142/S0129055X16500070
[35]  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
[36]  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
[37]  Munehisa, T. and Munehisa, Y. (2004) Numerical Study for an Equilibrium in the Recursive Stochastic State Selection Method. arXiv:cond-mat/0403626
[38]  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
[39]  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
[40]  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.
https://doi.org/10.1088/0953-8984/19/19/196202
[41]  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.
https://doi.org/10.1088/0953-8984/21/23/236008
[42]  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. arXiv:1008.1612
[43]  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
[44]  De Raedt, H. and von der Linden, W. (1995) The Monte Carlo Method in Condensed Matter Physics. Springer-Verlag, Berlin, Heidelberg, 249-284.
[45]  Kawashima, N. (2002) Quantum Monte Carlo Methods. Progress of Theoretical Physics Supplement, 145, 138-149.
https://doi.org/10.1143/PTPS.145.138
[46]  del Castillo, T. (2007) Spin-Weighted Spherical Harmonics and Their Applications. Revista Mexicana de Fisica S, 53, 125-134.
[47]  Podolsky, D. and Sachdev, S. (2012) Spectral Functions of the Higgs Mode Near Two-Dimensional Quantum Critical Points. Physics Review B, 86, 054508.
https://doi.org/10.1103/PhysRevB.86.054508
[48]  Bruckmann, F., Jansen, K. and Kühn, S. (2019) O(3) Nonlinear Sigma Model in 1 + 1 Dimensions with Matrix Product States. Physics Review D, 99, Article ID: 074501.
https://doi.org/10.1103/PhysRevD.99.074501
[49]  Miller, Jr. W. (1968) Lie Theory and Special Functions. Academic Press, New York.
[50]  Wasson, R.D. (2013) An Overview of the Relationship between Group Theory and Representation to the Special Functions in Mathematical Physics. arXiv:1309.2544
[51]  Newman, E.T. and Penrose, R. (1966) Note on the Bondi-Metzner-Sachs Group. Journal of Mathematical Physics, 7, 863-870.
https://doi.org/10.1063/1.1931221
[52]  Boyle, M. (2016) How Should Spin-Weighted Spherical Functions Be Defined? arXiv:1604.08140
[53]  Wu, T.T. and Yang, C.N. (1976) Dirac Monopole without Strings: Monopole Harmonics. Nuclear Physics B, 107, 365-380.
https://doi.org/10.1016/0550-3213(76)90143-7
[54]  Dray, T. (1985) The Relationship between Monopole Harmonics and Spin-Weighted Spherical Harmonics. Journal of Mathematical Physics, 26, 1030-1033.
https://doi.org/10.1063/1.526533
[55]  Prieto, C.T. (2001) Quantization and Spectral Geometry of a Rigid Body in a Magnetic Monopole Field. Differential Geometry and Its Applications, 14, 157-179.
https://doi.org/10.1016/S0926-2245(00)00044-9

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