Research PaperNo Access

Strict deformation quantization of the state space of $M_{k} (ℂ)$ with applications to the Curie–Weiss model

Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, Heyendaalseweg 135 6525 AJ Nijmegen, The Netherlands

E-mail Address: [email protected]

Search for more papers by this author

Valter Moretti

http://orcid.org/0000-0001-6182-6613

Department of Mathematics, University of Trento, and INFN-TIFPA, via Sommarive 14, I-38123 Povo (Trento), Italy

E-mail Address: [email protected]

Search for more papers by this author

, and

Christiaan J. F. van de Ven

Department of Mathematics, University of Trento, and INFN-TIFPA, via Sommarive 14, I-38123 Povo (Trento), Italy

Marie Skłodowska-Curie Fellow of the Istituto Nazionale di Alta Matematica, Italy

E-mail Address: [email protected]

Search for more papers by this author

https://doi.org/10.1142/S0129055X20500312Cited by:10 (Source: Crossref)

Abstract

Increasing tensor powers of the $k \times k$ matrices $M_{k} (ℂ)$ are known to give rise to a continuous bundle of $C^{*}$ -algebras over $I = {0} \cup 1 / ℕ \subset [0, 1]$ with fibers $A_{1 / N} = M_{k} {(ℂ)}^{\otimes N}$ and $A_{0} = C (X_{k})$ , where $X_{k} = S (M_{k} (ℂ))$ , the state space of $M_{k} (ℂ)$ , which is canonically a compact Poisson manifold (with stratified boundary). Our first result is the existence of a strict deformation quantization of $X_{k}$ à la Rieffel, defined by perfectly natural quantization maps $Q_{1 / N} : Ã_{0} \to A_{1 / N}$ (where $Ã_{0}$ is an equally natural dense Poisson subalgebra of $A_{0}$ ).

We apply this quantization formalism to the Curie–Weiss model (an exemplary quantum spin with long-range forces) in the parameter domain where its $ℤ_{2}$ symmetry is spontaneously broken in the thermodynamic limit $N \to \infty$ . If this limit is taken with respect to the macroscopic observables of the model (as opposed to the quasi-local observables), it yields a classical theory with phase space $X_{2} ≅ B^{3}$ (i.e. the unit three-ball in $ℝ^{3}$ ). Our quantization map then enables us to take the classical limit of the sequence of (unique) algebraic vector states induced by the ground state eigenvectors $Ψ_{N}^{(0)}$ of this model as $N \to \infty$ , in which the sequence converges to a probability measure $μ$ on the associated classical phase space $X_{2}$ . This measure is a symmetric convex sum of two Dirac measures related by the underlying $ℤ_{2}$ -symmetry of the model, and as such the classical limit exhibits spontaneous symmetry breaking, too. Our proof of convergence is heavily based on Perelomov-style coherent spin states and at some stage it relies on (quite strong) numerical evidence. Hence the proof is not completely analytic, but somewhat hybrid.

Keywords:

AMSC: 46L65, 81R30, 82B20

References

1. A. E. Allahverdyana, R. Balian and Th. M. Nieuwenhuizen , Understanding quantum measurement from the solution of dynamical models, Phys. Rep. 525 (2013) 1–166. Crossref, Web of Science, ADS, Google Scholar
2. N. L. Balazs and B. K. Jennings , Wigner’s function and other distribution functions in mock phase spaces, Phys. Rep. 104 (1984) 347–391. Crossref, Web of Science, ADS, Google Scholar
3. I. Bengtsson and K. Zyczkowski , Geometry of Quantum States: An Introduction to Quantum Entanglement (Cambridge University Press, 2006). Crossref, Google Scholar
4. P. Bona , The dynamics of a class of mean-field theories, J. Math. Phys. 29 (1988) 2223–2235. Crossref, Web of Science, ADS, Google Scholar
5. M. Bordemann, E. Meinrenken and M. Schlichenmaier , Toeplitz quantization of Kähler manifolds and $g l (N)$ , $N \to \infty$ limits, Comm. Math. Phys. 165 (1994) 281–296. Crossref, Web of Science, ADS, Google Scholar
6. O. Bratteli and D. W. Robinson , Operator Algebras and Quantum Statistical Mechanics. Vol. II: Equilibrium States, Models in Statistical Mechanics (Springer, 1981). Crossref, Google Scholar
7. E. Brüning, H. Mäkelä, A. Messina and F. Petruccione , Parametrizations of density matrices, J. Modern Opt. 59 (2011) 1–20. Crossref, Web of Science, ADS, Google Scholar
8. L. Chayes, N. Crawford, D. Ioffe and A. Levit , The phase diagram of the quantum Curie–Weiss model, J. Stat. Phys. 133 (2008) 131–149. Crossref, Web of Science, ADS, Google Scholar
9. J. Dixmier , C*-Algebras (North-Holland, 1977). Google Scholar
10. N. G. Duffield and R. F. Werner , Local dynamics of mean-field quantum systems, Helv. Phys. Acta 65 (1992) 1016–1054. Google Scholar
11. S. Friedli and Y. Velenik , Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction (Cambridge University Press, 2017). Crossref, Google Scholar
12. J. Grabowski, M. Kuś and G. Marmo , Geometry of quantum systems: Density states and entanglement, J. Phys. A: Math. Gen. 38 (2005) 10217–10244. Crossref, ADS, Google Scholar
13. M. Hillery, R. F. O’Connell, M. O. Scully and E. P. Wigner , Distribution functions in physics: Fundamentals, Phys. Rep. 106 (1984) 121–167. Crossref, Web of Science, ADS, Google Scholar
14. D. Ioffe and A. Levit , Ground states for mean field models with a transverse component, J. Stat. Phys. 151 (2013) 1140–1161. Crossref, Web of Science, ADS, Google Scholar
15. N. P. Landsman , Mathematical Topics Between Classical and Quantum Theory (Springer, 1998). Crossref, Google Scholar
16. N. P. Landsman , Foundations of Quantum Theory: From Classical Concepts to Operator Algebras (Springer, 2017); Open Access at http://www.springer.com/gp/book/9783319517766. Crossref, Google Scholar
17. J. E. Marsden and T. S. Ratiu , Introduction to Mechanics and Symmetry (Springer, 1994). Crossref, Google Scholar
18. V. Moretti , Spectral Theory and Quantum Mechanics, 2nd edn. (Springer, 2018). Google Scholar
19. A. M. Perelomov , Coherent states for arbitrary Lie groups, Comm. Math Phys. 26 (1972) 222–236. Crossref, Web of Science, ADS, Google Scholar
20. M. J. Pflaum , Analytic and Geometric Study of Stratified Spaces (Springer, 2001). Google Scholar
21. G. A. Raggio and R. F. Werner , Quantum statistical mechanics of general mean field systems, Helv. Phys. Acta 62 (1989) 980–1003. Google Scholar
22. M. A. Rieffel , Deformation quantization of Heisenberg manifolds, Comm. Math. Phys. 121 (1989) 531–562. Crossref, Web of Science, ADS, Google Scholar
23. M. A. Rieffel , Quantization and $C^{*}$ -algebras, Contemp. Math. 167 (1994) 67–97. Google Scholar
24. B. Simon , The Statistical Mechanics of Lattice Gases. Vol. I (Princeton University Press, 1993). Crossref, Google Scholar
25. M. Takesaki , Theory of Operator Algebras I, 2nd edn. (Springer, 2002). Google Scholar
26. C. J. F. van de Ven, Properties of quantum spin systems and their classical limit, M.Sc. Thesis, Radboud University (2018); https://www.math.ru.nl/ $^{\sim}$ landsman/ Chris2018.pdf. Google Scholar
27. C. J. F. van de Ven, G. C. Groenenboom, R. Reuvers and N. P. Landsman, Quantum spin systems versus Schrödinger operators: A case study in spontaneous symmetry breaking, arXiv:1811.12109 (v3); to appear in SciPost. Google Scholar
28. H. Weyl , Gruppentheorie und Quantenmechanik (Hirzel, 1928). Google Scholar