Abstract
In a recent paper [17] we conjectured that for ferromagnetic Heisenberg models the smallest eigenvalues in the invariant subspaces of fixed total spin are monotone decreasing as a function of the total spin and called this property ferromagnetic ordering of energy levels (FOEL). We have proved this conjecture for the Heisenberg model with arbitrary spins and coupling constants on a chain [17, 20]. In this paper we give a pedagogical introduction to this result and also discuss some extensions and implications. The latter include the property that the relaxation time of symmetric simple exclusion processes on a graph for which FOEL can be proved, equals the relaxation time of a random walk on the same graph with jump rates given by the coupling constants, i.e., the relaxation time is independent of the number of particles. Therefore, our results also provide a proof of Aldous’ Conjecture in one dimension.
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References
D. Aldous, http://stat-www.berkeley.edu/users/aldous/problems.ps.
R. Bacher, Valeur propre minimale du laplacien de Coxeter pour le groupe symétrique, J. Algebra 167 (1994), 460–472.
P. Diaconis and L. Saloff-Coste, Comparison techniques for random walk on finite groups, Ann. Prob. 21 (1993), 2131–2156.
P. Diaconis and M. Shahshahani, Generating a random permutation with random transpositions, Z. Wahrsch. Verw. Geb. 57 (1981), 159–179.
L. Flatto, A.M. Odlyzko, and D.B. Wales, Random shuffles and group representations, Ann. Prob. 13 (1985), 154–178.
I.B. Frenkel and M.G. Khovanov, Canonical bases in tensor products and graphical calculus for uq(sl2), Duke Math. J. 87 (1997), 409–480.
E. Gutkin, Plancherel formula and critical spectral behaviour of the infinite XXZ chain, Quantum symmetries (Clausthal, 1991), World Scientific, River Edge, NJ, 1993, pp. 84–98.
S. Handjani and D. Jungreis, Rate of convergence for shuffing cards by transpositions, J. Theor. Prob. 9 (1996), 983–993.
M. Jimbo and T. Miwa, Algebraic analysis of solvable lattice models, Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, 1995.
L.H. Kauffman and S.L. Lins, Temperley-Lieb recoupling theory and invariants of 3-manifolds, Princeton University Press, 1994.
T. Kennedy, Expansions for droplet states in the ferromagnetic XXZ Heisenberg chain, Markov Processes and Rel. Fields 11 (2005) 223–236.
T. Koma and B. Nachtergaele, The spectral gap of the ferromagnetic XXZ chain, Lett. Math. Phys. 40 (1997), 1–16.
V.E. Korepin, N.M. Bogoliubov, and A.G. Izergin, Quantum inverse scattering method and correlation functions, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, England, 1993.
E.H. Lieb, Two theorems on the Hubbard model, Phys. Rev. Lett. 62 (1989), 1201–1204.
E.H. Lieb and D. Mattis, Ordering energy levels of interacting spin systems, J. Math. Phys. 3 (1962), 749–751.
B. Nachtergaele, W. Spitzer, and S. Starr, Droplet Excitations for the Spin-1/2 XXZ Chain with Kink Boundary Conditions, arXiv::math-ph/0508049.
B. Nachtergaele, W. Spitzer, and S. Starr, Ferromagnetic ordering of energy levels, J. Stat. Phys. 116 (2004), 719–738.
B. Nachtergaele and S. Starr, in preparation.
B. Nachtergaele and S. Starr, Droplet states in the XXZ Heisenberg model, Commun. Math. Phys. 218 (2001), 569–607, math-ph/0009002.
B. Nachtergaele and S. Starr, Ferromagnetic Lieb-Mattis theorem, Phys. Rev. Lett. 94 (2005), 057206, arXiv:math-ph/0408020.
V. Pasquier and H. Saleur, Common structures between finite systems and conformal field theories through quantum groups, Nucl. Phys. B330 (1990), 523–556.
H.N.V. Temperley and E.H. Lieb, Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the “percolation” problem, Proc. Roy. Soc. A322 (1971), 252–280.
H. Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Z. 52 (1950), 642–648.
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Nachtergaele, B., Starr, S. (2006). Ordering of Energy Levels in Heisenberg Models and Applications. In: Asch, J., Joye, A. (eds) Mathematical Physics of Quantum Mechanics. Lecture Notes in Physics, vol 690. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-34273-7_13
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DOI: https://doi.org/10.1007/3-540-34273-7_13
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