Quantum spin chains and Riemann zeta function with odd arguments. (English) Zbl 1059.82006
Summary: The Riemann zeta function is an important object of number theory. We argue that it is related to the Heisenberg spin-1/2 anti-ferromagnet. In the \(XXX\) spin chain we study the probability of formation of a ferromagnetic string in the anti-ferromagnetic ground state in the thermodynamics limit. We prove that for short strings the probability can be expressed in terms of the Riemann zeta function with odd arguments.
MSC:
82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
11Z05 | Miscellaneous applications of number theory |
33E20 | Other functions defined by series and integrals |