The Coprime Quantum Chain

Abstract

In this paper we introduce and study the coprime quantum chain, i.e. a strongly correlated quantum system defined in terms of the integer eigenvalues $n_i$ of the occupation number operators at each site of a chain of length $M$. The $n_i$'s take value in the interval $[2,q]$ and may be regarded as $S_z$ eigenvalues in the spin representation $j=(q-2)/2$. The distinctive interaction of the model is based on the coprimality matrix $\mathbf{\Phi }$: for the ferromagnetic case, this matrix assigns lower energy to configurations where occupation numbers $n_i$ and $n_{i+1}$ of neighbouring sites share a common divisor, while for the anti-ferromagnetic case it assigns lower energy to configurations where $n_i$ and $n_{i+1}$ are coprime. The coprime chain, both in the ferro and anti-ferromagnetic cases, may present an exponential number of ground states whose values can be exactly computed by means of graph theoretical tools. In the ferromagnetic case there are generally also frustration phenomena. A fine tuning of local operators may lift the exponential ground state degeneracy and, according to which operators are switched on, the system may be driven into different classes of universality, among which the Ising or Potts universality class. The paper also contains an appendix by Don Zagier on the exact eigenvalues and eigenvectors of the coprimality matrix in the limit $q\to \infty$.

38+17 pages, 25+1 figures, Appendix C written by Don Zagier

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