Some correlation functions in matrix product ground states of one-dimensional two-state chains. (English) Zbl 1315.82007
Summary: Consider one-dimensional chains with nearest neighbour interactions, for which to each site correspond two independent states (say up and down), and the ground state is a matrix product state. It has been shown [A. H. Fatollahi et al., “Classification of matrix-product states corresponding to one-dimensional chains of two-state sites of nearest neighbor interaction”, Phys. Rev. A83, 042108 (2011)] that for such systems, the ground states are linear combinations of specific vectors which are essentially direct products of specific numbers of ups and downs, symmetrized in a generalized manner. By a generalized manner, it is meant that the coefficient corresponding to the interchange of states of two sites, in not necessarily plus one or minus one, but a phase which depends on the Hamiltonian and the position of the two sites. Such vectors are characterized by a phase \(\chi\), the \(N\)-th power of which is one (where \(N\) is the number of sites), and an integer. Corresponding to \(\chi\), there is another integer \(M\) which is the smallest positive integer that \(\chi^{M}\) is one. Two classes of correlation functions for such systems (basically correlation functions for such vectors) are calculated. The first class consists of correlation functions of tensor products of one-site diagonal observables; the second class consists of correlation functions of tensor products of less than \(M\) one-site observables (but not necessarily diagonal).
MSC:
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 81V70 | Many-body theory; quantum Hall effect |
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