Abstract
We exhibit simple lattice systems, motivated by recently proposed cold atom experiments, whose continuum limits interpolate between real and $p$-adic smoothness as a spectral exponent is varied. A real spatial dimension emerges in the continuum limit if the spectral exponent is negative, while a $p$-adic extra dimension emerges if the spectral exponent is positive. We demonstrate Holder continuity conditions, both in momentum space and in position space, which quantify how smooth or ragged the two-point Green's function is as a function of the spectral exponent. The underlying discrete dynamics of our model is defined in terms of a Gaussian partition function as a classical statistical mechanical lattice model. The couplings between lattice sites are sparse in the sense that as the number of sites becomes large, a vanishing fraction of them couple to one another. This sparseness property is useful for possible experimental realizations of related systems.
Highlights
The p-adic numbers are an alternative way of filling in the “gaps” between rational numbers in order to form a complete set, or continuum
Just as the real numbers R are the completion of the rationals Q with respect to j·j∞, so the p-adic numbers Qp are the completion of Q with respect to j·jp for any fixed p
II, we describe the class of statistical mechanical, onedimensional spin chains that we will study, and we give a general account of how to compute Green’s functions before treating in turn four models within this class: Nearest neighbor interactions, power-law interactions, p-adic interactions, and sparse couplings, which interpolate between nearest neighbor and p-adic behavior
Summary
The p-adic numbers (for any fixed choice of a prime number p) are an alternative way of filling in the “gaps” between rational numbers in order to form a complete set, or continuum. Green’s functions of spins in the furthest neighbor Ising model depend on the locations i and j of the spins only through the 2-adic norm ji − jj. When we turn to sparse coupling patterns, we will recognize that we are recovering 2-adic continuity precisely when the two-point Green’s function is well approximated by a locally constant. Position space continuity is more complex, with different Hölder exponents depending on whether one is looking at global or local smoothness properties
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