### 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 Hölder 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.

Original language | English (US) |
---|---|

Article number | 045009 |

Journal | Physical Review D |

Volume | 98 |

Issue number | 4 |

DOIs | |

State | Published - Aug 15 2018 |

### All Science Journal Classification (ASJC) codes

- Physics and Astronomy (miscellaneous)

## Fingerprint Dive into the research topics of 'Continuum limits of sparse coupling patterns'. Together they form a unique fingerprint.

## Cite this

*Physical Review D*,

*98*(4), [045009]. https://doi.org/10.1103/PhysRevD.98.045009