# Continuum limits of sparse coupling patterns

## 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.

- Authors:

- Publication Date:

- Research Org.:
- Princeton Univ., NJ (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1464303

- Alternate Identifier(s):
- OSTI ID: 1498933

- Grant/Contract Number:
- [FG02-91ER40671]

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- [Journal Name: Physical Review D Journal Volume: 98 Journal Issue: 4]; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

### Citation Formats

```
Gubser, Steven S., Jepsen, Christian, Ji, Ziming, and Trundy, Brian. Continuum limits of sparse coupling patterns. United States: N. p., 2018.
Web. doi:10.1103/PhysRevD.98.045009.
```

```
Gubser, Steven S., Jepsen, Christian, Ji, Ziming, & Trundy, Brian. Continuum limits of sparse coupling patterns. United States. doi:10.1103/PhysRevD.98.045009.
```

```
Gubser, Steven S., Jepsen, Christian, Ji, Ziming, and Trundy, Brian. Mon .
"Continuum limits of sparse coupling patterns". United States. doi:10.1103/PhysRevD.98.045009.
```

```
@article{osti_1464303,
```

title = {Continuum limits of sparse coupling patterns},

author = {Gubser, Steven S. and Jepsen, Christian and Ji, Ziming and Trundy, Brian},

abstractNote = {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.},

doi = {10.1103/PhysRevD.98.045009},

journal = {Physical Review D},

number = [4],

volume = [98],

place = {United States},

year = {2018},

month = {8}

}

DOI: 10.1103/PhysRevD.98.045009

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