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Multipole groups and fracton phenomena on arbitrary crystalline lattices
by Daniel Bulmash, Oliver Hart, Rahul Nandkishore
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|Authors (as registered SciPost users):||Daniel Bulmash · Oliver Hart · Rahul Nandkishore|
|Preprint Link:||https://arxiv.org/abs/2301.10782v1 (pdf)|
|Date submitted:||2023-05-08 07:06|
|Submitted by:||Hart, Oliver|
|Submitted to:||SciPost Physics|
Multipole symmetries are of interest in multiple contexts, from the study of fracton phases, to nonergodic quantum dynamics, to the exploration of new hydrodynamic universality classes. However, prior explorations have focused on continuum systems or hypercubic lattices. In this work, we systematically explore multipole symmetries on arbitrary crystal lattices. We explain how, given a crystal structure (specified by a space group and the occupied Wyckoff positions), one may systematically construct all consistent multipole groups. We focus on two-dimensional crystal structures for simplicity, although our methods are general and extend straightforwardly to three dimensions. We classify the possible multipole groups on all two-dimensional Bravais lattices, and on the kagome and breathing kagome crystal structures to illustrate the procedure on general crystal lattices. Using Wyckoff positions, we provide an in-principle classification of all possible multipole groups in any space group. We explain how, given a valid multipole group, one may construct an effective Hamiltonian and a low-energy field theory. We then explore the physical consequences, beginning by generalizing certain results originally obtained on hypercubic lattices to arbitrary crystal structures. Next, we identify two seemingly novel phenomena, including an emergent, robust subsystem symmetry on the triangular lattice, and an exact multipolar symmetry on the breathing kagome lattice that does not include conservation of charge (monopole), but instead conserves a vector charge. This makes clear that there is new physics to be found by exploring the consequences of multipolar symmetries on arbitrary lattices, and this work provides the map for the exploration thereof, as well as guiding the search for emergent multipolar symmetries and the attendant exotic phenomena in real materials based on nonhypercubic lattices.
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1 Clearly formulated mathematical framework to explore multipole symmetries on arbitrary lattices.
2 Novel findings based on the provided framework: (i) finite number of global multipole symmetries can induce emergent subsystem symmetries and (ii) translationally invariant system can conserve certain multipole moments without conserving the monopole charge.
In this work, the authors provide a framework to explore multipole symmetries. While previous studies have mostly focused on continuum systems and hyper cubic lattices, the present framework can be applied to on arbitrary crystal lattices. The manuscript is clearly written and the results are sound. Interestingly, an exact multipolar symmetry on the breathing kagome lattice is found that does not include conservation of charge but instead conserves a vector charge.
I recommend publication in SciPost Physics.
Minor "historical" comment: Refs.  all appeared within a couple of days on the arXiv and independently derived subdhiffusive transport for dipole conserving systems.
This is a clear and mathematically sound paper that provides a framework for determining sets of multipole symmetries that are consistent with any given crystal lattice. In particular, the paper answers the question "given a crystal lattice and a specific conservation of a multipole moment, which other multipole moments must be conserved?" (akin to the well-know fact that the dipole moment conservation implies charge conservation in translationally invariant systems). The authors' framework is generic and can be applied to both Bravais lattices and lattices with more than one site in the unit cell. The authors apply their framework to several 2D lattices, explicitly summarizing the results. It is noteworthy that, in addition to explaining certain known results within their framework, the authors have discovered two new peculiar phenomena: 1) a finite number of global multipole symmetries can induce emergent subsystem symmetries (if the interaction range is small enough); 2) a translationally invariant system can conserve certain multipole moments without conserving the monopole charge (but instead conserves a vector charge).
In my opinion, the paper satisfies all the required criteria, and I recommend it for publication in SciPost Physics.
I would be happy if the authors could clarify several points for me, though:
1) A lattice only allows for finite translations. Why is it then appropriate to consider infinitesimal translations (with the procedure from Appendix C) to get the lower-degree irreps?
2) If an irrep of the extended point group is more-than-1-dimensional, what does it mean physically in terms of the conservation laws of the model?
3) In Fig. 3, what happens if we impose conservation of, say, $x^4+y^4$ and $x^3y+y^3x$, i.e., the ones pointing to different gray clouds. Which cubic terms would be conserved then? This remained unclear to me both from the text and from the figure.
4) Is it possible that conservation of a certain multipole moment on some lattice is completely forbidden? Or one can always enforce conservation of any given multipole moment on any given lattice (and then find all the descendant multipole moments that must be conserved)?
5) Is it possible that conservation of certain combinations of lower-degree multipole moments on some lattice would imply conservation of a higher-degree moment?
I also have two minor cosmetic suggestions (but I leave the decision up to the authors):
a) In Fig. 1, it would be more informative to also specify the angles between the lattice vectors in the picture (i.e., $\gamma= \pi/2$, $\gamma \neq \pi/2$, or $\gamma = \pi/3$).
b) In Eq. (8), $C$ denotes both a rotation and a basis site. Perhaps one could slightly change the notation by, e.g., changing the font of one of the $C$'s.