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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 
Submission information  

Preprint Link:  https://arxiv.org/abs/2301.10782v1 (pdf) 
Date submitted:  20230508 07:06 
Submitted by:  Hart, Oliver 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
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 twodimensional crystal structures for simplicity, although our methods are general and extend straightforwardly to three dimensions. We classify the possible multipole groups on all twodimensional 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 inprinciple 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 lowenergy 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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Strengths
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.
Report
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. [32][19][40] all appeared within a couple of days on the arXiv and independently derived subdhiffusive transport for dipole conserving systems.
Report
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 wellknow 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 lowerdegree irreps?
2) If an irrep of the extended point group is morethan1dimensional, 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 lowerdegree multipole moments on some lattice would imply conservation of a higherdegree 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.
Author: Oliver Hart on 20230803 [id 3869]
(in reply to Report 1 on 20230708)
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 wellknow 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.
Reply: We thank the referee for their summary and positive evaluation.
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 lowerdegree irreps?
Reply: Appendix C is not a procedure for considering infinitesimal translations; it is a proof that infinitesimal translations and finite translations generate the same lowerdegree irreps. The proof is somewhat technical, but the outline is roughly the following. For a polynomial of degree $n$, translating by each (discrete) lattice vector up to $n+1$ times produces lowerdegree polynomials which are distinct and, as it turns out, linearly independent. Similarly, taking infinitesimal translations along each lattice vector directions produces some set of linearly independent lowerdegree polynomials at each degree. Certainly repeated infinitesimal translations will eventually produce finite translations, so the polynomials produced via infinitesimal translations include those produced by finite translations. Showing the opposite inclusion is the main result of Appendix C.
2) If an irrep of the extended point group is morethan1dimensional, what does it mean physically in terms of the conservation laws of the model?
Reply: This means that in the presence of the space group symmetry, certain multipole moments have multiple components that are either all conserved, or none are (as long as the space group symmetry is present). For an intuitive example, consider dipole moment conservation on the square lattice. The fact that $x$ belongs to the 2D irrep $\lbrace x, y\rbrace$ of $\mathsf{D}_4$ means that if the $x$ component of dipole moment is conserved, point group symmetry requires the $y$ component to be conserved as well.
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.
Reply: The span of $\lbrace x^3, y^3\rbrace$ and $\lbrace x^3 + 3xy^2, y^3+3x^2y\rbrace$ would also be conserved. The clouds do not change this interpretation; they just indicate that the list of cubic terms in the figure is redundant, i.e., the lefthand cloud spans all possible cubic terms, but so does the righthand cloud.
The reason for the clouds is that if we choose a fixed basis for the cubic polynomials irreps, some of the quartic polynomials will conserve not one, not both, but a specific nontrivial linear combination of the (degenerate) cubic polynomial irreps. The clouds were the best visual representation we had available to show that fact. We have added a short description of this to Sec. II.B.3.
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)?
Reply: It is not possible to completely forbid the conservation of a multipole moment via lattice symmetries. Given any fixed multipole moment, one can just find its orbit under the point group and then find the translation descendants.
5) Is it possible that conservation of certain combinations of lowerdegree multipole moments on some lattice would imply conservation of a higherdegree moment?
Reply: We are unaware of any reason for higherdegree moments to be exactly conserved; certainly lattice symmetries cannot be responsible for this. However, as our examples show, lowerdegree moments can lead to the emergent conservation of higherdegree moments, i.e., at low enough energies higherdegree moments will be approximately conserved. This is how, e.g., we obtain emergent subsystem symmetries from a fixed list of lowdegree moments.
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.
Reply: We thank the referee for their useful suggestions. We have updated Fig. 1 and its caption accordingly, and added a comment on overloading the letter $C$.
Author: Oliver Hart on 20230803 [id 3868]
(in reply to Report 2 on 20230711)Reply: We thank the referee for their summary and positive evaluation.
Reply: The references in question have now become numbers [35], [20] and [36], respectively, in the new version. Reference numbers in what follows refer to the new version with these updated numbers.
We thank the referee for this comment. In regards to the chronology of subdiffusion in fractonic systems, we note that Ref. [34] was posted to the arXiv on 24 July 2019, Ref. [35] on 20 March 2020, Ref. [20] on 31 March, 2020, and Ref. [36] on 1 April 2020. Certainly all papers are important, and we have ensured that all are cited prominently in the context of fracton hydrodynamics.