REVIEW
Space-dependent symmetries and fractons
High Energy Physics - Theory · Submitted 1 Dec 2021 · 101 references
Pith's verdict: A solid, well-organized review of the symmetry-first view of gapless fractons; its abstract promises slightly more than the body delivers, but the body is honest about it.
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Referee report
The paper
The paper claims
The review argues that coordinate-dependent symmetries—polynomial shift symmetries and the multipole algebra—are the organizing principle for gapless fractons, and that elasticity dualizes to a tensor gauge theory whose charges are lattice.
The reading
What carries the argument
The load-bearing object is the multipole algebra, realized as polynomial shift symmetry and geometrically as the Heisenberg group. In the elastic dual, the stress tensor is written in terms of a symmetric tensor gauge field, and the dual gauge transformations are $��delta A_{ij}=��partial_i��partial_j��alpha$ and $��delta A_0=��partial_0��alpha$. The Heisenberg group's Maurer-Cartan form supplies the invariant one-form $v=d��phi+z��cdot dx$ and the symplectic form $J=e^i��wedge��omega_i$, which reproduces $[P_i,Q_j]=��delta_{ij}Q$; embedding physical spacetime into the group manifold and imposing the inverse Higgs constraint $z_i=-��partial_i��phi$ reduces the Goldstone mode to one field $��phi$ and produces the fracton metric $B=(��partial_0��phi��,dx^0)^2+(��partial_0��partial_i��phi��,dx^0+��partial_i��partial_j��phi��,dx^j)^2$. This machinery is what lets the review treat ordinary elasticity, quasicrystal elasticity, and Cosserat elasticity as instances of one symmetry principle.
Extended reading notes
Core claim
The central claim is that fracton behavior is not an assortment of lattice accidents but the consequence of coordinate-dependent symmetries. In the simplest case, invariance under $��phi��to��phi+��alpha+��beta_i x^i$ yields conservation of total charge and dipole moment; the continuity equation uses a symmetric tensor current, and a gauged version uses a scalar $A_0$ and symmetric tensor $A_{ij}$ transforming as $��delta A_0=-��partial_0��alpha$ and $��delta A_{ij}=��partial_i��partial_j��alpha$. The review's main exhibit is the fracton-elasticity duality: in two dimensions the elastic stress tensor is resolved by tensor gauge fields, $T^{i��mu}=��epsilon^{��mu��nu��rho}��partial_��nu A^i{}_{��rho}$, so dislocations and disclinations become charged sources in a tensor gauge theory. The Heisenberg group encodes the non-Abelian algebra between dipole charge and translations, $[P_i,Q_j]=��delta_{ij}Q$, and gives the Goldstone mode a geometric interpretation as a breathing mode of an enlarged space. The review then argues that hydrodynamics with conserved multipole moments produces subdiffusion rather than ordinary diffusion.
Load-bearing premise
The load-bearing premise is that the Heisenberg-group embedding with the inverse Higgs constraint is the correct geometric description of spontaneously broken monopole-dipole-momentum symmetry, rather than merely a convenient parametrization, and that the resulting tensor gauge symmetry survives coupling to curved space.
Editorial extensions
If this is right
- If the symmetry principle is right, the fracton-elasticity duality gives a dictionary in which dislocation and disclination densities are the charge and current sources of a tensor gauge theory, so defect dynamics can be computed in the dual language.
- Quasicrystal elasticity adds a phason field, and its matching faults become vector charges in the dual theory, extending the same dictionary beyond ordinary crystals.
- Cosserat elasticity with independent rotations dualizes to a tensor gauge theory coupled to an ordinary $U(1)$ gauge field, with defect charges receiving contributions from both displacement and orientation singularities.
- Conserved multipole moments make density spreading subdiffusive, $��Delta x��sim t^{1/(n+1)}$ for the highest conserved $n$-multipole, which is a concrete transport signature.
- The Heisenberg-group construction predicts that a dipole-symmetric system on an arbitrary curved background generically spontaneously breaks the monopole-dipole-momentum symmetry, which would constrain any curved-space realization.
Reading between the lines
- The review's unification suggests that the right experimental target is a transport exponent: measuring $��Delta x��sim t^{1/(n+1)}$ in a dipole-conserving cold-atom or circuit platform would test the multipole symmetry more directly than observing a single immobile defect.
- If the Heisenberg-group geometry is more than a parametrization, the number of Goldstone modes in a dipole-symmetric phase should be fixed by the inverse Higgs constraint; a microscopic model with a different count would be evidence against the geometric picture.
- The quasicrystal and Cosserat extensions imply that materials with internal rotational or phason degrees of freedom are natural places to look for fractonic defect charges, even when the underlying lattice is not a simple periodic crystal.
- The unresolved curved-space coupling suggests a test: put a dipole-conserving system on a curved or strained background and look for symmetry-breaking terms that vanish in flat space; their coefficient would measure how seriously the tensor gauge symmetry is a genuine gauge symmetry rather than a flat-space accident.
Original abstract
There has been a surge of interest in effective non-Lorentzian theories of excitations with restricted mobility, known as fractons. Examples include defects in elastic materials, vortex lattices or spin liquids. In the effective theory novel coordinate-dependent symmetries emerge that shape the properties of fractons. In this review we will discuss these symmetries, cover the effective description of gapless fractons via elastic duality, and discuss their hydrodynamics.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
There has been a surge of interest in effective non-Lorentzian theories of excitations with restricted mobility, known as fractons. Examples include defects in elastic materials, vortex lattices or spin liquids. In the effective theory novel coordinate-dependent symmetries emerge that shape the properties of fractons. In this review we will discuss these symmetries, cover the effective description of gapless fractons via elastic duality, and discuss their hydrodynamics.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | There has been a surge of interest in effective non-Lorentzian theories of excitations with restricted mobility, known as fractons. | background 1.00 | 0.03 | 0.04 | asserted 1.00 |
| s1 | Examples include defects in elastic materials, vortex lattices or spin liquids. | background 1.00 | 0.04 | 0.03 | asserted 1.00 |
| s2 | In the effective theory novel coordinate-dependent symmetries emerge that shape the properties of fractons. | background 0.87 | 0.12 | 0.19 | asserted 1.00 |
| s3 | In this review we will discuss these symmetries, cover the effective description of gapless fractons via elastic duality, and discuss their hydrodynamics. | contribution 0.99 | 0.39 | 0.15 | asserted 0.67 |
What was examined, and what was not
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References
How the literature uses it
44 arXiv papers cite this in 64 sentences
Papers citing it · 24 sentences without a confident label
Builds on it 1 paper
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Our results are a dissipative completion of the theory constructed in [ref] , where the zero temperature ideal constitutive relations were found using the Poisson bracket formalism.
Hydrodynamics of dipole-conserving fluids · Zeroth order hydrodynamics · 1.00
Cites it as background 36 papers
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From a purely theoretical perspective, fractons are very puzzling objects, as they seem to defy the standard methods of quantum field theory (see the reviews [ref] for (original) references, further applications and details).
Carroll/fracton particles and their correspondence · Fracton symmetry and Carroll/fracton correspondence · 1.00
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(See [ref] for a review on this subject.) The simplest theory in this class, commonly known as the “rank-2 scalar charge theory” [ref] , involves $U(1)$ gauge fields $(A_τ,A_ij)$ with gauge symmetry A_τ∼A_τ+ ∂_τα , A_ij ∼A_ij + ∂_i ∂_j α .
Global Dipole Symmetry, Compact Lifshitz Theory, Tensor Gauge Theory, and Fractons · Introduction · 1.00
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Along this line of thinking and motivated by the field of fracton physics [ref] , recently, an exotic class of SPT orders dubbed as subsystem symmetry protected topological (SSPT) orders was proposed [ref] , and shows a series of intriguing properties beyond aforementioned SPT phases protected by global symmetries, such as spurious topological entanglement entropy [ref] and duality into fracton topological orders [ref] .
Detecting Subsystem Symmetry Protected Topological Order Through Strange Correlators · Introduction · 0.81
What it cites · 69 sentences without a confident label
Cites as background 12 papers
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In a pair of groundbreaking papers [ref] , Pretko demonstrated how the simultaneous conservation of charge and dipole moment (and even a particular component of the quadrupole moment) naturally leads to fractons, and he pioneered the study of symmetric tensor gauge theories as models containing gapless fractonic excitations.
Subdimensional Particle Structure of Higher Rank U(1) Spin Liquids · Introduction · 0.90
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In a pair of groundbreaking papers [ref] , Pretko demonstrated how the simultaneous conservation of charge and dipole moment (and even a particular component of the quadrupole moment) naturally leads to fractons, and he pioneered the study of symmetric tensor gauge theories as models containing gapless fractonic excitations.
Generalized Electromagnetism of Subdimensional Particles: A Spin Liquid Story · Introduction · 0.90
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by Griffin, Grosvenor, Hořava and Yan (GGHY) in the context of technical naturalness in non-relativistic quantum field theories [ref] .
Multicritical Symmetry Breaking and Naturalness of Slow Nambu-Goldstone Bosons · Introduction · 0.86
Discussion
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