# Dual gauge theory formulation of planar quasicrystal elasticity and fractons

Piotr Surówka<EMAIL_ADDRESS>Max Planck Institute for the Physics of
Complex Systems, 01187 Dresden, Germany  Department of Theoretical Physics,
Wrocław University of Science and Technology, 50-370 Wrocław, Poland

###### Abstract

Elastic description of planar quasicrystals can be formulated as an interplay
between two Goldstone fields corresponding to phonon and phason degrees of
freedom. We reformulate this description as a gauge theory with one gauge
field that is symmetric under exchange of indices and one that is not. We also
show which topological defects in quasicrystals can be succinctly incorporated
in the dual description and interpret them as fractonic excitations. Finally
we calculate the static interaction potential between defects in a
quasicrystal with fivefold symmetry. This is done in the limit of a small
coupling between phonon and phason stresses.

Solid materials are most commonly represented by crystals, whose atomic
constituents are arranged in a highly ordered microscopic structure, forming a
lattice. Macroscopic description of crystals is provided by the theory of
elasticity that deals with mechanics of bodies modelled as a continuous object
rather than a crystalline lattice of atoms Landau et al. (1986). An important
point is, however, that not all solids are crystals. In fact there are various
classes of solid materials, whose microscopic structure is not a periodic
lattice. Examples include policrystals, glasses and quasicrystals. We do not
have such a detailed understanding of these materials as for crystals. Both of
the microscopic structure and macroscopic descriptions are an active field of
study.

Topological defects play a key role in the physics of elastic solids Nabarro
(1987). They are characterized by a discontinuity in the order parameter. In
the context of classical elasticity there are two types of such
discontinuities: dislocations and disclinations. They are crucial in the two-
dimensional phase transitions as first shown by Kosterlitz and Thouless and
applied to solids by Nelson, Halperin and Young José (2013); Nelson and
Halperin (1979); Young (1979). The transition happens through the
proliferation of topological defects at finite temperature that results in a
thermal melting of a crystal. In elasticity such melting occurs in two stages.
First the dislocations condense, while the disclinations are still
energetically too costly. This is the hexatic state, which is an example of
the quantum nematic order Nelson (2002). In the next stage both dislocations
and disclinations are condensing leading to the isotropic state. Although
defects is a classic subject in the science of materials, the intricate
geometric constructions are quite far from an effective field theory framework
usually employed to study phase transition. To circumvent this issue a major
theoretical insight, that allows one to easily incorporate defects as sources
in the field theory formulation, is provided by Kleinert Kleinert (1982, 1983)
(for a review see Kleinert (1989, 1995); Beekman et al. (2017a)). In a
complete analogy to the particle vortex duality he rewrites elasticity as a
symmetric gauge field. This duality maps defects to matter fields charged by
the dual gauge fields.

Symmetric tensor gauge fields emerge also as a low-energy description of
certain spin liquids Xu (2006); Xu and Hořava (2010); Pretko (2017a, b); You
et al. (2020). Such gauge fields are sourced by matter fields with restricted
mobility, whose presence indicate a new type of topological phase of matter
Nandkishore and Hermele (2019); Pretko et al. (2020). This similarity with
quantum elasticity can be made precise in the language of dualities formulated
earlier. Dislocations are vector charges that can move along their Burgers
vector, while disclinations are fully immobile scalar charges. This behavior
leads to the conclusion that elastic defects are in fact fractons Pretko and
Radzihovsky (2018a). As a result elastic dualities serve to expand our
knowledge on the classical and quantum elasticity as well as to give insights
into new fractonic phases of matter Zaanen et al. (2004); Beekman et al.
(2017b); Pretko and Radzihovsky (2018b); Gromov (2019); Kumar and Potter
(2019); Pretko et al. (2019); Zhai and Radzihovsky (2019); Gromov and Surówka
(2020); Nguyen et al. (2020); Nampoothiri et al. (2020); Fruchart and Vitelli
(2020); Manoj et al. (2020); Zhai and Radzihovsky (2020).

In this paper we intend to study the connection between fractons and
elasticity of quasicrystals. Quasicrystals are solids, with long-range
positional order and no periodicity Levine and Steinhardt (1984); Levine et
al. (1985); Socolar et al. (1986); Baggioli and Landry (2020) (See Fan (2016)
for a review). This means that the elastic description of quasicrystals is
fundamentally different than the one for crystals. The free energy of ordinary
crystals is unchanged under a discrete translation corresponding to the
lattice vectors defining the unit cell of the periodic structure. When the
translation is allowed to vary slowly as a function of the position the free
energy increases. This can be parameterized by a displacement field
$u_{i}(x)$, which describes phonons, Goldstone bosons that are low-energy
collective excitations of the crystal. In quasicrystals the free energy also
remains constant under the global rearrangements of atomic positions that can
be parameterized by an additional vector $w_{i}(x)$. Small fluctuations of
this vector introduce the so-called phason field or strain, capturing the low-
energy collective excitations of quasicrystal called phasons.

The main goal of the present work is to reformulate the elastodynamics of
quasicrystals Ding et al. (1993) in terms of gauge theories. Such a
formulation allows for a systematic study of defects and their interactions as
well as it opens up a possibility to investigate two-dimensional defect-
mediated phase transitions present in quasicrystals. It also provides a
theoretical background for quantum phases with quasicrystalline symmetries.

Elasticity of quasicrystals.$-$Quasicrystals are characterized by two types of
displacement fields $u_{i}(x)$ and $w_{i}(x)$. The first is analogous to the
phonon displacement and leads to the symmetric strain tensor $u_{ij}=u_{ji}$,
where $u_{ij}=\frac{1}{2}(\partial_{i}u_{j}+\partial_{j}u_{i}).$ Contrary to
the phonon field $u_{ij}$ the phason displacement tensor
$w_{ij}=\partial_{i}w_{j}$ is not symmetric $w_{ij}\neq w_{ji}$. Following the
usual procedure of writing a free energy as an expansion around the zero
displacement $u_{ij}=w_{ij}=0$ one can write down the potential part of the
effective action $S=\int dtd^{2}x\mathcal{L}$ as

$\displaystyle S_{\text{pot}}[u_{i},w_{i}]$ $\displaystyle=\int
dtd^{2}x\frac{-1}{2}\Big{(}C^{ijkl}u_{ij}u_{kl}\Big{)}\,$ $\displaystyle+\int
dtd^{2}x\frac{-1}{2}\Big{(}K^{ijkl}w_{ij}w_{kl}\Big{)}$ (1)
$\displaystyle+\int dtd^{2}x\frac{-1}{2}\Big{(}R^{ijkl}w_{ij}u_{kl}+R^{\prime
ijkl}w_{ij}u_{kl}\Big{)}$
$\displaystyle\equiv\int\frac{-1}{2}\Big{[}\left(u_{ij}\,w_{ij}\right)\begin{pmatrix}C_{ijkl}&R_{ijkl}\\\
R^{\prime}_{ijkl}&K_{ijkl}\end{pmatrix}\begin{pmatrix}u_{kl}\\\
w_{kl}\end{pmatrix}\Big{]},\,$

where we have introduced four tensors of elastic coefficients parameterizing
different couplings between fluctuations. We always assume a summation over
repeated indices, which run over spatial coordinates $x$ and $y$ as we focus
on two-dimensional systems. It is important to note that the phonon elastic
tensor $C_{ijkl}$ possesses both minor and major symmetries, the couplings
between phonons and phasons represented $R^{ijkl}$ and $R^{\prime ijkl}$ have
a minor symmetry of indices contracted with the phonon field. Finally the
$K^{ijkl}$ has neither minor nor major symmetries. Potential energy in the
effective action can be supplemented by the kinetic part

$S_{\text{kin}}[u_{i},w_{i}]=\int
dtd^{2}x\Big{[}\dot{u}_{i}\dot{u}_{i}+\dot{w}_{i}\dot{w}_{i}\Big{]}\,.$ (2)

It is well known that the phason field is diffusive Lubensky et al. (1985);
Francoual et al. (2003), therefore the above elastodynamics describes either
short time behavior of classical quasicrystals or systems at zero temperature.
Several realizations of quantum quasicrystalline symmetry have been proposed
Gopalakrishnan et al. (2013); Kraus et al. (2013); Tran et al. (2015); Sagi
and Nussinov (2016); Bandres et al. (2016); Huang and Liu (2018); Ahn et al.
(2018); Varjas et al. (2019). The partition function for quasicrystals
elasticity reads

$Z=\int
Du^{i}Dw^{i}e^{iS_{\text{kin}}[u_{i},w_{i}]+iS_{\text{pot}}[u^{i},w^{i}]}\,.$
(3)

Our intention is to perform the duality transformation on the above action. In
order to do that we need to write it first in terms of stress variables. There
are two equivalent ways of doing that. One can directly perform the Hubbard-
Stratonovich transformation or write down generalized Hooke’s laws

$\displaystyle T_{ij}$ $\displaystyle=-\frac{\partial\mathcal{L}}{\partial
u_{ij}}=C_{ijkl}u_{kl}+R_{ijkl}w_{kl}$ (4a) $\displaystyle H_{ij}$
$\displaystyle=-\frac{\partial\mathcal{L}}{\partial
w_{ij}}=K_{ijkl}w_{kl}+R^{\prime}_{ijkl}u_{kl},$ (4b)

solve for displacements in terms of stresses $T_{ij}$ and $H_{ij}$ and express
the action in terms of these variables. To write down (4a) and (4b) we use the
fact that $R^{\prime}_{klij}=R_{ijkl}$. These transformations bring the action
to the following form

$\displaystyle S[T_{ij},H_{ij},u_{i},w_{j}]$ $\displaystyle=\int
dtd^{2}x\frac{1}{2}\Big{[}P_{i}P^{i}+\mathcal{P}_{i}\mathcal{P}^{i}\,$
$\displaystyle+\left(T_{ij}\,H_{ij}\right)\begin{pmatrix}C_{ijkl}&R_{ijkl}\\\
R^{\prime}_{ijkl}&K_{ijkl}\end{pmatrix}^{-1}\begin{pmatrix}T_{kl}\\\
H_{kl}\end{pmatrix}$
$\displaystyle+2u_{i}(\partial_{\mu}T^{i\mu})+2w_{i}(\partial_{\mu}H^{i\mu})\Big{]},$
(5)

where we have introduced momentum operators $P^{i}=T^{i0}$ and
$\mathcal{P}^{i}=H^{i0}$. Greek letters run over spacetime indices. After the
transformation to stress variables displacements act as Lagrange multipliers
for the conservation of momentum.

Elastic duality for quasicrystals.$-$Dualities offer new insights into non-
perturbative physics of strongly correlated systems. In $2+1$ dimensions the
core of dualities lies in the mapping of the Goldstone bosons fluctuations
onto appropriate gauge fields. For example particle-vortex duality maps scalar
fields onto $U(1)$ gauge fields and crystal elasticity maps strain
fluctuations into symmetric tensor gauge fields. Elasticity of quasicrystals
generalizes this mapping and introduces two sets of distinct elastic gauge
fields. In order to see this we note that integrating out $u_{i}(x)$ and
$w_{i}(x)$ we obtain two constraints coming from rewriting the action in terms
of stress variables $\delta\left(\partial_{\mu}T^{i\mu}\right)$ and
$\delta\left(\partial_{\mu}H^{i\mu}\right)$. In order to have a dual action
for quasicrystals we resolve these constraints by two tensor gauge fields

$T^{i\mu}=\epsilon^{\mu\nu\rho}\partial_{\nu}A^{i}_{\rho}\,,\qquad
H^{i\mu}=\epsilon^{\mu\nu\rho}\partial_{\nu}\mathcal{A}^{i}_{\rho}\,.$ (6)

We note that a symmetric tensor such as $T^{i\mu}$ can be resolved by a
symmetric tensor field $A_{ij}=A_{ji}$ and a scalar $\phi$ in a complete
analogy with crystal elasticity

$P^{i}=\epsilon^{kl}\partial_{k}A^{i}_{l}\,,\qquad
T^{ij}=\epsilon^{jk}(-\partial_{0}A^{i}_{k}+\partial_{k}\partial_{i}\phi)\,,$
(7)

however, the resolution of the constraint for $H_{ij}$ leads to a tensor field
$\mathcal{A}_{ij}\neq\mathcal{A}_{ji}$ that is not symmetric under exchange of
indices and a vector potential $\Phi_{i}$

$\mathcal{P}^{i}=\epsilon^{kl}\partial_{k}A^{i}_{l}\,,\qquad
H^{ij}=\epsilon^{jk}(-\partial_{0}\mathcal{A}^{i}_{k}+\partial_{k}\Phi^{i})\,.$
(8)

In analogy with Maxwell electrodynamics we can define electric and magnetic
fields

$B^{i}=\epsilon^{kl}\partial_{k}A^{i}_{l}\,,\qquad
E^{i}_{j}=\epsilon^{i}{}_{k}(-\partial_{0}A^{k}_{j}+\partial_{j}\partial_{k}\phi)\,.$
(9)
$\mathcal{B}^{i}=\epsilon^{kl}\partial_{k}\mathcal{A}^{i}_{l}\,,\qquad\mathcal{E}^{i}_{j}=\epsilon^{i}{}_{k}(-\partial_{0}\mathcal{A}^{k}_{j}+\partial_{j}\Phi^{k})\,.$
(10)

These fields are invariant under the following gauge transformations

$\delta
A_{ij}=\partial_{i}\partial_{j}\alpha\,,\qquad\delta\phi=\dot{\alpha}\,,$ (11)
$\delta\mathcal{A}_{ij}=\partial_{j}\beta_{i}\,,\qquad\delta\Phi_{i}=\dot{\beta}_{i}\,.$
(12)

Already at this stage we see that the gauge structure for the symmetric stress
tensor is the same as in scalar fracton theories and the asymmetric stress
field leads to a vector gauge theory Pretko (2018). Thus quasicrystal
elasticity combines these two degrees of freedom. We can now write the
effective action in the dual formulation

$\displaystyle S_{\text{dual}}$ $\displaystyle=\int
dtd^{2}x\frac{1}{2}\Bigg{[}B_{i}B^{i}+\mathcal{B}_{i}\mathcal{B}^{i}\,$
$\displaystyle+\left(E_{ij}\,\mathcal{E}_{ij}\right)\begin{pmatrix}\tilde{\mathcal{C}}_{ijkl}&\tilde{\mathcal{R}}_{ijkl}\\\
\tilde{\mathcal{R}}^{\prime}_{ijkl}&\tilde{\mathcal{K}}_{ijkl}\end{pmatrix}\begin{pmatrix}E_{kl}\\\
\mathcal{E}_{kl}\end{pmatrix}\Bigg{]}$ (13) $\displaystyle+\int
dtd^{2}x\mathcal{L}_{\text{sources}}$

We have introduced a set of tensors that are yet to be fixed by calculating
the proper inverse defined in (Dual gauge theory formulation of planar
quasicrystal elasticity and fractons). We note that the entries are not just
the inverses of the original elastic coefficients as we need to invert the
whole matrix of tensors. Tilde denotes index rotations, e.g.
$\tilde{\mathcal{C}}_{ijkl}=\epsilon_{ii^{\prime}}\epsilon_{jj^{\prime}}\epsilon_{kk^{\prime}}\epsilon_{ll^{\prime}}\mathcal{C}^{i^{\prime}j^{\prime}k^{\prime}l^{\prime}}$.
The dual charges, that we later map to defects, couple to the gauge potentials
in the following way

$\mathcal{L}_{\text{sources}}=\phi\rho+A_{ij}J_{ij}+\Phi_{i}\varrho_{i}+\mathcal{A}_{ij}\mathcal{J}_{ij}\,.$
(14)

The last ingredient that we need is the inverse matrix of elastic tensors. The
explicit form of this matrix is in general complicated and depends on the
symmetries of the quasicrystal that we want to study. In order to have a
better intuition about the structure of this matrix w focus on a sub-class,
for which the tensors of elastic coefficients can be decomposed into a set of
projectors

$\displaystyle C_{ijkl}$
$\displaystyle=c_{0}P^{(0)}_{ijkl}+c_{1}P^{(1)}_{ijkl}+c_{2}P^{(2)}_{ijkl},$
(15a) $\displaystyle K_{ijkl}$
$\displaystyle=k_{0}P^{(0)}_{ijkl}+k_{1}P^{(1)}_{ijkl}+k_{2}P^{(2)}_{ijkl},$
(15b) $\displaystyle R_{ijkl}$
$\displaystyle=r_{0}P^{(0)}_{ijkl}+r_{1}P^{(1)}_{ijkl}+r_{2}P^{(2)}_{ijkl},$
(15c) $\displaystyle R^{\prime}_{ijkl}$
$\displaystyle=r^{\prime}_{0}P^{(0)}_{ijkl}+r^{\prime}_{1}P^{(1)}_{ijkl}+r^{\prime}_{2}P^{(2)}_{ijkl},$
(15d)

where the basis tensors are given by

$\displaystyle P^{(0)}_{ijkl}$
$\displaystyle=\frac{1}{2}\delta_{ij}\delta_{kl},$ (16a) $\displaystyle
P^{(1)}_{ijkl}$
$\displaystyle=\frac{1}{2}(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk}),$
(16b) $\displaystyle P^{(2)}_{ijkl}$
$\displaystyle=\frac{1}{2}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})-\frac{1}{2}\delta_{ij}\delta_{kl}.$
(16c)

One can check that $P^{m}_{ijab}P^{n}_{abkl}=P^{m}_{ijkl}$ if $m=n$ and zero
otherwise. Such a choice is of course an oversimplification, which will not
apply to all systems. However, it illustrates the proof of concept behind this
construction and the result can be presented in a compact form. In general, in
order to construct a dual formulation for a specific quasicrystal, one has to
specify elastic tensors and then explicitly construct an inverse. The simplest
way to achieve this in two dimensions, for a given set of elastic parameters,
is to pass to the Pauli matrix representation (see e.g. Scheibner et al.
(2020); Banerjee et al. (2020); Fruchart and Vitelli (2020)) and then apply an
algorithm for the inversion of a block matrix Lu and Shiou (2002).

The inverse matrix is determined by the following matrix equation

$\begin{pmatrix}\mathcal{C}_{ijmn}&\mathcal{R}_{ijmn}\\\
\mathcal{R}^{\prime}_{ijmn}&\mathcal{K}_{ijmn}\end{pmatrix}\begin{pmatrix}C_{ijkl}&R_{ijkl}\\\
R^{\prime}_{ijkl}&K_{ijkl}\end{pmatrix}=\begin{pmatrix}\text{Id}_{ijmn}&0\\\
0&\text{Id}_{ijmn}\end{pmatrix},$ (17)

where appropriate index contractions are understood after the multiplication
of matrices. $\text{Id}_{ijmn}=\delta_{im}\delta_{jn}$ is the identity
operator for four-tensors. For the case considered this is a system of linear
equations for twelve coefficients. The solution in the basis of projectors
reads

$\displaystyle\mathcal{C}_{ijkl}$
$\displaystyle=\frac{k_{0}}{\Delta_{0}}P^{(0)}_{ijkl}+\frac{k_{1}}{\Delta_{1}}P^{(1)}_{ijkl}+\frac{k_{2}}{\Delta_{2}}P^{(2)}_{ijkl},$
(18a) $\displaystyle\mathcal{K}_{ijkl}$
$\displaystyle=\frac{c_{0}}{\Delta_{0}}P^{(0)}_{ijkl}+\frac{c_{1}}{\Delta_{1}}P^{(1)}_{ijkl}+\frac{c_{2}}{\Delta_{2}}P^{(2)}_{ijkl},$
(18b) $\displaystyle\mathcal{R}_{ijkl}$
$\displaystyle=-\frac{r_{0}}{\Delta_{0}}P^{(0)}_{ijkl}-\frac{r_{1}}{\Delta_{1}}P^{(1)}_{ijkl}-\frac{r_{2}}{\Delta_{2}}P^{(2)}_{ijkl},$
(18c) $\displaystyle\mathcal{R}^{\prime}_{ijkl}$
$\displaystyle=-\frac{r^{\prime}_{0}}{\Delta_{0}}P^{(0)}_{ijkl}-\frac{r^{\prime}_{1}}{\Delta_{1}}P^{(1)}_{ijkl}-\frac{r^{\prime}_{2}}{\Delta_{2}}P^{(2)}_{ijkl},$
(18d)

where $\Delta_{m}=c_{m}k_{m}+r_{m}r^{\prime}_{m}$ for each coefficient
labelled by $i\in\\{0,1,2\\}$. We note that it may happen the matrix of
elastic coefficients is not invertible. In this case it is enough to construct
the inverse in the invertible subspace as in the classical elasticity.
Moreover, in analogy with block matrices not all individual entries not have
to be invertible for the existence of the total inverse matrix.

Defects.$-$Soon after the discovery of quasicrystals questions about the
nature of defects, their dynamics and interactions became relevant Levine et
al. (1985); Socolar et al. (1986); Lubensky et al. (1986); Bohsung and Trebin
(1987). Several intricate geometric constructions are available, however, the
intrinsic difficulty of these methods prevents us from having definite answers
to all relevant questions about defects. Below we argue that the duality
offers a natural, simple language for a systematic study of topological
singularities in quasicrystals. In the dual language the elastic defects are
mapped to the charges of the gauge theory. In order to see this mapping one
can decompose the phonon and phason displacement fields into regular and
singular parts $u^{i}=u^{i}_{\rm reg}+u^{i}_{\rm sing}$, $w^{i}=w^{i}_{\rm
reg}+w^{i}_{\rm sing}$. Phonon displacement singularities couple to the
conservation of the stress tensor

$\delta S=\int dtd^{2}x\,\,\,\Big{[}u^{i}_{\rm
sing}\partial_{\mu}T^{i\mu}\Big{]}=\int
dtd^{2}x\,\,\,\Big{[}\rho\phi+J^{ij}A_{ij}\Big{]}\,.$ (19)

where $\rho=\partial^{i}\rho_{i}$,
$\rho^{i}=\epsilon^{i}{}_{j}\epsilon^{kl}\partial_{k}\partial_{l}u^{j}_{\rm
sing}$ and $J^{ij}=\epsilon^{i}{}_{n}\epsilon^{\mu\nu
j}\partial_{\mu}\partial_{\nu}u^{n}_{\rm sing}$. In the last equality we
employ an integration by parts. The charge $\rho$ is mapped to the
disclination density

$\rho_{\rm
disc}=\frac{1}{2}\epsilon^{k}{}_{l}\epsilon^{ij}\partial_{i}\partial_{j}\partial_{k}u^{l}_{\rm
sing}\,.$ (20)

We now study the coupling of the phason singularities $w^{i}_{\rm{sing}}$.
Following the same logic as above we find

$\delta S=\int dtd^{2}x\,\,\,\Big{[}w^{i}_{\rm
sing}\partial_{\mu}H^{i\mu}\Big{]}=\int
dtd^{2}x\,\,\,\Big{[}\varrho_{i}\Phi^{i}+\mathcal{J}^{ij}\mathcal{A}_{ij}\Big{]}\,,$
(21)

where we have introduced matching or stacking faults as the phason defects are
sometimes dubbed in the literature

$\varrho^{i}=\epsilon^{i}{}_{j}\epsilon^{kl}\partial_{k}\partial_{l}w^{j}_{\rm
sing}$ (22)

and the current

$\mathcal{J}^{ij}=\epsilon^{i}{}_{n}\epsilon^{\mu\nu
j}\partial_{\mu}\partial_{\nu}w^{n}_{\rm sing}.\,$ (23)

We can now identify the duality mapping between charges and defects for
phasons. Vector charges in the dual theory map to the rotated matching faults
$\epsilon^{i}{}_{j}\varrho^{j}$. With these mappings we can study the Gauss
laws in the theory. They follow from the gauge transformations in the
Hamiltonian formulation of the theory

$\mathcal{H}=\Pi_{ij}\dot{A}_{ij}+\Pi^{H}_{ij}\mathcal{A}^{ij}-\mathcal{L},$
(24)

where $\Pi_{ij}=\frac{\delta S}{\delta\dot{A}_{ij}}$ and
$\Pi^{H}_{ij}=\frac{\delta S}{\delta\dot{\mathcal{A}}_{ij}}$ are the canonical
momenta of the phonon and phason fields respectively. Invariance with respect
to the gauge transformations leads to the following generalized Gauss laws

$\partial_{i}\partial_{j}\Pi^{ij}=\rho,$ (25)
$\partial_{i}\Pi^{ij}_{H}=\varrho^{j}.$ (26)

We see that these Gauss laws are exactly the same as the ones constructed for
scalar and vector gauge theories Pretko (2018). As a result we can now
identify defects in quasicrystals as different types of fractons. As far as
the phonon field is concerned the defects are the same as in classical
elasticity. We have disclinations that are immobile and dislocations
corresponding to the disinclination dipoles that can move only along their
Burgers vector. In addition to that the matching faults correspond to vector
charges. The matching faults conserve the dipole moment $D_{i}$, where
$D_{i}=\int x^{i}\partial_{j}\varrho^{j}=0$. Therefore the matching faults are
fractons with restricted mobility. Given this quasicrystals offer a natural
playground to investigate scalar and vector fracton theories.

Defect potential.$-$We now apply the duality to study the static defect
potential in a simple illustrative example of quasicrystals with fivefold
symmetry in the limit of negligible phonon-phason coupling. The relevant part
of the action reads

$\displaystyle S$ $\displaystyle=\int
dtd^{2}x\frac{1}{2}\Bigg{[}\left(\partial_{i}\partial_{j}\phi\,\partial_{i}\Phi_{j}\right)\begin{pmatrix}\tilde{\mathcal{C}}_{ijkl}&0\\\
0&\tilde{\mathcal{K}}_{ijkl}\end{pmatrix}\begin{pmatrix}\partial_{k}\partial_{l}\phi\\\
\partial_{k}\Phi_{l}\end{pmatrix}$ (27)
$\displaystyle+\phi\rho+\Phi_{i}\varrho_{i}\Big{]}\,$

In the case of planar fivefold symmetry the elastic tensors are given by Ding
et al. (1993)

$\displaystyle C_{ijkl}$
$\displaystyle=\lambda\delta_{ij}\delta_{kl}+\mu(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}),$
(28a) $\displaystyle K_{ijkl}$
$\displaystyle=K_{1}\delta_{ik}\delta_{jl}+K_{2}(\delta_{ij}\delta_{kl}-\delta_{il}\delta_{jk}).$
(28b)

The above tensors can be expanded in the bassis of projectors upon the
following identifications: $c_{0}=2(\lambda+\mu)$, $c_{1}=0$, $c_{2}=2\mu$,
$k_{1}=K_{1}+K_{2}$, $k_{2}=K_{1}+K_{2}$, $k_{3}=K_{1}-K_{2}$. $\lambda$ and
$\mu$ are the first and second Lamé parameters. Integrating out the gauge
fields $\phi$ and $\Phi_{i}$ one arrives at the following expression,

$\mathcal{L}=-\frac{1}{2}\rho_{\text{vec}}^{\text{T}}(-q)\mathcal{V}\rho_{\text{vec}}(q),$
(29)

where
$\rho_{\text{vec}}^{\text{T}}=\left(\rho\,\,\varrho_{1}\,\,\varrho_{2}\right)$.
It gives the static potential between the defects. The disclination potential
is $\mathcal{V}_{\rho\rho}=\frac{4\mu(\lambda+\mu)}{q^{4}(\lambda+2\mu)}$ and
the explicit form of the matching fault potential potential is given by

$\displaystyle\mathcal{V_{\varrho\varrho}}=$
$\displaystyle\begin{pmatrix}\frac{(K_{1}-K_{2})\left[(K_{1}+K_{2})q_{1}^{2}+2K_{1}q_{2}^{2}\right]}{q^{4}K_{1}}&-\frac{(K_{1}-K_{2})^{2}q_{1}q_{2}}{q^{4}K_{1}}\\\
-\frac{(K_{1}-K_{2})^{2}q_{1}q_{2}}{q^{4}K_{1}}&\frac{(K_{1}-K_{2})\left[2K_{1}q_{1}^{2}+(K_{1}+K_{2})q_{2}^{2}\right]}{q^{4}K_{1}}\end{pmatrix}.$
(30)

In the limit we consider there is no potential between matching faults and
defects in the phonon field. Our computation shows the power of the duality
that greatly simplifies the analysis by matching the defects into charges of
appropriate gauge fields that can be easily dealt with using field theory
methods.

Discussion.$-$We have constructed a dual formulation of quasicrystal
elasticty. It has two stress tensors, one symmetric and one not, which
ultimately leads to the dual description in terms of the two tensor gauge
fields. Both these fields couple to fractonic charges dual to dislocations and
disclinations in the phonon sector and to matching faults for phasons. It
follows that quasicrystal elasticity is a natural place where scalar and
vector fracton theories coexist.

The duality offers a way to address open questions about the phase structure
of quasicrystals. It was speculated in the past that a brittle-ductile
transition can be related to the BKT-type of transition Kleman (2003). The
dual formulation simplifies an analysis of phase transitions due to defect
proliferation. Thus the construction presented here can be used as a starting
point for a detailed study.

Acknowledgements.$-$PS was supported by the Deutsche Forschungsgemeinschaft
through the Leibniz Program, the cluster of excellence ct.qmat (EXC 2147,
project-id 39085490) and the National Science Centre Sonata Bis grant
2019/34/E/ST3/00405.

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