Volume-preserving diffeomorphism as nonabelian higher-rank gauge symmetry

Yi-Hsien Du 1, Umang Mehta 1, Dung Xuan Nguyen2,*, Dam Thanh Son1

1 Kadanoff Center for Theoretical Physics, University of Chicago, Illinois
60637, USA

2 Brown Theoretical Physics Center and Department of Physics, Brown
University, 182 Hope Street, Providence, RI 02912, USA

*<EMAIL_ADDRESS>
## Abstract

We propose nonabelian higher-rank gauge theories in 2+1D and 3+1D. The gauge
group is constructed from the volume-preserving diffeomorphisms of space. We
show that the intriguing physics of the lowest Landau level (LLL) limit can be
interpreted as the consequences of the symmetry. We derive the renowned
Girvin-MacDonald-Platzman (GMP) algebra as well as the topological Wen-Zee
term within our formalism. Using the gauge symmetry in 2+1D, we derive the LLL
effective action of vortex crystal in rotating Bose gas as well as Wigner
crystal of electron in an applied magnetic field. We show that the nonlinear
sigma models of ferromagnets in 2+1D and 3+1D exhibit the higher-rank gauge
symmetries that we introduce in this paper. We interpret the fractonic
behavior of the excitations on the lowest Landau level and of skyrmions in
ferromagnets as the consequence of the higher-rank gauge symmetry.

###### Contents

  1. 1 Introduction
  2. 2 Review of the traceless scalar charge theory
  3. 3 Nonlinear higher-ranked symmetry
    1. 3.1 Traceless scalar charge theory in (2+1)D as a theory of linearized gravity
    2. 3.2 Nonlinear higher-rank symmetry
  4. 4 Connection to quantum Hall effect
    1. 4.1 Higher-rank symmetry in the lowest Landau level limit
    2. 4.2 The Wen-Zee term
  5. 5 Generalization to (3+1) dimensions
    1. 5.1 Construction of the (3+1)D nonlinear higher-rank symmetry
    2. 5.2 Linearized higher-ranked symmetries and conservation laws
  6. 6 Examples of theories with volume-preserving diffeomorphism invariance
    1. 6.1 Composite fermions
    2. 6.2 Crystal on the lowest Landau level
    3. 6.3 Vortex crystal
    4. 6.4 Ferromagnets
      1. 6.4.1 Ferromagnets in 2+1 D
      2. 6.4.2 Ferromagnets in 3+1 D
  7. 7 Conclusion
  8. A The uniqueness of the nonlinear transformation (24)

##  1 Introduction

Recently, considerable interest has been drawn to “higher-rank gauge
theories,” i.e., theories where the gauge potential is not a one-form, but a
tensor of higher rank [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. The
physical motivation of the higher-rank gauge theories is the discovery of a
new class of topological matter known as fractons [15, 16, 17, 18, 19], one
feature of which is the existence of excitations with restricted mobility. The
excitations either cannot move at all or can only move in lower-dimensional
sub-spaces, while composites of elementary excitations can move freely. The
restricted mobility of the fractonic excitations can be interpreted as the
consequence of the higher-rank theories’ conservation laws. In particular, the
tensor Gauss’s law leads to the conservation of not only the electric charge
but also the electric dipole moment and, in some cases, higher moments of the
charge distribution [1, 2, 19]. This has the physical effect of rendering the
electric charge immobile but leaving the dipoles mobile or partially mobile.
One representative example is the so-called “traceless scalar charge theory,”
in which the electric charges are immobile and the electric dipole can move
only in the direction perpendicular to the dipole moment [2]. The higher-rank
gauge theories have also been applied to describe defects in solids [20, 21,
22, 23, 24], supersolids [25], superfluid vortices [26, 27], smectics [27, 5,
28], and other systems.

All the higher-rank gauge symmetries in the physical models mentioned above
are abelian. There were attempts to generalize these symmetries to nonabelian
symmetries [6, 29, 30, 31]; however, physical systems that realize those
symmetries have not been explicitly proposed. In this paper, we propose
nonabelian higher-rank symmetry theories for condensed matter systems of
physical relevance. We reformulate the tensor gauge transformation in the
traceless scalar charge theory in 2+1D [1, 2], showing that the gauge
transformation is nothing but the volume-preserving diffeomorphism (VPD) in
the linearized form. We then construct nonlinear higher-rank gauge theories
with VPDs as the symmetry group. The nonabelian nature of the gauge symmetry
has a nontrivial consequence: the charge density operators at different
positions do not commute. Instead, they form the long-wavelength limit of the
Girvin-MacDonald-Platzman (GMP) algebra [32, 33, 34], which reveals a
connection to the lowest Landau level (LLL).

It is easy to notice that many features of the physics on the LLL bear a close
resemblance to that of the field-theory models with higher rank symmetries
[35]: for example, electric charges are pinned to one place by the large
magnetic field, and neutral excitations (e.g., the composite fermion in the
half-filled Landau level [36]) carry an electric dipole moment and can move in
the direction perpendicular to the direction of motion. In this paper, we show
that this resemblance is not accidental; in fact, the nonlinear higher-rank
symmetry is realized as a symmetry of the problem of charged particles on the
LLL. The relation between the dipole moment and momentum of an excitation in
models with higher-rank symmetry was noticed in Refs. [13, 26].

We also will present several physical systems that enjoy the nonabelian
higher-rank symmetry. In 2+1D systems, the symmetry originates from the
lowest-Landau-level limit, where the volume-preserving nature of the
diffeomorphisms comes from the restriction that the transformations should
preserve the background magnetic field. In addition to the derivation of the
GMP algebra, we draw a connection between the topological Wen-Zee term [37]
with the Chern-Simons term in a higher-rank gauge theory.

Furthermore, we will use the gauge symmetry to derive the effective theories
of the Wigner crystal of electrons in a strong magnetic field and the vortex
crystal in a rotating Bose gas.

Finally, we find that the nonlinear sigma models describing ferromagnetism in
2+1D and 3+1D also exhibit the global higher-rank symmetry. The higher rank
gauge symmetry provides a new interpretation of the conservation of multipole
moments in ferromagnets [38]; it also explains the close resemblance between
the behaviors of skyrmions in ferromagnets and charged particles in a magnetic
field [39].

##  2 Review of the traceless scalar charge theory

For the paper to be self-contained, in this Section we will review the
symmetric tensor gauge theory proposed by Pretko [1, 2]. We consider a higher-
rank gauge theory called “traceless scalar charge theory,” where the gauge
potential is a symmetric rank-2 tensor $A_{ij}$: $A_{ij}=A_{ji}$. Its
conjugate momentum is the electric field $E_{ij}$. They satisfy the canonical
commutation relation

$[E_{ij}(\mathbf{x}),\,A_{kl}(\mathbf{y})]=i(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})\delta(\mathbf{x}-\mathbf{y}).$
(1)

One imposes the Gauss law and the traceless constraint:

$\displaystyle\partial_{i}\partial_{j}E_{ij}$ $\displaystyle=\rho,$ (2)
$\displaystyle E\equiv E_{ii}$ $\displaystyle=0,$ (3)

which lead to the conservation of the following charges:

$\int\\!d\mathbf{x}\,\rho(\mathbf{x}),\quad\int\\!d\mathbf{x}\,\mathbf{x}\rho(\mathbf{x}),\quad\int\\!d\mathbf{x}\,x^{2}\rho(\mathbf{x}).$
(4)

The conservation of the quantities listed in in Eq. (4) imply that a charge
cannot move, and a dipole can move only along the direction perpendicular to
the dipole moment [1].

The constraints (2) and (3) generate the gauge transformations on the gauge
potential

$\displaystyle A_{ij}$ $\displaystyle\to
A_{ij}+\partial_{i}\partial_{j}\lambda,$ (5) $\displaystyle A_{ij}$
$\displaystyle\to A_{ij}+\delta_{ij}\mu.$ (6)

For our purpose, it is convenient to use the second gauge transformation (6)
to explicitly fix the gauge $A\equiv A_{ii}=0$ and eliminate the trace of
$A_{ij}$ from the set of dynamical degrees of freedom. Since the two
constraints $E=0$ and $A=0$ do not commute according to the commutation
relation (1), following Dirac one should modify the commutators, replacing
them by the Dirac brackets [40], which in our case is

$[O_{1},\,O_{2}]\to[O_{1},\,O_{2}]_{\text{D}}=[O_{1},\,O_{2}]+[O_{1},\,E][E,\,A]^{-1}[A,\,O_{2}]-[O_{1},\,A][E,\,A]^{-1}[E,\,O_{2}]$
(7)

The new commutator is then

$[E_{ij}(\mathbf{x}),\,A_{kl}(\mathbf{y})]=i\left(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-\frac{2}{d}\delta_{ij}\delta_{kl}\right)\delta(\mathbf{x}-\mathbf{y}).$
(8)

The Gauss constraint $\partial_{i}\partial_{j}E_{ij}=\rho$ generates now the
gauge transformation

$A_{ij}\to
A_{ij}+\partial_{i}\partial_{j}\lambda-\frac{1}{d}\delta_{ij}\partial^{2}\lambda,\qquad\partial^{2}\equiv\partial^{k}\partial_{k}.$
(9)

To construct a gauge-invariant Lagrangian, we introduce the field strengths.
The electric field

$E_{ij}=\partial_{i}\partial_{j}A_{0}-\frac{1}{d}\delta_{ij}\partial^{2}A_{0}-\partial_{t}A_{ij},$
(10)

is obviously gauge invariant. One notices that

$\omega_{i}=-\partial_{j}A_{ij}$ (11)

transforms like a $U(1)$ vector potential,

$\omega_{i}\to\omega_{i}-\partial_{i}\tilde{\lambda},\qquad\tilde{\lambda}=\left(1-\frac{1}{d}\right)\partial^{2}\lambda,$
(12)

using which one can define the magnetic field,

$H_{ij}=\partial_{i}\omega_{j}-\partial_{j}\omega_{i}=-\partial_{i}\partial_{k}A_{jk}+\partial_{j}\partial_{k}A_{ik},$
(13)

that is manifestly gauge invariant. The simplest Lagrangian for the gauge
field is then the “Maxwell theory,”

$L=c_{1}E_{ij}^{2}-c_{2}H_{ij}H_{ij}.$ (14)

However, as noticed in Ref. [41], in (2+1)D, another possible term in the
Lagrangian is the Chern-Simons term which, up to an overall coefficient, reads

$L_{\rm CS}=\varepsilon^{ij}(A_{0}H_{ij}-A_{ik}\dot{A}_{jk}).$ (15)

As for a Chern-Simons term, the Lagrangian density is gauge-invariant only up
to a total derivative. This term is more relevant than the Maxwell term. The
higher-rank Chern-Simons theory in 3 spatial dimensions was also discussed
previously in Ref. [42].

Note that one does not have to require the theory to contain dynamical gauge
fields in order to have the conserved quantities (4). A theory coupled to
background gauge fields $(A_{0},A_{ij})$ and is invariant under the gauge
symmetry under which the gauge fields transform as

$A_{0}\to A_{0}+\dot{\lambda},\qquad
A_{ij}+\partial_{i}\partial_{j}\lambda-\frac{1}{d}\delta_{ij}\partial^{2}\lambda,$
(16)

will have the conservation law

$\partial_{t}\rho-\partial_{i}\partial_{j}J_{ij}=0,$ (17)

where $\rho$ and $J_{ij}$ are the operators that couple to $A_{0}$ and
$A_{ij}$, respectively, and $J_{ii}=0$. This is sufficient to derive the
conservation of the quantities (4). In fact, in most of this paper, we will
consider theories coupled to nondynamical background gauge fields.

##  3 Nonlinear higher-ranked symmetry

###  3.1 Traceless scalar charge theory in (2+1)D as a theory of linearized
gravity

We now show that the theory presented in the previous section can be
interpreted as a theory of linearized gravity, and the higher-rank symmetry is
the linearized version of VPD. Instead of $A_{ij}$ we introduce an equivalent
field $h_{ij}$ defined as

$h_{ij}=-\ell^{2}(\varepsilon_{ik}A_{jk}+\varepsilon_{jk}A_{ik}),$ (18)

where $\ell$ is some constant of the dimension of length111We assume $A_{0}$
has dimension 1 and $A_{ij}$ is of dimension 2, so $h_{ij}$ is dimensionless..
Note that $h_{ij}$ is also symmetric and traceless. The gauge transformation
for $h_{ij}$ is inherited from (16)

$h_{ij}\to
h_{ij}-\ell^{2}(\varepsilon_{ik}\partial_{j}\partial_{k}+\varepsilon_{jk}\partial_{i}\partial_{k})\lambda.$
(19)

If we define

$\xi^{i}=\ell^{2}\varepsilon^{ik}\partial_{k}\lambda,$ (20)

then the transformation law of $h_{ij}$ can be reformulated as

$h_{ij}\to h_{ij}-\partial_{i}\xi_{j}-\partial_{j}\xi_{i}.$ (21)

The transformation (21) is nothing but the transformation of the metric under
the volume-preserving (or in 2D, area-preserving) diffeomorphism $x^{i}\to
x^{i}+\xi^{i}$, since $\partial_{i}\xi^{i}=0$ due to the definition (20). The
connection between a higher-rank gauge theory and a linearized gravity theory
was proposed previously in Ref. [43].

###  3.2 Nonlinear higher-rank symmetry

The fact that the gauge symmetry resembles the transformation law of a metric
under VPDs allows one to devise a nonlinear version of the gauge symmetry.
Namely, in our nonlinear theory, instead of a gauge field $h_{ij}$ (or
$A_{ij}$ for that matter), the degree of freedom is the metric $g_{ij}$. The
tracelessness of $h_{ij}$ translates into the statement that the metric is
unimodular: $\det g=1$. The linear theory is restored when one expands the
metric around the flat metric: $g_{ij}=\delta_{ij}+h_{ij}+O(h^{2})$.

Under an infinitesimal VPD $x^{i}\to
x^{i}+\xi^{i}=x^{i}+\ell^{2}\varepsilon^{ij}\partial_{j}\lambda$, the metric
transforms as

$\delta_{\lambda}g_{ij}=-\xi^{k}\partial_{k}g_{ij}-g_{kj}\partial_{i}\xi^{k}-g_{ik}\partial_{j}\xi^{k}=-\ell^{2}\varepsilon^{kl}(\partial_{k}g_{ij}+g_{kj}\partial_{i}+g_{ik}\partial_{j})\partial_{l}\lambda.$
(22)

Now we need to write down the nonlinear version of the transformation laws for
$A_{0}$. One notices that the VPDs do not commute: from Eq. (22) one reads

$[\delta_{\alpha},\delta_{\beta}]=\delta_{[\alpha,\beta]},$ (23)

with
$[\alpha,\beta]=\ell^{2}\varepsilon^{ij}\partial_{i}\alpha\partial_{j}\beta$.
This means that our gauge symmetry is nonabelian; in this paper, we will use
“nonlinear” and “nonabelian” interchangeably. The transformation of $A_{0}$
must satisfy the commutation relation (23). One can check that this can be
accomplished by the following simple modification

$\delta_{\lambda}A_{0}=\partial_{t}\lambda-\xi^{k}\partial_{k}A_{0}=\partial_{t}\lambda-\ell^{2}\varepsilon^{kl}\partial_{k}A_{0}\,\partial_{l}\lambda.$
(24)

The transformation of $A_{0}$ in Eq. (24) was motivated by the symmetries of
the lowest Landau level that will be discussed subsequently. Nonetheless, it
is the unique nonlinear generalization of (16) given that the transformation
is at most linear in $A_{0}$ and respects rotational invariance. Furthermore,
the second term of (24) means that $A_{0}$ transforms as a scalar field under
time-independent spatial diffeomorphism, which is expected. We leave the
detailed discussion on the uniqueness of the nonlinear transformation (24) to
Appendix A. Equations (24) and (22) give the transformation laws of a
nonlinear higher-rank symmetry, which we collect here for convenience:

$\displaystyle\delta_{\lambda}A_{0}$
$\displaystyle=\partial_{t}\lambda-\ell^{2}\varepsilon^{kl}\partial_{k}A_{0}\,\partial_{l}\lambda,$
(25) $\displaystyle\delta_{\lambda}g_{ij}$
$\displaystyle=-\ell^{2}\varepsilon^{kl}(\partial_{k}g_{ij}+g_{kj}\partial_{i}+g_{ik}\partial_{j})\partial_{l}\lambda.$
(26)

A nonlinear transformation similar to Eq. (26) was considered in Ref. [14]
within a dynamical gauge model of the traceless scalar charge theory. One can
derive the Ward identity from Eqs. (25) and (26). Let us define the charge
density $\rho$ and the stress tensor $T^{ij}$ by varying the logarithm of the
partition function

$\delta\ln Z=\int\\!d^{3}x\left(\rho\delta A_{0}+\frac{1}{2}T^{ij}\delta
h_{ij}\right).$ (27)

The Ward identity is then

$\dot{\rho}-\ell^{2}\varepsilon^{kl}\partial_{l}\left[\rho\partial_{k}A_{0}+\tfrac{1}{2}T^{ij}\partial_{k}g_{ij}+\partial_{i}(T^{ij}g_{jk})\right]=0$
(28)

In the presence of the background field, only the total charge is conserved,
but not the higher multipoles in (4).

Since the total charge is conserved, it is possible to introduce a vector
potential $A_{i}$ so that the theory is invariant under the usual $U(1)$ gauge
symmetry $A_{\mu}\to A_{\mu}+\partial_{\mu}\alpha$. In this case $A_{0}$ plays
a double role: it is the temporal component of a U(1) gauge field
$(A_{0},A_{i})$, and also as the scalar component of the gauge potential of a
higher-spin symmetry, $(A_{0},g_{ij})$. The two sets of gauge potentials share
one scalar component, see Fig. 1. We will see an example when we consider
ferromagnets (Sec. 6.4).

Figure 1: $A_{0}$ is shared by two sets of gauge potentials.

The nonabelian nature of the gauge symmetry, in some cases, allows us to
derive the algebra satisfied by the charge density of the matter coupled to
the gauge field. Imagine that the action $S_{\text{m}}(\psi,A_{0},g_{ij})$
describing the coupling of the matter fields $\psi$ with the gauge fields
$(A_{0},g_{ij})$ does not contain the time derivatives of any fields, $A_{0}$,
$g_{ij}$, or $\psi$. In this case, if one promotes the gauge fields to
dynamical fields by adding to the action a pure gauge action $S_{\text{g}}$,

$S=S_{\text{g}}[A_{0},g_{ij}]+S_{\text{m}}[\psi,A_{0},g_{ij}],$ (29)

then upon quantization, the canonical commutation relations in the gauge
sector are set by $S_{\text{g}}$ and in the matter sector by $S_{\text{m}}$.
The left-hand side of the Gauss constraint,

$\frac{\delta S_{\text{g}}}{\delta A_{0}}+\frac{\delta S_{\text{m}}}{\delta
A_{0}}=0,$ (30)

is the generator that generates gauge transformations. In particular, in the
matter sector, the commutator of the charge density $\rho(\mathbf{x})$ with
any matter field $O(\mathbf{x})$ will give the change of $O$ under
infinitesimal gauge transformation:

$\left[\int\\!d\mathbf{y}\,\lambda(\mathbf{y})\rho(\mathbf{y}),\,O(\mathbf{x})\right]=i\delta_{\lambda}O(\mathbf{x}).$
(31)

But diffeomorphisms do not commute, so we conclude that the charge density at
different points does not commute in our theory. We find

$[\rho(\mathbf{x}),\,\rho(\mathbf{y})]=i\ell^{2}\varepsilon^{ij}\partial_{i}\rho(\mathbf{x})\partial_{j}\delta(\mathbf{x}-\mathbf{y}).$
(32)

Here we recover the long-wavelength version of the Girvin–MacDonald–Platzman
(GMP) algebra [32] (or the $w_{\infty}$ algebra), which suggests that the
symmetry described here is related to the physics of the LLL.

##  4 Connection to quantum Hall effect

To establish the connection with the physics of the LLL, we recall the
symmetry of the problem. A system of particles interacting with an
electromagnetic field can also be put in curved space:

$S=\int\\!dt\,d\mathbf{x}\,\sqrt{g}\left(\frac{i}{2}\psi^{\dagger}\overset{\leftrightarrow}{\partial}_{t}\psi+A_{0}\psi^{\dagger}\psi-\frac{g_{ij}}{2m}D_{i}\psi^{\dagger}D_{j}\psi+\cdots\right),$
(33)

where $\cdots$ includes interaction terms. (Strictly speaking, the discussion
here corresponds to the $g=2$, $s=1$ version of the LLL symmetry [44].) One
can check that the classical action is invariant with respect to time-
dependent spatial diffeomorphisms (i.e., all diffeomorphism transformations
which preserve the time slices) [45, 44]:

$\displaystyle\delta A_{0}$
$\displaystyle=-\xi^{k}\partial_{k}A_{0}-A_{k}\dot{\xi}^{k},$ (34)
$\displaystyle\delta A_{i}$
$\displaystyle=-\xi^{k}\partial_{k}A_{i}-A_{k}\partial_{i}\xi^{k}-mg_{ik}\dot{\xi}^{k},$
(35) $\displaystyle\delta g_{ij}$
$\displaystyle=-\xi^{k}\partial_{k}g_{ij}-g_{ik}\partial_{j}\xi^{k}-g_{ik}\partial_{j}\xi^{k}.$
(36)

The LLL limit corresponds to taking $m\to 0$. The term proportional to $m$
disappears from the transformation law for $A_{i}$; now $A_{\mu}$ simply
transforms like a one-form under spatial diffs:

$\delta
A_{\mu}=-\xi^{k}\partial_{k}A_{\mu}-A_{\lambda}\partial_{\mu}\xi^{\lambda},\qquad\xi^{\lambda}=(0,\xi^{i}).$
(37)

The metric $g_{ij}$ also transforms like a covariant tensor. The nontrivial
feature of the states on the LLL that sets it apart from other states in a
magnetic field is that, although a time-varying diffeomorphism generates the
$g_{0i}$ components of the metric tensor from $g_{ij}$: $\delta
g_{0i}=\cdots+g_{ij}\dot{\xi}^{j}$, the partition function of the theory does
not depend at all on $g_{0i}$.

The fractional quantum Hall effect exists in a finite magnetic field
$B=\partial_{1}A_{2}-\partial_{2}A_{1}$. Suppose one is not interested in
computing the electric current by varying the partition function with respect
to $A_{i}$. In that case, one can assume that $A_{i}$ has some fixed value,
for example, $A_{i}=-\frac{1}{2}B\varepsilon_{ij}x^{j}$, and only consider
$A_{0}$ and $g_{ij}$ as external backgrounds. Then it is natural to ask if the
partition function of the theory is symmetric under any gauge transformation
that touches only $A_{0}$ and $g_{ij}$, and explore the Ward-Takahashi
identities that follow. To keep $B$ unchanged, we need to restrict ourselves
to VPDs. These correspond to
$\xi^{k}=\ell^{2}\varepsilon^{kl}\partial_{l}\lambda$, where we chose $\ell$
to be the magnetic length $\ell=1/\sqrt{B}$. For VPDs, the change of the
spatial components of gauge potential $A_{i}$,

$\delta
A_{i}=-\ell^{2}\varepsilon^{kl}\partial_{l}\lambda\partial_{k}A_{i}-\ell^{2}A_{k}\varepsilon^{kl}\partial_{i}\partial_{l}\lambda=-\ell^{2}\varepsilon^{kl}\partial_{l}\lambda(\partial_{k}A_{i}-\partial_{i}A_{k})-\ell^{2}\partial_{i}(\varepsilon^{kl}A_{k}\partial_{l}\lambda)\\\
=-\partial_{i}(\lambda+\ell^{2}\varepsilon^{kl}A_{k}\partial_{l}\lambda),$
(38)

can be compensated by a gauge transformation $A_{\mu}\to
A_{\mu}+\partial_{\mu}\alpha$ with
$\alpha=\lambda+\ell^{2}\varepsilon^{kl}A_{k}\partial_{l}\lambda$. Under this
combination of coordinates and gauge transformations, $A_{0}$ transforms as

$\delta
A_{0}=-\ell^{2}\varepsilon^{kl}\partial_{l}\lambda\partial_{k}A_{0}-\ell^{2}A_{k}\partial_{t}(\varepsilon^{kl}\partial_{l}\lambda)+\dot{\lambda}+\ell^{2}\partial_{t}(\varepsilon^{kl}A_{k}\partial_{l}\lambda)=\dot{\lambda}-\ell^{2}\varepsilon^{kl}\partial_{k}A_{0}\partial_{l}\lambda.$
(39)

We see that the transformation law for $A_{0}$ has exactly the form that we
have postulated in Eq. (24). The metric, of course, transforms as in Eq. (22).

###  4.1 Higher-rank symmetry in the lowest Landau level limit

Within the context of the LLL, it is possible to give an intuitive
interpretation of the higher-rank conservation law. We write down the current
conservation

$\frac{\partial\rho}{\partial t}+\bm{\nabla}\cdot\mathbf{j}=0,$ (40)

and the law of conservation of momentum,

$\frac{\partial\pi_{i}}{\partial
t}+\partial_{j}T_{ij}=E_{i}\rho+\varepsilon_{ik}j_{k}B,$ (41)

where $\pi_{i}$ is the momentum density. In a Galilean-invariant theory with
particles of mass $m$, the momentum density is proportional to the particle
number flux $\pi_{i}=mj_{i}$, and vanishes in the LLL limit $m\to 0$. Now the
conservation of momentum becomes simply the equation of balance of force,
which in the absence of the electric field simply reads

$\partial_{j}T_{ij}=\varepsilon_{ik}j_{k}B,$ (42)

and can be solved to yield for the current

$j_{i}=-\frac{1}{B}\varepsilon_{ij}\partial_{k}T_{jk}.$ (43)

The equation for the conservation of charge is now

$\frac{\partial\rho}{\partial
t}-\frac{1}{2B}\partial_{i}\partial_{j}(\varepsilon_{ik}T_{kj}+\varepsilon_{jk}T_{ik})=0.$
(44)

One notices that the conservation law (44) is in the same form as (17) in the
earlier version of the symmetric tensor gauge theory of fracton. Thus, the
conservation of charge has a “higher-rank” form due to the fact that, on the
LLL, the current density is no longer independent but can be expressed through
the derivative of the stress tensor. The same conservation law was derived
previously in Ref. [46] using a LLL field theory formalism. The connection
between volume-preserving diffeomorphism and quantum Hall physics was also
noticed in Refs. [14, 26, 33, 34, 47, 48, 49, 50].

Some comments are in order. We began with two independent Ward’s identities,
(40) and (41). The conservation law (44) can be considered as the linear
combination of the charge conservation (40) and the massless limit of (41).
One then recognizes that we end up with two independent conservation laws, one
being (44) and the other being the original charge conservation. It is the
same conclusion that we arrived at in Section 3.

###  4.2 The Wen-Zee term

One possible term in the effective action for the fractional quantum Hall is
the Wen-Zee term [37]. To introduce this term, we need to define the Newton-
Cartan geometry and the spin connection that comes with it. We only give the
relevant formulas here; for details, see, e.g., Refs. [51, 44]. The Newton-
Cartan geometry structure is given by a one-form $n_{\mu}$ (in the simplest
version of the geometry $dn=0$), a vector $v^{\mu}$, and a symmetric
contravariant metric tensor $g^{\mu\nu}$ satisfying $n_{\mu}v^{\mu}=1$,
$g^{\mu\nu}n_{\nu}=0$. In our case,

$n_{\mu}=(1,\mathbf{0}),\qquad v^{\mu}=\begin{pmatrix}1\\\
v^{i}\end{pmatrix},\qquad g^{\mu\nu}=\begin{pmatrix}0&0\\\
0&g^{ij}\end{pmatrix}.$ (45)

where

$v^{i}=\ell^{2}\varepsilon^{ij}\partial_{j}A_{0},$ (46)

and $g^{ij}$ is the inverse matrix of $g_{ij}$. One then defines the covariant
metric tensor $g_{\mu\nu}$ so that $g_{\mu\nu}v^{\nu}=0$ and
$g_{\mu\nu}g^{\nu\lambda}=\delta_{\mu}^{\lambda}-n_{\mu}v^{\lambda}$,

$g_{\mu\nu}=\begin{pmatrix}g_{ij}v^{i}v^{j}&-v_{j}\\\
-v_{i}&g_{ij}\end{pmatrix},$ (47)

where $v_{i}\equiv g_{ij}v^{j}$, together with the Christoffel symbol which
can be used to define covariant derivatives

$\Gamma^{\mu}_{\nu\lambda}=v^{\mu}\partial_{\nu}n_{\lambda}+\frac{1}{2}g^{\mu\rho}(\partial_{\nu}g_{\rho\lambda}+\partial_{\lambda}g_{\rho\nu}-\partial_{\rho}g_{\nu\lambda}).$
(48)

One further defines the vielbein $e^{a}_{\mu}$ so that

$g^{\mu\nu}=e^{a\mu}e^{a\nu},$ (49)

and the spin connection is defined as

$\omega_{\mu}=\frac{1}{2}\varepsilon^{ab}e^{a\nu}\nabla_{\mu}e^{b}_{\nu},$
(50)

which, in components, reads

$\displaystyle\omega_{0}$
$\displaystyle=\frac{1}{2}(\varepsilon^{ab}e^{aj}\partial_{0}e^{b}_{j}+\varepsilon^{ij}\partial_{i}v_{j}),$
(51) $\displaystyle\omega_{i}$
$\displaystyle=\frac{1}{2}(\varepsilon^{ab}e^{aj}\partial_{i}e^{b}_{j}-\varepsilon^{jk}\partial_{j}g_{ik}).$
(52)

The spin connection transforms like the gauge potential under the local O(2)
rotation of the vielbein: $e^{a}(x)\to
e^{a}(x)+\alpha(x)\varepsilon^{ab}e^{b}(x)$. The Wen-Zee term [37] is given by

$\frac{\kappa}{4\pi}\varepsilon^{\mu\nu\lambda}\omega_{\mu}\partial_{\nu}A_{\lambda}=\frac{\kappa}{4\pi}\left(\omega_{0}B+A_{0}\varepsilon^{ij}\partial_{i}\omega_{j}-\varepsilon^{ij}\omega_{i}\partial_{0}A_{j}\right)=\frac{\kappa}{4\pi}\left(\frac{\omega_{0}}{\ell^{2}}+\frac{1}{2}A_{0}R\right),$
(53)

where we have put $\varepsilon^{ij}\partial_{i}A_{j}=\ell^{-2}$ and
$\partial_{0}A_{i}=0$. The coefficient $\kappa$ is related to the filling
fraction $\nu$ and the Wen-Zee shift $\mathcal{S}$ of a fraction quantum Hall
(FQH) system by the relation

$\kappa=\nu\mathcal{S}.$ (54)

Up to quadratic order and ignoring the total derivative terms, we can rewrite
the Wen-Zee term as

$\frac{\kappa}{8\pi}\left(A_{0}R-\frac{1}{4\ell^{2}}\varepsilon^{ij}h_{ik}\dot{h}_{jk}\right),$
(55)

which is exactly the Chern-Simons term (15).

A remark can be made here. We demonstrated that the Wen-Zee term in the FQH
literature is nothing but the Chern-Simon term in the higher-rank gauge
theory. One can think of this in the reversed order. The higher-rank gauge
symmetry dictates the relationship between the Wen-Zee shift $\mathcal{S}$ and
the Hall viscosity $\eta_{H}$, which enter two separated components of the
Chern-Simons Lagrangian in the higher-rank gauge theory (55).

##  5 Generalization to (3+1) dimensions

###  5.1 Construction of the (3+1)D nonlinear higher-rank symmetry

This section will generalize the nonabelian higher-rank symmetry to (3+1)
dimensions. To do that, we imagine a three-dimensional version of the LLL.
Instead of a background magnetic field, we imagine a background Kalb-Ramond
field. Concretely, we imagine a nonrelativistic theory living in background
metric $g_{ij}$ and a Kalb-Ramond field $B_{\mu\nu}=-B_{\nu\mu}$. The field
strength of the latter is

$H_{\mu\nu\lambda}=\partial_{\mu}B_{\nu\lambda}+\partial_{\nu}B_{\lambda\mu}+\partial_{\lambda}B_{\mu\nu},$
(56)

and we assume that there is gauge symmetry with one-form gauge parameter
$\alpha_{\mu}$ under which

$\delta B_{\mu\nu}=\partial_{\mu}\alpha_{\nu}-\partial_{\nu}\alpha_{\mu}.$
(57)

and $H_{\mu\nu\lambda}$ is invariant. Most crucially, we assume that our
theory is invariant symmetry under time-dependent spatial diffeomorphisms,

$\displaystyle\delta g_{ij}$
$\displaystyle=-\xi^{k}\partial_{k}g_{ij}-g_{kj}\partial_{i}\xi^{k}-g_{ik}\partial_{j}\xi^{k},$
(58) $\displaystyle\delta B_{ij}$
$\displaystyle=-\xi^{k}\partial_{k}B_{ij}-B_{kj}\partial_{i}\xi^{k}-B_{ik}\partial_{j}\xi^{k},$
(59) $\displaystyle\delta B_{i0}$
$\displaystyle=-\xi^{k}\partial_{k}B_{i0}-B_{k0}\partial_{i}\xi^{k}-B_{ik}\dot{\xi}^{k}.$
(60)

which is a 3D version of the $m\to 0$ (i.e., LLL) limit of the nonrelativistic
diffeomorphism (34). We do not have a concrete example of a well-defined
theory with the symmetry (58), (59) and (60). As the LLL can be thought of as
the massless limit for particles in a magnetic field, one can imagine a theory
of massless strings coupled to a Kalb-Ramond field. The details (or even the
existence) of such a theory are not important for our further discussion.

Following our discussion of the LLL in (2+1)D, we assume that our system lives
in a finite Kalb-Ramond field $H_{ijk}=\ell^{-3}\varepsilon_{ijk}$, and
restrict ourselves to VPDs with $\partial_{k}\xi^{k}=0$ or

$\xi^{k}=\ell^{3}\varepsilon^{klm}\partial_{l}\lambda_{m}.$ (61)

The change of $B_{ij}$ under this VPD,

$\delta
B_{ij}=-\ell^{3}\varepsilon^{klm}(\partial_{k}B_{ij}+B_{kj}\partial_{i}+B_{ik}\partial_{j})\partial_{l}\lambda_{m},$
(62)

again can be compensated by a gauge transformation (57) with the gauge
parameter

$\alpha_{i}=\lambda_{i}-\ell^{3}\varepsilon^{klm}\
B_{ik}\partial_{l}\lambda_{m}.$ (63)

The combined VPD and gauge transformation changes only the $B_{0i}$ components
of the Kalb-Ramond potential and the metric. This is a higher-rank symmetry
that transforms our set of gauge fields $(B_{0i},g_{ij})$ like

$\displaystyle\delta B_{0i}$
$\displaystyle=\dot{\lambda}_{i}-\ell^{3}\varepsilon^{klm}(\partial_{k}B_{0i}+B_{0k}\partial_{i})\partial_{l}\lambda_{m},$
(64a) $\displaystyle\delta g_{ij}$
$\displaystyle=-\ell^{3}\varepsilon^{klm}(\partial_{k}g_{ij}+g_{kj}\partial_{i}+g_{ik}\partial_{j})\partial_{l}\lambda_{m}.$
(64b)

In addition, we also inherit from the gauge symmetry (57) those
transformations which leave $B_{ij}$ invariant. These correspond to gauge
parameters with vanishing spatial components:
$\alpha_{\mu}=(\alpha_{0},\mathbf{0})$. Under these gauge transformations,

$\displaystyle\delta B_{0i}$ $\displaystyle=-\partial_{i}\alpha_{0},$ (65a)
$\displaystyle\delta g_{ij}$ $\displaystyle=0.$ (65b)

Equations (64) and (65) represent the full group of higher-rank symmetries.
Though we have used an analogy with the LLL physics in 2D as a motivation, the
resulting transformation laws do not require any LLL-type microscopic physics.
In fact, we will see that the higher-rank symmetry appears in the context of
3D ferromagnets.

As in 2D, it is possible to “complete” $B_{0i}$ by adding the spatial
components $B_{ij}$ so that $B_{\mu\nu}$ form a set of Kalb-Ramond gauge
potentials. This would make $B_{0i}$ the shared components of the two sets of
gauge potentials: the Kalb-Ramond gauge potentials and the gauge potentials of
the higher-rank symmetry of VPDs (see Fig. 2).

Figure 2: $B_{0i}$ is shared by two sets of gauge potentials.

###  5.2 Linearized higher-ranked symmetries and conservation laws

At the linearized level, the higher-rank gauge invariance is

$\displaystyle\delta B_{0i}$
$\displaystyle=\dot{\lambda}_{i}-\partial_{i}\alpha_{0},$ (66)
$\displaystyle\delta h_{ij}$
$\displaystyle=-\ell^{3}(\varepsilon_{jkl}\partial_{i}+\varepsilon_{ikl}\partial_{j})\partial_{k}\lambda_{l}.$
(67)

As far as we know, this linear higher-rank symmetry has not been considered
previously in the literature.

Let us now assume that the currents coupled to $B_{0i}$ and $h_{ij}$ are
$J^{i}$ and $T^{ij}$,

$\delta\ln Z=\int\\!d^{4}x\left(J^{i}\delta B_{0i}+\frac{1}{2}T^{ij}\delta
h_{ij}\right).$ (68)

Two conservation laws follow from Eqs. (66) and (67). First, the gauge
invariance with gauge parameter $\alpha_{0}$ implies that the “current”
$J^{i}$ is divergence-free:

$\bm{\nabla}\cdot\mathbf{J}=0.$ (69)

On the other hand the VPD invariance, generated by $\lambda_{i}$, leads to

$\frac{\partial J_{i}}{\partial
t}-\ell^{3}\varepsilon_{ijk}\partial_{j}\partial_{l}T_{kl}=0.$ (70)

This means the following quantities are conserved

$\displaystyle\mathbf{I}_{1}$ $\displaystyle=\int\\!d\mathbf{x}\,\mathbf{J},$
(71) $\displaystyle I_{2}$
$\displaystyle=\int\\!d\mathbf{x}\,(\mathbf{x}\cdot\mathbf{J}),\quad\mathbf{I}_{3}=\int\\!d\mathbf{x}\,(\mathbf{x}\times\mathbf{J}),\quad
I_{4}^{ij}=\int\\!d\mathbf{x}\,x^{\\{i}J^{j\\}},$ (72)
$\displaystyle\mathbf{I}_{5}$
$\displaystyle=\int\\!d\mathbf{x}\,\mathbf{x}(\mathbf{x}\cdot\mathbf{J}),\qquad\mathbf{I}_{6}=\int\\!d\mathbf{x}\,\frac{x^{2}}{2}\mathbf{J},$
(73) $\displaystyle I_{7}^{ijk}$
$\displaystyle=\int\\!d\mathbf{x}\,x^{\\{i}x^{j}J^{k\\}}.$ (74)

where the $\\{ij\cdots\\}$ denotes symmetrization over the indices
$i,j,\cdots$ 222 Explicitly
$A^{\\{i}B^{j\\}}=\frac{1}{2}\left(A^{i}B^{j}+A^{j}B^{i}\right),$ (75) and
$A^{\\{i}B^{j}C^{k\\}}=\frac{1}{3}\left(A^{i}B^{\\{j}C^{k\\}}+B^{i}C^{\\{j}A^{k\\}}+C^{i}A^{\\{j}B^{k\\}}\right).$
(76) . On the other hand,

$I_{8}^{ij}=\int\\!d\mathbf{x}\,x^{\\{i}(\mathbf{x}\times\mathbf{J})^{j\\}}$
(77)

is not conserved; its time derivative is proportional to
$\int\\!d\mathbf{x}\,T^{ij}$.

Assume that the “current” $\mathbf{J}$ is nonzero only in a finite region of
space, then because it is divergence-free, one can express it as the curl of a
vector field: $\mathbf{J}=\bm{\nabla}\times\bm{\mu}$, where $\bm{\mu}$
vanishes at infinity. Then among the conserved quantities, only the following
are nonzero: $\mathbf{I}_{3}$, $\mathbf{I}_{5}$, and $\mathbf{I}_{6}$, and the
last two quantities are not independent:

$\displaystyle\mathbf{I}_{3}$ $\displaystyle=\int\\!d\mathbf{x}\,\bm{\mu},$
(78) $\displaystyle\mathbf{I}_{5}$
$\displaystyle=-\mathbf{I}_{6}=\int\\!d\mathbf{x}\,(\mathbf{x}\times\bm{\mu}).$
(79)

One can think of $\bm{\mu}$ as the magnetic moment density and $\mathbf{J}$ as
the magnetization current. Let us assume that there is a quasiparticle that
carries a magnetic moment. Then the conservation of $\mathbf{I}_{3}$ means
that the total magnetic moment does not change its value. The conservation of
$\mathbf{I}_{5}$ implies that a particle can move only along the direction of
its magnetic moment, but not along the two perpendicular directions.

Note that the (3+1)D higher-rank symmetry presented above, even in the
linearized version, differs from that of the vector charge theory proposed in
Ref. [2]. Our motivation was to generalize the area-preserving diffeomorphism
of the LLL in (2+1)D to volume-preserving diffeomorphism of (3+1)D. We have
generalized the charge density $\rho$, the one-form gauge potential $A_{\mu}$
and the constant magnetic field $B$ to the vector charge density $J^{i}$, the
two-form gauge potential $B_{\mu\nu}$ and the constant Kalb-Ramond field
strength $H_{ijk}$. We then arrive at a different gauge transformation rather
than the one in Ref. [2]. One can define the electric field and magnetic field
that are invariant under the gauge transformations and write down a
generalized Maxwell action. We will not do that here, as the aim of this paper
is to investigate the conservation laws and their physical
consequences333Since the gauge transformations (66) and (67) differ from the
ones in Ref. [2], the definitions of the gauge-invariant electric field and
magnetic field should be modified accordingly, namely
$E^{ij}=\left(\varepsilon_{jkl}\partial_{i}+\varepsilon_{ikl}\partial_{j}\right)\partial_{k}B_{0l}+\partial_{t}h_{ij}$
and $H=\partial_{i}\partial_{j}h_{ij}$. We will not discuss them further..

The conservation law (70), though similar to the one of vector charge theory
in Ref [2], is not the same because of the different gauge transformations.
Recall that the gauge transformation of the symmetry tensor gauge in the
vector charge version is [2, 41]

$\delta
A_{ij}=\partial_{i}\lambda_{j}+\partial_{j}\lambda_{i},\quad\delta\phi_{i}=\partial_{t}\lambda_{i}$
(80)

which leads to a conservation law with only one spatial derivative acting on
the current density $J^{ij}$

$\partial_{t}\rho^{i}+\partial_{j}J^{ij}=0$ (81)

instead of an equation with two spatial derivative. Furthermore, inherited
from a system of vector matter field coupled with the Kalb-Ramond field, we
have an extra “conservation law” $\vec{\nabla}\cdot\vec{J}=0$, which does not
have a counterpart in the vector charge theory proposed in Ref. [2].

We will show in Section 6.4 that the symmetry that has been proposed is
realized in the nonlinear sigma model describing 3D ferromagnets.

##  6 Examples of theories with volume-preserving diffeomorphism invariance

For the abelian higher-rank symmetry, one of the simplest ways to couple a
matter field to the gauge field $(A_{0},h_{ij})$ is to introduce a Goldstone
boson $\varphi$ which transforms under the gauge transformation as
$\varphi\to\varphi+\lambda$ and write

$\mathcal{L}=\frac{c_{1}}{2}(\partial_{0}\varphi-
A_{0})^{2}-\frac{c_{2}}{2}(\partial_{i}\partial_{j}\varphi-h_{ij})^{2}.$ (82)

However, we were not able to find a nonlinear version of this transformation.
For example, if one postulates

$\delta_{\lambda}\varphi=\lambda-\ell^{2}\varepsilon^{kl}\partial_{k}\varphi\partial_{l}\lambda,$
(83)

then one can check by direct calculation that
$[\delta_{\alpha},\delta_{\beta}]\varphi\neq\delta_{[\alpha,\beta]}\varphi$,
so such transformation law would be inconsistent. We have to devise other ways
to couple the matter field to the gauge field.

###  6.1 Composite fermions

In the fractional quantum Hall effect at filling fractions $\nu=1/2$,
$\nu=1/4$ etc., the quasiparticle is electrically neutral. One can
consistently couple such a particle to the higher-rank gauge field. For
example, a Lagrangian for a nonrelativistic particle with dispersion relation
$\omega=k^{2}/2m$ would be

$L=\frac{i}{2}v^{\mu}\psi^{\dagger}\overset{\leftrightarrow}{\partial}_{\mu}\psi-\frac{1}{2m}g^{\mu\nu}\partial_{\mu}\psi^{\dagger}\partial_{\nu}\psi=\frac{i}{2}\psi^{\dagger}\overset{\leftrightarrow}{\partial}_{0}\psi+\frac{i}{2}\ell^{2}\varepsilon^{ij}\partial_{j}A_{0}\,\psi^{\dagger}\overset{\leftrightarrow}{\partial}_{i}\psi-\frac{g^{ij}}{2m}\partial_{i}\psi^{\dagger}\partial_{j}\psi.$
(84)

One can interpret the coupling of the electric field $\partial_{i}A_{0}$ with
the particle momentum as a dipole moment, perpendicular to the direction of
its motion. This is consistent with the constraints that follow from
conservation laws. In fact, that is how a composite fermion in the fractional
quantum Hall state at $\nu=1/2$ or $\nu=1/4$ should couple to the external
potential: the composite fermion is neutral but possesses an electric dipole
moment.

###  6.2 Crystal on the lowest Landau level

Another way to realize the higher-rank symmetry is through an effective theory
of a solid. Such a solid may be realized as a Wigner crystal, which is
expected to be the ground state of electrons on the LLL at small filling
fractions. A solid is parametrized by a map from the physical coordinates
$x^{i}$ to the coordinate system $X^{a}$ frozen into the solid:
$X^{a}=X^{a}(x^{i})$ [52]. In the ground state $X^{a}=x^{i}\delta_{ai}$, the
displacement $u^{a}$ are defined as $X^{a}=x^{a}-u^{a}$. On the LLL the
coordinates of a lattice site do not commute: $[x,\,y]=-i\ell^{2}$. The
Lagrangian should thus contain the following term

$S_{\text{Berry}}=\frac{n_{0}}{2\ell^{2}}\varepsilon^{ab}X^{a}\partial_{t}X^{b},$
(85)

where $n_{0}$ is the equilibrium particle number density. Under a VPD, $X^{a}$
transforms as

$\delta_{\lambda}X^{a}=-\ell^{2}\varepsilon^{ij}\partial_{i}X^{a}\partial_{j}\lambda,$
(86)

and so

$\delta_{\lambda}S_{\text{Berry}}=-\frac{n_{0}}{2}\varepsilon^{ij}\varepsilon^{ab}X^{a}\partial_{i}X^{b}\partial_{j}\dot{\lambda}=-\frac{n_{0}}{2}\dot{\lambda}\varepsilon^{ij}\varepsilon^{ab}\partial_{i}X^{a}\partial_{j}X^{b}.$
(87)

This change of the action can be compensated by including a term proportional
to $A_{0}$ into the Lagrangian. The full Lagrangian is then

$\mathcal{L}=\frac{n_{0}}{2\ell^{2}}\varepsilon^{ab}X^{a}\partial_{t}X^{b}+\frac{n_{0}}{2}A_{0}\varepsilon^{ab}\varepsilon^{ij}\partial_{i}X^{a}\partial_{j}X^{b}-\varepsilon(O^{ab}),$
(88)

with $O^{ab}=g^{ij}\partial_{i}X^{a}\partial_{j}X^{b}$ and
$\varepsilon(O^{ab})$ is the energy associated with elastic deformations. The
spectrum of this theory can be obtained by expanding the action to quadratic
order over the displacement $u^{a}$. The presence of a term with a first time
derivative implies that the dispersion relation of the lattice sound wave has
the quadratic form $\omega\sim q^{2}$ rather than the linear form.

Thus, we have been able to construct the effective theory of a Wigner crystal
on the LLL starting from the higher-rank symmetry.

###  6.3 Vortex crystal

Another type of matter is the “vortex crystal,” which is realized, for
example, in a rotating Bose gas [53]. The crystal is formed by the zeros of
the condensate wave function. In this case, the lattice fields $X^{a}$ do not
couple to $A_{0}$ directly, but through a dynamical gauge field $a_{\mu}$,
which is the dual of the superfluid phonon. Under VPD $a_{\mu}$ transforms as
a one-form,

$\delta
a_{\mu}=-\ell^{2}\varepsilon^{kl}(\partial_{k}a_{\mu}-a_{k}\partial_{\mu})\partial_{l}\lambda.$
(89)

The field $a_{\mu}$ couples to the background $A_{0}$ through the following
term

$\frac{1}{2\pi}\int\\!d^{3}x\left(\frac{a_{0}}{\ell^{2}}+A_{0}b\right),$ (90)

where $b=\varepsilon^{ij}\partial_{i}a_{j}$ is the emergent magnetic field.
One can check directly that this term is invariant under VPDs. It is also
obviously invariant under gauge transformations $a_{\mu}\to
a_{\mu}+\partial_{\mu}\alpha$. In fact, (90) can be obtained from the Chern-
Simons term
$\frac{1}{2\pi}\varepsilon^{\mu\nu\lambda}a_{\mu}\partial_{\nu}A_{\lambda}$ by
setting $A_{i}$ to be the static background with
$\varepsilon^{ij}\partial_{i}A_{j}=\ell^{-2}$.

The Lagrangian of the vortex crystal is then determined by the symmetry and
reads

$\mathcal{L}=-\varepsilon^{\mu\nu\lambda}\varepsilon^{ab}a_{\mu}\partial_{\nu}X^{a}\partial_{\lambda}X^{b}-\varepsilon(O^{ab})-\varepsilon_{b}(b)+\frac{1}{2\pi}\left(\frac{a_{0}}{\ell^{2}}+A_{0}b\right),$
(91)

where $\varepsilon(O^{ab})$ and $\varepsilon_{b}(b)$ represent the energies of
the lattice and the condensate, respectively. Assuming $a_{\mu}$ transforms
like a one-form under VPD, it can be checked that the Lagrangian above is
invariant with respect to this symmetry. The eigenmode of this theory is again
a Tkachenko mode with a quadratic dispersion relation [53].

Note that the above Lagrangian contains only leading-derivative terms and does
not include next-to-leading terms considered in Ref. [53].

###  6.4 Ferromagnets

####  6.4.1 Ferromagnets in 2+1 D

We now show that the ferromagnet in 2+1D secretly possesses a higher-rank
symmetry similar to the models with particles on the LLL 444In fact, one can
show that the dynamical equation of a single skyrmion in a 2-dimensional
ferromagnet is the same as the equation of motion of a charged particle in a
constant magnetic field [54]. . At the long-wavelength limit, a ferromagnet is
described by a nonlinear sigma (NLS) model [55], written in terms of an $O(3)$
unit vector $n^{a}$, $n^{a}n^{a}=1$, with the action

$S=S_{\text{Berry}}+S_{\text{NLS}}=S_{0}\\!\int\limits_{0}^{1}\\!d\sigma\\!\int\\!dt\,d\mathbf{x}\,\varepsilon^{abc}n^{a}\partial_{t}n^{b}\partial_{\sigma}n^{c}-\frac{J}{2}\\!\int\\!dt\,d\mathbf{x}\,\delta^{ij}\partial_{i}n^{a}\partial_{j}n^{a}.$
(92)

The first term is a Wess-Zumino topological term in spin’s action, which is
induced by the Berry phase [55]. The second term is the energy term of the
nonlinear sigma model and can be made invariant under VPD by replacing
$\delta^{ij}$ with $g^{ij}$. The first term is, however not invariant:

$\delta_{\lambda}S_{\text{Berry}}=-S_{0}\ell^{2}\\!\int\limits_{0}^{1}\\!d\sigma\\!\int\\!dt\,d\mathbf{x}\,\varepsilon^{abc}\varepsilon^{ij}n^{a}\partial_{i}n^{b}\partial_{\sigma}n^{c}\partial_{j}\dot{\lambda}.$
(93)

Integrating by parts, taking into account that
$\epsilon^{abc}\partial_{j}n^{a}\partial_{i}n^{b}\partial_{\sigma}n^{c}=0$
(this is because all the three O(3) vectors $\partial_{i}n^{a}$,
$\partial_{j}n^{a}$, $\partial_{\sigma}n^{a}$ are perpendicular to $n^{a}$ and
hence are linearly dependent) we find

$\delta_{\lambda}S_{\text{Berry}}=S_{0}\ell^{2}\\!\int\limits_{0}^{1}\\!d\sigma\\!\int\\!dt\,d\mathbf{x}\,\varepsilon^{abc}\varepsilon^{ij}n^{a}\partial_{i}n^{b}\partial_{j}\partial_{\sigma}n^{c}\dot{\lambda}\\\
=\frac{S_{0}}{2}\ell^{2}\\!\int\limits_{0}^{1}\\!d\sigma\\!\int\\!dt\,d\mathbf{x}\,\varepsilon^{abc}\varepsilon^{ij}\partial_{\sigma}(n^{a}\partial_{i}n^{b}\partial_{j}n^{c})\dot{\lambda}=\frac{S_{0}}{2}\ell^{2}\\!\int\\!dt\,d\mathbf{x}\,\varepsilon^{abc}\varepsilon^{ij}n^{a}\partial_{i}n^{b}\partial_{j}n^{c}\dot{\lambda}.$
(94)

We now choose

$\ell^{2}=\frac{1}{4\pi S_{0}}\,,$ (95)

and add the following term to the action

$S_{\\!A_{0}}=-\frac{1}{8\pi}\\!\int\\!dt\,d\mathbf{x}\,A_{0}\varepsilon^{abc}\varepsilon^{ij}n^{a}\partial_{i}n^{b}\partial_{j}n^{c}.$
(96)

Then $\delta_{\lambda}(S_{\text{Berry}}+S_{\\!A_{0}})=0$. Thus if we couple
the ferromagnetic order parameter with the gauge fields $A_{0}$, $g_{ij}$ in
the following way

$S=S_{\text{Berry}}+S_{\\!A_{0}}-\frac{J}{2}\\!\int\\!dt\,d\mathbf{x}\,g^{ij}\partial_{i}n^{a}\partial_{j}n^{a},$
(97)

then the action is invariant under VPD with $\ell^{2}$ defined in Eq. (95).

One can further introduce into the theory the vector gauge potential $A_{i}$,
promoting Eq. (96) to

$S_{\\!A_{\mu}}=-\frac{1}{8\pi}\\!\int\\!dt\,d\mathbf{x}\,A_{\mu}\varepsilon^{abc}\varepsilon^{\mu\nu\lambda}n^{a}\partial_{\nu}n^{b}\partial_{\lambda}n^{c}.$
(98)

with $A_{i}$ transforming as a one-form under VPDs. In this case, the
potential $A_{0}$ is simultaneously the scalar component of the U(1) gauge
field $(A_{0},A_{i})$ and the scalar component of the gauge potential of a
higher-spin symmetry, $(A_{0},g_{ij})$ (Fig. 1).

Now the scalar potential $A_{0}$ is coupled to the topological charge density
$\rho(\mathbf{x})$. This means that a skyrmion behaves like a particle in an
effective magnetic field with the magnitude

$B_{\text{eff}}=-4\pi S_{0}q,$ (99)

where $q=\int\\!d\mathbf{x}\,\rho(\mathbf{x})$ is the topological charge of
the skyrmion. This fact can be derived by calculating the Berry phase
associated with the motion of a skyrmion [39]. One also finds that the
following quantities

$\int\\!d\mathbf{x}\,\rho,\quad\int\\!d\mathbf{x}\,x^{i}\rho,\qquad\int\\!d\mathbf{x}\,x^{2}\rho,$
(100)

are conserved. This fact is again well known [38].555In the presence of the
Dyaloshinskii-Morya interaction, which breaks the higher-rank symmetry, only
the first two quantities are conserved [56].

It is instructive to rewrite the ferromagnet in the $\mathbb{CP}^{1}$
parametrization, where

$n^{a}=z^{\dagger}\sigma^{a}z,\qquad z=\begin{pmatrix}z_{1}\\\
z_{2}\end{pmatrix},\qquad z^{\dagger}z=1.$ (101)

The action of the ferromagnet is then [54]

$S=S_{Berry}+S_{NLS}=2iS_{0}\\!\int\\!dt\,d\mathbf{x}\,z^{\dagger}\partial_{t}z-2J\\!\int\\!dt\,d\mathbf{x}\,D_{i}z^{\dagger}D_{i}z,$
(102)

where $D_{i}z\equiv(\partial_{i}-ia_{i})z$, and
$a_{i}=-iz^{\dagger}\partial_{i}z$ is promoted to a dynamical field.

Now we couple the $\mathbb{CP}^{1}$ model to the external probes
$(g_{ij},A_{0})$. We assume that under VPD, $z$ transforms as

$\delta_{\lambda}z=-\ell^{2}\varepsilon^{kl}\partial_{k}z\partial_{l}\lambda,$
(103)

therefore,

$\delta_{\lambda}S_{\text{Berry}}=-2iS_{0}\ell^{2}\\!\int\\!dt\,d\mathbf{x}\,\varepsilon^{ij}z^{\dagger}\partial_{i}z\partial_{j}\dot{\lambda}=-2iS_{0}\ell^{2}\\!\int\\!dt\,d\mathbf{x}\,\dot{\lambda}\varepsilon^{ij}\partial_{i}z^{\dagger}\partial_{j}z.$
(104)

We now add the following term to the action:

$S_{A_{0}}=2iS_{0}\ell^{2}\\!\int\\!dt\,d\mathbf{x}\,A_{0}\varepsilon^{ij}\partial_{i}z^{\dagger}\partial_{j}z.$
(105)

Then $\delta_{\lambda}(S_{Berry}+S_{A_{0}})=0$, therefore the coupling of the
ferromagnetic model with the gauge field $g_{ij}$, $A_{0}$ is

$S=S_{\text{Berry}}+S_{A_{0}}-2J\\!\int\\!dt\,d\mathbf{x}\,g^{ij}D_{i}z^{\dagger}D_{j}z,$
(106)

and respects the higher-rank symmetry.

####  6.4.2 Ferromagnets in 3+1 D

For a ferromagnet in (3+1)D, the term $S_{\\!A_{0}}$ that couples $A_{0}$ to
the topological charge density is replaced by the coupling of $B_{0i}$ to the
density of a one-form current,

$S_{B_{0i}}=-\frac{1}{8\pi}\int\\!dt\,d\mathbf{x}\,\varepsilon^{abc}\varepsilon^{ijk}B_{0i}n^{a}\partial_{j}n^{b}\partial_{k}n^{c},$
(107)

and this has the (3+1)D version of VPD invariance with $\ell^{3}=(4\pi
S_{0})^{-1}$. Again one can promote (107) to

$S_{B_{\mu\nu}}=-\frac{1}{8\pi}\int\\!dt\,d\mathbf{x}\,\varepsilon^{abc}\varepsilon^{\mu\nu\lambda\rho}B_{\mu\nu}n^{a}\partial_{\lambda}n^{b}\partial_{\rho}n^{c}.$
(108)

In this case, $B_{0i}$ are the shared components of a Kalb-Ramond gauge field
and the set of gauge potentials of a higher-rank gauge symmetry
$(B_{0i},g_{ij})$ (see Fig. 2).

It was found in Ref. [38] that $\mathbf{I}_{3}$ and
$\mathbf{I}_{5}=-\mathbf{I}_{6}$ are conserved. We have given this fact a new
interpretation in terms of a hidden higher-rank symmetry.

##  7 Conclusion

We have presented a nonlinear version of a higher-rank gauge symmetry. The
symmetry is basically that of volume-preserving diffeomorphism. We show
several examples of coupling of matter with the higher-rank gauge potential
that respects the symmetry. Many examples are taken from the physics of the
LLL, which we show to naturally have volume-preserving diffeomorphism
invariance. We also show that the nonlinear sigma models of ferromagnetism
also exhibit this symmetry if one couples a gauge potential of the higher-rank
symmetry with the topological charge density.

We have shown that, under certain conditions, the charge densities satisfy the
commutation relation of the VPD, i.e., the $w_{\infty}$ algebra. In this way,
one can easily understand why this algebra is realized in the bimetric model
of the FQH effect [57], without explicit calculations. One interesting
question is whether the $w_{\infty}$ symmetry can be “upgraded” to the quantum
$W_{\infty}$ symmetry [33, 34]. This question, relevant for the fractional
quantum Hall effect, is deferred to future work.

## Acknowledgements

The authors thank Adrey Gromov and Sergej Moroz for discussions and comments
on the earlier draft of this paper. This work is supported, in part, by the
U.S. DOE grant No. DE-FG02-13ER41958, a Simons Investigator grant and by the
Simons Collaboration on Ultra-Quantum Matter, which is a grant from the Simons
Foundation (651440, DTS). DXN is supported by the Brown Theoretical Physics
Center.

##  Appendix A The uniqueness of the nonlinear transformation (24)

In this Appendix, we will briefly argue that the nonlinear transformation (24)
is the unique generalization of (16), given the following assumptions:

  * •

The nonlinear transformation satisfies the area-preserving diffeomorphism
algebra (23).

  * •

In the background $(A_{0},g_{ij})$, $A_{0}$ is a scalar, therefore the
transformation $\delta_{\lambda}A_{0}$ should be a scalar.

  * •

The transformation of $(A_{0},g_{ij})$ is at most linear in $(A_{0},g_{ij})$.

  * •

The rotational symmetry is preserved.

Due to the algebra (23), the transformation $\delta_{\lambda}A_{0}$ has to be
linear in $\lambda$. We consider the general transformation that is linear in
$A_{0}$ with the form666One can even generalize the argument with the more
general transformation
$\delta_{\lambda}A_{0}\sim\sum_{m,n}c_{mn}(\partial_{i_{1}}\cdots\partial_{i_{m}})A_{0}(\partial_{j_{1}}\cdots\partial_{j_{n}})\lambda$
(109) and ends up the same conclusion. By couting the number of time
derivatives, we also see that one can’t add time derivatives to (110) in order
to satisfy the algbera (23).

$\delta_{\lambda}A_{0}\sim(\partial_{i_{1}}\cdots\partial_{i_{m}})A_{0}(\partial_{j_{1}}\cdots\partial_{j_{n}})\lambda,$
(110)

with $\partial_{k}$ is the derivative in the spartial directions. By counting
the number of derivative in the spartial directions, in order to satisfy the
algebra (23), both $m$ and $n$ have to be 1. Furthermore,
$\delta_{\lambda}A_{0}$ has to be a scalar and invariant under rotation. We
need to contract all the spatial indices of the derivatives,
$\varepsilon^{ij}$ is the only rotational invariant two indices tensor that
helps us to satisfy the algebra (23)

$\displaystyle\delta_{\lambda}A_{0}=-\ell^{2}\varepsilon^{ij}\partial_{i}A_{0}\partial_{j}\lambda.$
(111)

We also consider the term that is independent of $A_{0}$

$\displaystyle\delta_{\lambda}A_{0}=\alpha^{{i_{1}}\cdots{i_{s}}}(\partial_{t})^{u}(\partial_{i_{1}}\cdots\partial_{i_{s}})\lambda-\ell^{2}\varepsilon^{ij}\partial_{i}A_{0}\partial_{j}\lambda,$
(112)

with $\partial_{t}$ is the time derivative and $\alpha^{{i_{1}}\cdots{i_{s}}}$
is a rotational symmetric tensor. By counting the derivatives, to satisfy (23)
then $u+s=1$. There is no rotational symmetric tensor with just one spartial
index, thereofre $s=0$ and $u=1$. We then arrive at the transformation (24).

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