# Dipolar BF theory and dipolar braiding statistics

Jung Hoon Han  Department of Physics, Sungkyunkwan University, Suwon 16419,
South Korea<EMAIL_ADDRESS>
###### Abstract

We analyze the recently proposed dipolar BF theory with couplings to charge
and dipole currents. The quasiparticles of the theory are either charge-like
or dipole-like, and the mutual braiding statistics between charge-like and
dipole-like quasiparticles are dipolar, meaning that it depends on the
position of the quasiparticle being encircled. The braiding statistics between
two dipole-like quasiparticles is that of ordinary anyons. We further prove
that the dipolar BF theory is equivalent to the rank-2 tensor BF theory
developed earlier as an effective theory for the rank-2 toric code. Although
the two theories are equivalent, the dipolar BF formulation embodies the
dipole symmetry explicitly and gives a clean insight into the way the dipole
symmetry manifests itself in various conservation laws and the dipolar
braiding statistics.

##  I Introduction

Extending the horizon of existing many-body models by incorporating dipolar,
multipolar, or general modulated symmetries has been a fruitful theme of
condensed matter theory in recent years. It was driven in part by a desire to
incorporate fractonic behavior [1, 2] into field-theoretic formulations [3, 4,
5], and was further fueled by the experimental realization of tilted optical
lattice and some interesting phenomena found in such systems [6, 7, 8, 9, 10,
11, 12, 13, 14, 15]. Now the converging view is that these extensions can be
systematically categorized by the order of the multipole symmetry [16, 17, 18,
19] being incorporated in the theory. Examples of extensions include rank-2
gauge theory coupled to fractonic quasiparticles [3, 20, 4], Hubbard models
with pair-hopping fermions or bosons [12, 13, 14], Chern-Simons theory with
dipolar symmetry [20, 21, 22, 23], and an 1D SPT model protected by multipolar
symmetry [24], to name a few.

Following an unrelated line of reasoning, we have written down a variant of
the toric code called the rank-2 toric code (R2TC) [25] by building on some
earlier works [26, 27]. Anyonic excitations in the R2TC have mobility
restrictions not unlike the fractons and are forced to hop only by $N$ sites
at a time in a $\mathbb{Z}_{N}$ stabilizer model of R2TC [25, 28]. Several
other stabilizer models with similar mobility restrictions on anyonic
excitations were discovered [29, 30, 31]. The braiding statistics among the
anyons in R2TC displayed unusual position dependence, distinct from the
ordinary anyon braiding statistics that are purely topological. Attempts to
explain the position-dependent braiding in the field theory framework were
made by several authors [28, 32]. In particular, the dipolar BF theory
proposed in [32] had rank-2 tensor gauge fields coupled to rank-2 tensor
matter currents, and embodied dipole symmetry in its symmetry structure
although the R2TC model had no such symmetry to begin with. In that regard the
dipole symmetry appeared to be ‘emergent’ rather than imprinted on the lattice
Hamiltonian of R2TC from the outset.

Quite recently, topological Chern-Simons and BF theories incorporating
multipolar symmetries explicitly at the level of the effective action have
been put forward [33, 34]. They are indeed lucid representations of
topological systems embodying both charge and dipole symmetries and are
claimed to be equivalent to certain rank-2 Chern-Simons or BF theories after
suitable transformations [20, 21, 22]. Here we look into this issue further
and prove that the dipolar BF theory put forward in [34] is in fact equivalent
to the effective theory of the R2TC previously suggested in [32], which was
also dubbed the dipolar BF theory. To avoid the confusion, here we dub the new
action the dipolar BF (dBF) theory, and rename the rank-2 BF theory of [32] as
the rank-2 BF (R2BF) theory. The origin of dipolar braiding statistics, first
discussed field-theoretically in [28, 32], is given a more transparent
interpretation in the framework of dBF theory thanks to the dipole symmetry
being embedded in the action from the outset.

In Sec. II we re-visit the dBF action [34] and analyze it in greater depth
than [34] with additional coupling to matter fields. A complete set of
conservation laws related to charge and dipole symmetries are derived. In Sec.
III we show that the dBF action can be transformed into the R2BF action and
prove their equivalence. Some aspect of this analysis has been done in [34],
but we perform a more complete analysis with the full set of gauge-matter
coupling allowed by the theory. Then in Sec. IV we re-visit the issue of
dipolar braiding statistics and re-derive the results on the basis of insights
gained from the analysis of dBF action. The paper is summarized in Sec. V.

##  II Dipolar BF theory

The purpose of this section is to introduce the dipolar BF theory and outline
its basic properties. Some outstanding features of dBF theory is shared by the
dipolar Chern-Simons (dCS) theory [33], which is simpler and discussed first
as a way of illustrating the main principle.

The dCS theory consists of two parts ${\cal L}={\cal L}_{\rm dCS}+{\cal
L}_{AJ}$ where

$\displaystyle{\cal L}_{{\rm dCS}}$
$\displaystyle=\frac{N}{4\pi}\varepsilon_{\lambda\mu\nu}A_{\lambda}^{a}\partial_{\mu}A_{\nu}^{a}$
$\displaystyle{\cal L}_{AJ}$
$\displaystyle=-J_{\mu}A_{\mu}-J_{\mu}^{a}A_{\mu}^{a}.$ (1)

For each spatial direction $a=x,y$ we have the gauge field $A^{a}_{\mu}$
described by the Chern-Simons (CS) action with level $N$. Two kinds of matter
fields $(J_{\mu},J^{a}_{\mu})$ are introduced, corresponding to the charge and
the dipole current associated with each dipole moment orientation $a$ and
coupled to the gauge fields $A_{\mu}$ and $A^{a}_{\mu}$, respectively. The
summation over the spacetime indices $\mu,\nu=t,x,y$ and the spatial index
$a=x,y$ is implicit.

The gauge fields $A_{\mu},A^{a}_{\mu}$ are not totally independent. Their
gauge transformations are linked by the dipole conservation, and follow [23,
33, 34]

$\displaystyle A^{\nu}_{\mu}$ $\displaystyle\rightarrow
A^{\nu}_{\mu}+\partial_{\mu}\beta^{\nu},$ $\displaystyle A_{\mu}$
$\displaystyle\rightarrow A_{\mu}+\partial_{\mu}\alpha+\beta^{\mu}.$ (2)

Here $\beta^{\mu}$ is nonzero only for the spatial components $\mu=x,y$. There
is no temporal component in the dipolar gauge fields, i.e. $A^{t}_{\mu}=0$.
The dCS action given above is invariant under the transformation (2). There
can be no CS term for $A_{\mu}$ or a mixture of $A_{\mu}$ and $A^{a}_{\mu}$
since such terms fail to remain invariant under (2). Although $A_{\mu}$ does
not participate in the CS action, it can still couple to the charge current
$J_{\mu}$ and have non-trivial effect. Throughout the paper we refer to
quantities without the superscript $a$ as ‘charge’ and those with the
superscript as ‘dipole’ or ‘dipolar’.

Requiring the invariance of the dCS action under the transformation (2) gives
the continuity equations

$\displaystyle\partial_{\mu}J_{\mu}=0,~{}~{}\partial_{\mu}J_{\mu}^{x}=J_{x},~{}~{}\partial_{\mu}J_{\mu}^{y}=J_{y}.$
(3)

While the charge current satisfies the usual continuity equation, the dipole
currents $J_{\mu}^{x},J_{\mu}^{y}$ satisfy modified continuity equations with
source terms coming from the charge current. Physically, it means that when a
charge moves in a certain direction it will violate the dipole moment
conservation in that direction and must act as a source term in the dipole
continuity equation. One can re-write (3) in a more suggestive form

$\displaystyle\partial_{\mu}[J_{\mu}^{x}-(x-x_{0})J_{\mu}]=\partial_{\mu}[J_{\mu}^{y}-(y-y_{0})J_{\mu}]=0$
(4)

where $x_{0},y_{0}$ are some constants. We will make heavy use of this form
when we discuss dipolar braiding in Sec. IV.

We move to examine the dipolar BF theory given by [34]

$\displaystyle{\cal L}={\cal L}_{\rm dBF}+{\cal L}_{AJ}+{\cal L}_{bK},$ (5)

where

$\displaystyle{\cal L}_{{\rm dBF}}$
$\displaystyle=\frac{N}{2\pi}\varepsilon_{\lambda\mu\nu}b_{\lambda}\left(\partial_{\mu}A_{\nu}+A^{\mu}_{\nu}\right)+\frac{N}{2\pi}\varepsilon_{\lambda\mu\nu}b_{\lambda}^{a}\partial_{\mu}A^{a}_{\nu}$
$\displaystyle{\cal L}_{AJ}$
$\displaystyle=-J_{\mu}A_{\mu}-J_{\mu}^{a}A_{\mu}^{a}$ $\displaystyle{\cal
L}_{bK}$ $\displaystyle=-b_{\mu}K_{\mu}-b_{\mu}^{a}K_{\mu}^{a}.$ (6)

The last term ${\cal L}_{bK}$ was not included in [34] but is necessary for a
complete analysis. The gauge transformation properties of the dBF action are
as follows:

$\displaystyle A^{\nu}_{\mu}$ $\displaystyle\rightarrow
A^{\nu}_{\mu}+\partial_{\mu}\beta^{\nu}$ $\displaystyle A_{\mu}$
$\displaystyle\rightarrow A_{\mu}+\partial_{\mu}\alpha+\beta^{\mu}$
$\displaystyle b_{\mu}$ $\displaystyle\rightarrow
b_{\mu}+\partial_{\mu}\theta$ $\displaystyle b_{\mu}^{a}$
$\displaystyle\rightarrow b_{\mu}^{a}+\partial_{\mu}\phi^{a}-\theta\delta_{\mu
a}.$ (7)

It is again assumed that $\beta^{t}=0=A^{t}_{\mu}$.

Requiring the invariance of the dBF action under the transformation (7) gives
several continuity equations:

$\displaystyle\partial_{\mu}J_{\mu}=0,~{}~{}\partial_{\mu}K_{\mu}^{a}$
$\displaystyle=0$
$\displaystyle\partial_{\mu}[J_{\mu}^{x}-(x-x_{0})J_{\mu}]=\partial_{\mu}[J_{\mu}^{y}-(y-y_{0})J_{\mu}]$
$\displaystyle=0$
$\displaystyle\partial_{\mu}[K_{\mu}+(x-x_{0})K^{x}_{\mu}+(y-y_{0})K^{y}_{\mu}]$
$\displaystyle=0.$ (8)

These equations play vital roles in deducing braiding statistics among charge
and dipoles, as discussed in Sec. IV.

##  III equivalence of dipolar BF theory to rank-2 BF theory

We show how to gradually morph the dBF action in (6) into the R2BF theory,
written in terms of rank-2 tensor gauge fields and rank-2 tensor currents. In
the end, we find that dBF action is in complete agreement with the R2BF action
constructed in [32] as an effective field theory of R2TC. Some aspect of this
transformation was worked out in [34] for the pure gauge part ${\cal L}_{\rm
dBF}$ without the coupling terms ${\cal L}_{AJ}+{\cal L}_{bK}$.

The proof proceeds by first integrating out $b_{t}$ from the dBF action,
resulting in the constraint:

$\displaystyle A_{xy}-A_{yx}=(2\pi/N)K_{t}.$ (9)

We define the rank-2 gauge fields $A_{ab}$,

$\displaystyle A_{ab}=\partial_{a}A_{b}-A_{a}^{b},$ (10)

which transforms under (7) as

$\displaystyle A_{ab}\rightarrow A_{ab}+\partial_{a}\partial_{b}\alpha.$ (11)

Only the charge sector shows up in the transformation of $A_{ab}$. This is the
well-known transformation associated with the tensor gauge fields previously
used in fractonic field theories [3, 20, 21, 22]. The new insight here is that
the tensor fields $A_{ab}$ are not the fundamental objects themselves, but are
constructed from the underlying vector gauge fields $A_{\mu}^{a}$ and
$A_{\mu}$. Although both $A_{xy},A_{yx}$ transform in the same way, they are
not the same fields when $K_{t}\neq 0$. The definition of $A_{ab}$ in (10)
does not guarantee such symmetry either. We will proceed for the time being by
treating $A_{xy}$ and $A_{yx}$ as two distinct fields.

Among the remaining terms in the action ${\cal L}$ (after integrating out
$b_{t}$), terms containing $A^{a}_{t}$ can be collected into

$\displaystyle\frac{N}{2\pi}\left[A_{t}^{a}(\partial_{x}b_{y}^{a}-\partial_{y}b_{x}^{a})-b_{y}A^{x}_{t}+b_{x}A^{y}_{t}\right]-A^{a}_{t}J_{t}^{a}.$
(12)

Integrating out $A_{t}^{a}$ from (12) enforces two relations:

$\displaystyle\partial_{x}b_{y}^{x}-\partial_{y}b_{x}^{x}-b_{y}$
$\displaystyle=(2\pi/N)J_{t}^{x}$
$\displaystyle\partial_{x}b_{y}^{y}-\partial_{y}b_{x}^{y}+b_{x}$
$\displaystyle=(2\pi/N)J_{t}^{y}.$ (13)

Since $b_{t}$ has been integrated out already,
$\varepsilon_{\lambda\mu\nu}b_{\lambda}\partial_{\mu}A_{\nu}$ becomes
$b_{x}(\partial_{y}A_{t}-\partial_{t}A_{y})+b_{y}(\partial_{t}A_{x}-\partial_{x}A_{t})$.
We can use (13) to write $b_{x},b_{y}$ in this action in terms of other
fields:

$\displaystyle\frac{N}{2\pi}\Bigl{[}b_{x}(\partial_{y}A_{t}-\partial_{t}A_{y})+b_{y}(\partial_{t}A_{x}-\partial_{x}A_{t})\Bigr{]}\rightarrow$
$\displaystyle\frac{N}{2\pi}\Bigl{[}b_{x}^{x}(\partial_{y}\partial_{t}A_{x}-\partial_{x}\partial_{y}A_{t})-b_{y}^{y}(\partial_{x}\partial_{t}A_{y}-\partial_{x}\partial_{y}A_{t})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-b_{x}^{y}(\partial_{y}^{2}A_{t}-\partial_{y}\partial_{t}A_{y})+b_{y}^{x}(\partial_{x}^{2}A_{t}-\partial_{x}\partial_{t}A_{x})\Bigr{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-A_{t}(\partial_{x}J_{t}^{x}+\partial_{y}J_{t}^{y})+A_{x}\partial_{t}J_{t}^{x}+A_{y}\partial_{t}J_{t}^{y}.$
(14)

In the meantime, the coupling $-J^{a}_{x}A_{x}^{a}-J^{a}_{y}A_{y}^{a}$ is re-
written, using $A_{ab}=\partial_{a}A_{b}-A_{a}^{b}$, as

$\displaystyle-(J^{a}_{x}A_{x}^{a}+J^{a}_{y}A_{y}^{a})\rightarrow$
$\displaystyle
A_{x}(\partial_{x}J_{x}^{x}+\partial_{y}J_{y}^{x})+A_{y}(\partial_{x}J^{y}_{x}+\partial_{y}J^{y}_{y})$
$\displaystyle+A_{xx}J_{x}^{x}+A_{yy}J_{y}^{y}+A_{xy}J_{x}^{y}+A_{yx}J_{y}^{x}.$
(15)

(Note that $A_{t}^{a}$ is already integrated out and no longer present in the
action.) Collecting terms proportional to $A_{\mu}$ from the last line of (14)
and the first line of (15) and combining them with $-A_{\mu}J_{\mu}$, we
conclude that $A_{\mu}J_{\mu}$ is modified to $A_{\mu}\tilde{J}_{\mu}$ where

$\displaystyle\tilde{J}_{t}$
$\displaystyle=J_{t}+\partial_{x}J^{x}_{t}+\partial_{y}J^{y}_{t}$
$\displaystyle\tilde{J}_{x}$ $\displaystyle=J_{x}-\partial_{\mu}J_{\mu}^{x}=0$
$\displaystyle\tilde{J}_{y}$
$\displaystyle=J_{y}-\partial_{\mu}J_{\mu}^{y}=0.$ (16)

The last two lines come from invoking the dipole current conservation (3). We
thus arrive at the very simple result that only the temporal coupling
$-A_{t}\tilde{J}_{t}$ survives in the charge sector of ${\cal L}_{AJ}$ after
integrating out some fields. In the end we find

$\displaystyle{\cal L}_{AJ}\rightarrow-
A_{t}\tilde{J}_{t}+A_{xx}J_{xx}+A_{xy}J_{xy}+A_{yx}J_{yx}+A_{yy}J_{yy}$ (17)

with

$\displaystyle\tilde{J}_{t}$
$\displaystyle=J_{t}+\partial_{x}J^{x}_{t}+\partial_{y}J^{y}_{t}$
$\displaystyle J_{xx}$
$\displaystyle=J_{x}^{x},~{}J_{yy}=J_{y}^{y},~{}J_{xy}=J_{x}^{y},~{}J_{yx}=J_{y}^{x}.$
(18)

Other terms in the dBF action can be re-arranged as well, for instance

$\displaystyle
b_{t}^{a}(\partial_{x}A_{y}^{a}-\partial_{y}A_{x}^{a})\rightarrow$
$\displaystyle~{}~{}~{}~{}b_{t}^{x}(-\partial_{x}A_{yx}+\partial_{y}A_{xx})+b_{t}^{y}(\partial_{y}A_{xy}-\partial_{x}A_{yy})$
(19)

using

$\displaystyle\partial_{x}A_{y}^{x}-\partial_{y}A_{x}^{x}$
$\displaystyle=-\partial_{x}A_{yx}+\partial_{y}A_{xx},$
$\displaystyle\partial_{x}A_{y}^{y}-\partial_{y}A_{x}^{y}$
$\displaystyle=-\partial_{x}A_{yy}+\partial_{y}A_{xy}.$ (20)

The other terms,
$b_{y}^{a}\partial_{t}A_{x}^{a}-b_{x}^{a}\partial_{t}A_{y}^{a}$, can be
combined with (14) to give

$\displaystyle
b_{x}^{x}(\partial_{t}A_{yx}-\partial_{x}\partial_{y}A_{t})-b_{y}^{y}(\partial_{t}A_{xy}-\partial_{x}\partial_{y}A_{t})$
$\displaystyle+b_{x}^{y}(\partial_{t}A_{yy}-\partial_{y}^{2}A_{t})-b_{y}^{x}(\partial_{t}A_{xx}-\partial_{x}^{2}A_{t}).$
(21)

Finally, adding (19) and (21) gives

$\displaystyle{\cal L}_{\rm dBF}$
$\displaystyle=\frac{N}{2\pi}\Bigl{[}b_{t}^{x}(\partial_{y}A_{xx}-\partial_{x}A_{yx})+b_{t}^{y}(\partial_{y}A_{xy}-\partial_{x}A_{yy})$
$\displaystyle~{}~{}~{}~{}~{}~{}+b_{x}^{x}(\partial_{t}A_{yx}-\partial_{x}\partial_{y}A_{t})-b_{y}^{y}(\partial_{t}A_{xy}-\partial_{x}\partial_{y}A_{t})$
$\displaystyle~{}~{}~{}~{}~{}~{}+b_{x}^{y}(\partial_{t}A_{yy}-\partial_{y}^{2}A_{t})-b_{y}^{x}(\partial_{t}A_{xx}-\partial_{x}^{2}A_{t})\Bigr{]}.$
(22)

In this form, $b_{\mu}$ has been completely integrated out and some second
derivatives of the tensor gauge fields $A_{ab}$ appear in the action.

So far we have avoided the assumption of symmetry $A_{xy}=A_{yx}$ and the
action (22) is valid even for $A_{xy}\neq A_{yx}$. To proceed further, we
solve the constraint (9) by re-writing

$\displaystyle A_{xy}\rightarrow
A_{xy}+\frac{\pi}{N}K_{t},~{}~{}A_{yx}\rightarrow A_{yx}-\frac{\pi}{N}K_{t}.$
(23)

The new tensor is symmetric, $A_{xy}=A_{yx}$. We can re-write the action (22)
in light of the substitution (23). In addition to the second line of (22)
simplifying to
$(b_{x}^{x}-b_{y}^{y})(\partial_{t}A_{xy}-\partial_{x}\partial_{y}A_{t})$,
there are additional terms showing up in the action due to $K_{t}$:

$\displaystyle-\frac{1}{2}b_{x}^{x}\partial_{t}K_{t}-\frac{1}{2}b_{y}^{y}\partial_{t}K_{t}+\frac{1}{2}b_{t}^{x}\partial_{x}K_{t}+\frac{1}{2}b_{t}^{y}\partial_{y}K_{t}.$
(24)

Also invoking (13), we can transform the $bK$ coupling

$\displaystyle b_{x}K_{x}+b_{y}K_{y}\rightarrow
b_{y}^{y}\partial_{x}K_{x}+b_{x}^{x}\partial_{y}K_{y}-b_{x}^{y}\partial_{y}K_{x}-b_{y}^{x}\partial_{x}K_{y}.$
(25)

We can add (24) and (25) to $b_{\mu}^{a}K_{\mu}^{a}$ to get

$\displaystyle
b_{t}^{x}\left(K_{t}^{x}-\frac{1}{2}\partial_{x}K_{t}\right)+b_{t}^{y}\left(K_{t}^{y}-\frac{1}{2}\partial_{y}K_{t}\right)$
$\displaystyle+b_{x}^{x}\left(K_{x}^{x}+\partial_{y}K_{y}+\frac{1}{2}\partial_{t}K_{t}\right)+b_{y}^{x}\left(K_{y}^{x}-\partial_{x}K_{y}\right)$
$\displaystyle+b_{x}^{y}\left(K_{x}^{y}-\partial_{y}K_{x}\right)+b_{y}^{y}\left(K_{y}^{y}+\partial_{x}K_{x}+\frac{1}{2}\partial_{t}K_{t}\right)$
$\displaystyle\equiv b^{a}_{\mu}\tilde{K}_{\mu}^{a},$ (26)

and conclude the action ${\cal L}_{bK}$ is modified to

$\displaystyle{\cal L}_{bK}\rightarrow-b^{a}_{\mu}\tilde{K}_{\mu}^{a}.$ (27)

The charge part $-b_{\mu}K_{\mu}$ has been integrated out. Meanwhile, the
${\cal L}_{AJ}$ action in (17) is modified by $K_{t}\neq 0$,

$\displaystyle{\cal L}_{AJ}\rightarrow-
A_{t}\tilde{J}_{t}+A_{xx}J_{xx}+A_{xy}J_{xy}+A_{yy}J_{yy}+\frac{\pi}{N}(J_{x}^{y}-J_{y}^{x})K_{t},$
(28)

with $A_{xy}=A_{yx}$ and $J_{xy}=J_{x}^{y}+J_{y}^{x}$.

It turns out $K_{t}$ appears linearly in (27) and (28) but does not appear
elsewhere. Integrate out $K_{t}$ gives

$\displaystyle(\partial_{y}b_{t}^{y}-\partial_{t}b_{y}^{y})-(\partial_{t}b_{x}^{x}-\partial_{x}b_{t}^{x})=\frac{2\pi}{N}(J_{x}^{y}-J_{y}^{x}).$
(29)

This relation is nothing new, and can be derived from the constraint that
arises from integrating out $A^{a}_{x},A^{a}_{y}$ from the full action:

$\displaystyle\partial_{y}b_{t}^{a}-\partial_{t}b_{y}^{a}$
$\displaystyle=(2\pi/N)J^{a}_{x}$
$\displaystyle\partial_{t}b_{x}^{a}-\partial_{x}b_{t}^{a}$
$\displaystyle=(2\pi/N)J^{a}_{y}.$ (30)

Finally, we can arrange the two actions ${\cal L}_{AJ}$ and ${\cal L}_{bK}$ as

$\displaystyle{\cal L}_{AJ}$
$\displaystyle=-A_{t}\tilde{J}_{t}+A_{xx}J_{xx}+A_{xy}J_{xy}+A_{yy}J_{yy}$
$\displaystyle{\cal L}_{bK}$ $\displaystyle=-b_{\mu}^{a}\tilde{K}_{\mu}^{a}$
(31)

with

$\displaystyle\tilde{J}_{t}=J_{t}+\partial_{x}J^{x}_{t}+\partial_{y}J^{y}_{t},$
$\displaystyle
J_{xx}=J_{x}^{x},~{}~{}J_{xy}=J_{x}^{y}+J_{y}^{x},~{}~{}J_{yy}=J_{y}^{y},$
(32)

and

$\displaystyle\tilde{K}_{t}^{a}$ $\displaystyle=K_{t}^{a}$
$\displaystyle\tilde{K}_{x}^{x}$
$\displaystyle=K_{x}^{x}+\partial_{y}K_{y},~{}~{}~{}\tilde{K}_{y}^{x}=K_{y}^{x}-\partial_{x}K_{y}$
$\displaystyle\tilde{K}_{x}^{y}$
$\displaystyle=K_{x}^{y}-\partial_{y}K_{x},~{}~{}~{}\tilde{K}_{y}^{y}=K_{y}^{y}+\partial_{x}K_{x}.$
(33)

We note that ${\cal L}_{\rm dBF}$ and ${\cal L}_{AJ}$ are by now written in
terms of tensorial gauge fields and currents, but ${\cal L}_{bK}$ is not.

To remedy this problem, we re-define certain variables

$\displaystyle(b_{t}^{x},b_{t}^{y})$
$\displaystyle\rightarrow(E_{t}^{y},-E_{t}^{x}),$
$\displaystyle(b_{x}^{x}-b_{y}^{y},b_{x}^{y},b_{y}^{x})$
$\displaystyle\rightarrow(-E_{xy},-E_{yy},E_{xx}),$ (34)

and further re-write

$\displaystyle(E_{xx},E_{yy},E_{xy})$
$\displaystyle\rightarrow(E_{yy},E_{xx},-E_{xy}),$
$\displaystyle(A_{xx},A_{yy},A_{xy})$
$\displaystyle\rightarrow(A_{yy},A_{xx},-A_{xy}).$ (35)

This gives

$\displaystyle{\cal L}_{\rm dBF}$ $\displaystyle\rightarrow
A_{xy}\partial_{t}E_{xy}+A_{yy}\partial_{t}E_{yy}+A_{xx}\partial_{t}E_{xx}$
$\displaystyle~{}~{}+E_{t}^{x}(\partial_{x}A_{xx}+\partial_{y}A_{xy})+E_{t}^{y}(\partial_{y}A_{yy}+\partial_{x}A_{xy})$
$\displaystyle~{}~{}+A_{t}(\partial_{y}^{2}E_{xx}+\partial_{x}^{2}E_{yy}-\partial_{x}\partial_{y}E_{xy}),$
(36)

and

$\displaystyle{\cal
L}_{AJ}=-A_{t}\tilde{J}_{t}+A_{yy}J_{xx}+A_{xx}J_{yy}-A_{xy}J_{xy}.$ (37)

Meanwhile, re-writing ${\cal L}_{bK}$ with these new variables gives

$\displaystyle{\cal L}_{bK}\rightarrow$ $\displaystyle-
E_{t}^{y}\tilde{K}_{t}^{x}+E_{t}^{x}\tilde{K}_{t}^{y}+E_{xx}\tilde{K}_{x}^{y}-E_{yy}\tilde{K}_{y}^{x}-\frac{1}{2}E_{xy}(\tilde{K}_{x}^{x}-\tilde{K}_{y}^{y})$
$\displaystyle-\frac{1}{2}(b_{x}^{x}+b_{y}^{y})(\tilde{K}_{x}^{x}+\tilde{K}_{y}^{y})$
(38)

The last line cannot be written in terms of any of the variables in (34).
Luckily, $b_{x}^{x}+b_{y}^{y}$ does not appear elsewhere in the action and can
be integrated out to yield

$\displaystyle\tilde{K}_{x}^{x}+\tilde{K}_{y}^{y}=K_{x}^{x}+K_{y}^{y}+\partial_{\mu}K_{\mu}=0,$
(39)

which is nothing but the $K$-current conservation already derived in (8).

Combining all three remaining actions, the full action ${\cal L}={\cal L}_{\rm
dBF}+{\cal L}_{AJ}+{\cal L}_{bK}$ becomes

$\displaystyle{\cal L}_{\rm R2BF}=$
$\displaystyle\frac{N}{2\pi}\Bigl{[}A_{xy}\partial_{t}E_{xy}+A_{yy}\partial_{t}E_{yy}+A_{xx}\partial_{t}E_{xx}$
$\displaystyle~{}~{}~{}~{}~{}~{}+E_{t}^{x}(\partial_{x}A_{xx}+\partial_{y}A_{xy}-\rho^{x})$
$\displaystyle~{}~{}~{}~{}~{}~{}+E_{t}^{y}(\partial_{y}A_{yy}+\partial_{x}A_{xy}-\rho^{y})$
$\displaystyle+A_{t}(\partial_{y}^{2}E_{xx}+\partial_{x}^{2}E_{yy}-\partial_{x}\partial_{y}E_{xy}-\tilde{J}_{t})\Bigr{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}-A_{xy}J_{xy}+A_{yy}J_{xx}+A_{xx}J_{yy}$
$\displaystyle~{}~{}~{}~{}~{}~{}-E_{xx}K_{xx}-E_{xy}K_{xy}-E_{yy}K_{yy},$ (40)

with

$\displaystyle(\rho^{x},\rho^{y})$
$\displaystyle=(-\tilde{K}_{t}^{y},\tilde{K}_{t}^{x})$
$\displaystyle(K_{xx},K_{xy},K_{yy})$
$\displaystyle=\left(-\tilde{K}_{x}^{y},\frac{1}{2}(\tilde{K}_{x}^{x}-\tilde{K}_{y}^{y}),\tilde{K}_{y}^{x}\right).$
(41)

The final action ${\cal L}_{\rm R2BF}$ is in complete accord with the R2BF
action derived in [32] [Eq. (5.2)]. The R2BF action supports a scalar charge
$\tilde{J}_{t}$ and the associated tensor current $(J_{xx},J_{xy},J_{yy})$, as
well as vector charges $(\rho^{x},\rho^{y})$ and the associated tensor current
$(K_{xx},K_{xy},K_{xy})$. Although there are two vector charges, only one
tensor current $K_{ab}$ exists for both of them.

There are three conservation laws, one for each of the three charges. One of
them is given by $(\tilde{J}_{t},J_{xx},J_{xy},J_{yy})$ satisfying the
equation

$\displaystyle\partial_{t}\tilde{J}_{t}+\partial_{x}^{2}J_{xx}+\partial_{x}\partial_{y}J_{xy}+\partial_{y}^{2}J_{yy}$
$\displaystyle=\partial_{t}J_{t}+\partial_{x}(\partial_{\mu}J^{x}_{\mu})+\partial_{y}(\partial_{\mu}J^{y}_{\mu})$
$\displaystyle=\partial_{t}J_{t}+\partial_{x}J_{x}+\partial_{y}J_{y}=0.$ (42)

The relation $\partial_{\mu}J^{a}_{\mu}=J_{a}$ was used to reach the second
line. The charge density $\tilde{J}_{t}$ is the sum of $J_{t}$, the bare
charge density, and the dipole-induced charge density
$\partial_{x}J^{x}_{t}+\partial_{y}J^{y}_{t}$. A charge ($J_{t}$) by itself
cannot move due to the dipole moment conservation and must be accompanied by
dipole motion. This forces charge to form a composite object with a dipole in
order to move freely. Although the observation that a free object in dipole-
constrained theory is a charge-dipole composite has been made in [32] already,
the dBF formulation is helpful in clarifying the nature of this composite
charge as the sum of the bare charge ($J_{t}$) and the dipole-induced charge
($\partial_{x}J^{x}_{t}+\partial_{y}J^{y}_{t}$) through the formula (32). The
tensor currents $(J_{xx},J_{xy},J_{yy})$ and $(K_{xx},K_{xy},K_{yy})$,
introduced through heuristic reasoning in [32], are also given concrete
definitions in terms of underlying dipole currents.

The second continuity equation follows from defining a three-current

$(\rho^{x},K_{xx},K_{xy})=(-\tilde{K}^{y}_{t},-\tilde{K}^{y}_{x},(\tilde{K}^{xx}-\tilde{K}^{yy})/2)$

and working out

$\displaystyle\partial_{t}\rho^{x}+\partial_{x}K_{xx}+\partial_{y}K_{xy}$
$\displaystyle=\partial_{t}\rho^{x}+\partial_{x}K_{xx}+\partial_{y}K_{xy}-\frac{1}{2}\partial_{y}(\tilde{K}_{x}^{x}+\tilde{K}_{y}^{y})$
$\displaystyle=-\partial_{\mu}\tilde{K}_{\mu}^{y}=-\partial_{\mu}K^{y}_{\mu}=0.$
(43)

Note that in the second line we added an extra term
$\partial_{y}(\tilde{K}_{x}^{x}+\tilde{K}_{y}^{y})/2$ to complete the
continuity equation. This is allowed since the added term is zero by the
constraint (39). It also implies a degree of flexibility in defining $K_{xy}$,
as

$\displaystyle
K_{xy}=\frac{1}{2}(\tilde{K}_{x}^{x}-\tilde{K}_{y}^{y})+\frac{\theta}{2}(\tilde{K}_{x}^{x}+\tilde{K}_{y}^{y})$
(44)

with $\theta$ being an arbitrary real number.

A third conservation law exists for the three-current

$(\rho^{y},K_{xy},K_{yy})=(\tilde{K}^{x}_{t},(\tilde{K}_{x}^{x}-\tilde{K}_{y}^{y})/2,\tilde{K}^{x}_{y})$

obeying the continuity equation

$\displaystyle\partial_{t}\rho^{y}+\partial_{x}K_{xy}+\partial_{y}K_{yy}$
$\displaystyle=\partial_{t}\rho^{y}+\partial_{x}K_{xy}+\partial_{y}K_{yy}+\frac{1}{2}\partial_{x}(\tilde{K}_{x}^{x}+\tilde{K}_{y}^{y})$
$\displaystyle=\partial_{\mu}\tilde{K}_{\mu}^{x}=\partial_{\mu}K^{x}_{\mu}=0.$
(45)

In conclusion, we have derived all three conservation laws as well as the
exact form of the R2BF given in [32] by starting from the dBF action of [32]
and integrating out some degrees of freedom. In the course of the
transformation we found explicit representations of the various charge
densities and tensor currents in the R2BF in terms of the charge and dipole
currents of the original dBF - see (32) and (41). These expressions will play
a vital role in deriving dipolar braiding statistics.

##  IV Dipolar Braiding

A key feature of the dBF (or R2BF) theory is that its quasiparticles show
braiding statistics that depends on the location of the particle being braided
around [25, 32]. This point can be highlighted already in the dCS theory,
which we discuss first due to its simplicity.

It is well known that $\partial_{\mu}J_{\mu}=0$ admits a solution

$\displaystyle J^{c}_{\mu}=(1,\dot{x}_{c},\dot{y}_{c})\delta^{2}({\bf r}-{\bf
r}_{c}(t))$ (46)

where ${\bf r}_{c}(t)$ is the position of the charge at time $t$, and
$(\dot{x}_{c},\dot{y}_{c})$ is its velocity. By virtue of continuity equation
$\partial_{\mu}[J_{\mu}^{x}-(x-x_{0})J_{\mu}]=0$ and
$\partial_{\mu}[J_{\mu}^{y}-(y-y_{0})J_{\mu}]=0$ [Eq. (4)], one has solutions
for the dipole currents

$\displaystyle J_{\mu}^{x}$
$\displaystyle=(x_{c}-x_{0})J_{\mu}^{c}+J_{\mu}^{d,x}$ $\displaystyle
J_{\mu}^{y}$ $\displaystyle=(y_{c}-y_{0})J_{\mu}^{c}+J_{\mu}^{d,y}$ (47)

with $J_{\mu}^{c}$ given in (46) and

$\displaystyle J^{d,x}_{\mu}$
$\displaystyle=(1,\dot{x}_{x},\dot{y}_{x})\delta^{2}({\bf r}-{\bf r}_{x}(t))$
$\displaystyle J^{d,y}_{\mu}$
$\displaystyle=(1,\dot{x}_{y},\dot{y}_{y})\delta^{2}({\bf r}-{\bf r}_{y}(t))$
(48)

representing currents associated with a single $x$\- or $y$-dipole located at
${\bf r}_{x}(t)$ and ${\bf r}_{y}(t)$, respectively. We will refer to
solutions given in (46) and (48) as single-charge and single-dipole solutions,
respectively. Current due to a single-charge motion $J_{\mu}^{c}$ generates
accompanying dipole current $J^{x}_{\mu}=(x-x_{0})J^{c}_{\mu}$ and
$J^{y}_{\mu}=(y-y_{0})J^{c}_{\mu}$. To distinguish them from the intrinsic
dipole currents $J^{d,x}_{\mu}$ and $J^{d,y}_{\mu}$, we refer to them as
orbital dipole currents of $a$-orientation. The name was suggeested in [33].

We examine the consequence of inserting various solutions of the charge and
the dipole currents into the action ${\cal L}_{AJ}$. First, inserting the
single-charge solution $J_{\mu}=J^{c}_{\mu}$ into $J_{\mu}A_{\mu}$ (and
integrating over spacetime) gives

$\displaystyle\oint[A_{x}({\bf r}_{c})dx_{c}+A_{y}({\bf
r}_{c})dy_{c}]=\int(\partial_{x}A_{y}-\partial_{y}A_{x})dx_{c}dy_{c}$ (49)

where the integral is over the area encompassed by the closed trajectory of
${\bf r}_{c}$. Since there is no Chern-Simons term for $A_{\mu}$, this flux is
not quantized. In fact, $\partial_{x}A_{y}-\partial_{y}A_{x}$ is not even
gauge-invariant under the dipolar gauge transformation by $\beta$ in (2), and
will soon be combined with another term to make it fully gauge-invariant.

When we insert the single-dipole solution (48) into the action
$J_{\mu}^{a}A^{a}_{\mu}$, we find for $a=x$

$\displaystyle\int
J_{\mu}^{d,x}A_{\mu}^{x}=\int(\partial_{x}A^{x}_{y}-\partial_{y}A^{x}_{x}).$
(50)

Invoking the flux attachment rule

$\displaystyle\varepsilon_{\lambda\mu\nu}\partial_{\mu}A^{a}_{\nu}=\frac{2\pi}{N}J^{a}_{\lambda},$
(51)

which can be derived by integrating out $A^{a}_{\mu}$ from the dCS action, one
can further write

$\displaystyle\int(\partial_{x}A^{x}_{y}-\partial_{y}A^{x}_{x})=\frac{2\pi}{N}\int
J_{t}^{x}$ (52)

showing that the geometric phase is $2\pi/N$ per $x$-dipole enclosed within
the path. We conclude that an $a$-dipole braiding around another dipole of the
same orientation results in the fractional phase of $2\pi/N$.

When a charge is being braided, there is an accompanying orbital dipole
current $(x-x_{0})J_{\mu}^{c}$ contributing to $J^{x}_{\mu}$. Its contribution
to the phase is

$\displaystyle\oint(x-x_{0})J_{\mu}^{c}A_{\mu}^{x}=\int(x_{c}-x_{0})[A_{x}^{x}({\bf
r}_{c})dx_{c}+A_{y}^{x}({\bf r}_{c})dy_{c}]$
$\displaystyle=\oint(x-x_{0})[\partial_{x}A_{y}^{x}-\partial_{y}A_{x}^{x}]dxdy+\int
A_{y}^{x}dxdy.$ (53)

There is also the orbital $y$-dipole current $(y-y_{0})J_{\mu}^{c}$
contributing the phase

$\displaystyle\oint(y-y_{0})J_{\mu}^{c}A_{\mu}^{y}=\oint(y-y_{0})[\partial_{x}A_{y}^{y}-\partial_{y}A_{x}^{y}]dxdy-\int
A_{x}^{y}dxdy.$ (54)

Overall, there are three different contributions to the phase, given in (49),
(53), (54), as a charge is moved adiabatically around. Combining all three
contributions gives

$\displaystyle\oint(x-x_{0})[\partial_{x}A_{y}^{x}-\partial_{y}A_{x}^{x}]dxdy+\oint(y-y_{0})[\partial_{x}A_{y}^{y}-\partial_{y}A_{x}^{y}]dxdy$
$\displaystyle+\int(\partial_{x}A_{y}-\partial_{y}A_{x}+A_{y}^{x}-A_{x}^{y})dxdy.$
(55)

The second line represents a contribution from the gauge-invariant flux 111The
author thanks Xiaoyang Huang for pointing out the gauge invariance of this
combination. $F_{xy}=\partial_{x}A_{y}-\partial_{y}A_{x}+A_{y}^{x}-A_{x}^{y}$,
but the value is not tied to the charge density as there is no CS term
involving $A_{\mu}$ in the dCS action. The first line, after substituting
$\partial_{x}A_{y}^{a}-\partial_{y}A_{x}^{a}=(2\pi/N)J_{t}^{a}$, gives

$\displaystyle-\frac{2\pi}{N}\Bigl{[}\sum_{i}(x_{i}-x_{0})+\sum_{i}(y_{i}-y_{0})\Bigr{]}$
(56)

where $x_{i}$ ($y_{i}$) is the $a$-coordinate of the $i$-th $a$-dipole inside
the loop. The minus sign occurs because the action is ${\cal
L}_{AJ}=-A_{\mu}J_{\mu}-A_{\mu}^{a}J_{\mu}^{a}$. The braiding statistics
between a charge and the dipoles are therefore dipolar, in the sense that the
phase acquired in the braiding process depends linearly on the coordinates of
the particles.

In summary, when an $a$-dipole is adiabatically moved, it picks up the
fractional phase of $2\pi/N$ by braiding around other $a$-dipoles. Meanwhile a
charge braiding around dipoles picks up dipolar phase because the charge
motion induces orbital dipole current. The orbital dipole current accompanying
the charge current is necessitated by the need to satisfy the dipole symmetry.

Next we examine the braiding statistics in the dBF (or R2BF) theory. From the
various conservation laws of $J$-currents and $K$-currents derived in (8), we
can write down the solutions

$\displaystyle J_{\mu}=J^{c}_{\mu}$ $\displaystyle
J^{x}_{\mu}=J^{d,x}_{\mu}+(x-x_{0})J^{c}_{\mu},~{}~{}J^{y}_{\mu}=J^{d,y}_{\mu}+(y-y_{0})J^{c}_{\mu}$
$\displaystyle K^{a}_{\mu}=K^{d,a}_{\mu}$ $\displaystyle
K_{\mu}=K^{c}_{\mu}-(x-x_{0})K^{d,x}_{\mu}-(y-y_{0})K^{d,y}_{\mu},$ (57)

where $J^{c}_{\mu},K^{c}_{\mu},J^{d,a}_{\mu},K^{d,a}_{\mu}$ are the single-
particle solutions of (46) and (48).

Since our focus is on the dipolar braiding statistics, we can “turn off” the
intrinsic dipolar current $J^{d,a}_{\mu}$ and write the orbital dipole
currents as

$\displaystyle J^{x}_{\mu}$
$\displaystyle=(x-x_{0})J^{c}_{\mu}=(x-x_{0})(1,\dot{x}_{c},\dot{y}_{c})\delta^{2}({\bf
r}-{\bf r}_{c})$ $\displaystyle J^{y}_{\mu}$
$\displaystyle=(y-x_{0})J^{c}_{\mu}=(y-x_{0})(1,\dot{x}_{c},\dot{y}_{c})\delta^{2}({\bf
r}-{\bf r}_{c}).$ (58)

This gives

$\displaystyle\tilde{J}_{t}$
$\displaystyle=J_{t}+\partial_{x}J^{x}_{t}+\partial_{y}J^{y}_{t}$
$\displaystyle=[1+(x-x_{0})\partial_{x}+(y-y_{0})\partial_{y}]\delta^{2}({\bf
r}-{\bf r}_{c})$ $\displaystyle J_{xx}$
$\displaystyle=J_{x}^{x}=(x-x_{0})\dot{x}_{c}\delta^{2}({\bf r}-{\bf r}_{c})$
$\displaystyle J_{xy}$
$\displaystyle=J_{x}^{y}+J_{y}^{x}=[(x-x_{0})\dot{y}_{c}+(y-y_{0})\dot{x}_{c}]\delta^{2}({\bf
r}-{\bf r}_{c})$ $\displaystyle J_{yy}$
$\displaystyle=J_{y}^{y}=(y-y_{0})\dot{y}_{c}\delta^{2}({\bf r}-{\bf r}_{c}).$
(59)

Inserting $(J_{xx},J_{xy},J_{yy})$ obtained here into the action
$\int(A_{xx}J_{yy}+A_{yy}J_{xx}-A_{xy}J_{xy})$ gives

$\displaystyle\oint{\bf a}({\bf r}_{c})\cdot d{\bf r}_{c}$ (60)

where

$\displaystyle{\bf
a}=\Bigl{(}(x-x_{0})A_{yy}-(y-y_{0})A_{xy},(y-y_{0})A_{xx}-(x-x_{0})A_{xy}\Bigr{)}.$

Taking the curl of ${\bf a}$ gives

$\displaystyle{\bm{\nabla}}\times{\bf a}$
$\displaystyle=(y-y_{0})(\partial_{x}A_{xx}+\partial_{y}A_{xy})-(x-x_{0})(\partial_{x}A_{xy}+\partial_{y}A_{yy})$
$\displaystyle=\frac{2\pi}{N}\left[(y-y_{0})\rho^{x}-(x-x_{0})\rho^{y}\right],$
(61)

where we used the relations $\partial_{x}A_{xx}+\partial_{y}A_{xy}=\rho^{x}$
and $\partial_{y}A_{yy}+\partial_{x}A_{xy}=\rho^{y}$ obtained from integrating
out $E_{t}^{x},E_{t}^{y}$ in the R2BF action. Since we further have
$\rho^{x}=-K^{y}_{t}$ and $\rho^{y}=K_{t}^{x}$, we can re-write the curl

$\displaystyle{\bm{\nabla}}\times{\bf
a}=-\frac{2\pi}{N}\left[(x-x_{0})K_{t}^{x}+(y-y_{0})K_{t}^{y}\right].$ (62)

The Berry phase acquired by moving a $J$-charge along a closed path is
therefore

$\displaystyle-\frac{2\pi}{N}\left[\sum_{i}(x_{i}-x_{0})+\sum_{i}(y_{i}-y_{0})\right],$
(63)

where $x_{i}$ ($y_{i}$) are the $x$\- ($y$-) coordinate of the $x$-dipole
($y$-dipole) enclosed inside the path. Adiabatic motion of $J$-charge is
interpreted as that of a bare $J$-charge and $J$-dipole composite.

We next turn to single-dipole solutions for $\partial_{\mu}K_{\mu}^{a}=0$:

$\displaystyle K_{\mu}^{d,x}$
$\displaystyle=(1,\dot{x}_{x},\dot{y}_{x})\delta^{2}({\bf r}-{\bf r}_{x})$
$\displaystyle K_{\mu}^{d,y}$
$\displaystyle=(1,\dot{x}_{y},\dot{y}_{y})\delta^{2}({\bf r}-{\bf r}_{y}).$
(64)

The $K$-dipoles, being exempt from dipole constraint, can move freely. The
solutions for
$\partial_{\mu}[K_{\mu}+(x-x_{0})K_{\mu}^{x}+(y-y_{0})K_{\mu}^{y}]=0$ is

$\displaystyle K_{\mu}=-(x-x_{0})K_{\mu}^{d,x}-(y-y_{0})K_{\mu}^{d,y}.$ (65)

It is the sum of contributions from $x$-dipoles and $y$-dipoles. One can keep
only one of the terms on the right when there are dipoles of one orientation
present but not the other.

The two vector charges $\rho^{x}=-K^{y}_{t}=-\delta^{2}({\bf r}-{\bf r}_{y})$
and $\rho^{y}=K^{x}_{t}=\delta^{2}({\bf r}-{\bf r}_{x})$ refer to situations
where there is only a single $y$-dipole or a single $x$-dipole. When we work
out the associated $K$-currents $(K_{xx},K_{xy},K_{yy})$, we can separately
consider situations with only the $x$-dipole or the $y$-dipole present. For
instance, we choose $\rho^{x}=-\delta^{2}({\bf r}-{\bf r}_{y})$ and the
associated $K$-currents are

$\displaystyle K_{xx}=-\tilde{K}^{y}_{x}$ $\displaystyle\rightarrow-
K_{x}^{d,y}-(y-y_{0})\partial_{y}K_{x}^{d,y},$ $\displaystyle
K_{xy}=\tilde{K}_{x}^{x}$
$\displaystyle\rightarrow-(y-y_{0})\partial_{y}K_{y}^{d,y},$ $\displaystyle
K_{yy}=\tilde{K}_{y}^{x}$
$\displaystyle\rightarrow(y-y_{0})\partial_{x}K_{y}^{d,y}.$ (66)

That is, from the generic definition of
$(\tilde{K}^{y}_{x},\tilde{K}_{x}^{x},\tilde{K}_{y}^{x})$ given in (41), we
extract those single-dipole solutions associated with the $y$-dipole (terms on
the right side of the arrow).

Inserting (66) into the R2BF action gives

$\displaystyle-\int(E_{xx}K_{xx}+E_{xy}K_{xy}+E_{yy}K_{yy})\rightarrow\oint{\bf
a}^{y}\cdot d{\bf r}_{y}$ (67)

where

$\displaystyle{\bf
a}^{y}=\Bigl{(}E_{xx}-(y-y_{0})\partial_{y}E_{xx},(y-y_{0})(\partial_{x}E_{yy}-\partial_{y}E_{xy})\Bigr{)}.$
(68)

Taking the curl of ${\bf a}^{y}$ gives the dipolar flux attachment rule:

$\displaystyle{\bm{\nabla}}\times{\bf a}^{y}$
$\displaystyle=(y-y_{0})(\partial_{x}^{2}E_{yy}+\partial_{y}^{2}E_{xx}-\partial_{x}\partial_{y}E_{xy})$
$\displaystyle=\frac{2\pi}{N}(y-y_{0})\tilde{J}_{t}.$ (69)

The second line follows from integrating out $A_{t}$ in the R2BF action which
gives
$\partial_{x}^{2}E_{yy}+\partial_{y}^{2}E_{xx}-\partial_{x}\partial_{y}E_{xy}=(2\pi/N)\tilde{J}_{t}$.
The phase picked up by a $y$-dipole is

$\displaystyle\frac{2\pi}{N}\sum_{i}(y_{i}-y_{0})$ (70)

from braiding around the composite $J$-particles enclosed inside, in addition
to the fractional phase $2\pi/N$ acquired by braiding around other
$y$-dipoles.

A similar conclusion is reached for the braiding of $x$-dipole. For
$\rho^{y}=\delta^{2}({\bf r}-{\bf r}_{x})$, we find

$\displaystyle K_{xx}=\tilde{K}^{y}_{x}$
$\displaystyle\rightarrow-(x-x_{0})\partial_{y}K_{x}^{d,x},$ $\displaystyle
K_{xy}=-\tilde{K}_{y}^{y}$
$\displaystyle\rightarrow(x-x_{0})\partial_{x}K_{x}^{d,x},$ $\displaystyle
K_{yy}=\tilde{K}_{y}^{x}$ $\displaystyle\rightarrow
K_{y}^{d,x}+(x-x_{0})\partial_{x}K_{y}^{d,x}$ (71)

as the $K$-current associated with the adiabatic motion of $x$-dipole.
Inserting (71) into the R2BF action gives

$\displaystyle-\int(E_{xx}K_{xx}+E_{xy}K_{xy}+E_{yy}K_{yy})\rightarrow\oint{\bf
a}^{x}\cdot d{\bf r}_{x}$ (72)

where

$\displaystyle{\bf
a}^{x}=\Bigl{(}(x-x_{0})(\partial_{x}E_{xy}-\partial_{y}E_{xx}),(x-x_{0})\partial_{x}E_{yy}-E_{yy}\Bigr{)}.$
(73)

Taking the curl of ${\bf a}^{x}$ results in the dipolar flux attachment rule:

$\displaystyle{\bm{\nabla}}\times{\bf
a}^{x}=\frac{2\pi}{N}(x-x_{0})\tilde{J}_{t}.$ (74)

Braiding an $x$-dipole around the composite $J$-particles results in the
dipolar phase

$\displaystyle\frac{2\pi}{N}\sum_{i}(x_{i}-x_{0}),$ (75)

in addition to $2\pi/N$ from braiding around other $x$-dipoles.

To conclude, dipolar braiding occurs between a composite $J$-charge and
$K$-dipoles and are proportional to the $x$ ($y$) coordinate of the particle
being braided around when the $K$-dipole involved is an $x$-dipole
($y$-dipole). The braiding statistics between $K$-dipoles of the same
orientation is that of the usual anyons with fractional phases.

##  V Discussion

The picture underlying the rank-2 toric code was that of two layers of
$\mathbb{Z}_{N}$ lattice gauge theories being coupled through some constraint
[25, 36]. The constraint was not related to the dipole symmetry in any way, or
at least the connection to dipole symmetry was not obvious. The dipolar
braiding statistics among the anyonic excitations in the rank-2 toric code was
therefore an unexpected surprise, and its understanding prompted the
formulation of BF-like theory employing rank-2 tensor gauge fields [32]. The
dipolar braiding among the quasiparticles was derived from the rank-2 BF
theory, but the precise relation between the dipolar braiding statistics and
the dipole symmetry of the action remained obscure. In this work, the origin
of dipolar braiding among the anyons is clarified as a direct consequence of
the dipole symmetry built into the dipolar BF theory.

Analysis of the present paper suggests that dipolar braiding is a natural
consequence of topological models with dipole symmetry. Extension of the ideas
and techniques explored in this paper to the BF theory with quadrupolar
symmetry [34] with the resulting quadrupolar braiding statistics should be
straightforward, albeit technically challenging.

###### Acknowledgements.

I am grateful to Xiaoyang Huang and H. Ebisu for an insightful exchange on the
dipolar Chern-Simons and dipolar BF theories. Many discussions on the rank-2
toric code and dipolar braiding statistics with Jintae Kim, Yun-Tak Oh, and
Yizhi You over the years are gratefully acknowledged. He was supported by the
National Research Foundation of Korea(NRF) grant funded by the Korea
government(MSIT) (No. 2023R1A2C1002644). This research was also supported in
part by grants NSF PHY-1748958 and PHY-2309135 to the Kavli Institute for
Theoretical Physics (KITP).

## References

  * Nandkishore and Hermele [2019] R. M. Nandkishore and M. Hermele, Fractons, Annual Review of Condensed Matter Physics 10, 295 (2019), https://doi.org/10.1146/annurev-conmatphys-031218-013604 .
  * Pretko  _et al._ [2020] M. Pretko, X. Chen, and Y. You, Fracton Phases of Matter, International Journal of Modern Physics A 35, 2030003 (2020), arXiv:2001.01722 .
  * Pretko [2018] M. Pretko, The fracton gauge principle, Phys. Rev. B 98, 115134 (2018).
  * Seiberg and Shao [2021] N. Seiberg and S.-H. Shao, Exotic symmetries, duality, and fractons in 2+1-dimensional quantum field theory, SciPost Phys. 10, 027 (2021).
  * Gorantla  _et al._ [2022] P. Gorantla, H. T. Lam, N. Seiberg, and S.-H. Shao, Global dipole symmetry, compact lifshitz theory, tensor gauge theory, and fractons, Phys. Rev. B 106, 045112 (2022).
  * Guardado-Sanchez  _et al._ [2020] E. Guardado-Sanchez, A. Morningstar, B. M. Spar, P. T. Brown, D. A. Huse, and W. S. Bakr, Subdiffusion and heat transport in a tilted two-dimensional fermi-hubbard system, Physical Review X 10, 011042 (2020).
  * Scherg  _et al._ [2021] S. Scherg, T. Kohlert, P. Sala, F. Pollmann, B. H. Madhusudhana, I. Bloch, and M. Aidelsburger, Observing non-ergodicity due to kinetic constraints in tilted fermi-hubbard chains, Nature Communications 12, 1 (2021).
  * Zahn  _et al._ [2022] H. Zahn, V. Singh, M. Kosch, L. Asteria, L. Freystatzky, K. Sengstock, L. Mathey, and C. Weitenberg, Formation of spontaneous density-wave patterns in dc driven lattices, Physical Review X 12, 021014 (2022).
  * Sala  _et al._ [2020] P. Sala, T. Rakovszky, R. Verresen, M. Knap, and F. Pollmann, Ergodicity breaking arising from hilbert space fragmentation in dipole-conserving hamiltonians, Phys. Rev. X 10, 011047 (2020).
  * Feldmeier  _et al._ [2020] J. Feldmeier, P. Sala, G. De Tomasi, F. Pollmann, and M. Knap, Anomalous diffusion in dipole- and higher-moment-conserving systems, Phys. Rev. Lett. 125, 245303 (2020).
  * Sala  _et al._ [2022] P. Sala, J. Lehmann, T. Rakovszky, and F. Pollmann, Dynamics in systems with modulated symmetries, Phys. Rev. Lett. 129, 170601 (2022).
  * Lake  _et al._ [2022] E. Lake, M. Hermele, and T. Senthil, Dipolar bose-hubbard model, Phys. Rev. B 106, 064511 (2022).
  * Lake  _et al._ [2023] E. Lake, H.-Y. Lee, J. H. Han, and T. Senthil, Dipole condensates in tilted bose-hubbard chains, Phys. Rev. B 107, 195132 (2023).
  * Zechmann  _et al._ [2023] P. Zechmann, E. Altman, M. Knap, and J. Feldmeier, Fractonic luttinger liquids and supersolids in a constrained bose-hubbard model, Phys. Rev. B 107, 195131 (2023).
  * Han  _et al._ [2023] J. H. Han, E. Lake, and S. Ro, Scaling and localization in kinetically constrained diffusion, arXiv preprint arXiv:2304.03276 (2023).
  * Gromov [2019] A. Gromov, Towards classification of fracton phases: the multipole algebra, Physical Review X 9, 031035 (2019).
  * Iaconis  _et al._ [2021] J. Iaconis, A. Lucas, and R. Nandkishore, Multipole conservation laws and subdiffusion in any dimension, Phys. Rev. E 103, 022142 (2021).
  * Gliozzi  _et al._ [2023] J. Gliozzi, J. May-Mann, T. L. Hughes, and G. De Tomasi, Hierarchical hydrodynamics in long-range multipole-conserving systems, Phys. Rev. B 108, 195106 (2023).
  * Gromov and Radzihovsky [2024] A. Gromov and L. Radzihovsky, Colloquium: Fracton matter, Rev. Mod. Phys. 96, 011001 (2024).
  * Prem  _et al._ [2018] A. Prem, M. Pretko, and R. M. Nandkishore, Emergent phases of fractonic matter, Physical Review B 97, 085116 (2018).
  * You  _et al._ [2020] Y. You, T. Devakul, S. L. Sondhi, and F. J. Burnell, Fractonic chern-simons and bf theories, Phys. Rev. Res. 2, 023249 (2020).
  * You  _et al._ [2021] Y. You, F. J. Burnell, and T. L. Hughes, Multipolar topological field theories: Bridging higher order topological insulators and fractons, Phys. Rev. B 103, 245128 (2021).
  * Ohmori and Shimamura [2023] K. Ohmori and S. Shimamura, Foliated-exotic duality in fractonic BF theories, SciPost Phys. 14, 164 (2023).
  * Han  _et al._ [2024] J. H. Han, E. Lake, H. T. Lam, R. Verresen, and Y. You, Topological quantum chains protected by dipolar and other modulated symmetries, Phys. Rev. B 109, 125121 (2024).
  * Oh  _et al._ [2022a] Y.-T. Oh, J. Kim, E.-G. Moon, and J. H. Han, Rank-2 toric code in two dimensions, Phys. Rev. B 105, 045128 (2022a).
  * Bulmash and Barkeshli [2018] D. Bulmash and M. Barkeshli, Higgs mechanism in higher-rank symmetric u(1) gauge theories, Phys. Rev. B 97, 235112 (2018).
  * Ma  _et al._ [2018] H. Ma, M. Hermele, and X. Chen, Fracton topological order from the higgs and partial-confinement mechanisms of rank-two gauge theory, Phys. Rev. B 98, 035111 (2018).
  * Pace and Wen [2022] S. D. Pace and X.-G. Wen, Position-dependent excitations and uv/ir mixing in the ${\mathbb{z}}_{N}$ rank-2 toric code and its low-energy effective field theory, Phys. Rev. B 106, 045145 (2022).
  * Delfino  _et al._ [2023a] G. Delfino, W. B. Fontana, P. R. S. Gomes, and C. Chamon, Effective fractonic behavior in a two-dimensional exactly solvable spin liquid, SciPost Phys. 14, 002 (2023a).
  * Watanabe  _et al._ [2023] H. Watanabe, M. Cheng, and Y. Fuji, Ground state degeneracy on torus in a family of ZN toric code, Journal of Mathematical Physics 64, 051901 (2023), https://pubs.aip.org/aip/jmp/article-pdf/doi/10.1063/5.0134010/17614211/051901_1_5.0134010.pdf .
  * Delfino  _et al._ [2023b] G. Delfino, C. Chamon, and Y. You, 2d fractons from gauging exponential symmetries (2023b), arXiv:2306.17121 [cond-mat.str-el] .
  * Oh  _et al._ [2022b] Y.-T. Oh, J. Kim, and J. H. Han, Effective field theory of dipolar braiding statistics in two dimensions, Phys. Rev. B 106, 155150 (2022b).
  * Huang [2023] X. Huang, A Chern-Simons theory for dipole symmetry, SciPost Phys. 15, 153 (2023).
  * Ebisu  _et al._ [2024] H. Ebisu, M. Honda, and T. Nakanishi, Foliated bf theories and multipole symmetries (2024), arXiv:2310.06701 [cond-mat.str-el] .
  * Note [1] The author thanks Xiaoyang Huang for pointing out the gauge invariance of this combination.
  * Oh  _et al._ [2023] Y.-T. Oh, S. D. Pace, J. H. Han, Y. You, and H.-Y. Lee, Aspects of ${\mathbb{z}}_{N}$ rank-2 gauge theory in $(2+1)$ dimensions: Construction schemes, holonomies, and sublattice one-form symmetries, Phys. Rev. B 107, 155151 (2023).


# Uniform solvability for families of linear systems on time scales

Sergey Kryzhevich  Institute of Applied Mathematics, Faculty of Applied
Physics and Mathematics, Gdańsk University of Technology, 80-233 Gdańsk,
Poland and BioTechMed Center, Gdańsk University of Technology, 80-233 Gdańsk,
Poland<EMAIL_ADDRESS>
(Date: March 17, 2022)

###### Abstract.

We give explicit criteria of solvability for families of linear systems on
time scales. We introduce a new method of embedding a time scale into a non-
autonomous system of ODEs. This will be the first step to implementing the
structural stability result obtained by one of the co-authors together with V.
A. Pliss to time scale dynamics.

###### Key words and phrases:

time scale dynamics, structural stability, solvability, linear systems,
hyperbolicity

The work of the first author was supported by Gdańsk University of Technology
by the DEC 14/2021/IDUB/I.1 grant under the Nobelium - ’Excellence Initiative
- Research University’ program.

##  1\. Introduction

Non-autonomous systems of ordinary differential equations are usually regarded
as more sophisticated compared to classical flows engendered by autonomous
systems of ordinary differential equations. The principal trouble is that
periodic (and even recurrent) solutions do not play an important role in the
dynamics of non-autonomous systems unless the right-hand side of the studied
system is periodic.

For flows, engendered by autonomous systems of ODEs, the structural stability,
and $\Omega$ – stability theorems were obtained by Robinson [12] and Mañé [6].
Both of them are related to the so-called Axiom A which is also the core
assumption for various further results in Stability Theory.

For Axiom-A flows, the non-wandering set can be represented as a disjoint
union of a finite number of closed transitive invariant subsets. For any of
these subsets, there is a neighborhood where the behavior of solutions is
hyperbolic (dimensions of stable and unstable manifolds can, however, vary).

Moreover, all solutions of Axiom-A systems spend a finite (and uniformly
bounded) amount of time out of the union of those neighborhoods. In other
words, one can say that linear approximations are hyperbolic on big intervals
of time, depending on a particular solution.

The above conditions can be reformulated for non-autonomous systems, see [5]
and [8]-[11] where sufficient conditions for structural stability were
obtained. However, it is still unclear if the mentioned conditions are close
to necessary ones.

We are going to translate these results to the language of time scale
dynamics. This is a relatively new area in dynamical systems, first introduced
by [1]. Firstly, this can be regarded as a combination of discrete and
continuous dynamics and, secondly, as a generalization of numerical methods
with non-uniform steps. A survey on the theory of such systems including
Stability Theory is given in [2] and the relatively recent book [7], see also
references therein. However, no global structural stability conditions for
generic time scale systems have been established yet. It should be highlighted
that we do not just copy-paste the result from the ODE theory. Some additional
conditions respecting the geometry of the time scale have to be added. For
instance, we consider a particular case of a non-syndetic time scale (i.e.
that with an unbounded graininess function).

The main objective of this paper is to prove analogs of mentioned results for
time scale systems. For this purpose, we introduce a renormalization of time
scales and, re-define the concept of hyperbolicity. for time scale systems.
Moreover, we show how linear systems on time scales can be reduced to ordinary
differential equations.

For the sake of convenience, we deal with a dynamical system on a Euclidean
space (unlike the classical case of ordinary differential equations, in the
case of non-periodic time scales there is no principle difference between
autonomous and non-autonomous systems).

The paper is organized as follows.

In Section 2 we recall the results of the classical structural stability
theory. Next, in Section 3, we recall the basic concepts of time scale
dynamics. Section 4 is the core of the paper. There, we discuss how a linear
system on a time scale can be represented as a system of ordinary differential
equations (and also, how the concept of hyperbolicity can be translated into
the language of time scale dynamics). Finally, in Section 5, we formulate the
structural stability conjecture which will be the main objective of further
research.

In a nutshell, we follow the main idea of the quoted papers by V. A. Pliss.
First of all, we obtain some conditions on uniform solvability for families of
linear systems. Thereafter, we are going to develop Perron’s approximations
for time scale systems to be able to find solutions for nonlinear systems.

##  2\. Structural stability for systems of ODEs.

Let $V$ be a $C^{1}$ – smooth vector field on a $C^{1}$ – smooth Riemannian
manifold $M$, and $\Omega_{V}$ be the non-wandering set of the considered
autonomous system of ordinary differential equations

$\dot{x}=V(x).$

We recall the famous Axiom $A^{\prime}$ for flows/vector fields that claims
that

  1. (1)

$\Omega_{V}$ can be represented as a union of two disjoint closed sets
$\Omega^{1}_{V}$ and $\Omega^{2}_{V}$ where $\Omega^{1}_{V}$ consists of
finitely many isolated fixed points and $\Omega^{2}_{V}$ does not contain any
stationary point;

  2. (2)

the set $\Omega_{V}$ is hyperbolic.

  3. (3)

periodic orbits are dense in $\Omega^{2}_{V}$.

Then the following statement is well-known in Hyperbolic Theory.

Smale’s spectral decomposition theorem. Let the vector field $V$ satisfy Axiom
$A^{\prime}$. Then the non-wandering set $\Omega_{V}$ admits the unique
representation:

$\Omega_{f}=\Omega_{1}\bigcup\Omega_{2}\bigcup\ldots\bigcup\Omega_{N}$

with all components being closed, disjoint, invariant, and transitive.

Structural stability requires an additional condition:

Geometric strong transversality condition. An $A^{\prime}$ flow, defined by a
vector field $V$, satisfies the strong transversality condition if for any
$p,q\in\Omega_{V}$ the manifolds $W^{s}(p)$ and $W^{u}(q)$ intersect
transversally.

Now let us recall the famous result by C. Robinson [12].

Structural Stability Theorem. _For $C^{1}$ – smooth vector fields, Axiom
$A^{\prime}$ and Geometric Strong Transversality Condition imply structural
stability._

In this section we recall a ’non-autonomous’ analog of the above result. Now
let us consider a non-autonomous system

(2.1) $\dot{x}=V(t,x),\qquad t\in{\mathbb{R}},\quad x\in{\mathbb{R}}^{n};$

where

$|V(t,x)|\leq M,\qquad|D_{x}V(t,x)|\leq M.$

Let $x(t,t_{0},x_{0})$ be a solution of Eq. (2.1) corresponding to initial
conditions $x(t_{0})=x_{0}$.

Let us also consider a perturbed system:

(2.2)
$\dot{x}=V(t,x)+Y(t,x),\qquad|Y(t,x)|\leq\delta,\qquad|D_{x}Y(t,x)|\leq\delta.$

with $y(t,t_{0},x_{0})$ being a solution corresponding to the same initial
conditions.

Let us recall the main definitions of the Hyperbolic Theory for non-autonomous
systems of ordinary differential equations.

Consider a linear system

(2.3) $\dot{x}=A(t)x,\qquad t\in[t_{-},t_{+}],$

Let $\Phi(t,t_{0})=\Phi(t)\Phi^{-1}(t_{0})$ be the Cauchy matrix of that
system.

We say that system (2.3) is hyperbolic on the segment $[t_{-},t_{+}]$ if for
any $t\in[t_{-},t_{+}]$ there exists a decomposition

${\mathbb{R}}^{n}=E^{s}(t)\oplus
E^{u}(t):\qquad\Phi(t,\tau)E^{s,u}(\tau)=E^{s,u}(t)$

such that

$\begin{array}[]{c}|\Phi(t,\tau)x_{0}|\leq
a\exp(-\lambda(t-\tau))|x_{0}|,\qquad t\geq\tau,x_{0}\in E^{s}(\tau);\\\
|\Phi(t,\tau)x_{0}|\leq a\exp(\lambda(t-\tau))|x_{0}|,\qquad
t\leq\tau,x_{0}\in E^{u}(\tau).\end{array}$

for some $a>0$ and $\lambda>0.$

Now we consider the set of linear systems

(2.4) $\dot{x}=A(t,x_{0})x,\qquad x_{0}\in{\mathbb{R}}^{n},\qquad
A(t,x_{0})=\dfrac{\partial V(t,x(t,0,x_{0}))}{\partial x}$

We make the following assumptions. Let $a_{0}>0$, $\lambda>0$, $\alpha>0$, be
such that for any $x_{0}\in{\mathbb{R}}^{n}$:

  1. (1)

There exists a number $a\in(0,a_{0})$ and instants of time

$-\infty=\tau_{0}<\tau_{1}<\ldots<\tau_{s}<\tau_{s+1}=\infty,\qquad 0<s<n$

such that on any of the segments $[\tau_{j},\tau_{j+1}]$, $j=0,\dots,s$ system
(2.4) is hyperbolic with constants $a$ and $\lambda$. Let $E^{s}_{j}$ and
$E^{u}_{j}$ be the corresponding stable and unstable spaces.

  2. (2)

$\dim E_{j}^{s}(\tau_{j+1})<\dim E_{j+1}^{s}(\tau_{j+1}),\qquad j=0,\ldots
s-1.$

  3. (3)

Spaces $E_{j}^{u}(\tau_{j+1})$ and $E_{j+1}^{s}(\tau_{j+1})$ intersect
transversally and angles between them satisfy inequalities $\sphericalangle
E_{j}^{u}(\tau_{j+1}),E_{j+1}^{s}(\tau_{j+1})\geq\alpha$.

It was proved [5] that such ’uniform hyperbolicity’ of family Eq. (2.4) with
appropriate implies structural stability for nonlinear system (2.1).

Let the function $T(a,\lambda,\alpha)$ satisfy inequalities

$36a^{2}\exp\left(\dfrac{-\lambda
T}{3}\right)<\dfrac{\alpha}{8}\sin\dfrac{\alpha}{4},\qquad
3a\left(\dfrac{2}{\sin(\alpha/2)}+1\right)\exp\left(\dfrac{-\lambda
T}{3}\right)<1.$

Theorem 1 [5]._Let the system ( 2.4) satisfy the above conditions I-III and
for any $j$ the conditions_

$\tau_{j+1}-\tau_{j}>T(a,\lambda,\alpha),\qquad j=0,\ldots s.$

_Then for any $\varepsilon>0$ there exists a $\delta>0$ such that if condition
(2.2) is satisfied, there exists a homeomorphism $\varphi$ of the space
${\mathbb{R}}^{n}$ such that_

$|x(t,0,x_{0})-y(t,0,\varphi(x_{0}))|<\varepsilon$

_for any $x_{0}\in{\mathbb{R}}^{n}$._

Our main objective (not limited to that paper) is to translate the mentioned
result into the language of time scale dynamics.

##  3\. Timescale dynamics

In this section, we recall some basic definitions of time scale dynamics.

Definition 1. We say that a subset ${\mathbb{T}}\subset{\mathbb{R}}$ is a
_timescale_ if it is closed and

$\inf{\mathbb{T}}=-\infty,\qquad\sup{\mathbb{T}}=+\infty$

We always assume for convenience that $0\in{\mathbb{T}}$.

Given a time scale, we consider the _forward jump operator_

$\sigma:{\mathbb{T}}\to{\mathbb{T}}:\sigma(t)=\inf\\{s\in{\mathbb{T}}:s>t\\}$

and consider the so-called _graininess function_ , see Fig. 1:

$\mu(t):=\sigma(t)-t.$

We say that a point $t\in{\mathbb{T}}$ is _right-dense_ if $\mu(t)=0$ and
_left-dense_ is $\sigma(s)=t$ implies $s=t$. We say that a time-scale is
_syndetic_ if $\sup_{t\in{\mathbb{T}}}\mu(t)<+\infty$.

Figure 1. A sample of a time scale.

The classic concepts of Analysis may also be defined for time scales.

Definition 2. A function $f:{\mathbb{T}}\mapsto{\mathbb{R}}$ is called _rd-
continuous_ provided it is continuous at right-dense points in ${\mathbb{T}}$
and finite left-sided limits exist at left-dense points in ${\mathbb{T}}$.
Denote the class of rd-continuous functions by
${\mathcal{C}}_{rd}={\mathcal{C}}_{rd}({\mathbb{T}},{\mathbb{R}})$.

Definition 3. The function $f:{\mathbb{T}}\mapsto{\mathbb{R}}$ is called
$\Delta$-differentiable at a point $t\in{\mathbb{T}}$ if there exists
$\gamma\in{\mathbb{R}}$ such that for any $\varepsilon>0$ there exists a
neighborhood $W$ of $t$ satisfying

$|[f(\sigma(t))-f(s)]-\gamma[\sigma(t)-s]|\leq\varepsilon|\sigma(t)-s|$

for all $s\in W$. In this case we write $f^{\Delta}(t)=\gamma.$

For example, when ${\mathbb{T}}={\mathbb{R}}$, $f^{\Delta}(t)=\dot{f}(t)$.
When ${\mathbb{T}}={\mathbb{Z}}$, $f^{\Delta}(t)$ is the standard forward
difference operator $f(n+1)-f(n)$.

In time scale dynamics, there is a dual concept of the so-called $\nabla$ \-
derivative, corresponding to the left jump operator. We do not consider such a
derivative in this paper. Probably, this will be done in the future.

Also, we define integrals over time scales.

Definition 4. If $F^{\Delta}(t)=f(t)$, $t\in{\mathbb{T}}$, then $F$ is a
_$\Delta$ \- antiderivative_ of $f$, and the Cauchy $\Delta$ \- integral is
given by formula

$\int_{a}^{b}f(t)\,\Delta t=F(b)-F(a),\quad\mbox{for all}\quad
a,b\in{\mathbb{T}}.$

We recall a simple formula which relates integrals over time-scale with
classic ones (this formula can also be used as a definition of integrals and
derivatives over time scales)

$\int_{a}^{b}f(t)\,\Delta
t=\int\limits_{[a,b]\cap{\mathbb{T}}}f(t)d\mathbf{m}(t)+\sum\limits_{\mu(t_{i})>0}f(t_{i})\mu(t_{i}).$

Definition 5. A function $p:{\mathbb{T}}\mapsto{\mathbb{R}}$ is called
_regressive_ provided that

$1+\mu(t)p(t)\neq 0$

for all $t\in{\mathbb{T}}$ and _positively regressive_ if $1+\mu(t)p(t)>0$ for
all $t\in{\mathbb{T}}$. A matrix mapping ${\mathcal{A}}:{\mathbb{T}}\mapsto
M^{n}({\mathbb{R}})$ is called _regressive_ if for each $t\in{\mathbb{T}}$ an
$n\times n$ matrix $E_{n}+\mu(t){\mathcal{A}}(t)$ is invertible, and
_uniformly regressive_ if in addition the matrix function
$(E_{n}+\mu(t){\mathcal{A}}(t))^{-1}$ is bounded.

Regressivity is equivalent to the backward uniqueness of solutions of the
system $x^{\Delta}=A(t)x$.

We give a counterexample of a scalar equation, which is not regressive:

$x^{\Delta}=-x,\quad{\mathbb{T}}={\mathbb{N}}.$

This equation has only one solution, defined on the whole time scale
${\mathbb{T}}$ that is $x\equiv 0$.

For a linear system

(3.1) $x^{\Delta}=A(t)x,\qquad t\in{\mathbb{T}},$

stability and asymptotic stability are defined similarly to ODEs.

The following result is an analog of that of the theory of linear ODEs.

Theorem (Choi and DaCunha), [3], [4]. Let an $rd$ \- continuous matrix
function $A$ be regressive. Linear system (3.1) is stable if and only if all
its solutions are bounded on ${\mathbb{T}}\bigcap[0,+\infty)$. It is uniformly
stable if and only if there exists a positive constant $\gamma$, such that
$|\Phi_{A}(t,t_{0})|\leq\gamma$, $\forall t_{0},t\in{\mathbb{T}},t\geq t_{0}$.

Definition 6. For $p\in{\mathcal{R}}$, we define the generalized exponential
function $e_{p}(t,s)$ by

$e_{p}(t,s)=\exp\left(\int_{s}^{t}\xi_{\mu(\tau)}p(\tau)\,\Delta\tau\right).$

where $\xi_{h}$ is the cylinder transformation given by formula
$\xi_{h}(z)=\log(1+zh)/h$ if $h\neq 0$, $\xi_{h}(z)=z$ if $h=0$.

This is the solution of equation $x^{\Delta}=p(t)x$ with initial conditions
$x(s)=1$.

Other techniques, e.g. Grönwall–Bellmann Inequality, Lyapunov exponents theory
can be developed for systems on time scales.

However, stability, Lyapunov exponents, and even uniqueness of solutions
depend significantly on a time scale. This is a serious obstacle to applying
the classic Hyperbolic Theory for time scale dynamics. For example, if we
consider ${\mathbb{T}}=\\{2^{n}:n\in{\mathbb{N}}\\}$, for any bounded matrix
$A(t)$ all solutions have non-positive Lyapunov exponents.

##  4\. Hyperbolicity for time scale systems

1\. Renormalization

Let ${\mathbf{m}}$ be the Lebesgue measure on the real line. Recall that
$\mu(t)$ is the graininess function of the time scale. We fix

$t_{0}=\inf\\{{\mathbb{T}}\bigcap[0,\infty)\\}$

and define the function $s:{\mathbb{R}}\to{\mathbb{R}}$ as follows:

  1. (1)

$s(t)={\mathbf{m}}([t_{0},t])+\sum_{\tau\in[t_{0},t):\mu(\tau)>0}\ln(1+\mu(\tau))$
if $t\in{\mathbb{T}},t\geq t_{0}$.

  2. (2)

$s(t)=s(t_{1})+\ln(1+t-t_{1})$ if $t\notin{\mathbb{T}},t\geq t_{0}$. Here
$t_{1}=\max({\mathbb{T}}\bigcap(-\infty,t])$.

  3. (3)

$s(t)=-{\mathbf{m}}([t_{0},t])-\sum_{\tau\in[t_{0},t):\mu(\tau)>0}\ln(1+\mu(\tau))$
if $t\in{\mathbb{T}},t<t_{0}$.

  4. (4)

$s(t)=s(t_{1})-\ln(1+t_{2}-t)$ if $t\notin{\mathbb{T}},t\geq t_{0}$. Here
$t_{2}=\min({\mathbb{T}}\bigcap[t,\infty))$.

Evidently $s(t_{0})=0$, $s(t)$ is strictly monotonous, continuous, unbounded
in both directions and $|s(t)|\leq|t-t_{0}|$ for any $t\in{\mathbb{R}}$. Given
a time scale $\mathbb{T}$ we introduce the renormalized time scale
${\mathbb{S}}:=s(\mathbb{T})$, Fig. 2.

Figure 2. A sample of renormalization procedure.

2\. Equivalent linear system. For any system

(4.1) $x^{\Delta}=A(t)x+f(t)$

on the time scale $\mathbb{T}$ with $A$ and $f$ being bounded uniformly
$rd$-continuous there exists a system

(4.2) $\dot{x}=B(t)x+g(t)$

such that $\Phi_{A}(t,\tau)=\Phi_{B}(s(t),s(\tau))$ for all
$t,s\in{\mathbb{T}}$. Here $\Phi_{A}$ and $\Phi_{B}$ are fundamental matrices
of systems (4.1) and (4.2), respectively. Besides, given the right-hand side
$f$ we construct the function $g$ so that for any solution $\varphi(t)$ of
system (4.1) there exists a solution $\psi(t)$ of Eq. (4.2) such that
$\varphi(t)=\psi(s(t))$ for any $t\in{\mathbb{T}}$.

Here we may assume that $B$ is constant on all connected components of the
completion of ${\mathbb{T}}$.

Definition 7. Let $(t_{1},t_{2})$ be an interval such that
$t_{1,2}\in{\mathbb{T}}$, $(t_{1},t_{2})\bigcap{\mathbb{T}}=\emptyset$.

Lemma 1. _The matrix $B$ and the vector function $f$ can be found by the
following rule._

  1. (1)

$A(t)=B(s(t))$_,_ $f(t)=g(s(t))$ for all $t\in{\mathbb{T}}$.

  2. (2)

_Let_ $t\notin{\mathbb{T}}$. We consider
$t_{0}=\sup_{\tau\in{\mathbb{T}}:\tau\leq t}$ and define

(4.3)
$\begin{array}[]{c}B(s(t))=\dfrac{{\mathrm{Ln}}\,(E+\mu(t_{0})A(t_{0}))}{\ln(1+\mu(t_{0}))},\\\\[5.0pt]
g(s(t))=\dfrac{A^{-1}(t_{0}){\mathrm{Ln}}\,(E+\mu(t_{0})A(t_{0}))}{1+\mu(t_{0})}f(t)\end{array}$

Remark 1. Here $\mathrm{Ln}$ stands for the matrix logarithm. This function is
multi-valued and it could be imaginary even if the matrix $A$ is real.
However, it can be canonically defined as a real value provided
$\sup_{t}\mu(t)\|A(t)\|<1$ where the matrix norm is taken in an appropriate
basis. For this we can use the classic Taylor decomposition

$\mathrm{Ln}\,(E+\mu A)=\sum_{k=1}^{\infty}(-1)^{k+1}\mu^{k}A^{k}.$

Meanwhile, the second formula of (4.3) can be correctly defined even in the
case when $A(t_{0})$ is non-invertible. This follows from the fact that the
function $\ln(1+\mu x)/x$ is holomorphic in a neighborhood of zero.

The latter expression, being well-defined for small values of $\mu$ and a
fixed matrix $A$ can be extended for all positive values of $\mu$ unless the
matrix $A$ has negative eigenvalues.

If matrix $A$ has negative eigenvalues we still link a negative value $-\mu$
with zero going around the ’problematic’ values $\lambda_{j}$ if any from
above. Here $\lambda_{j}$ are negative eigenvalues of $A$ (Fig. 3) and
continue the logarithm along that contour.

Figure 3. The appropriate contour for the function ${\mathrm{Ln}}(E+\mu A)$ in
case of negative eigenvalues.

Proof of Lemma 1.

1. First of all, we consider the case $f(t)\equiv 0$ that is linear homogenuous system. Fix a $t_{0}\in{\mathbb{T}}$. Consider the fundamental matrix $\Phi(t)$ of the time scale system

$x^{\Delta}=A(t)x$

with initial conditions $\Phi(t_{0})=E$. Then

(4.4)
$\begin{array}[]{l}\Phi(t)=E+\int_{t_{0}}^{t}A(\tau)\Psi(\tau)\,\Delta\,\tau=\\\
E+\int\limits_{[t_{0},s(t)]\cap{\mathbb{T}}}A(\tau)\Phi(\tau)\,d\,\tau+\sum\limits_{\tau\in[t_{0},t]\cap{\mathbb{T}}:\mu(\tau)>0}A(\tau)\Phi(\tau)\mu(\tau).\end{array}$

On the other hand, for any $\tau\in{\mathbb{T}}$ with $\mu(\tau)>0$, we have

(4.5)
$\begin{array}[]{l}\Phi(\sigma(\tau))=(E+\mu(\tau)A(\tau))\Phi(\tau)=\exp(B(s(\tau))(\ln(1+\mu(\tau)))\Phi(\tau)=\\\
\Phi(\tau)+\int_{0}^{\ln(1+\mu(\tau))}B(s(\tau))\Phi(\tau+\theta)\,d\theta\end{array}$

And now, we introduce $\Psi(t)$ as a fundamental matrix of the system

$\dot{x}=B(t)x$

such that $\Psi(t_{0})=E$.

Obviously, this matrix satisfies the equation

(4.6) $\Psi(s)=E+\int_{t_{0}}^{s}A(\tau)\Psi(\tau)\,d\,\tau$

Given a $t\in{\mathbb{T}}$, we obtain from (4.4), (4.5) and (4.6) that
matrices $\Phi(t)$ and $\Psi(s(t))$ coincide being solutions of the same
integral equation (4.4).

Now, we apply a similar approach to a non-homogeneous system.

Fix a $\tau$: $\mu(\tau)>0$. Let $x(t)$ be a solution of Eq, (4.2). By
definition,

$x(\sigma(\tau))=(E+A(\tau)\mu(\tau))x(\tau))+f(\tau)=\exp(\ln(1+\mu(\tau))B(s(\tau)))x(\tau)+f(\tau)\mu(\tau).$

On the other hand, in order to obtain a solution of Eq. (4.2) we must have

$x(\sigma(\tau))=\exp(\ln(1+\mu(\tau))B(s(\tau)))x(\tau)+\int_{0}^{\ln(1+\mu(\tau))}\exp(B(\mu(\tau)-\theta))g(s(\tau))\,d\theta$

which implies the second equality of Eq. (4.3). $\square$

Later on, considering linear systems (4.1) on time scales we assume the
following.

Condition I. __

  1. (1)

_Matrices_ $A(t)$ and $(E+\mu(t)A(t))^{-1}$ are continuous and bounded on
$\mathbb{T}$.

  2. (2)

_Either the time scale is syndetic or_ $A(t)$ is invertible for all
$t\in{\mathbb{T}}$ and $\limsup_{t\in{\mathbb{T}}}\|A^{-1}(t)\|<+\infty$.

Remark 2. The matrix

$A^{-1}(t_{0}){\mathrm{Ln}}\,(E+\mu(t_{0})A(t_{0}))$

can be well-defined for non-invertible matrices $A$. If $A$ has zero
eigenvalues only, this matrix can be defined as

$\sum_{k=0}^{\infty}(-1)^{k}A^{k}\mu^{k+1}/(k+1).$

The sum is finite which is evident for a Jordan block. For other non-
invertible matrices, the above expression may be calculated block-wise. If $A$
and $f$ are continuous on $\mathbb{T}$ then $B$ and $g$ are continuous on
$\mathbb{S}$ (however, they may be discontinuous on all the axis
$\mathbb{R}$).

Lemma 2. _Let Condition I be satisfied and $f(t)$ be bounded and continuous.
Then the matrix function $B(s)$ and the vector function $g(s)$ defined by Eq.
(4.3) are bounded and piece-wise continuous on the real axis._

Proof. First of all, let us prove this statement for $B(s)$. Assume that there
is a sequence $t_{k}$ such that

(4.7) $\|B(s_{k})\|\to\infty$

where $s_{k}=s(t_{k})$. Let $A_{k}=A(t_{k})$, $\mu_{k}=\mu(t_{k})$. We may
assume that the matrices $A_{k}$ converge to $A_{0}$, and
$\mu_{k}\to\mu_{0}\in[0,\infty]$.

If $\mu_{0}\notin\\{0,\infty\\}$ we obtain that the matrix $E+\mu_{0}A_{0}$ is
degenerate that contradicts to our assumptions.

Let $\mu_{0}=0$. We easily obtain that $\lim B(s_{k})=A_{0}$ that contradicts
to (4.7).

Let $\mu_{0}=+\infty$. Then

$\begin{array}[]{c}\lim\dfrac{{\mathrm{Ln}}(E+\mu_{k}A_{k})}{\ln(1+\mu_{k})}=\lim\dfrac{{\mathrm{Ln}}[(\mu_{k}E)(E\mu_{k}^{-1}+A_{k})]}{\ln(1+\mu(t_{k}))}=\\\
\lim\dfrac{{\mathrm{Ln}}[(\mu_{k}E)(E\mu_{k}^{-1}+A_{k})]}{\ln(1+\mu_{k})}=\lim\dfrac{({\mathrm{Ln}}\mu_{k}E)+{\mathrm{Ln}}(E\mu_{k}^{-1}+A_{k})]}{\ln(1+\mu_{k})}.\end{array}$

We consider two cases. If $A_{0}$ is non-degenerate, the above limit equals
$E$. In the opposite case, we appeal to the second assumption of Condition I.
$\square$

Remark 3. The last statement justifies the renormalization procedure,
introduced above. Indeed, if we leave the time scale ’as is’, all bounded
systems become non-hyperbolic. However, non-homogeneous systems still might
have bounded solutions.

Definition 8. We say that a system (3.1) is _hyperbolic_ on an interval
$(t_{1},t_{2})$ which can be unbounded, if the equivalent system

$\dot{x}=B(t)x$

is hyperbolic on $(s(t_{1}),s(t_{2})$.

Now, let us consider the continuous family of matrices on a time scale
$\mathbb{T}$:

$A(t,\nu),\qquad\nu\in{\mathbb{R}}^{n}.$

We consider a non-homogenuous system

(4.8) $x^{\Delta}=A(t,\nu)x+f(t)$

with the function $f$ being uniformly bounded $|f(t)|\leq 1$ and the
corresponding family of homogenuous systems

(4.9) $x^{\Delta}=A(t,\nu)x$

which is assumed to be uniformly hyperbolic (the constants $a$ and $\lambda$
may be selected the same) on families of segments.

We assume that the above Condition I is satisfied uniformly to the parameter
$\nu$. However, we would like some other conditions to be fulfilled.

Condition II. There exist positive constants $a$ and $\lambda$ such that for
any $\nu\in{\mathbb{R}}^{m}$ there exist

$-\infty=\tau_{0}<\tau_{1}<\ldots<\tau_{k}=+\infty$

such that for any $\nu$ the system (4.9) is hyperbolic on all segments
$[\tau_{j-1},\tau_{j}]$, $j=1,\ldots,k$. Here the number $k$ and all values
$\tau_{j}$ may depend on the parameter $\nu$. Let $E^{s}_{j}(t,\nu)$ and
$E^{u}_{j}(t,\nu)$ be the corresponding stable and unstable spaces (any, if
the segment is finite and multiple choice is possible).

Condition III. $\dim E^{s}_{j}(\tau_{j},\nu)<\dim E^{s}_{j+1}(\tau_{j},\nu)$
for any $j$ and $\nu$.

Condition IV. There exists a value $\alpha>0$ that does not depend on $\nu$
and such that
$\angle(E^{u}_{j}(\tau_{j},\nu),E^{s}_{j+1}(\tau_{j},\nu))>\alpha$.

It follows from Lemma 2 that for non-syndetic time scales conditions I-IV
cannot be true unless all systems are uniformly unstable hyperbolic
($E^{u}={\mathbb{R}}^{n}$ and the constants can be taken the same) over the
real line.

Theorem 2._Let the homogeneous system ( 4.9) satisfy conditions I-IV
uniformly. Then there exists a value $K>0$ such that for any right hand side
$f$ with $|f(t)|\leq 1$ there exists a solution $\varphi(t,\nu)$ of Eq. (4.8)
such that $|\varphi(t,\nu)|\leq K$ for any $\nu$. Moreover, increasing the
value $K$, we may obtain a linear operator ${\mathcal{L}}_{\nu}$ such that
$\varphi(\cdot,\nu)={\mathcal{L}_{\nu}}f$ and, for any fixed value $t$ the
function $\varphi(t,\nu)$ continuously depends on $\nu$._

Proof. Lemma 1 demonstrates that there is a continuous embedding of continuous
functions on a time scale to ${\mathbb{L}}^{\infty}({\mathbb{R}})$. Due to
uniform hyperbolicity of system (4.9), there exists uniformly bounded systems
of linear operators that map right-hand sides of
${\mathbb{L}}^{\infty}({\mathbb{R}})$ to bounded solutions of corresponding
linear non-homogeneous systems of ordinary differential equations.

Then, the reduction of these solutions to the time scale gives us a family of
bounded solutions on the time scale. $\square$

The proof of the above theorem follows from the reduction, introduced above
and, also, from the similar result for real line obtained by V. A. Pliss [11].

##  5\. Structural stability for time scale systems

Now we proceed to formulate a conjecture on structural stability for the time
scale dynamics.

Consider a system

(5.1) $x^{\Delta}\vskip 12.0pt plus 4.0pt minus 4.0pt=V(t,x),\qquad
t\in{\mathbb{T}}.$

We suppose that

(5.2) $|V(t,x)|\vskip 12.0pt plus 4.0pt minus 4.0pt<M\qquad\dfrac{\partial
V}{\partial x}(t,x)<M$

for some $M>0$ and that the above partial derivative is uniformly continuous.
Suppose that all solutions of the considered system satisfy the forward
uniqueness condition which can be provided by $rd$ – continuity condition. Let
$\varphi(t,x_{0})$ be the solution of (11) with initial conditions
$x(0)=x_{0}$.

Consider the perturbed system

$x^{\Delta}=V(t,x)+Y(t,x);$

with

$|Y(t,x)|\leq\delta,\qquad|D_{x}Y(t,x)|\leq\delta.$

defined on a time scale ${\mathbb{T}}_{1}$ such that

$d_{H}({\mathbb{T}},{\mathbb{T}}_{1})<\delta.$

Here $d_{H}$ is the Hausdorff distance between two closed subsets of
$\mathbb{R}$ (of course, this distance may be infinite). Take the
linearization of system (5.1) in a neighborhood of the solution
$\varphi(t,x_{0})$

$u^{\Delta}=A(t,x_{0})u$

where

$A(t,x_{0})=\dfrac{\partial V}{\partial x}(t,\varphi(t,x_{0})).$

Conjecture._Let system ( 5.2) satisfy the above conditions I-IV and for any
$j$ the conditions_

$\tau_{j+1}-\tau_{j}>T(a,\lambda,\alpha),\qquad j=0,\ldots s.$

_Then for any $\varepsilon>0$ there exists a $\delta>0$ such that if condition
(16) is satisfied, there exists a homeomorphism $\varphi$ of the space
${\mathbb{R}}^{n}$ such that for any $x_{0}\in{\mathbb{R}}^{n}$_

$|x(t,0,x_{0})-y(t,0,\varphi(x_{0}))|<\varepsilon.$

We hope that the results of this paper will be a keystone that allows us to
prove the above conjecture.

## References

  * [1] B. Aulbach, S. Hilger, _Linear Dynamic Processes with Inhomogenous Time Scale_ , In Nonlinear Dynamics and Quantum Dynamical Systems (Gaussig, 1990), volume 59 of Math. Res., pages 9–20. Akademie Verlag, Berlin, 1990.
  * [2] M. Bohner, A. Peterson, _Advances in Dynamic Equations on Time Scales_. Birkhäuser Boston Inc., Boston, MA, 2003.
  * [3] S. K. Choi, D. M. Im, N. Koo, _Stability of Linear Dynamic Systems on Time Scales_ , Advances in Difference Equations, Article ID 670203 2008 (2008), 1–12.
  * [4] J. J. DaCunha, _Stability for Time Varying Linear Dynamic Systems on Time Scales_ , J. Comput. Appl. Math., 176:2 (2005): 381–410.
  * [5] S. Kryzhevich and V. Pliss, _Structural stability of nonautonomous systems_ , Differential equations, 39 (2003), no. 10, 1395 – 1403.
  * [6] R. Mañé, _A proof of the $C^{1}$ stability conjecture_, Publications Mathématiques de l’IHÉS, Tome 66 (1987), pp. 161-210.
  * [7] A. A Martynyuk, _Stability Theory for Dynamic Equations on Time Scales_ , Birkhäuser, 2016.
  * [8] V. A. Pliss, _Uniformly bounded solutions of linear systems of differential equations_ , Differents. Uravn., 1977, vol. 13, no. 5, pp. 883–891.
  * [9] V. A. Pliss, _Sets of linear systems with uniformly bounded solutions_ , Differents. Uravn., 1980, vol. 16, no. 9, pp. 1599–1616.
  * [10] V. A. Pliss, _On stability of an arbitrary system with respect to $C^{1}$-small perturbations_, Differents. Uravn., 1980, vol. 16, no. 10, pp. 1891–1892.
  * [11] V. A. Pliss, _Relationship between various conditions of structural stability_ , Differents. Uravn., 1981, vol. 17, no. 5, pp. 828–835.
  * [12] C. Robinson, _Structural stability of $C^{1}$ flows_, Lecture Notes in Math., volume 468, Springer-Verlag, 262-277, 1975.