multi-site flip terms, one may use values of these coefficients to track down
how multi-site flip terms are renormalized to a very small magnitude.

Recall that at each step of the RG, we select a local term with the largest
coefficient, reduce number of degrees of freedom by projecting the system onto
the low-energy subspace of the local term, and derive the couplings in the
effective Hamiltonian using perturbation theory. We identified six different
types of decimations depending on which local term is chosen for the RG
transformation in the main text: Ferromagnetic/antiferromagnetic bond
decimations, single/double-site decimations, and double/triple bond
decimations. In the main text, we did not give full details of how the
couplings in the effective Hamiltonian change after each RG step.

Here we explicitly write down and derive how the parameters
$\\{m^{\prime}_{i},h^{\prime}_{i},J^{\prime}_{i},t^{\prime}_{i},s^{\prime}_{i}\\}$
that determine effective couplings of the Hamiltonian after each RG step are
given in terms of the parameters $\\{m_{i},h_{i},J_{i},t_{i},s_{i}\\}$ before
the RG step of our interest. In each of the six subsections, we specify and
justify the transformation rules for six different types of decimations. These
rules allow one to implement the strong disorder RG iteratively on a computer
and obtain the results presented in the main text.

Before diving into details, we sketch here the general protocol for
determining effective couplings of the Hamiltonian after a given
transformation:

  * •

We incorporate all terms generated under first and second-order degenerate
perturbation theory (for all bond decimations except for the case covered in
Sec. B.5.1) and first-order perturbation theory (for site decimations and the
bond decimations for the case in Sec. B.5.1). Incorporating these results
suffices to provide leading order contributions on how $h_{i}$’s and $J_{i}$’s
transform, and in some cases $t_{i}$’s as well.

  * •

To track couplings between Majorana fermions with coefficients $t_{i}$ and
$s_{i}$, we incorporate some higher-order perturbation theory results compared
to those specified in the previous bullet point.

The coupling $t_{i}$ is set to zero when $m_{i-1}=m_{i}=1$: In this case, none
of the two bonds involved in the $-t_{i}T_{i}$ term is strongly
antiferromagnetic. One expects that because $t_{i}$ is generated from higher-
order perturbation theory, it will be much smaller than the Ising couplings
$J_{i}$ and flip terms $h_{i}$’s generated from the second-order perturbation
theory, and that its effect is presumably negligible. However, within a spin
cluster linked by strongly antiferromagnetic $m_{i}=0$ bonds, $t_{i}$ mediates
nearest-neighbor Majorana fermion hoppings, and to capture the physics
correctly within such a cluster, $t_{i}$ should be kept track of whenever
$m_{i-1}$ or $m_{i}$ is 0.

Similarly, $s_{i}$ is only non-zero when $m_{i-1}=m_{i+1}=0$ and $m_{i}=1$
because, as we saw in Sec. III.3 and Appendix A, this term only plays a role
of keeping track of the nearest-neighbor Majorana fermion hopping upon
encountering this specific configuration of $m$’s mid-RG. It involves
incorporating higher-order perturbation results than keeping track of any
other couplings, and consequently its effect should be negligible except for
the specific case in which we set $s_{i}$ to be non-zero.

  * •

We drop any contributions from multi-site flip terms involving three or more
sites. We incorporate contributions from double-site flip terms when there is
no bond with $m_{i}=0$ within the second nearest-neighbor of the double site-
flip term of our interest, on the grounds that when two antiferromagnetic
bonds are close to each other, the flip terms are already renormalized to a
very small magnitude. (See Appendix D for a more detailed argument.)

While the RG rules are admittedly complicated, having this protocol in mind
helps one to understand the underlying organizing principles and its
derivation.

Finally, we introduce the following notation that will be useful for writing
down the RG rules:

$\text{amax}\left(a,b\right)=\begin{cases}a&|a|>|b|\\\ b&|b|>|a|\end{cases},$
(49)

i.e., the function above outputs the number with largest absolute value. We
will use this notation to implement the “max” rule for the RG transformation
in which, for example, a term with coefficient $a+b+c$ after the
transformation receives a modified coefficient $\text{amax}(a,b,c)$, owing to
the fact that with the strong-disorder assumption, the coefficients $a$, $b$,
and $c$ are likely to have wildly different magnitudes.

###  B.1 Decimating an antiferromagnetic bond at $j$

The transformation rule in this case reads:

$m_{j}=1\rightarrow m_{j}=0$ (50)
$\begin{split}&J_{j-1}\rightarrow\text{amax}\left(J_{j-1},m_{j-1}\frac{(\sqrt{2}-1)h_{j}^{2}}{8|J_{j}|}\right)\\\
&J_{j+1}\rightarrow\text{amax}\left(J_{j+1},m_{j+1}\frac{(\sqrt{2}-1)h_{j+1}^{2}}{8|J_{j}|}\right)\end{split}$
(51)
$\begin{split}&t_{j}\rightarrow\text{max}\left(t_{j},m_{j-1}\frac{h_{j}^{2}}{8|J_{j}|}\right)\\\
&t_{j+1}\rightarrow\text{max}\left(t_{j+1},m_{j+1}\frac{h_{j+1}^{2}}{8|J_{j}|}\right)\end{split}$
(52) $\begin{split}&s_{j}\rightarrow 0\\\ &s_{j-1}\rightarrow
m_{j-1}\frac{1}{4\sqrt{2}}\frac{t_{j-2}h_{j}^{2}}{|J_{j}|^{2}}\\\
&s_{j+1}\rightarrow
m_{j+1}\frac{1}{4\sqrt{2}}\frac{t_{j+1}h_{j+1}^{2}}{|J_{j}|^{2}}.\end{split}$
(53)

(For this case only, the system length does not change, and the Hamiltonian is
determined by the same set of parameters before/after the transformation.
Hence, for brevity, above we simply list parameters that undergo non-trivial
transformations; we adopt different notation for the transformation rules in
other cases.) The transformation rule for $J_{i}$ and $t_{i}$ is captured by
the second-order perturbation theory mostly identical to the one we saw in
Eqs. (32), (33) and (34). Key differences are:

  * •

In the fully general strong diorder RG we consider here, antiferromagnetic
Ising interactions are also random. So $\Lambda_{A}$ in those equations should
be $|J_{j}|$ for application to this case.

  * •

In Eq. (34), we dropped the contributions to the Ising interaction due to the
special assumption on the energy scales of each term made for the setup in
Appendix A. Here, however, one has to incorporate this nearest-neighbor Ising
interaction as one can see in Eq. (51).

  * •

$h_{j}$, $F_{j}$, $h_{j+1}$ and $F_{j+1}$ can be a part of multi-site flip
terms and corresponding coefficients; meanwhile, in the setup of Sec. III and
Appendix A, by assuming that antiferromagnetic Ising interactions dominate and
that antiferromagnetic bonds do not appear in the close neighborhood of each
other, we avoided this situation. As explained in the main text and the
beginning of this appendix, we mostly ignore terms generated from multi-site
flip terms. Note that most coefficients generated in the above equations have
$m_{j-1}$ or $m_{j+1}$ in the coefficients, denoting that if ($h_{j}$,
$F_{j}$) or ($h_{j+1}$, $F_{j+1}$) is a part of a multi-site flip term, the
corresponding coefficients vanish.

Also, we remark that the transformation rule $m_{j}=1\rightarrow m_{j}=0$
suffices to keep track of the multi-site flip term generated from decimating
an antiferromagnetic bond at $j$—which can be shown by employing associativity
and commutativity of the $\odot$ operation introduced in Eqs. (29) and (30).

The transformation rule for $s_{i}$’s in Eq. (53) is a new feature and is
derived as follows. By marking $m_{j}=0$, the term $-2s_{j}S_{j}$ cannot act
non-trivially anymore because $S_{j}$ is only non-zero when
$\sigma_{j}^{z}=\sigma_{j+1}^{z}$; yet by decimating the antiferromagnetic
bond $j$, we are assuming $\sigma_{j}^{z}=-\sigma_{j+1}^{z}$ to be a hard
constraint. Hence, we just set $s_{j}$ to be zero when decimating the strong
antiferromagnetic bond $j$. Meanwhile, if $m_{j-2}=0$ and $m_{j-1}=1$,
decimating the antiferromagnetic bond at site $j$ creates a configuration
where we should assign non-zero $s_{j-1}$ values. We generate $s_{j-1}$ from
the third-order perturbation theory. Observe that since $m_{j-1}=1$, the site
$j$ has a single-site flip term before decimation. In the third-order formula
in Eq. (43), one can show that selecting the first and the third $H_{p}$ to be
$-h_{j}F_{j}$ and the second $H_{p}$ to be $-2t_{j-2}T_{j-2}$ generates the
term we desire, corresponding to the second line in Eq. (53). One can
similarly show that there is a third-order perturbation theory term that
generates $s_{j+1}$ in the third line of Eq. (53).

###  B.2 Decimating a ferromagnetic bond at $j$

In this case, the parameters
$\\{m^{\prime}_{i},h^{\prime}_{i},J^{\prime}_{i},t^{\prime}_{i},s^{\prime}_{i}\\}$
after the decimation, in terms of the parameters
$\\{m_{i},h_{i},J_{i},t_{i},s_{i}\\}$ before the decimation, is given by:

$m^{\prime}_{i}=\begin{cases}m_{i}&i<j\\\ m_{i+1}&i\geq j\end{cases}$ (54)
$h^{\prime}_{i}=\begin{cases}h_{i}&i<j\\\
\frac{2^{-1/4}h_{j}h_{j+1}}{J_{j}}&i=j\\\ h_{i+1}&i>j\end{cases}$ (55)
$J^{\prime}_{i}=\begin{cases}J_{i}&i<j-2\\\
\text{amax}\left(J_{j-2},m_{j-2}(1-m_{j-1})\frac{h_{j-1}^{2}h_{j}^{2}}{16\sqrt{2}J_{j}J_{j-1}^{2}}\right)&i=j-2\\\
\text{amax}\left(J_{j-1},m_{j-1}\frac{(\sqrt{2}-1)h_{j}^{2}}{4\sqrt{2}J_{j}}\right)&i=j-1\\\
\text{amax}\left(J_{j+1},m_{j+1}\frac{(\sqrt{2}-1)h_{j+1}^{2}}{4\sqrt{2}J_{j}}\right)&i=j\\\
\text{amax}\left(J_{j+2},m_{j+2}(1-m_{j+1})\frac{h_{j+1}^{2}h_{j+2}^{2}}{16\sqrt{2}J_{j}J_{j+1}^{2}}\right)&i=j+1\\\
J_{j+1}&j>i+1\end{cases}$ (56) $t^{\prime}_{i}=\begin{cases}t_{i}&i<j-1\\\
\max\left(t_{j-1},m_{j-2}(1-m_{j-1})\frac{(1+\sqrt{2})h_{j-1}^{2}h_{j}^{2}}{32J_{j}J_{j-1}^{2}}\right)&i=j-1\\\
\max\left(m_{j+1}\frac{1}{4\sqrt{2}}\frac{t_{j}h_{j+1}^{2}}{J_{j}^{2}},m_{j-1}\frac{1}{4\sqrt{2}}\frac{t_{j+1}h_{j}^{2}}{J_{j}^{2}},s_{j}\right)&i=j\\\
\max\left(t_{j+2},m_{j+2}(1-m_{j+1})\frac{(1+\sqrt{2})h_{j+1}^{2}h_{j+2}^{2}}{32J_{j}J_{j+1}^{2}}\right)&i=j+1\\\
t_{i+1}&i>j+1\end{cases}$ (57) $s^{\prime}_{i}=\begin{cases}s_{i}&j<i-1\\\
m_{j-1}\frac{t_{j-1}t_{j+1}h_{j}^{2}}{2J_{j}^{3}}&i=j-1\\\
m_{j+1}\frac{t_{j}t_{j+2}h_{j+1}^{2}}{2J_{j}^{3}}&i=j\\\
s_{i+1}&j>i\end{cases}.$ (58)

Notice that most of the coefficients are simply identical to those before the
transformations – only coefficients corresponding to the terms around the bond
$j$ are non-trivially modified. Hereafter, for other types of decimation rules
as well, we will simply focus on how the non-trivial transformation rules
around the site/bond of interest are obtained.

Appendix A.2 already covered the essential details regarding the derivation of
the above rule, so here we only highlight the key differences:

  * •

The $\Lambda_{A}$’s in Appendix A.2 should be substituted for the Ising
interaction coefficients $|J_{i}|$’s for the corresponding bonds.

  * •

In Appendix A.2, we ignored all contributions from
$-\frac{c_{l_{j}}^{2}}{2J_{j}}C_{l_{j}}^{2}$ and
$-\frac{c_{r_{j}}^{2}}{2J_{j}}C_{r_{j}+1}^{2}$ in Eq. (37). However, in the
full strong-disorder RG, if these flip terms involve a single site or double
site, we include these contributions in the decimation rule. Contributions
from these terms are computed in Eq. (41) and Eq. (42). We drop these
contributions if flip terms involve three or more sites; in this case, these
contributions may lead to non-nearest neighbor Majorana fermion interactions.
As mentioned in deriving the rule for the antiferromagnetic bond decimation,
we argue in Appendix D that these terms are much smaller than $-2t_{i}T_{i}$’s
and may be dropped.

###  B.3 Single-site decimation at site $j$

For this decimation, we project onto eigenstates of $h_{j}F_{j}$ with the two
lowest eigenvalues. These eigenstates exhibit a disordered Ising spin at $j$,
i.e., $\langle\sigma_{j}^{z}\rangle=0$, and the states with two different
eigenvalues are distinguished by whether two Ising spins at $j-1$ and $j+1$
are aligned ($\sigma_{j-1}^{z}=\sigma_{j+1}^{z}$) or anti-aligned
($\sigma_{j-1}^{z}=-\sigma_{j+1}^{z}$). One may treat this “projection” as
removing the disordered Ising spin $\sigma_{j}^{z}$ from the
system—effectively decreasing the system size by 1. The new parameters are
given as follows, incorporating first-order as well as some second-order
corrections:

$m^{\prime}_{i}=\begin{cases}m_{i}&i<j\\\ 0&i=j\\\ m_{i+1}&i>j\end{cases}$
(59) $h^{\prime}_{i}=\begin{cases}h_{i}&i<j-1\\\
\frac{(1+2^{-1/4})h_{j-1}}{2}&i=j-1\\\ \frac{(1+2^{-1/4})h_{j+1}}{2}&i=j\\\
h_{i+1}&i>j\end{cases}$ (60) $J^{\prime}_{i}=\begin{cases}J_{i}&i<j-1\\\
\frac{(1-2^{-1/4})}{2}h_{j}&i=j-1\\\ J_{i+1}&i>j-1\end{cases}$ (61)
$t^{\prime}_{i}=\begin{cases}t_{i}&i<j-1\\\ \frac{t_{j-1}}{2}&i=j-1\\\
\frac{t_{j+1}}{2}&i=j\\\ t_{i+1}&i>j\end{cases}$ (62)
$s^{\prime}_{i}=\begin{cases}s_{i}&i<j-1\\\
\frac{2t_{j-1}t_{j+1}}{h_{j}}&i=j-1\\\ s_{i+1}&i>j-1\end{cases}.$ (63)

We set up the perturbation theory leading to this transformation by writing
the Hamiltonian as

$\begin{split}&H=H_{0}+H_{p},\\\ &H_{0}=-h_{j}F_{j}\\\
&H_{p}=-c_{j-1}C_{j-1}-c_{j+1}C_{j+1}-2t_{j-1}T_{j-1}-2t_{j+1}T_{j+1}+\cdots\end{split}$
(64)

Here $-c_{j-1}C_{j-1}$ and $-c_{j+1}C_{j+1}$ are multi-site flip terms
associated with sites $j-1$ and $j+1$, respectively; terms that commute with
$-h_{j}F_{j}$ and hence do not provide perturbative corrections are contained
in the ellipsis. All the results except for Eq. (63) are derived from the
first-order perturbation theory, i.e., by projecting the existing terms onto
the low-energy eigenstates we specified. Equation (59) simply reflects the
fact that after removing site $j$, the bond connecting what was originally
site $j-1$ and site $j+1$ should carry a label $0$. Equation (61) comes from
the difference in the energy between states with
$\sigma_{j-1}^{z}=\sigma_{j+1}^{z}$ and states with
$\sigma_{j-1}^{z}=-\sigma_{j+1}^{z}$. Equations (60) and (62) are simply
obtained by projecting the terms in $H_{p}$ of Eq. (64).

After showing that Eq. (60) holds when $C_{j-1}$ and $C_{j+1}$ are single-site
flip terms, one can show through the trick employed in Eqs. (38) and (39) that
this rule also correctly describes how multi-site flip terms evolve after the
transformation. Generally, for any transformation rules for $h_{i}$’s in other
types of decimations as well, using a similar trick shows that the same
transformation rule applies regardless of whether $h_{i}$’s are parts of
multi-site flip terms or single-site flip terms.

Equation (63) requires going to second-order perturbation theory. If
$m_{j-2}=m_{j+1}=0$, decimating site $j$ produces a configuration where
$m^{\prime}_{j-2}=m^{\prime}_{j}=0$ and $m^{\prime}_{j-1}=1$. Hence, one must
go beyond first-order perturbation theory employed elsewhere in the rule to
generate $-2S_{j-1}$. Recall that degenerate perturbation theory gives the
following energy correction at second order,

$E^{(2)}_{ab}=\sum_{k,E_{k}\neq E_{0}}\frac{\langle a|H_{p}|k\rangle\langle
k|H_{p}|b\rangle}{E_{0}-E_{k}}.$ (65)

Choosing the first and second $H_{p}$ to be $-2t_{j-1}T_{j-1}$ and
$-2t_{j+1}T_{j+1}$, respectively, and vice versa, yields the nonzero
$s^{\prime}_{j-1}$ in the second line of Eq. (63).

###  B.4 Decimating a double site covering $j$ and $j+1$

This transformation rule is given by:

$m^{\prime}_{i}=\begin{cases}m_{i}&i<j-1\\\ 0&i=j-1\\\
m_{i+2}&i>j-1\end{cases}$ (66) $h^{\prime}_{i}=\begin{cases}h_{i}&i<j-1\\\
\frac{2^{-1/4}+\sqrt{\frac{1+\sqrt{2}}{2\sqrt{2}}}}{2}h_{j-1}&i=j-1\\\
\frac{2^{-1/4}+\sqrt{\frac{1+\sqrt{2}}{2\sqrt{2}}}}{2}h_{j+2}&i=j\\\
h_{i+2}&i>j\end{cases}$ (67) $J^{\prime}_{i}=\begin{cases}J_{i}&j<i-1\\\
\frac{1}{2}\left(1-\frac{\sqrt{2}}{\sqrt{1+\sqrt{2}}}\right)d_{j}&i=j-1\\\
J_{i+2}&j>i-1\end{cases}$ (68) $t^{\prime}_{i}=\begin{cases}t_{i}&i<j-1\\\
\frac{1}{2\sqrt{2}}t_{j-1}&i=j-1\\\ \frac{1}{2\sqrt{2}}t_{j+2}&i=j\\\
t_{i+2}&i>j\end{cases}$ (69) $s^{\prime}_{i}=\begin{cases}s_{i}&i<j-1\\\
\frac{t_{j-1}t_{j+2}}{d_{j}}&i=j-1\\\ s_{i+2}&i>j-1\end{cases}.$ (70)

The derivation is almost identical to that presented for the single-site
decimation case and is hence omitted.

###  B.5 Double-bond decimation associated with $T_{j}$

####  B.5.1 $m_{j}=m_{j-1}=0$

For this case the rule is:

$m^{\prime}_{i}=\begin{cases}m_{i}&i<j-1\\\ m_{i+2}&i\geq j-1\end{cases}$ (71)
$h^{\prime}_{i}=\begin{cases}h_{i}&i<j-1\\\
\frac{(1+2^{1/2})h_{j-1}h_{j}h_{j+1}}{4J_{j}J_{j+1}}&i=j-1\\\
h_{i+2}&i>j-1\end{cases}$ (72) $J^{\prime}_{i}=\begin{cases}J_{i}&i\leq j-2\\\
J_{i+2}&i>j-2\end{cases}$ (73) $t^{\prime}_{i}=\begin{cases}t_{i}&j<i-1\\\
(1-m_{j-2}m_{j+1})\frac{t_{j-1}t_{j+1}}{t_{j}}&i=j-1\\\
t_{i+2}&i>j-1\end{cases}$ (74) $s^{\prime}_{i}=\begin{cases}s_{i}&i<j-1\\\
(1-m_{j+1})\frac{s_{j-2}t_{j+1}}{t_{j}}&i=j-2\\\
(1-m_{j-2})\frac{s_{j+1}t_{j-1}}{t_{j}}&i=j-1\\\ s_{i+2}&i>j-1\end{cases}.$
(75)

To this end we project onto the lowest-energy state of $H_{0}=-2t_{j}T_{j}$.
The rule for $m_{i}$’s and $J_{i}$’s corresponds to simply removing the two
bond terms at $j-1$ and $j$, while the rule for $h_{i}$’s follows by
projecting the multi-site flip terms involving sites $j-1$, $j$, and $j+1$
into the low-energy subspace, equivalent to applying the first-order
perturbation theory. For the $h_{i}$’s, one may derive the rule when sites
$j-1$, $j$, and $j+1$ are not linked by any other $m_{i}=0$ bonds, i.e., when
$m_{j-2}=m_{j+2}=0$; the usual trick can be employed to show that this rule
applies in more general cases where $m_{j-2}\neq 0$ or $m_{j+2}\neq 0$ as
well.

To keep track of $t_{i}$ and $s_{i}$, we employ second-order perturbation
theory. From the second-order formula in Eq. (65), choosing two $H_{p}$’s to
be $-2t_{j-1}T_{j-1}$ and $-2t_{j+1}T_{j+1}$ (in either order) leads to the
non-trivial rule for $t^{\prime}_{j}$ in the second line of Eq. (79). Note
that when $m_{j-2}=m_{j+1}=1$, we assume $t^{\prime}_{j-1}=0$ after the
transformation – hence the factor $(1-m_{j-2}m_{j+1})$. Similarly, when
$m_{j-3}=0$, $m_{j-2}=1$, and $m_{j+1}=0$, after the decimation we encounter
the situation in which $m^{\prime}_{j-3}=m^{\prime}_{j-1}=0$ and
$m^{\prime}_{j-2}=1$. Hence, one should keep track of $s^{\prime}_{j-2}$. In
this case, from Eq. (65), choosing two $H_{p}$’s to be $-2s_{j-2}S_{j-2}$
(convince yourself that this term is non-zero in our case) and
$-2t_{j+1}T_{j+1}$ in either order leads to the non-trivial renormalization of
$s^{\prime}_{j-2}$ in the second line of Eq. (75). Similar considerations for
the case $m_{j-1}=m_{j+2}=0$, $m_{j+1}=1$ lead to the third line of Eq. (75).

####  B.5.2 $m_{j}=1$ and $m_{j-1}=0$

In this case, the rule is given by:

$m^{\prime}_{i}=\begin{cases}m_{i}&i<j-1\\\ m_{i+2}&i\geq j-1\end{cases}$ (76)
$h^{\prime}_{i}=\begin{cases}h_{i}&i<j-1\\\
\frac{(1+2^{1/2})h_{j-1}h_{j}h_{j+1}}{4t_{j}J_{j}}&i=j-1\\\
h_{i+2}&i>j-1\end{cases}$ (77) $J^{\prime}_{i}=\begin{cases}J_{i}&i<j-2\\\
\text{amax}\left(J_{j-1},m_{j-2}\frac{(\sqrt{2}+1)h_{j-1}^{2}h_{j}^{2}}{32\sqrt{2}t_{j}J_{j-1}^{2}},-m_{j-2}\frac{t_{j-1}^{2}}{2t_{j}}\right)&i=j-2\\\
\text{amax}\left(J_{j+2},m_{j+1}\frac{(\sqrt{2}-1)h_{j+1}^{2}}{8t_{j}}\right)&i=j-1\\\
J_{i+2}&i>j-1\end{cases}$ (78) $t^{\prime}_{i}=\begin{cases}t_{i}&i<j-1\\\
0&i=j-1,m_{j-2}m_{j+1}=1\\\
\frac{h_{j+1}^{2}}{4t_{j}^{2}}t_{j-1}&i=j-1,\,m_{j-2}=0,\,m_{j+1}=1\\\
\frac{t_{j-1}t_{j+1}}{t_{j}}&i=j-1,\text{all other cases}\\\
t_{i+2}&i>j-1\end{cases}$ (79) $s^{\prime}_{i}=\begin{cases}s_{i}&i<j-1\\\
(1-m_{j+1})\frac{s_{j-2}t_{j+1}}{t_{j}}&i=j-2\\\
m_{j+1}(1-m_{j-2})\frac{3t_{j-1}t_{j+2}h_{j+1}^{2}}{4\sqrt{2}t_{j}^{3}}&i=j-1\\\
s_{i+2}&i>j-1\end{cases}.$ (80)

We set up the degenerate perturbation leading to the above by writing

$\begin{split}&H=H_{0}+H_{p},\,H_{0}=-2t_{j}T_{j},\\\
&H_{p}=-c_{l_{j}}C_{l_{j}}-c_{r_{j}}C_{r_{j}}\\\
&\quad-2s_{j-2}S_{j-2}-2s_{j+1}S_{j+1}-2t_{j-1}T_{j-1}-2t_{j+1}T_{j+1}+\cdots.\end{split}$
(81)

Here, $-c_{l_{j}}C_{l_{j}}$ and $-c_{r_{j}}C_{r_{j}}$ are multi-site flip
terms associated with sites $j-1$ (or equivalently $j$ since $m_{j-1}=0$) and
$j+1$ respectively. As before, the ellipsis encodes Hamiltonian terms that
commute with $H_{0}$. We will incorporate all the second-order perturbation
theory results and some terms generated from higher order for the second line
in Eq. (79) and the third line in Eq. (80). For the second-order perturbation
theory formula in Eq. (65), if we consider a term with one $H_{p}$ chosen to
be a flip term $-c_{l_{j}}C_{l_{j}}$, the other $H_{p}$ should be also be a
flip term (either $-c_{r_{j}}C_{r_{j}}$ or $-c_{l_{j}}C_{l_{j}}$) for the
matrix elements to be non-vanishing. Hence, one can separate purely flip-term
contributions, and non-flip-term contributions wherein none of $H_{p}$ is
chosen to be flip term.

Consider the flip-term contribution when $m_{j-2}=m_{j+1}=1$, where
$-c_{l_{j}}C_{l_{j}}$ is simply a double-site flip term and
$c_{r_{j}}C_{r_{j}}$ is a single-site flip term. One can therefore write

$\begin{split}&c_{l_{j}}C_{l_{j}}=\frac{h_{j-1}h_{j}}{|J_{j-1}|}(F_{j-1}\odot
F_{j}),\\\ &c_{r_{j}}C_{r_{j}}=h_{j+1}F_{j+1}.\end{split}$ (82)

Using similar logic as in deriving the decimation rule for the ferromagnetic
and antiferromagnetic bonds, one can show that flip-term contributions to the
second-order perturbation theory are

$\begin{split}H^{(2)}_{F}&=\frac{h_{j-1}h_{j}h_{j+1}}{|J_{j-1}|t_{j}}\mathcal{P}_{T_{j}}(F_{j-1}\odot
F_{j}\odot F_{j+1})\mathcal{P}_{T_{j}}\\\
&\quad+\frac{h_{j-1}^{2}h_{j}^{2}}{2|J_{j-1}|^{2}t_{i}}\mathcal{P}_{T_{j}}(F_{j-1}\odot
F_{j})^{2}\mathcal{P}_{T_{j}}+\frac{h_{j+1}^{2}}{2t_{i}}\mathcal{P}_{T_{j}}F_{j+1}^{2}\mathcal{P}_{T_{j}}\end{split}$
(83)

with $\mathcal{P}_{T_{j}}$ the projection operator to the lowest-energy
eigenstates of $H_{0}=-2t_{j}T_{j}$. The first term on the right side of Eq.
(83) simultaneously flips spins at sites $j-1$, $j$, and $j+1$, and after the
projection and the decimation procedure, it becomes a flip term for the super-
site in which the aforementioned three sites are combined. This contribution
is given in the second line of Eq. (77), with the numerical factor
$\frac{(1+2^{1/2})}{4}$ coming from the projection $\mathcal{P}_{T_{j}}$. Note
that while we derived the rules in Eq. (77) for the special case where
$m_{j-2}=m_{j+1}=1$, using a trick similar to that used for Eqs. (38) and (39)
shows that this rule applies to other cases in which $m_{j-2}$ or $m_{j+1}$ is
0, and the super-site formed after the decimation is still linked by some bond
with $m_{i}=0$. The remaining two terms in Eq. (83) do not flip any spins;
rather, they can be shown to generate Ising interaction:

$\frac{h_{j-1}^{2}h_{j}^{2}}{2|J_{j-1}|^{2}t_{i}}\mathcal{P}_{T_{j}}(F_{j-1}\odot
F_{j})^{2}\mathcal{P}_{T_{j}}\equiv\frac{(\sqrt{2}+1)h_{j-1}^{2}h_{j}^{2}}{32\sqrt{2}t_{j}J_{j-1}^{2}}\sigma_{j-2}^{z}\sigma_{j-1}^{z}$
(84)
$\frac{h_{j+1}^{2}}{2t_{i}}\mathcal{P}_{T_{j}}F_{j+1}^{2}\mathcal{P}_{T_{j}}\equiv\frac{(\sqrt{2}-1)h_{j+1}^{2}}{8t_{i}}\sigma_{j+1}^{z}\sigma_{j+2}^{z}.$
(85)

We included these terms in the transformation rule for $J^{\prime}_{i}$’s. If
$m_{j-2}=0$, the flip term containing sites $j-1$ and $j$ flips three or more
spins. We will ignore this flip-term-squared contribution as mentioned at the
beginning of this Appendix. Similarly, if $m_{j+1}=0$, we ignore flip-term-
squared contributions associated with site $j+1$.

Finally, we discuss second-order perturbation theory for non-flip-term
contributions:

  * •

Choosing both $H_{p}$’s in Eq. (65) to be $-2t_{j-1}T_{j-1}$ leads to the
Ising interaction

$\frac{t_{i-1}^{2}}{2t_{i}}\sigma_{j-2}^{z}\sigma_{j-1}^{z}.$ (86)

We included this contribution in the second line of Eq. (78), in the third
argument of amax. We ignore this contribution when $m_{j-2}=0$; in this case,
the bond at $j-2$ is already decimated, and $|J_{j-2}|$ is likely much larger
than this newly generated Ising interaction. An analogue of this term
generated upon choosing $H_{p}$’s to be $-t_{j+1}T_{j+1}$ can be thrown out
for similar reasons: For $t_{j+1}\neq 0$ in our scheme, $m_{j+1}=0$ always.
The generated Ising interaction mediates coupling between two spins joined by
the bond $j+1$, which already has been decimated, and hence is likely to have
a much larger energy scale than the generated Ising interaction.

  * •

Choosing one $H_{p}$ to be $-2t_{j-1}T_{j-1}$ and the other to be
$-2t_{j+1}T_{j+1}$, in either order, yields the correction

$\frac{t_{j-1}t_{j+1}}{t_{j}}T^{\prime}_{j-1}.$ (87)

This contribution is reflected in the fourth line of Eq. (79). Note that if
$m_{j-2}=m_{j+1}=1$, this contribution vanishes because $t_{j+1}=0$, as
intended.

  * •

Choosing one $H_{p}$ to be $-2t_{j+1}T_{j+1}$ and the other to be
$-2s_{j-2}S_{j-2}$ leads to the contribution in the second line of Eq. (80).

  * •

We drop a term generated from choosing both $H_{p}$’s to be $s_{j-2}S_{j-2}$.
One can show that this term corresponds to an interaction between spins at
sites $j-3$, $j-2$ and $j-1$. The rationale is that $S_{j}$ is generated
through employing higher-order perturbation theory in other decimations, and
although it generate three-body Ising interactions, its contribution is
generally much smaller than two-body nearest-neighbor Ising interactions
generated through lower-order perturbation theory.

We have now exhausted the possible terms that appear in second-order
perturbation theory, and explained which terms should be retained versus
dropped. Finally, we discuss higher-order contributions. Recall that second-
order perturbation theory does not generate a nonzero $t^{\prime}_{j-1}$ when
$m_{j+1}=1$. Similarly, second-order perturbation theory fails to generate
non-zero $s^{\prime}_{j-1}$ when $m_{j+1}=1$, $m_{j+2}=0$ and $m_{j-2}=0$. For
the first instance, we generate nontrivial $t^{\prime}_{j-1}$ at third-order
in perturbation theory by choosing the first and the third $H_{p}$’s in Eq.
(43) to be $-h_{j+1}F_{j+1}$ (due to the condition on the $m_{i}=0$ mark, the
flip term associated with site $j+1$ is a single-site flip term), and the
second $H_{p}$ to be $-2t_{j-1}T_{j-1}$. To generate non-trivial
$s^{\prime}_{j-1}$, we go to fourth order. The following three choices for
$H_{p}$’s in Eq. (LABEL:eq:4thpgen) are relevant:

  * •

The first and third $H_{p}$’s are set as $-h_{j+1}F_{j+1}$, the second is
chosen to be $-2t_{j+2}T_{j+2}$, and the last one is $-2t_{j-1}T_{j-1}$.

  * •

The second and fourth $H_{p}$’s are set as $-h_{j+1}F_{j+1}$, the third is
$-2t_{j+2}T_{j+2}$, and the first is $-2t_{j-1}T_{j-1}$.

  * •

The first and fourth $H_{p}$’s are $-h_{j+1}F_{j+1}$, while the second and
third are chosen to be $-2t_{j+2}T_{j+2}$ and $-2t_{j-1}T_{j-1}$, in either
order.

These contributions together generate the term in the third line of Eq. (80).

####  B.5.3 Other cases

The transformation rule when $m_{j-1}=1$ and $m_{j}=0$ is simply a spatially
inverted version of the rule for $m_{j-1}=0$ and $m_{j}=1$. Finally, if
$m_{j-1}=m_{j}=1$, we assume that $t_{j}=0$, and there is no decimation rule
in this case.

###  B.6 Triple bond decimation

The triple bond decimation induced by $-s_{j}S_{j}$ brings the following
changes to the Hamiltonian parameters:

$m^{\prime}_{i}=\begin{cases}m_{i}&i\leq j-2\\\ m_{i+3}&i>j-2\end{cases}$ (88)
$h^{\prime}_{i}=\begin{cases}h_{i}&i<j-1\\\
\frac{2^{-1/4}+2^{1/4}}{4}\frac{h_{j-1}h_{j}h_{j+1}h_{j+2}}{s_{j}J_{j-1}J_{j+1}}&i=j-1\\\
h_{i+3}&i>j-1\end{cases}$ (89) $J^{\prime}_{i}=\begin{cases}J_{i}&i\leq j-2\\\
J_{i+3}&i>j-2\end{cases}$ (90) $t^{\prime}_{i}=\begin{cases}t_{i}&i<j-1\\\
(1-m_{j-2}m_{j+1})\frac{t_{j-2}t_{j+1}}{s_{j}}&i=j-1\\\
t_{i+3}&i>j-1\end{cases}$ (91) $s^{\prime}_{i}=\begin{cases}s_{i}&i<j-2\\\
m_{j+2}\frac{s_{j-2}t_{j+1}}{s_{j}}&i=j-2\\\
m_{j-2}\frac{s_{j+2}t_{j-2}}{s_{j}}&i=j-1\\\ s_{i+3}&i>j-1\end{cases}.$ (92)

All these results are derived from second-order perturbation theory. The
procedure shares many details with the case covered in Appendix B.5.2. We
start with the setup

$\begin{split}&H=H_{0}+H_{p},\,\\\ &H_{0}=-2s_{j}S_{j},\\\
&H_{p}=-c_{l_{j}}C_{l_{j}}-c_{r_{j}}C_{r_{j}}-2t_{j-1}T_{j-1}-2t_{j+2}T_{j+2}\\\
&\quad\quad\,\,-2s_{j-2}S_{j-2}-2s_{j+1}S_{j+1}+\cdots.\end{split}$ (93)

Similarly as before, $-c_{l_{j}}C_{l_{j}}$ and $-c_{r_{j}}C_{r_{j}}$ are
multi-site flip terms that cover spins on the left and right of the bond $j$
respectively. In our setup, if $s_{j}\neq 0$, then $m_{j-1}=m_{j+1}=0$. Hence,
both flip terms are multi-site flip terms that cover at least two sites. As in
Appendix B.5.2, one can divide terms generated at second order into flip-term
contributions and non-flip-term contributions.

The flip-term contributions are given by

$\begin{split}H^{(2)}_{F}=&\frac{c_{l_{j}}c_{r_{j}}}{s_{j}}\mathcal{P}_{S_{j}}(C_{l_{j}}\odot
C_{r_{j}})\mathcal{P}_{S_{j}}\\\
&\quad\quad+\frac{c_{l_{j}}^{2}}{2s_{j}}\mathcal{P}_{S_{j}}C_{l_{j}}^{2}\mathcal{P}_{S_{j}}+\frac{c_{r_{j}}^{2}}{2s_{j}}\mathcal{P}_{S_{j}}C_{r_{j}}^{2}\mathcal{P}_{S_{j}},\end{split}$
(94)

where $\mathcal{P}_{S_{j}}$ is analogous to $\mathcal{P}_{T_{j}}$ but instead
projects onto the ground states of $H_{0}=-2s_{j}S_{j}$. One can show that
when $m_{j-2}=m_{j+3}=1$, the first term is equivalent to a single-site flip
term after the decimation, whose coefficient appears in the second line of Eq.
(89). While derived for a simple case of $m_{j-2}=m_{j+3}=1$, with the trick
we employed, this rule works even when either or both of $m_{j-2}$ or
$m_{j+3}$ are $0$. As for other pieces, upon following the argument in
Appendix D, one may discard the flip-term-squared contributions in the second
line.

For non-flip-term contributions, we have the option to choose either
$-2t_{j-1}T_{j-1}$, $-2t_{j+2}T_{j+2}$, $-2s_{j-2}S_{j-2}$ or
$-2s_{j+1}S_{j+1}$ for each $H_{p}$ in Eq. (65) to generate them. We do the
following:

  * •

We drop any contributions that are generated from choosing the same term for
two $H_{p}$’s in Eq. (65). Plugging in $-2t_{j-1}T_{j-1}$ in both $H_{p}$’s
leads to a small antiferromagnetic interaction we ignore; the same is true for
plugging in $-2t_{j+2}T_{j+2}$ in both $H_{p}$’s. Similarly, choosing
$-2s_{j-2}S_{j-2}$ or $-2s_{j+1}S_{j+1}$ for both $H_{p}$’s lead to a non-
nearest neighbor term we chose to ignore before as well.

  * •

Choosing $-2t_{j-1}T_{j-1}$ and $-2t_{j+2}T_{j+2}$ for the two $H_{p}$’s gives
rise to the term $-2t^{\prime}_{j-1}T_{j-1}$ after decimation, as in the
second line of Eq. (91). Note that we drop this term if $m_{j-2}=m_{j+2}=0$

  * •

Choosing $-2s_{j-2}S_{j-2}$ and $-2t_{j+2}T_{j+2}$ leads to the correction in
the second line of Eq. (92), while choosing $-2t_{j+2}T_{j+2}$ and
$-2s_{j+1}S_{j+1}$ gives the correction in the third line of Eq. (92).

  * •

We ignore the contribution from choosing $-2s_{j-2}S_{j-2}$ and
$-2s_{j+1}S_{j+1}$ due to the similar reasoning presented in Appendix. B.5.2.

##  Appendix C The fixed point values of $p_{i}$’s

In the main text and in Appendices A and B, we assumed that $p=2^{-1/4}$. We
here show that $p_{i}=\pm 2^{-1/4}$ are the only stable fixed-point values of
$p_{i}$ in the strong-disorder limit. To this end, we let individual $F_{i}$’s
have different $p_{i}$ values, as opposed to one uniform value across the
whole system. We then explore how $p_{i}$’s renormalize after each decimation.

Decimating the ferromagnetic bond at $j$ transforms $p_{i}$’s before the
decimation to $p^{\prime}_{i}$’s as follows:

$p^{\prime}_{i}=\begin{cases}p_{i}&i<j\\\
\frac{2^{-1/2}+p_{j}p_{j+1}}{p_{j+1}+p_{j}}&i=j\\\ p_{i+1}&i>j\end{cases}.$
(95)

Assuming that $p_{i}$ and $p^{\prime}_{i}$ are all given by the same value
$p$, the only two consistent values of $p$ are $\pm 2^{-1/4}$. We investigate
the stability of this solution under the decimation procedure by expanding
$p_{i}=2^{-1/4}+\delta_{i}$ and $p^{\prime}_{i}=2^{-1/4}+\delta^{\prime}_{i}$,
assuming $\delta_{i}\ll 1$. By Taylor expanding one can show that

$\delta^{\prime}_{j}=O(\delta_{j}^{2})+O(\delta_{j}\delta_{j+1})+O(\delta_{j+1}^{2}).$
(96)

That is, to lowest order, $\delta^{\prime}_{j}$ is quadratic in
$\delta_{i}$’s. Hence, assuming that initially $\delta_{j}$ and $\delta_{j+1}$
are sufficiently small, $\delta^{\prime}_{j}$ is much smaller than
$\delta_{j}$ or $\delta_{j+1}$! Hence, the fixed point value
$p_{i}=p=2^{-1/4}$ is stable to small perturbations. The solution
$p_{i}=p=-2^{-1/4}$ can be similarly shown to be stable under ferromagnetic
bond decimations.

Antiferromagnetic bond decimations mark the corresponding bond $j$ with
$m_{j}=0$. In our schemes, $m_{j}$ labels direct how to form multi-site flip
terms with individual $F_{i}$’s but do not modify information about $F_{i}$’s.
Hence, in our formulation, antiferromagnetic bond decimations do not modify
$p_{i}$’s.

We conduct a similar exercise for double-bond and triple-bond decimations. The
decimations associated with $T_{j}$ and $S_{j}$ respectively transform
$p_{i}$’s as

$p^{\prime}_{i}=\begin{cases}p_{i}&i<j-1\\\
\frac{p_{j-1}+p_{j+1}+p_{j}^{*}\left(2^{-1/2}+p_{j-1}p_{j+1}\right)}{1+p_{j}^{*}(p_{j-1}+p_{j+1})+\sqrt{2}p_{j-1}p_{j+1}}&i=j-1\\\
p_{i+2}&i>j-1\end{cases}$ (97)
$\begin{split}&p^{\prime}_{i}=\begin{cases}p_{i}&i<j-1\\\
\frac{p_{j-1}+p_{j+2}+p_{j,j+1}^{*}\left(2^{-1/2}+p_{j-1}p_{j+2}\right)}{1+p_{j,j+1}^{*}(p_{j-1}+p_{j+2})+\sqrt{2}p_{j-1}p_{j+2}}&i=j-1\\\
p_{i+3}&i>j-1\end{cases}\\\
&p_{j,j+1}=\frac{2^{-1/2}+p_{j}p_{j+1}}{p_{j+1}+p_{j}}.\end{split}$ (98)

In both cases, one can expand $p_{i}=\pm 2^{-1/4}+\delta_{i}$,
$p^{\prime}_{j-1}=\pm 2^{-1/4}+\delta^{\prime}_{j-1}$ to show that
$\delta^{\prime}_{j-1}$ is, to lowest order, quadratic in $\delta_{i}$.

Finally, we comment on how a single-site and double-site decimation modifies
$p_{i}$. There, as we saw in the previous Appendices and the main text,
$p_{i}=\pm 2^{-1/4}$ is a fixed point of the transformation. One can perform
the same exercise of expanding $p_{i}=\pm 2^{-1/4}+\delta_{i}$ and investigate
how $p^{\prime}_{i}=\pm 2^{-1/4}+\delta_{i}^{\prime}$ evolves. Unfortunately,
in this case, the $\delta^{\prime}_{i}$’s that non-trivially renormalize are,
to the lowest order, linear in $\delta_{i}$. However, we also observed that
site decimations generate a ferromagnetic interaction of order $\Omega$, and
in the strong-disorder limit site decimations are immediately followed by
ferromagnetic bond decimations. If one combines the effect of ferromagnetic
bond decimations, once more the non-trivially renormalized
$\delta^{\prime}_{i}$’s are quadratic in $\delta_{i}$’s. Hence, in any strong-
disorder fixed point $p_{i}=\pm 2^{-1/4}$ persist as stable fixed-point
values.

As a closing remark, there may be some strong-disorder fixed points with
$p_{i}$ themselves randomly distributed. We reserve exploring such strong-
disorder fixed points for future work.

##  Appendix D More on the irrelevancy of the multi-site flip terms

Here, we lay out the rationale for dropping the contributions from multi-site
flip terms in our strong-disorder RG scheme. We specifically argue that $(i)$
multi-site flip terms are renormalized to a very small magnitude compared to
the RG energy scale at each step and $(ii)$ non-local Majorana fermion
interactions generated from multi-site flip terms are much smaller than the
nearest-neighbor Majorana fermion hopping terms we keep track of in the
$-t_{i}T_{i}$ and $-s_{i}S_{i}$ terms from Eq. (23) and can be ignored. Point
$(i)$ justifies ignoring any possible decimation procedure coming from the
multi-site flip terms, while $(ii)$ justifies ignoring terms generated due to
multi-site flip terms during the decimation procedure associated with local
terms we do keep track of in our RG procedure.

The key observation behind the argument is that antiferromagnetic bonds are
typically very far from each other in the limit we are considering, and up to
some energy scale $\Lambda_{FG}$ ($FG$ standing for Ferromagnetic Griffiths),
the RG flow is “unaware” of the presence of antiferromagnetic bonds. Hence,
when the RG energy scale satisfies $\Omega>\Lambda_{FG}$, the ferromagnetic
Griffiths behavior characterized by the distribution of couplings given in Eq.
(20) governs the scaling behavior. In Eq. (20), when $\Omega$ is small enough,
we have $\alpha\gg\beta$, and ferromagnetic Ising interactions are on average
much larger than flip terms. Hence, when antiferromagnetic bonds appear rarely
enough that $\Lambda_{FG}$ is sufficiently small, one can generally expect
flip terms to be renormalized to very small values at
$\Omega\sim\Lambda_{FG}$, especially compared to ferromagnetic interactions
present or generated mid-RG. Meanwhile, up to $\Omega>\Lambda_{FG}$, there is
only a small number of clusters connected by $m_{i}=0$ bonds whose sizes are
larger than two; most sites are not flanked by $m_{i}=0$ bonds at all, or are
flanked by one bond due to the presence of rare antiferromagnetic bonds.

Recall that in the RG flow we presented in the main text, there is also an
energy scale $\Lambda_{MF}$ below which spins are mostly pinned by strongly
antiferromagnetic bonds with $m_{i}=0$ and the physics of the system is
governed by domain-wall Majorana fermions. When
$\Lambda_{MF}<\Omega<\Lambda_{FG}$, multi-site clusters form, and multi-site
flip terms potentially disrupt the RG procedure we presented in the main text.
The high-level idea is that since flip terms are renormalized to a very small
magnitude at $\Omega\sim\Lambda_{FG}$, their effect will be suppressed during
$\Lambda_{MF}<\Omega<\Lambda_{FG}$ as well; when $\Omega<\Lambda_{MF}$, all
spins in the system effectively form a single cluster, and the notion of
individual flip terms, whether multi-site or single-site, effectively
disappears.

Let us spell out more detailed arguments in favor of points $(i)$ and $(ii)$
from the beginning of the Appendix. Even when
$\Lambda_{MF}<\Omega<\Lambda_{FG}$, the majority of decimations at RG steps
are ferromagnetic bond decimations. Another key observation regarding the
nearest-neighbor Ising interaction is that, according to the rule we derived
in Appendix B, when the nearest-neighbor Ising interaction $J_{i}$ obeys a
non-trivial RG transformation rule, the rule always involves the “amax” symbol
introduced at the beginning of Appendix B. The rule essentially chooses the
dynamically generated Ising interaction only when it is larger than the Ising
interaction $J_{i}$ already present. In other words, the magnitude of the
nearest-neighbor Ising interaction does not decrease under the RG
transformation rule. Hence, he relevant RG energy scales in the regime
$\Lambda_{MF}<\Omega<\Lambda_{FG}$ are effectively lower-bounded by the
ferromagnetic Ising interactions present/generated up to the scale
$\Omega\sim\Lambda_{FG}$, and these ferromagnetic Ising interactions are much
larger than $h_{i}$’s that parameterize the magnitude of spin flip terms. This
conclusion implies that the possibility that any flip terms—including multi-
site ones that start to appear frequently in the regime
$\Lambda_{MF}<\Omega<\Lambda_{FG}$—enter as the RG energy scale is greatly
suppressed in the regime of our interest, justifying $(i)$.

To justify $(ii)$, we observe that in our RG, hopping or interaction between
domain wall Majorana fermions are essentially generated by the physical
processes of flipping spins back and forth. Using this observation, one may
estimate and compare contributions to Majorana fermion hoppings from single-
site flip terms and multi-site flip terms. To do so, assume that at the middle
of an RG step with RG energy scale $\Omega\sim\Lambda_{FG}$, there are two
bonds at $j$ and $j+n$ which become two neighboring bonds, say $j^{\prime}$
and $j^{\prime}+1$ with labels $m_{j^{\prime}}=m_{j^{\prime}+1}=0$ at a later
RG step. Denote the amplitude for Majorana fermion hopping between bond $j$
and bond $j+n$ at the later RG step of the RG as $t_{j,j+n}$. When only the
single-site flip terms are utilized, the simplest way to generate $t_{j,j+n}$
is to flip spins between bond $j$ and bond $j+n$ back and forth. Then, one may
estimate $t_{j,j+n}$ to be

$t_{j,j+n}\sim\frac{\displaystyle\prod_{i=j+1}^{j+n}h_{i}^{2}}{\displaystyle\prod_{k=1}^{2n-1}\Omega_{k}}.$
(99)

The fact that one flips spins back and forth is reflected in the $h_{i}^{2}$
in the numerator, while $\Omega_{k}$ in the denominator is the energy scale
lying between $\Lambda_{P}$ and $\Lambda_{FG}$ and characterizes the virtual
spin-flip energy cost. Note that numerator is a product of $2n$ energy scales,
while the denominator is a product of $2n-1$ energy scales, so the expression
on the right side has the correct dimensions. Finally, invoking the same logic
used to justify $(i)$, generally $\Omega_{k}\gg h_{i}$ for any $k$ and $i$.
Hence, when $n$, the number of spins involved to generate $t_{j,j+n}$, is
larger, the generated $t_{j,j+n}$ will be smaller.

Multi-site flip term contributions for the same Majorana fermion hoppings
appear at a later step of the RG, when the bond $j$ and $j+n$ becomes two
neighboring bonds $j^{\prime}$ and $j^{\prime}+1$, by flipping the cluster
including the bonds $j^{\prime}$ and $j^{\prime}+1$ back and forth. From the
perspective of the renormalized system at the RG energy scale
$\Omega\sim\Lambda_{FG}$, spins comprising the cluster at the later step are
not only those between bond $j$ and $j+n$ – a large number of spins to the
left of the bond $j$ and to the right of the bond $j+n$ are also involved. One
may write down a formula analogous to Eq. (99) for the multi-site flip term
contributions; however, it will involve a number of $h_{i}$’s much larger than
$2n$ in the numerator and a number of $\Omega_{k}$’s much larger than $2n-1$
in the denominators. Hence, the generated Majorana fermion hoppings are
expected to be much smaller than the single-site contributions that involve
fewer spins. Furthermore, all multi-site fermion terms are presumably
irrelevant (as claimed in the original paper by Fisher [26], supported by
recent numerics in [46]; see also [47]) near the infinite randomness fixed
point of our consideration. Hence, as long as Majorana fermion hopping terms
generated from multi-site flip terms are kept to be sufficiently small as we
suggested here, they cannot alter the IR physics. This argument justifies
cutting off terms generated from multi-site flip terms in our decimation
procedure.

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# Benedicks-Amrein-Berthier theorem for the Heisenberg motion group and
quaternion Heisenberg group

Somnath Ghosh and R.K. Srivastava  Department of Mathematics, Indian Institute
of Technology, Guwahati, India 781039<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this article, we prove that if the group Fourier transform of a certain
finitely supported function on the Heisenberg motion group is of arbitrary
finite rank, then the function must be zero. We also prove an analogous result
on the quaternion Heisenberg group. In the end, we discuss the concept of
strong annihilating pair for the Weyl transform.

###### Key words and phrases:

Fourier transform, Heisenberg motion group, Quaternion Heisenberg group,
Uncertainty principle.

###### 2000 Mathematics Subject Classification:

Primary 42A38; Secondary 44A35

##  1\. Introduction

In an interesting article [2] Benedicks’ proved that if $f\in
L^{1}(\mathbb{R}^{n}),$ then both the sets $\\{x\in\mathbb{R}^{n}:f(x)\neq
0\\}$ and $\\{\xi\in\mathbb{R}^{n}:\hat{f}(\xi)\neq 0\\}$ cannot have finite
Lebesgue measure, unless $f=0.$ Concurrently, in the article [1], Amrein-
Berthier reached the same conclusion via the Hilbert space theory. The
aforesaid fundamental result got further attention in general Lie groups.

Let $G$ be a locally compact group and $\hat{m}$ denotes the Plancherel
measure on the unitary dual group $\hat{G}.$ Then $G$ is said to satisfy
qualitative uncertainty principle (QUP) if for each $f\in L^{2}(G)$ with
$m\\{x\in G:f(x)\neq 0\\}<m(G)$ and

(1.1) $\int_{\hat{G}}\text{rank}\hat{f}(\lambda)\,d\hat{m}(\lambda)<\infty$

implies $f=0.$ In [11], QUP was proved for the unimodular groups of type I. A
brief survey of QUP is presented in [7]. In the case of the Heisenberg group,
the condition (1.1) of QUP implies $\hat{f}$ should be supported on a set of
finite Plancherel measure together with $\text{rank}\hat{f}(\lambda)$ is
finite for almost all $\lambda.$

In [12], Narayanan and Ratnakumar proved that if $f\in L^{1}(\mathbb{H}^{n})$
is supported on $B\times\mathbb{R}$, where $B$ is a compact subset of
$\mathbb{C}^{n}$, and $\hat{f}(\lambda)$ has finite rank for each $\lambda,$
then $f=0.$ Thereafter, Vemuri [15] replaced the compact support condition on
the set $B$ by finite measure. In [5] authors consider $B$ as a rectangle in
$\mathbb{R}^{2n}$ while proving an analogous result on step two nilpotent Lie
groups and a version of this result, with a strong assumption on rank, derived
therein for the Heisenberg motion group. Later in the article [9], this result
is extended to arbitrary set $B$ of finite measure for general step two
nilpotent Lie groups. We prove the result on the Heisenberg motion group and
quaternion Heisenberg group when $B$ is an arbitrary set of finite measure
using the Hilbert space theory. However, specifying the appropriate set of
projections in the setups of Heisenberg motion group and quaternion Heisenberg
group was a major bottleneck. This result, as of now, is the most general
analogue of Benedicks-Amrein-Berthier theorem in these set ups. Finally, we
define a strong annihilating pair for the Weyl transform and present some
comparison with the above result.

##  2\. Heisenberg motion group

The Heisenberg group $\mathbb{H}^{n}=\mathbb{C}^{n}\times\mathbb{R}$ is a step
two nilpotent Lie group having center $\mathbb{R}$ that equipped with the
group law

$(z,t)\cdot(w,s)=\left(z+w,t+s+\frac{1}{2}\text{Im}(z\cdot\bar{w})\right).$

By Stone-von Neumann theorem, the infinite dimensional irreducible unitary
representations of $\mathbb{H}^{n}$ can be parameterized by
$\mathbb{R}^{\ast}=\mathbb{R}\smallsetminus\\{0\\}.$ That is, each
$\lambda\in\mathbb{R}^{\ast}$ defines a Schrödinger representation
$\pi_{\lambda}$ of $\mathbb{H}^{n}$ via

$\pi_{\lambda}(z,t)\varphi(\xi)=e^{i\lambda
t}e^{i\lambda(x\cdot\xi+\frac{1}{2}x\cdot y)}\varphi(\xi+y),$

where $z=x+iy$ and $\varphi\in L^{2}(\mathbb{R}^{n}).$

Having chosen sublaplacian $\mathcal{L}$ of the Heisenberg group
$\mathbb{H}^{n}$ and its geometry, there is a larger group of isometries that
commute with $\mathcal{L}$, known as Heisenberg motion group. The Heisenberg
motion group $G$ is the semidirect product of $\mathbb{H}^{n}$ with the
unitary group $K=U(n).$ Since $K$ defines a group of automorphisms on
$\mathbb{H}^{n},$ via $k\cdot(z,t)=(kz,t),$ the group law on $G$ can be
expressed as

$(z,t,k_{1})\cdot(w,s,k_{2})=\left(z+k_{1}w,t+s-\frac{1}{2}\text{Im}(k_{1}w\cdot\bar{z}),k_{1}k_{2}\right).$

Since a right $K$-invariant function on $G$ can be thought as a function on
$\mathbb{H}^{n},$ the Haar measure on $G$ is given by $dg=dzdtdk,$ where
$dzdt$ and $dk$ are the normalized Haar measure on $\mathbb{H}^{n}$ and $K$
respectively.

For $k\in K$ define another set of representations of the Heisenberg group
$\mathbb{H}^{n}$ by $\pi_{\lambda,k}(z,t)=\pi_{\lambda}(kz,t).$ Since
$\pi_{\lambda,k}$ agrees with $\pi_{\lambda}$ on the center of
$\mathbb{H}^{n},$ it follows by Stone-Von Neumann theorem for the Schrödinger
representation that $\pi_{\lambda,k}$ is equivalent to $\pi_{\lambda}.$ Hence
there exists an intertwining operator $\mu_{\lambda}(k)$ satisfying

$\pi_{\lambda}(kz,t)=\mu_{\lambda}(k)\pi_{\lambda}(z,t)\mu_{\lambda}(k)^{\ast}.$

Then $\mu_{\lambda}$ can be thought of as a unitary representation of $K$ on
$L^{2}(\mathbb{R}^{n}),$ called metaplectic representation. For details, we
refer to [3]. Let $(\sigma,\mathcal{H}_{\sigma})$ be an irreducible unitary
representation of $K$ and
$\mathcal{H}_{\sigma}=\text{span}\\{e_{j}^{\sigma}:1\leq j\leq d_{\sigma}\\}.$
For $k\in K,$ the matrix coefficients of the representation $\sigma\in\hat{K}$
are given by

$\varphi_{ij}^{\sigma}(k)=\langle\sigma(k)e_{j}^{\sigma},e_{i}^{\sigma}\rangle,$

where $i,j=1,\ldots,d_{\sigma}.$

Let
$\phi_{\alpha}^{\lambda}(x)=|\lambda|^{\frac{n}{4}}\phi_{\alpha}(\sqrt{|\lambda|}x);~{}\alpha\in\mathbb{Z}_{+}^{n},$
where $\phi_{\alpha}$’s are the Hermite functions on $\mathbb{R}^{n}.$ Since
for each $\lambda\in\mathbb{R}^{\ast},$ the set
$\\{\phi_{\alpha}^{\lambda}:\alpha\in\mathbb{N}^{n}\\}$ forms an orthonormal
basis for $L^{2}(\mathbb{R}^{n}),$ letting
$P_{m}^{\lambda}=\text{span}\\{\phi_{\alpha}^{\lambda}:~{}|\alpha|=m\\},$
$\mu_{\lambda}$ becomes an irreducible unitary representation of $K$ on
$P_{m}^{\lambda}.$ Hence, the action of $\mu_{\lambda}$ can be realized on
$P_{m}^{\lambda}$ by

(2.1)
$\mu_{\lambda}(k)\phi_{\gamma}^{\lambda}=\sum_{|\alpha|=|\gamma|}\eta_{\alpha\gamma}^{\lambda}(k)\phi_{\alpha}^{\lambda},$

where $\eta_{\alpha\gamma}^{\lambda}$’s are the matrix coefficients of
$\mu_{\lambda}(k).$ Define a bilinear form $\phi_{\alpha}^{\lambda}\otimes
e_{j}^{\sigma}$ on $L^{2}(\mathbb{R}^{n})\times\mathcal{H}_{\sigma}$ by
$\phi_{\alpha}^{\lambda}\otimes
e_{j}^{\sigma}=\phi_{\alpha}^{\lambda}e_{j}^{\sigma}.$ Then
$\\{\phi_{\alpha}^{\lambda}\otimes
e_{j}^{\sigma}:\alpha\in\mathbb{N}^{n},1\leq j\leq d_{\sigma}\\}$ forms an
orthonormal basis for $L^{2}(\mathbb{R}^{n})\otimes\mathcal{H}_{\sigma}.$
Denote
$\mathcal{H}_{\sigma}^{2}=L^{2}(\mathbb{R}^{n})\otimes\mathcal{H}_{\sigma}.$

Define a representation $\rho_{\sigma}^{\lambda}$ of $G$ on the space
$\mathcal{H}_{\sigma}^{2}$ by

$\rho_{\sigma}^{\lambda}(z,t,k)=\pi_{\lambda}(z,t)\mu_{\lambda}(k)\otimes\sigma(k).$

In the article [13], it is shown that $\rho_{\sigma}^{\lambda}$ are all
possible irreducible unitary representations of $G$ those participate in the
Plancherel formula. Thus, in view of the above discussion, we shall denote the
partial dual of the group $G$ by
$G^{\prime}\cong\mathbb{R}^{\ast}\times\hat{K}.$

The Fourier transform of $f\in L^{1}(G)$ defined by

$\hat{f}(\lambda,\sigma)=\int_{K}\int_{\mathbb{R}}\int_{\mathbb{C}^{n}}f(z,t,k)\rho_{\sigma}^{\lambda}(z,t,k)dzdtdk,$

is a bounded linear operator on $\mathcal{H}_{\sigma}^{2}.$ As the Plancherel
formula [13]

$\int_{K}\int_{\mathbb{H}^{n}}|f(z,t,k)|^{2}dzdtdk=(2\pi)^{-n}\sum_{\sigma\in\hat{K}}d_{\sigma}\int_{\mathbb{R}\setminus\\{0\\}}\|\hat{f}(\lambda,\sigma)\|^{2}_{HS}|\lambda|^{n}d\lambda$

holds for $f\in L^{2}(G),$ it follows that $\hat{f}(\lambda,\sigma)$ is a
Hilbert-Schmidt operator on $\mathcal{H}_{\sigma}^{2}.$

In order to prove our main result on the Heisenberg motion group $G,$ it is
enough to consider a similar proposition for the Weyl transform on
$G^{\times}=\mathbb{C}^{n}\times K.$ For that, we require to set some
preliminaries about the Weyl transform on $G^{\times}.$

Let $f^{\lambda}$ be the inverse Fourier transform of the function $f$ in the
$t$ variable defined by

(2.2) $\displaystyle f^{\lambda}(z,k)=\int_{\mathbb{R}}f(z,t,k)e^{i\lambda
t}dt.$

Then

$\hat{f}(\lambda,\sigma)=\int_{K}\int_{\mathbb{C}^{n}}f^{\lambda}(z,k)\rho_{\sigma}^{\lambda}(z,k)dzdk,$

where $\rho_{\sigma}^{\lambda}(z,k)=\rho_{\sigma}^{\lambda}(z,0,k).$

For $(\lambda,\sigma)\in G^{\prime},$ define the Weyl transform
$W_{\sigma}^{\lambda}$ on $L^{1}(G^{\times})$ by

$W_{\sigma}^{\lambda}(g)=\int_{K}\int_{\mathbb{C}^{n}}g(z,k)\rho_{\sigma}^{\lambda}(z,k)dzdk.$

Then $\hat{f}(\lambda,\sigma)=W_{\sigma}^{\lambda}(f^{\lambda}),$ and hence
$W_{\sigma}^{\lambda}(g)$ is a bounded operator if $g\in L^{1}(G^{\times}).$
On the other hand, if $g\in L^{2}(G^{\times})$ then $W_{\sigma}^{\lambda}(g)$
becomes a Hilbert-Schmidt operator satisfying the Plancherel formula

(2.3)
$\displaystyle\int_{K}\int_{\mathbb{C}^{n}}|g(z,k)|^{2}dzdk=(2\pi)^{-n}|\lambda|^{n}\sum_{\sigma\in\hat{K}}d_{\sigma}\left|\left|W_{\sigma}^{\lambda}(g)\right|\right|_{HS}^{2}.$

Next, we prove the inversion formula for the Weyl transform
$W_{\sigma}^{\lambda}$ which is a key ingredient while proving our main
result. For this, we need the fact that

(2.4)
$\rho_{\sigma}^{\lambda}(z,k_{1})\rho_{\sigma}^{\lambda}(w,k_{2})=e^{-\frac{i\lambda}{2}\text{Im}(k_{1}w.\bar{z})}\rho_{\sigma}^{\lambda}(z+k_{1}w,k_{1}k_{2}),$

where $(z,k),(w,s)\in G^{\times}.$

######  Theorem 2.1.

$(\textbf{Inversion formula})$ Let $g\in L^{1}\cap L^{2}(G^{\times}).$ Then

(2.5)
$g(z,k)=(2\pi)^{-n}\sum\limits_{\sigma\in\hat{K}}d_{\sigma}\text{tr}(W_{\sigma}^{\lambda}(g)(\rho_{\sigma}^{\lambda})^{*}(z,k)),$

where the series converges in $L^{2}$\- sense.

###### Proof.

For $(z,k_{1})\in G^{\times},$ we have

$\displaystyle W_{\sigma}^{\lambda}(g)(\rho_{\sigma}^{\lambda})^{*}(z,k_{1})$
$\displaystyle=\int_{G^{\times}}g(w,k_{2})\rho_{\sigma}^{\lambda}(w,k_{2})\rho_{\sigma}^{\lambda}(-k_{1}^{-1}z,k_{1}^{-1})dwdk_{2}$
$\displaystyle=\int_{G^{\times}}g(w,k_{2})e^{\frac{i\lambda}{2}\text{Im}(k_{2}k_{1}^{-1}z.\bar{w})}\rho_{\sigma}^{\lambda}(w-k_{2}k_{1}^{-1}z,k_{2}k_{1}^{-1})dwdk_{2}.$

Hence
$\text{tr}\left(W_{\sigma}^{\lambda}(g)(\rho_{\sigma}^{\lambda})^{*}(z,k_{1})\right)$
is equal to

$\displaystyle\sum\limits_{\begin{subarray}{c}\gamma\in\mathbb{N}^{n}\\\ 1\leq
j\leq
d_{\sigma}\end{subarray}}\int_{G^{\times}}g(w,k_{2})e^{\frac{i\lambda}{2}\text{Im}(k_{2}k_{1}^{-1}z.\bar{w})}\left\langle\rho_{\sigma}^{\lambda}(w-k_{2}k_{1}^{-1}z,k_{2}k_{1}^{-1})(\phi_{\gamma}^{\lambda}\otimes
e_{j}^{\sigma}),\phi_{\gamma}^{\lambda}\otimes e_{j}^{\sigma}\right\rangle
dwds.$

By (2.1), the above expression takes the form

$\displaystyle\sum\limits_{\begin{subarray}{c}\gamma\in\mathbb{N}^{n}\\\ 1\leq
j\leq
d_{\sigma}\end{subarray}}\sum_{|\alpha|=|\gamma|}\int_{K}\eta_{\gamma\alpha}^{\lambda}(k_{2}k_{1}^{-1})\int_{\mathbb{C}^{n}}g(w,k_{2})e^{\frac{i\lambda}{2}\text{Im}(k_{2}k_{1}^{-1}z.\bar{w})}\phi_{\alpha\gamma}^{\lambda}(w-k_{2}k_{1}^{-1}z)\varphi_{jj}^{\sigma}(k_{2}k_{1}^{-1})dwdk_{2},$

where
$\phi_{\alpha\gamma}^{\lambda}(x)=\langle\pi_{\lambda}(x)\phi_{\alpha}^{\lambda},\phi_{\gamma}^{\lambda}\rangle.$
Then by the Peter-Weyl theorem for the compact groups, we derive that

$\displaystyle\sum\limits_{\sigma\in\hat{K}}d_{\sigma}\text{tr}(W_{\sigma}^{\lambda}(g)(\rho_{\sigma}^{\lambda})^{*}(z,k_{1}))$
$\displaystyle=\sum\limits_{\gamma\in\mathbb{N}^{n}}\sum_{|\alpha|=|\gamma|}\eta_{\gamma\alpha}^{\lambda}(I)\int_{\mathbb{C}^{n}}g(w,k_{1})e^{\frac{i\lambda}{2}\text{Im}(z.\bar{w})}\phi_{\alpha\gamma}^{\lambda}(w-z)dw$
$\displaystyle=\sum\limits_{\gamma\in\mathbb{N}^{n}}\int_{\mathbb{C}^{n}}g(w,k_{1})e^{\frac{i\lambda}{2}\text{Im}(z.\bar{w})}\phi_{\gamma\gamma}^{\lambda}(w-z)dw,$

where $I$ is the identity element in $K.$ Thus in view of the inversion
formula for the Weyl transform on the Heisenberg group, we infer that

(2.6)
$\displaystyle\sum\limits_{\sigma\in\hat{K}}d_{\sigma}\text{tr}(W_{\sigma}^{\lambda}(g)(\rho_{\sigma}^{\lambda})^{*}(z,k_{1}))=(2\pi)^{n}g(z,k_{1}).$

∎

For simplicity, we assume $\lambda=1$ and denote $W_{\sigma}=W_{\sigma}^{1}.$
Define the Fourier-Wigner transform $V_{\zeta}^{\eta}$ of the functions
$\zeta,\eta\in\mathcal{H}_{\sigma}^{2}$ on $G^{\times}$ by

$V_{\zeta}^{\eta}(z,k)=\left\langle\rho_{\sigma}(z,k)\zeta,\eta\right\rangle.$

The following orthogonality relation is derived in [5]. A version of this
result also appeared in [13].

######  Lemma 2.2.

For $\zeta_{l},\eta_{l}\in\mathcal{H}_{\sigma}^{2},~{}l=1,2,$ the following
orthogonality relation holds true.

$\int_{K}\int_{\mathbb{C}^{n}}V_{\zeta_{1}}^{\eta_{1}}(z,k)\overline{V_{\zeta_{2}}^{\eta_{2}}(z,k)}dzdk=(2\pi)^{n}\left\langle\zeta_{1},\zeta_{2}\right\rangle\overline{\left\langle\eta_{1},\eta_{2}\right\rangle}.$

In particular, $V_{\zeta}^{\eta}\in L^{2}(G^{\times}).$ Let
$V_{\sigma}=\overline{\text{span}}\\{V_{\zeta}^{\eta}:\zeta,\eta\in\mathcal{H}_{\sigma}^{2}\\}$
and set $\Psi_{\alpha,j}^{\sigma}=\phi_{\alpha}\otimes e_{j}^{\sigma}.$ Since
$B_{\sigma}=\\{\Psi_{\alpha,j}^{\sigma}~{}:\alpha\in\mathbb{N}^{n},1\leq j\leq
d_{\sigma}\\}$ forms an orthonormal basis for $\mathcal{H}_{\sigma}^{2},$ by
Lemma 2.2, we infer that

$V_{B_{\sigma}}=\left\\{V_{\Psi_{\alpha_{1},j_{1}}^{\sigma}}^{\Psi_{\alpha_{2},j_{2}}^{\sigma}}:~{}\Psi_{\alpha_{1},j_{1}}^{\sigma},\Psi_{\alpha_{2},j_{2}}^{\sigma}\in
B_{\sigma}\right\\}$

is an orthonormal basis for $V_{\sigma}.$ Next, we recall the Peter-Weyl
theorem, which as a corollary, gives Proposition 2.4.

######  Theorem 2.3.

[14] (Peter-Weyl). Let $\hat{K}$ be the unitary dual of the compact group $K.$
Then $\\{\sqrt{d_{\sigma}}\phi_{ij}^{\sigma}:1\leq i,j\leq
d_{\sigma},\sigma\in\hat{K}\\}$ is an orthonormal basis for $L^{2}(K).$

######  Proposition 2.4.

The set $\bigcup\limits_{\sigma\in\hat{K}}V_{B_{\sigma}}$ is an orthonormal
basis for $L^{2}(G^{\times}).$

Moreover, $V_{B_{\sigma}}$ is an orthonormal basis for $V_{\sigma},$ and hence
by Proposition 2.4, we infer that
$L^{2}(G^{\times})=\bigoplus\limits_{\sigma\in\hat{K}}V_{\sigma}.$

For convenience, throughout this section, we shall assume $A$ is a Lebesgue
measurable subset of $\mathbb{C}^{n}$ with finite measure. Next, we define a
set of orthogonal projection operators which is core in formulating a problem
analogous to the Benedicks-Amrein-Berthier theorem.

Let $\mathcal{B}_{\sigma}^{N}$ be an $N$ dimensional subspace of
$\mathcal{H}_{\sigma}^{2}.$ Then there exists an orthonormal basis
$\\{\psi_{l}^{\sigma}:l\in\mathbb{N}\\}$ of $\mathcal{H}_{\sigma}^{2}$ such
that $\mathcal{B}_{\sigma}^{N}=\left\\{\psi_{l}^{\sigma}:1\leq l\leq
N\right\\}.$ For $\sigma_{o}\in\hat{K},$ define an orthogonal projection
$P_{N}$ of $\mathcal{H}_{\sigma_{o}}^{2}$ onto
$\mathcal{R}(P_{N})=\mathcal{B}_{\sigma_{o}}^{N}.$ Further, we define a pair
of orthogonal projections $E_{A}$ and $F_{N}$ of $L^{2}(G^{\times})$ by

$E_{A}g=\chi_{A\times K}~{}g\quad\mbox{and}\quad
W_{\sigma}(F_{N}g)=\begin{cases}P_{N}W_{\sigma_{o}}(g)&\text{if
}\sigma=\sigma_{o},\\\ 0&\text{otherwise,}\end{cases}$

where $\chi_{A\times K}$ denotes the characteristic function of $A\times K.$
Then it is easy to see that $\mathcal{R}(E_{A})=\\{g\in
L^{2}(G^{\times}):g=g~{}\chi_{A\times K}\\}$ and

$\mathcal{R}(F_{N})=\\{g\in
L^{2}(G^{\times}):\mathcal{R}(W_{\sigma_{o}}(g))\subseteq\mathcal{B}_{\sigma_{o}}^{N}\text{
and }\mathcal{R}(W_{\sigma}(g))=0,\,\sigma(\neq\sigma_{o})\in\hat{K}\\}.$

Now, we derive a key lemma that enables us to recognize $E_{A}F_{N}$ as an
integral operator.

######  Lemma 2.5.

$E_{A}F_{N}$ is an integral operator on $L^{2}(G^{\times}).$

###### Proof.

Let $g\in L^{2}(G^{\times})$. By inversion formula (2.5) we get

$\displaystyle(F_{N}g)(z,k_{1})$
$\displaystyle=a_{\sigma_{o}}\text{tr}(\rho_{\sigma_{o}}^{*}(z,k_{1})W_{\sigma_{o}}(F_{N}g))$
$\displaystyle=a_{\sigma_{o}}\text{tr}(P_{N}W_{\sigma_{o}}(g)\rho_{\sigma_{o}}^{*}(z,k_{1}))$
$\displaystyle=a_{\sigma_{o}}\int_{K}\int_{\mathbb{C}^{n}}g(w,k_{2})tr\left(P_{N}\rho_{\sigma_{o}}(w,k_{2})\rho_{\sigma_{o}}^{*}(z,k_{1})\right)dwdk_{2},$

where $a_{\sigma_{o}}=(2\pi)^{-n}d_{\sigma_{o}}.$ Hence

$\displaystyle(E_{A}F_{N}g)(z,k_{1})$
$\displaystyle=a_{\sigma_{o}}\chi_{A\times K}(z,k_{1})(F_{N}g)(z,k_{1})$
$\displaystyle=a_{\sigma_{o}}\chi_{A\times
K}(z,k_{1})\int_{K}\int_{\mathbb{C}^{n}}g(w,k_{2})\text{tr}\left(P_{N}\rho_{\sigma_{o}}(w,k_{2})\rho_{\sigma_{o}}^{*}(z,k_{1})\right)dwdk_{2}$
$\displaystyle=a_{\sigma_{o}}\int_{K}\int_{\mathbb{C}^{n}}g(w,k_{2})\mathcal{K}((z,k_{1}),(w,k_{2}))dwdk_{2},$

where $\mathcal{K}\left((z,k_{1}),(w,k_{2})\right)=a_{\sigma_{o}}\chi_{A\times
K}(z,k_{1})\text{tr}\left(P_{N}\rho_{\sigma_{o}}(w,k_{2})\rho_{\sigma_{o}}^{*}(z,k_{1})\right).$
∎

Further, the integral operator $E_{A}F_{N}$ is a Hilbert-Schmidt operator and
satisfies the following dimension condition.

######  Lemma 2.6.

$E_{A}F_{N}$ is a Hilbert-Schmidt operator with
$\|E_{A}F_{N}\|_{HS}^{2}=c_{\sigma_{o}}m(A)N$, where
$c_{\sigma_{o}}=(2\pi)^{n}m(K)a_{\sigma_{o}}^{2}.$

###### Proof.

From Lemma 2.5 it follows that

$\displaystyle\|E_{A}F_{N}\|_{HS}^{2}$
$\displaystyle=\int_{G^{\times}}\int_{G^{\times}}|\mathcal{K}((z,k_{1}),(w,k_{2}))|^{2}dwdk_{2}dzdk_{1}$
$\displaystyle=a_{\sigma_{o}}^{2}\int_{G^{\times}}|\chi_{A\times
K}(z,k_{1})|^{2}\int_{G^{\times}}|\text{tr}\left(P_{N}\rho_{\sigma_{o}}(w,k_{2})\rho_{\sigma_{o}}^{*}(z,k_{1})\right)|^{2}dwdk_{2}dzdk_{1}.$

Now,

$\displaystyle\int_{G^{\times}}|\text{tr}\left(P_{N}\rho_{\sigma_{o}}(w,k_{2})\rho_{\sigma_{o}}^{*}(z,k_{1})\right)|^{2}dwdk_{2}$
$\displaystyle=$ $\displaystyle\int_{G^{\times}}\Big{|}\sum\limits_{1\leq
l\leq
N}\langle\rho_{\sigma_{o}}(w,k_{2})\rho_{\sigma_{o}}^{*}(z,k_{1})\psi_{l}^{\sigma_{o}},\psi_{l}^{\sigma_{o}}\rangle\Big{|}^{2}dwdk_{2}$
$\displaystyle=$ $\displaystyle\int_{G^{\times}}\Big{|}\sum\limits_{1\leq
l\leq
N}\langle\rho_{\sigma_{o}}(w,k_{2})\eta_{l}^{\sigma_{o}},\psi_{l}^{\sigma_{o}}\rangle\Big{|}^{2}dwdk_{2},$

where
$\eta_{l}^{\sigma_{o}}=\rho_{\sigma_{o}}^{*}(z,k_{1})\psi_{l}^{\sigma_{o}}\in\mathcal{H}_{\sigma_{o}}^{2}.$
Since,

$\langle\eta_{l_{1}}^{\sigma_{o}},\eta_{l_{2}}^{\sigma_{o}}\rangle=\langle\rho_{\sigma_{o}}^{*}(z,k_{1})\psi_{l_{1}}^{\sigma_{o}},\rho_{\sigma_{o}}^{*}(z,k_{1})\psi_{l_{2}}^{\sigma_{o}}\rangle=\langle\psi_{l_{1}}^{\sigma_{o}},\psi_{l_{2}}^{\sigma_{o}}\rangle=\delta_{l_{1}l_{2}},$

the above integral becomes

$\displaystyle\int_{G^{\times}}\Big{|}\sum\limits_{1\leq l\leq
N}\langle\rho_{\sigma_{o}}(w,k_{2})\eta_{l}^{\sigma_{o}},\psi_{l}^{\sigma_{o}}\rangle\Big{|}^{2}dwdk_{2}=\int_{G^{\times}}\Big{|}\sum\limits_{1\leq
l\leq
N}V_{\eta_{l}^{\sigma_{o}}}^{\psi_{l}^{\sigma_{o}}}(w,k_{2})\Big{|}^{2}dwdk_{2}$
$\displaystyle=$ $\displaystyle\sum\limits_{1\leq l_{1},l_{2}\leq
N}\int_{G^{\times}}V_{\eta_{l_{1}}^{\sigma_{o}}}^{\psi_{l_{1}}^{\sigma_{o}}}(w,k_{2})\overline{V_{\eta_{l_{2}}^{\sigma_{o}}}^{\psi_{l_{2}}^{\sigma_{o}}}(w,k_{2})}dwdk_{2}$
$\displaystyle=$ $\displaystyle(2\pi)^{n}\sum\limits_{1\leq l_{1},l_{2}\leq
N}\langle\eta_{l_{1}}^{\sigma_{o}},\eta_{l_{2}}^{\sigma_{o}}\rangle\overline{\langle\psi_{l_{1}}^{\sigma_{o}},\psi_{l_{2}}^{\sigma_{o}}\rangle}=(2\pi)^{n}N.$

Thus, $\|E_{A}F_{N}\|_{HS}^{2}=a_{\sigma_{o}}^{2}m(A)m(K)(2\pi)^{n}N.$ ∎

We need the following result that describes an interesting property of
Lebesgue measurable sets [1]. Denote $wA=\\{z\in\mathbb{C}^{n}:z-w\in A\\}.$

######  Lemma 2.7.

[1] Let $B$ be a measurable set in $\mathbb{C}^{n}$ with $0<m(B)<\infty.$ If
$B_{0}$ is a measurable subset of $B$ with $m(B_{0})>0,$ then for each
$\epsilon>0$ there exists $w\in\mathbb{C}^{n}$ such that

$m(B)<m(B\cup wB_{0})<m(B)+\epsilon.$

We also need the following basic fact about the orthogonal projection, which
help in deciding the disjointness of the projections $E_{A}$ and $F_{N}$ while
$m(A)<\infty.$

For given orthogonal projections $E$ and $F$ of a Hilbert space $\mathcal{H},$
let $E\cap F$ denote the orthogonal projection of $\mathcal{H}$ onto
$\mathcal{R}(E)\cap\mathcal{R}(F).$ Then

(2.7) $\displaystyle\|E\cap F\|_{HS}^{2}=\dim\mathcal{R}(E\cap
F)\leq\|EF\|_{HS}^{2}.$

Let $F_{N}^{\perp}=I-F_{N},$ and $A^{\prime}$ be the complement of $A.$

######  Proposition 2.8.

Let $A$ be a measurable subset of $\mathbb{C}^{n}$ of finite Lebesgue measure.
Then the projection $E_{A}\cap F_{N}=0$.

###### Proof.

Assume towards a contradiction that there exists a non-zero function $g_{0}$
in $\mathcal{R}(E_{A}\cap F_{N})$. Then
$\mathcal{R}(W_{\sigma_{o}}(g))\subseteq\mathcal{B}_{\sigma_{o}}^{N}$ and
$\mathcal{R}(W_{\sigma}(g))=0$ for $\sigma(\neq\sigma_{o})\in\hat{K}$.
Consider $A_{0}=\\{z\in A:g_{0}(z,k)\neq 0\text{ for some }k\in K\\}$. Then
$0<m(A_{0})<\infty$. Choose $s\in\mathbb{N}$ such that
$s>2c_{\sigma_{o}}m(A_{0})N.$ Now, we construct an increasing sequence of sets
$\\{A_{l}:l=1,\ldots,s\\}.$ Using Lemma 2.7 with
$\epsilon=\frac{1}{2c_{\sigma_{o}}N}$, $B_{0}=A_{0}$ and $B=A_{l-1},$ there
exists $w_{l}\in\mathbb{C}^{n}$ such that

$m(A_{l-1})<m(A_{l-1}\cup w_{l}A_{0})<m(A_{l-1})+\frac{1}{2c_{\sigma_{o}}N}.$

Denote $A_{l}=A_{l-1}\cup w_{l}A_{0}$. Then from (2.7), we get

(2.8) $\displaystyle\dim\mathcal{R}(E_{A_{s}}\cap F_{N})\leq
c_{\sigma_{o}}m(A_{s})N<\left\\{m(A_{0})+\frac{s}{2c_{\sigma_{o}}N}\right\\}c_{\sigma_{o}}N<s.$

On the other hand, we construct $s+1$ linearly independent functions in the
space $\mathcal{R}(E_{A_{s}}\cap F_{N}),$ after verifying $\mathcal{R}(F_{N})$
is a twisted translation invariant space.

Let $g_{l}(z,k)=e^{\frac{i}{2}Im(z.\bar{w_{l}})}g_{0}(z-w_{l},k)$. Then for
$\eta^{\sigma_{o}}\in\mathcal{H}_{\sigma_{o}}^{2}$ and $p>N,$ we have

$\displaystyle\langle
W_{\sigma_{o}}(g_{l})\eta^{\sigma_{o}},\psi_{p}^{\sigma_{o}}\rangle$
$\displaystyle=\int_{G^{\times}}g_{l}(z,k)\langle\rho_{\sigma_{o}}(z,k)\eta^{\sigma_{o}},\psi_{p}^{\sigma_{o}}\rangle
dzdk$
$\displaystyle=\int_{G^{\times}}e^{\frac{i}{2}Im(z.\bar{w_{l}})}g_{0}(z-w_{l},k)\langle\rho_{\sigma_{o}}(z,k)\eta^{\sigma_{o}},\psi_{p}^{\sigma_{o}}\rangle
dzdk$
$\displaystyle=\int_{G^{\times}}e^{\frac{i}{2}Im(z.\bar{w_{l}})}g_{0}(z,k)\langle\rho_{\sigma_{o}}(z+w_{l},k)\eta^{\sigma_{o}},\psi_{p}^{\sigma_{o}}\rangle
dzdk.$

Since
$\rho_{\sigma_{o}}(z,k)\rho_{\sigma_{o}}(k^{-1}w,I)=e^{\frac{i}{2}Im(z.\bar{w})}\rho_{\sigma_{o}}(z+w,k),$
we get

$\displaystyle\langle
W_{\sigma_{o}}(g_{l})\eta^{\sigma_{o}},\psi_{p}^{\sigma_{o}}\rangle$
$\displaystyle=\int_{G^{\times}}g_{0}(z,k)\langle\rho_{\sigma_{o}}(z,k)\rho_{\sigma_{o}}(k^{-1}w_{l},I)\eta^{\sigma_{o}},\psi_{p}^{\sigma_{o}}\rangle
dzdk$
$\displaystyle=\int_{G^{\times}}g_{0}(z,k)\langle\rho_{\sigma_{o}}(z,k)\zeta^{\sigma_{o}},\psi_{p}^{\sigma_{o}}\rangle
dzdk$ $\displaystyle=\langle
W(g_{0})\zeta^{\sigma_{o}},\psi_{p}^{\sigma_{o}}\rangle=0.$

Thus
$\mathcal{R}(W_{\sigma_{o}}(g_{l}))\subseteq\mathcal{B}_{\sigma_{o}}^{N}.$
Similarly, it can be shown that $\mathcal{R}(W_{\sigma}(g_{l}))=0$ for
$\sigma(\neq\sigma_{o})\in\hat{K}.$ Since $A_{m}=A_{0}\cup
w_{1}A_{0}\cup\cdots\cup w_{m}A_{0}$ and $g_{l}=0$ on $(w_{l}A_{0})^{\prime}$,
we have $E_{A_{m}}g_{l}=g_{l}$ for $l=0,1,\ldots,m$. Furthermore,
$E_{A_{m}\setminus A_{m-1}}g_{l}=0$ for $l=0,\ldots,m-1$ and
$E_{A_{m}\setminus A_{m-1}}g_{m}\neq 0$. Therefore, it shows that $g_{m}$ is
not a linear combination of $g_{0},\ldots,g_{m-1}$. Hence,
$g_{0},\ldots,g_{s}$ are $s+1$ linearly independent functions in
$\mathcal{R}(E_{A_{s}}\cap F_{N})$ that contradicts (2.8). This completes the
proof. ∎

Now, we prove Benedicks-Amrein-Berthier theorem for the Weyl transform
$W_{\sigma}.$ For this, we recall the following fact. Let $g\in
L^{2}(G^{\times}).$ By Proposition 2.4 we get
$g=\bigoplus\limits_{\sigma\in\hat{K}}g_{\sigma}.$

######  Proposition 2.9.

Let $g\in L^{2}(G^{\times})$ and $\\{(z,k)\in G^{\times}:g_{\sigma}(z,k)\neq
0\\}\subseteq A_{\sigma}\times K,$ where $m(A_{\sigma})<\infty,$ whenever
$\sigma\in\hat{K}.$ If $W_{\sigma}(g)$ is a finite rank operator for each
$\sigma,$ then $g=0.$

###### Proof.

For $\varphi,\psi\in\mathcal{H}_{\sigma}^{2}$ we have

$\displaystyle\left\langle W_{\sigma}(g)\varphi,\psi\right\rangle$
$\displaystyle=\int_{K}\int_{\mathbb{C}^{n}}g(z,k)\left\langle\rho_{\sigma}(z,k)\varphi,\psi\right\rangle
dzdk$
$\displaystyle=\int_{K}\int_{\mathbb{C}^{n}}{g_{\sigma}}(z,k)\left\langle\rho_{\sigma}(z,k)\varphi,\psi\right\rangle
dzdk$ $\displaystyle=\left\langle
W_{\sigma}(g_{\sigma})\varphi,\psi\right\rangle.$

Hence,
$\mathcal{R}(W_{\sigma_{o}}(g_{\sigma_{o}}))=\mathcal{R}(W_{\sigma_{o}}(g))$
be a finite dimensional subspace of $\mathcal{H}_{\sigma_{o}}^{2}$ and
$\mathcal{R}(W_{\sigma}(g_{\sigma_{o}}))=0$ for
$\sigma(\neq\sigma_{o})\in\hat{K}$. Thus by Proposition 2.8 we get
$g_{\sigma_{o}}=0.$ Since $\sigma_{o}\in\hat{K}$ is arbitrary, we infer that
$g=0.$ ∎

As a consequence of Proposition 2.9, we obtained the following Benedicks-
Amrein-Berthier theorem for the Heisenberg motion group. If $f\in L^{2}(G),$
then from Proposition 2.4 we get
$f^{\lambda}=\bigoplus\limits_{\sigma\in\hat{K}}f^{\lambda}_{\sigma}.$

######  Theorem 2.10.

Let $f\in L^{2}(G)$ be such that $\\{(z,k)\in
G^{\times}:f^{\lambda}_{\sigma}(z,k)\neq 0\\}\subseteq A_{\sigma}\times K,$
where $m(A_{\sigma})<\infty$ for each $\sigma\in\hat{K}.$ If
$\hat{f}(\lambda,\sigma)$ is a finite rank operator for each
$(\lambda,\sigma)\in\mathbb{R}^{*}\times\hat{K},$ then $f=0.$

##  3\. Quaternion Heisenberg group

In this section, we prove an analogue of Benedicks-Amrein-Berthier theorem for
the quaternion Heisenberg group. For this, we describe notations and
preliminary facts about the quaternion Heisenberg group. For more details, see
[4, 6].

The quaternion Heisenberg group is step two nilpotent Lie group with centre
$\mathbb{R}^{3}.$ Let $\mathbb{Q}$ be the set of all quaternions. For
$q=q_{0}+iq_{1}+jq_{2}+kq_{3}\in\mathbb{Q}$, the conjugate of $q$ is defined
by $\bar{q}=q_{0}-iq_{1}-jq_{2}-kq_{3}.$ The inner product in $\mathbb{Q}$ is
defined by $\langle q,\tilde{q}\rangle=\text{Re}(\bar{q}\tilde{q}).$ This
leads to $|q|^{2}=\langle q,q\rangle=\sum\limits_{l=0}^{3}q_{l}^{2},$ and we
get the relations $\overline{q\tilde{q}}=\bar{\tilde{q}}\bar{q}$ and
$|q\tilde{q}|=|q\|\tilde{q}|.$ The set
$\mathcal{Q}=\mathbb{Q}\times\mathbb{R}^{3}=\\{(q,t):q\in\mathbb{Q},t\in\mathbb{R}^{3}\\}$
becomes a non-commutative group when equipped with the group law

$(q,t)(\tilde{q},\tilde{t})=(q+\tilde{q},t+\tilde{t}-2~{}\text{Im}(\bar{\tilde{q}}q)).$

It is easy to see that the Lebesgue measure $dqdt$ on
$\mathbb{Q}\times\mathbb{R}^{3}$ is the Haar measure on $\mathcal{Q}.$ For
$1\leq p\leq\infty,$ $L^{p}(\mathcal{Q})$ denotes the usual $L^{p}$ space of
all complex-valued functions on $\mathbb{Q}\times\mathbb{R}^{3}.$

Let $a\in\text{Im }\mathbb{Q}\smallsetminus\\{0\\}.$ Then $J_{a}:q\mapsto
q\cdot\frac{a}{|a|}$ defines a complex structure on $\mathbb{Q}.$ Let
$\mathcal{F}_{a}$ be the Fock space of all holomorphic functions $F$ with
quaternion values on $(\mathbb{Q},J_{a})$ such that

$\|F\|_{2}^{2}=\int_{\mathbb{Q}}|F(q)|^{2}e^{-2|a\|q|^{2}}dq<\infty.$

An irreducible unitary representation $\pi_{a}$ of $\mathcal{Q}$ realized on
$\mathcal{F}_{a}$ is given by

$\pi_{a}(q,t)F(\tilde{q})=e^{i\langle
a,t\rangle-|a|\left(|q|^{2}+2\langle\tilde{q},q\rangle-2i\langle\tilde{q}.\frac{a}{|a|},q\rangle\right)}F(\tilde{q}+q),$

where $F\in\mathcal{F}_{a}.$ Up to unitary equivalence, $\pi_{a}$’s are all
the infinite dimensional irreducible unitary representations of $\mathcal{Q}.$
For $f\in L^{1}(\mathcal{Q})$ the group Fourier transform can be expressed as

$\hat{f}(a)=\int_{\mathcal{Q}}f(q,t)\pi_{a}(q,t)dqdt.$

For $f\in L^{2}(\mathcal{Q}),$ the following Plancheral formula holds.

(3.1) $\|f\|_{2}^{2}=\frac{1}{2\pi^{5}}\int_{\text{Im
}\mathbb{Q}\setminus\\{0\\}}\|\hat{f}(a)\|_{\text{HS}}^{2}|a|^{2}da.$

Let

$f^{a}(q)=\int_{\mathbb{R}^{3}}f(q,t)\pi_{a}(q,t)dt,$

the inverse Fourier transform of $f$ in the $t$ variable. If we denote
$\pi_{a}(q)=\pi_{a}(q,0),$ then the Weyl transform of $g\in L^{1}(\mathbb{Q})$
can be define by

$\displaystyle W_{a}(g)=\int_{\mathbb{Q}}g(q)\pi_{a}(q)dq.$

Hence it follows that $\hat{f}(a)=W_{a}(f^{a}).$ Further, $W_{a}(g)$ is a
bounded operator if $g\in L^{1}(\mathbb{Q})$ and a Hilbert-Schmidt operator
when $g\in L^{2}(\mathbb{Q}).$ In addition, when $g\in L^{2}(\mathbb{Q}),$ the
Plancheral formula for the Weyl transform $W_{a}$ is given by

(3.2)
$\displaystyle\|W_{a}(g)\|_{\text{HS}}^{2}=\frac{\pi^{2}}{4|a|^{2}}\|g\|^{2}.$

Finally, inversion formula for the Weyl transform is given by

(3.3) $\displaystyle
g(q)=\frac{4|a|^{2}}{\pi^{2}}\text{tr}\left(\pi_{a}^{*}(q)W_{a}(g)\right).$

In order to prove an orthogonality relation for $\pi_{a},$ we need to recall
the following Schur’s orthogonality relation for the unimodular groups, see
[8].

######  Proposition 3.1.

[8] Let $G$ be a unimodular group and $\pi$ be an irreducible unitary
representation of $G$ on a Hilbert space $H.$ Then $\langle\pi(g)h,h\rangle\in
L^{2}(G)$ for some non-zero $h\in H$ if and only if

$\displaystyle\int_{G}\langle\pi(g)h_{1},k_{1}\rangle\overline{\langle\pi(g)h_{2},k_{2}\rangle}dg=c\langle
h_{1},h_{2}\rangle\overline{\langle k_{1},k_{2}\rangle}$

for all $h_{l},k_{l}\in H,l=1,2,$ where $c$ is a constant depending on $\pi.$

######  Lemma 3.2.

Let $\varphi_{l},\psi_{l}\in\mathcal{F}_{a}$ for $l=1,2.$ Then,

(3.4)
$\displaystyle\int_{\mathbb{Q}}\langle\pi_{a}(q)\varphi_{1},\psi_{1}\rangle\overline{\langle\pi_{a}(q)\varphi_{2},\psi_{2}}\rangle
dq=c_{a}\langle\varphi_{1},\varphi_{2}\rangle\overline{\langle\psi_{1},\psi_{2}\rangle},$

where $c_{a}$ is some constant.

###### Proof.

For $q=q_{0}+iq_{1}+jq_{2}+kq_{3}\in\mathbb{Q},$ write $z_{1}=q_{0}+iq_{1}$
and $z_{2}=q_{3}+iq_{4}.$ Then
$\varphi(q)=\frac{4|a|^{2}}{\pi}z_{1}z_{2}\in\mathcal{F}_{a}$ and
$\|\varphi\|_{2}=1.$ Further,

$\displaystyle\int_{\mathbb{Q}}|\langle\pi_{a}(q)\varphi,\varphi\rangle|^{2}dq$
$\displaystyle=\int_{\mathbb{Q}}\Big{|}\int_{\mathbb{Q}}\pi_{a}(q)\varphi(\tilde{q})\overline{\varphi(\tilde{q})}e^{-2|a||\tilde{q}|^{2}}d\tilde{q}\Big{|}^{2}dq$
$\displaystyle=\int_{\mathbb{Q}}\Big{|}\int_{\mathbb{Q}}e^{-|a|(|q|^{2}+2\langle\tilde{q},q\rangle-2i\langle\tilde{q}\frac{a}{|a|},q\rangle}\varphi(\tilde{q}+q)\overline{\varphi(\tilde{q})}e^{-2|a||\tilde{q}|^{2}}d\tilde{q}\Big{|}^{2}dq.$

By Minkowski’s integral inequality, it follows that

$\displaystyle\left(\int_{\mathbb{Q}}|\langle\pi_{a}(q)\varphi,\varphi\rangle|^{2}dq\right)^{\frac{1}{2}}$
$\displaystyle\leq\int_{\mathbb{Q}}\left(\int_{\mathbb{Q}}|\varphi(\tilde{q}+q)|^{2}e^{-2|a||\tilde{q}+q|^{2}}|\varphi(\tilde{q})|^{2}e^{-2|a||\tilde{q}|^{2}}dq\right)^{\frac{1}{2}}d\tilde{q}$
$\displaystyle=\|\varphi\|_{2}\int_{\mathbb{Q}}|\varphi(\tilde{q})|e^{-|a||\tilde{q}|^{2}}d\tilde{q}$
$\displaystyle=\frac{4|a|^{2}}{\pi}\int_{\mathbb{C}^{2}}|z_{1}\|z_{2}|e^{-|a|(|z_{1}|^{2}+|z_{2}|^{2})}dz_{1}dz_{2}<\infty.$

Hence by Proposition 3.1, we get

$\displaystyle\int_{\mathbb{Q}}\langle\pi_{a}(q)\varphi_{1},\psi_{1}\rangle\overline{\langle\pi_{a}(q)\varphi_{2},\psi_{2}}\rangle
dq=c_{a}\langle\varphi_{1},\varphi_{2}\rangle\overline{\langle\psi_{1},\psi_{2}\rangle}.$

∎

Let $g\in L^{2}(\mathbb{Q})$ and $W_{a}(g)$ be a finite rank operator. Then
there exists an orthonormal basis, say, $\\{e_{1},e_{2},\ldots\\}$ of
$\mathcal{F}_{a}$ such that $\mathcal{R}(W_{a}(g))=\mathcal{B}_{N},$ where
$\mathcal{B}_{N}=\text{span}\\{e_{1},\ldots,e_{N}\\}.$ Define an orthogonal
projection $P_{N}$ of $\mathcal{F}_{a}$ onto $\mathcal{B}_{N}.$ Let $A$ be a
measurable subset of $\mathbb{Q}$. Define a pair of orthogonal projections
$E_{A}$ and $F_{N}$ of $L^{2}(\mathbb{Q})$ by

(3.5) $\displaystyle E_{A}g=\chi_{A}g\qquad\text{ and }\qquad
W_{a}(F_{N}g)=P_{N}W_{a}(g),$

where $\chi_{A}$ denotes the characteristic function of $A.$ Then
$\mathcal{R}(E_{A})=\\{g\in L^{2}(\mathbb{Q}):g=\chi_{A}g\\}$ and
$\mathcal{R}(F_{N})=\\{g\in
L^{2}(\mathbb{Q}):\mathcal{R}(W_{a}(g))\subseteq\mathcal{B}_{N}\\}.$

Next, we prove that $E_{A}F_{N}$ is a Hilbert-Schmidt operator satisfying the
condition $\|E_{A}F_{N}\|_{HS}^{2}=c^{\prime}m(A)N.$ Throughout this section,
we shall assume that $A$ is a measurable subset of $\mathbb{Q}$ with finite
measure.

######  Lemma 3.3.

$E_{A}F_{N}$ is an integral operator on $L^{2}(\mathbb{Q}).$

###### Proof.

For $g\in L^{2}(\mathbb{Q})$, we have $W_{a}(F_{N}g)=P_{N}W_{a}(g)$. By
inversion formula for the Weyl transform

$\displaystyle(F_{N}g)(q)$
$\displaystyle=\tilde{c}_{a}\text{tr}(\pi_{a}(q)^{*}W_{a}(F_{N}g))=\tilde{c}_{a}\text{tr}(\pi_{a}(-q)P_{N}W_{a}(g))$
$\displaystyle=\tilde{c}_{a}\text{tr}(P_{N}W_{a}(g)\pi_{a}(-q))$
$\displaystyle=\tilde{c}_{a}\int_{\mathbb{Q}}g(\tilde{q})\text{tr}\left(P_{N}\pi_{a}(\tilde{q})\pi_{a}(-q)\right)d\tilde{q}.$

Hence, it follows that

$\displaystyle(E_{A}F_{N}g)(q)$
$\displaystyle=\chi_{A}(q)(F_{N}g)(q)=\tilde{c}_{a}\chi_{A}(q)\int_{\mathbb{Q}}g(\tilde{q})\text{tr}\left(P_{N}\pi_{a}(\tilde{q})\pi_{a}(-q)\right)d\tilde{q}$
$\displaystyle=\tilde{c}_{a}\int_{\mathbb{Q}}g(w)K(q,\tilde{q})d\tilde{q},$

where
$K(q,\tilde{q})=\tilde{c}_{a}~{}\chi_{A}(q)\text{tr}\left(P_{N}\pi_{a}(\tilde{q})\pi_{a}(-q)\right)$
and $\pi^{2}\tilde{c}_{a}=4|a|^{2}$. Thus, we infer that $E_{A}F_{N}$ is an
integral operator with kernel $K.$ ∎

######  Lemma 3.4.

$E_{A}F_{N}$ is a Hilbert-Schmidt operator and
$\|E_{A}F_{N}\|_{HS}^{2}=c^{\prime}m(A)N$, for some constant $c^{\prime},$
independent of the choice of $A$ and $N$.

###### Proof.

From Lemma 3.3, it follows that

$\displaystyle\|E_{A}F_{N}\|_{HS}^{2}$
$\displaystyle=\tilde{c}_{a}^{2}\int_{\mathbb{Q}}\int_{\mathbb{Q}}|K(q,\tilde{q})|^{2}d\tilde{q}dq$
$\displaystyle=\tilde{c}_{a}^{2}\int_{\mathbb{Q}}|\chi_{A}(q)|^{2}\left(\int_{\mathbb{Q}}|\text{tr}\,\left(P_{N}\pi_{a}(\tilde{q})\pi_{a}(-q)\right)|^{2}d\tilde{q}\right)dq$
$\displaystyle=\tilde{c}_{a}^{2}\int_{\mathbb{Q}}\chi_{A}(q)\left(\int_{\mathbb{Q}}\Big{|}\sum\limits_{l=1}^{N}\langle\pi_{a}(\tilde{q})\pi(-q)e_{l},e_{l}\rangle\Big{|}^{2}d\tilde{q}\right)dq$
$\displaystyle=\tilde{c}_{a}^{2}\int_{\mathbb{Q}}\chi_{A}(q)\left(\int_{\mathbb{Q}}\Big{|}\sum\limits_{l=1}^{N}e^{-2i\langle\tilde{q}a,q\rangle}\langle\pi_{a}(\tilde{q}-q)e_{l},e_{l}\rangle\Big{|}^{2}d\tilde{q}\right)dq.$

Since
$\pi_{a}(\tilde{q})\pi_{a}(q)=e^{2i\langle\tilde{q}a,q\rangle}\pi_{a}(\tilde{q}+q)),$
we get

$\displaystyle\|E_{A}F_{N}\|_{HS}^{2}=\tilde{c}_{a}^{2}\int_{\mathbb{Q}}\chi_{A}(q)\left(\int_{\mathbb{Q}}\Big{|}\sum\limits_{l=1}^{N}\langle\pi_{a}(\tilde{q})e_{l},e_{l}\rangle\Big{|}^{2}d\tilde{q}\right)dq.$

Hence, from orthogonality relation (3.4) it follows that

$\displaystyle\|E_{A}F_{N}\|_{HS}^{2}=\tilde{c}_{a}^{2}m(A)c_{a}N=c^{\prime}m(A)N<\infty.$

∎

Let $S$ be a closed subspace of $\mathcal{F}_{a}.$ Define $F_{S}$ by
$W_{a}(F_{S}g)=P_{S}W_{a}(g),$ where $P_{S}$ is the orthogonal projection of
$\mathcal{F}_{a}$ onto $S$ and $g\in L^{2}(\mathbb{Q}).$ In particular, if
$S=\mathcal{B}_{N},$ then $F_{S}=F_{N}.$

######  Proposition 3.5.

Let $A\subseteq\mathbb{Q}$ with finite measure, and $S$ be a closed subspace
of $\mathcal{F}_{a}.$ Then, either $E_{A}\cap F_{S}=0$ or for each
$\epsilon^{\prime}>0$ there exists $\tilde{A}\supset A$ with
$m(\tilde{A}\smallsetminus A)<\epsilon^{\prime}$ such that
$\dim\mathcal{R}(E_{\tilde{A}}\cap F_{S})=\infty.$

###### Proof.

If $E_{A}\cap F_{S}\neq 0,$ then there exists a non-zero function
$g_{0}\in\mathcal{R}(E_{A}\cap F_{S})$. Let $A_{0}=\\{x\in A:g_{0}(x)\neq
0\\}$ and $\tilde{A}_{1}=A.$ By Lemma 2.7, for
$\epsilon=\frac{\epsilon^{\prime}}{2^{l}}$, $B_{0}=A_{0}$ and
$B=\tilde{A}_{l},$ there exists $\tilde{q}_{l}\in\mathbb{Q}$ such that

(3.6)
$m(\tilde{A}_{l})<m(\tilde{A}_{l}\cup\tilde{q}_{l}A_{0})<m(\tilde{A}_{l})+\frac{\epsilon^{\prime}}{2^{l}}$

where $l\in\mathbb{N}.$ Put
$\tilde{A}_{l+1}=\tilde{A}_{l}\cup\tilde{q}_{l}A_{0}$ and
$\tilde{A}=\bigcup\limits_{l=1}^{\infty}\tilde{A}_{l}.$ Then $\tilde{A}_{l}$
is a non-decreasing sequence, and hence from (3.6) it follows that
$m(\tilde{A}\smallsetminus A)<\epsilon^{\prime}.$ For $l\in\mathbb{N},$ define
$g_{l}(q)=e^{2i\langle qa,\tilde{q}_{l}\rangle}g_{0}(q-\tilde{q}_{l}).$ We
show that $g_{l}\in\mathcal{R}(E_{\tilde{A}}\cap F_{S})$ for each $l$ and they
are linearly independent. Let $\mathcal{B}_{S}$ be an orthonormal basis of
$S.$ Then we can extend $\mathcal{B}_{S}$ to an orthonormal basis
$\mathcal{B}$ of $\mathcal{F}_{a}.$ For $\varphi\in\mathcal{F}_{a}$ and
$e_{\beta}\in\mathcal{B}\smallsetminus\mathcal{B}_{S},$ we have

$\displaystyle\langle W_{a}(g_{l})\varphi,e_{\beta}\rangle$
$\displaystyle=\int_{\mathbb{Q}}g_{l}(q)\langle\pi_{a}(q)\varphi,e_{\beta}\rangle
dq$ $\displaystyle=\int_{\mathbb{Q}}e^{2i\langle
qa,\tilde{q}_{l}\rangle}g_{0}(q-\tilde{q}_{l})\langle\pi_{a}(q)\varphi,e_{\beta}\rangle
dq.$

Since $\langle(q+\tilde{q}_{l})a,\tilde{q}_{l}\rangle=\langle
qa,\tilde{q}_{l}\rangle,$ by the change of variables

$\displaystyle\langle W_{a}(g_{l})\varphi,e_{\beta}\rangle$
$\displaystyle=\int_{\mathbb{Q}}e^{2i\langle
qa,\tilde{q}_{l}\rangle}g_{0}(q)\langle\pi_{a}(q+\tilde{q}_{l})\varphi,e_{\beta}\rangle
dq.$

We know that $\pi_{a}(q)\pi_{a}(\tilde{q}_{l})=e^{2i\langle
qa,\tilde{q}_{l})\rangle}\pi_{a}(q+\tilde{q}_{l}).$ Therefore,

$\displaystyle\langle W_{a}(g_{l})\varphi,e_{j}\rangle$
$\displaystyle=\int_{\mathbb{Q}}g_{0}(q)\langle\pi_{a}(q)\pi_{a}(\tilde{q}_{l})\varphi,e_{\beta}\rangle
dq$
$\displaystyle=\int_{\mathbb{Q}}g_{0}(q)\langle\pi_{a}(q)\psi,e_{\beta}\rangle
dq$ (3.7) $\displaystyle=\langle W_{a}(g_{0})\psi,e_{\beta}\rangle=0.$

Hence $\mathcal{R}(W_{a}(g_{l}))\subseteq S.$ Let
$A_{l}=A_{l-1}\cup\tilde{q}_{l}A_{0}.$ Then $\tilde{A}_{l+1}=\tilde{A}_{l}\cup
A_{l}.$ Thus, we have $m(A_{l}\smallsetminus A_{l-1})\geq
m(\tilde{A}_{l+1}\smallsetminus\tilde{A}_{l})>0.$ Let $s\in\mathbb{N}.$ Since,
$A_{s}=A_{0}\cup\tilde{q}_{1}A_{0}\cup\cdots\cup\tilde{q}_{s}A_{0}$ and
$g_{l}=0$ on $(\tilde{q}_{l}A_{0})^{\prime}$, we have $E_{A_{s}}g_{l}=g_{l}$
for $l=0,1,\ldots,s$. Furthermore, $E_{A_{s}\smallsetminus A_{s-1}}g_{l}=0$
for $l=0,\ldots,s-1$ and $E_{A_{s}\smallsetminus A_{s-1}}g_{s}\neq 0$.
Therefore, it shows that $g_{s}$ is not a linear combination of
$g_{0},\ldots,g_{s-1}.$ Since $s$ is arbitrary, $\\{g_{l}:l\in\mathbb{N}\\}$
is a linearly independent set in $\mathcal{R}(E_{\tilde{A}}\cap F_{S}).$ ∎

######  Proposition 3.6.

Let $A$ be a measurable subset of $\mathbb{Q}$ having finite measure. Then the
projection $E_{A}\cap F_{N}=0$.

###### Proof.

In view of (2.7) and Lemma 3.4, we obtain the relations

$\dim\mathcal{R}(E_{\tilde{A}}\cap F_{N})\leq c^{\prime}m(\tilde{A})N<\infty.$

Therefore, as a corollary of Proposition 3.5, we get $E_{A}\cap F_{N}=0.$ ∎

The following theorem is our main result of this section, which is an analogue
of Benedicks-Amrein-Berthier theorem on the quaternion Heisenberg group.

######  Theorem 3.7.

Let $A\subseteq\mathbb{Q}$ be a set of finite measure. Suppose $f\in
L^{1}(\mathcal{Q})$ and $\\{(q,t)\in\mathcal{Q}:f(q,t)\neq 0\\}\subseteq
A\times\mathbb{R}^{3}.$ If $\hat{f}(a)$ is a finite rank operator for each
$a\in\text{Im }\mathbb{Q}\setminus\\{0\\}$, then $f=0$.

Proof of Theorem 3.7 follows from the following result for the Weyl transform
on $\mathbb{Q}.$

######  Proposition 3.8.

Let $g\in L^{1}(\mathbb{Q})$ and $\\{q\in\mathbb{Q}:g(q)\neq 0\\}\subseteq A,$
where $m(A)$ is finite. If there exists $a\in\text{Im
}\mathbb{Q}\setminus\\{0\\}$ such that $W_{a}(g)$ has finite rank, then $g=0$.

Since $W_{a}(g)$ is a finite rank operator, by the Plancherel theorem for the
Weyl transform on $\mathbb{Q},$ it follows that $g\in L^{2}(\mathbb{Q})$.
Hence, it is enough to prove Proposition 3.8 for $g\in L^{2}(\mathbb{Q}).$ The
prove of Proposition 3.8 follows from Proposition 3.6.

######  Remark 3.9.

If $0<m(A)<\infty,$ then $\dim\mathcal{R}(E_{A})=\infty.$ In view of
Proposition 3.6 and the fact that $E_{A}=(E_{A}\cap F_{N})+(E_{A}\cap
F_{N}^{\perp})=(E_{A}\cap F_{N}^{\perp}),$ it follows that
$\dim\mathcal{R}(E_{A}\cap F_{N}^{\perp})=\infty.$ Since
$m(A^{\prime})=\infty,$ there exists a measurable set $B\subseteq A^{\prime}$
satisfying $0<m(B)<\infty.$ Hence $\mathcal{R}(E_{A^{\prime}}\cap
F_{N}^{\perp})\supseteq\mathcal{R}(E_{B}\cap F_{N}^{\perp})$. This implies
$\dim\mathcal{R}(E_{A^{\prime}}\cap F_{N}^{\perp})=\infty$. Similarly it can
be shown that $\dim\mathcal{R}(E_{A^{\prime}}\cap F_{N})=\infty$.

The following result can be thought of as a dual problem of Proposition 3.8
and a similar result also holds for the Heisenberg motion group.

######  Proposition 3.10.

Let $A_{g}=\\{q\in\mathbb{Q}:g(q)\neq 0\\}.$ Then $S=\\{g\in
L^{2}(\mathbb{Q}):m(A_{g})=\infty,\text{ and }\text{rank }W_{a}(g)=\infty\\}$
is dense in $L^{2}(\mathbb{Q}).$

###### Proof.

By the Hahn-Banach theorem, it is enough to show that $S^{\perp}=\\{0\\}.$ On
the contrary, suppose there exists $h\in L^{2}(\mathbb{Q})$ such that $\langle
g,h\rangle=0$ for all $g\in S.$ For $\tilde{q}\in\mathbb{Q}$ define
$g_{\tilde{q}}(q)=g(q-\tilde{q})e^{-2i\langle\tilde{q}a,q\rangle}.$ In view of
(3), $W_{a}(g_{\tilde{q}})$ is not a finite rank operator unless $W_{a}(g)$ is
a finite rank operator. Thus $g\in S,$ implies $g_{\tilde{q}}\in S.$ Define
$\tilde{h}(q)=\bar{h}(-q)$ and $(\tilde{h}\times_{a}g)(\tilde{q})=\int
g(\tilde{q}-q)\tilde{h}(q)e^{-2i\langle\tilde{q}a,q\rangle}dq.$ Then it
follows that

$(\tilde{h}\times_{a}g)(-\tilde{q})=\int
g(q-\tilde{q})e^{-2i\langle\tilde{q}a,q\rangle}\tilde{h}(-q)dq=\int
g_{\tilde{q}}(q)\bar{h}(q)dq=0$

for all $g\in S.$ From the definition of Weyl transform $W_{a},$ it is
straightforward to see that
$W_{a}(\tilde{h})W_{a}(g)=W_{a}(\tilde{h}\times_{a}g)=0.$ This implies that
$\mathcal{R}W_{a}(g)\subseteq\text{ker}\,W_{a}(\tilde{h}),$ for each $g\in S.$
Hence we infer that $W_{a}(\tilde{h})=0.$ Thus, $h=0.$ ∎

Strong annihilating pair: Let $A\subseteq\mathbb{R}$ and
$\Sigma\subseteq\hat{\mathbb{R}}$ be measurable sets. Then the pair
$(A,\Sigma)$ is called weak annihilating pair if $\text{supp}\,f\subseteq A$
and $\text{supp}\,\hat{f}\subseteq\Sigma,$ implies $f=0.$ The pair
$(A,\Sigma)$ is called strong annihilating pair if there exists a positive
number $C=C(A,\Sigma)$ such that

(3.8) $\displaystyle\|f\|_{2}^{2}\leq
C\left(\int_{A^{\prime}}|f|^{2}+\int_{\Sigma^{\prime}}|\hat{f}|^{2}\right)$

for every $f\in L^{2}(\mathbb{R}).$ It is obvious that every strong
annihilating pair is a weak annihilating pair. In [2], Benedicks had proved
that $(A,\Sigma)$ is a weak annihilating pair when $A$ and $\Sigma$ both have
finite measure. In [1], Amrein-Berthier had proved that $(A,\Sigma)$ is a
strong annihilating under the identical assumption as in [2].

Since Fourier transform on the quaternion Heisenberg group is an operator
valued function, we could not expect a similar conclusion as (3.8), though we
can define strong annihilating pair in an analogous way.

######  Definition 3.11.

Let $A$ be a measurable subset of $\mathbb{Q},$ and $S$ be a closed subspace
of $\mathcal{F}_{a}$. We say that the pair $(A,S)$ is a strong annihilating
pair for the Weyl transform $W_{a}$ if there exists a positive number
$C=C(A,S)$ such that for every $g\in L^{2}(\mathbb{Q}),$

$\displaystyle\|g\|_{2}^{2}\leq
C\left(\int_{A^{\prime}}|g|^{2}+\|P_{S}^{\perp}W_{a}(g)\|_{HS}^{2}\right),$

where $P_{S}$ is the projection of $\mathcal{F}_{a}$ onto $S.$

We prove that if $A$ has finite measure and dimension of $S$ is finite, then
$(A,S)$ is a strong annihilating pair. For this, we need the following basic
result.

######  Lemma 3.12.

[10] Let $P$ and $Q$ be two orthogonal projections on a complex Hilbert space
$H.$ Then $\|PQ\|<1$ if and only if there exists a positive number $C$ such
that for each $x\in H$

$\|x\|^{2}\leq C\left(\|P^{\perp}x\|^{2}+\|Q^{\perp}x\|^{2}\right).$

Consider the projections $E_{A}$ and $F_{N}$ as defined by (3.5). By Lemma 3.4
and Proposition 3.6, $E_{A}F_{N}$ is a compact operator and $E_{A}\cap
F_{N}=0.$ Therefore, we must have $\|E_{A}F_{N}\|<1.$ Since
$\mathcal{R}(F_{N})^{\perp}=\\{g\in
L^{2}(\mathbb{Q}):\mathcal{R}(W_{a}(g))\subseteq\mathcal{B}_{N}^{\perp}\\},$
it follows that $W_{a}F_{N}^{\perp}=P_{N}^{\perp}W_{a}.$ Thus, by Lemma 3.12,
$(A,S)$ is a strong annihilating pair, whenever $m(A)$ and $\dim S$ are
finite.

Acknowledgements: The first author gratefully acknowledges the support
provided by IIT Guwahati, Government of India.

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