# Maple Leaf Antiferromagnet in a Magnetic Field

Pratyay Ghosh<EMAIL_ADDRESS>Institut für Theoretische
Physik und Astrophysik and Würzburg-Dresden Cluster of Excellence ct.qmat,
Julius-Maximilians-Universität Würzburg, Am Hubland Campus Süd, Würzburg
97074, Germany  Jannis Seufert  Institut für Theoretische Physik und
Astrophysik and Würzburg-Dresden Cluster of Excellence ct.qmat, Julius-
Maximilians-Universität Würzburg, Am Hubland Campus Süd, Würzburg 97074,
Germany  Tobias Müller  Institut für Theoretische Physik und Astrophysik and
Würzburg-Dresden Cluster of Excellence ct.qmat, Julius-Maximilians-Universität
Würzburg, Am Hubland Campus Süd, Würzburg 97074, Germany  Frédéric Mila
Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015
Lausanne, Switzerland  Ronny Thomale<EMAIL_ADDRESS>Institut für Theoretische Physik und Astrophysik and Würzburg-Dresden Cluster
of Excellence ct.qmat, Julius-Maximilians-Universität Würzburg, Am Hubland
Campus Süd, Würzburg 97074, Germany

###### Abstract

We analyze the quantum antiferromagnet on the maple leaf lattice in the
presence of a magnetic field. Starting from its exact dimer ground state and
for a magnetic field strength of the order of the local dimer spin exchange
coupling, we perform a strong coupling expansion and extract an effective
hardcore boson model. The interplay of effective many-body interactions,
suppressed single-particle dynamics, and correlated hopping gives way to an
intriguing series of superfluid to insulator transitions which correspond to
magnetization plateaux in terms of the maple leaf spin degrees of freedom.
While we find plateaux at intermediate magnetization to be dominated by
bosonic density wave order, we conjecture plateau formation from multi-boson
bound states due to correlated hopping for lower magnetization.

Introduction. Frustrated quantum antiferromagnets have become a
crystallization seed for a plethora of correlated quantum many-body phenomena
Diep (2012); Lacroix  _et al._ (2011). Among them, dimer states take a
particular role, as they allow for the formulation of a natural local basis
encoding quantum entangled degrees of freedom. Singlet dimer states were the
first discovered exact ground states of quantum antiferromagnets Majumdar and
Ghosh (1969), and have taken a pivotal role in connecting frustrated magnetism
to contemporary frontiers of condensed matter physics such as
fractionalization, topological order, and lattice gauge theories Rokhsar and
Kivelson (1988); Moessner and Sondhi (2001); Misguich  _et al._ (2002); Wen
(1991). A major direction of research on dimer models relates to quantum
magnetization plateaux, where a quantum plateau does not originate from semi-
classical order by disorder phenomena, but in its essence traces back to
quantum entanglement Takigawa and Mila (2011). This hints at the formidable
suitability of dimer ground states to address such quantum plateaux, since
their quantum nature is already engraved in the very basis of a dimer singlet
state such as formed by two adjacent spin-1/2 degrees of freedom.

Recently, the quantum antiferromagnet on the maple leaf lattice has been found
by three of us to host an exact dimer ground state with exceptional stability
Ghosh  _et al._ (2022). As compared to the Shastry-Sutherland model (SSM)
Shastry and Sutherland (1981) which has been the dominant cornerstone with an
exact dimer ground state for decades Miyahara and Ueda (2003); Ghosh (2023),
the Maple Leaf Model (MLM) exhibits a larger dimer ground state domain in
terms of minimal bond anisotropy, along with intriguing adjacent magnetic and
paramagnetic regimes that are under active investigation Ghosh  _et al._
(2022); Schmalfuß  _et al._ (2002); Farnell  _et al._ (2011). A particularly
interesting domain of a model with a dimer ground state in a magnetic field is
reached when the magnetic field strength is of the order of the dimer singlet-
triplet gap, i.e., the exchange coupling scale $J$ (Fig. 1). At low energies,
this then allows one to perform the hardcore boson projection onto a two-
dimensional basis formed by the dimer state and the field-aligned triplet
state component. The effective hardcore boson model naturally shows a highly
exotic arrangement of terms entering the effective Hamiltonian, which are hard
to realize in other condensed matter contexts: the single particle dynamics
are typically rather suppressed, promoting the relevance of interactions
Miyahara and Ueda (1999, 2000). Depending on the bosonic filling, i.e., the
magnetization, and the bond anisotropy, long-range two-body and higher-body
interactions arise, combined with correlated hopping terms that can overcome
the suppressed single-particle dynamics. There is in principle three major
ways for such a bosonic model to develop a phase which, in terms of its
underlying spin degrees of freedom, would correspond to a magnetization
plateau. First, the system could break translation symmetry and form density
waves. Second, the correlated hopping can give rise to bound state formation,
which might be particularly relevant for low magnetization Momoi and Totsuka
(2000); Totsuka  _et al._ (2001); Dorier  _et al._ (2008); Corboz and Mila
(2014); Schneider  _et al._ (2016). Third, topological ordered states of
bosons with a finite condensation energy could turn out to be preferred
energetically, which would show no sign of translation symmetry breaking
Misguich  _et al._ (2001); Moessner and Raman (2010). This is where a slight
analogy to fractional quantum Hall effect can be drawn, where the Wigner
crystal state at low densities is replaced by a topologically ordered state at
higher densities whose quantum fluctuations reinstall translation symmetry.

Figure 1: Maple leaf model in (1). The dotted blue and thick purple bonds
relate to an exchange coupling $J^{\prime}$ and $J$. The 6-site unit cell has
three dimer sublattices a, b, and c (dashed triangle frame). The effective
triplet-triplet repulsion (reference triplet in black) is expanded into
pseudopotentials $V_{1,\dots,4}$ (3). The bonds related to $V_{1,\dots,3}$
($V_{4}$) unfold from one (three) bond(s) plus lattice symmetry operations.

In this Letter, we analyze the MLM in a magnetic field. The Hamiltonian is
given by

$\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}^{\prime}-B\sum_{i}S_{i}^{z}$ (1)

with

$\mathcal{H}_{0}=J\sum_{\langle ij\rangle}\vec{S}_{i}\cdot\vec{S}_{j}\text{,
and }\mathcal{H}^{\prime}=J^{\prime}\sum_{\langle
kl\rangle}\vec{S}_{k}\cdot\vec{S}_{l},$

where $\vec{S}_{i}$ denotes a spin vector operator acting on a spin-$1/2$
degree of freedom at site $i$. Aside from a Zeeman term, there are two
separate summations $ij$, and $kl$ over the nearest neighbor bonds denoted
purple (thick) and blue (dotted), respectively (Fig. 1). Upon a hardcore boson
projection for $B\sim J$, we perform a perturbative expansion in the non-dimer
exchange bond strength $J^{\prime}$. In terms of single-boson self-energy
expansion, we find the single-particle dynamics to cancel up to ninth order in
$J^{\prime}/J$, hinting at the exceptional suppression of triplet kinetic
energy in the MLM. We extract the triplet-triplet repulsive interactions up to
fourth order in $J^{\prime}$, and perform Monte Carlo simulations(see
Supplemental Material sup  which also contains Refs. Betts (1995); Gelfand
(1991); Caspers  _et al._ (1984); Chung  _et al._ (2001); Senthil  _et al._
(2004); Mambrini  _et al._ (2006)) to obtain density wave type plateaux at
$m=1/6$, $2/9$, $2/7$, and $1/3$. We identify the formation of two-particle
and, possibly, three-particle bound states from correlated hopping which may
lead to additional plateaux at lower magnetization.

Spin gap and single-particle dynamics. The ground state of $\mathcal{H}_{0}$
is a product state of spin singlets on all the purple bonds for
$J^{\prime}/J\lesssim 0.74$ Ghosh  _et al._ (2022). The eigenstates of an
isolated purple dimer are the singlet state $|s\rangle$ and the three triplet
states $|t_{1}\rangle,|t_{0}\rangle$, and $|t_{-1}\rangle$, where the
subscript denotes the total $S^{z}$. The effect of $\mathcal{H}^{\prime}$ is
included perturbatively in the basis of the dimer eigenstates Sakurai and
Napolitano (2020). The spin, i.e., singlet-triplet gap reads

$\Delta=J-B-\frac{J^{\prime 2}}{J}-\frac{J^{\prime
3}}{2J^{2}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-\frac{5J^{\prime
4}}{8J^{3}}}+\mathcal{O}(J^{\prime 5}).$ (2)

The expression of $\Delta$ in MLM is identical to the SSM up to third order
Miyahara and Ueda (1999).

The parity of the singlets and the matrix elements of $\mathcal{H}^{\prime}$
impose strong constraints on the hopping processes of the triplets, namely, a
triplet can only move to two of its four neighboring dimers, and when it does,
it leaves another triplet behind. Consequently, we see that the hopping of
single triplets starts in fourth-order perturbation theory, where a triplet
can hop to any of the neighbouring dimers(Fig. 2(a)) sup . For SSM, such a
hopping appears only at sixth-order Miyahara and Ueda (1999). The enhancement
of triplet hopping for MLM is rooted in its lattice geometry sup .

Figure 2: (a) Hopping of a single triplet (marked by the dimer filled in red)
at fourth-order in $J^{\prime}$. There are two symmetry-inequivalent ways a
triplet can move between two equal-sublattice dimers. (b) For correlated
hopping, one triplet (filled in red) hops assisted by another triplet in its
vicinity (filled in black). Such a hopping emerges from second order in
$J^{\prime}$ and implies triplet bound states.

Two-particle dynamics. An important property of the dimer model is that it
allows nontrivial two-triplet hopping called _correlated hopping_ , where one
triplet can hop assisted by a second triplet in its proximity. Note that a
process like that can neither be interpreted as single-triplet hopping nor a
triplet pair creation-annihilation. Correlated hopping motion of two triplets
occurs from a second order perturbation (Fig. 2(b)) sup . Since the single
triplet excitations occur at further two-orders of perturbation, the
correlated hopping plays a crucial role. That has likewise been noticed in the
SSM, where the correlated hopping results in two-triplets bound states Momoi
and Totsuka (2000); Totsuka  _et al._ (2001); Dorier  _et al._ (2008); Corboz
and Mila (2014); Schneider  _et al._ (2016).

Hardcore boson projection. For $B>0$, the lowest energy states with
magnetization $m\geq 0$ for the whole system consist of $|s\rangle$ and
$|t_{1}\rangle$, which we project to as the low-energy physical degrees of
freedom. Considering $|t_{1}\rangle$ as a hard-core boson and $|s\rangle$ as a
vacancy, we project our system onto an effective hard-core boson model along
with a perturbative treatment around the limit $J^{\prime}/J\ll 1$. Whereas
the $|t_{1}\rangle$ triplets are treated as on-shell magnetic particles, the
rest of the dimer states, i.e., $|t_{0}\rangle$ and $|t_{-1}\rangle$, are
considered as intermediate virtual states in perturbation theory. We estimate
the interaction energy between two hard-core bosons for which we calculate the
energy required to create two triplet excitations from the ground state as a
sum of (i) the spin gap energy required to generate a triplet excitation
($2\Delta$), which includes the self-energy-type corrections outlined in (2),
and (ii) the interaction energy between the excited triplets which we call
pseudopotentials (PPs) $V_{n}$. The $n$ in the subscript refers to the fact
that two triplets can interact in different orders in perturbation theory due
to their different relative positions. In Fig. 1, we depict the interactions
(symmetry reduced) between different triplet pairs that interact within third-
order perturbation theory. We thus find the PPs

$\displaystyle V_{1}$ $\displaystyle=\frac{J^{\prime}}{2}+\frac{J^{\prime
2}}{2J}+\mathcal{O}(J^{\prime 4})$ (3) $\displaystyle V_{2}$
$\displaystyle=\frac{J^{\prime 2}}{2J}+\frac{3J^{\prime
3}}{4J^{2}}+\mathcal{O}(J^{\prime 4})$ $\displaystyle V_{3}$
$\displaystyle=\frac{J^{\prime 3}}{8J^{2}}+\mathcal{O}(J^{\prime 4})$

Clearly, the triplet-triplet interactions do not truncate at this order, but
the higher order calculations are analytically rather cumbersome. This is
where extensive numerical methods, e.g. perturbative continuous unitary
transformations Dorier  _et al._ (2008), are ideal to take over, which we keep
as a future endeavor.

Density wave magnetization plateaux.

Figure 3: (a) The density wave structures at magnetization plateaux. The
triplet excitations (singlets) are shown by filled (open) dimers. (b)
Magnetization processes of the MLM as a function of $J^{\prime}$ and $h$ upto
third order in perturnation.

We first discuss the magnetization plateaux of (1) through Monte Carlo
simulations sup  of the effective hard-core boson model, with PPs defined in
(3), in the limit of negligible kinetic energy. We determine magnetization
curves upto third-order in perturbation for different values of $J^{\prime}/J$
which yield plateaux at $m/m_{sat}=2/9$, $2/7$, and $1/3$ at sufficiently low
temperatures, where $m_{sat}$ denotes the saturation magnetization of full
triplet occupancy. The $1/3$ plateau has also been predicted via exact
diagonalization Farnell  _et al._ (2011). These plateaux only appear when the
spin gap $\Delta$ is closed by $B$. The strong repulsive interactions,
however, will not allow for all the singlets to become triplets upon gap
closing. Due to the competition between the energy gained by triplet
excitations and the repulsive energy, one would expect that for a given $B$
the triplets would assume a particular superstructure (density wave) to
minimize the energy. Such a density wave with a fixed magnetization is a
stable configuration for a finite range in $B$, long-range correlated, and
exhibits a spin gap. We construct all possible density waves of hardcore
bosons with different unit cells, and identify the particular arrangement that
minimizes the energy for a given field. This approach turns out to agree well
with our Monte Carlo results. The various bosonic density waves and their
corresponding magnetization phase diagram at $T=0$ are presented in Fig. 3.
The inclusion of higher-order calculations, however, would produce PPs beyond
the ones mentioned in (3) and can produce plateaux with lower magnetization.
For example, we depict the PPs that would appear in fourth-order perturbation
theory,

$V_{4}=\frac{J^{\prime 4}}{8J^{3}}+\mathcal{O}(J^{\prime 5}),$ (4)

in Fig. 1. The inclusion of $V_{4}$ would introduce a $m/m_{sat}=1/6$ plateau
(see superstructure in Fig. 3). We, also, do not explore the magnetization
beyond the $1/3$ plateau, as from there on we would encounter more than one
hard-core boson per unit cell, and hence three-particle interactions would
need to be considered to properly investigate such plateaux of higher
magnetization.

While the Monte Carlo simulations of the hardcore boson projection is suitable
to capture the static physics of the density wave magnetization plateaux,
understanding how the system rearranges itself across a critical field
requires the inclusion of correlated hopping. When a plateau state is
destabilized by a field, the system would still maintain the density wave of
the preceding phase, but would have extra bosons in the system. These bosons
can perform superfluid motion in the lattice through correlated hoppings
assisted by the bosonic density wave. Thus, as a combination of the density
wave and the superfluid, the resulting state near criticality is a supersolid
phase Matsuda  _et al._ (2013); Shi  _et al._ (2022), where the magnetization
should increase smoothly with the increasing field and finally reach the next
plateau state. In addition to the plateau states depicted in Fig. 3, we thus
also conjecture supersolid phases to appear near the plateau transitions. Not
all of these supersolid phases, however, should a priori be expected to be
stable Momoi and Totsuka (2000); Gan (2011).

Triplet bound states.

Figure 4: (a) Two- and (c) possible three-particle bound states, and the
constituent states with dominant contributions. The filled dimer end-points
reprents a triplet. The light red paralellograms in (a) and (b), and light
orange hexagons in (c) and (d) represent the bound states of two and three
triplets, respectively. Speculative crystal of (b) two- and (d) three-particle
bound states giving rise to $m/m_{sat}=2/19$ and $1/7$. The bound states of
two and three triplets are arranged to interact only via $V_{4}$.

When two excited triplets are far apart from each other, individual triplets
are nearly localized and gain only minuscule energy from kinematics. When they
are adjacent to each other, however, correlated hopping makes the coherent
motion of two triplets possible, and hence the gain of kinetic energy (Fig.
2(b)). The balance between PPs and correlated hopping can give rise to
$S^{z}=2$ bound states, i.e. a bound state of two on-shell $|t_{1}\rangle$
triplets: When we place two triplets next to each other, the interaction
between them is linear in $J^{\prime}$ and typically dominant. They can,
however, employ correlated hopping $\propto J^{\prime 2}$ and move away from
each other in such a way that their PP energy penalty reduces to $V_{3}$,
which is cubic in $J^{\prime}$, and thus as a whole form a two-particle bound
state.

The hopping processes of two triplets can be decomposed into the center-of-
mass motion and the relative motion, which leaves us with twelve unique states
with different two-triplet configurations. Note that, for SSM, there are only
four such states, as the effective lattice of dimer bonds is a square lattice.
Applying the same rationale to MLM, the effective dimer lattice is of kagomé
type, which enhances the complexity of the problem. Upon calculating the
energies $e_{2P}$ of a two-triplet excitation sup , we find the energetically
favoured two-triplet bound state, which locates in the light red region shown
in Fig. 4 (a), and is mostly composed of a linear combination of two triplet
states interacting via $V_{3}$. The binding energy of this bound state,
$e_{2P}-2\Delta$, is found to be $-3J^{\prime 4}/16J^{3}$ (in lowest order)
per particle sup . Because the two-particle bound states thus have lower
energy than two separate triplets, the gap of the bound states collapses
faster than the singlet-triplet gap upon increasing $B$, thus enforcing the
bound states to condense first rather than the isolated single triplets.

For SSM, it is shown that the $S^{z}=2$ bound states give rise to additional
magnetization plateaux near $m=0$ by forming crystals of bound states Corboz
and Mila (2014); Schneider  _et al._ (2016). The same phenomenon should be
expected for MLM. In Fig. 4 (b), we speculate on the existence of such a bound
state crystal corresponding to $m/m_{sat}=2/19$. Moreover, in the $2/7$ and
$1/3$ plateaux depicted in Fig. 3 (a), the relative position of the triplets
favors the onset of correlated hopping. Therefore, upon inclusion of dynamics,
these are likely to gain stability from becoming a crystal of bound states.
The $1/6$ and $2/9$ plateaux, on the other hand, can be destabilized by other
competing bound state crystals. The small size of the binding energy further
suggests that these intricacies are likely to only appear for larger values of
$J^{\prime}/J$.

There is, however, much more to explore with regard to bound states in the
MLM. Drawing a comparison to SSM, the lowest-energy bound state is formed
around a square on the effective lattice Corboz and Mila (2013). In our case,
the effective lattice is made of triangles and hexagons. No two-particle bound
state can be formed around a triangle, as the interaction between the triplet
would always be $V_{1}$, and not around hexagons either due to the range of
correlated hopping. Therefore, it is quite natural to expect the appearance of
three-triplet bound states, where three particles themselves perform a
coherent motion around a hexagon and overcome the mutual repulsion sup . We
investigate this regime within a truncated three-particle basis. The binding
energy corresponding to the lowest energy state, which is predominantly a
linear combination of the three particle states in Fig. 4 (c), is found to be
$J^{\prime 3}/24J^{2}-J^{\prime 4}/8J^{3}+\mathcal{O}(J^{\prime 5})$ per
particle, where the binding energy is positive for small $J^{\prime}$. A more
elaborate analysis is required to capture the stability of such three-particle
bound states. Under the assumption that such a bound state is stable, we
present a speculative bound state crystal in Fig. 4 (d) producing a plateau at
$m/m_{sat}=1/7$.

Conclusions and Outlook. We have investigated the magnetization phase diagram
of the MLM by using a strong-coupling expansion of a hard-core boson projected
effective model. We discover that finite single-particle dynamics only occur
by tenth order perturbation theory in $J^{\prime}$, suggesting a remarkable
flatness of the single triplet dispersion. We extract the triplet-triplet
repulsive PPs up to fourth order in $J^{\prime}$, and find density wave type
plateaux at $m/m_{sat}=1/6$, $2/9$, $2/7$, and $1/3$. We predict the
appearance of supersolid phases within this unfolding magnetization phase
diagram. Furthermore, we emphasize the phenomenological step towards the
development of two- and perhaps three-particle bound states from correlated
hopping, which might result in an amended stability of density wave
magnetization plateaux, and further plateaux at lower magnetization.

Several pressing issues about the MLM in a magnetic field remain unanswered
and require further investigation. First, we truncate our evaluation of two-
particle PPs at fourth-order despite that higher-order expansions will produce
longer-range repulsion between particles and may induce other lower
magnetization plateaux. There, another significant open question is to
investigate the crystallization of two-particle, or even three-body, bound
states at lower magnetic fields, and the impact of three-body interactions at
higher magnetization.

Acknowledgments. We thank Kai Phillip Schmidt for pointing out a fourh-order
hopping process for the triplets that was missed in an early version of the
manuscript. The work in Würzburg is supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) through Project-ID
258499086-SFB 1170 and the Würzburg-Dresden Cluster of Excellence on
Complexity and Topology in Quantum Matter – ct.qmat Project-ID 390858490-EXC
2147. FM has been supported by the Swiss National Science Foundation, Grant
No, 182179. PG and JS contributed equally in this work.

## References

  * Diep (2012) H. T. Diep,  _Frustrated Spin Systems_ (WORLD SCIENTIFIC, 2012).
  * Lacroix  _et al._ (2011) C. Lacroix, P. Mendels, and F. Mila,  _Introduction to Frustrated Magnetism: Materials, Experiments, Theory_, Springer Series in Solid-State Sciences (Springer Berlin Heidelberg, 2011).
  * Majumdar and Ghosh (1969) C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, 1388 (1969).
  * Rokhsar and Kivelson (1988) D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988).
  * Moessner and Sondhi (2001) R. Moessner and S. L. Sondhi, Phys. Rev. Lett. 86, 1881 (2001).
  * Misguich  _et al._ (2002) G. Misguich, D. Serban, and V. Pasquier, Phys. Rev. Lett. 89, 137202 (2002).
  * Wen (1991) X. G. Wen, Phys. Rev. B 44, 2664 (1991).
  * Takigawa and Mila (2011) M. Takigawa and F. Mila, in  _Introduction to Frustrated Magnetism_ (Springer Berlin Heidelberg, Berlin, Heidelberg, 2011) pp. 241–267.
  * Ghosh  _et al._ (2022) P. Ghosh, T. Müller, and R. Thomale, Phys. Rev. B 105, L180412 (2022).
  * Shastry and Sutherland (1981) B. S. Shastry and B. Sutherland, Physica B+C 108, 1069 (1981).
  * Miyahara and Ueda (2003) S. Miyahara and K. Ueda, Journal of Physics: Condensed Matter 15, R327 (2003).
  * Ghosh (2023) P. Ghosh, Phys. Rev. B 108, 014426 (2023).
  * Schmalfuß  _et al._ (2002) D. Schmalfuß, P. Tomczak, J. Schulenburg, and J. Richter, Phys. Rev. B 65, 224405 (2002).
  * Farnell  _et al._ (2011) D. J. J. Farnell, R. Darradi, R. Schmidt, and J. Richter, Phys. Rev. B 84, 104406 (2011).
  * Miyahara and Ueda (1999) S. Miyahara and K. Ueda, Phys. Rev. Lett. 82, 3701 (1999).
  * Miyahara and Ueda (2000) S. Miyahara and K. Ueda, Phys. Rev. B 61, 3417 (2000).
  * Momoi and Totsuka (2000) T. Momoi and K. Totsuka, Phys. Rev. B 61, 3231 (2000).
  * Totsuka  _et al._ (2001) K. Totsuka, S. Miyahara, and K. Ueda, Phys. Rev. Lett. 86, 520 (2001).
  * Dorier  _et al._ (2008) J. Dorier, K. P. Schmidt, and F. Mila, Phys. Rev. Lett. 101, 250402 (2008).
  * Corboz and Mila (2014) P. Corboz and F. Mila, Phys. Rev. Lett. 112, 147203 (2014).
  * Schneider  _et al._ (2016) D. A. Schneider, K. Coester, F. Mila, and K. P. Schmidt, Phys. Rev. B 93, 241107(R) (2016).
  * Misguich  _et al._ (2001) G. Misguich, T. Jolicoeur, and S. M. Girvin, Phys. Rev. Lett. 87, 097203 (2001).
  * Moessner and Raman (2010) R. Moessner and K. S. Raman, in  _Introduction to Frustrated Magnetism_ (Springer Berlin Heidelberg, 2010) pp. 437–479.
  * (24) Supplementary Information at [URL to be inserted by the publisher].
  * Betts (1995) D. D. Betts, Proc. N. S. Inst. Sci. 40, 95 (1995).
  * Gelfand (1991) M. P. Gelfand, Phys. Rev. B 43, 8644 (1991).
  * Caspers  _et al._ (1984) W. J. Caspers, K. M. Emmett, and W. Magnus, Journal of Physics A: Mathematical and General 17, 2687 (1984).
  * Chung  _et al._ (2001) C. H. Chung, J. B. Marston, and S. Sachdev, Phys. Rev. B 64, 134407 (2001).
  * Senthil  _et al._ (2004) T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303, 1490 (2004).
  * Mambrini  _et al._ (2006) M. Mambrini, A. Läuchli, D. Poilblanc, and F. Mila, Phys. Rev. B 74, 144422 (2006).
  * Sakurai and Napolitano (2020) J. J. Sakurai and J. Napolitano,  _Modern Quantum Mechanics -_ (Cambridge University Press, Cambridge, 2020).
  * Matsuda  _et al._ (2013) Y. H. Matsuda, N. Abe, S. Takeyama, H. Kageyama, P. Corboz, A. Honecker, S. R. Manmana, G. R. Foltin, K. P. Schmidt, and F. Mila, Phys. Rev. Lett. 111, 137204 (2013).
  * Shi  _et al._ (2022) Z. Shi, S. Dissanayake, P. Corboz, W. Steinhardt, D. Graf, D. M. Silevitch, H. A. Dabkowska, T. F. Rosenbaum, F. Mila, and S. Haravifard, Nature Communications 13 (2022), 10.1038/s41467-022-30036-w.
  * Gan (2011) J.-Y. Gan, Journal of the Physical Society of Japan 80, 034604 (2011).
  * Corboz and Mila (2013) P. Corboz and F. Mila, Phys. Rev. B 87, 115144 (2013).


# Conjectured $DXZ$ decompositions
of a unitary matrix

Alexis De Vos, Martin Idel, and Stijn De Baerdemacker

###### Abstract

For any unitary matrix there exists a ZXZ decomposition, according to a
theorem by Idel and Wolf. For any even-dimensional unitary matrix there exists
a block-ZXZ decomposition, according to a theorem by Führ and Rzeszotnik. We
conjecture that these two decompositions are merely special cases of a set of
decompositions, one for every divisor of the matrix dimension. For lack of a
proof, we provide an iterative Sinkhorn algorithm to find an approximate
numerical decomposition.

Keywords: unitary matrix; matrix decomposition; biunimodular vector; Arnold
conjecture.

MSC: 15A21; 15A51; 53D12.

##  1 Introduction

Recently, two decompositions of an arbitrary $n\times n$ unitary matrix $U$
into a matrix product $DXZ$ of three unitary matrices have been proposed:

  * •

For arbitrary $n$, Idel and Wolf [1] present a decomposition where $D$ and $Z$
are diagonal matrices, whereas $X$ is a matrix with all linesums equal to
unity.

  * •

For arbitrary even $n$, Führ and Rzeszotnik [2] present a decomposition where
$D$ and $Z$ are block-diagonal matrices, whereas $X$ is a matrix with all
block-linesums equal to the $n/2\times n/2$ unit matrix.

These matrix decompositions have been applied in quantum optics [1], quantum
computing [3] [4] [5], and quantum memory [6].

The two matrix decompositions have been proved in a very different way.
Whereas the proof of the Idel–Wolf decomposition (based on symplectic
topology) is not constructive, the proof of the Führ–Rzeszotnik decomposition
(based on linear algebra) is constructive. In the present paper, we conjecture
that nevertheless the two decompositions belong to a same set of similar
decompositions. We conjecture that there exist as many such decompositions as
there are divisors of the number $n$. We present no proof, as neither the
Idel–Wolf proof nor the Führ–Rzeszotnik proof can be easily extrapolated.

##  2 Conjecture

We introduce the following three positive integers:

  * •

$n$, an arbitrary integer greater than 1,

  * •

$m$, a divisor of $n$, distinct111The restriction $m\neq n$ is merely
introduced for convenience. The reader may easily investigate the case $m=n$.
E.g., if $m=n$, then Conjecture 1 is trivially true: suffice it to choose $D$
equal to $U$ and both $X$ and $Z$ equal to the $n\times n$ unit matrix. from
$n$, and

  * •

$q$, equal to $n-m$.

We write $n=rm$ and $q=(r-1)m$. Hence, both $m$ and $r$ are divisors of $n$.
They satisfy $1\leq m<n$ and $1<r\leq n$. For convenience, $n\times n$
matrices will be called ‘great matrices’, $m\times m$ matrices will be called
‘small matrices’, and $q\times q$ matrices will be called ‘intermediate
matrices’.

###### Conjecture 1

Every great unitary matrix $U$ can be decomposed into three great unitary
matrices:

$U=DXZ\ ,$

where

  * •

$D$ consists of $r$ small matrices on its diagonal:

$D=\mbox{diag}\,(D_{11},D_{22},D_{33},...,D_{rr})\ ,$

  * •

$Z$ consists of $r$ small matrices on its diagonal, the upper-left small
matrix being equal to the $m\times m$ unit matrix $I$:

$Z=\mbox{diag}\,(I,Z_{22},Z_{33},...,Z_{rr})\ ,$

and

  * •

$X$ consists of $r^{2}$ small matrices $X_{jk}$, such that all row sums
$\sum_{k=1}^{r}X_{jk}$ and all column sums $\sum_{j=1}^{r}X_{jk}$ are equal to
the small matrix $I$.

Because $D$ is unitary, automatically all its blocks $D_{jj}$ are unitary;
because $Z$ is unitary, automatically all its blocks $Z_{jj}$ are unitary. In
contrast, the blocks $X_{jk}$ are not necessarilly unitary.

We define the $n\times n$ transformation matrix

$T=F_{r}\otimes I\ ,$

where the matrix $F_{r}$ is the $r\times r$ discrete Fourier transform. We can
easily demonstrate that the product $T^{-1}XT$ is of the form

$\left(\begin{array}[]{cc}I&\\\ &G\end{array}\right)\ .$

We thus have the following property:

$X=T\left(\begin{array}[]{cc}I&\\\ &G\end{array}\right)T^{-1}\ .$

Because both $X$ and $T$ are unitary, automatically $G$ is a unitary
intermediate matrix.

We summarize that the decomposition of $U$ corresponds to finding the
appropriate $2r-1$ small unitary matrices and a single appropriate
intermediate unitary matrix. This corresponds to find the appropriate

$(2r-1)m^{2}+q^{2}=(2r-1)m^{2}+[(r-1)m]^{2}=r^{2}m^{2}$

real parameters, a number which exactly matches $n^{2}$, i.e. the number of
degrees of freedom of the given matrix $U$.

##  3 Three special cases

If $m=1$ (and thus $q=n-1$), then all small matrices are, in fact, just
complex numbers. Both $D$ and $Z$ are diagonal unitary matrices and $X$ is a
unit linesum unitary matrix. The transformation matrix $T$ equals the great
Fourier matrix $F_{n}$. In this particular case, the above conjecture has been
proposed by De Vos and De Baerdemacker [7] and subsequently proved by Idel and
Wolf [1]. The proof is by symplectic topology. Unfortunately, the proof is not
constructive and therefore only provides the guarantee that the numbers
$D_{11},D_{22},...,D_{nn}$ and $Z_{22},Z_{33},...,Z_{nn}$ exist, without
providing their values. De Vos and De Baerdemacker [7] give a Sinkhorn
algorithm that yields numerical approximations of these numbers. Finally, we
note that examples of the case $m=1$ demonstrate that the $DXZ$ decomposition
is not always unique.

If $n$ is even and $m$ equals $n/2$ (and thus $q=n/2$), then intermediate
matrices are, in fact, small matrices. The transformation matrix $T$ equals
$H\otimes I$, where $H=F_{2}$ is the $2\times 2$ Hadamard matrix. In this
particular case, the above conjecture has been proved by Führ and Rzeszotnik
[2]. The proof is constructive and thus gives explicit values for the small
matrices $D_{11}$, $D_{22}$, $Z_{22}$, and $G$. Also in this special case
decomposition is not unique [4] [8].

Finally, if both $m=1$ and $m=n/2$, i.e. if $n=r=2$ and $m=q=1$, then the
decomposition is well-known. An arbitrary matrix from U(2) looks like

$U=\left(\begin{array}[]{rl}\cos(\varphi)e^{i(\theta+\psi)}&\sin(\varphi)e^{i(\theta+\chi)}\\\
-\sin(\varphi)e^{i(\theta-\chi)}&\cos(\varphi)e^{i(\theta-\psi)}\end{array}\right)\
.$ (1)

One possible decomposition is

$U=\left(\begin{array}[]{cc}e^{i(\theta+\varphi+\psi)}&\\\
&ie^{i(\theta+\varphi-\chi)}\end{array}\right)\ \frac{1}{2}\
\left(\begin{array}[]{cc}1+e^{-2i\varphi}&1-e^{-2i\varphi}\\\
1-e^{-2i\varphi}&1+e^{-2i\varphi}\end{array}\right)\left(\begin{array}[]{cc}1&\\\
&-ie^{i(-\psi+\chi)}\end{array}\right)\ .$ (2)

##  4 Group hierarchy

The matrices $D$ form a group isomorphic to U($m$)r, of dimension $rm^{2}=nm$.
The matrices $Z$ form a group isomorphic to U($m$)r-1, of dimension
$(r-1)m^{2}=(n-m)m$. Finally, the matrices $X$ form a group isomorphic to
U($q$), of dimension $q^{2}=(n-m)^{2}$. We denote these three matrix groups by
DU($n,m$), ZU($n,m$), and XU($n,m$), respectively. In particular, the groups
XU($n,1$) and ZU($n,1$) are the groups XU($n$) and ZU($n$), extensively
studied in the past [5] [9].

According to the conjecture, for any $m$, the closure of the groups DU($n,m$),
ZU($n,m$), and XU($n,m$) is the unitary group of $U$ matrices. Of course,
because ZU($n,m$) is a subgroup of DU($n,m$), the closure of DU($n,m$) and
XU($n,m$) is also U($n$). In fact, the closure of merely ZU($n,m$) and
XU($n,m$) already equals U($n$). Indeed, any DU($n,m$) matrix can be
decomposed into two ZU($n,m$) matrices and two XU($n,m$) matrices:

$\displaystyle\left(\begin{array}[]{ccccc}D_{11}&&&&\\\ &D_{22}&&&\\\
&&D_{33}&&\\\ &&&\ddots&\\\ &&&&D_{rr}\end{array}\right)=$
$\displaystyle\left(\begin{array}[]{ccccc}&I&&&\\\ I&&&&\\\ &&I&&\\\
&&&\ddots&\\\ &&&&I\end{array}\right)\left(\begin{array}[]{ccccc}I&&&&\\\
&D_{11}&&&\\\ &&I&&\\\ &&&\ddots&\\\
&&&&I\end{array}\right)\left(\begin{array}[]{ccccc}&I&&&\\\ I&&&&\\\ &&I&&\\\
&&&\ddots&\\\ &&&&I\end{array}\right)\left(\begin{array}[]{ccccc}I&&&&\\\
&D_{22}&&&\\\ &&D_{33}&&\\\ &&&\ddots&\\\ &&&&D_{rr}\end{array}\right)\ .$

If $m>1$, then we have the following group hierarchy:

$\displaystyle\mbox{U}(n)\supset$
$\displaystyle\mbox{DU}(n,m)\supset\mbox{DU}(n,1)=\mbox{DU}(n)$
$\displaystyle\mbox{XU}(n,m)\subset\mbox{XU}(n,1)=\mbox{XU}(n)$
$\displaystyle\subset\mbox{U}(n)\ .$

In fact, we have the following isomorphisms:

$\displaystyle\mbox{DU}(n,m)$ $\displaystyle\cong$
$\displaystyle\mbox{U}(m)^{n/m}$ $\displaystyle\mbox{ZU}(n,m)$
$\displaystyle\cong$ $\displaystyle\mbox{U}(m)^{n/m-1}$
$\displaystyle\mbox{XU}(n,m)$ $\displaystyle\cong$
$\displaystyle\mbox{U}(n-m)\ .$

If $n$ is a power of a prime, say $p^{w}$, then $m$ necessarilly is also a
prime power, say $p^{u}$ (with $0\leq u<w$). The XU($n,m$) groups with all
possible values of $m$ (i.e. $1,p,p^{2},...,p^{w-1}$) form an elegant subgroup
chain according to

$\mbox{XU}(p^{w})=\mbox{XU}(p^{w},1)\supset\mbox{XU}(p^{w},p)\supset\mbox{XU}(p^{w},p^{2})\supset...\supset\mbox{XU}(p^{w},p^{w-1})\
,$

with successive dimensions

$(p^{w}-1)^{2}>(p^{w}-p)^{2}>(p^{w}-p^{2})^{2}>...>(p^{w}-p^{w-1})^{2}\ .$

##  5 Conjugate conjecture

If Conjecture 1 is true, then automatically a second conjecture is also true:

###### Conjecture 2

Every great unitary matrix $U$ can be decomposed into three great unitary
matrices:

$U=C\left(\begin{array}[]{cc}I&\\\ &A\end{array}\right)Y\ ,$

where

  * •

$C$ is a circulant $n\times n$ matrix, i.e. a unitary matrix consisting of
$m\times m$ small blocks, such that two $C_{jk}$ are identical if their two
$j-k$ are equal,

  * •

$A$ is a $q\times q$ unitary matrix, and

  * •

$Y$ is an $n\times n$ circulant matrix, the upper row sum222As the matrix is
circulant, all row sums and column sums are equal. Hence $Y$ is a member of
XU($n,m$). being equal to the $m\times m$ unit matrix $I$.

Indeed, if we apply Conjecture 1, not to the given matrix $U$, but instead to
its conjugate

$u=T^{-1}UT\ ,$

then we obtain the decomposition

$u=dxz\ .$

This leads to

$U=T\,d\,T^{-1}T\,x\,T^{-1}T\,zT^{-1}\ .$

One can easily verify that

  * •

$T\,d\,T^{-1}$ is a circulant great matrix,

  * •

$T\,x\,T^{-1}$ is of the form $\left(\begin{array}[]{cc}I&\\\
&A\end{array}\right)$, and

  * •

$T\,z\,T^{-1}$ is a circulant XU($n,m$) matrix.

Such conjugate decomposition was already noticed before, in both the $m=1$
case and the $m=n/2$ case [1] [5] [9].

##  6 Unitary and biunitary vectors

For $m=1$, the DXZ decomposition involves unit-modulus numbers $d_{jj}$ and
$z_{jj}$:

$U=\left(\begin{array}[]{rrrr}d_{11}&&&\\\ &d_{22}&&\\\ &&\ddots&\\\
&&&d_{nn}\end{array}\right)X\left(\begin{array}[]{rrrr}1&&&\\\ &z_{22}&&\\\
&&\ddots&\\\ &&&z_{nn}\par\end{array}\right)\ .$

If we multiply both sides of the equation by the $n\times 1$ matrix (i.e.
column vector) $v=(1,z_{22}^{-1},z_{33}^{-1},...,z_{nn}^{-1})^{T}$, then we
obtain

$Uv=\left(\begin{array}[]{rrrr}d_{11}&&&\\\ &d_{22}&&\\\ &&\ddots&\\\
&&&d_{nn}\end{array}\right)X\left(\begin{array}[]{r}1\\\ 1\\\ \vdots\\\
1\end{array}\right)\ .$

Taking into account that all row sums of $X$ equal unity, we find

$Uv=w\ ,$

where $w=(d_{11},d_{22},d_{33},...,d_{nn})^{T}$. Both $v$ and $w$ are vectors
with all entries having unit modulus. Therefore, they are called unimodular
vectors. The unimodular vector $v$ is called biunimodular for the matrix $U$,
as $Uv$ is unimodular as well [1] [2]. We say that the Idel–Wolf DXZ
decomposition implies the fact that any unitary matrix has at least one
biunimodular vector. Moreover, it possesses a biunimodular vector with leading
entry 1. As an example, decomposition (2) of the matrix (1) corresponds with
the following biunimodular vector:

$U\ \left(\begin{array}[]{c}1\\\
i\,e^{i(\psi-\chi)}\end{array}\right)=\left(\begin{array}[]{r}e^{i(\varphi+\theta+\psi)}\\\
i\,e^{i(\varphi+\theta-\chi)}\end{array}\right)\ .$

If Conjecture 1 is true for $m>1$, then we can draw a similar conclusion
$UV=W$, however with $V$ and $W$ matrices of size $n\times m$. These matrices
consist of $r$ blocks, each a unitary $m\times m$ matrix. Because such blocks
have no modulus, we cannot call $V$ and $W$ unimodular vectors. We will
instead call them unitary vectors and $V$ a biunitary vector. These unitary
vectors reside in an $nm$-dimensional vector space
$\mathbb{C}^{n}\otimes\mathbb{C}^{m}$, isomorphic to $\mathbb{R}^{2nm}$. A
basis for this space consists e.g. of the $nm$ following basis vectors:
$a_{i}\otimes b_{j}^{T}$, where the $a_{i}$ are the $n$ standard basis vectors
of $\mathbb{C}^{n}$ and the $b_{j}$ are the $m$ standard basis vectors of
$\mathbb{C}^{m}$. We note that a unitary vector $V$ has the property
$V^{\dagger}V=rI$, with $I$ once again the $m\times m$ unit matrix.

If Conjecture 1 is true, then also the following conjecture is true:

###### Conjecture 3

Every great unitary matrix $U$ has at least one biunitary vector $V$:

$UV=W,$

where

  * •

both $V$ and $W$ consist of $n/m$ unitary $m\times m$ entries and

  * •

$V$ has leading entry equal to the small unit matrix $I$.

Suffice it to repeat the above reasoning with $m=1$ for $m>1$, the vector
$E=(I,I,I,...,I)^{T}$ taking over the role of the vector $e=(1,1,1,...,1)^{T}$
above.

Important is the fact that not only Conjecture 3 is a consequence of
Conjecture 1, but Conjecture 1 is equally a consequence of Conjecture 3.
Indeed, if $UV=W$, with both $V$ and $W$ being unitary vectors and $V$ having
the unit matrix $I$ as leading entry, then the matrix

$A=\mbox{diag}\,(W_{1}^{-1},W_{2}^{-1},...,W_{r}^{-1})\ U\
\mbox{diag}\,(I,V_{2}^{-1},...,V_{r}^{-1})$

belongs to XU($n$, $m$). Proof of this fact consists of two parts:

  * •

Taking into account that $UV=W$, we find that $AE$ equals $E$, such that $A$
has unit row sums.

  * •

Because of $E=AE$, we have $E=\overline{E}=\overline{AE}=\overline{A}E$.
Taking into account that $A$ is unitary, we have $A^{T}\,\overline{A}$ equal
to the $n\times n$ unit matrix. Hence $E=A^{T}\,\overline{A}E=A^{T}E$. Because
$A^{T}E$ thus turns out to equal $E$, we conclude that $A$ has unit column
sums.

As $A$ belongs to XU($n$, $m$) and $U$ has decomposition

$\mbox{diag}\,(W_{1},W_{2},...,W_{r})\ A\ \mbox{diag}\,(I,V_{2},...,V_{r})\ ,$

Conjecture 1 is fulfilled.

We finally note that Conjecture 2 leads to the same Conjecture 3, according to
a similar proof, where however the vector $(I,0,0,...,0)^{T}$ takes over the
role of the vector $E=(I,I,I,...,I)^{T}$ above. We conclude:

###### Theorem 1

The Conjectures 1, 2, and 3 are equivalent: if one is proved, then all three
are proved.

##  7 Group topology

In order to prove the three conjectures, it suffices to prove Conjecture 3.
For that purpose, we first give a lemma:

###### Lemma 1

If an $n\times n$ unitary matrix $U$ possesses a biunitary $n\times m$ vector,
then it possesses a biunitary $n\times m$ vector with leading entry equal to
the $m\times m$ unit matrix $I$.

Indeed, let us suppose that

$U\left(\begin{array}[]{c}V_{1}\\\ V_{2}\\\ V_{3}\\\ \vdots\\\
V_{r}\end{array}\right)=\left(\begin{array}[]{c}W_{1}\\\ W_{2}\\\ W_{3}\\\
\vdots\\\ W_{r}\end{array}\right)\ ,$

with all $V_{j}$ and $W_{j}$ are unitary blocks. We multiply to the right with
the small matrix $V_{1}^{-1}$ and thus obtain

$U\left(\begin{array}[]{c}I\\\ V_{2}V_{1}^{-1}\\\ V_{3}V_{1}^{-1}\\\ \vdots\\\
V_{r}V_{1}^{-1}\end{array}\right)=\left(\begin{array}[]{c}W_{1}V_{1}^{-1}\\\
W_{2}V_{1}^{-1}\\\ W_{3}V_{1}^{-1}\\\ \vdots\\\
W_{r}V_{1}^{-1}\end{array}\right)\ ,$

a result which proves the lemma.

We consider the vector space ${\bf M}$ of vectors
$(M_{1},M_{2},...,M_{r})^{T}$, where each $M_{j}$ is a complex $m\times m$
matrix. Let ${\bf S}$ be the the following submanifold of ${\bf M}$:

${\bf S}=\\{(V_{1},V_{2},...,V_{r})^{T}\ |\ V_{j}\in{\mbox{U}}(m)\\}\ .$

The Lie group U($m$)r behaves as if it were the following topological product
of odd-dimensional spheres [10] [11]:

$\left(S^{1}\times S^{3}\times S^{5}\times...\times S^{2m-1}\right)^{r}\ ,$

where $S^{k}$ denotes the $k$-sphere. In fact, the Poincaré polynomial of the
manifold $\bf S$ is

$P(x)=[\ (1+x)(1+x^{3})(1+x^{5})...(1+x^{2m-1})\ ]^{r}\ .$

Therefore, the sum of its Betti numbers is

$P(1)=(2^{m})^{r}=2^{n}\ ,$

where $n=mr$.

It is clear that, if

${\bf S}\cap\,U{\bf S}\neq\emptyset\ ,$ (5)

then there exists at least one unitary vector in ${\bf S}$ which is a
biunitary vector an which, because of Lemma 1, has a unit leading entry.

One promising approach is to reduce the problem to the Arnold conjecture [12],
as has been done in the $m=1$ case [1]. If ${\bf S}$ was a Lagrangian
submanifold for a symplectic form on $\mathbb{C}^{n}$ such that $U$ was still
a Hamiltonian symplectomorphism, then eqn (5) would be true, provided the
Arnold conjecture is true for this particular manifold.

Direct computation suggests that ${\bf S}$ is no Lagrangian submanifold of
$\mathbb{C}^{n}$ with the standard symplectic form. There are two possible
roads to still prove a relation to the Arnold conjecture:

  * •

show that ${\bf S}$ is a submanifold for some other symplectic structure on
$\mathbb{C}^{n}$ and $U$ is a Hamiltonian symplectomorphism for that
particular structure, too

  * •

find a Lagrangian embedding of ${\bf S}$ into some other manifold such that
the mapping of $U$ results in a Hamiltonian symplectomorphism for this other
manifold.

Let us start with the first idea: we note that ${\bf S}$ is a Cartesian
product of odd-dimensional spheres and that the Cartesian product of
Lagrangian manifolds is a Lagrangian manifold, it might be possible to
consider each sphere separately. However this is not true, as no sphere
$S^{n}$ with $n>1$ can be embedded into $\mathbb{C}^{n}$ as a Lagrangian
manifold according to [13], as no simply-connected manifold can be embedded
into $\mathbb{C}^{n}$. Since ${\bf S}$ is not simply connected as it contains
a product factor of $S^{1}$, this does not yet rule out the possibility of
finding a symplectic structure such that it is a Lagrangian submanifold, but
there is no argument we know of.

This leaves us with attempting the second idea: Indeed, using [14], who
attributes this idea to [15], we can find a Lagrangian embedding of every odd-
dimensional space via:

$\displaystyle S^{2n+1}$ $\displaystyle\to$ $\displaystyle{\bf
P}^{n}(\mathbb{C})\times{\bf P}^{n}(\mathbb{C})$ $\displaystyle z$
$\displaystyle\mapsto$ $\displaystyle(\,[z],[\overline{z}]\,)$

where ${\bf P}^{n}(\mathbb{C})$ denotes the complex projective space and $[z]$
being the canonical projection. Since $S^{1}$ is a Lagrangian submanifold of
$\mathbb{C}$, and products of Lagrangian embeddings are Lagrangian embeddings
in the product manifold, we can embed ${\bf S}$ as a Lagrangian submanifold.
To be of help, we also would need $U$ to be mapped to a symplectomorphism. To
do that, we note that, if we decompose $z\in{\bf S}$ as
$(z_{1}^{1},z_{3}^{1},z_{5}^{1},\ldots,z_{2m+1}^{r})$, then $U$ acts on any
factor $z_{i}^{r}$ as $U(0,\ldots,0,z_{i}^{r},0,\ldots,0)$, which explains how
it must act on $([z],[\overline{z}])$. But this implies that $U$ will mix
factors of ${\bf P}^{n}(\mathbb{C})$ in our product manifold, which in turn
results in $U$ not being a symplectomorphism after direct computation. This
does not rule out the second idea either, but shows where the difficulties
lie. It is still unclear whether the applicability of symplectic topology to
the original problem of a Sinkhorn-like decomposition was a mere coincidence
or whether there is a deeper link to unitary decompositions so it seems
worthwile to consider this problem.

We summarize: if the Arnold conjecture is applicable, then the above
Conjectures 1, 2, and 3 are true.

##  8 Numerical approximation

We note that the above three conjectures are not constructive. Only in the
case $m=n/2$, do we have explicit expressions for the matrices $D$, $X$, $Z$,
$C$, $A$, and $Y$ and for the vectors $V$ and $W$. For other cases, we can
only find numerical approximations. Therefore, in the present section, we give
a numerical procedure to find, given the matrix $U$, an approximation of the
matrices $D$, $X$, and $Z$. It is similar to the Sinkhorn-like method
presented earlier for the $m=1$ case [7].

The successive approximations $X_{t}$ of $X$ are given by

$X_{0}=U$

and

$X_{t}=L_{t}X_{t-1}R_{t}\ .$

The diagonal of the left great matrix $L_{t}$ consists of $r$ small matrices
$(L_{t})_{jj}$, equal to $\Phi_{j}^{-1}$, i.e. the inverse of the unitary
factor in the polar decomposition $\Phi_{j}P_{j}$ of the row sum
$r_{j}=\sum_{k=1}^{r}(X_{t-1})_{jk}$. The right great matrix $R_{t}$ consists
of $r$ small matrices $(R_{t})_{kk}$, equal to
$\Upsilon_{k}^{-1}\Upsilon_{1}$, with $\Upsilon_{k}^{-1}$ equal to the inverse
of the unitary factor in the polar decomposition $Q_{k}\Upsilon_{k}$ of the
column sum $c_{k}=\sum_{j=1}^{r}(L_{t}X_{t-1})_{jk}$. The extra factor
$\Upsilon_{1}$ in the expression of $(R_{t})_{kk}$ guarantees that
$(R_{t})_{11}$ equals $I$. After a sufficient number (say, $\tau$) of
iterations, the product $L=L_{\tau}.L_{\tau-1}...L_{1}$ and the product
$R=R_{1}.R_{2}...R_{\tau}$ yield the desired great matrix $X$:

$X\approx X_{\tau}=LUR\ .$

The fact that all $(R_{t})_{11}=I$ guarantees that $R_{11}=I$ and thus that
$R$ belongs to ZU($n,m$) instead of merely to DU($n,m$). We have

$\displaystyle D$ $\displaystyle\approx$ $\displaystyle L^{-1}$ $\displaystyle
Z$ $\displaystyle\approx$ $\displaystyle R^{-1}\ .$

Exceptionally, a particular row sum $r_{j}$ might be singular. Then its polar
decomposition is not unique, such that the corresponding matrix $\Phi_{j}$ is
not determined. In that case, we choose $(L_{t})_{jj}$ equal to the unit
matrix $I$. Analogously we choose $(R_{t})_{kk}=I$ whenever a particular
column sum $c_{k}$ is singular.

The progress of the iteration process can be monitored by the following
property of a great matrix $M$:

$\Psi(M)=n^{2}-|\mbox{Btr}(M)|^{2}$

where we call $\mbox{Btr}(M)$ the ‘block trace’ of $M$:

$\mbox{Btr}(M)=\sum_{j=1}^{r}\sum_{k=1}^{r}\,\mbox{Tr}(M_{jk})\ .$

Indeed, the quantity $\Psi(M)$ is zero iff $M\in$ XU($n,m$). During the
iteration process, $\Psi(X_{t})$ becomes smaller and smaller, approaching zero
in the limit. See Appendix for details. We note that, if $m=1$, then Btr is
simply the sum of all $n^{2}$ matrix entries [7].

##  9 Example

As an example, we choose the following U(6) matrix:

$U=\frac{1}{12}\ \left(\begin{array}[]{cccccc}-5&6+2i&-5-5i&-4+2i&2&-2-i\\\
2+2i&-2-4i&-4i&-3-i&5+5i&2+6i\\\ -6-3i&-2-2i&1+3i&-6&-4-2i&3+4i\\\
-2-4i&-1-7i&2-6i&4+3i&-1-2i&-2i\\\ 3-i&4&-4-2i&2-2i&-6+2i&7+i\\\
-6i&-1+3i&-2+2i&3+6i&5i&-2+4i\end{array}\right)\ .$

Hence $n=6$. For $m$, we invesitigate all different possibilities, i.e. $m=1$,
$m=2$, and $m=3$. During the numerical procedure, the progress parameter
$\Psi$ diminishes according to Table 1. We see that, after 36 iterations,
$\Psi$ already approaches 0. Therefore, below we give results for $\tau=36$.

Table 1: Progress parameter $\Psi$ as a function of the number $t$ of iteration steps. | $m=1$ | $m=2$ | $m=3$
---|---|---|---
$t$ | $\Psi_{t}$
0 | 34.889 | 32.000 | 33.743
1 | 4.407 | 9.517 | 6.643
2 | 2.573 | 4.332 | 2.533
3 | 1.381 | 2.680 | 1.023
4 | 0.586 | 1.627 | 0.513
5 | 0.213 | 0.868 | 0.375
6 | 0.084 | 0.577 | 0.318
7 | 0.042 | 0.492 | 0.277
8 | 0.027 | 0.461 | 0.240
9 | 0.020 | 0.442 | 0.206
10 | 0.016 | 0.423 | 0.174
11 | 0.014 | 0.400 | 0.147
12 | 0.012 | 0.372 | 0.122
13 | 0.010 | 0.339 | 0.101
14 | 0.009 | 0.303 | 0.083
15 | 0.008 | 0.264 | 0.067
… |  |  |
36 | 0.001 | 0.001 | 0.001

We thus find, after 36 iterations333Each iteration, in turn, needs $2r$ polar
decompositions. These are performed by Hero’s iterative method (a.k.a. Heron’s
method). For each, we applied only ten iterations.:

  * •

for $m=1$:

$X=\left(\begin{array}[]{rrrrrr}0.27-0.31i&-0.27+0.45i&0.58+0.13i&0.28-0.24i&0.07+0.15i&0.07-0.17i\\\
0.04+0.23i&-0.12-0.35i&0.26-0.21i&-0.01-0.26i&0.57+0.13i&0.25+0.46i\\\
0.51+0.22i&0.23+0.06i&-0.02-0.26i&0.42+0.27i&0.27-0.25i&-0.42-0.03i\\\
0.37-0.03i&0.57-0.14i&0.28+0.45i&-0.42-0.04i&0.06-0.18i&0.16-0.06i\\\
0.26+0.01i&0.33-0.04i&-0.16-0.33i&0.23+0.05i&-0.23+0.47i&0.57-0.16i\\\
-0.49-0.12i&0.26+0.02i&0.06+0.23i&0.51+0.22i&0.26-0.33i&0.37-0.03i\end{array}\right)\
,$

with the following row sums and column sums:

$\displaystyle r_{1}$ $\displaystyle=$ $\displaystyle 1.002+0.000i$
$\displaystyle r_{2}$ $\displaystyle=$ $\displaystyle 0.998-0.001i$
$\displaystyle r_{3}$ $\displaystyle=$ $\displaystyle 1.007+0.001i$
$\displaystyle r_{4}$ $\displaystyle=$ $\displaystyle 1.007+0.002i$
$\displaystyle r_{5}$ $\displaystyle=$ $\displaystyle 0.998-0.000i$
$\displaystyle r_{6}$ $\displaystyle=$ $\displaystyle 0.988-0.002i$
$\displaystyle c_{1}$ $\displaystyle=$ $\displaystyle 0.989-0.002i$
$\displaystyle c_{2}$ $\displaystyle=$ $\displaystyle 0.998-0.001i$
$\displaystyle c_{3}$ $\displaystyle=$ $\displaystyle 0.997-0.000i$
$\displaystyle c_{4}$ $\displaystyle=$ $\displaystyle 1.006+0.001i$
$\displaystyle c_{5}$ $\displaystyle=$ $\displaystyle 1.002+0.001i$
$\displaystyle c_{6}$ $\displaystyle=$ $\displaystyle 1.008+0.001i\ ;$

  * •

for $m=2$:

$X=\left(\begin{array}[]{rrrrrr}0.33-0.05i&-0.39-0.07i&0.61+0.34i&0.32+0.19i&0.06-0.29i&0.08-0.12i\\\
-0.30-0.16i&0.35+0.36i&0.07+0.16i&0.08+0.09i&0.23-0.01i&0.57-0.45i\\\
0.41-0.38i&0.51-0.02i&0.34+0.14i&-0.30-0.12i&0.26+0.24i&-0.21+0.13i\\\
0.36-0.08i&0.26-0.27i&-0.17-0.26i&0.56+0.34i&-0.19+0.34i&0.17-0.07i\\\
0.26+0.43i&-0.12+0.08i&0.06-0.48i&-0.02-0.06i&0.68+0.04i&0.13-0.02i\\\
-0.06+0.24i&0.39-0.09i&0.10+0.09i&0.35-0.43i&-0.04-0.34i&0.26+0.52i\end{array}\right)\
,$

with row sums and column sums

$\displaystyle r_{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{rr}0.994-0.001i&0.003-0.002i\\\
0.001+0.000i&0.997-0.000i\end{array}\right)$ $\displaystyle r_{2}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{rr}1.003+0.002i&-0.001-0.005i\\\
-0.003-0.000i&1.003+0.002i\end{array}\right)$ $\displaystyle r_{3}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{rr}1.002+0.002i&-0.001+0.001i\\\
0.001-0.005i&1.000+0.001i\end{array}\right)$ $\displaystyle c_{1}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{rr}1.002+0.001i&-0.004-0.002i\\\
-0.004-0.001i&1.001+0.001i\end{array}\right)$ $\displaystyle c_{2}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{rr}1.002+0.001i&0.001+0.002i\\\
0.002-0.001i&0.998-0.001i\end{array}\right)$ $\displaystyle c_{3}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{rr}0.995-0.002i&0.002-0.000i\\\
0.002+0.002i&1.001+0.000i\end{array}\right)\ ;$

  * •

for $m=3$:

$X=\left(\begin{array}[]{rrrrrr}0.54+0.37i&-0.10-0.25i&0.16-0.13i&0.45-0.36i&0.10+0.24i&-0.16+0.13i\\\
-0.23-0.13i&0.47-0.06i&-0.07-0.41i&0.23+0.13i&0.53+0.06i&0.07+0.41i\\\
-0.13-0.16i&-0.37-0.19i&0.53+0.17i&0.13+0.16i&0.37+0.19i&0.47-0.17i\\\
0.46-0.37i&0.10+0.24i&-0.16+0.13i&0.54+0.36i&-0.10-0.24i&0.16-0.13i\\\
0.23+0.13i&0.53+0.06i&0.07+0.41i&-0.23-0.14i&0.47-0.06i&-0.07-0.41i\\\
0.13+0.16i&0.37+0.19i&0.47-0.17i&-0.13-0.16i&-0.37-0.19i&0.52+0.17i\end{array}\right)\
,$

with row sums and column sums

$\displaystyle r_{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{rrr}0.997+0.001i&-0.004-0.001i&-0.002-0.002i\\\
-0.001+0.002i&0.999+0.000i&0.001-0.001i\\\
0.001+0.001i&0.002-0.001i&1.003-0.001i\end{array}\right)$ $\displaystyle
r_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{rrr}1.002-0.001i&0.002+0.000i&-0.000+0.001i\\\
0.003-0.003i&1.000-0.000i&-0.003+0.000i\\\
0.001-0.002i&-0.001+0.001i&0.997+0.001i\end{array}\right)$ $\displaystyle
c_{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{rrr}1.000+0.000i&-0.000-0.003i&0.001-0.003i\\\
0.002+0.002i&1.000+0.000i&0.000-0.002i\\\
0.003+0.001i&0.001+0.001i&1.000-0.000i\end{array}\right)$ $\displaystyle
c_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{rrr}1.000-0.000i&0.000+0.003i&-0.001+0.003i\\\
-0.001-0.002i&1.000-0.000i&-0.000+0.002i\\\
-0.003-0.001i&-0.002-0.001i&1.000+0.000i\end{array}\right)\ .$

For $m=2$, we also give the corresponding biunitary vector:

$U\left(\begin{array}[]{rr}1.00-0.00i&0.00+0.00i\\\
0.00-0.00i&1.00+0.00i\\\\[2.84526pt] 0.81-0.31i&0.23+0.43i\\\
-0.20+0.44i&0.84+0.25i\\\\[2.84526pt] -0.34+0.77i&-0.37+0.38i\\\
-0.29-0.45i&0.19+0.83i\end{array}\right)=\left(\begin{array}[]{rr}-0.95-0.16i&0.24-0.14i\\\
-0.14-0.24i&-0.91-0.31i\\\\[2.84526pt] 0.06-0.70i&-0.71+0.01i\\\
-0.28-0.65i&0.62-0.34i\\\\[2.84526pt] -0.12-0.73i&0.67-0.03i\\\
-0.48-0.46i&-0.57+0.47i\end{array}\right)\ .$

##  10 Permutation matrices

Although we lack a proof of Conjecture 1 in the case of an arbitrary unitary
matrix $U$, we can say that Conjecture 1 is certainly true for the case where
$U$ is an arbitrary $n\times n$ permutation matrix. Indeed, any permutation
matrix of size of $n\times n=mr\times mr$ can be decomposed as a product of
three permutation matrices $D$, $X$, and $Z$, the matrix $D$ belonging to the
group DU($n,m$), the matrix $X$ belonging to the group XU($n,m$), and the
matrix $Z$ belonging to the group ZU($n,m$). In fact, $D$ belongs to a finite
subgroup of DU($n,m$), of order $(m!)^{r}$ and isomorphic to the product
S${}_{m}^{r}$ of symmetric goups, $X$ belongs to a finite subgroup of
XU($n,m$), of order $(r!)^{m}$ and isomorphic to the product S${}_{r}^{m}$ of
symmetric goups, and $Z$ belongs to a finite subgroup of ZU($n,m$), of order
$(m!)^{r-1}$ and isomorphic to the product S${}_{m}^{r-1}$. The fact that such
a decomposition is always possible [16] [17], is a consequence of Birkhoff’s
theorem [18] on doubly stochastic matrices (with rational entries). The
decomposition has been applied both in Clos networks of telephone switching
systems [19] [20] and in reversible computing [21].

As an example, we choose the following $6\times 6$ permutation matrix:

$U=\left(\begin{array}[]{rrrrrr}0&0&0&0&1&0\\\ 1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\
0&0&0&1&0&0\\\ 0&0&0&0&0&1\\\ 0&0&1&0&0&0\end{array}\right)\ .$

For $m$, we investigate all different non-trivial444We note that, for
permuation matrices, not only the case $m=n$ is trivial, but also the case
$m=1$: suffice it to choose both $D$ and $Z$ equal to the $n\times n$ unit
matrix and to choose $X$ equal to $U$. possibilities: $m=2$ and $m=3$. We
have:

  * •

for $m=2$:

$U=\left(\begin{array}[]{rrrrrr}0&1&&&&\\\ 1&0&&&&\\\ &&0&1&&\\\ &&1&0&&\\\
&&&&1&0\\\ &&&&0&1\end{array}\right)\left(\begin{array}[]{rrrrrr}1&0&\ 0&0&\
0&0\\\ 0&0&0&0&0&1\\\\[2.84526pt] 0&0&1&0&0&0\\\ 0&1&0&0&0&0\\\\[2.84526pt]
0&0&0&0&1&0\\\
0&0&0&1&0&0\end{array}\right)\left(\begin{array}[]{rrrrrr}1&0&&&&\\\
0&1&&&&\\\ &&0&1&&\\\ &&1&0&&\\\ &&&&0&1\\\ &&&&1&0\end{array}\right)\ ,$

where indeed the middle matrix has six unit line sums
$r_{1}=r_{2}=r_{3}=c_{1}=c_{2}=c_{3}={\tiny\left(\begin{array}[]{rr}1&\\\
&1\end{array}\right)}$;

  * •

for $m=3$:

$U=\left(\begin{array}[]{rrrrrr}0&0&1&&&\\\ 1&0&0&&&\\\ 0&1&0&&&\\\
&&&1&0&0\\\ &&&0&1&0\\\
&&&0&0&1\end{array}\right)\left(\begin{array}[]{rrrrrr}1&0&0&\ 0&0&0\\\
0&1&0&0&0&0\\\ 0&0&0&0&0&1\\\\[2.84526pt] 0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\
0&0&1&0&0&0\end{array}\right)\left(\begin{array}[]{rrrrrr}1&0&0&&&\\\
0&1&0&&&\\\ 0&0&1&&&\\\ &&&1&0&0\\\ &&&0&0&1\\\ &&&0&1&0\end{array}\right)\ ,$

where indeed the middle matrix has four unit line sums
$r_{1}=r_{2}=c_{1}=c_{2}={\tiny\left(\begin{array}[]{rrr}1&&\\\ &1&\\\
&&1\end{array}\right)}$.

Because Conjecture 1 is true for any $n\times n$ permutation matrix, it also
is true for any $n\times n$ complex permutation matrix (i.e. unitary matrix
with only one non-zero entry in every row and column). Such matrices form an
$n$-dimensional non-connected subgroup of the $n^{2}$-dimensional group U($n$)
(consisting of $n!$ components, each $n$-dimensional). We can indeed decompose
such matrix as $D^{\prime}P$, where $D^{\prime}$ is a diagonal unitary matrix
and $P$ is a permutation matrix. We decompose $P$ as $D^{\prime\prime}XZ$,
leading to the decomposition $D^{\prime}D^{\prime\prime}XZ$ of the complex
permutation matrix. Introducing $D=D^{\prime}D^{\prime\prime}$, we obtain a
desired decomposition $DXZ$.

##  11 Conclusion

Every $n\times n$ unitary matrix has an Idel–Wolf decomposition. If $n$ is
even, then it also has a Führ–Rzeszotnik decomposition. We conjecture that, if
$n$ is a composed integer, it has as many similar decompositions as $n$ has
divisors. We offer no proof, as generalization of either the Idel–Wolf proof
(based on symplectic topology) or the Führ–Rzeszotnik proof (based on linear
algebra) is not straightforward. We provide an iterative algorithm for finding
a numerical approximation of each of the conjectured decompositions. Finally,
we demonstrate that the conjecture is true for $n\times n$ (complex)
permutation matrices.

## Appendix

In Section 8, multiplying $X_{t-1}$ to the left with $L_{t}$ increases its
block trace:

$|\mbox{Btr}(L_{t}X_{t-1})|=\left|\
\sum_{j=1}^{r}\sum_{k=1}^{r}\,\mbox{Tr}((L_{t}X_{t-1})_{jk})\ \right|=\left|\
\sum_{j=1}^{r}\sum_{k=1}^{r}\,\mbox{Tr}(\Phi_{j}^{-1}\,(X_{t-1})_{jk})\
\right|$ $=\left|\ \sum_{j=1}^{r}\,\mbox{Tr}(P_{j})\ \right|\geq\left|\
\sum_{j=1}^{r}\,\mbox{Tr}(\Phi_{j}P_{j})\ \right|=\left|\
\sum_{j=1}^{r}\sum_{k=1}^{r}\,\mbox{Tr}((X_{t-1})_{jk})\
\right|=|\mbox{Btr}(X_{t-1})|\ .$

Analogously, multiplying $L_{t}X_{t-1}$ to the right with $R_{t}$ increases
its block trace. Hence, we have

$|\mbox{Btr}(X_{t})|=|\mbox{Btr}(L_{t}X_{t-1}R_{t})|\geq|\mbox{Btr}(L_{t}X_{t-1})|\geq|\mbox{Btr}(X_{t-1})|\
.$

The increasing value of $|\mbox{Btr}(X_{t})|$ is bounded by the value $n$. An
$n\times n$ unitary matrix $A$ has $|\mbox{Btr}(A)|=n$ iff it is a member of
the group $e^{i\alpha}$ XU($n,m$). These two facts are proved by reasoning as
in Appendix A of De Vos and De Baerdemacker [7], by considering the following
property of the row sums $r_{a}$ and column sums $c_{b}$ of $A$:

$\sum_{a=1}^{r}\sum_{j=1}^{m}\sum_{k=1}^{m}\
\left|(r_{a})_{jk}\right|^{2}=\sum_{b=1}^{r}\sum_{j=1}^{m}\sum_{k=1}^{m}\
\left|(c_{b})_{jk}\right|^{2}=n\ ,$

a fact which, in turn, is proved by reasoning as in Appendix A of De Vos, Van
Laer, and Vandenbrande [22].

Acknowledgements. SDB acknowleddes the Canada Research Chair program and the
New Brunswick Innovation Foundation.

## References

  * [1] M. Idel and M. Wolf: Sinkhorn normal form for unitary matrices. Linear Algebra and its Applications, volume 471 (2015), 76-84.
  * [2] H. Führ and Z. Rzeszotnik: On biunimodular vectors for unitary matrices. Linear Algebra and its Applications, volume 484 (2015), 86-129.
  * [3] A. De Vos and S. De Baerdemacker: The synthesis of a quantum circuit. Proceedings of the 11th International Workshop on Boolean Problems, Freiberg (17-19 September 2014), 129-136.
  * [4] A. De Vos and S. De Baerdemacker: Block-$ZXZ$ synthesis of an arbitrary quantum circuit. Physical Review A, volume 94 (2016), 052317.
  * [5] A. De Vos, S. De Baerdemacker, and Y. Van Rentergem: Synthesis of quantum circuits versus synthesis of classical reversible circuits. Synthesis Lectures on Digital Circuits and Systems, volume 54, Morgan & Claypool, La Porte (2018), ISBN 9781681733814.
  * [6] T. Simnacher, N. Wyderka, C. Spee, X. Yu, and O. Gühhe: Certifying quantum memories with coherence. Physical Review A, volume 99 (2019), 062319.
  * [7] A. De Vos and S. De Baerdemacker: Scaling a unitary matrix. Open Systems & Information Dynamics, volume 21 (2014), 1450013.
  * [8] H. Führ and Z. Rzeszotnik: A note on factoring unitary matrices. Linear Algebra and its Applications, volume 547 (2018), 32-44.
  * [9] A. De Vos and S. De Baerdemacker: On two subgroups of U($n$), useful for quantum computing. Journal of Physics: Conference Series: Proceedings of the 30th International Colloquium on Group-theoretical Methods in Physics, Gent, 14-18 July 2014, volume 597 (2015), 012030.
  * [10] L. Pontrjagin: Homologies in compact Lie groups. Recueil Mathématique, volume 6 (1939), 389-422.
  * [11] H. Samelson: Topology of Lie groups. Bulletin of the American Mathematical Society, volume 58 (1952), 2-37.
  * [12] V. Arnold: Sur une propriété topologique des applications globalement canoniques de la mécanique classique. Comptes Rendues de l’Académie des Sciences, volume 261 (1965), 3719-3722.
  * [13] M. Gromov: Pseudo-holomorphic curves in symplectic manifolds. Inventiones Mathematicae, volume 82 (1985), 307-347.
  * [14] M. Audin: On the topology of Lagrangian submanifolds examples and counter-examples. Portugaliae Mathematica, volume 62 (2005), 375-419.
  * [15] M. Audin, F. Lalonde, and L. Polterovich: Symplectic rigidity: Lagrangian submanifolds. In: Holomorphic curves in symplectic geometry, M. Audin and J. Lafontaine, editors, Progress in Mathematics, Birkhäuser (1994), 271-321.
  * [16] A. De Vos and Y. Van Rentergem: Young subgroups for reversible computers. Advances in Mathematics of Communications, volume 2 (2008), 183-200.
  * [17] A. De Vos and Y. Van Rentergem: Networks for reversible logic. Proceedings of the 8 th International Workshop on Boolean Problems, Freiberg, 18-19 September 2008, 41-47.
  * [18] G. Birkhoff: Tres observationes sobre el algebra lineal. Universidad Nacional de Tucaman: Revista Matematicas y Fisica Teorica, volume 5 (1946), 147-151.
  * [19] C. Clos: A study of non-blocking switching networks. Bell Systems Technical Journal, volume 32 (1953), 406-424.
  * [20] F. Hwang: Control algorithms for rearrangeable Clos networks. IEEE Transactions on Communications, volume 31 (1983), 952-954.
  * [21] A. De Vos: Reversible computing. Wiley - VCH, Weinheim, 2010, ISBN 9783527409921.
  * [22] A. De Vos, R. Van Laer, and S. Vandenbrande: The group of dyadic unitary matrices. Open Systems & Information Dynamics, volume 19 (2012), 1250003.


# Broadband Continuous Spectral Control of a Single Wavelength Polymer-Based
Solid-State Random Laser

Bhupesh Kumar  1,2 Sebastian A. Schulz  2 and Patrick Sebbah1  1Department of
Physics, The Jack and Pearl Resnick Institute for Advanced Technology, Bar-
Ilan University, Ramat-Gan, 5290002 Israel
2SUPA, School of Physics and Astronomy, University of St. Andrews, Fife, KY16
9SS, UK
<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

We demonstrate temperature-controlled spectral tunability of a partially-
pumped single-wavelength random laser in a solid-state random laser based on
DCM (4-dicyanomethylene-2-methyl-6-(p-dimethylaminostyryl)-4H-pyran) doped
PMMA (polymethyl methacrylate) dye. By carefully shaping the spatial profile
of the pump, we first achieve low-threshold, single-mode random lasing with
excellent side lobes rejection. Notably, we show how temperature-induced
changes in the refractive index of the PMMA-DCM layer result in a blue-shift
of this single lasing mode. Continuous tunability of the lasing wavelength is
demonstrated over a 8nm-wide bandwidth.

##  1 Introduction

Random lasers are unconventional laser sources in which feedback is provided
by randomly-distributed scattering particles. In the past two decades, random
lasers have been the subject of intense theoretical and experimental study [1,
2, 3, 4, 5, 6, 7, 8, 9]. Tunability and directionality are important features
that determine the application scope of any laser device in fields such as
integrated spectroscopy, remote sensing, and optical communication[10]. Single
wavelength random laser tunability is challenging due to the random and
multimode nature of the emission spectrum, fluctuations in the emission
spectrum, and lack of precise, non-invasive and reversible tuning mechanism.
Spectral tunability in random lasers has been demonstrated via multiple
mechanisms, including optical fiber-based random lasers[11], pump size control
or scatterer concentration variation [12], gain medium thickness variation
[13], dye molecule selection [14, 15], or mechanical stretching [16, 17].
Other tuning mechanisms include engineering absorption of light emission[18],
switching modes associated with different lengths of silver nanorods in
plasmonic random lasers[19], as well as electric-field-induced tunability
[20]. However, all these mechanisms have been limited to the spectral tuning
of broad linewidth or multimode random lasers. Tunable single-mode random
laser has been reported in rare-earth-doped fiber random laser, emitting
exclusively in the mid-near-infrared [22, 23, 24, 25]. Single-mode random
lasing temperature-based tunability in the visible has been demonstrated in
liquid crystal-embedded random lasers, but tunability was found to be limited
to a few nanometers by the nematic-isotropic transition temperature [21]. To
the best of our knowledge, continuous broadband tuning of a singlemode random
laser in the visible as not yet been reported.
Recently, we have demonstrated that using an iterative pump shaping
optimization method, selective excitation of a particular mode of the
multimode emission spectrum and single-mode operation with high side-lobe
rejection can be achieved [26, 27, 28]. However, it is not possible to achieve
lasing at any arbitrary desired wavelength, but only at wavelengths
corresponding to lasing modes of the discrete multimode emission spectrum.
Overcoming this limitation would add tunability to this technique and offer
full spectral control of the random laser. In this paper, we report single-
wavelength continuous broadband spectral tuning in a dye-doped solid-state
random laser. Random lasers (RLs) typically produce multimode coherent light
due to the random distribution of scattering particles. By combining the
technique of spatial pump shaping together with the negative thermal
coefficient of the refractive index of PMMA polymer, we are able to achieve
single wavelength spectral control of a random laser on a single device.
Specifically, we show continuous broadband tunability over a bandwidth of 8
nm.

##  2 Sample fabrication

Disordered structures were fabricated using e-beam lithography on a 600 nm
layer of poly(methyl methacrylate) or PMMA that was doped with 5$\%$ weight of
DCM laser dye. The fluorescence spectrum of the dye is centered around 600 nm;
it has a high fluorescence quantum yield and a large stoke shift (100 nm),
which means it does not reabsorb the emitted light. The PMMA used had a
molecular weight of 49500 and was used at a concentration of 6$\%$ weight in
anisole.

Figure 1: (a) High resolution SEM image of the small section of the sample
from the top showing air grooves carved into the PMMA-DCM layer. (b) Schematic
of the experimental setup. BE1 and BE2: Beam expanders; P: Polarizer, SLM:
Spatial light modulator; M2: Mirror; S: Sample; TC: Temperature controller;
CP: Copper plate;

The active layer was obtained by spin-coating a doped-polymer solution at 1000
rpm for 60 seconds on a fused silica wafer (Edmund optics) and post-baking it
at 120°C for 2 hours in an oven[28]. The silica wafer has a refractive index
of 1.45, compared to 1.54 for the PMMA-DCM layer. To fabricate one
dimensional(1D)-disorder samples, 125 randomly distributed parallel grooves,
each 200 nm wide and 50 µm long, covering a total length of 1000 $\mu$m, were
carved into the PMMA-DCM layer using e-beam lithography (CRESTEC/CABL-9000C)
(see Fig. 1(a)), resulting in dielectric sections with an average length of 8
µm separated by air gaps of 200 nm. The fabrication method ensures a high
refractive index contrast of 0.54 between air grooves and the polymer layer,
which helps in achieving random lasing action at a low threshold. We require
manuscripts to identify a single corresponding author.

##  3 Experimental Setup

A schematic of the experimental setup is shown in Fig.1(b). The setup consists
of a frequency-doubled mode-locked Nd:YAG laser (EXPLA PL2230: 532 nm, 20 ps,
maximum output energy 28 mJ, repetition rate 1-50 Hz). The beam is expanded 5X
and spatially modulated by a 1952 x 1088 pixel reflective spatial light
modulator (SLM) (Holoeye HES 6001, pixel size 8.0 µm). The SLM is placed in
the object plane of a telescope with 4X reduction and is imaged on the sample.
This setup is used to create a laser strip whose width and length are adjusted
according to the sample dimensions. The disorder structure is precisely
aligned with the laser strip under a fixed stage Zeiss microscope
(AxioExaminer A1) and imaged using an Andor Zyla sCMOS camera (22 mm diagonal
view, 6.5 µm pixel size) using a 10X objective. The laser emission is
collected through a multimode optical fiber connected to a Horiba iHR550
imaging spectrometer equipped with a 2400 $mm^{-1}$ grating and Synapse CCD
detection system (sampling rate 1 MHz, 1024 x 256 pixels, 26 µm pixel pitch).
The entrance slit is 50 µm, resulting in a spectral resolution of 20 pm.

Figure 2:  (a) Multimode Random laser emission spectra obtained after pumping
the sample uniformly above the threshold, sample length 1000 $\mu$m, pump size
1000 $\mu$m X 50 $\mu$m . (b) Emission spectra were recorded for 3 different
samples of identical disorder configuration at same pump power. (c) The
integrated intensity of the emission spectrum with increasing Pump energy
(nJ). inset: Angular distribution of the integrated output intensity of random
laser emission at a distance of 10 cm from the sample edge, the blue line is a
Gaussian fit. (d) Integrated intensity as a function of the number of pump
pulses for the random laser. The pump energy is about 50 nJ for a uniform pump
of size 450 $\mu$m X 50 $\mu$m at a repetition rate of 10 Hz. The blue line is
an exponential fit. Figure 3: (a) Optical microscope image of field intensity
distribution near the sample surface when pumped above the threshold. (b)
Multimode emission spectra for a uniform pump strip of size 450 $\mu$m X 50
$\mu$m. (c) Singlemode lasing at $\lambda$= 602.60 nm after an iterative pump
optimization process. Inset shows the optimized pump profile in grayscale. (d)
Iterative optimization of pump shaping has been applied to select 50 lasing
modes. The sidelobe rejection ratio(SR) for each individually selected lasing
mode w.r.t maximum noise count in the emission spectrum is plotted in log
representation (dB).

##  4 Results

When the sample undergoes optical pumping, amplified spontaneous emission
experiences multiple scattering. This provides in turn coherent optical
feedback, enabling random lasing oscillations with sharp emission linewidth.
Once the pump intensity reaches above the lasing threshold, the output
intensity increases manifold. This nonlinear increase in output intensity
confirms the onset of random lasing action. By linear fit, the lasing
threshold value is found to be 35 nJ for a pump size of 450 $\mu$m X 50 $\mu$m
(Fig.2(c)). Above the threshold, multimode lasing is achieved (illustrated in
Fig.2(a)). The resulting spectrum displays randomly positioned, distinct
lasing peaks, each exhibiting a typical linewidth of 0.2 nm, constrained by
the resolution of the spectrometer.

Figure 4:  (a) The spectral blue shift of 1 nm in the single mode lasing
spectrum at $\lambda$ = 602.60 nm with increasing temperature of the polymer
layer. The figure in the inset shows the temperature vs wavelength plot.(b)
Emission spectrum stability at constant temperature. The emission spectrum was
recorded at an interval of 15, 30, 45 minutes when the sample was heated to a
constant temperature of 55°C. (c) Number of lasing modes for three different
sample sets plotted as a function of increasing deviation (D) from the mean
periodic position of 8 $\mu m$ in a sample of length 1000 $\mu m$ having 125
air grooves. (d) Continuous tunability over a bandwidth of 8 nm by tuning of
10 different individually selected lasing modes M1= 600.2 nm, M2= 600.9 nm,
M3= 601.50 nm, M4= 602.6 nm, M5= 603.3 nm, M6= 604.1 nm, M7= 605.00 nm, M8=
605.800 nm, M9= 606.900 nm, M10= 608.00 nm.

The directivity of our random laser is assessed, with half of the emission
intensity confined within $\pm$ 2.5 degrees from the center of the sample.
This is found by scanning a 20X microscope objective placed at a distance of
10 cm from the sample edge to collect emission in a direction orthogonal to
the sample length. Directional measurements are shown in inset Fig.2(c).
We demonstrate spectral reproducibility by manufacturing three samples with
identical disorder configurations on the same substrate. These samples are
then subjected to uniform pumping at the same pump power, and their respective
emission spectra are recorded. Remarkably, all three samples yield identical
emission spectra, as depicted in Fig. 2(b). The high degree of spectral
reproducibility is attributed to the highly precise e-beam nanolithography
fabrication technique.
We also examined the photostability of our quasi-1D random laser by subjecting
the sample to uniform pumping using the same experimental conditions as
described earlier. The pump energy was held constant at 50 nJ, employing a
pump size of 450 µm x 50 µm, and a pump laser repetition rate of 10 Hz. In
Figure 2(c), we present the integrated emission intensity graphed against the
laser pulse number. The photostability of the laser was observed to be 28380
pulses, equivalent to approximately 47.3 minutes, during which the integrated
laser emission intensity reduced to half of its original integrated lasing
intensity.
To achieve singlemode lasing at any of the lasing wavelengths of the multimode
emission spectrum(Fig.3(b)), we apply a nonuniform intensity profile to the
pump beam by modulating the SLM. We implement an iterative optimization method
[26, 28] to find the optimal pump profile for which modes other then the
target mode are suppressed. Here we use Nelder-Mead simplex algorithm
implemented in the fminsearch function of MATLAB to optimize the pump profile.
We slightly modified the fminsearch function by setting the initial step
(usually the $\delta$ parameter in fminsearch) to 1.0 in order to explore a
larger region of the 32-dimensional space. The number of pixels (32) was
chosen as a compromise between sensitivity and computation time. The Matlab-
generated image that is sent to the SLM is made up of 32 intensity blocks,
each of which is encoded on 256 levels of grey and projected onto the sample.
To start the optimization process, we choose 32 column-vectors $V_{i}$ from
the 32 x 32 binary Hadamard matrix as the initial vertices.

The pump profile P(x) is therefore written as

$\Huge{P(x)}=\Huge{\frac{1}{255}\sum_{i=1}^{32}\beta_{i}V_{i}}$

where $\beta_{i}$ takes discrete values in the range [0,255]. Each vector
$\beta_{i}$ corresponds to a particular pump profile associated with a
particular emission spectrum $I\left(\lambda\right)$. The optimization
algorithm uses the inverse of the extinction ratio, $\eta=I_{t}/I_{o}$, as its
cost function and aims for its minimization, where $I_{t}$ is the peak
intensity of the targeted mode and $I_{o}$ is the highest intensity among all
other modes except the targeted mode. To find the optimized pump profile, the
algorithm iteratively generates a new pump profile, applies it to the pump
stripe, acquires the emission spectrum averaged over 10 shots, and computes
the cost function. Mode selection and single mode operation at the lasing
wavelength of $\lambda$ = 602.60 nm are achieved after the convergence of the
iterative optimization, as shown in Figure 3(c). After 250 iterations, the
algorithm converges to a pump profile that suppresses modes other than the
targeted mode. The selected lasing mode has a sidelobe rejection ratio
(measured as the ratio of the peak intensity count of the target mode w.r.t.
maximum noise level in the spectrum) of 800, which corresponds to 53 dB. The
next highest intensity count, except for the target mode, in the emission
spectrum is 4 counts, which is close to the average noise level present in the
spectrum. We also tested the robustness of our pump profile optimization
method to achieve single mode lasing by selecting 50 different lasing modes on
different samples. Figure 3(d) shows the sidelobe rejection of selected modes.
All the modes are selected with an excellent side-lobe rejection of more than
40 dB.
Next, we demonstrate how the emission wavelength of the selected lasing mode
can be tuned by changing the temperature of the sample. RL sample was
positioned on a copper plate featuring a rectangular hole. Placement of the
sample over the hole, enabled us to pump the RL sample from the bottom. The
copper plate was connected to a heating probe to heat the sample and a sensor
to monitor the temperature. Figure 4(a) shows the blue shift of single-mode
random laser emission initially centered at $\lambda$ = 602.60 nm, when
temperature is increased. The inset in figure 4(a) shows the linear
temperature dependence of the spectral shift. A linear fit gives a slope of
d$\lambda$/dT= -0.02 nm/°C., which means that a 5 °C increase results in a
spectral shift of 0.1 nm. Overall, we obtained a total shift of 1 nm between
25 °C to 75 °C. When the sample is cooled down to 25 °C, the initial spectral
position is recovered and the process is perfectly reversible. This behavior
is easily explained by the fact that PMMA polymer has a negative thermal
coefficient of refractive index[29, 30, 31]. Its refractive index therefore
decreases with increasing temperature, which results in a decrease in the
optical path length within the polymer layer and a blue-shift. This process is
reversible.
We also performed a spectral stability test at a constant temperature higher
than room temperature in a minimum airflow lab environment. An optimally-
pumped lasing sample emitting light at $\lambda$= 602.60 nm is supplied with
constant heat to achieve a wavelength shift of 0.60 nm at sample temperature
of 55 °C. The emission spectra are recorded at 3 intervals of 15 minutes each.
Lasing emission at @$\lambda$= 602.00 nm remains stable within $\pm$ 0.1 nm as
shown in Fig.4(b).
Since all lasing modes within the multimode spectrum can be selected
individually [28], the tuning range can be in principle extended over the
whole gain curve by hopping from mode to mode. To demonstrate continuous
tunability by mode hopping over a broader frequency range, we need first to
identify disordered structures that provide a free spectral range smaller than
the frequency shift we can achieve with a single lasing mode (typically 1 nm).
Ideally, we need 10-15 modes within 8 to 10 nm spectral bandwidth of the gain
curve. Interestingly, by varying the degree of disorder, the spectral density
of lasing modes varies, as shown in Fig. 4c where the average number of lasing
modes is plotted as a function of disorder deviation from the mean periodic
position. Here, a 1000 $\mu$m sample with 125 air grooves has been considered,
with increasing disorder ranging from 0.1 $\mu$m to 3 $\mu$m deviation from
the mean spatial period of 8 $\mu$m. The reason for it is that the spatial
confinement of the eigenmodes increases with increasing disorder, resulting in
an increasing number of lasing modes able to reach threshold. We found that a
disorder pattern with a deviation of $\pm$ 0.5 $\mu$m is enough to yield an
average of 13 lasing modes within the wavelength range of 600 nm to 608 nm. We
choose a pump length of 450 $\mu$m and run iterative optimization of the pump
profile to select 10 lasing modes distributed over the range of 600-608 nm.
The thermal-induced spectral shift is recorded for all the modes. Single-
wavelength continuous broadband tunability by mode hopping is achieved over 8
nm, as shown in Fig.4(d).

##  5 Conclusion

In this paper, we have used a stable solid-state dye-based random laser to
demonstrate temperature-induced tunability in the visible. By enforcing
singlemode operation using pump shaping method, we have shown how temperature-
induced change of the refractive index can achieve spectral tunability. In
contrast to other mechanisms reported in the literature, our proposed method
ensures singlemode tuning, it is non-invasive and does not require any
modification of the sample. This random laser offers the freedom to control
the free spectral range (FSR) by changing the degree of disorder. We have
demonstrated that the emission wavelength of any individually selected lasing
mode can be continuously blue-shifted by up to 1 nm when increasing the
temperature of the PMMA-DCM layer. By hoping from one mode to another, we have
demonstrated remarkable broadband tunability-range of 8 nm. This tunability-
range can in principle be further increased by exploring different disorder
configurations and dyes.

Such a tunable random laser offers the advantage of simplicity of fabrication,
large tunable bandwidth, compact in size, and the ability to operate in harsh
environments compared to conventional tunable lasers that might be bulky, or
require precise tuning and control mechanisms, and have limited spectral
bandwidth. This single-wavelength tunable random laser holds promising
potential for future applications for on-chip tunable laser sources as well as
wearable temperature sensors.

##  6 Acknowledgement

We extend our heartfelt appreciation to Dr. Leonid Wolfson for his unwavering
dedication to the lab, and to Dr. Yossi Abulafia for his valuable assistance
with the fabrication process. We are grateful to the Bar-Ilan Institute of
Nanotechnology and Advanced Materials for providing us access to their
fabrication facilities.

##  7 Disclosures

The authors declare no conflicts of interest.

##  8 Data Availability Statement

Data underlying the results presented in this paper are not publicly available
at this time but may be obtained from the authors upon reasonable request.

## References

  * [1] H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, "Random laser action in semiconductor powder." Phys. Rev. Lett. 82, 2278-2281 (1999).
  * [2] C. Vanneste, P. Sebbah, "Selective Excitation of Localized Modes in active random media" Phys. Rev. Lett. 87, 183903 (2001).
  * [3] C. J. S. de Matos, L. S. Menezes, A. M. Brito-Silva, M. A. M. Gámez, A. S. L. Gomes, and C. B. de Araújo, "Random Fiber Laser", Phys. Rev. Lett. 99, 153903 (2007).
  * [4] M. Gaio, D. Saxena, J. Bertolotti, D. Pisignano, A. Camposeo, and R. Sapienza, "A nanophotonic laser on a graph", Nat. Comm. 10, 226 (2019).
  * [5] J. Liu, P. D. Garcia, S. Ek, N. Gregersen, T. Suhr, M. Schubert, J. Mørk, S. Stobbe, and P. Lodahl, "Random nanolasing in the Anderson localized regime", Nature Nanotechnology , 9 285-289 (2014).
  * [6] S. Schönhuber, M. Brandstetter, T. Hisch, C. Deutsch, M. Krall, H. Detz, A. M. Andrews, G. Strasser, S. Rotter, and K. Unterrainer, "Random lasers for broadband directional emission", Optica. 3,1035-1038 (2016).
  * [7] DN. Dirin , RT. Lechner, G. Fritz-Popovski G,M. Sytnyk, S. Yakunin, S. Rotter, A. AAY, C. Vidal , C. Hrelescu, TA. Klar TA, MV. Kovalenko, W. Heiss, "Random lasing with systematic threshold behavior in films of CdSe/CdS core/thick-shell colloidal quantum dots", ACS Nano. 9,9792-9801 (2015).
  * [8] H. Liang, B. Meng, G. Liang, J. Tao, Y. Chong, Q. J. Wang, and Y. Zhang, " Electrically Pumped Mid-Infrared Random Lasers." Adv. Mater., 25, 5859 (2013).
  * [9] "Random lasing in an Anderson localizing optical fiber." B. Abaie, E. Mobini, S. Karbasi, et al., Light Sci Appl. 6, 17041 (2017).
  * [10] D. Wiersma, "The physics and applications of random lasers." Nature Phys. 4, 359–367 (2008).
  * [11] Z. Hu, J. Xia, Y. Liang, J. Wen, E. Miao, J. Chen, S. Wu, X. Qian, H. Jiang, and K. Xie, "Tunable random polymer fiber laser." Opt. Express 25, 18421-18430 (2017).
  * [12] B. Kumar, R. Homri, and P. Sebbah. "2D tunable all-solid-state random laser in the visible." Sci. Rep. 13, 8337 (2023).
  * [13] J. Tong, X. Shi, S. Li, C. Chen, T. Zhai, X. Zhang, "Tunable plasmonic random laser based on a wedge shaped resonator." Org. Electron., 75, 105337(2019)
  * [14] J. Ziegler, C. Wörister, C. Vidal, C. Hrelescu, T. A. Klar. "Plasmonic Nanostars as Efficient Broadband Scatterers for Random Lasing" ACS Photonics, 3, 919(2016).
  * [15] Z. Wang, X. Meng, A. V. Kildishev, A. Boltasseva, V. M. Shalaev, " Nanolasers Enabled by Metallic Nanoparticles: From Spasers to Random Lasers", Laser Photonics Rev, 11, 1700212(2017)
  * [16] T. Zhai, J. Chen, L. Chen, J. Wang, L. Wang, D. Liu, S. Li, H. Liu, and X. Zhang, “A plasmonic random laser tunable through stretching silver nanowires embedded in a flexible substrate”, Nanoscale 7(6), 2235–2240 (2015).
  * [17] Y. Lee, C. Chou, Z. Yang, T. B. H. Nguyen, Y. Yao, T. Yeh, M. Tsai, and H. Kuo, “Flexible random lasers with tunable lasing emissions”, Nanoscale 10(22), 10403–10411 (2018).
  * [18] R. G. S. EI-Dardiry and A. Lagendijk, “Tuning random lasers by engineered absorption”, Appl. Phys. Lett. 98,(16), 161106 (2011).
  * [19] J. Zhang, F. Wang, S. Ghafoor, H. Wang, W. Zhang, K. Xie, R. Cheng, J. Yan, L. Niu, P. Wang, L. Zhang, Z. Hu. "Tunable Plasmonic Random Laser Based on Emitters Coupled to Plasmonic Resonant Nanocavities of Silver Nanorod Arrays." Adv. Opt. materials, 10(10), 2200426 (2022).
  * [20] Q. Song, L. Liu, L. Xu, Y. Wu, and Z. Wang, "Electrical tunable random laser emission from a liquid-crystal infiltrated disordered planar microcavity," Opt. Lett. 34, 298-300 (2009).
  * [21] Q. Song, S. Xiao, X. Zhou, L. Liu, L. Xu, Y. Wu, and Z. Wang, "Liquid-crystal-based tunable high-Q directional random laser from a planar random microcavity," Opt. Lett. 32, 373-375 (2007).
  * [22] R. Ma, X. Quan, H. Wu, W. C. Gao, D. J. Huang, X. C. Wang, S. X. Xu, D. Y. Fan, and J. Liu, "20-watt-level single transverse mode narrow linewidth and tunable random fiber laser at 1.5 µm band," Opt. Express 30, 28795-28804 (2022).
  * [23] J. Liu, Z. Tong, W. Zhang, J. Li, Y. Li, "Switchable and tunable multi-wavelength erbium-doped random distributed feedback fiber laser based on a compound filter.", Optik, 241, 167015(2021).
  * [24] H. Wu, B. Han, and Y. Liu, "Tunable narrowband cascaded random Raman fiber laser," Opt. Express 29, 21539-21550 (2021).
  * [25] X. Du, H. Zhang, X. Wang, and P. Zhou, "Tunable random distributed feedback fiber laser operating at 1$\mu$m," Appl. Opt. 54, 908-911 (2015).
  * [26] N. Bachelard, S. Gigan, X. Noblin, P. Sebbah, "Adaptive pumping for spectral control of random lasers" Nature Physics , 10, 426-431(2014).
  * [27] N. Bachelard, J. Andreasen, S. Gigan, and P. Sebbah "Taming Random Lasers through Active Spatial Control of the Pump" Phys. Rev. Lett. 109, 033903(2012).
  * [28] B. Kumar, R. Homri, Priyanka, S. K. Maurya, M. Lebental, and P. Sebbah, "Localized modes revealed in random lasers," Optica, 8, 1033-1039 (2021).
  * [29] X.-l. Zhu and D. Lo, "Temperature tuning of output wavelength for solid-state dye lasers." J. Opt. A: Pure Appl. Opt. 3, 225 (2001)
  * [30] F. J. Duarte and R. O. James, "Tunable solid-state lasers incorporating dye-doped, polymer–nanoparticle gain media." Opt. Lett. 28, 2088-2090, (2003).
  * [31] F. J. Duarte, A. Costela, I. Garcia-Moreno, and R. Sastre, "Measurements of $\delta n/\delta T$ in Solid-State Dye-Laser Gain Media," Appl. Opt. 39, 6522(2000)