# Weyl semimetallic state in the Rashba-Hubbard model

Katsunori Kubo  Advanced Science Research Center, Japan Atomic Energy Agency,
Tokai, Ibaraki 319-1195, Japan

###### Abstract

We investigate the Hubbard model with the Rashba spin-orbit coupling on a
square lattice. The Rashba spin-orbit coupling generates two-dimensional Weyl
points in the band dispersion. In a system with edges along [11] direction,
zero-energy edge states appear, while no edge state exists for a system with
edges along an axis direction. The zero-energy edge states with a certain
momentum along the edges are predominantly in the up-spin state on the right
edge, while they are predominantly in the down-spin state on the left edge.
Thus, the zero-energy edge states are helical. By using a variational Monte
Carlo method for finite Coulomb interaction cases, we find that the Weyl
points can move toward the Fermi level by the correlation effects. We also
investigate the magnetism of the model by the Hartree-Fock approximation and
discuss weak magnetic order in the weak-coupling region.

##  I Introduction

In a two-dimensional system without inversion symmetry, such as in an
interface of a heterostructure, a momentum-dependent spin-orbit coupling is
allowed. It is called the Rashba spin-orbit coupling [1]. The Rashba spin-
orbit coupling lifts the spin degeneracy and affects the electronic state of
materials.

Several interesting phenomena originating from the Rashba spin-orbit coupling
have been proposed and investigated. By considering the spin precession by the
Rashba spin-orbit coupling, Datta and Das proposed the spin transistor [2], in
which electron transport between spin-polarized contacts can be modulated by
the gate voltage. After this proposal, the tunability of the Rashba spin-orbit
coupling by the gate voltage has been experimentally demonstrated [3, 4, 5].
Such an effect may be used in a device in spintronics. The possibility of the
intrinsic spin Hall effect, which is also important in the research field of
spintronics, by the Rashba spin-orbit coupling has been discussed for a long
time [6, 7, 8, 9, 10, 11, 12]. Another interesting phenomenon with the Rashba
spin-orbit coupling is superconductivity. When the Rashba spin-orbit coupling
is introduced in a superconducting system, even- and odd-parity
superconducting states are mixed due to the breaking of the inversion symmetry
[13, 14, 15]. This mixing affects the magnetic properties of the
superconducting state, such as the Knight shift.

While the above studies have mainly focused on the one-electron states in the
presence of the Rashba spin-orbit coupling, the effects of the Coulomb
interaction between electrons have also been investigated. The Hubbard model
with the Rashba spin-orbit coupling on a square lattice called the Rashba-
Hubbard model is one of the simplest models to investigate such effects. In
this study, we investigate the ground state of this model at half-filling,
i.e., electron number per site $n=1$, by the variational Monte Carlo method
and the Hartree-Fock approximation.

In the strong coupling limit, an effective localized model is derived and the
possibility of long-period magnetic order is discussed [16, 17, 18]. The long-
period magnetism is a consequence of the Dzyaloshinskii-Moriya interaction
caused by the Rashba spin-orbit coupling.

Such long-period magnetic order is also discussed by the Hartree-Fock
approximation for the Rashba-Hubbard model [19, 20, 21]. However, there is a
contradiction among these studies even within the Hartree-Fock approximation.
In the weak-coupling region with a finite Rashba spin-orbit coupling, an
antiferromagnetic order is obtained in Ref. [19], but a paramagnetic phase is
obtained in Refs. [20] and [21]. We will discuss this point in Sec. V.

The knowledge of the electron correlation beyond the Hartree-Fock
approximation is limited. The electron correlation in the Rashba-Hubbard model
is studied by a dynamical mean-field theory mainly focusing on magnetism [22]
and by a cluster perturbation theory investigating the Mott transition in the
paramagnetic state [23]. We will study the electron correlation in the
paramagnetic phase by using the variational Monte Carlo method in Sec. IV. The
results concerning the Mott transition are consistent with Ref. [23]. In
addition, we find a transition to a Weyl semimetallic state by the electron
correlation.

Even without the Coulomb interaction, the band structure of this model is
intriguing. When the Rashba spin-orbit coupling is finite, the upper and lower
bands touch each other at Weyl points. In the large Rashba spin-orbit coupling
limit, all the Weyl points locate at the Fermi level for half-filling.
Topological aspects of the Weyl points and corresponding edge states of this
simple model are discussed in Sec. III.

##  II Model

The model Hamiltonian is given by $H=H_{\text{kin}}+H_{R}+H_{\text{int}}$. The
kinetic energy term is given by

$H_{\text{kin}}=-t\sum_{(\bm{r},\bm{r}^{\prime})\sigma}(c_{\bm{r}\sigma}^{\dagger}c_{\bm{r}^{\prime}\sigma}+c_{\bm{r}^{\prime}\sigma}^{\dagger}c_{\bm{r}\sigma})=\sum_{\bm{k}\sigma}\epsilon_{\bm{k}}c_{\bm{k}\sigma}^{\dagger}c_{\bm{k}\sigma},$
(1)

where $c_{\bm{r}\sigma}$ is the annihilation operator of the electron at site
$\bm{r}$ with spin $\sigma$ and $c_{\bm{k}\sigma}$ is the Fourier transform of
it. $(\bm{r},\bm{r}^{\prime})$ denotes a pair of nearest-neighbor sites, $t$
is the hopping integral, and the kinetic energy is $\epsilon_{\bm{k}}=-2t(\cos
k_{x}+\cos k_{y})$, where the lattice constant is set as unity.

The Rashba spin-orbit coupling term is given by [24]

$\begin{split}H_{R}&=i\lambda_{R}\sum_{\bm{r}\sigma\sigma^{\prime}a=\pm
1}a\left(\sigma^{x}_{\sigma\sigma^{\prime}}c_{\bm{r}\sigma}^{\dagger}c_{\bm{r}+a\hat{\bm{y}}\sigma^{\prime}}-\sigma^{y}_{\sigma\sigma^{\prime}}c_{\bm{r}\sigma}^{\dagger}c_{\bm{r}+a\hat{\bm{x}}\sigma^{\prime}}\right)\\\
&=-2\lambda_{R}\sum_{\bm{k}\sigma\sigma^{\prime}}\left(\sin
k_{y}\sigma^{x}_{\sigma\sigma^{\prime}}-\sin
k_{x}\sigma^{y}_{\sigma\sigma^{\prime}}\right)c_{\bm{k}\sigma}^{\dagger}c_{\bm{k}\sigma^{\prime}}\\\
&=\sum_{\bm{k}\sigma\sigma^{\prime}}\left[h_{x}(\bm{k})\sigma^{x}_{\sigma\sigma^{\prime}}+h_{y}(\bm{k})\sigma^{y}_{\sigma\sigma^{\prime}}\right]c_{\bm{k}\sigma}^{\dagger}c_{\bm{k}\sigma^{\prime}}\\\
&=\sum_{\bm{k}\sigma\sigma^{\prime}}H_{R\sigma\sigma^{\prime}}(\bm{k})c_{\bm{k}\sigma}^{\dagger}c_{\bm{k}\sigma^{\prime}},\end{split}$
(2)

where $\hat{\bm{x}}$ ($\hat{\bm{y}}$) is the unit vector along the $x$ ($y$)
direction, $\bm{\sigma}$ are the Pauli matrices, $\lambda_{R}$ is the coupling
constant of the Rashba spin-orbit coupling, $h_{x}(\bm{k})=-2\lambda_{R}\sin
k_{y}$, and $h_{y}(\bm{k})=2\lambda_{R}\sin k_{x}$. We can assume $t\geq 0$
and $\lambda_{R}\geq 0$ without loss of generality. We parametrize them as
$t=\tilde{t}\cos\alpha$ and $\lambda_{R}=\sqrt{2}\,\tilde{t}\sin\alpha$.

The band dispersion of $H_{0}=H_{\text{kin}}+H_{R}$ is

$E_{\pm}(\bm{k})=-2t(\cos k_{x}+\cos k_{y})\pm|\bm{h}(\bm{k})|,$ (3)

where
$|\bm{h}(\bm{k})|=\sqrt{h_{x}^{2}(\bm{k})+h_{y}^{2}(\bm{k})}=2\lambda_{R}\sqrt{\sin^{2}k_{x}+\sin^{2}k_{y}}$.
The bandwidth is $W=8\tilde{t}$. Due to the electron-hole symmetry of the
model, the Fermi level is zero at half-filling. For $\alpha=0$, that is,
without the Rashba spin-orbit coupling, the band is doubly degenerate [Fig.
1(a)].

Figure 1:  Energy dispersion along high symmetry directions for (a)
$\alpha=0$, (b) $\alpha=0.3\pi$, and (c) $\alpha=0.5\pi$. (d) Energy
dispersion in the entire Brillouin zone for $\alpha=0.5\pi$.

For a finite $\lambda_{R}$, the spin degeneracy is lifted except at the time-
reversal invariant momenta $\bm{X}^{(0)}=(0,0)$, $\bm{X}^{(1)}=(\pi,0)$,
$\bm{X}^{(2)}=(0,\pi)$, and $\bm{X}^{(3)}=(\pi,\pi)$ [Figs. 1(b) and 1(c)].
These are two-dimensional Weyl points. The energies at the Weyl points
$\bm{X}^{(1)}$ and $\bm{X}^{(2)}$ are always zero. By increasing $\alpha$ to
$0.5\pi$ ($t=0$), the energies at the other Weyl points $\bm{X}^{(0)}$ and
$\bm{X}^{(3)}$ also move to zero. In Fig. 1(d), we show the energy dispersion
in the entire Brillouin zone for $\alpha=0.5\pi$. We can see the linear
dispersions around the Weyl points.

The Coulomb interaction term is given by

$H_{\text{int}}=U\sum_{\bm{r}}n_{\bm{r}\uparrow}n_{\bm{r}\downarrow},$ (4)

where $n_{\bm{r}\sigma}=c_{\bm{r}\sigma}^{\dagger}c_{\bm{r}\sigma}$ and $U$ is
the coupling constant of the Coulomb interaction.

##  III Topology and edge states of the non-interacting Hamiltonian

The energy bands degenerate when $\bm{h}(\bm{k})=\bm{0}$, i.e., at the Weyl
points. In the vicinity of these points, we set $\bm{k}=\bm{X}^{(l)}+\bm{p}$
and obtain

$\begin{split}H_{R}(\bm{k})&=\sum_{j}h_{j}(\bm{k})\sigma^{j}\\\
&\simeq\sum_{ij}\left.\frac{\partial h_{j}(\bm{k})}{\partial
k_{i}}\right|_{\bm{k}=\bm{X}^{(l)}}p_{i}\sigma^{j}\\\
&=\sum_{ij}v^{(l)}_{ij}p_{i}\sigma^{j}.\end{split}$ (5)

The chirality of each Weyl point $\bm{X}^{(l)}$ is defined as
$\chi_{l}=\text{sgn}[\det v^{(l)}]$ [25] and we obtain $\chi_{0}=\chi_{3}=1$
and $\chi_{1}=\chi_{2}=-1$.

The winding number of a normalized two-component vector field
$\hat{\bm{h}}(\bm{k})=\bm{h}(\bm{k})/|\bm{h}(\bm{k})|$ is [26, 25]

$w_{l}=\oint_{C_{l}}\frac{d\bm{k}}{2\pi}\cdot\left[\hat{h}_{x}(\bm{k})\bm{\nabla}\hat{h}_{y}(\bm{k})-\hat{h}_{y}(\bm{k})\bm{\nabla}\hat{h}_{x}(\bm{k})\right],$
(6)

where $C_{l}$ is a loop enclosing $\bm{X}_{l}$. We obtain $w_{l}=\chi_{l}$.
Figure 2 shows $\hat{\bm{h}}(\bm{k})$ around $\bm{k}=\bm{X}^{(0)}$ and
$\bm{X}^{(1)}$ as examples.

Figure 2:  $\hat{\bm{h}}(\bm{k})$ (a) around $\bm{k}=\bm{X}^{(0)}=(0,0)$ and
(b) around $\bm{k}=\bm{X}^{(1)}=(\pi,0)$.

We can recognize the winding numbers 1 and $-1$, respectively, from this
figure.

These topological numbers are related to the Berry phase [27]. The eigenvector
of $H_{R}(\bm{k})$ with eigenvalue $-|\bm{h}(\bm{k})|$ is
$|\bm{k}\rangle=(1/\sqrt{2})(-1,\hat{h}_{x}(\bm{k})+i\hat{h}_{y}(\bm{k}))^{\text{T}}$.
The Berry connection is

$\begin{split}\bm{a}(\bm{k})&=-i\langle\bm{k}|\bm{\nabla}|\bm{k}\rangle\\\
&=\frac{1}{2}\left[\hat{h}_{x}(\bm{k})\bm{\nabla}\hat{h}_{y}(\bm{k})-\hat{h}_{y}(\bm{k})\bm{\nabla}\hat{h}_{x}(\bm{k})\right].\end{split}$
(7)

Then, the Berry phase is

$\gamma_{l}=\int_{C_{l}}d\bm{k}\cdot\bm{a}(\bm{k})=w_{l}\pi.$ (8)

From the existence of such topological defects like the Weyl points, we expect
edge states as in graphene with Dirac points [28, 29, 30]. We consider two
types of edges: the edges along an axis direction [straight edges, Fig. 3(a)]
and the edges along [11] direction [zigzag edges, Fig. 3(b)].

Figure 3:  Lattices with edges. (a) Straight edges. (b) Zigzag edges. $i$ is
the label denoting the position perpendicular to the edges, and $N$ is the
number of these positions.

We denote the momentum along the edges as $k$ and the momentum perpendicular
to the edges as $k_{\perp}$. To discuss the existence of the edge states, the
chiral symmetry and the winding number for a fixed $k$ are important [29, 30].
The Rashba term has a chiral symmetry:
$\\{H_{R}(\bm{k}),\sigma^{z}\\}=H_{R}(\bm{k})\sigma^{z}+\sigma^{z}H_{R}(\bm{k})=0$
and $\sigma^{z}\sigma^{z\dagger}=I$ with $I$ being the unit matrix. The
winding number for a fixed $k$ is given by

$w(k)=\int_{0}^{2\pi}\frac{dk_{\perp}}{2\pi}\left[\hat{h}_{x}(\bm{k})\frac{\partial}{\partial
k_{\perp}}\hat{h}_{y}(\bm{k})-\hat{h}_{y}(\bm{k})\frac{\partial}{\partial
k_{\perp}}\hat{h}_{x}(\bm{k})\right].$ (9)

For the straight edges, we find $w(k)=0$ and we expect that the edge states
are probably absent. For the zigzag edges,
$h_{x}(\bm{k})=-2\lambda_{R}\sin(k-k_{\perp})$ and
$h_{y}(\bm{k})=2\lambda_{R}\sin(k+k_{\perp})$, where we have set $1/\sqrt{2}$
times the bond length as unity, and we find $w(k)=-\text{sgn}[\sin(2k)]$
except for $k=0$, $\pm\pi/2$, and $\pm\pi$ (projected Weyl points). At the
projected Weyl points, $w(k)=0$. Thus, the edge states should exist except for
the projected Weyl points at least without $t$.

We note that the edge states can be understood as those of a one-dimensional
topological insulator. The model only with the Rashba term with fixed $k$ is a
one-dimensional model. When this one-dimensional system has a gap with a non-
zero topological number, the system can be regarded as a one-dimensional
topological insulator and has edge states. This one-dimensional system is of
symmetry class BDI and can possess a topological number of $\mathbb{Z}$ [31,
32, 33].

To explicitly demonstrate the existence of the edge states, we numerically
evaluate the band energy for lattices with finite widths. We denote the number
of lattice sites perpendicular to the edges as $N$ (see Fig. 3) and obtain
$2N$ bands. The obtained energy bands are shown in Fig. 4.

Figure 4:  The upper panels show the band dispersion with straight edges: (a)
for $\alpha=0$, (b) for $\alpha=0.3\pi$, and (c) for $\alpha=0.5\pi$. The
lower panels show the band dispersion with zigzag edges: (d) for $\alpha=0$,
(e) for $\alpha=0.3\pi$, and (f) for $\alpha=0.5\pi$. $k$ is the momentum
parallel to the edges. The number of lattice sites perpendicular to the edges
is $N=51$.

For the straight edges [Figs. 4(a)–(c)], we do not find the edge states. It is
consistent with $w(k)=0$. For the zigzag edges [Figs. 4(d)–(f)], we obtain
isolated zero-energy states except for $\lambda_{R}=0$ [Fig. 4(d)]. In
particular, for $\alpha=0.5\pi$, the zero-energy states appear at all the $k$
points except for the projected Weyl points as is expected from $w(k)\neq 0$.
We find that the zero-energy states remain even for finite $t$ as shown in
Fig. 4(e). For an even number of $N$, the energy of the zero-energy states
shifts from zero around the projected Weyl points when $N$ is small. For an
odd number of $N$, we obtain zero energy even for a small $N$. Thus, we set
$N=51$ in the calculations.

We discuss the characteristics of the zero-energy edge states. We define
$c_{ik\sigma}$ as the Fourier transform of $c_{\bm{r}\sigma}$ along the edges,
where $i$ labels the site perpendicular to the edges (see Fig. 3). For the
lattice with the zigzag edges, we can show that the states
$c_{-(N-1)/2,\,\pi/4,\,\downarrow}^{\dagger}|0\rangle$ and
$c_{(N-1)/2,\,\pi/4,\,\uparrow}^{\dagger}|0\rangle$ do not have matrix
elements of $H_{R}$, where $|0\rangle$ is the vacuum state. Thus, these states
are the zero-energy states for $\alpha=0.5\pi$ completely localized on the
left and right edges, respectively, with opposite spins. This helical
character of the edge states is natural since the system lacks inversion
symmetry due to the Rashba spin-orbit coupling. For other momenta and
$\alpha$, we calculate the spin density of the zero-energy edge states
$n_{0k\sigma}(i)=\langle 0k|c_{ik\sigma}^{\dagger}c_{ik\sigma}|0k\rangle$,
where $|0k\rangle$ denotes the zero-energy state at momentum $k$. The zero-
energy states are doubly degenerate, and we take the average of the two
states. We show $n_{0k\sigma}(i)$ for $\alpha=0.3\pi$, as an example, in Fig.
5.

Figure 5:  Spin density of the zero-energy edge states $n_{0k\sigma}(i)$ for
$\alpha=0.3\pi$ on the lattice with zigzag edges with $N=51$: (a) for
$k=0.19\pi$, (b) for $k=0.21\pi$, (c) for $k=0.3\pi$, (d) for $k=0.4\pi$, (e)
for $k=0.45\pi$, (f) for $k=0.5\pi$, (g) for $k=-0.4\pi$, and (h) for
$k=-0.45\pi$. (i) Schematic view of the spin density on the real-space lattice
for $k\simeq 0.4\pi$. The sizes of the circles represent the magnitude of the
spin density.

At $k$ where the bulk band gap is sufficiently large, the zero-energy states
are localized well on the edges [Figs. 5(c) and 5(d)]. As the bulk band gap
becomes small, the zero-energy states penetrate inner sites [Figs. 5(b) and
5(e)] and the zero-energy states extend in the entire lattice when the gap
closes [Figs. 5(a) and 5(f)]. The spin components are opposite between the
edges. For example, for $k=0.4\pi$ and $0.45\pi$, the up-spin state dominates
on the right edge while the down-spin state dominates on the left edge. Thus,
the edge states are helical. The spin components are exchanged between states
at $k$ and $-k$ [compare Fig. 5(d) with Fig. 5(g) and Fig. 5(e) with Fig.
5(h)]. In Fig. 5(i), we show a schematic view of the spin density
corresponding to $k\simeq 0.4\pi$ on the real-space lattice.

##  IV Weyl semimetallic state induced by the correlation effects

In this section, we investigate the effects of the Coulomb interaction $U$ at
half-filling, i.e., the electron number per site $n=1$, within the
paramagnetic phase by applying the variational Monte Carlo method [34]. To
achieve this objective, it is necessary to select a wave function capable of
describing the Mott insulating state, as a Mott transition is anticipated, at
least in the ordinary Hubbard model without the Rashba spin-orbit coupling. In
this study, we employ a wave function with doublon-holon binding factors
[doublon-holon binding wave function (DHWF)] [35, 36]. A doublon means a
doubly occupied site and a holon means an empty site. Such intersite factors
like doublon-holon binding factors are essential to describe the Mott
insulating state [37, 38]. Indeed, the DHWF has succeeded in describing the
Mott transition for the single-orbital [37, 39, 40, 41] and two-orbital [42,
43, 44, 45] Hubbard models.

The DHWF is given by

$|\Psi(\alpha_{\text{eff}})\rangle=P_{d}P_{h}P_{G}|\Phi(\alpha_{\text{eff}})\rangle.$
(10)

The Gutzwiller projection operator

$P_{G}=\prod_{\bm{r}}[1-(1-g)P_{d\,\bm{r}}],$ (11)

describes onsite correlations, where
$P_{d\,\bm{r}}=n_{\bm{r}\uparrow}n_{\bm{r}\downarrow}$ is the projection
operator onto the doublon state at $\bm{r}$ and $g$ is a variational
parameter. The parameter $g$ tunes the population of the doubly occupied
sites. When the onsite Coulomb interaction is strong and $n=1$, most sites
should be occupied by a single electron each. In this situation, if a doublon
is created, a holon should be around it to reduce the energy by using singly
occupied virtual states. $P_{d}$ and $P_{h}$ describe such doublon-holon
binding effects. $P_{d}$ is an operator to include intersite correlation
effects concerning the doublon states. This is defined as follows [44, 46,
45]:

$P_{d}=\prod_{\bm{r}}\left[1-(1-\zeta_{d})P_{d\,\bm{r}}\prod_{\bm{a}}(1-P_{h\,\bm{r}+\bm{a}})\right],$
(12)

where $P_{h\,\bm{r}}=(1-n_{\bm{r}\uparrow})(1-n_{\bm{r}\downarrow})$ is the
projection operator onto the holon state at $\bm{r}$ and $\bm{a}$ denotes the
vectors connecting the nearest-neighbor sites. $P_{d}$ gives factor
$\zeta_{d}$ when site $\bm{r}$ is in the doublon state and there is no holon
at nearest-neighbor sites $\bm{r}+\bm{a}$. Similarly, $P_{h}$ describing the
intersite correlation effects on the holon state is defined as

$P_{h}=\prod_{\bm{r}}\left[1-(1-\zeta_{h})P_{h\,\bm{r}}\prod_{\bm{a}}(1-P_{d\,\bm{r}+\bm{a}})\right].$
(13)

Factor $\zeta_{h}$ appears when a holon exists without a nearest-neighboring
doublon. For the half-filled case, we can use the relation
$\zeta_{d}=\zeta_{h}$ due to the electron-hole symmetry of the model.

The one-electron part $|\Phi(\alpha_{\text{eff}})\rangle$ of the wave function
is given by the ground state of the non-interacting Hamiltonian
$H_{0}(\alpha_{\text{eff}})$ in which $\alpha$ in $H_{0}$ is replaced by
$\alpha_{\text{eff}}$. We can choose $\alpha_{\text{eff}}$ different from the
original $\alpha$ in the model Hamiltonian. Such a band renormalization effect
of the one-electron part is discussed for a Hubbard model with next-nearest-
neighbor hopping [47]. We define the normal state as
$|\Psi_{\text{N}}\rangle=|\Psi(\alpha_{\text{eff}}=\alpha)\rangle$, i.e.,
$\alpha_{\text{eff}}$ remains the bare value. We also define the Weyl
semimetallic state as
$|\Psi_{\text{Weyl}}\rangle=|\Psi(\alpha_{\text{eff}}=0.5\pi)\rangle$, i.e.,
all the Weyl points are at the Fermi level and the Fermi surface disappears.
In addition, we can choose other values of $\alpha_{\text{eff}}$, but in a
finite-size lattice, a slight change of $\alpha_{\text{eff}}$ does not change
the set of the occupied wave numbers and the wave function
$|\Phi(\alpha_{\text{eff}})\rangle$. Thus, we have limited choices for
$\alpha_{\text{eff}}$ as the band renormalization in the Hubbard model with
the next-nearest-neighbor hopping [47].

We use the antiperiodic-periodic boundary conditions since the closed shell
condition is satisfied, i.e., no $\bm{k}$ point is exactly on the Fermi
surface for a finite-size lattice and there is no ambiguity to construct
$|\Phi(\alpha_{\text{eff}})\rangle$. The calculations are done for $L\times L$
lattices with $L=12$, 14, and 16.

We evaluate the expectation value of energy by the Monte Carlo method. We
optimize the variational parameters $g$ and $\zeta_{d}=\zeta_{h}$ to minimize
the energy. We denote the optimized energy of
$|\Psi(\alpha_{\text{eff}})\rangle$ as $E(\alpha_{\text{eff}})$. In
particular, we denote $E_{\text{N}}=E(\alpha_{\text{eff}}=\alpha)$ and
$E_{\text{Weyl}}=E(\alpha_{\text{eff}}=0.5\pi)$. By using the Monte Carlo
method, we also evaluate the momentum distribution function
$n(\bm{k})=\sum_{\sigma}\langle
c_{\bm{k}\sigma}^{\dagger}c_{\bm{k}\sigma}\rangle$, where
$\langle\cdots\rangle$ represents the expectation value in the optimized wave
function.

In Fig. 6(a), we show $n(\bm{k})$ in the normal state at $\alpha=0.25\pi$ for
$L=16$.

Figure 6:  (a) Momentum distribution function $n(\bm{k})$ for $\alpha=0.25\pi$
and $L=16$ at $U=10\tilde{t}$ ($<U_{\text{MIT}}$) and $U=14\tilde{t}$
($>U_{\text{MIT}}$). The quasiparticle renormalization factor $Z$ is evaluated
by the jump between $(\pi,0)$ and $(\pi,\pi)$. Due to the antiperiodic
boundary condition for the $x$ direction, we shift $k_{x}$ by $\pi/L$; for
example, $(\pi,\pi)$ denoted in this figure actually means the point
$(\pi-\pi/L,\pi)$. (b) Quasiparticle renormalization factor $Z$ as a function
of $U$ for $\alpha=0.25\pi$ and $L=16$. The Mott metal-insulator transition
point $U_{\text{MIT}}$ is determined by extrapolating the data to $Z=0$.

For $U/\tilde{t}=10$, $n(\bm{k})$ has clear discontinuities at the Fermi
momenta. On the other hand, for $U/\tilde{t}=14$, $n(\bm{k})$ does not have
such a discontinuity; that is, the system is insulating and a Mott metal-
insulator transition takes place between $U/\tilde{t}=10$ and
$U/\tilde{t}=14$. To determine the Mott metal-insulator transition point
$U_{\text{MIT}}$, we evaluate the quasiparticle renormalization factor $Z$,
which is inversely proportional to the effective mass and becomes zero in the
Mott insulating state, by the jump in $n(\bm{k})$. Except for $\alpha=0$, we
evaluate $Z$ by the jump between $(\pi,0)$ and $(\pi,\pi)$ as shown in Fig.
6(a). For $\alpha=0$, the above path does not intersect the Fermi surface and
we use the jump between $(\pi,\pi)$ and $(0,0)$ instead. In Fig. 6(b), we show
the $U$ dependence of $Z$ for $\alpha=0.25\pi$ and $L=16$. By extrapolating
$Z$ to zero, we determine $U_{\text{MIT}}/\tilde{t}\simeq 12.9$. We note that
for a small $\alpha$ with a large $L$, the Mott transition becomes first-order
consistent with a previous study for $\alpha=0$ [40].

We have also evaluated energies for some values of
$\alpha_{\text{eff}}\neq\alpha$. Figure 7(a) shows energies for
$\alpha_{\text{eff}}=0.18\pi$ and $0.22\pi$ measured from the normal state
energy at $\alpha=0.2\pi$ for $L=16$.

Figure 7:  (a) Energy measured from that of the normal state
[$E_{\text{N}}=E(\alpha_{\text{eff}}=\alpha)$] for
$\alpha_{\text{eff}}=0.18\pi$ (circles) and $\alpha_{\text{eff}}=0.22\pi$
(squares) at $\alpha=0.2\pi$ for $L=16$. (b) Energy of the Weyl semimetallic
state [$E_{\text{Weyl}}=E(\alpha_{\text{eff}}=0.5\pi)$] measured from that of
the normal state at $\alpha=0.4\pi$ for $L=16$. The arrow indicates the Weyl
transition point $U_{\text{Weyl}}$.

The normal state has the lowest energy, at least for $U/\tilde{t}\leq 20$.
Thus, the renormalization of $\alpha$, even if it exists, is weak for a system
distant from the Weyl semimetallic state ($\alpha=0.5\pi$). A similar
conclusion is obtained for a small intersite spin-orbit coupling case of the
Kane-Mele-Hubbard model [48]. It is in contrast to the onsite spin-orbit
coupling case [49, 48, 46, 45, 50], where the effective spin-orbit coupling is
enhanced by the Coulomb interaction even when the bare spin-orbit coupling is
small. On the other hand, the renormalization of $\alpha$ becomes strong
around $\alpha=0.5\pi$. In Fig. 7(b), we show the energy $E_{\text{Weyl}}$ of
the Weyl semimetallic state measured from that of the normal state for
$\alpha=0.4\pi$ for $L=16$. $E_{\text{Weyl}}$ becomes lower than the normal
state energy at $U>U_{\text{Weyl}}\simeq 9.4\tilde{t}$. There is a possibility
that the normal state changes to the Weyl semimetallic state gradually by
changing $\alpha_{\text{eff}}$ continuously. However, for a finite lattice,
the choices of $\alpha_{\text{eff}}$ are limited between
$\alpha_{\text{eff}}=\alpha$ and $\alpha_{\text{eff}}=0.5\pi$. For example, at
$\alpha=0.4\pi$, there is no choice for $L=12$ and $L=14$ and only one choice
$0.4017<\alpha_{\text{eff}}/\pi<0.4559$ for $L=16$. For this reason, we
evaluate $U_{\text{Weyl}}$ by comparing the energies of the normal and the
Weyl semimetallic states to show the tendency toward the Weyl semimetallic
state by the renormalization effect on $\alpha$.

Figure 8 shows a phase diagram without considering magnetic order.

Figure 8:  Phase diagram obtained with the variational Monte Carlo method
without considering magnetic order. The Mott transition points for $L=12$
(open triangles), $L=14$ (open circles), and $L=16$ (solid circles) and the
Weyl transition points for $L=12$ (open diamonds), $L=14$ (open squares), and
$L=16$ (solid squares) are plotted.

The size dependence of the phase boundaries is weak. For a weak Rashba spin-
orbit coupling region, i.e., for a small $\alpha$, the Rashba spin-orbit
coupling stabilizes the metallic phase. It is consistent with a previous study
by a cluster perturbation theory [23]. Around $\alpha=0.5\pi$, we obtain a
wide region of the Weyl semimetallic phase. Thus, we expect phenomena
originating from the Weyl points can be realized even away from
$\alpha=0.5\pi$ with the aid of electron correlations. In the Weyl
semimetallic state, the density of states at the Fermi level vanishes, and
thus, energy gain is expected similar to the energy gain by a gap opening in
an antiferromagnetic transition. We note that such a renormalization effect on
$\alpha$ cannot be expected within the Hartree-Fock approximation and is a
result of the electron correlations beyond the Hartree-Fock approximation.

##  V Hartree-Fock approximation for magnetism

In this section, we discuss the magnetism of the model by the Hartree-Fock
approximation. The energy dispersion given in Eq. (3) has the following
property: $E_{\pm}(\bm{k}+\bm{Q})=-E_{\mp}(\bm{k})$ for $\bm{Q}=(\pi,\pi)$.
When $E_{a}(\bm{k})=0$, in particular,
$E_{-a}(\bm{k}+\bm{Q})=E_{a}(\bm{k})=0$. Thus, the Fermi surface is perfectly
nested for half-filling (the Fermi energy is zero) with the nesting vector
$\bm{Q}=(\pi,\pi)$ [see Figs. 9(a)–(c)].

Figure 9:  Fermi surface (a) for $\alpha=0$, (b) for $\alpha=0.3\pi$, and (c)
for $\alpha=0.45\pi$. The arrows represent the nesting vector
$\bm{Q}=(\pi,\pi)$. Density of states (d) for $\alpha=0$, (e) for
$\alpha=0.3\pi$, and (f) for $\alpha=0.45\pi$. The dashed line in (d) is the
asymptotic form $-[1/(2\pi^{2}\tilde{t})]\ln[|\epsilon|/(16\tilde{t})]$.

Due to this nesting, the magnetic susceptibility at $\bm{Q}=(\pi,\pi)$
diverges at zero temperature [19]. It indicates that the magnetic order occurs
with an infinitesimally small value of the Coulomb interaction $U$ at zero
temperature. However, some recent Hartree-Fock studies argue the existence of
the paramagnetic phase with finite $U$ [20, 21]. To resolve this contradiction
and gain insights into magnetism, we apply the Hartree-Fock approximation to
the model within two-sublattice magnetic order, i.e., with ordering vector of
$\bm{Q}=(\pi,\pi)$ or $\bm{Q}=(\pi,0)$.

The Hartree-Fock Hamiltonian is given by

$H_{\text{HF}}=\sum_{\bm{k}}\begin{pmatrix}c_{\bm{k}}^{\dagger}c_{\bm{k}+\bm{Q}}^{\dagger}\end{pmatrix}\begin{pmatrix}\hat{\epsilon}(\bm{k})&-\bm{\Delta}\cdot\bm{\sigma}\\\
-\bm{\Delta}\cdot\bm{\sigma}&\hat{\epsilon}(\bm{k}+\bm{Q})\end{pmatrix}\begin{pmatrix}c_{\bm{k}}\\\
c_{\bm{k}+\bm{Q}}\end{pmatrix},$ (14)

where $\bm{k}$-summation runs over the folded Brillouin zone of the
antiferromagnetic state,
$c_{\bm{k}}=(c_{\bm{k}\uparrow},c_{\bm{k}\downarrow})^{\text{T}}$,
$\hat{\epsilon}(\bm{k})=\epsilon_{\bm{k}}I+H_{R}(\bm{k})$, and
$\bm{\Delta}=U\bm{m}_{\text{AF}}$. Here,
$\bm{m}_{\text{AF}}=[1/(2L^{2})]\sum_{\bm{r}\sigma\sigma^{\prime}}e^{-i\bm{Q}\cdot\bm{r}}\langle
c_{\bm{r}\sigma}^{\dagger}\bm{\sigma}_{\sigma\sigma^{\prime}}c_{\bm{r}\sigma^{\prime}}\rangle_{\text{HF}}$,
where $\langle\cdots\rangle_{\text{HF}}$ represents the expectation value in
the ground state of $H_{\text{HF}}$. We solve the gap equation
$\bm{\Delta}=U\bm{m}_{\text{AF}}$ self-consistently.

First, we consider the magnetic order for $\bm{Q}=(\pi,\pi)$. Without the
Rashba spin-orbit coupling, the asymptotic form
$m_{\text{AF}}=|\bm{m}_{\text{AF}}|\sim(\tilde{t}/U)e^{-2\pi\sqrt{\tilde{t}/U}}$
for the weak-coupling region $\Delta=|\bm{\Delta}|\ll W$ was obtained by
Hirsch analyzing the gap equation [51]. If we take into consideration the fact
that the asymptotic form of the density of states
$\rho(\epsilon)\simeq-[1/(2\pi^{2}\tilde{t})]\ln[|\epsilon|/(16\tilde{t})]$
for $\epsilon\simeq 0$ [52] is a good approximation even up to the band edge
[see Fig. 9(d)], we obtain
$m\simeq(32\tilde{t}/U)e^{-2\pi\sqrt{\tilde{t}/U}}$. Indeed, this approximate
form reproduces the numerical data well in the weak-coupling region as shown
in Fig. 10(a).

Figure 10:  Antiferromagnetic moment $m_{\text{AF}}$ with $\bm{Q}=(\pi,\pi)$
for (a) $\alpha=0$, (b) $\alpha=0.2\pi$, (c) $\alpha=0.4\pi$, and (d)
$\alpha=0.5\pi$. The dashed line in (a) is
$(32\tilde{t}/U)e^{-2\pi\sqrt{\tilde{t}/U}}$. The dashed lines in (b) and (c)
are fitted curves with $(a\tilde{t}/U)e^{-2/[U\rho(0)]}$. We used an $L\times
L$ lattice with $L=200$ except for $\alpha=0$ with $U<\tilde{t}$. For
$\alpha=0$ with $U<\tilde{t}$, the finite size effect is severe and we
extrapolated the magnetization for $L\rightarrow\infty$ by using data of
$L=400$, 1000, and 2000.

For a finite $\lambda_{R}$, we find numerically that $\bm{m}_{\text{AF}}$ is
parallel to the $x$ or $y$ direction. It is expected from the effective
Hamiltonian in the strong coupling limit we will discuss later. By assuming
$\Delta\ll\lambda_{R}$ and $\Delta\ll W$, we obtain
$m_{\text{AF}}\sim(\tilde{t}/U)e^{-2/[U\rho(0)]}$ for a finite $\rho(0)$,
where $\rho(0)$ is the density of states at the Fermi level. The coefficient
to $m_{\text{AF}}$ is determined by the entire behavior of the density of
states up to the band edge [see Figs. 9(e) and 9(f)] and we cannot obtain it
analytically in general. Figures 10(b) and 10(c) show the numerically obtained
$m_{\text{AF}}$ for $\alpha=0.2\pi$ and $0.4\pi$, respectively, along with the
fitted curves of $(a\tilde{t}/U)e^{-2/[U\rho(0)]}$, where $a$ is the fitting
parameter. The fitted curves reproduce well the numerical data in the weak-
coupling region.

From the obtained asymptotic form and the numerical data supporting it, we
conclude that the magnetic order occurs by an infinitesimally small $U$ for
$0\leq\alpha<0.5\pi$ consistent with the divergence of the magnetic
susceptibility [19]. We cannot apply this asymptotic form for $\alpha=0.5\pi$
since $\rho(0)=0$ there. The numerical result shown in Fig. 10(d) indicates a
first-order transition for $\alpha=0.5\pi$.

Here, we discuss previous papers indicating the existence of the paramagnetic
phase with finite $U$. In Ref. [20], the authors introduced a threshold
$\varepsilon$ for the magnetization $m_{\text{AF}}$. Then, the authors
determined the magnetic transition point when $m_{\text{AF}}$ becomes smaller
than $\varepsilon$. However, $m_{\text{AF}}$ becomes exponentially small in
the weak-coupling region, as understood from the above analysis. In Ref. [20],
$\varepsilon$ is not sufficiently small to discuss exponentially small
$m_{\text{AF}}$ and a finite region of the paramagnetic phase was obtained. In
Ref. [21], the authors calculated the energy difference $\Delta E$ between the
paramagnetic state and the antiferromagnetic state. Then, the authors
introduced a scaling between $\Delta E$ and $U-U_{\text{AF}}$, where
$U_{\text{AF}}$ is the antiferromagnetic transition point. They tuned
$U_{\text{AF}}$ to collapse the data with different $\alpha$ onto a single
curve in a large-$U$ region. Then, they obtained finite $U_{\text{AF}}$ for
$\alpha\neq 0$. However, this scaling analysis does not have a basis. In
particular, if such a scaling holds for critical behavior, the data collapse
should occur for $U\simeq U_{\text{AF}}$, not for a large-$U$ region.

We have also solved the gap equation for $\bm{Q}=(\pi,0)$ and obtained
$\bm{m}_{\text{AF}}$ parallel to the $y$ direction. By comparing energies for
$\bm{Q}=(\pi,\pi)$ and $\bm{Q}=(\pi,0)$, we construct a phase diagram shown in
Fig. 11.

Figure 11:  Magnetic phase diagram obtained with the Hartree-Fock
approximation within two-sublattice magnetic order.

As noted, the antiferromagnetic state with $\bm{Q}=(\pi,\pi)$ occurs at
infinitesimally small $U$ except for $\alpha=0.5\pi$. The Weyl semimetallic
state remains for $U/~{}\tilde{t}\lesssim 4.4$ at $\alpha=0.5\pi$. The
antiferromagnetic state with $\bm{Q}=(\pi,0)$ appears at large $U$ for
$\alpha/\pi\gtrsim 0.2$.

This phase boundary can be understood from the effective Hamiltonian in the
strong coupling limit. The effective Hamiltonian is derived from the second-
order perturbation theory concerning $t$ and $\lambda_{R}$ and is given by
[16, 17, 19, 18]

$H_{\text{eff}}=\sum_{\bm{r}\bm{a}\mu}\left[J^{\mu}_{\bm{a}}S_{\bm{r}}^{\mu}S_{\bm{r}+\bm{a}}^{\mu}+D_{\bm{a}}^{\mu}(\bm{S}_{\bm{r}}\times\bm{S}_{\bm{r}+\bm{a}})^{\mu}\right],$
(15)

where $\bm{a}=\hat{\bm{x}}$ or $\hat{\bm{y}}$, $\mu=x$, $y$, or $z$,
$\bm{S}_{\bm{r}}$ is the spin operator at site $\bm{r}$,
$J_{\hat{\bm{x}}}^{x}=J_{\hat{\bm{x}}}^{z}=J_{\hat{\bm{y}}}^{y}=J_{\hat{\bm{y}}}^{z}=4(t^{2}-\lambda_{R}^{2})/U$,
$J_{\hat{\bm{x}}}^{y}=J_{\hat{\bm{y}}}^{x}=4(t^{2}+\lambda_{R}^{2})/U$,
$D_{\hat{\bm{x}}}^{y}=-D_{\hat{\bm{y}}}^{x}=8t\lambda_{R}/U$, and the other
components of $\bm{D}_{\bm{a}}$ are zero. From the anisotropy in the
interaction, we expect the ordered moments along the $x$ or $y$ direction for
$\bm{Q}=(\pi,\pi)$ and along the $y$ direction for $\bm{Q}=(\pi,0)$. Thus, the
directions of the ordered moments obtained with the Hartree-Fock approximation
are in accord with the effective Hamiltonian. For $t\ll\lambda_{R}$
($\alpha\simeq 0$), the magnetic order with $\bm{Q}=(\pi,\pi)$ is stable as in
the ordinary Heisenberg model. For $t\gg\lambda_{R}$ ($\alpha\simeq 0.5\pi$),
the magnetic order with $\bm{Q}=(\pi,0)$ has lower energy than that with
$\bm{Q}=(\pi,\pi)$ due to the anisotropic interaction. For $t=\lambda_{R}$
($J_{\hat{\bm{x}}}^{x}=J_{\hat{\bm{x}}}^{z}=J_{\hat{\bm{y}}}^{y}=J_{\hat{\bm{y}}}^{z}=0$),
if we ignore the Dzyaloshinskii-Moriya interaction $\bm{D}_{\bm{a}}$, the
model is reduced to the compass model [53]. It is known as a highly frustrated
model. The condition $t=\lambda_{R}$ corresponds to
$\alpha=\tan^{-1}(1/\sqrt{2})=0.1959\pi$. Thus, the phase boundary
$\alpha\simeq 0.2\pi$ obtained with the Hartree-Fock approximation at a
large-$U$ region is corresponding to the highly frustrated region of the
model.

However, in a large-$U$ region, we expect longer-period magnetic order due to
the Dzyaloshinskii-Moriya interaction. It is out of the scope of the present
study and has already been investigated by previous studies using the
effective Hamiltonian [16, 17, 18]. Our important finding in this section is
the absence of the paramagnetic phase except for $\alpha=0.5\pi$ in the weak-
coupling region. However, the ordered moment and the energy gain of the
antiferromagnetic state in the weak-coupling region are exponentially small.
Thus, the effects of this magnetic order should be weak. In addition, this
magnetic order would be easily destroyed by perturbations such as the next-
nearest-neighbor hopping breaking the nesting condition [15]. Thus, the
discussions in the previous sections without considering magnetic order are
still meaningful.

##  VI Summary

We have investigated the Rashba-Hubbard model on a square lattice. The Rashba
spin-orbit coupling generates the two-dimensional Weyl points, which are
characterized by non-zero winding numbers. We have investigated lattices with
edges and found zero-energy states on a lattice with zigzag edges. The zero-
energy states are localized around the edges and have a helical character. The
large density of states due to the flat zero-energy band may result in
magnetic polarization at edges, similar to graphene [28].

We have also examined the effects of the Coulomb interaction $U$. The Coulomb
interaction renormalizes the ratio of the coupling constant of the Rashba
spin-orbit coupling $\lambda_{R}$ to the hopping integral $t$ effectively. As
a result, the Weyl points can move to the Fermi level by the correlation
effects. Thus, the Coulomb interaction can enhance the effects of the Weyl
points and assist in observing phenomena originating from the Weyl points even
if the bare Rashba spin-orbit coupling is not large.

We have also investigated the magnetism of the model by the Hartree-Fock
approximation. We have found that the antiferromagnetic state with the
ordering vector $\bm{Q}=(\pi,\pi)$ occurs at infinitesimally small $U$ due to
the perfect nesting of the Fermi surface even for a finite $\lambda_{R}$.
However, the density of states at the Fermi level becomes small for a large
$\lambda_{R}$ and as a result, the energy gain by the antiferromagnetic order
is small in the weak-coupling region. Therefore, the effects of the magnetic
order should be weak in such a region. In addition, this magnetic order would
be unstable against perturbations, such as the inclusion of next-nearest-
neighbor hopping [15]. Thus, we conclude that the discussions on the Weyl
semimetal without assuming magnetism are still meaningful.

###### Acknowledgements.

This work was supported by JSPS KAKENHI Grant Number JP23K03330.

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# Bijective Approaches for Schmidt-Type Theorems

Hunter Waldron

###### Abstract

We provide new Schmidt-type results through an investigation of two
bijections, which are results involving partitions with parts counted only at
given indices. Mork’s bijection, the first of these, was originally given as a
proof of Schmidt’s theorem. We show that a version of Sylvester’s bijection is
equivalent to Mork’s bijection applied to 2-modular diagrams, which implies
refinements of existing results and new generating function identities. We
then develop a bijection based on an idea appearing in a recent paper of
Andrews and Keith, that places partitions counted at the indices $r$, $t+r$,
$2t+r,\dots$ in correspondence with $t$-colored partitions. This leads to a
substantial generalization of an identity of Bridges and Uncu, and complements
a similar investigation of Li and Yee.

##  1 Introduction

A partition is a sequence
$\lambda=(\lambda_{1},\lambda_{2},\lambda_{3},\dots)$ of weakly decreasing
non-negative integers that eventually become zero. The parts of $\lambda$ are
the non-zero terms, the number of which is written $\ell(\lambda)$. When
convenient, we write $\lambda=(\lambda_{1},\dots,\lambda_{\ell(\lambda)})$,
ignoring the trailing zeroes. The size of $\lambda$, written
$\lvert\lambda\rvert$, is the sum of all parts,
$\lambda_{1}+\lambda_{2}+\lambda_{3}+\cdots$. If $\lambda$ has size $n$, we
say $\lambda$ is a partition of $n$. For example, the 7 partitions are 5 are

(5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), and (1,1,1,1,1).

The first 3 of these partitions are the partitions of 5 with distinct parts,
meaning no part appears more than once.

There has been rapidly growing interest [andrews_keith, li_yee, bridges_uncu,
berkovich_uncu, ji, andrews_paule, uncu, chern_yee] in the research of
partitions counted with Schmidt-weights, where parts are only counted at
specified indices. These are so named after Frank Schmidt who in [schmidt]
made the remarkable observation:

######  Theorem 1.1 (Schmidt [schmidt]).

The number of partitions of $n$ is equal to the number of partitions $\lambda$
with distinct parts such that $\lambda_{1}+\lambda_{3}+\lambda_{5}+\cdots=n$.

For example, the 7 partitions with distinct parts corresponding to the
partitions listed above are

(5), (5,1), (5,2), (5,3), (5,4), (4, 2, 1), and (4, 3, 1).

Relationships between families of partitions like Schmidt’s often translate
into identities for generating functions. Let $\mathcal{P}$ be the set of all
partitions and $\mathcal{D}$ be the set of partitions with distinct parts.
Then Schmidt’s theorem is equivalent to the equality

$\sum_{\lambda\in\mathcal{D}}q^{\displaystyle\lambda_{1}+\lambda_{3}+\lambda_{5}+\cdots}=\sum_{\lambda\in\mathcal{P}}q^{\displaystyle\lvert\lambda\rvert}.$

The sum on the right is well-known to be equal to $1/(q;q)_{\infty}$, giving
the identity

$\sum_{\lambda\in\mathcal{D}}q^{\displaystyle\lambda_{1}+\lambda_{3}+\lambda_{5}+\cdots}=\frac{1}{(q;q)_{\infty}}.$

This is written using the standard notation for the q-Pochhammer symbol, shown
below. When possible, we use standard definitions and notation such as this
throughout the paper. More can be found in section 2.

$(z;q)_{n}=\prod_{k=0}^{n-1}(1-zq^{k})\hskip 7.11317pt\text{for integers
$n\geq 0$},\hskip 14.22636pt(z;q)_{\infty}=\prod_{k=0}^{\infty}(1-zq^{k}).$

One of the many known proofs of Schmidt’s theorem is a bijection given by Mork
[mork], which matches certain hooks from a given partition to the distinct
parts of another. Beyond an application by Chern and Yee in [chern_yee], this
map has not been studied in much depth.

We show in section 3 that a well-known version of Sylvester’s bijection by
Bessenrodt [bessenrodt], which proves a refinement of Euler’s classic theorem
on the equinumerosity of partitions of equal size with distinct and odd parts,
is essentially equivalent to Mork’s bijection. Indeed, Mork’s bijection
becomes identical when applied to 2-modular diagrams of partitions with odd
parts, forgetting the numbers. We use the equivalence of these maps to add a
Schmidt-type refinement to the already long list of properties that
Bessenrodt’s bijection is known to possess.

######  Proposition 1.2.

Let $\lambda$ be a partition of $n$ with distinct parts and first part $k$,
such that $\lambda_{1}+\lambda_{3}+\lambda_{5}+\cdots=m$, and let $\mu$ be the
preimage of $\lambda$ under Bessenrodt’s bijection. Then $\mu$ has $2m-n$
parts and first part $1+2k+2n-4m$.

######  Remark.

This result shows that there is a strong linear relationship between the
Schmidt-weight $\lambda_{1}+\lambda_{3}+\lambda_{5}+\cdots$ and $\ell(\mu)$
for partitions of fixed size, which is illustrated in table 1 below.

$\lambda_{1}+\lambda_{3}+\lambda_{5}+\cdots$ |  $\lambda$ with distinct parts |  $\mu$ with odd parts
---|---|---
7 | (7) | (1,1,1,1,1,1,1)
6 | (6,1) | (3,1,1,1,1)
5 | (5,2) | (5,1,1)
5 | (4, 2, 1) | (3,3,1)
4 | (4,3) | (7)
Table 1: The partitions of 7 with distinct and odd parts.

As a byproduct of the work leading to Proposition 1.2, we also produce a
refinement of Schmidt’s theorem:

######  Corollary 1.3.

The number of partitions of $n$ with $\ell$ parts is equal to the number of
partitions $\lambda$ with distinct parts such that
$\lambda_{1}+\lambda_{3}+\lambda_{5}+\cdots=n$ and
$\lvert\lambda\rvert=2n-\ell$.

The following generating function identities are also implied:

######  Theorem 1.4.

$\displaystyle\sum_{\lambda\in\mathcal{D}}$ $\displaystyle
z^{\displaystyle\lvert\lambda\rvert}q^{\displaystyle\lambda_{1}+\lambda_{3}+\lambda_{5}+\cdots}=\frac{1}{(qz;qz^{2})_{\infty}}$
(1) $\displaystyle\sum_{\lambda\in\mathcal{D}}$ $\displaystyle
z^{\displaystyle\lvert\lambda\rvert}q^{\displaystyle\lambda_{2}+\lambda_{4}+\lambda_{6}+\cdots}=\frac{1}{(z;qz^{2})_{\infty}}$
(2)

######  Remark.

If we factor (1) as $\sum_{n=0}^{\infty}p_{n}(z)q^{n}$, $p_{n}(z)$ will be a
polynomial, and indeed a $z$-analog of the partition function.

$\frac{1}{(qz;qz^{2})_{\infty}}=1+zq+(z^{2}+z^{3})q^{2}+(z^{3}+z^{4}+z^{5})q^{3}+\cdots$
(3)

These polynomials have two interpretations based on this work,

$p_{n}(z)\hskip 14.22636pt=\hskip
14.22636pt\smashoperator[]{\sum_{\begin{subarray}{c}\lambda\in\mathcal{D},\\\
\lambda_{1}+\lambda_{3}+\lambda_{5}+\cdots=n\end{subarray}}^{}}z^{\displaystyle\lvert\lambda\rvert}\hskip
14.22636pt=\hskip
14.22636pt\smashoperator[]{\sum_{\begin{subarray}{c}\lambda\in\mathcal{P},\\\
\lvert\lambda\rvert=n\end{subarray}}^{}}z^{\displaystyle 2n-\ell(\lambda)}.$
(4)

The sum on the right is simply a (shifted and reflected) distribution of the
partitions of $n$ by number of parts, giving a connection between the Schmidt-
weight and length similar to what we have with Proposition 1.2. This is also
the algebraic equivalent of Corollary 1.3.

For (2), the inclusion of $z$ is necessary for the coefficients to be finite,
since for example any partition with one part will have Schmidt-weight zero.

Our final application of Mork’s bijection uses a different approach, counting
only some of the hook lengths used in order to give generating function
identities for partitions $\lambda$ with distinct parts $\lambda_{i}$ counted
when $i\equiv 1\pmod{4}$. This technique is probably not difficult to
generalize.

######  Theorem 1.5.

$\displaystyle\smashoperator[]{\sum_{\begin{subarray}{c}\lambda\in\mathcal{D},\\\
\ell(\lambda)\equiv 0\;\mathrm{or}\;3\\\ (\mathrm{mod}\;4)\end{subarray}}^{}}$
$\displaystyle
z^{\displaystyle\lambda_{1}}q^{\displaystyle\lambda_{1}+\lambda_{5}+\lambda_{9}+\cdots}=1+\sum_{n=1}^{\infty}\frac{q^{2n+1\choose
2}z^{4n-1}}{(qz;q)_{n}^{4}}$ (5)
$\displaystyle\smashoperator[]{\sum_{\begin{subarray}{c}\lambda\in\mathcal{D},\\\
\ell(\lambda)\equiv 1\;\mathrm{or}\;2\\\ (\mathrm{mod}\;4)\end{subarray}}^{}}$
$\displaystyle
z^{\displaystyle\lambda_{1}}q^{\displaystyle\lambda_{1}+\lambda_{5}+\lambda_{9}+\cdots}=\sum_{n=1}^{\infty}\frac{q^{2n\choose
2}z^{4n-3}}{(qz;q)_{n}^{2}(qz;q)_{n-1}^{2}}$ (6)

In section 4, we change our focus to investigating Schmidt-type results for
unrestricted partitions. Several such results have already been found. For
instance, in [bridges_uncu] Bridges and Uncu gave the following two generating
function identites.

######  Theorem 1.6 (Bridges-Uncu [bridges_uncu]).

$\displaystyle\sum_{\lambda\in\mathcal{P}}z^{\displaystyle\lambda_{1}}q^{\displaystyle\lambda_{1}+\lambda_{3}+\lambda_{5}+\cdots}$
$\displaystyle=\frac{1}{(qz;q)_{\infty}^{2}}$ (7)
$\displaystyle\sum_{\lambda\in\mathcal{P}}z^{\displaystyle\lambda_{1}}q^{\displaystyle\lambda_{2}+\lambda_{4}+\lambda_{6}+\cdots}$
$\displaystyle=\frac{1}{(1-z)(qz;q)_{\infty}^{2}}$ (8)

Uncu had found the $z=1$ specialization of (7) previously in [uncu], which was
also independently found by Andrews and Paule in [andrews_paule]. They added a
combinatorial interpretation in terms of 2-colored partitions, which Ji [ji]
later proved, and indeed refined, bijectively.

A more general connection with colored partitions was more recently given by
Li and Yee in [li_yee], which they proved bijectively as well. We have
translated their result into the language of colored partitions, which they
call $t$-multipartitions. From here, let $t$ and $r$ be arbitrary positive
integers.

######  Theorem 1.7 (Li-Yee [li_yee]).

The number of partitions $\lambda$ such that $\ell(\lambda)=(s-1)t+j$ and
$\lambda_{1}+\lambda_{t+1}+\lambda_{2t+1}+\cdots=n$ is equal to the number of
$t$-colored partitions of $n$ where $s$ is the most number of times any color
appears, and $j$ is the largest such color.

By developing an idea that was remarked on by Andrews and Keith in
[andrews_keith], we give a bijection, which we call the color-conjugate map,
that implies a similar result.

######  Theorem 1.8.

The number of partitions $\lambda$ such that $\lambda_{1}=k_{1}$,
$\lambda_{r}=k_{r}$, $\lambda_{r}+\lambda_{t+r}+\lambda_{2t+r}+\cdots=n$, and
for each $1\leq i\leq t$,

$c_{i}(\lambda)=\sum_{k\geq 0}(\lambda_{kt+r+i-1}-\lambda_{kt+r+i}),$

is equal to the number of pairs $(\nu,\mu)$ such that $\nu$ is a partition
with first part $k_{1}-k_{r}$ and at most $r-1$ parts, and $\mu$ is a
$t$-colored partition with $k_{r}$ parts where each color $i$ appears
$c_{i}(\lambda)$ times.

######  Remark.

Both of these theorems imply that the partitions $\lambda$ such that
$\lambda_{1}+\lambda_{t+1}+\lambda_{2t+1}+\cdots=n$ are equinumerous with the
$t$-colored partitions of $n$, however the refinements are distinct. Indeed,
when $r=1$, it can be easily checked that the color-conjugate map places
partitions with $(s-1)t+j$ parts in bijection with $t$-colored partitions
where, instead of the condition in Li and Yee’s theorem, $s$ is the largest
part size, and $j$ is the largest color used on the largest part size.

For example, when $t=3$, the color-conjugate bijection sends $(4,4,3,3,3,3)$
to the 3-colored partition $(2^{3},2^{3},2^{3},1^{2})$, while the map given by
Li and Yee sends this same partition to $(3^{3},3^{3},1^{2})$.

There are two notable interpretations of Theorem 1.8 as generating function
identities. Now, Theorem 1.6 becomes the special case of (9) with $t=2$ and
$r\in\\{1,2\\}$.

######  Theorem 1.9.

$\displaystyle\sum_{\lambda\in\mathcal{P}}z^{\displaystyle\lambda_{1}}q^{\displaystyle\lambda_{r}+\lambda_{t+r}+\lambda_{2t+r}+\cdots}$
$\displaystyle=\frac{1}{(1-z)^{r-1}(qz;q)_{\infty}^{t}}$ (9)
$\displaystyle\sum_{\lambda\in\mathcal{P}}\prod_{i=1}^{t}z_{i}^{\displaystyle
c_{i}(\lambda)}\times
q^{\displaystyle\lambda_{1}+\lambda_{t+1}+\lambda_{2t+1}+\cdots}$
$\displaystyle=\prod_{i=1}^{t}\frac{1}{(qz_{i};q)_{\infty}}$ (10)

We can actually improve (9) by adding a third variable $s$ that tracks the
ordinary size of the partitions summed over, if we abandon our purely
bijective approach. This gives us our main result:

######  Theorem 1.10.

$\sum_{\lambda\in\mathcal{P}}z^{\displaystyle\lambda_{1}}s^{\displaystyle\lvert\lambda\rvert}q^{\displaystyle\lambda_{r}+\lambda_{t+r}+\lambda_{2t+r}+\cdots}=\frac{1}{(sz;s)_{r-1}}\prod_{n=0}^{\infty}\frac{1}{(s^{nt+r}q^{n+1}z;s)_{t}}$
(11)

Replacing $s$ with $q$ and $q$ with $q^{-1}$ in Theorem 11, then simplifying
the right hand side gives us another interesting generating function identity,
with a straight forward combinatorial interpretation in terms of $2$-colored
partitions.

######  Corollary 1.11.

$\sum_{\lambda\in\mathcal{P}}z^{\displaystyle\lambda_{1}}q^{\displaystyle|\lambda|-\lambda_{r}-\lambda_{t+r}-\lambda_{2t+r}-\cdots}=\frac{1}{(qz;q)_{\infty}(q^{r-1}z;q^{t-1})_{\infty}}.$
(12)

######  Corollary 1.12.

If $r>1$, the number of partitions with first part $k$ such that all parts
except at the indices $r,t+r,2t+r,\dots$ sum to $n$ is equal to the number of
$2$-colored partitions of $n$ with $k$ parts where the second color only
appears on parts of size $r-1,t-1+r-1,2(t-1)+r-1,\dots$

######  Remark.

We can also prove Corollary 1.12 by modifying color-conjugate map such that
the parts $\lambda_{r},\lambda_{t+r},\lambda_{2t+r},\dots$ are used to color
the parts $\lambda_{r-1}$, $\lambda_{t-1+r-1}$, $\lambda_{2(t-1)+r-1}$, ….

## Acknowledgments

The author would like to thank his advisor William Keith for introducing him
to this area of research and for critiquing early versions of this work.

##  2 Preliminaries

Here we provide additional background and the notation that will be used in
the proofs of section 3 and 4.

For any positive integer $t$, a $t$-colored partition is a partition $\lambda$
with each part $\lambda_{i}$ marked by a color $c_{i}\in\\{1,\dots,t\\}$, with
the order of colors across parts of equal size weakly decreasing. That is, if
$1\leq i<j$ and $\lambda_{i}=\lambda_{j}$, then $c_{i}\geq c_{j}$. The
notation we use for $t$-colored partitions places the color as a superscript:
$\lambda=(\lambda_{1}^{c_{1}},\lambda_{2}^{c_{2}},\lambda_{3}^{c_{3}},\dots)$.
The size of $\lambda$ is simply $\lambda_{1}+\lambda_{2}+\lambda_{3}+\cdots$,
ignoring the colors. For example, the 2-colored partitions of 3 are

$\begin{gathered}(3^{1}),\;(3^{2}),\;(2^{1},1^{1}),\;(2^{2},1^{1}),\;(2^{1},1^{2}),\;(2^{2},1^{2}),\;(1^{1},1^{1},1^{1}),\\\
(1^{2},1^{1},1^{1}),\;(1^{2},1^{2},1^{1}),\;\textrm{and}\;(1^{2},1^{2},1^{2}).\end{gathered}$

The Young diagram of a partition $\lambda$ is a visual representation of
$\lambda$ as a grid of squares arranged into rows that are aligned on the
left, with the number of squares in the $j$th row down corresponding to the
value of $\lambda_{j}$. The cell $(i,j)$ of $\lambda$ is the $i$th square down
in the $j$th column from the left in the Young diagram of $\lambda$. The cells
$(i,i)$ form the diagonal of $\lambda$. Swapping $i$ and $j$ in all the cells
of $\lambda$ produces another partitions $\lambda^{\prime}$, the conjugate of
$\lambda$.

The Durfee square of $\lambda$ is the largest square of cells that can be
placed inside the Young diagram of $\lambda$. Its size is the number of cells
in the width, or equivalently in the height.

To any cell $(i,j)$ in the Young diagram of $\lambda$ we can associate a hook
which consists of the cell $(i,j)$ and any below and to the right. The hook
length is the number of cells in the hook.

boxsize=1.25em *(CELL_C) & *(CELL_C)
*(CELL_C) *(CELL_C)

&
*(CELL_C)6 *(CELL_C) *(CELL_C)
*(CELL_C)
*(CELL_C)
*(CELL_C)

Figure 1: The partitions $(5,4,2,1)$ and $(4,3,2,1,1)$ are conjugates. The
Durfee square is shown in $(5,4,2,1)$, and the hook of $(2,1)$ is shown in
$(4,3,2,1,1).$

boxsize=1.25em 2 & 2 2 2 2 1
2 2 2 2 2
2 2 2
2 1
1

Figure 2: The 2-modular diagram of the partition $(11,10,6,3,1).$

For any integer $m\geq 2$, if we write the parts of a partition $\lambda$ as
$\lambda_{i}=m\nu_{i}+r_{i}$ where $1\leq r_{i}\leq m$ and
$\nu=(\nu_{1},\nu_{2},\nu_{3},\dots)$ is a partition, the m-modular diagram of
$\lambda$ is the Young diagram of $\nu$ with the cell at the end of the $i$th
row filled with $r_{i}$, and all others with $m$.

The $q$-binomial coefficient is defined using the $q$-Pochhammer symbol for
all integers $n,k\geq 0$ as

$\left[\begin{matrix}n\\\ k\end{matrix}\right]_{\displaystyle
q}=\frac{(q;q)_{n}}{(q;q)_{k}(q;q)_{n-k}},$

which is useful for giving us an identity for partitions with Young diagrams
whose cells fit inside an $n\times k$ rectangle.

$\sum_{\begin{subarray}{c}\lambda\in\mathcal{P},\\\ \lambda_{1}\leq
n,\;\mathrm{and}\;\\\ \ell(\lambda)\leq
k\end{subarray}}\\!\\!\\!\\!\\!\\!\\!q^{\displaystyle|\lambda|}=\left[\begin{matrix}n+k\\\
k\end{matrix}\right]_{\displaystyle q}.$

##  3 Mork’s bijection

Mork’s bijection [mork] $\mathrm{M}\colon\mathcal{P}\to\mathcal{D}$ is defined
on the partition $\mu$ as follows. For $i\geq 1$, $\mathrm{M}(\mu)_{2i-1}$ is
the $i$th diagonal hook length of $\mu$, and $\mathrm{M}(\mu)_{2i}$ is the
hook length of the cell to the right of that. Since the diagonal hook lengths
of $\mu$ sum to $\lvert\mu\rvert$, the odd indexed parts of $\mathrm{M}(\mu)$
sum to $\lvert\mu\rvert$, which implies Schmidt’s theorem. This is easily
understood by example.

boxsize=1.25em [*(CELL_C)] 12 & 10
7 5
3 2
1

[12] & *(CELL_C) *(CELL_C) *(CELL_C) *(CELL_C) *(CELL_C) *(CELL_C) *(CELL_C)
*(CELL_C) *(CELL_C) *(CELL_C) *(CELL_C)
[10]
[7] *(CELL_C) *(CELL_C) *(CELL_C) *(CELL_C) *(CELL_C) *(CELL_C) *(CELL_C)
[5]
[3] *(CELL_C) *(CELL_C) *(CELL_C)
[2]
[1] *(CELL_C)

Figure 3: Mork’s bijection maps the partition $(7,5,4,4,2,1)\in\mathcal{P}$ to
$(12,10,7,5,3,2,1)\in\mathcal{D}$.

By similar reasoning, the even indexed parts of $\mathrm{M}(\mu)$ sum to
$\lvert\mu\rvert-\ell(\mu)$, so
$|\mathrm{M}(\mu)\rvert=2\lvert\mu\rvert-\ell(\mu)$. Therefore if we fill the
cells of $\mu$ at the end of each row with a 1, and all others with 2s, we
construct the 2-modular diagram of a partition with odd parts that has size
equal to $\mathrm{M}(\mu)$. Since this construction defines a bijection
$\mathrm{D}\colon\mathcal{P}\to\mathcal{O}$, $\mathrm{M}\circ\mathrm{D^{-1}}$
is a size preserving bijection from $\mathcal{O}$ to $\mathcal{D}$. This also
immediately implies Corollary 1.3.

Bessenrodt’s bijection, given in [bessenrodt] as a version of Sylvester’s
bijection, similarly acts on a 2-modular diagram’s diagonal hooks to give a
size preserving bijection $\mathrm{B}\colon\mathcal{O}\to\mathcal{D}$. For
$\mu\in\mathcal{O}$ and $i\geq 1$, Bessenrodt’s map assigns to
$\mathrm{B}(\mu)_{2i-1}$ the number of 2s and 1s in the $i$th diagonal hook of
$\mu$’s 2-modular diagram, and to $\mathrm{B}(\mu)_{2i}$ the number of 1s.
Although defined differently, we can easily see that
$\mathrm{B}=\mathrm{M}\circ\mathrm{D}^{-1}$. For, from their respective
definitions, we immediately see that $\mathrm{B}(\mu)$ and
$\mathrm{M}\circ\mathrm{D}^{-1}(\mu)$ coincide at the odd indices.

As for even indices $2i$, the number of 2s in the cells at and below $(i,i)$
in the 2-modular diagram of $\mu$ must be equal to the number of cells at and
below $(i,i+1)$ since any 2 must have a cell to the right, but a 1 may not.
Similarly the cells to the right of $(i,i)$ with 2s will be one less than the
number of all such cells. Thus both maps coincide at the even indices. We use
this equivalence to prove the following.

Proof of Proposition 1.2. Let $\mu^{*}=\mathrm{D}^{-1}(\mu)$. Then Mork’s
bijection maps $\mu^{*}$ to $\lambda$. So, as above,
$2\lvert\mu^{*}\rvert-\ell(\mu^{*})=\lvert\lambda\rvert$. Since
$\lvert\mu^{*}\rvert=m$ and $\ell(\mu^{*})=\ell(\mu)$ the first property
follows.

For the second property, clearly the first diagonal hook length of $\mu^{*}$
is $\lambda_{1}$, and $\mu^{*}_{1}=(\mu_{1}-1)/2$, so

$\displaystyle\lambda_{1}=\mu^{*}_{1}+\ell(\mu^{*})-1$
$\displaystyle=\frac{\mu_{1}-1}{2}+\ell(\mu)-1$ (13)
$\displaystyle=\frac{\mu_{1}-1}{2}+2m-n-1.$

Solving for $\mu_{1}$ completes the proof.

∎

### Identities implied by Mork’s bijection

Proof of Theorem 1.4. From the work above, Mork’s bijection implies the
equality

$\sum_{\lambda\in\mathcal{D}}z^{\displaystyle\lvert\lambda\rvert}q^{\displaystyle\lambda_{1}+\lambda_{3}+\lambda_{5}+\cdots}=\sum_{\mu\in\mathcal{P}}z^{\displaystyle
2\lvert\mu\rvert-\ell(\mu)}q^{\displaystyle\lvert\mu\rvert}.$ (14)

The sum on the right has a simple interpretation. For any partition
$\mu\in\mathcal{P}$ with $n$ parts, the first $n$ cells on left in $\mu$’s
Young diagram are counted as $(qz)^{n}$, and to the right of that may be any
partition with at most $n$ parts, with each cell contributing $qz^{2}$.
Summing over $n$ gives

$\sum_{\mu\in\mathcal{P}}z^{\displaystyle
2\lvert\mu\rvert-\ell(\mu)}q^{\displaystyle\lvert\mu\rvert}=\sum_{n=0}^{\infty}\frac{(qz)^{n}}{(qz^{2};qz^{2})_{n}}=\sum_{n=0}^{\infty}\frac{(qz^{2})^{n}}{(qz^{2};qz^{2})_{n}}(z^{-1})^{n}.$
(15)

Using the well-known identity

$\sum_{n=0}^{\infty}\frac{q^{n}}{(q;q)_{n}}z^{n}=\frac{1}{(qz;q)_{\infty}}$

with $q$ and $z$ replaced with $qz^{2}$ and $z^{-1}$, respectively, this
becomes

$\frac{1}{(z^{-1}qz^{2};qz^{2})_{\infty}}=\frac{1}{(qz;qz^{2})_{\infty}}$ (16)

proving (1).

To prove (2), we can use a similar argument, starting instead with.

$\sum_{\lambda\in\mathcal{D}}z^{\displaystyle\lvert\lambda\rvert}q^{\displaystyle\lambda_{2}+\lambda_{4}+\lambda_{6}+\cdots}=\sum_{\mu\in\mathcal{P}}z^{\displaystyle
2\lvert\mu\rvert-\ell(\mu)}q^{\displaystyle\lvert\mu\rvert-\ell(\mu)}.$ (17)

∎

Proof of Theorem 1.5. From the definition of Mork’s bijection, if
$\lambda\in\mathcal{D}$, then $\ell(\lambda)\equiv 0\;\textrm{or}\;3\pmod{4}$
exactly when the Durfee square of $\mathrm{M}^{-1}(\lambda)$ is has even
length. Moreover, we have that the parts
$\lambda_{1},\lambda_{5},\lambda_{9},\dots$ are given by the 1st, 3rd, 5th,
…diagonal hook lengths of $\mathrm{M}^{-1}(\lambda)$, which here we will
denote $h_{1}^{\mu}$, $h_{3}^{\mu}$, $h_{5}^{\mu},\dots$. Using these facts,
the following equality must hold.

$\smashoperator[]{\sum_{\begin{subarray}{c}\lambda\in\mathcal{D},\\\
\ell(\lambda)\equiv 0\;\mathrm{or}\;3\\\
(\mathrm{mod}\;4)\end{subarray}}^{}}z^{\displaystyle\lambda_{1}}q^{\displaystyle\lambda_{1}+\lambda_{5}+\lambda_{9}+\cdots}=\smashoperator[]{\sum_{\begin{subarray}{c}\mu\in\mathcal{P}\;\textrm{with}\\\
\textrm{even Durfee length}\end{subarray}}^{}}z^{\displaystyle
h_{1}^{\mu}}q^{\displaystyle h_{1}^{\mu}+h_{3}^{\mu}+h_{5}^{\mu}+\cdots}$ (18)

Consider a partition $\mu$ with Durfee length $2n$. The cells that are both in
$\mu$’s Durfee square and in the hooks counted by the above sum are in total
the $2n$th triangular number, contributing $q^{2n+1\choose 2}z^{4n-1}$. This
can be seen by deleting the other cells, then sliding these hooks up and to
the left.

To the right of $\mu$’s Durfee square and, in conjugate, below, may be any
partition with at most $2n$ parts where only the odd indexed parts contribute
to $q$, since these are the remainder of the hook lengths, and the first part
contributes to $z$. The identity

$\smashoperator[]{\sum_{\begin{subarray}{c}\lambda\in\mathcal{P},\\\
\ell(\lambda)\leq
2n\end{subarray}}^{}}z^{\displaystyle\lambda_{1}}q^{\displaystyle\lambda_{1}+\lambda_{3}+\lambda_{5}+\cdots}=\frac{1}{(qz;q)_{n}^{2}}$
(19)

can easily be derived from the color-conjugate bijection in section 4. This
all together implies that

$\smashoperator[]{\sum_{\begin{subarray}{c}\mu\in\mathcal{P}\;\textrm{with}\\\
\textrm{even Durfee length}\end{subarray}}^{}}z^{\displaystyle
h_{1}^{\mu}}q^{\displaystyle
h_{1}^{\mu}+h_{3}^{\mu}+h_{5}^{\mu}+\cdots}=1+\sum_{n=1}^{\infty}\frac{q^{2n+1\choose
2}z^{4n-1}}{(qz;q)_{n}^{4}}$ (20)

which gives (5). The proof of (6) is similar.

∎

##  4 The color-conjugate map

Let $\mathcal{P}^{t}$ be the set of all $t$-colored partitions, and
$\mathcal{P}_{\leq r-1}$ be the set of partitions with at most $r-1$ parts. We
define the color-conjugate map
$\mathrm{C}_{t,r}\colon\mathcal{P}\to\mathcal{P}_{\leq
r-1}\times\mathcal{P}^{t}$ as follows for given $t,r\geq 1$ on a partition
$\lambda$.

Let $\nu$ be the partition given by $\nu_{i}=\lambda_{i}-\lambda_{r}$ for
$1\leq i\leq r$. The $t$-colored partition $\mu$ is constructed as follows.
The parts of $\mu$ are given by the conjugate of the partition
$(\lambda_{r},\lambda_{t+r},\lambda_{2t+r},\dots)$. The color of $\mu_{i}$ is
given by subtracting $r-1$ from $\lambda$’s $i$th column, then counting the
number of cells from the bottom of the column up, stopping at the first
counted part. We then define $\mathrm{C}_{t,r}(\lambda)=(\nu,\mu)$.

boxsize=1.25em

$\ytableau\none&*(CELL_{B})*(CELL_{B})*(CELL_{B})*(CELL_{B})\\\
\none*(CELL_{B})*(CELL_{B})\\\ \none*(CELL_{B})\\\
\none[\lambda_{r}]*(CELL_{C})*(CELL_{C})*(CELL_{C})*(CELL_{C})*(CELL_{C})\\\
\none\\\ \none\\\
\none[\\!\\!\\!\\!\\!\lambda_{t+r}]*(CELL_{C})*(CELL_{C})*(CELL_{C})*(CELL_{C})\\\
\none\\\ \none\\\
\none[\\!\\!\\!\\!\\!\\!\\!\lambda_{2t+r}]*(CELL_{C})*(CELL_{C})\\\ \none$

$\ytableau[*(CELL_{B})]\;&\\\ \;\\\ \;$

$\ytableau\none[2]&*(CELL_{C})*(CELL_{C})*(CELL_{C})\none\none\\\
\none[1]*(CELL_{C})*(CELL_{C})*(CELL_{C})\\\ \none[3]*(CELL_{C})*(CELL_{C})\\\
\none[2]*(CELL_{C})*(CELL_{C})\\\ \none[1]*(CELL_{C})$

Figure 4: $\mathrm{C}_{3,4}$ maps the partition
$\lambda=(9,7,6,5,4,4,4,4,3,2,1)$ to $\nu=(4,2,1)$ and
$\mu=(3^{2},3^{1},2^{3},2^{2},1^{1})$.

The claimed properties immediately follow from the construction. We can give
$\mathcal{C}_{t,r}^{-1}$ just as easily, showing that $\mathcal{C}_{t,r}$ is
indeed a bijection.

Given $\nu$ and $\mu$, let $\lambda_{i}=\ell(\mu)+\nu_{i}$ for $1\leq i\leq
r-1$. For $1\leq i\leq\ell(\mu)$, append $(t-1)\mu_{i}+c_{i}$ to $\lambda$’s
$i$th column where $c_{i}$ is the color of $\mu$’s $i$th part.

Proof of Theorem 1.9. Using the well-known identities

$\smashoperator[]{\sum_{\lambda\in\mathcal{P}_{\leq
r-1}}^{}}z^{\displaystyle\lambda_{1}}=\frac{1}{(1-z)^{r-1}}\hskip
14.22636pt\text{and}\hskip
14.22636pt\sum_{\lambda\in\mathcal{P}^{t}}z^{\displaystyle\lambda_{1}}q^{\displaystyle\ell(\lambda)}=\frac{1}{(qz;q)_{\infty}^{t}},$
(21)

(9) is immediately implied from the properties of the color-conjugate map, and
(10) similarly.

∎

### Proof of the main result

We first prove Theorem 11 in the $r=1$ case. That result will then be used to
find the general case. For integers $n\geq 0$ define

$f_{n}(q,s)=\sum_{\begin{subarray}{c}\lambda\in\mathcal{P},\\\
\lambda_{1}=n\end{subarray}}s^{\displaystyle|\lambda|}q^{\displaystyle\lambda_{1}+\lambda_{t+1}+\lambda_{2t+1}+\cdots}.$
(22)

The first step of the proof is finding a recurrence relation for $f_{n}(q,s)$.

Consider any partition $\lambda$ with first part $n$. The first part will be
counted as $(qs)^{n}$ by $f_{n}(q,s)$. For each possibility $0\leq
k=\lambda_{t+1}\leq n$, in $\lambda$’s Young diagram we are guaranteed a
$k\times(t-1)$ rectangle of cells between $\lambda_{1}$ and $\lambda_{t+1}$
that is counted as $s^{k(t-1)}$, and an $(n-k)\times(t-1)$ rectangle to the
right that may contain any partition. This is counted as

$\left[\begin{matrix}n-k+t-1\\\ t-1\end{matrix}\right]_{\displaystyle s}$

and finally, the part $\lambda_{t+1}$ and beyond is counted by $f_{k}(q,s)$.

$f_{k}(s,q)$$(n-k)\times(t-1)$$\lambda_{1}$$\lambda_{t+1}$ Figure 5: A
visualization of the cells counted by $f_{n}(q,s)$. In this example $n=6$,
$t=3$, and $k=2$.

Some elementary combinatorial reasoning then gives us the recurrence

$f_{n}(q,s)=(qs)^{n}\sum_{k=0}^{n}s^{k(t-1)}\left[\begin{matrix}n-k+t-1\\\
t-1\end{matrix}\right]_{\displaystyle s}f_{k}(q,s).$ (23)

Now let $F(s,q,z)$ be the generating function for $f_{n}(q,s)$:

$F(s,q,z)=\sum_{n=0}^{\infty}f_{n}(q,s)z^{n}.$

This is indeed the sum appearing on the left in the statement of Theorem 11
since the power of $z$ coincides with the first part of the partitions
counted.

Replacing $f_{n}(q,s)$ with the recurrence found above in $F(s,q,z)$,

$\begin{split}F(s,q,z)&=\sum_{n=0}^{\infty}\left((qs)^{n}\sum_{k=0}^{n}s^{k(t-1)}\left[\begin{matrix}n-k+t-1\\\
t-1\end{matrix}\right]_{\displaystyle s}f_{k}(q,s)\right)z^{n}\\\
&=\sum_{n=0}^{\infty}\sum_{k=0}^{n}s^{k(t-1)}\left[\begin{matrix}n-k+t-1\\\
t-1\end{matrix}\right]_{\displaystyle s}f_{k}(q,s)(sqz)^{n}.\end{split}$ (24)

Recognizing this as the Cauchy product of two power series in the variable
$sqz$, we may factor the above to get

$F(s,q,z)=\left(\sum_{n=0}^{\infty}s^{n(t-1)}f_{n}(q,s)(sqz)^{n}\right)\left(\sum_{n=0}^{\infty}\left[\begin{matrix}n+t-1\\\
t-1\end{matrix}\right]_{\displaystyle s}(sqz)^{n}\right).$ (25)

The factor on the left is plainly $F(s,q,s^{t}qz)$ after combining powers. The
factor on the right may be interpreted as an instance of the generating
function

$\sum_{\begin{subarray}{c}\lambda\in\mathcal{P},\\\ \ell(\lambda)\leq
t\end{subarray}}\\!(qz)^{\displaystyle\lambda_{1}}s^{\displaystyle|\lambda|}.$
(26)

A simpler identity for this is $1/(sqz;s)_{t}$. Thus

$F(s,q,z)=\frac{1}{(sqz;s)_{t}}F(s,q,s^{t}qz)$ (27)

which is a functional equation that can easily be solved for $F(s,q,z)$.
Applying this relation $N$ times produces

$F(s,q,z)=\prod_{n=0}^{N}\frac{1}{(s^{nt+1}q^{n+1}z;s)_{t}}F(s,q,s^{Nt}q^{N}z).$
(28)

Letting $N\to\infty$ scales the terms of $F(s,q,s^{Nt}q^{N}z)$ by increasingly
large powers of $s^{t}q$. In the limit, we obtain

$F(s,q,z)=\prod_{n=0}^{\infty}\frac{1}{(s^{nt+1}q^{n+1}z;s)_{t}}$ (29)

which proves the $r=1$ case.

From here, assume $r>1$ and define

$F_{r}(s,q,z)=\sum_{\lambda\in\mathcal{P}}z^{\displaystyle\lambda_{1}}s^{\displaystyle|\lambda|}q^{\displaystyle\lambda_{r}+\lambda_{t+r}+\lambda_{2t+r}+\cdots}.$
(30)

For any integer $n\geq 0$ consider the partitions $\lambda$ with
$\lambda_{r}=n$. The $n\times(r-1)$ rectangle of cells above $\lambda_{r}$ is
counted by $F_{r}(s,q,z)$ as $s^{n(r-1)}z^{n}$ and the $\infty\times(r-1)$
rectangle to the right of that may contain any partition with at most $r-1$
parts, counted as

$\sum_{\begin{subarray}{c}\lambda\in\mathcal{P},\\\ \ell(\lambda)\leq
r-1\end{subarray}}\\!\\!\\!\\!\\!\\!z^{\displaystyle\lambda_{1}}s^{\displaystyle|\lambda|},$
(31)

which is simply $1/(sz;s)_{r-1}$. The parts $\lambda_{r}$ and below are
counted by $f_{n}(s,q)$.

$s^{n(r-1)}z^{n}$$f_{n}(s,q)$$1/(sz;s)_{r-1}$ Figure 6: A visualization of
what is counted by $F_{r}(s,q,z)$ for a given $n$.

Summing over $n$ produces

$\begin{split}F_{r}(s,q,z)&=\sum_{n=0}^{\infty}s^{n(r-1)}z^{n}\frac{1}{(sz;s)_{r-1}}f_{n}(s,q)=\frac{1}{(sz;s)_{r-1}}F(s,q,s^{r-1}z)\\\
&=\frac{1}{(sz;s)_{r-1}}\prod_{n=0}^{\infty}\frac{1}{(s^{nt+r}q^{n+1}z;s)_{t}}\end{split}$
(32)

which is the desired result. ∎

## Further work

The research of Schmidt-weighted partitions is still a very new idea in
partition theory, and new results and techniques are appearing rapidly. Much
of the low hanging fruit has already been picked, but there are still many
unexplored ideas.

As for what we considered in this paper, generalizing Mork’s bijection in any
way would be interesting. Maybe the most obvious approach is to first
generalize Bessenrodt’s map, perhaps to $m$-modular diagrams with multiple
allowed remainders. The natural way to this while preserving size is to count
the number of 1s, 2s, 3s, …, $m$s in the first diagonal hook, then 2s, 3s, …,
$m$s, then 3s, …, $m$s, and so on, giving the parts of the resulting
partition, and repeating for each diagonal hook. This will not in general be a
bijection unless the partitions considered are restricted in some way. For
example, the following partitions written as 3-modular diagrams both map to
(5,4,3,1):

boxsize=1.25em 3 & 3 2
3 1
1 3 & 3 1
3 1
2

The color-conjugate map can be used to describe partitions with restrictions
on individual colors, since on the Schmidt-weight side these become the
alternating sums in Theorem 1.8. This can be used to find results similar to
those considered in [andrews_keith]. One could also attempt to modify the
proof of Theorem 11 to work with different Schmidt-weights or to sum over a
different set of partitions.

## References