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This section provides additional information about magnon-holon representation. We will cover here details of the transformation to magnon-holon basis, including:

• sublattice-wise transformation of standard operators,
• standard $t$-$J$ Hamiltonian, its magnon-holon representation and generalization to $t$-$J_{xx}$-$J_{z}(\lambda)$ model,
• quantities preserved by the generalized Hamiltonian and commutation relation with translation operator,
• definition and numerical representation of momentum states and representative states in magnon-holon basis,
• example derivations for operators action on magnon-holon representative basis states.

Let us start by denoting annihilation operators for magnon (hard-core boson) as $\hat{a}$ and holon (spinless fermion) as $\hat{h}$. The hard-core nature of the boson comes from $s = 1/2$ spin in the considered model. What we call the magnon-holon transformation is the procedure of rewriting operators in terms of $\hat{a}$ and $\hat{h}$ operators. Accordingly, we say that operator $\hat{O}$ expressed in terms of $\hat{a}$ and $\hat{h}$ is in magnon-holon representation (or in magnon-holon basis). Considering magnons as excitations around a perfect antiferromagnetic spin background, the below set of equations defines the magnon-holon representation of the standard operators in the tJMagnonHolon. We assume the system is one-dimensional with $L = 2N$ sites, 0-based indexed, and $N$ is a positive integer.

On sublattice A (sites with even indices $\{0, 2, 4, ..., L - 2\}$):

\[\begin{aligned} \hat{\tilde{c}}_{i\uparrow}^\dag &= \hat{P}_i \hat{h}_i, &\quad \hat{\tilde{c}}_{i\uparrow} &= \hat{h}_i^\dag \hat{P}_i, \\ \hat{\tilde{c}}_{i\downarrow}^\dag &= \hat{a}_i^\dag \hat{P}_i \hat{h}_i, &\quad \hat{\tilde{c}}_{i\downarrow} &= \hat{h}_i^\dag \hat{P}_i \hat{a}_i, \\ \hat{S}_i^+ &= \hat{h}_i \hat{h}_i^\dag \hat{a}_i, &\quad \hat{S}_i^z &= \left(\frac{1}{2} - \hat{a}_i^\dag \hat{a}_i \right) \hat{h}_i \hat{h}_i^\dag, \\ \hat{S}_i^- &= \hat{a}_i^\dag \hat{h}_i \hat{h}_i^\dag, &\quad \hat{\tilde{n}}_i &= 1 - \hat{h}_i^\dag \hat{h}_i = \hat{h}_i \hat{h}_i^\dag. \end{aligned}\]

On sublattice B (sites with odd indices $\{1, 2, 5, ..., L - 1\}$):

\[\begin{aligned} \hat{\tilde{c}}_{i\uparrow}^\dag &= \hat{a}_i^\dag \hat{P}_i \hat{h}_i, &\quad \hat{\tilde{c}}_{i\uparrow} &= \hat{h}_i^\dag \hat{P}_i \hat{a}_i, \\ \hat{\tilde{c}}_{i\downarrow}^\dag &= \hat{P}_i \hat{h}_i, &\quad \hat{\tilde{c}}_{i\downarrow} &= \hat{h}_i^\dag \hat{P}_i, \\ \hat{S}_i^+ &= \hat{a}_i^\dag \hat{h}_i \hat{h}_i^\dag, &\quad \hat{S}_i^z &= \left(\hat{a}_i^\dag \hat{a}_i - \frac{1}{2}\right) \hat{h}_i \hat{h}_i^\dag, \\ \hat{S}_i^- &= \hat{h}_i \hat{h}_i^\dag \hat{a}_i, &\quad \hat{\tilde{n}}_i &= 1 - \hat{h}_i^\dag \hat{h}_i = \hat{h}_i \hat{h}_i^\dag. \end{aligned}\]

On both sublattices, $\hat{P}_i = 1 - \hat{a}_i^\dag \hat{a}_i$.

With the above definition we can take the standard $t$-$J$ Hamiltonian formulation,

\[\hat{H} = -t \sum_{\langle i,j \rangle, \sigma} \left( \hat{\tilde{c}}_{i\sigma}^{\dag} \hat{\tilde{c}}_{j\sigma} + \mathrm{H.c.} \right) + J \sum_{\langle i,j \rangle} \left( \hat{S}^x_i \hat{S}^x_j + \hat{S}^y_i \hat{S}^y_j + \hat{S}^z_i \hat{S}^z_j - \frac{1}{4} \hat{\tilde{n}}_i \hat{\tilde{n}}_j \right),\]

and rewrite it to magnon-holon basis,

\[\hat{H} = \hat{H}_t + \hat{H}_{J},\]

\[\hat{H}_t = t \sum_{\langle i,j \rangle} \hat{P}_i \left( \hat{h}_i^\dag \hat{h}_j \hat{a}_i + \hat{h}_i^\dag \hat{h}_j \hat{a}_j^\dag \right) \hat{P}_j + \textrm{H.c.},\]

\[\hat{H}_{J} = \frac{J}{2} \sum_{\langle i,j \rangle} \hat{h}_i \hat{h}_i^\dag \left( \hat{a}_i \hat{a}_j + \hat{a}_i^\dag \hat{a}_j^\dag + \hat{a}_i^\dag \hat{a}_i + \hat{a}_j^\dag \hat{a}_j - 2 \hat{a}_i^\dag \hat{a}_i \hat{a}_j^\dag \hat{a}_j - 1 \right) \hat{h}_j \hat{h}_j^\dag.\]

We extend the above model by introducing two additional parameters:

• $\textcolor{orange}{\alpha}$ - anisotropy between quantization z-axis and perpendicular axes (XXZ anisotropy),
• $\textcolor{orange}{\lambda}$ - scaling parameter for interactions between neighboring magnons (magnon-magnon interaction).

\[\hat{H} = \hat{H}_t + \hat{H}_{xx} + \hat{H}_z\]

\[\hat{H}_t = t \sum_{\langle i,j \rangle} \hat{P}_i \left( \hat{h}_i^\dag \hat{h}_j \hat{a}_i + \hat{h}_i^\dag \hat{h}_j \hat{a}_j^\dag \right) \hat{P}_j + \textrm{H.c.},\]

\[\hat{H}_{xx} = \frac{\textcolor{orange}{\alpha} J}{2} \sum_{\langle i,j \rangle} \hat{h}_i \hat{h}_i^\dag \left( \hat{a}_i \hat{a}_j + \hat{a}_i^\dag \hat{a}_j^\dag \right) \hat{h}_j \hat{h}_j^\dag\]

\[\hat{H}_{z} = \frac{J}{2} \sum_{\langle i,j \rangle} \hat{h}_i \hat{h}_i^\dag \left(\hat{a}_i^\dag \hat{a}_i + \hat{a}_j^\dag \hat{a}_j - 2 \textcolor{orange}{\lambda} \hat{a}_i^\dag \hat{a}_i \hat{a}_j^\dag \hat{a}_j - 1 \right) \hat{h}_j \hat{h}_j^\dag\]

For numerical diagonalization of an abstract model like the one above, it is important to determine its symmetries. Those symmetries generate quantities that are conserved by the model. When writing the matrix of the model Hamiltonian in a basis, the basis may incorporate those symmetries, i.e. the basis states can be taken as eigen-states of the symmetry operators. Eigen-states that correspond to different eigenvalues of the symmetry operator are orthogonal. Thus, Hamiltonian written in such basis will consist of orthogonal blocks that can be enumerated by the eigenvalues of the symmetry operators used for the basis construction. Then, each block can be diagonalized separately, decreasing the amount of memory needed to store the Hamiltonian matrix and resulting eigen-states. But what constitutes a symmetry of our model? The word symmetry may be misleading in this case, since the symmetry does not have to reflect only geometrical properties. What we really look for when asking about the symmetries of the Hamiltonian, are operators that commute with the Hamiltonian,

\[[\hat{H}, \hat{O}] = 0.\]

At the beginning, let us observe that the size $L$ of the system is conserved by the model Hamiltonian.

Next, we can observe that our Hamiltonian does not change the number of electrons (holes). The corresponding operator is $\hat{N} = \sum_i \hat{\tilde{n}}_i$. One can check that,

\[[\hat{H}, \hat{N}] = 0.\]

The eigenvalues of $\hat{N}$ are simply $N_e \in \{0, 1, 2, ..., L\}$. This operator does not influence the system size either. Thus, subspace for given $L$ can be split into further subspaces with different electron occupancy $N_e$.

In similar way, once we set a system size $L$ and number of electrons $N_e$, such subspace can be split into subspaces with different magnetization $S^z_{\mathrm{tot}}$. The corresponding operator is $\hat{S}^z_{\mathrm{tot}} = \sum_i \hat{S}^z_i$. It indeed commutes with the Hamiltonian and does not break considered so far symmetries (it commutes with their operators too),

\[[\hat{H}, \hat{S}^z_{\mathrm{tot}}] = 0,\]

\[[\hat{N}, \hat{S}^z_{\mathrm{tot}}] = 0.\]

The standard Hamiltonian (and other previously mentioned operators) commute also with translation operator $\hat{T}$. But it is not true for the generalized Hamiltonian. Generalized model commutes with $\hat{T}$ only for $\lambda = 1$. This is related to distinguished sublattices in the transformation to magnon-holon basis. At the same time, the square of the translation operator, $\hat{D} = \hat{T}^2$ commutes with the generalized Hamiltonian for all values of $\lambda$,

\[[\hat{H}, \hat{D}] = 0.\]

We will call $\hat{D}$ an even translation operator. It of course commute with all the operators that $\hat{T}$ commutes with.

All the above symmetries are incorporated in the tJMagnonHolon. But there is still a room for improvements. For example, parity symmetry is another symmetry that can be taken into account. It is important to remember that additional symmetries will make the basis in itself more complex. This can make the obtained results difficult to analyze or interpret. Thus stopping at the translation operator is optimal. Allowed system sizes are only few sites smaller than maximum for available computers. At the same time, basis construction is still easy to understand.

Let us now construct the basis for our problem. Our requirements are that all the basis states are simultaneously eigen-states of all the mentioned symmetry operators: system size, number of electrons $\hat{N}$, magnetization $\hat{S}^z_{\mathrm{tot}}$ and even translation $\hat{D}$. Let us start with representation that will automatically include the first three of them. We will use the following notation:

• system size: $L = 2N, N \in \mathbb{N}$,
• number of electrons: $N_e \in \{0, 1, ..., L\}$,
• number of spins up: $n_\uparrow \in \{0, 1, ..., N_e\}$.

Notice that for given $N_e$ the magnetization $S^z_{\mathrm{tot}}$ is uniquely defined by $n_\uparrow$,

\[ S^z_{\mathrm{tot}} \propto n_\uparrow - n_\downarrow = n_\uparrow - (N_e - n_\uparrow) = 2n_\uparrow - N_e,\]

where $n_\downarrow$ stands for number of spins down in the system.

We will use two 64-bit numbers to represent configuration of electrons and spins. This will set a limit of 64 to available system sizes, but apart from trivial cases we won't be able to calculate more than around 20-30 sites anyway. Let us name the two numbers as charges and spins respectively. We will use $L$ first bits of each of the number to represent presence (absence) of electrons and spins up. Our rules are simple. We connect bit values of $\{0, 1\}$ with occupancy.

• for charges:
  • 1 at $i$-th bit $\to$ electron at $i$-th site,
  • 0 at $i$-th bit $\to$ hole at $i$-th site,
• for spins:
  • 1 at $i$-th bit $\to$ spin up at $i$-th site,
  • 0 at $i$-th bit $\to$ spin down $i$-th site.

Since bits are indexed from 0, we use the same indexing for sites. It may not be the most intuitive choice for physicists, but it is the most practical one (and natural for most programmers). It is also important to specify that only electron can carry a spin (holons are spinless fermions). To disambiguate the definition of the site occupied by a hole, we always set the bit of the spin variable to $0$ if corresponding bit of charges variable is $0$. Therefore, only 3 out of 4 combinations are meaningful for us ($i$-th bit of charges, $i$-th bit of spins) $\in\{(0,0), (1,0), (1,1)\}$.

The above representation determines physical states uniquely for bosonic systems. But electrons are fermions, and therefore the order of operators matters. It means the above basis determines states only with respect to a phase shift by $\pi$. To disambiguate the phase multipliers, we have to connect each configuration of charges and spins with one chosen order of fermionic operators. To make it simple, we take a standard approach and order operators with respect to the site index they act on.

\[\begin{aligned} \hat{c}^\dag_{0\sigma_0} \hat{c}^\dag_{1\sigma_1} \hat{c}^\dag_{2\sigma_2} ... \hat{c}^\dag_{n\sigma_n} \vert \varnothing \rangle = \vert \sigma_0 \sigma_1 \sigma_2 ... \sigma_n \rangle \end{aligned}\]

Before we follow with translation operator, let us transform the above representation to magnon-holon basis. We look at the details of the transformation to magnon-holon basis (operators defined at the beginning), and see that the transformation does not alter the electron/hole configuration. We therefore use the above defined charges without changes. (Of course one can swap the meaning of $0$ and $1$ bit values, but using $0$ for hole seems more intuitive). On the other hand, we have no spins in magnon-holon basis, but magnons instead. Let us define magnons - number that represents configuration of magnons in the system. On odd sites, spins up directly transform to magnons. If we chose $1$s to represent presence of magnons then there's nothing to do there. On even sites on the other hand, spins down transform to magnons, so we have to transform all zeros to ones and all ones to zeros on even sites. We call this last step a rotation of sublattice A (sublattice of even-indexed sites assuming 0-based indexing). There is of course a deeper meaning behind calling it a rotation, but it won't be important here.

After the rotation of sublattice A is performed, we land at the following representation,

• for charges:
  • 1 at $i$-th bit $\to$ electron at $i$-th site,
  • 0 at $i$-th bit $\to$ hole at $i$-th site,
• for magnons:
  • 1 at $i$-th bit $\to$ magnon at $i$-th site,
  • 0 at $i$-th bit $\to$ no magnon $i$-th site.

We disambiguate the hole representation in the same way as before (there cannot be both the hole and the magnon at the same site). The above corresponds to the State structure in the tJMagnonHolon code. Since holons are fermions, we again chose the standard order of growing site index to represent 'charges' configuration. For bosons, like magnons, the order does not have to be specified, but we can assume the same scheme.

\[\begin{aligned} \hat{h}^\dag_{0} \hat{h}^\dag_{1} \hat{h}^\dag_{2} ... \hat{h}^\dag_{n} \hat{a}^\dag_{0} \hat{a}^\dag_{1} \hat{a}^\dag_{2} ... \hat{a}^\dag_{n} \vert \varnothing \rangle = \left\vert \begin{matrix} h_0 & h_1 & h_2 & ... & h_n \\ a_0 & a_1 & a_2 & ... & a_n \end{matrix} \right\rangle \end{aligned}\]

Let us now follow with introduction of momentum states - eigen-states of the even translation operator $\hat{D}$. Let $\vert s, L, N_e, n_\uparrow \rangle$ stand for a state with certain configuration of charges and magnons denoted by s for a system with $L = 2N, N \in \mathbb{N}$ sites, $N_e$ electrons, and $n_\uparrow$ spins up. Let us also shortly write $\vert s \rangle$ when exact values of $L, N_e, n_\uparrow$ are not important. Notice that set of all s can be well-ordered since s is represented by a pair of integers. We define the magnon-holon momentum states in a standard way (we just use $\hat{D}$ instead of $\hat{T}$),

\[\vert s(k) \rangle = \frac{1}{\sqrt{N_s}} \sum_{r = 0}^{N - 1} \exp\left(-\frac{2\pi i}{N}kr\right) \hat{D}^r \vert s \rangle.\]

One can see that indeed $\vert s(k) \rangle$ are eigen-states of $\hat{D}$,

\[\hat{D} \vert s(k) \rangle = \exp\left(\frac{2\pi i}{N}k\right) \vert s(k) \rangle,\]

with eigenvalues $\exp\left(\frac{2\pi i}{N}k\right)$, where $k \in \{0, 1, ..., N-1\}$. This $k$ variable enumerates the subspaces generated by even translation operator. When system variable of type System is created, $k$ stands for system.momentum field.

Two more steps are required to form a basis out of the above defined momentum states.

• First: we need to make sure that they have unique representation (for linear independence of basis states).

For this we need to pick a representative state $\vert \tilde{s} \rangle$ out of the set of even translations $\hat{D}^r \vert s \rangle$ of particular configuration $s$. Now the fact that set of all $s$ can be well-ordered comes in handy. We define our ordering relation between $s_1$ and $s_2$. It will allow us to answer e.g. if $s_1$ is greater than $s_2$. For example, we can compare charges of the two configurations, and if charges are the same we compare magnons. With this we can pick the representative state $\vert \tilde{s} \rangle$ such that $\tilde{s}$ it the smallest out of all translations of particular configuration $s$.

• Second: we need to make sure that state $\vert s \rangle$ is compatible with momentum subspace $k \in \{0, ..., N-1\}$.

This is connected to normalizability of momentum states. Let us define periodicity of state $\vert s \rangle$, as the minimal number $R_s > 0$ of even translations that transform $\vert s \rangle$ onto itself,

\[R_s = \min \{ r \in \mathbb{N} \setminus \{0\} : \hat{D}^{r} \vert s \rangle = \xi \vert s \rangle \},\]

where $\xi \in \{-1, 1\}$ represents possible phase shift from anticommutation relations when pushing fermions over the PBC. The compatibility of $\vert s \rangle$ with $k$ requires that for $\xi = 1$,

\[kR_s\mod N = 0,\]

and for $\xi = -1$,

\[(k - \frac{N}{2R_s}) R_s \mod N = 0.\]

If the above is fulfilled, the normalization factor $N_s = \frac{N^2}{R_s}$. If the above is not fulfilled, then $\vert s(k) \rangle \equiv 0$ is not normalizable.

Set of representative states compatible with $k$ form a momentum basis for subspace with given $L, N_e, S^z_{\mathrm{tot}}, k$.

Equipped with the introduced notation, we are ready to derive some expressions for operators action on magnon-holon (momentum) basis states. To shorten the notation, from now on, we write $\exp(-ikr)$ instead of $\exp\left(-\frac{2\pi i}{N}kr\right)$. It should be clear to which momentum subspace we refer even if we use redefined $k$ to enumerate it. Just remember that in the code momentum field of System is an integer from set $\{0, ..., N-1\}$.

We start with the simplest case of $\hat{S}^z_k$ operator,

\[\hat{S}^z_k = \frac{1}{\sqrt{L}} \sum_{r=0}^{L-1} \exp(-ikr) \hat{S}^z_r.\]

We evaluate its action on a magnon-holon momentum basis state $\vert \tilde{s}(p) \rangle = \vert \tilde{s}(p), L, N_e, n_\uparrow \rangle$,

\[\begin{aligned} \hat{S}^z_k \vert \tilde{s}(p) \rangle &= \frac{1}{\sqrt{N_s}} \sum_{r=0}^{N-1} \exp(-ipr) \hat{S}^z_k \hat{D}^r \vert \tilde{s} \rangle = \frac{1}{\sqrt{L N_s}} \sum_{r=0}^{N-1} \sum_{r'=0}^{L-1} \exp(-ipr - ikr') \hat{S}^z_{r'} \hat{D}^r \vert \tilde{s} \rangle \\ &= \frac{1}{\sqrt{L N_s}} \sum_{r,r'} \exp(-ipr -ikr') (-1)^{r'} \left( \frac{1}{2} - \hat{a}^{\dag}_{r'} \hat{a}_{r'} \right) \hat{h}_{r'} \hat{h}^{\dag}_{r'} \hat{D}^r \vert \tilde{s} \rangle \\ &= \frac{1}{\sqrt{L N_s}} \sum_{r,r'} \exp(-ipr -ikr') \hat{D}^r (-1)^{r'} \left( \frac{1}{2} - \hat{a}^{\dag}_{2r + r'} \hat{a}_{2r + r'} \right) \hat{h}_{2r + r'} \hat{h}^{\dag}_{2r + r'} \vert \tilde{s} \rangle \\ &= \frac{1}{\sqrt{L N_s}} \sum_{r,r'} \exp[-i(p - 2k)r -ik(2r + r')] \hat{D}^r (-1)^{r'} \left( \frac{1}{2} - \hat{a}^{\dag}_{2r + r'} \hat{a}_{2r + r'} \right) \hat{h}_{2r + r'} \hat{h}^{\dag}_{2r + r'} \vert \tilde{s} \rangle \\ &= \frac{1}{\sqrt{L N_s}} \sum_{r,R} \exp[-i(p - 2k)r -ikR] \hat{D}^r (-1)^{R - 2r} \left( \frac{1}{2} - \hat{a}^{\dag}_{R} \hat{a}_{R} \right) \hat{h}_{R} \hat{h}^{\dag}_{R} \vert \tilde{s} \rangle \\ &= \frac{1}{\sqrt{L N_s}} \sum_{r,R} \exp[-i(p - 2k)r -ikR] \hat{D}^r (-1)^{R} \left( \frac{1}{2} - \hat{a}^{\dag}_{R} \hat{a}_{R} \right) \hat{h}_{R} \hat{h}^{\dag}_{R} \vert \tilde{s} \rangle \\ &= \frac{1}{\sqrt{L}} \sum_{R=0}^{L-1} \exp(-ikR) \alpha_R(\tilde{s}) \vert \tilde{s}(p-2k) \rangle, \end{aligned}\]

where $(-1)^{R} \left( \frac{1}{2} - \hat{a}_{R} \hat{a}^{\dag}_{R} \right) \hat{h}^{\dag}_{R} \hat{h}_{R} \vert \tilde{s} \rangle = \alpha_R(\tilde{s}) \vert \tilde{s} \rangle$. We provide all transformations step by step to showcase the derivation procedure. We use the definition of the momentum representation (Fourier transform) for both the operator and the state. We follow with the transformation of the operators to magnon-holon basis. Then we commute the even translation operator $\hat{D}$ with other operators. Next 3 lines are probably least transparent. We change the summation order, rendering the action of the operators (other than even translation operator) independent of the Fourier transform to momentum space. Finally, we evaluate the action of operators on the state $\vert \tilde{s} \rangle$, and we write the state back in the momentum space. It is immediately visible that $\alpha_R(\tilde{s})$ evaluates to $0$ when site $R$ in state $\vert \tilde{s} \rangle$ is occupied by a hole. When site $R$ is not occupied by the hole, possible values of $\alpha_R(\tilde{s})$ are given according to the table below,

Using the result for $\hat{S}^z_k \vert \tilde{s}(p) \rangle$, we easily obtain,

\[\hat{S}^z_r \vert \tilde{s}(p) \rangle = \frac{1}{\sqrt{L}} \sum_{q} \exp(iqr) \hat{S}^z_q \vert \tilde{s}(p) \rangle = \frac{1}{L} \sum_{q,R} \exp[-iq(R-r)]\alpha_R(\tilde{s}) \vert \tilde{s} (p-2q) \rangle.\]

In practice, we assert the periodicity-momentum match between state $\vert \tilde{s} \rangle$ and momentum $p-2q$ by excluding from the above sum terms that do not fulfill periodicity-momentum relation.

Let us now work out the ladder operator $\hat{S}^+_k$. Compared to the previous example, it will require few more steps, since it affects the number of spins up in the system.

\[\begin{aligned} \hat{S}^+_k \vert \tilde{s}(p) \rangle &= \frac{1}{\sqrt{N_s}} \sum_{r=0}^{N-1} \exp(-ipr) \hat{S}^+_k \hat{D}^r \vert \tilde{s} \rangle = \frac{1}{\sqrt{L N_s}} \sum_{r=0}^{N-1} \sum_{r'=0}^{L-1} \exp(-ipr - ikr') \hat{S}^+_{r'} \hat{D}^r \vert \tilde{s} \rangle \\ &= \frac{1}{\sqrt{L N_s}} \sum_{r,r'} \exp(-ipr -ikr') (\delta_{r' \in A} \hat{a}_{r'} +\delta_{r' \in B} \hat{a}^{\dag}_{r'} ) \hat{h}_{r'} \hat{h}^{\dag}_{r'} \hat{D}^r \vert \tilde{s} \rangle \\ &= \frac{1}{\sqrt{L N_s}} \sum_{r,r'} \exp(-ipr -ikr') \hat{D}^r (\delta_{r' \in A} \hat{a}_{2r + r'} + \delta_{r' \in B} \hat{a}^{\dag}_{2r + r'} ) \hat{h}_{2r + r'} \hat{h}^{\dag}_{2r + r'} \vert \tilde{s} \rangle \\ &= \frac{1}{\sqrt{L N_s}} \sum_{r,R} \exp[-i(p-2k)r -ikR] \hat{D}^r (\delta_{R-2r \in A} \hat{a}_{R} + \delta_{R-2r \in B} \hat{a}^{\dag}_{R} ) \hat{h}_{R} \hat{h}^{\dag}_{R} \vert \tilde{s} \rangle \\ &= \frac{1}{\sqrt{L N_s}} \sum_{r,R} \exp[-i(p-2k)r -ikR] \hat{D}^r \left( \frac{1+(-1)^R}{2} \hat{a}_{R} + \frac{1-(-1)^R}{2} \hat{a}^{\dag}_{R} \right) \hat{h}_{R} \hat{h}^{\dag}_{R} \vert \tilde{s} \rangle \\ &= \frac{1}{\sqrt{L N_s}} \sum_{r,R} \exp[-i(p-2k)r -ikR] \hat{D}^r \frac{1}{2}(\hat{a}_{R} + \hat{a}^{\dag}_{R}) \hat{h}_{R} \hat{h}^{\dag}_{R} \vert \tilde{s} \rangle \\ &+ \frac{1}{\sqrt{L N_s}} \sum_{r,R} \exp[-i(p-2k)r -ikR] \hat{D}^r \frac{(-1)^R}{2}(\hat{a}_{R} - \hat{a}^{\dag}_{R}) \hat{h}_{R} \hat{h}^{\dag}_{R} \vert \tilde{s} \rangle \\ &= \frac{1}{\sqrt{L N_s}} \sum_{r,R} \exp[-i(p-2k)r -ikR] \hat{D}^r [\alpha^+_R(\tilde{s}) \vert s^+_R \rangle + \alpha^-_R(\tilde{s}) \vert s^-_R \rangle] \\ &+ \frac{1}{\sqrt{L N_s}} \sum_{r,R} \exp[-i(p-2k)r -ikR] \hat{D}^r [\beta^+_R(\tilde{s}) \vert s^+_R \rangle + \beta^-_R(\tilde{s}) \vert s^-_R \rangle] \\ &= \frac{1}{\sqrt{L N_s}} \sum_{R} \sqrt{N_{s^+_R}} \exp(-ikR) [\alpha^+_R(\tilde{s}) + \beta^+_R(\tilde{s})] \vert s^+_R (p - 2k) \rangle \\ &+ \frac{1}{\sqrt{L N_s}} \sum_{R} \sqrt{N_{s^-_R}} \exp(-ikR) [\alpha^-_R(\tilde{s}) + \beta^-_R(\tilde{s})] \vert s^-_R (p - 2k) \rangle \\ &= \frac{1}{\sqrt{L N_s}} \sum_{R} \sqrt{N_{s^+_R}} \exp[-ikR -i(p-2k)d_{s^+_R}] [\alpha^+_R(\tilde{s}) + \beta^+_R(\tilde{s})] \vert \tilde{s}^+_R (p - 2k) \rangle \\ &+ \frac{1}{\sqrt{L N_s}} \sum_{R} \sqrt{N_{s^-_R}} \exp[-ikR -i(p-2k)d_{s^-_R}] [\alpha^-_R(\tilde{s}) + \beta^-_R(\tilde{s})] \vert \tilde{s}^-_R (p - 2k) \rangle, \end{aligned}\]

with $\frac{1}{2}(\hat{a}_{R} + \hat{a}^{\dag}_{R}) \hat{h}_{R} \hat{h}^{\dag}_{R} \vert \tilde{s} \rangle = [\alpha^+_R(\tilde{s}) \vert s^+_R \rangle + \alpha^-_R(\tilde{s}) \vert s^-_R \rangle]$ and $\frac{(-1)^R}{2}(\hat{a}_{R} - \hat{a}^{\dag}_{R}) \hat{h}_{R} \hat{h}^{\dag}_{R} \vert \tilde{s} \rangle = [\beta^+_R(\tilde{s}) \vert s^+_R \rangle + \beta^-_R(\tilde{s}) \vert s^-_R \rangle]$, where $\vert s^{\pm}_R \rangle \equiv \vert s^{\pm}_R, L, N_e, n_\uparrow \pm 1 \rangle$ comes from flipping a spin at site $R$ in state $\vert \tilde{s} \rangle$. The normalization factors $\sqrt{N_{s^{\pm}_R}}$ appear since new states $\vert s^{\pm}_R \rangle$ may in general have periodicity different from $\vert \tilde{s} \rangle$. These states may also not be representative states, so we need to find their representatives $\vert \tilde{s}^{\pm}_R \rangle$ and include proper phase shifts, as determined by the distance $d_{s^{\pm}_R}$ to the representative. Of course, $\beta^-_R(\tilde{s}) = -\alpha^-_R(\tilde{s})$ meaning that we can only increase the number of spins up (as expected from $\hat{S}^+_k$). On the other hand, $\beta^+_R(\tilde{s}) = \alpha^+_R(\tilde{s})$ with $\alpha^+_R(\tilde{s})$ equal to $0$ when $R$ is occupied by a hole, or otherwise it is given according to the table below.

where we ignored sign changes due to anticommutation relations. Whether the $\alpha^+_R(\tilde{s})$ should be multiplied by an additional $-1$ depends on the order of fermions in state $\vert \tilde{s} \rangle$, position R, the resulting state $\vert s^{\pm}_R \rangle$ after the spin flip, and the distance to its representative. From the above considerations, we see that the code for this operator shall first check if the value of $\alpha^+_R(\tilde{s})$ is non-zero and only proceed if that's the case. Then the operators action on state $\vert \tilde{s} \rangle$ simply alternates the $R$-th bit of magnons variable, which equates to XORing magnons with $2^R$.

As one can expect, derivation for $\hat{S}^-_k$ is analogical, with the same result form and following relations between coefficients,

\[\alpha^-_R(\tilde{s}) = \beta^-_R(\tilde{s}) = \frac{1}{2} - \alpha^+_R(\tilde{s}), \\ \beta^+_R(\tilde{s}) = -\alpha^+_R(\tilde{s}).\]

Also, $\hat{S}^{\pm}_r$ can be obtained the same way as $\hat{S}^z_r$.

Creation and annihilation operators for electrons follow the same scheme as spin ladder operators. For example,

\[\begin{aligned} \hat{\tilde{c}}_{k\downarrow} \vert \tilde{s}(p) \rangle &= \frac{1}{\sqrt{N_s}} \sum_{r=0}^{N-1} \exp(-ipr) \hat{\tilde{c}}_{k\downarrow} \hat{D}^r \vert \tilde{s} \rangle = \frac{1}{\sqrt{L N_s}} \sum_{r=0}^{N-1} \sum_{r'=0}^{L-1} \exp(-ipr - ikr') \hat{\tilde{c}}_{r'\downarrow} \hat{D}^r \vert \tilde{s} \rangle \\ &= \frac{1}{\sqrt{L N_s}} \sum_{r,r'} \exp(-ipr -ikr') (\delta_{r' \in A} \hat{h}^{\dag}_{r'} \hat{P}_{r'} \hat{a}_{r'} + \delta_{r' \in B} \hat{h}^{\dag}_{r'} \hat{P}_{r'}) \hat{D}^r \vert \tilde{s} \rangle \\ &= \frac{1}{\sqrt{L N_s}} \sum_{r,r'} \exp(-ipr -ikr') \hat{D}^r (\delta_{r' \in A} \hat{h}^{\dag}_{2r + r'} \hat{P}_{2r + r'} \hat{a}_{2r + r'} + \delta_{r' \in B} \hat{h}^{\dag}_{2r + r'} \hat{P}_{2r + r'}) \vert \tilde{s} \rangle \\ &= \frac{1}{\sqrt{L N_s}} \sum_{r,R} \exp[-i(p - 2k)r -ikR] \hat{D}^r (\delta_{R-2r \in A} \hat{h}^{\dag}_{R} \hat{P}_{R} \hat{a}_{R} + \delta_{R-2r \in B} \hat{h}^{\dag}_{R} \hat{P}_{R}) \vert \tilde{s} \rangle \\ &= \frac{1}{\sqrt{L N_s}} \sum_{r,R} \exp[-i(p - 2k)r -ikR] \hat{D}^r \frac{1}{2} \hat{h}^{\dag}_{R} \hat{P}_{R} [\hat{a}_{R} + 1 + (-1)^R (\hat{a}_{R} - 1)] \vert \tilde{s} \rangle \\ &= \frac{1}{\sqrt{L N_s}} \sum_{r,R} \exp[-i(p-2k)r -ikR] \hat{D}^r [\alpha^+_R(\tilde{s}) \vert s^+_R \rangle + \alpha^-_R(\tilde{s}) \vert s^-_R \rangle] \\ &+ \frac{1}{\sqrt{L N_s}} \sum_{r,R} \exp[-i(p-2k)r -ikR] \hat{D}^r [\beta^+_R(\tilde{s}) \vert s^+_R \rangle + \beta^-_R(\tilde{s}) \vert s^-_R \rangle] \\ &= \frac{1}{\sqrt{L N_s}} \sum_{R} \sqrt{N_{s^+_R}} \exp[-ikR -i(p-2k)d_{s^+_R}] [\alpha^+_R(\tilde{s}) + \beta^+_R(\tilde{s})] \vert \tilde{s}^+_R (p - 2k) \rangle \\ &+ \frac{1}{\sqrt{L N_s}} \sum_{R} \sqrt{N_{s^-_R}} \exp[-ikR -i(p-2k)d_{s^-_R}] [\alpha^-_R(\tilde{s}) + \beta^-_R(\tilde{s})] \vert \tilde{s}^-_R (p - 2k) \rangle. \end{aligned}\]

where the hole is added at site R in state $\vert s^{\pm}_R \rangle$ resulting in the increased(+)/decreased(-) magnetization with respect to $\vert \tilde{s} \rangle$. The only non-zero contributions come from sites not occupied by the holes. Of course, even then the terms for decreased magnetization state must vanish, and indeed we have $\beta^-_R(\tilde{s}) = -\alpha^-_R(\tilde{s})$. In general, the coefficients in the above result follow the same relations as for $\hat{S}^+_k$. On the other hand, for creation operator $\hat{\tilde{c}}^{\dag}_{k\downarrow}$, only sites occupied by a hole give contribution. Then we are guaranteed that there are no magnons and in such case, $\beta^+_R(\tilde{s}) = -\alpha^+_R(\tilde{s})$ and simply $\beta^-_R(\tilde{s}) = \alpha^-_R(\tilde{s}) = \frac{1}{2}$.

In the end, let us discuss operators with more than one index. Such operators may appear when calculating n-particle correlation functions. For example, let $\hat{O}_{k,q} = \hat{\tilde{c}}_{k\uparrow}\hat{\tilde{c}}_{q\downarrow}$. Applying previous results, we obtain,

\[\begin{aligned} \hat{O}_{k,q} \vert \tilde{s}(p) \rangle &= \hat{\tilde{c}}_{k\uparrow} \sum_{R} \sqrt{\frac{N_{s^+_R}}{L N_s}} \exp[-iqR -i(p-2q)d_{s^+_R}] [\alpha^+_R(\tilde{s}) + \beta^+_R(\tilde{s})] \vert \tilde{s}^+_R (p - 2q) \rangle \\ &= \sum_{R,R'} \sqrt{\frac{N_{s^{+-}_{R,R'}}}{L^2 N_s}} \exp[-iqR -i(p-2q)d_{s^+_R}] \exp[-ikR' -i(p-2q-2k)d_{s^{+-}_{R,R'}}] \times \\ &\times [\alpha^+_R(\tilde{s}) + \beta^+_R(\tilde{s})] [\alpha^-_{R'}(\tilde{s}^+_R) + \beta^-_{R'}(\tilde{s}^+_R)] \vert \tilde{s}^{+-}_{R,R'} (p - 2q - 2k) \rangle. \end{aligned}\]

It is important to mention that periodicity-momentum correspondence has to be checked for all the intermediate states, not only for the final state. This means that the sum over $R'$ only makes sense if state $\vert \tilde{s}^+_R \rangle$ is compatible with momentum $p-2q$. Otherwise, one may introduce false non-zero contributions to the final result. Other than that, there are no surprises compared to previous examples.

This summarizes our derivations. If you plan to introduce your own custom operators, read through Custom Operators subsection of the Guide. You will learn there how the operators functions are designed in tJMagnonHolon. Then you may want to compare presented here derivations, with corresponding operators functions in the operators.jl file in .../tJMagnonHolon/src/modules/.