Multi-state Asymmetric Simple Exclusion Processes

Abstract

It is known that the Markov matrix of the asymmetric simple exclusion process (ASEP) is invariant under the $U_q(sl_2)$ algebra. This is the result of the fact that the Markov matrix of the ASEP coincides with the generator of the Temperley–Lieb (TL) algebra, the dual algebra of the $U_q(sl_2)$ algebra. Various types of algebraic extensions have been considered for the ASEP. In this paper, we considered the multi-state extension of the ASEP, by allowing more than two particles to occupy the same site. We constructed the Markov matrix by dimensionally extending the TL generators and derived explicit forms of particle densities and currents on steady states. Then we showed how decay lengths differ from the original two-state ASEP under closed boundary conditions.

Appendices

Appendix 1: The $U_q(sl_2)$ Algebra

The $U_q(sl_2)$ algebra is generated by $S^{\pm }$ and $q^{S^z}$ which satisfy the following commutation relations:

$$\begin{aligned} q^{S^z} S^{\pm } q^{-S^z} = q^{\pm } S^{\pm }, \qquad [S^+,\, S^-] = \frac{q^{2S^z} - q^{-2S^z}}{q - q^{-1}}. \end{aligned}$$

(87)

As the Hopf algebra, the following comultiplication holds for the $U_q(sl_2)$ generators $X \in \{S^{\pm }, q^{S^z}\}$:

$$\begin{aligned} ({\varvec{1}} \otimes \Delta ) \circ \Delta (X) = (\Delta \otimes {\varvec{1}}) \circ \Delta (X). \end{aligned}$$

(88)

By choosing $\Delta (S^{\pm }) = S^{\pm } \otimes q^{-S^z} + q^{S^z} \otimes S^{\pm }$ and defining $\Delta ^{(N)}$ by

$$\begin{aligned} \Delta ^{(N)}&= (\underbrace{{\varvec{1}} \circ \cdots \circ {\varvec{1}}}_{N-2} \circ \Delta ) \cdots ({\varvec{1}} \circ \Delta ) \Delta , \nonumber \\&= \cdots \nonumber \\&= (\Delta \circ \underbrace{{\varvec{1}} \circ \cdots \circ {\varvec{1}}}_{N-2}) \cdots (\Delta \circ {\varvec{1}}) \Delta , \end{aligned}$$

(89)

we have the spacially extended generators:

$$\begin{aligned}&\Delta ^{(N)}(S^{\pm }) = \sum _{j=1}^N q^{S_1^z} \otimes \cdots \otimes q^{S_{j-1}^z} \otimes S^{\pm }_j \otimes q^{-S_{j+1}^z} \otimes \cdots \otimes q^{-S_N^z},\end{aligned}$$

(90)

$$\begin{aligned}&\Delta ^{(N)}(q^{S^z}) = q^{S_1^z} \otimes \cdots \otimes q^{S_N^z}. \end{aligned}$$

(91)

Appendix 2: Graphical Representations of the TL Algebra

The basis of the TL algebra is known to be described by the link patterns. The link pattern is made by connecting two distinct sites with non-crossing arches. Each of sites is identified with a space of the TL algebra. Link patterns with different shapes are orthogonal to each other, which form the basis of the TL algebra. On this basis, the identity operator just map the original spaces to themselves which is graphically represented as in the left figure of Fig. 8. On the other hand, the TL generator mixes two spaces and then its graphical representation is given by the right one of Fig. 8. Then the TL algebraic relations (10) are graphically given by Fig. 7.

Fig. 7

Graphical representations of the algebraic relations (10). If a circle shows up, a weight $(q+q^{-1})$ is added

As an illustration, we show the action of $e_2$ on the basis of link patterns in the case of $N=6$ (Fig. 9).

Fig. 8

Graphical representations of the identity operator and the TL generator

Fig. 9

The action of $e_2$ on the basis of link patterns with $N=6$. Each shape can be topologically deformed. A weight $(q+q^{-1})$ is added coming from the first relation of (10)

Taking into consideration of the definitions (19) and (20), it is now naturally obtained that the generators $e_i^{(2;1)}$ and $e_i^{(2;2)}$ are graphically represented as in Fig. 10. The projection operator $Y^{(2)}$ is denoted by a red circle.

Fig. 10

The graphical representations of the three-dimensional fused TL generators $e_i^{(2;1)}$ and $e_i^{(2;2)}$. The projection operator $Y^{(2)}$ is denoted by a red circle (Color figure online)

Appendix 3: $SO(3)$ Birman–Murakami–Wenzl Algebra

Here we give algebraic relations between $e_i^{(2;1)}$ and $e_i^{(2;2)}$:

$$\begin{aligned} \left( e_i^{(2;2)}\right) ^2&= (q^2 + 1 + q^{-2}) e_i^{(2;2)},\end{aligned}$$

(92)

$$\begin{aligned} \left( e_i^{(2;1)}\right) ^2&= \frac{q^2 + q^{-2}}{q + q^{-1}} e_i^{(2;1)} + \frac{1}{q^2 + q^{-2}} e_i^{(2;2)},\end{aligned}$$

(93)

$$\begin{aligned} e_i^{(2;1)} e_i^{(2;2)}&= \frac{q^2 + 1 + q^{-2}}{q + q^{-1}} e_i^{(2;2)}, \end{aligned}$$

(94)

and

$$\begin{aligned} e_i^{(2;2)} e_{i+1}^{(2;2)} e_i^{(2;2)} = e_i^{(2;2)},\end{aligned}$$

(95)

$$\begin{aligned} e_i^{(2;1)} e_{i+1}^{(2;2)} e_i^{(2;1)} = e_{i+1}^{(2;1)} e_i^{(2;2)} e_{i+1}^{(2;1)},\end{aligned}$$

(96)

$$\begin{aligned} e_i^{(2;2)} e_{i+1}^{(2;1)} e_i^{(2;2)} = \tfrac{q^2 + 1 + q^{-2}}{q + q^{-1}} e_i^{(2;2)},\end{aligned}$$

(97)

$$\begin{aligned} e_i^{(2;1)} e_{i+1}^{(2;2)} e_i^{(2;2)} = e_{i+1}^{(2;1) e_i^{(2;2)}},\end{aligned}$$

(98)

$$\begin{aligned} e_i^{(2;1)} e_{i+1}^{(2;1)} e_i^{(2;2)} = \tfrac{q^2 + q^{-2}}{q + q^{-1}} e_{i+1}^{(2;1)} e_i^{(2;2)} + \tfrac{1}{(q + q^{-1})^2} e_i^{(2;2)}, \end{aligned}$$

(99)

$$\begin{aligned} e_i^{(2;1)} e_{i+1}^{(2;1)} e_i^{(2;1)}&- e_{i+1}^{(2;1)} e_i^{(2;1)} e_{i+1}^{(2;1)}\nonumber \\&= \tfrac{1}{(q + q^{-1})^2} e_{i+1}^{(2;1)} e_i^{(2;2)} + \tfrac{1}{(q + q^{-1})^2} e_i^{(2;2)} e_{i+1}^{(2;1)} \nonumber \\&\quad + \tfrac{1}{(q + q^{-1})^2} e_i^{(2;1)} - \tfrac{1}{(q + q^{-1})^3} e_i^{(2;2)} - \tfrac{1}{(q + q^{-2})^2} e_i^{(2;1)} e_{i+1}^{(2;2)} \nonumber \\&\quad - \tfrac{1}{(q + q^{-1})^2} e_{i+1}^{(2;2)} e_i^{(2;1)} - \tfrac{1}{(q + q^{-1})^3} e_{i+1}^{(2;2)} + \tfrac{1}{(q + q^{-1})^2} e_{i+1}^{(2;1)}. \end{aligned}$$

(100)

These are known as the $SO(3)$ BMW algebra. This kind of algebraic relations have not been found yet for the fused TL generators with $\ell \ge 3$.

Appendix 4: Matrix Elements of the $e_i^{(\ell ;r)}$ Generator

In this appendix, we compute explicit matrix elements of the $e_i^{(\ell ;r)}$ generator. First of all, we give $e_i^{(\ell ;r)}$ in terms of two-dimensional representations. Let us introduce the following vector notations:

$$\begin{aligned} |m_1,\ldots ,m_r \rangle \rangle = |\underset{1}{0}\, \ldots \, \underset{m_1}{1}\, \underset{\ldots }{\ldots }\, \underset{m_r}{1}\, \ldots \, \underset{2r}{0} \rangle , \end{aligned}$$

(101)

with the definitions (1) and (8). Then, $e_i^{(\ell ;r)}$ is expressed as

$$\begin{aligned} e_i^{(\ell ;r)}&= \underbrace{\varvec{1} \otimes \cdots \otimes \varvec{1}}_{\ell - r} \otimes \sum _{\begin{array}{c} 1 \le m_1 < \cdots < m_r \le 2r \\ 1 \le n_1 < \cdots < n_r \le 2r \end{array}} q^{\sum _{k=1}^r \Theta (\ell -m_k)-\frac{1}{2}} q^{\sum _{k=1}^r \Theta (\ell -n_k)-\frac{1}{2}}\nonumber \\&\quad \cdot \prod _{k=1}^r (1 - \delta _{m_{r-k+1}, 2r-m_k+1})\, |m_1 \ldots m_r \rangle \rangle \langle \langle n_1 \ldots n_r| \otimes \underbrace{\varvec{1} \otimes \cdots \otimes \varvec{1}}_{\ell - r}, \end{aligned}$$

(102)

where

$$\begin{aligned} \delta _{ab} = {\left\{ \begin{array}{ll} 1 &{} a = b \\ 0 &{} a\ne b. \end{array}\right. } \end{aligned}$$

(103)

The $(\ell +1)$-dimensional representation of this generator is obtained by using the two-dimensional vector basis of the $U_q(sl_2)$ algebra. Since the $s$-particle state is expressed in terms of the notation (101) by

$$\begin{aligned} |s \rangle _\mathrm{norm}&= \begin{bmatrix} \ell \\ s \end{bmatrix}^{-1} q^{-\frac{s}{2} (\ell +1)} \sum _{1 \le n_1 < \cdots < n_s \le \ell } q^{\sum _{k=1}^s n_k} |n_1 \ldots n_s \rangle \rangle ,\end{aligned}$$

(104)

$$\begin{aligned} \langle s|&= q^{-\frac{s}{2} (\ell +1)} \sum _{1 \le n_1 < \cdots < n_s \le \ell } q^{\sum _{k=1}^s n_k} \langle \langle n_1 \ldots n_s|, \end{aligned}$$

(105)

we have

$$\begin{aligned}&\langle r_L| \otimes \langle s_L|\; e_i^{(\ell ;r)}\; |r_R \rangle \otimes |s_R \rangle \nonumber \\&\!=\! \begin{bmatrix} \ell \\ r_R \end{bmatrix}^{-1} \begin{bmatrix} \ell \\ s_R \end{bmatrix}^{-1} q^{-\frac{1}{2}(r_L + s_L + r_R + s_R) (\ell +1)} \sum \limits _{\begin{array}{c} 1 \le m_1<\cdots <m_{r_L} \le \ell \\ 1 \le n_1<\cdots <n_{s_L} \le \ell \\ 1 \le m^{\prime }_1<\cdots <m^{\prime }_{r_R} \le \ell \\ 1 \le n^{\prime }_1<\cdots <n^{\prime }_{s_R} \le \ell \end{array}} q^{\sum _{k=1}^{r_L} m_k} q^{\sum _{j=1}^{s_L} n_j} q^{\sum _{k^{\prime }\!=\!1}^{r_R} m^{\prime }_{k^{\prime }}} q^{\sum _{j^{\prime }=1}^{s_R} n^{\prime }_{j^{\prime }}}\nonumber \\&\,\,\cdot \langle \langle m_1 \ldots m_{r_L}| \otimes \langle \langle n_1 \ldots n_{s_L}|\ e_i^{(\ell ;r)} \;|m_1^{\prime } \ldots m_{r_R}^{\prime } \rangle \rangle \otimes |n_1^{\prime } \ldots n_{s_R}^{\prime } \rangle \rangle . \end{aligned}$$

(106)

Here we abbreviated the subscription “norm”. If one sets $m_j$ and $n_j$ in (102) by

$$\begin{aligned}&\varvec{\alpha } = \{m_1+\ell -r,\ldots ,m_r+\ell -r\} = \{a_1,\ldots ,a_r\},\end{aligned}$$

(107)

$$\begin{aligned}&\varvec{\beta } = \{n_1+\ell -r,\ldots ,n_r+\ell -r\} = \{b_1,\ldots ,b_r\}, \end{aligned}$$

(108)

(106) has a non-zero value only when $\varvec{\alpha } \in \{m_1, \ldots , m_{r_L}, n_1+\ell , \ldots , n_{s_L}+\ell \}$ and $\varvec{\beta } \in \{m^{\prime }_1, \ldots , m^{\prime }_{r_R}, n^{\prime }_1+\ell , \ldots , n^{\prime }_{s_R}+\ell \}$. We also introduce the following notations:

$$\begin{aligned}&{\bar{\varvec{\alpha }}} = \{m_1, \ldots , m_{r_L}, n_1+\ell , \ldots , n_{s_L}+\ell \}\; \backslash \; \varvec{\alpha } = \{\bar{a}_1, \ldots , \bar{a}_{r_L+s_L-r}\},\end{aligned}$$

(109)

$$\begin{aligned}&{\bar{\varvec{\beta }}} = \{m^{\prime }_1, \ldots , m^{\prime }_{r_R}, n^{\prime }_1+\ell , \ldots , n^{\prime }_{s_R}+\ell \}\; \backslash \; \varvec{\beta } = \{\bar{b}_1, \ldots , \bar{b}_{r_R+s_R-r}\}. \end{aligned}$$

(110)

Taking into account the following relations:

$$\begin{aligned} \sum _{k=1}^{r_L} m_k&= \sum _{k=1}^{r_L - \mathrm{card}(a_k \le \ell )} \bar{a}_k + \sum _{k=1}^{\mathrm{card}(a_k \le \ell )} a_k,\end{aligned}$$

(111)

$$\begin{aligned} \sum _{j=1}^{s_L} n_j&= \sum _{j=\mathrm{card}(a_k \le \ell )+1}^r (a_j-\ell ) + \sum _{j=\mathrm{card}(a_k > \ell )+1}^{r_L + s_L - r} (\bar{a}_j - \ell ),\end{aligned}$$

(112)

$$\begin{aligned} \sum _{k^{\prime }=1}^{r_R} m^{\prime }_{k^{\prime }}&= \sum _{k=1}^{r_R - \mathrm{card}(b_{j} \le \ell )} \bar{b}_{k} + \sum _{k=1}^{\mathrm{card}(b_{j} \le \ell )+1} b_{k},\end{aligned}$$

(113)

$$\begin{aligned} \sum _{j^{\prime }=1}^{s_R} n^{\prime }_{j^{\prime }}&= \sum _{j=\mathrm{card}(b_{j} > \ell )+1}^r (b_{j}-\ell ) + \sum _{j=\mathrm{card}(b_{j} > \ell )+1}^{r_R + s_R - r} (\bar{b}_{j} - \ell ), \end{aligned}$$

(114)

we have

$$\begin{aligned}&\langle r_L| \otimes \langle s_L|\; e_i^{(\ell ;r)}\; |r_R \rangle \otimes |s_R \rangle \nonumber \\&= \begin{bmatrix} \ell \\ r_R \end{bmatrix}^{-1} \begin{bmatrix} \ell \\ s_R \end{bmatrix}^{-1} \delta _{r_L+s_L, r_R+s_R} q^{-\frac{1}{2} (r_L+s_L+r_R+s_R)(\ell +1)} q^{\sum _{k=1}^r \Theta (\ell -m_k)-\frac{1}{2}} q^{\sum _{k=1}^r \Theta (\ell -n_k)-\frac{1}{2}} \nonumber \\&\quad \cdot \sum \limits _{\begin{array}{c} \ell -r+1 \le a_1 < \cdots < a_r \le \ell +r\\ \ell -r+1 \le b_1 < \cdots < b_r \le \ell +r \end{array}} (-1)^{\mathrm{card}(m_j \ne n_k)} \prod _{k=1}^r (1 - \delta _{a_{r-k+1}-\ell +r, \ell +r-a_k+1}) (1 - \delta _{b_{r-k+1}-\ell +r, \ell +r-b_k+1}) \nonumber \\&\quad \cdot \sum \limits _{\begin{array}{c} 1 \le \bar{a}_1 < \cdots < \bar{a}_{r_L - \mathrm{card}(a_k \le \ell )} \le \ell -r\\ \ell +r+1 \le \bar{a}_{r_L - \mathrm{card}(a_k \le \ell )+1 < \cdots < \bar{a}_{r_L+s_L-r} \le 2\ell } \end{array}} q^{-\sum _{k=1}^{r_L+s_L-r} \bar{a}_k} \sum \limits _{\begin{array}{c} 1 \le \bar{b}_1 < \cdots < \bar{b}_{r_R - \mathrm{card}(b_k \le \ell )} \le \ell -r\\ \ell +r+1 \le \bar{b}_{r_R - \mathrm{card}(b_k \le \ell )+1 < \cdots < \bar{b}_{r_L+s_L-r} \le 2\ell } \end{array}} q^{-\sum _{k=1}^{r_R+s_R-r} \bar{b}_k} \nonumber \\&\quad \cdot \prod _{k=1}^{r_L+s_L-r} \delta _{\bar{a}_k\bar{b}_k} \cdot q^{-\ell \cdot \mathrm{card}(a_k>\ell )} q^{-\ell \cdot \mathrm{card}(\bar{a}_k>\ell )} q^{-\ell \cdot \mathrm{card}(b_k>\ell )} q^{-\ell \cdot \mathrm{card}(\bar{b}_k>\ell )}. \end{aligned}$$

(115)

Especially, non-zero elements of $e_i^{(\ell ;1)}$ are obtained from explicit calculation as

$$\begin{aligned}&\langle r| \otimes \langle s|\; e_i^{(\ell ;1)} \;|\; r \rangle \otimes |s \rangle =q^{-r+s+\ell +1} \frac{[r] [\ell -s]}{[\ell ]^2} + q^{-r+s-\ell -1} \frac{[\ell -r] [s]}{[\ell ]^2} \end{aligned}$$

(116)

$$\begin{aligned}&\quad \langle r+1| \otimes \langle s-1|\; e_i^{(\ell ;1)} \;|r \rangle \otimes |s \rangle = -q^{-r+s-1} \frac{[\ell -r] [s]}{[\ell ^2]} \end{aligned}$$

(117)

$$\begin{aligned}&\langle r-1| \otimes \langle s+1|\; e_i^{(\ell ;1)} \;|r \rangle \otimes |s \rangle =-q^{-r+s+1} \frac{[r] [\ell -s]}{[\ell ^2]} \end{aligned}$$

(118)

$r$ and $s$ run from $0$ to $\ell$ for (116), while they run from $1$ to $\ell -1$ for (117) and (118). The expression (115) naturally leads to the particle-conservation law given by $r_L + s_L = r_R + s_R$.

Appendix 5: Derivation of the Norms

In derivation of the norms of an $n$-particle steady state of the $(\ell +1)$-state ASEP, we use the commutation relation of the $U_q(sl_2)$ generators:

$$\begin{aligned}{}[\Delta (S^+),\, \Delta (S^-)] = \frac{\Delta \left( q^{2S^z}\right) - \Delta \left( q^{-2S^z}\right) }{q - q^{-1}}. \end{aligned}$$

(119)

This relation leads to

$$\begin{aligned} (\Delta (S^+))^n (\Delta (S^-))^n&= (\Delta (S^+))^{n-1} \left\{ \Delta (S^-) \Delta (S^+) + \frac{\Delta (q^{2S^z}) - \Delta (q^{-2S^z})}{q - q^{-1}} \right\} (\Delta (S^-))^{n-1} \nonumber \\&= (\Delta (S^+))^{n-1} (\Delta (S^-))^n \Delta (S^+)\nonumber \\&\quad \!+\! (\Delta (S^+))^{n-1} (\Delta (S^-))^{n-1} \sum _{j = 1}^n \frac{q^{-2(n-j)} \Delta (q^{2S^z}) \!-\! q^{2(n-j)} \Delta (q^{-2S^z})}{q \!-\! q^{-1}}. \end{aligned}$$

(120)

Using the fact that $\Delta (S^+) |0\rangle = 0$ and $\Delta (q^{\pm 2S^z}) | 0\rangle = q^{\pm N}$, we obtain the following recursion relation:

$$\begin{aligned}&\langle 0| (\Delta ^{(\ell N)}(S^+))^n (\Delta ^{(\ell N)}(S^-))^n |0 \rangle \nonumber \\&\quad = \sum _{j = 1}^n \frac{q^{\ell N-2(n-j)} - q^{-\ell N+2(n-j)}}{q - q^{-1}} \langle 0| (\Delta ^{(\ell N)}(S^+))^{n-1} (\Delta ^{(\ell N)}(S^-))^{n-1} |0 \rangle . \end{aligned}$$

(121)

Taking into account that the initial condition:

$$\begin{aligned} \langle 0| \Delta ^{(\ell N)}(S^+) \Delta ^{(\ell N)}(S^-) |0 \rangle = \frac{q^{\ell N} - q^{-\ell N}}{q - q^{-1}}, \end{aligned}$$

(122)

we obtain the expression (54) for the norm.

Appendix 6: Correlation Functions

We introduce a particle-counting operator defined on a two-dimensional vector space:

$$\begin{aligned} n_j = S_j^+ S_j^- = \begin{pmatrix} 0 &{} \\ &{} 1 \end{pmatrix}_j, \qquad 1 - n_j = S_j^- S_j^+ = \begin{pmatrix} 1 &{} \\ &{} 0 \end{pmatrix}_j. \end{aligned}$$

(123)

In the case of the two-state ASEP, important physical quantities such as particle densities and particle currents are expressed by $n_j$.

For instance, an $l$-point correlation function of the two-state ASEP is written by means of particle-counting operators in the following way:

$$\begin{aligned} \langle n|U n_{x_1} n_{x_2} \ldots n_{x_l} U^{-1}|n \rangle _\mathrm{norm} = \langle n|n_{x_1} n_{x_2} \ldots n_{x_l}|n \rangle _\mathrm{norm}. \end{aligned}$$

(124)

Useful formulae have been obtained in [49]: The one-point correlation functions is given by

$$\begin{aligned} \langle n|U^{-1} n_x U|n \rangle = \langle n| n_x |n \rangle = \begin{bmatrix} N \\ n \end{bmatrix}^{-1} \sum _{k=0}^{n-1} (-1)^{n-k+1} q^{-(n-k) (N+1-2x)} \begin{bmatrix} N \\ k \end{bmatrix}, \end{aligned}$$

(125)

which was derived using the following relations:

$$\begin{aligned} S^+_x |n\rangle \!=\! q^{(-N-1+2x)/2} (1-n_x) |n-1 \rangle , \qquad \langle n| S^-_x = q^{(-N-1+2x)/2} \langle n-1| (1-n_x). \end{aligned}$$

(126)

The relations (126) lead to a recursion relation for an $l$-point correlation function with respect to $n$:

$$\begin{aligned} \langle n | n_{x_1} \ldots n_{x_l} | n \rangle _\mathrm{norm} = \frac{[n] q^{- N - 1 + 2x}}{[N - n + 1]} \langle n-1 | n_{x_1} \ldots n_{x_l-1} (1 - n_{x_l}) | n-1 \rangle _\mathrm{norm}. \end{aligned}$$

(127)

In contrast to the recursion relation (127), by which one needs to compute correlation functions in basis of different particle-sectors, we found another recursion relation which does not change the number of particles:

Proposition 10

An $l$-point function is decomposed into one-point functions:

$$\begin{aligned} \langle n| n_{x_1} n_{x_2} \cdots n_{x_{l}} | n\rangle _\mathrm{norm} = \sum _{j=1}^{l} \prod \limits _{\begin{array}{c} k=1 \\ k\ne j \end{array}}^{\ell } \frac{q^{(x_k - x_j)}}{q^{(x_k - x_j)} - q^{-(x_k - x_j)}} \cdot \langle n| n_{x_j} |n \rangle _\mathrm{norm}. \end{aligned}$$

(128)

Proof

The proof is given by an induction on $l$. Before starting the proof, let us remark the following lemma: $\square$

Lemma 3

A two-point function is decomposed into one-point functions:

$$\begin{aligned} (q^{x_2-x_1}-q^{-(x_2-x_1)}) \cdot \langle n| n_{x_1} n_{x_2} |n \rangle _\mathrm{norm} = q^{x_2-x_1} \cdot \langle n| n_{x_1} |n \rangle _\mathrm{norm} - q^{-(x_2-x_1)} \cdot \langle n| n_{x_2} |n \rangle _\mathrm{norm}. \end{aligned}$$

(129)

Proof

This lemma can be proved by considering two expressions of the following function;

$$\begin{aligned} \langle n-1| (1 - n_{x_1}) (1 - n_{x_2}) |n-1 \rangle _\mathrm{norm}. \end{aligned}$$

(130)

First applying the formula (127) to the operator $(1-n_{x_2})$, one obtains

$$\begin{aligned}&\langle n-1| ( 1 - n_{x_1} )( 1 - n_{x_2} ) |n-1 \rangle _\mathrm{norm} \nonumber \\&\quad = \langle n-1| (1 - n_{x_2}) |n-1 \rangle _\mathrm{norm} - \langle n-1| n_{x_1} (1 - n_{x_2}) |n-1 \rangle _\mathrm{norm} \nonumber \\&\quad = \frac{[N-n+1]}{[n] q^{-N-1+2x_2}} \cdot \langle n| (n_{x_2} - n_{x_1} n_{x_2}) |n \rangle _\mathrm{norm}, \end{aligned}$$

(131)

and then applying to the operator $(1-n_{x_1})$, one has

$$\begin{aligned}&\langle n-1| ( 1 - n_{x_1} )( 1 - n_{x_2} ) |n-1 \rangle _\mathrm{norm} \nonumber \\&\quad = \langle n-1| (1 - n_{x_1}) |n-1 \rangle _\mathrm{norm} - \langle n-1| n_{x_2} (1 - n_{x_1}) |n-1 \rangle _\mathrm{norm} \nonumber \\&\quad = \frac{[N-n+1]}{[n] q^{-N-1+2x_1}} \cdot \langle n| (n_{x_1} - n_{x_1} n_{x_2}) |n \rangle _\mathrm{norm}. \end{aligned}$$

(132)

From (131) and (132), decomposition of two-point functions (129) is obtained. $\square$

Assume the relation (128) holds for an $l$-point function. Then an $(l + 1)$-point function is evaluated as

$$\begin{aligned} \langle n| n_{x_1} n_{x_2} \cdots n_{x_{l}} n_{x_{l + 1}} |n \rangle _\mathrm{norm}&= \sum _{j=1}^{l} \prod \limits _{\begin{array}{c} k=1 \\ k\ne j \end{array}}^{l} \frac{q^{x_k - x_j}}{q^{x_k - x_j} - q^{-(x_k - x_j)}} \cdot \langle n| n_{n_{x_j}} n_{x_{l+1}} |n \rangle _\mathrm{norm}. \end{aligned}$$

(133)

Using the decomposition formula of two-point functions into one-point functions (129) to the right-hand side, we obtain the relation (128) for an $(l+1)$-point function. $\square$

Substituting the expression for the one-point function (125) into (128), one obtains the expression for the $l$-point function.

Appendix 7: Currents in Terms of the Particle-Counting Operators

Here we give useful expressions for the seven types of expectation values in (74) in terms of particle-counting operators.

$$\begin{aligned}&\langle 2;n| \begin{pmatrix} 0&{}&{} \\ &{}1&{} \\ &{}&{}0 \end{pmatrix}_{x} \begin{pmatrix} 1&{}&{} \\ &{}0&{} \\ &{}&{}0 \end{pmatrix}_{x+1} |2;n \rangle _\mathrm{norm} \nonumber \\&= -\, \tfrac{q^{-5} (q+q^{-1})}{(q-q^{-1}) (q^2-q^{-2}) (q^3-q^{-3})} \langle n| n_{2x-1} |n \rangle _\mathrm{norm} - \tfrac{q^{-3} (q+q^{-1})}{(q^{-1}-q) (q-q^{-1}) (q^2-q^{-2})} \langle n| n_{2x} |n \rangle _\mathrm{norm} \nonumber \\& - \tfrac{q^{-1} (q+q^{-1})}{(q^{-2}-q^2) (q^{-1}-q) (q-q^{-1})} \langle n| n_{2x+1} |n \rangle _\mathrm{norm} - \tfrac{q (q+q^{-1})}{(q^{-3}-q^3) (q^{-2}-q^2) (q^{-1}-q)} \langle n| n_{2x+2} |n \rangle _\mathrm{norm},\nonumber \\&\langle 2;n| \begin{pmatrix} 0&{}&{} \\ &{}0&{} \\ &{}&{}1 \end{pmatrix}_{x} \begin{pmatrix} 0&{}&{} \\ &{}1&{} \\ &{}&{}0 \end{pmatrix}_{x+1} |2;n \rangle _\mathrm{norm} \nonumber \\&= -\,\tfrac{q (q+q^{-1})}{(q-q^{-1}) (q^2-q^{-2}) (q^3-q^{-3})} \langle n| n_{2x-1} |n \rangle _\mathrm{norm} - \tfrac{q^{-1} (q+q^{-1})}{(q^{-1}-q) (q-q^{-1}) (q^2-q^{-2})} \langle n| n_{2x} |n \rangle _\mathrm{norm}\nonumber \\& - \tfrac{q^{-3} (q+q^{-1})}{(q^{-2}-q^2) (q^{-1}-q) (q-q^{-1})} \langle n| n_{2x+1} |n \rangle _\mathrm{norm} - \tfrac{q^{-5} (q+q^{-1})}{(q^{-3}-q^3) (q^{-2}-q^2) (q^{-1}-q)} \langle n| n_{2x+2} |n \rangle _\mathrm{norm},\nonumber \\&\langle 2;n| \begin{pmatrix} 0&{}&{} \\ &{}0&{} \\ &{}&{}1 \end{pmatrix}_{x} \begin{pmatrix} 1&{}&{} \\ &{}0&{} \\ &{}&{}0 \end{pmatrix}_{x+1} |2;n \rangle _\mathrm{norm} \nonumber \\&= -\, \tfrac{q^{-4}}{(q-q^{-1}) (q^2-q^{-2}) (q^3-q^{-3})} \langle n| n_{2x-1} |n \rangle _\mathrm{norm} - \tfrac{q^{-4}}{(q^{-1}-q) (q-q^{-1}) (q^2-q^{-2})} \langle n| n_{2x} |n \rangle _\mathrm{norm}\nonumber \\& - \tfrac{q^{-4}}{(q^{-2}-q^2) (q^{-1}-q) (q-q^{-1})} \langle n| n_{2x+1} |n \rangle _\mathrm{norm} - \tfrac{q^{-4}}{(q^{-3}-q^3) (q^{-2}-q^2) (q^{-1}-q)} \langle n| n_{2x+2} |n \rangle _\mathrm{norm},\nonumber \\&\langle 2;n| \begin{pmatrix} 0&{}&{} \\ &{}1&{} \\ &{}&{}0 \end{pmatrix}_{x} \begin{pmatrix} 0&{}&{} \\ &{}1&{} \\ &{}&{}0 \end{pmatrix}_{x+1} |2;n \rangle _\mathrm{norm} \nonumber \\&= -\,\tfrac{(q+q^{-1})^2}{(q-q^{-1}) (q^2-q^{-2}) (q^3-q^{-3})} \langle n| n_{2x-1} |n \rangle _\mathrm{norm} - \tfrac{(q+q^{-1})^2}{(q^{-1}-q) (q-q^{-1}) (q^2-q^{-2})} \langle n| n_{2x} |n \rangle _\mathrm{norm}\nonumber \\& - \tfrac{(q+q^{-1})^2}{(q^{-2}-q^2) (q^{-1}-q) (q-q^{-1})} \langle n| n_{2x+1} |n \rangle _\mathrm{norm} - \tfrac{(q+q^{-1})^2}{(q^{-3}-q^3) (q^{-2}-q^2) (q^{-1}-q)} \langle n| n_{2x+2} |n \rangle _\mathrm{norm},\nonumber \\&\langle 2;n| \begin{pmatrix} 1&{}&{} \\ &{}0&{} \\ &{}&{}0 \end{pmatrix}_{x} \begin{pmatrix} 0&{}&{} \\ &{}1&{} \\ &{}&{}0 \end{pmatrix}_{x+1} |2;n \rangle _\mathrm{norm} \nonumber \\&= -\,\tfrac{q^{-1} (q+q^{-1})}{(q-q^{-1}) (q^2-q^{-2}) (q^3-q^{-3})} \langle n| n_{2x-1} |n \rangle _\mathrm{norm} - \tfrac{q (q+q^{-1})}{(q^{-1}-q) (q-q^{-1}) (q^2-q^{-2})} \langle n| n_{2x} |n \rangle _\mathrm{norm}\nonumber \\& - \tfrac{q^3 (q+q^{-1})}{(q^{-2}-q^2) (q^{-1}-q) (q-q^{-1})} \langle n| n_{2x+1} |n \rangle _\mathrm{norm} - \tfrac{q^5 (q+q^{-1})}{(q^{-3}-q^3) (q^{-2}-q^2) (q^{-1}-q)} \langle n| n_{2x+2} |n \rangle _\mathrm{norm},\nonumber \\&\langle 2;n| \begin{pmatrix} 0&{}&{} \\ &{}1&{} \\ &{}&{}0 \end{pmatrix}_{x} \begin{pmatrix} 0&{}&{} \\ &{}0&{} \\ &{}&{}1 \end{pmatrix}_{x+1} |2;n \rangle _\mathrm{norm}\nonumber \\&= -\, \tfrac{q^5 (q+q^{-1})}{(q-q^{-1}) (q^2-q^{-2}) (q^3-q^{-3})} \langle n| n_{2x-1} |n \rangle _\mathrm{norm} - \tfrac{q^3 (q+q^{-1})}{(q^{-1}-q) (q-q^{-1}) (q^2-q^{-2})} \langle n| n_{2x} |n \rangle _\mathrm{norm}\nonumber \\& - \tfrac{q (q+q^{-1})}{(q^{-2}-q^2) (q^{-1}-q) (q-q^{-1})} \langle n| n_{2x+1} |n \rangle _\mathrm{norm} - \tfrac{q^{-1} (q+q^{-1})}{(q^{-3}-q^3) (q^{-2}-q^2) (q^{-1}-q)} \langle n| n_{2x+2} |n \rangle _\mathrm{norm}, \nonumber \\&\langle 2;n| \begin{pmatrix} 1&{}&{} \\ &{}0&{} \\ &{}&{}0 \end{pmatrix}_{x} \begin{pmatrix} 0&{}&{} \\ &{}0&{} \\ &{}&{}1 \end{pmatrix}_{x+1} |2;n \rangle _\mathrm{norm}\nonumber \\&= -\, \tfrac{q^4}{(q-q^{-1}) (q^2-q^{-2}) (q^3-q^{-3})} \langle n| n_{2x-1} |n \rangle _\mathrm{norm} - \tfrac{q^4}{(q^{-1}-q) (q-q^{-1}) (q^2-q^{-2})} \langle n| n_{2x} |n \rangle _\mathrm{norm}\nonumber \\& - \tfrac{q^4}{(q^{-2}-q^2) (q^{-1}-q) (q-q^{-1})} \langle n| n_{2x+1} |n \rangle _\mathrm{norm} - \tfrac{q^4}{(q^{-3}-q^3) (q^{-2}-q^2) (q^{-1}-q)} \langle n| n_{2x+2} |n \rangle _\mathrm{norm}. \end{aligned}$$

(134)