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2 Introduction

The Matrix Product Ansatz [DEHP93] is a method for finding stationary measures in an interacting particle system with open boundary conditions. Here, we relate the Matrix Product Ansatz to quantum groups; much of this exposition was motivated by [CRV14].

Let the generator be written in the form \[ B_1 + \sum _{l=1}^{L-1} w_{l,l+1} + \bar {B}_L \] where we use the subscript notation and \(L\) is the number of lattice sites. In other words, \(w\) is a local two–site generator, while \(B\) acts on the left–most lattice site, and \(\bar {B}\) acts on the right–most lattice site.

Denote a particle configuration on \(L\) sites by \[ \mathcal {C} = (\tau _1,\ldots ,\tau _L), \] where each \(\tau _i\) is a particle variable equal to either 0 or 1, indicating the number of particles at site \(i\). The Ansatz takes the form \[ \mathcal {S}(\mathcal {C}) = \frac {1}{Z_L} \langle W | \prod _{i=1}^L ((1-\tau _i)E + \tau _i D) | V \rangle \] where \(E\) and \(D\) are elements of a non–commutative algebra, and the product is ordered from the index \(i=1\) on the left to \(i=L\) on the right.

In [KS97], it was proven that there is a solution of the form \begin{equation}\label {Bulk} w^T\left [ \binom {E}{D} \otimes \binom {E}{D} \right ] = \binom {E}{D} \otimes \binom {\bar {E}}{\bar {D}} - \binom {\bar {E}}{\bar {D}} \otimes \binom {E}{D}, \end{equation} and \begin{equation}\label {Boundary} \langle W | B^T \binom {E}{D} = \langle W | \binom {\bar {E}}{\bar {D}}, \quad \bar {B}^T | \binom {E}{D} \rangle = - \binom {\bar {E}}{\bar {D}} \vert V \rangle \end{equation} where \(\bar {E}\) and \(\bar {D}\) are new non–commuting elements, and the superscript \( ^T\) denotes transposition. We call these two equations the bulk relations and the boundary relations, respectively. The transposition is there to match terminology between probability papers and mathematical physics papers, since the latter often defines stochastic matrices as having columns summing to 1, rather than rows.

In the case of ASEP, we explicitly take the matrices as \[ B= \left ( \begin{array}{cc} -\alpha & \alpha \\ \gamma & -\gamma \end{array} \right ), \quad w = \left ( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & -q & q & 0 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right ), \quad \bar {B} = \left ( \begin{array}{cc} -\delta & \delta \\ \beta & - \beta \end{array} \right ). \] The algebraic elements \(D\) and \(E\) then solve the quadratic relations \[ DE-qED = (1-q)(D+E). \] This can be expressed in the formulation \[ w^T\left [ \binom {E}{D} \otimes \binom {E}{D} \right ] = \binom {E}{D} \otimes \binom {1}{-1} - \binom {1}{-1} \otimes \binom {E}{D}, \] so that \( \bar {E}=q-1\) and \(\bar {D}=1-q\). Plugging this into the boundary conditions, we obtain \[ \langle W \vert (\alpha E - \gamma D + q -1)=0, \quad (\delta E - \beta D + 1 - q)\vert V \rangle = 0. \] In [DEHP93], explicit representations were found, but only for up to three lattice sites.

The extension to arbitrary lattice sizes was done in [San94]. For these purposes, it is useful to define operators \(F\) and \(F^{\dagger }\) by (fixing a previous typo in [San94]) \[ D = F+1, \quad E = F^{\dagger }+1. \] Then the quadratic relation between \(D\) and \(E\) becomes \[ FF^{\dagger } - q F^{\dagger } F = 1-q \] The operators \(F\) and \(F^{\dagger }\) are more commonly known as creation and annihilation operators of the q–deformed harmonic oscillator. There is then an infinite–dimensional representation, with basis \( \vert k \rangle \), going back to [Mac89]. The representation is given by \[ F^{\dagger } \vert k \rangle = \{k+1\}_q^{1/2}\vert k + 1\rangle , \quad F\vert k \rangle = \{k\}_q^{1/2} \vert k -1\rangle \] where \[ \{m\}_q=1-q^m. \] The elements \(F\) and \(F^{\dagger }\) are adjoint, so \[ \langle k \vert F = \langle k +1 \vert \{k+1\}_q^{1/2} , \quad \langle k \vert F^{\dagger } = \langle k-1 \vert \{k\}_q^{1/2} \]

Next, we need to express \(\langle V \vert \) and \(\vert W \rangle \) in terms of the basis elements. Define the left coefficients \(l_k\) and right coefficients \(r_k\) by \[ \langle W \vert = \sum _{k=0}^{\infty } \langle k \vert l_k , \quad \vert V \rangle = \sum _{k=0}^{\infty } \vert r_k \rangle . \] Using that \(\langle W \vert (\alpha E - \gamma D + q -1)=0\), we obtain a recurrence relation for the left coefficients \[ l_{-1}=0, \quad l_0=1, \quad \alpha \{k+1\}_q^{1/2}l_{k+1} + (\alpha -\gamma +q-1)l_k + \gamma \{k\}_q^{1/2}l_{k-1} = 0 \text { for } k\geq 0 \] Using the adjointness property, then we obtain a similar recurrence relation for the right coefficients \[ r_{-1}=0, \quad r_0=1, \quad \beta \{k+1\}_q^{1/2}r_{k+1} + (\beta -\delta +q-1)r_k + \delta \{k\}_q^{1/2}r_{k-1} = 0 \text { for } k\geq 0 \]

3 Zamolodchikov Algebra

In this section, we explain how to obtain the bulk relations (1) for the Matrix Product Ansatz from the quantum group. Recall that these are the relations \[ w^T\left [ \binom {E}{D} \otimes \binom {E}{D} \right ] = \binom {E}{D} \otimes \binom {\bar {E}}{\bar {D}} - \binom {\bar {E}}{\bar {D}} \otimes \binom {E}{D}, \] where \(w\) is the local generator.

Recall that we assume that the R–matrix \(R\) satisfies the usual Yang–Baxter equation \[ R_{1,2}R_{1,3}R_{2,3} = R_{2,3}R_{1,3}R_{1,2}. \] We will need another property of the R–matrix, called the Markovian property. This states that there exists a vector \(v(x)\) such that \[ R_{1,2}^T(zw^{-1})v_1(z)v_2(w) = v_1(z)v_2(w) \] and the \(R\) matrix satisfies the normalization \[ \langle \lambda , \lambda \vert R^T(x) = \langle \lambda , \lambda \vert \] for some highest–weight vector \( \langle \lambda ,\lambda | \). We explain why this condition is natural.

The universal \(R\)–matrix of the quantum group has the form \[ R \in U^0 \otimes U^0 + U^{>0}\otimes U^{<0}. \] The highest weight vector \(v\) is defined by \(U^{>0}v=0\), so \(v \otimes v\) satisfies the second equality. The first equality essentially amounts to the condition that the \(R\)–matrix is stochastic, which was discussed in the main set of lecture notes.

The R–matrix needs to be related to the generator \(w\). The condition is that if we take the derivative of \(R(x)\) with respect to the spectral parameter \(x\) and evaluate at 1, \[ PR'(1) = \rho w^T \] where \(P\) is the permutation operator and \(\rho \) is some constant. Note that if \(R(x)\) is stochastic, then its derivative \(R'(x)\) will always have rows summing to 0, from the definition of derivatives.

We will also need the regularity property \[ R(1)=P. \] As explained in the main body of notes, since the permutation operator solves the braided Yang–Baxter equation, this is a natural condition.

Now, let \(T(x)\) be the matrix from the FRT construction. Those notes expressed \(T\) without a parameter; the parameter comes from the evaluation module in the affine Lie algebra. In this context, we have \[ R_{1,2}(zw^{-1})T_1(z)T_2(w) = T_2(w) T_1(z)R_{1,2}(zw^{-1}). \]

We can now define the Zamolodchikov algebra. Let \(v(x)\) denote \[ v(x) = \binom {v_1(x)}{v_2(x)}, \] referencing the vector in the Markovian property. Define \[ A(x) = \binom {\mathcal {X}_1(x)}{\mathcal {X}_2(x)} = T(x)v(x). \] The Zamolodchikov algebra is the subalgebra of the (dual) quantum group, generated by \(\mathcal {X}_1(x),\mathcal {X}_2(x)\).

Applying the RTT relation to \(v_1(z)v_2(w)\) we obtain \[ R_{1,2}(zw^{-1})A_1(z)A_2(w) = A_2(w)A_1(z). \] From here, take the derivative with respect to \(z\) and then evaluate at \(z=w=1\). Then we obtain \[ R_{1,2}'(1)A_1(1)A_2(1) + R_{1,2}(1)A_1'(1)A_2(1) =A_2(1)A_1'(1). \] Plugging in the regularity property \(R(1)=P\) and using that \(PR'(1)=\rho w\) we then get \[ \rho Pw A_1(1)A_2(1) + P A_1'(1)A_2(1) = A_2(1)A_1'(1). \] Multiplying by \(P\) on the left, we obtain \[ w^T A_1(1)A_2(1) = \rho ^{-1}( A_1'(1)A_2(1) - A_1(1)A_2'(1) ) \] Identifying \[ A_1(1) = \binom {E}{D}, \quad A_1'(1) = \rho \binom {\bar {E}}{\bar {D}}. \] we obtain the relation \[ w^T\left [ \binom {E}{D} \otimes \binom {E}{D} \right ] = \binom {E}{D} \otimes \binom {\bar {E}}{\bar {D}} - \binom {\bar {E}}{\bar {D}} \otimes \binom {E}{D}, \] defining the bulk relations in the Matrix Product Ansatz.

4 Reflection Equation and Ghoshal-Zamolodchikov relations

The reflection equation provides integrable boundary conditions (2) for a spin chain. The reflection equations reads \[ R_{1,2}(zw^{-1})K_1(z)R_{2,1}(zw)K_2(w) = K_2(w)R_{1,2}(zw)K_1(z)R_{2,1}(zw^{-1}) \] and the boundary conditions are \[ \langle W \vert ( BA(1) - \rho ^{-1}A'(1) ) = 0 , \quad ( \bar {B}A(1) - \rho ^{-1}A'(1) ) \vert V \rangle = 0. \] A visual representation is given in [CRV14]:

Given an R-matrix, solutions of these equations are called reflection matrices and denoted \(K(z)\). In general, these solutions are not unique. In our context, we will consider two solutions denoted \(K(z)\) and \(\bar {K}(z)\), for the left and right boundaries.

We make a few assumptions about the reflection matrix \(K(z)\). First, we assume the regularity condition \[ K(1) = \mathrm {Id}. \] This is a natural condition, as it corresponds to closed boundary conditions. Additionally, K should be related to the local generators at the boundaries: \[ K'(1) = 2\rho B, \quad \bar {K}'(1) = -2\rho \bar {B} \]

The Ghoshal-Zamolodchikov relations, which we will abbreviate as the GZ relations, are given by \[ \langle \Omega \vert (K(z)A(z^{-1}) - A(z))=0, \quad (\bar {K}(z)A(z^{-1}) - A(z))\vert \Omega \rangle =0. \] To explain the connection to the reflection equation, we calculate \( \langle \Omega \vert A_2(z_2)A_1(z_1) \) in two different ways. Assuming the GZ–relations, we first have \begin{align*} \langle \Omega \vert A_2(z_2)A_1(z_1) &= R_{1,2}(z_1z_2^{-1}) \langle \Omega \vert A_1(z_1) A_2(z_2) \\ &= R_{1,2}(z_1z_2^{-1}) K_1(z_1) \langle \Omega \vert A_1(z_1^{-1}) A_2(z_2) \\ &= R_{1,2}(z_1z_2^{-1}) K_1(z_1) R_{2,1}(z_1z_2)\langle \Omega \vert A_2(z_2) A_1(z_1^{-1}) \\ &= R_{1,2}(z_1z_2^{-1}) K_1(z_1) R_{2,1}(z_1z_2)K_2(z_2)\langle \Omega \vert A_2(z_2^{-1}) A_1(z_1^{-1}) \end{align*}

By a similar sequence of calculations, \[ \langle \Omega \vert A_2(z_2)A_1(z_1) = K_2(z_2)R_{1,2}(z_1z_2)K_1(z_1)R_{2,1}(z_1z_2^{-1})\langle \Omega \vert A_2(z_2^{-1}) A_1(z_1^{-1}) . \] Thus, the reflection equation is a sufficient condition to guarantee the consistency of the GZ relations.

If we take the derivative of the GZ–relations at \(z=1\) we obtain \[ \langle \Omega \vert ( K'(1)A(1) - K(1)A'(1) - A'(1))=0 \quad \langle ( \bar {K}'(1)A(1) - \bar {K}(1)A'(1) - A'(1)) \vert \Omega \rangle =0. \] Defining \( \langle W \vert = \langle \Omega \vert \) and \( \vert V\rangle = \vert \Omega \rangle \), we obtain \[ \langle W \vert ( BA(1) - \rho ^{-1}A'(1) ) = 0 , \quad ( \bar {B}A(1) - \rho ^{-1}A'(1) ) \vert V \rangle = 0. \] Thus, we obtain the boundary conditions in the Matrix Product Ansatz.

5 Explicit Formulas

In this section, we elaborate on explicit formulas for the spectral–dependent R–matrices and their relation to ASEP. We define the matrix entries \[ T(z) = \left ( \begin{array}{cc} T_{11}(z) & T_{12}(z) \\ T_{21}(z) & T_{22}(z) \end{array} \right ). \] We write these entries as a power series \[ T_{ij}(z) = \sum _{n=0}^{\infty } T_{ij}^{(n)}z^n. \] We will consider certain evaluation maps, which allow for the infinite–dimensional affine Lie algebra to be evaluated on finite–dimensional modules. We can truncate the infinite series: \[ \left ( \begin{array}{cc} T_{11}^{(0)} & 0 \\ T_{21}^{(0)} & 1 \end{array} \right ) + z \left ( \begin{array}{cc} 1 & T_{12}^{(0)} \\ 0 & T_{22}^{(0)} \end{array} \right ) \] For context on this evaluation map, one can refer to e.g. [ZG93], in particular the lemma on the evaluation module.

For ASEP, the R–matrix can be expressed as \[ \left ( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & \frac {(z-1)q}{qz-1} & \frac {q-1}{qz-1} & 0 \\ 0 & \frac {(q-1)z}{qz-1} & \frac {z-1}{qz-1} & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right ) \] Two solutions of the reflection equation are: \[ K(x)=\left ( \begin{array}{cc} \displaystyle {\frac {(-x \alpha +x \gamma +q+\alpha -\gamma -1)x}{x^2 \gamma +q x+x \alpha -x \gamma -x-\alpha }}&\displaystyle {\frac {(x^2-1)\gamma }{x^2\gamma +qx+x\alpha -x\gamma -x-\alpha }}\\[2ex] \displaystyle {\frac {\alpha (x^2-1)}{x^2\gamma +qx+x\alpha -x\gamma -x-\alpha }}&\displaystyle {-\frac {(-qx-x\alpha +x\gamma +x+\alpha -\gamma )}{x^2\gamma +qx+x\alpha -x\gamma -x-\alpha }} \end{array} \right ), \] and \[ \bar K(x)= \left ( \begin{array}{cc} \displaystyle {\frac {(x \delta -x \beta +q-\delta +\beta -1)x}{-x^2 \beta +qx-x \delta +x \beta -x+\delta }}&\displaystyle {-\frac {(x^2-1)\beta }{-x^2\beta +qx-x\delta +x\beta -x+\delta }}\\[2ex] \displaystyle {-\frac {(x^2-1)\delta }{-x^2\beta +qx-x\delta +x\beta -x+\delta }}&\displaystyle {\frac {qx-x\delta +x\beta -x+\delta -\beta }{-x^2\beta +qx-x\delta +x\beta -x+\delta }} \end{array} \right ). \] We also have \(\rho =(q-1)^{-1}\).

The Markovian property is satisfied by \[ v(z) = f(z) \binom {ax}{b} \] where \(f(z)\) is an arbitrary function, and \(a,b\) are arbitrary constants. Using the evalution map, the generators of the Zamolodchikov algebra are \[ A(x)=f(x)\left (\begin{array}{c} ax^2+x(a T_{11}^{(0)} + b {T_{12}}^{(1)})\\ x(a{T_{21}}^{(0)}+b{T_{22}}^{(1)})+b \end{array} \right ). \] Choosing \(a=b=1\) and \(f(x)=\frac {1}{x}\), one gets \[ E=1+{T_{11}}^{(0)} + {T_{12}}^{(1)}, \quad \overline {E}=q-1, \quad D={T_{21}}^{(0)}+{T_{22}}^{(1)}+1\, \quad \overline {D}=1-q \]

References