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Quantum Algebra Symmetry of the ASEP with Second-Class Particles

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Abstract

We consider a two-component asymmetric simple exclusion process (ASEP) on a finite lattice with reflecting boundary conditions. For this process, which is equivalent to the ASEP with second-class particles, we construct the representation matrices of the quantum algebra \(U_q[\mathfrak {gl}(3)]\) that commute with the generator. As a byproduct we prove reversibility and obtain in explicit form the reversible measure. A review of the algebraic techniques used in the proofs is given.

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Notes

  1. The occupation numbers can be formally regarded as families of mappings \(a_k: \mathbb {S}^L \mapsto \{0,1\}\), \(b_k: \mathbb {S}^L \mapsto \{0,1\}\) and should thus be understood as functions \(a_k(\eta )\), \(b_k(\eta )\) of \(\eta \). Since the functional argument \(\eta \) will always be clear from context [as is the case e.g. in (4)], we do not write it explicitly. However, we shall usually write explicitly the argument for the particle number functions \(N(\eta )\), \(M(\eta )\) to contrast them with their numerical values N, M.

  2. We shall use interchangeably the arguments \(\eta \), \(\mathbf {z}\), \(\{\mathbf {x},\mathbf {y}\}\) for functions of the configurations. When the argument is clear from context it may be omitted.

  3. We shall usually omit the set \(\mathbb {S}^L\) in the summation symbol and simply write \(\sum _{\eta }\).

  4. The connection to integrable models, in particular the parameter dependence of R, the construction of the associated statistical mechanics transfer matrix, and its quantum Hamiltonian limit, is not important for the purposes of this work. We refer the interested reader to [10, 11, 19] for more details and to [2] for an introduction to the field.

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Acknowledgments

This work was supported by DFG and by CNPq through the Grant 307347/2013-3. GMS thanks the University of São Paulo, where part of this work was done, for kind hospitality.

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Authors and Affiliations

  1. Instituto de Matemática e Estátistica, Universidade de São Paulo, Rua do Matão, 1010, São Paulo, SP, CEP 05508-090, Brazil

    V. Belitsky

  2. Institute of Complex Systems II, Forschungszentrum Jülich, 52425, Jülich, Germany

    G. M. Schütz

Authors
  1. V. Belitsky
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  2. G. M. Schütz
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Correspondence to G. M. Schütz.

Appendix

Appendix

We display some explicit results for unnormalized stationary distributions for small lattices \(L=2,3,4\) and also L arbitrary with small particle numbers \(N+M = 1,2,3,4\).

\(\underline{N+M=1:}\)

$$\begin{aligned} \pi ^*(\{x\},\emptyset )\propto & {} q^{2x-1} \\ \pi ^*(\emptyset ,\{y\})\propto & {} q^{-2y+1} \end{aligned}$$

\(\underline{N+M=2:}\)

$$\begin{aligned} \pi ^*(\{x_1,x_2\},\emptyset )\propto & {} q^{2 x_1 + 2 x_2 - 2} \\ \pi ^*(\{x_1\},\{y_1\})\propto & {} \left\{ \begin{array}{ll} q^{2 x_1 - 2 y_1 - 1} &{} \quad y_1<x_1 \\ q^{2 x_1 - 2 y_1 + 1} &{} \quad y_1>x_1 \end{array} \right. \\ \pi ^*(\emptyset ,\{y_1,y_2\})\propto & {} q^{-2 y_1 - 2 y_2 + 2} \end{aligned}$$

\(\underline{N+M=3:}\)

$$\begin{aligned} \pi ^*(\{x_1,x_2,x_3\},\emptyset )\propto & {} q^{2 x_1 + 2 x_2 + 2 x_3 - 3} \\ \pi ^*(\{x_1,x_2\},\{y_1\})\propto & {} \left\{ \begin{array}{ll} q^{2 x_1 + 2 x_2 - 2 y_1 - 3} &{} \quad y_1<x_1,x_2 \\ q^{2 x_1 + 2 x_2 - 2 y_1 - 1} &{} \quad x_1< y_1<x_2 \\ q^{2 x_1 + 2 x_2 - 2 y_1 + 1} &{} \quad x_1,x_2< y_1 \end{array} \right. \\ \pi ^*(\{x_1\},\{y_1,y_2\})\propto & {} \left\{ \begin{array}{ll} q^{2 x_1 - 2 y_1 - 2 y_2 - 1} &{} \quad y_1,y_2<x_1 \\ q^{2 x_1 - 2 y_1 - 2 y_2 + 1} &{} \quad y_1< x_1<y_2 \\ q^{2 x_1 - 2 y_1 - 2 y_2 + 3} &{} \quad x_1< y_1,y_2 \end{array} \right. \\ \pi ^*(\emptyset ,\{y_1,y_2,y_3\})\propto & {} q^{- 2 y_1 - 2 y_2 - 2 y_3 + 3} \end{aligned}$$

\(\underline{N+M=4:}\)

$$\begin{aligned} \pi ^*(\{x_1,x_2,x_3,x_4\},\emptyset )\propto & {} q^{2 x_1 + 2 x_2 + 2 x_3 + 2 x_4 - 4} \\ \pi ^*(\{x_1,x_2,x_3\},\{y_1\})\propto & {} \left\{ \begin{array}{ll} q^{2 x_1 + 2 x_2 + 2 x_3 - 2 y_1 - 5} &{} \quad y_1<x_1,x_2,x_3 \\ q^{2 x_1 + 2 x_2 + 2 x_3 - 2 y_1 - 3} &{} \quad x_1< y_1<x_2,x_3 \\ q^{2 x_1 + 2 x_2 + 2 x_3 - 2 y_1 - 1} &{} \quad x_1,x_2< y_1<x_3 \\ q^{2 x_1 + 2 x_2 + 2 x_3 - 2 y_1 + 1} &{} \quad x_1,x_2,x_3< y_1 \end{array} \right. \\ \pi ^*(\{x_1,x_2\},\{y_1,y_2\})\propto & {} \left\{ \begin{array}{ll} q^{2 x_1 + 2x_2 - 2 y_1 - 2 y_2 - 4} &{} \quad y_1,y_2<x_1,x_2 \\ q^{2 x_1 + 2x_2 - 2 y_1 - 2 y_2 - 2} &{} \quad y_1<x_1<y_2<x_2 \\ q^{2 x_1 + 2x_2 - 2 y_1 - 2 y_2} &{} \quad y_1<x_1,x_2<y_2 \\ q^{2 x_1 + 2x_2 - 2 y_1 - 2 y_2} &{} \quad x_1<y_1,y_2<x_2 \\ q^{2 x_1 + 2x_2 - 2 y_1 - 2 y_2 + 2} &{} \quad x_1<y_1<x_2<y_2 \\ q^{2 x_1 + 2x_2 - 2 y_1 - 2 y_2 + 4} &{} \quad x_1,x_2<y_1,y_2 \end{array} \right. \\ \pi ^*(\{x_1\},\{y_1,y_2,y_3\})\propto & {} \left\{ \begin{array}{ll} q^{2 x_1 - 2 y_1 - 2 y_2 - 2 y_3 - 1} &{} \quad y_1,y_2,y_3 < x_1\\ q^{2 x_1 - 2 y_1 - 2 y_2 - 2 y_3 + 1} &{} \quad y_1,y_2< x_1<x_3 \\ q^{2 x_1 - 2 y_1 - 2 y_2 - 2 y_3 + 3} &{} \quad y_1< x_1<y_2,y_3 \\ q^{2 x_1 - 2 y_1 - 2 y_2 - 2 y_3 + 5} &{} \quad x_1 < y_1,y_2,y_3 \end{array} \right. \\ \pi ^*(\emptyset ,\{y_1,y_2,y_3,y_4\})\propto & {} q^{- 2 y_1 - 2 y_2 - 2 y_3 - 2 y_4 + 4} \end{aligned}$$

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Belitsky, V., Schütz, G.M. Quantum Algebra Symmetry of the ASEP with Second-Class Particles. J Stat Phys 161, 821–842 (2015). https://doi.org/10.1007/s10955-015-1363-1

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