Towards a Matrix Product Ansatz in Two Dimensions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916028798205952 |
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| author | Ray, Chandraniva Guha Banerjee, Aikya Mohanty, P. K. |
| author_facet | Ray, Chandraniva Guha Banerjee, Aikya Mohanty, P. K. |
| contents | Matrix product ansatz (MPA) is a powerful framework for constructing exact steady state weights of one dimensional non-equilibrium stochastic processes; but its generalization to higher dimensions is limited. Here, we introduce the MPA formalism for two dimensions (2D). As a concrete application, we introduce and exactly solve a non-conserved assisted exclusion model (NAEM) in one and two dimensions with constrained hopping and local birth-death dynamics: a particle can hop to a neighbouring site only when exactly one of its neighbouring sites is vacant, while creation and annihilation occur exclusively at sites whose neighbours are all occupied. The MPA yields exact steady-state weights and provides a systematic method to compute observables such as density moments and particle currents. In the particle-conserving limit, the system undergoes an absorbing phase transition at the critical density $ρ_c=\frac12$ with order-parameter exponent $β=3$. We further show that the steady state of the NAEM maps exactly onto the well-studied hard-square lattice gas with nearest-neighbour exclusion, thereby providing a nonequilibrium dynamical route to realizing equilibrium states of constrained lattice gases. Our work generalizes matrix-product methods beyond one dimension, establishing a systematic approach to exact solutions of interacting stochastic systems in 2D. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_20160 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Towards a Matrix Product Ansatz in Two Dimensions Ray, Chandraniva Guha Banerjee, Aikya Mohanty, P. K. Statistical Mechanics Matrix product ansatz (MPA) is a powerful framework for constructing exact steady state weights of one dimensional non-equilibrium stochastic processes; but its generalization to higher dimensions is limited. Here, we introduce the MPA formalism for two dimensions (2D). As a concrete application, we introduce and exactly solve a non-conserved assisted exclusion model (NAEM) in one and two dimensions with constrained hopping and local birth-death dynamics: a particle can hop to a neighbouring site only when exactly one of its neighbouring sites is vacant, while creation and annihilation occur exclusively at sites whose neighbours are all occupied. The MPA yields exact steady-state weights and provides a systematic method to compute observables such as density moments and particle currents. In the particle-conserving limit, the system undergoes an absorbing phase transition at the critical density $ρ_c=\frac12$ with order-parameter exponent $β=3$. We further show that the steady state of the NAEM maps exactly onto the well-studied hard-square lattice gas with nearest-neighbour exclusion, thereby providing a nonequilibrium dynamical route to realizing equilibrium states of constrained lattice gases. Our work generalizes matrix-product methods beyond one dimension, establishing a systematic approach to exact solutions of interacting stochastic systems in 2D. |
| title | Towards a Matrix Product Ansatz in Two Dimensions |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2605.20160 |