Abstract
We present a general construction of matrix product states for stationary density matrices of one-dimensional quantum spin systems kept out of equilibrium through boundary Lindblad dynamics. As an application we review the isotropic Heisenberg quantum spin chain which is closely related to the generator of the simple symmetric exclusion process. Exact and heuristic results as well as numerical evidence suggest a local quantum equilibrium and long-range correlations reminiscent of similar large-scale properties in classical stochastic interacting particle systems that can be understood in terms of fluctuating hydrodynamics.
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Notes
- 1.
Due to the quantum mechanical phenomenon of entanglement, a quantum subsystem that has interacted with its environment in the past (until some time \(t_0\)) cannot be considered isolated for \(t \ge t_0\) even when there are no interactions from \(t_0\) onwards.
- 2.
Following quantum mechanical convention we use the short hand \(|\,{\cdot }\,\rangle \langle \,{\cdot }\,| \equiv |\,{\cdot }\,\rangle \otimes \langle \,{\cdot }\,|\) for the Kronecker product \(\otimes \) of a state vector \(|\,{\cdot }\,\rangle \in \mathfrak {H}\) and some dual state vector \(\langle \,{\cdot }\,| \in \mathfrak {H}^*\). We stress that by the rules of tensor calculus one has \(\langle \,{\varPsi }\,|\otimes |\,{\varPhi }\,\rangle = |\,{\varPsi }\,\rangle \otimes \langle \,{\varPhi }\,| \equiv |\,{\varPsi }\,\rangle \langle \,{\varPhi }\,|\) but \(\langle \,{\varPsi }\,|\otimes |\,{\varPhi }\,\rangle \ne \langle \,{\varPsi }\,|\,{\varPhi }\,\rangle \) since \(\langle \,{\varPsi }\,|\,{\varPhi }\,\rangle \) represents the scalar product.
- 3.
For this scenario, which we have in mind for applications, one often calls \(\rho \) the reduced density matrix, but we shall refrain doing so here.
- 4.
M is not uniquely defined. For a given M and arbitrary unitary U the product MU gives the same \(\rho \). This non-uniqueness seems to be exactly the point that makes M easier to treat than \(\rho \).
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Acknowledgements
VP and GMS thank T. Prosen for useful discussions and DFG for financial support.
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Karevski, D., Popkov, V., Schütz, G.M. (2017). Matrix Product Ansatz for Non-equilibrium Quantum Steady States. In: Gonçalves, P., Soares, A. (eds) From Particle Systems to Partial Differential Equations. PSPDE 2015. Springer Proceedings in Mathematics & Statistics, vol 209. Springer, Cham. https://doi.org/10.1007/978-3-319-66839-0_11
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