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arXiv:1310.4420 · Physical Review Letters

Exact Nonequilibrium Steady State of an Open Hubbard Chain

Tomaž Prosen

Published Jan 23, 2014• 103 citations
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Abstract

We discuss the current carrying nonequilibrium steady state of an open fermionic Hubbard chain that is strongly driven by Markovian incoherent processes localized at the chain ends. An explicit form of an exact many-body density operator for any value of the coupling parameter is presented. The structure of a matrix product form of the solution is encoded in terms of a novel diagrammatic technique that should allow for generalization to other integrable nonequilibrium models.

DOI
10.1103/physrevlett.112.030603

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We discuss the current carrying nonequilibrium steady state of an open fermionic Hubbard chain that is strongly driven by Markovian incoherent processes localized at the chain ends. An explicit form of an exact many-body density operator for any value of the coupling parameter is presented. The structure of a matrix product form of the solution is encoded in terms of a novel diagrammatic technique that should allow for generalization to other integrable nonequilibrium models.
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We demonstrate that the exact nonequilibrium steady state of the one-dimensional Heisenberg XXZ spin chain driven by boundary Lindblad operators can be constructed explicitly with a matrix product ansatz for the nonequilibrium density matrix where the matrices satisfy a quadratic algebra. This algebra turns out to be related to the quantum algebra U(q)[SU(2)]. Coherent state techniques are introduced for the exact solution of the isotropic Heisenberg chain with and without quantum boundary fields and Lindblad terms that correspond to two different completely polarized boundary states. We show that this boundary twist leads to nonvanishing stationary currents of all spin components. Our results suggest that the matrix product ansatz can be extended to more general quantum systems kept far from equilibrium by Lindblad boundary terms.
In the thesis we present an analytic approach towards exact description for steady state density operators of nonequilibrium quantum dynamics in the framework of open systems. We employ the so-called quantum Markovian semi-group evolution, i.e. a general form of time-autonomous positivity and trace-preserving dynamical equation for reduced density operators, by only allowing unitarity-breaking dissipative terms acting at the boundaries of a system. Such setup enables to simulate macroscopic reservoirs for different values of effective thermodynamic potentials, causing incoherent transitions between quantum states which are modeled with aid of the Lindblad operators. This serves as a simple minimalistic model for studying quantum transport properties, either in the linear response domain or in more general regimes far from canonical equilibrium. We are mainly exploring possibilities of identifying nonequilibrium situations which are amenable to exact description within matrix product state representation, by exclusively focusing on steady states, i.e. fixed points of the Lindblad equation, of certain prototypic interacting integrable spin chains driven by incoherent polarizing processes. Finally, we define a concept of pseudo-local extensive almost-conserved quantities by allowing a violation of time-invariance up to boundary-localized terms. We elucidate the role of such quantities on non-ergodic behaviour of temporal correlation functions rendering anomalous transport properties. It turns out that such conservation laws can be generated by means of boundary universal quantum transfer operators of the fundamental integrable models.
Using the Lindblad master equation approach, we investigate the structure of steady-state solutions of open integrable quantum lattice models, driven far from equilibrium by incoherent particle reservoirs attached at the boundaries. We identify a class of boundary dissipation processes which permits to derive exact steady-state density matrices in the form of graded matrix-product operators. All the solutions factorize in terms of vacuum analogues of Baxter’s Q-operators which are realized in terms of non-unitary representations of certain finite dimensional subalgebras of graded Yangians. We present a unifying framework which allows to solve fermionic models and naturally incorporates higher-rank symmetries. This enables to explain underlying algebraic content behind most of the previously-found solutions.
An exact and explicit ladder-tensor-network ansatz is presented for the nonequilibrium steady state of an anisotropic Heisenberg XXZ spin-1/2 chain which is driven far from equilibrium with a pair of Lindblad operators acting on the edges of the chain only. We show that the steady-state density operator of a finite system of size n is-apart from a normalization constant-a polynomial of degree 2n - 2 in the coupling constant. Efficient computation of physical observables is facilitated in terms of a transfer operator reminiscent of a classical Markov process. In the isotropic case we find cosine spin profiles, 1/n(2) scaling of the spin current, and long-range correlations in the steady state. This is a fully nonperturbative extension of a recent result [Phys. Rev. Lett. 106, 217206 (2011)].
We report a two-parametric irreducible infinitely dimensional representation of the Lax integrability condition for the Fermi Hubbard chain. In addition to being of fundamental interest, hinting at possible novel quantum symmetry of the model, our construction allows for an explicit representation of an exact steady state many-body density operator for a nonequilibrium boundary-driven Hubbard chain with arbitrary (asymmetric) particle source (sink) rates at the left (right) end of the chain and with arbitrary boundary values of chemical potentials.