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arXiv:1307.5546 · Journal of Statistical Mechanics Theory and Experiment

Quantum group approach to steady states of boundary-driven open quantum systems

Enej Ilievski, Bojan Žunkovič

Published Jan 16, 2014• 30 citations
arXivHTMLPDF

Abstract

We present a systematic approach for constructing steady state density operators of Markovian dissipative evolution for open quantum chain models with integrable bulk interaction and boundary incoherent driving. The construction is based on fundamental solutions of the quantum Yang–Baxter equation pertaining to quantum algebra symmetries and their quantizations ( q -deformations). In particular, we facilitate a matrix-product state description, by resorting to generic spin- s infinite-dimensional solutions associated with non-compact spins, serving as ancillary degrees of freedom. After formally deriving already known steady states for the anisotropic spin-1/2 Heisenberg chain from first symmetry principles, we obtain a class of solutions belonging to interacting quantum gases with SU ( N )-symmetric Hamiltonians, using a restricted set of incoherent boundary jump processes, and point out how new non-trivial generalizations emerge from twists of quantum group structures. Finally, we discuss the possibilities for the analytical calculation of observables by employing the algebraic properties of associated auxiliary vertex operators.

DOI
10.1088/1742-5468/2014/01/p01001

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We present a systematic approach for constructing steady state density operators of Markovian dissipative evolution for open quantum chain models with integrable bulk interaction and boundary incoherent driving. The construction is based on fundamental solutions of the quantum Yang–Baxter equation pertaining to quantum algebra symmetries and their quantizations ( q -deformations). In particular, we facilitate a matrix-product state description, by resorting to generic spin- s infinite-dimensional solutions associated with non-compact spins, serving as ancillary degrees of freedom. After formally deriving already known steady states for the anisotropic spin-1/2 Heisenberg chain from first symmetry principles, we obtain a class of solutions belonging to interacting quantum gases with SU ( N )-symmetric Hamiltonians, using a restricted set of incoherent boundary jump processes, and point out how new non-trivial generalizations emerge from twists of quantum group structures. Finally, we discuss the possibilities for the analytical calculation of observables by employing the algebraic properties of associated auxiliary vertex operators.
We review recent progress on constructing non-equilibrium steady state density operators of boundary driven locally interacting quantum chains, where driving is implemented via Markovian dissipation channels attached to the chain's ends. We discuss explicit solutions in three different classes of quantum chains, specifically, the paradigmatic (anisotropic) Heisenberg spin- chain, the Fermi–Hubbard chain, and the Lai–Sutherland spin-1 chain, and discuss universal concepts which characterize these solutions, such as matrix product ansatz and a more structured walking graph state ansatz. The central theme is the connection between the matrix product form of nonequilibrium states and the integrability structures of the bulk Hamiltonian, such as the Lax operators and the Yang–Baxter equation. However, there is a remarkable distinction with respect to the conventional quantum inverse scattering method, namely addressing nonequilibrium steady state density operators requires non-unitary irreducible representations of Yang–Baxter algebra which are typically of infinite dimensionality. Such constructions result in non-Hermitian, and often also non-diagonalisable families of commuting transfer operators which in turn result in novel conservation laws of the integrable bulk Hamiltonians. For example, in the case of the anisotropic Heisenberg model, quasi-local conserved operators which are odd under spin reversal (or spin flip) can be constructed, whereas the conserved operators stemming from orthodox Hermitian transfer operators (via logarithmic differentiation) are all even under spin reversal.
A new type of quantum transfer matrix, arising as a Cholesky factor for the steady-state density matrix of a dissipative Markovian process associated with the boundary-driven Lindblad equation for the isotropic spin-1/2 Heisenberg ( XXX ) chain, is presented. The transfer matrix forms a commuting family of non-Hermitian operators depending on the spectral parameter, which is essentially the strength of dissipative coupling at the boundaries. The intertwining of the corresponding Lax and monodromy matrices is performed by an infinitely dimensional Yang–Baxter R -matrix, which we construct explicitly and is essentially different from the standard 4 × 4 XXX R -matrix. We also discuss a possibility to construct Bethe ansatz for the spectrum and eigenstates of the non-equilibrium steady-state density operator. Furthermore, we indicate the existence of a deformed R -matrix in the infinite dimensional auxiliary space for the anisotropic XXZ spin-1/2 chain, which in general provides a sequence of new, possibly quasi-local, conserved quantities of the bulk XXZ dynamics.
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.
A new type of quantum transfer matrix, arising as a Cholesky factor for the steady-state density matrix of a dissipative Markovian process associated with the boundary-driven Lindblad equation for the isotropic spin-1/2 Heisenberg (XXX) chain, is presented. The transfer matrix forms a commuting family of non-Hermitian operators depending on the spectral parameter, which is essentially the strength of dissipative coupling at the boundaries. The intertwining of the corresponding Lax and monodromy matrices is performed by an infinitely dimensional Yang–Baxter R-matrix, which we construct explicitly and is essentially different from the standard 4 × 4 XXX R-matrix. We also discuss a possibility to construct Bethe ansatz for the spectrum and eigenstates of the non-equilibrium steady-state density operator. Furthermore, we indicate the existence of a deformed R-matrix in the infinite dimensional auxiliary space for the anisotropic XXZ spin-1/2 chain, which in general provides a sequence of new, possibly quasi-local, conserved quantities of the bulk XXZ dynamics.
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.
We present an explicit construction of a family of steady state density matrices for an open integrable spin-1 chain with bilinear and biquadratic interactions, also known as the Lai–Sutherland model, driven far from equilibrium by means of two oppositely polarizing Markovian dissipation channels localized at the boundary. The steady state solution exhibits n+1 fold degeneracy, for a chain of length n, due to existence of (strong) Liouvillian U(1) symmetry. The latter can be exploited to introduce a chemical potential and define a grand canonical nonequilibrium steady state ensemble. The matrix product form of the solution entails an infinitely-dimensional representation of a non-trivial Lie algebra (semidirect product of sl2 and a non-nilpotent radical) and hints to a novel Yang–Baxter integrability structure.