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Iranian Journal of Physics Research

Quantum transport in the Ising chain: global vs local approach

Azam Mani

Published Jun 1, 2023• 0 citations
Publisher

Abstract

We study energy and spin transport in a one-dimensional Ising chain which is connected to two separate heat baths on both sides. By applying the Born-Markov approximation, within the global approach, we derive the Markovian master equation of the system, and also the explicit form of the Lindblad operators and the steady state. Thereafter, we investigate the behavior of energy and spin dynamics of the system in the global regime. Finally, we solve the problem with the local approach, and we show that the results are not the same for both approaches.

DOI
10.47176/ijpr.23.1.11598

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We study energy and spin transport in a one-dimensional Ising chain which is connected to two separate heat baths on both sides. By applying the Born-Markov approximation, within the global approach, we derive the Markovian master equation of the system, and also the explicit form of the Lindblad operators and the steady state. Thereafter, we investigate the behavior of energy and spin dynamics of the system in the global regime. Finally, we solve the problem with the local approach, and we show that the results are not the same for both approaches.
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Abstract Open systems of coupled qubits are ubiquitous in quantum physics. Finding a suitable master equation to describe their dynamics is therefore a crucial task that must be addressed with utmost attention. In the recent past, many efforts have been made toward the possibility of employing local master equations, which compute the interaction with the environment neglecting the direct coupling between the qubits, and for this reason may be easier to solve. Here, we provide a detailed derivation of the Markovian master equation for two coupled qubits interacting with common and separate baths, considering pure dephasing as well as dissipation. Then, we explore the differences between the local and global master equation, showing that they intrinsically depend on the way we apply the secular approximation. Our results prove that the global approach with partial secular approximation always provides the most accurate choice for the master equation when Born–Markov approximations hold, even for small inter-system coupling constants. Using different master equations we compute the stationary heat current between two separate baths, the entanglement dynamics generated by a common bath, and the emergence of spontaneous synchronization, showing the importance of the accurate choice of approach.
We propose a simple, yet feasible, model for quantum transport of fermionic carriers across tight-binding chain connecting two reservoirs maintained at arbitrary temperatures and chemical potentials. The model allows for elementary derivation of the master equation for the reduced single particle density matrix (SPDM) in a closed form in both Markov and Born approximations. In the Markov approximation the master equation is solved analytically, whereas in the Born approximation the problem is reduced to an algebraic equation for the SPDM in the Redfield form. The non-Markovian equation is shown to lead to resonant transport similar to Landauer's conductance. It is shown that in the deep non-Markovian regime the transport current can be matched with that obtained by the non-equilibrium Green's function method.
This Letter makes progress on the issue of global vs local master equations. Global master equations like the Redfield master equation (following from standard Born and Markov approximation) require a full diagonalization of the system Hamiltonian. This is especially challenging for interacting quantum many-body systems. We discuss a short-bath-correlation-time expansion in reciprocal (energy) space, leading to a series expansion of the jump operator, which avoids a diagonalization of the Hamiltonian. For a bath that is coupled locally to one site, this typically leads to an expansion of the global Redfield jump operator in terms of local operators. We additionally map the local Redfield master equation to a novel local Lindblad form, giving an equation which has the same conceptual advantages of traditional local Lindblad approaches while being applicable in a much broader class of systems. Our ideas give rise to a nonheuristic foundation of local master equations, which can be combined with established many-body methods.
We analyse the dynamics of a qubit coupled to a dissipative impurity by comparing local and global derivation schemes of a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation within the Born-Markov and full secular (FS) approximations. We show that the local approach correctly captures a crossover in the dynamics of the qubit coherence, while the FS approximation restricts the validity of the global approach to regimes with well-separated energy scales. Our results clarify the domains of validity of the two approaches and show that the local scheme provides a better GKSL description of the qubit dynamics in the experimentally relevant parameter regime.