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arXiv:1612.01008 · New Journal of Physics

Energy current and its statistics in the nonequilibrium spin-boson model: Majorana fermion representation

Bijay Kumar Agarwalla, Dvira Segal

Published Apr 28, 2017• 33 citations
arXivHTMLPDF

Abstract

We study the statistics of thermal energy transfer in the nonequilibrium (two-bath) spin-boson model. This quantum many-body impurity system serves as a canonical model for quantum energy transport. Our method makes use of the Majorana fermion representation for the spin operators, in combination with the Keldysh nonequilibrium Green's function approach. We derive an analytical expression for the cumulant generating function of the model in the steady state limit, and show that it satisfies the Gallavotti–Cohen fluctuation symmetry. We obtain analytical expressions for the heat current and its noise, valid beyond the sequential and the co-tunneling regimes. Our results satisfy the quantum mechanical bound for heat current in interacting nanojunctions. Results are compared with other approximate theories, as well as with a non-interacting model, a fully harmonic thermal junction.

DOI
10.1088/1367-2630/aa6657

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We study the statistics of thermal energy transfer in the nonequilibrium (two-bath) spin-boson model. This quantum many-body impurity system serves as a canonical model for quantum energy transport. Our method makes use of the Majorana fermion representation for the spin operators, in combination with the Keldysh nonequilibrium Green's function approach. We derive an analytical expression for the cumulant generating function of the model in the steady state limit, and show that it satisfies the Gallavotti–Cohen fluctuation symmetry. We obtain analytical expressions for the heat current and its noise, valid beyond the sequential and the co-tunneling regimes. Our results satisfy the quantum mechanical bound for heat current in interacting nanojunctions. Results are compared with other approximate theories, as well as with a non-interacting model, a fully harmonic thermal junction.
We study the heat current flowing between two baths consisting of harmonic oscillators interacting with a qubit through a spin-boson coupling. An explicit expression for the generating function of the total heat flowing between the right and left baths is derived by evaluating the corresponding Feynman-Vernon path integral by performing the noninteracting blip approximation (NIBA). We recover the known expression, obtained by using the polaron transform. This generating function satisfies the Gallavotti-Cohen fluctuation theorem, both before and after performing the NIBA. We also verify that the heat conductance is proportional to the variance of the heat current, retrieving the well-known fluctuation dissipation relation. Finally, we present numerical results for the heat current.
We present a general theory to explore energy transfer in nonequilibrium spin-boson models within the framework of nonequilibrium Green's function (NEGF). In contrast to conventionally used NEGF methods based on a perturbation expansion in the system-bath coupling, we adopt the polaron transformation to the Hamiltonian and identify the tunneling term as a perturbation with the system-bath coupling being treated nonperturbatively, herein termed the polaron-transformed NEGF method. To evaluate terms in the Dyson series, we further utilize the Majorana-fermion representation. The proposed method not only allows us to deal with weak as well as strong coupling regime, but also enables an investigation on the role of bias. As an application of the method, we study the energy transfer between two Ohmic bosonic baths mediated by a spin. For a unbiased spin system, our energy current result smoothly bridges predictions of two benchmarks, namely, the quantum master equation and the nonequilibrium non-interacting blip approximation, thus our method is beyond existing theories. In case of a biased spin system, we reveal a bias-induced nonmomotonic behavior of the energy conductance in the intermediate coupling regime, due to the resonant character of the energy transfer. This finding may offer a nontrivial quantum control knob over energy transfer at the nanoscale.
To explore energy transfer in the nonequilibrium spin-boson model (NESB) from weak to strong system-bath coupling regimes, we propose a polaron-transformed nonequilibrium Green's function (NEGF) method. By combining the polaron transformation, we are able to treat the system-bath coupling nonperturbatively, thus in direct contrast to conventionally used NEGF methods which take the system-bath coupling as a perturbation. The Majorana-fermion representation is further utilized to evaluate terms in the Dyson series. This method not only allows us to deal with weak as well as strong coupling regimes but also enables an investigation on the role of bias in the energy transfer. As an application of the method, we study an Ohmic NESB. For an unbiased spin system, our energy current result smoothly bridges predictions of two benchmarks, namely, the quantum master equation and the nonequilibrium noninteracting blip approximation, a considerable improvement over existing theories. In case of a biased spin system, we found a bias-induced nonmonotonic behavior of the energy conductance in the intermediate coupling regime, resulting from the resonant character of the energy transfer. This finding may offer a nontrivial quantum control knob over energy transfer at the nanoscale.
The modelling of quantum heat transfer processes at the nanoscale is crucial for the development of energy harvesting and molecular electronic devices. Herein, we adopt a mixed quantum-classical description of a device, in which the open subsystem of interest is treated quantum mechanically and the surrounding heat baths are treated in a classical-like fashion. By introducing such a mixed quantum-classical description of the composite system, one is able to study the heat transfer between the subsystem and bath from a closed system point of view, thereby avoiding simplifying assumptions related to the bath time scale and subsystem-bath coupling strength. In particular, we adopt the full counting statistics approach to derive a general expression for the moment generating function of heat in systems whose dynamics are described by the quantum-classical Liouville equation (QCLE). From this expression, one can deduce expressions for the dynamics of the average heat and heat current, which may be evaluated using numerical simulations. Due to the approximate nature of the QCLE, we also find that the steady state fluctuation symmetry holds up to order ℏ for systems whose subsystem-bath couplings and baths go beyond bilinear and harmonic, respectively. To demonstrate the approach, we consider the nonequilibrium spin boson model and simulate its time-dependent average heat and heat current under various conditions.
We study the transfer of heat in the nonequilibrium spin-boson model with an Ohmic dissipation. In the nonadiabatic limit we derive a formula for the thermal conductance based on a rate equation formalism at the level of the noninteracting blip approximation, valid for temperatures T>T(K), with T(K) as the Kondo temperature. We evaluate this expression analytically assuming either weak or strong couplings, and demonstrate that our results agree with exact relations. Far-from-equilibrium situations are further examined, showing a close correspondence to the linear response limit.
To study the full counting statistics of quantum heat transfer in a driven nonequilibrium spin-boson model, we develop a generalized nonequilibrium polaron-transformed Redfield equation with an auxiliary counting field. This enables us to study the impact of qubit-bath coupling ranging from weak to strong regimes. Without external modulations, we observe maximal values of both steady-state heat flux and noise power in moderate coupling regimes, below which we find that these two transport quantities are enhanced by the finite-qubit-energy bias. With external modulations, the geometric-phase-induced heat flux shows a monotonic decrease upon increasing the qubit-bath coupling at zero qubit energy bias (without bias). While under the finite-qubit-energy bias (with bias), the geometric-phase-induced heat flux exhibits an interesting reversal behavior in the strong coupling regime. Our results unify the seemingly contradictory results in weak and strong qubit-bath coupling regimes and provide detailed dissections for the quantum fluctuation of nonequilibrium heat transfer.