Heat, work, and energy currents in the boundary-driven XXZ spin chain

Abstract

We address the detailed study of the energy current and its components, heat and work, in the boundary-driven one-dimensional XXZ quantum model. We carry out the investigation by considering two different approaches present in the literature. First, we take the repeated interaction scheme and derive the expressions for the currents of heat and work, exchanged between system and baths. Then we perform the derivation of the energy current by means of a Lindblad master equation together with a continuity equation, another approach which is recurrently used. A comparison between the obtained expressions allows us to show the consistency of both approaches, and, in the latter expression derived from the Lindblad equation, it allows us to split the energy, which comes from the baths to the system, into heat and work. The recognition of work in the process, which is recurrently ignored in studies of transport, enables us to understand thermodynamical aspects and to solve some imbroglios in the physics behind the energy current in the XXZ spin chain.

Reconciliation of quantum local master equations with thermodynamics

“…Such equations, which we shall henceforth refer to as LMEs (also frequently called boundarydriven master equations), are typically accurate when the dissipation rates are larger than the interaction between subsystems. Due to their computational simplicity, they have been extensively employed over the last two decades in the study of transport in non-equilibrium quantum systems [55][56][57][58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73].It turns out, however, that the nonlocal terms neglected in the LME may still lead to non-thermal steadystates [74] and play a significant role if the heat exchanges are small, even for weakly interacting parts. As a consequence, it has been found that LMEs may lead to apparent thermodynamic inconsistencies, as pointed out recently by Levy and Kosloff [75].…”

Thermodynamics of Weakly Coherent Collisional Models

“…A framework that is particularly suited for addressing the thermodynamics of engineered reservoirs is that of collisional models (also called repeated interactions) [13][14][15][16][17][18][19][20][21][22][23][24]. They draw inspiration from Boltzmann's original Stosszahlansatz: at any given interval of time, the system S will only interact with a tiny fraction of the environment.…”

“…We show that, despite their weakness, the implications of such residual coherence for both the first and second law of thermodynamics are striking, in that non-trivial contributions to the continuous-time open dynamics arise to affect the phenomenology of energy exchanges between system and environment [21]. In order to illustrate these features in a clear manner, we choose a scenario where no work is externally performed on the global system-ancilla compound [19,20,29], so that all changes in the energy of the system can be faithfully attributed to heat flowing from or into the environment. Despite this, we derive a bound showing how coherence in the ancillae (quantified by the relative entropy of coherence) is consumed to convert part of the heat into a coherent (worklike) term in the system.…”

“…As the global dynamics is unitary, the definition of work in this case is unambiguous, being associated with the cost of switching the S -A n interaction on and off [19,20,29]. This work cost will be strictly zero whenever the system satisfies the condition [26,27] [U S A n , H S + H A n ] = 0, which states strict energy conservation.…”

One-way street for the energy current: A ubiquitous phenomenon in boundary-driven quantum spin chains

“…See also Ref. [10] for the related analysis in these XXZ chains, and Ref. [11] for general considerations (in particular, responding the false inconsistency raised in Ref.…”