Flux rectification in the quantumXXZchain

Abstract

Thermal rectification is the phenomenon by which the flux of heat depends on the direction of the flow. It has attracted much interest in recent years due to the possibility of devising thermal diodes. In this paper, we consider the rectification phenomenon in the quantum XXZ chain subject to an inhomogeneous field. The chain is driven out of equilibrium by the contact at its boundaries with two different reservoirs, leading to a constant flow of magnetization from one bath to the other. The nonunitary dynamics of this system, which is modeled by a Lindblad master equation, is treated exactly for small sizes and numerically for larger ones. The functional dependence of the rectification coefficient on the model parameters (anisotropy, field amplitude, and out of equilibrium driving strength) is investigated in full detail. Close to the XX point and at small inhomogeneity and low driving, we have found an explicit expression for the rectification coefficient that is valid at all system sizes. In particular, it shows that the phenomenon of rectification persists even in the thermodynamic limit. Finally, we prove that in the case of the XX chain, there is no rectification.

Exact asymptotics of the current in boundary-driven dissipative quantum chains in large external fields

“…We can point out two mechanisms for enhancing current rectification: (i) rectification coefficient R → 1 in thermodynamic limit n → ∞, at fixed large gradient f/(n − 1), for any nonzero anisotropy (interaction) = 0, as a result of our asymptotic current formula; (ii) for resonant field (gradient) values where our asymptotic analysis fails, the current is of higher order in 1/f for one direction of the field f than in the other. This clearly yields rectification coefficient R → 1 even at fixed finite size n. Note that there is no rectification effect in the noninteracting chains = 0, in agreement with [33]. Our analysis can be used also to reveal large anisotropy asymptotics 030103-4…”

Energy rectification in quantum graded spin chains: Analysis of the XXZ model

“…Note that, from Eq. (20), in the steady state and in the presence of a homogeneous magnetic field we have F = F XXZ + B J , with F XXZ an even function of f , as argued above, and J an odd function of f (the spin current direction is determined by the direction of the magnetization imbalance). Hence, the energy rectification (change in the magnitude of F as we invert the baths) is clear: if we invert the sign of f , only one term of F changes the sign.…”

“…In Ref. [20], the authors show the existence of spin current rectification in the homogeneous XXZ chain (with α i,i+1 = α and ∆ i,i+1 = ∆, ∀i) under an inhomogeneous, linearly graded, magnetic field for small chains, up to N = 7. Moreover, they show the vanishing of rectification as ∆ → 0, i.e., for the XX chain, and emphasize the correspondence to the well known result of absence of (energy) rectification in classical harmonic chains: written in terms of fermionic creation and annihilation operators, the XX model contains only quadratic terms, whereas the XXZ model involves also a quartic term which is proportional to ∆.…”

“…For example, one may use the repeated interactions scheme as minutely described in appendices A and B of Ref. [20]. The procedure starts by taking an enlarged chain with two extra spins, labeled 0 and N + 1, coupled to the boundaries of the original chain, i.e., to spins 1 and N respectively.…”

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].…”