Dephasing enhanced transport in boundary-driven quasiperiodic chains Artur M. Lacerda,1, 2 John Goold,2 and Gabriel T. Landi1 1Instituto de Física, Universidade de São Paulo, CEP 05314-970, São Paulo, São Paulo, Brazil 2Department of Physics, Trinity College Dublin, Dublin 2, Ireland We study dephasing-enhanced transport in boundary-driven quasi-periodic systems. Specifically we consider dephasing modelled by current preserving Lindblad dissipators acting on the non-interacting Aubry-André- Harper (AAH) and Fibonacci bulk systems. The former is known to undergo a critical localization transition with a suppression of ballistic transport above a critical value of the potential. At the critical point, the presence of non-ergodic extended states yields anomalous sub-diffusion. The Fibonacci model, on the other hand, yields anomalous transport with a continuously varying exponent depending on the potential strength. By computing the covariance matrix in the non-equilibrium steady-state, we show that sufficiently strong dephasing always renders the transport diffusive. The interplay between dephasing and quasi-periodicity gives rise to a maximum of the diffusion coefficient for finite dephasing, which suggests the combination of quasi-periodic geometries and dephasing can be used to control noise-enhanced transport. I. INTRODUCTION to explore the many-body localised phase [37]. In the AAH, at fixed tunneling rate and below a critical value of the po- tential strength, all the energy eigenstates are delocalized, Non-equilibrium systems are characterized by the existence while above this value the entire spectrum is localized. This of macroscopic currents of energy or matter [1]. Understand- transition is clearly reflected in the non-equilibrium trans- ing these transport properties has, for more than a century, port properties, with particle transport going from ballistic to been a major field of research in physics. One of the fun- exponentially suppressed. At criticality, the eigenstates are damental issues is to ascertain the necessary ingredients in a neither delocalized nor localized, and the transport is sub- model to induce a certain transport regime. Low dimensional diffusive [30, 38]. Generalizations of the AAH model ex- systems, in particular, have received significant attention both hibiting mobility-edges have also been analyzed in the trans- in the classical [2,3] and quantum regimes [4]. For instance, port scenario [39]. A closely related model is the Fibonacci chains of harmonic oscillators present ballistic transport [5], model [40–43]. This model has a critical spectrum which which is fundamentally different from the diffusive behavior has been studied in detail [40, 42–44]. This unique spec- obtained in Fourier’s law of heat conduction [6]. However, trum gives rise to highly anomalous transport in the absence not all anharmonicities lead to diffusivity [7,8]. In addition of interactions where the transport exponent varies continu- to being fundamental, being able to understand and control ously with the potential strength. In fact, the transport in transport offers opportunities for potential device applications. the Fibonacci model can be tuned continuously, from ballistic Transport in low-dimensional devices, for instance, can be to sub-diffusive. While the quantum transport properties of manipulated for steady state thermal machines [9] and nano- quasi-periodic systems is well studied, recently studies have scale heat engineering [10–13]. highlighted that they can be exploited for thermal engineer- For low dimensional quantum many-body systems, the ing [45, 46]. technique of boundary-driving has been used widely to ex- In this work we are interested in how the anomalous trans- tract high temperature transport properties in non-equilibrium port, typically observed in quasi-periodic systems, is affected steady-states (NESSs) [14–22]. Boundary-driving is an open by dephasing noise from an environment. One phenomeno- systems technique where local Lindblad dissipators at the logical approach to model dephasing is by means means of edges of the chain induce a gradient in spin and or energy. self-consistent baths [47], also known as Büttiker probes [48]. Transport coefficients can then be extracted from finite size Another approach, which we will exploit here, is via current- scaling of the currents [4, 23]. Although the technique is lim- preserving Lindblad dissipators. For any non-zero dephas- ited to high temperature physics it has a distinct advantage in arXiv:2106.11406v1 [quant-ph] 21 Jun 2021 ing strengths, free tight-binding models typically become dif- so much as it can be extended to models with integrability fusive in the thermodynamic limit [16, 49]. Motivated by breaking perturbations [24, 25] by means of tensor networks. these results, we study the effects of dephasing in quasiperi- This technique has been instrumental in shedding light on the odic systems. We numerically compute the covariance ma- transport properties of the ergodic phase of interacting dis- trix in the NESS and study the finite size scaling of the par- ordered models [26–29] and systems with magnetic impuri- ticle current. Both the AAH model and the Fibonacci model ties [24, 25]. become diffuive in the presence of dephasing noise. Interest- Another class of models where boundary driving has been ingly, though, in certain regimes this introduces a competition, applied successfully is quasi-periodic chains [30–32]. A where the dephasing leads to noise-enhanced transport. The paradigmatic example is the Aubry-André-Harper (AAH) paper is divided as follows. The model is described in Sec.II model [33, 34], which is known to undergo a localization tran- and the analysis of the transport properties of both models, sition when the potential strength is increased. The model with and without dephasing, are discussed in Sec. III. Our is readily simulated in ultra-cold atomic physics experiments, main results are presented in Sec.IV, where we analyze the using bichromatic optical lattices [35, 36], and has been used interplay between quasi-periodicity and dephasing. Conclu- 2 sions are summarized in Sec.V. where γ is the coupling strength, fi is the Fermi-Dirac distri- bution of the bath and D is a Lindblad operator of the form 1n o II. THE MODEL D[L] = LρLy − LyL; ρ : (7) 2 A. Boundary-driven XX chain deph Similarly, Di in Eq. (5) describes the dephasing on site i, and is given by Consider a one-dimensional spin 1/2 (or free Fermion) sys- deph y tem with L sites, described by the XX (tight-binding) Hamil- Di (ρ) = ΓD[ci ci] (8) tonian where Γ is the dephasing strength. L−1 L X + − − + λ X z If the sites were uncoupled, a bath of the form (6) would H = − σ σ + σ σ + Viσ (1) i i+1 i i+1 2 i lead to the equilibrium state ρeq = f j0ih0j+(1− f ) j1ih1j, where i=1 i z D y E hσ i = 2 f − 1 and c c = f . Hence, the difference ∆ f = fL − L−1 L X X f1 can be interpreted as either a magnetization or a population − y y y = ci ci+1 + ci+1ci + λ Vici ci; (2) imbalance in the chain. As long as ∆ f , 0, the system will i=1 i converge to non-equilibrium steady state (NESS) with a non- zero magnetization (particle) current, given by which are equivalent via a Jordan-Wigner transformation ci = Qi−1 z − j=1(−σi )σi . We consider two models, defined by two D y y E Ji = i c ci − c ci+1 : (9) different choices of the onsite potential Vi, with strength λ i+1 i (Fig.1(a)). The first is the Aubry-André-Harper (AAH) For the internal sites, i = 2;:::; L − 1, these currents satisfy a model, in which the on-site potential is given by [33, 34] continuity equation V = 2 cos(2πgi + θ); (3) i d D y E p ci ci = Ji−1 − Ji; i = 2;:::; L − 1; (10) where g = (1 + 5)=2 is the golden ratio. This model un- dt dergoes a localization transition at λc = 1. When λ < 1, all which is obtained directly from Eq. (5). The sites at energy eigenstates are delocalized, and when λ > 1 they are the boundaries are subject to additional currents J0 = all localized. Similar results also follow when g is any other y y tr c1c1D1(ρ) and JL = tr cLcLDL(ρ) . Crucially, note that Diophantine number [50]. The second system is the Fibonacci the dephasing dissipators do not affect the continuity equa- model, with a potential is defined by [40–43] tion (10). There is, therefore, no particle exchange with them. D E " # " # In the NESS, d cyc =dt = 0. Hence, by Eq. (10) the cur- i + 1 i i i V = − ; (4) rent becomes homogeneous throughout the chain: i g2 g2 J = J = ··· = J ≡ J: (11) where [x] is the integer part of x. This represents the ith ele- 1 2 L−1 ment of a binary sequence called Fibonacci word, which can We can thus unambiguously refer to the particle current sim- be constructed by the recursive rule S n = S n−2 + S n−1, with ply as J. In the spin chain formulation, the particle current S 0 = 0, S 1 = 01 and “+” taken as concatenation. Successive naturally translates to a magnetization current, application of this rule generates the words: D E J = 2i σxσy − σyσx : (12) 0 ! 01 ! 010 ! 01001 ! 01001010 !··· i i i+1 i i+1 This definition can also be obtained by writing an explicit ex- Notice that each Fibonacci word is an extension of the previ- D E pression for d σz =dt and interpreting it as a continuity equa- ous one; Vi is then the ith digit of any word with size greater or i equal to i. Additionally, the length of each word is a Fibonacci tion, similarly to Eq.