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arXiv:2311.07375 · Physical review. B./Physical review. B

Superdiffusive magnetization transport in the XX spin chain with nonlocal dephasing

Marko Žnidarič

Published Feb 5, 2024• 9 citations
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Abstract

We study a recently discussed XX spin chain with nonlocal dephasing [Y.-P. Wang et al., arXiv:2310.03069] in a steady-state boundary-driven setting, confirming superdiffusive magnetization transport in the thermodynamic limit. The emergence of superdiffusion is rather interesting as the Lindblad operators causing it are a coherent sum of two terms, each of which would separately cause diffusion. One therefore has a quantum phenomenon where a coherent sum of two diffusive terms results in superdiffusion. We also study perturbations of the superdiffusive model, finding that breaking the exact form of dissipators, as well as adding interactions to the XX chain, results in superdiffusion changing into diffusion.

DOI
10.1103/physrevb.109.075105

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We study a recently discussed XX spin chain with nonlocal dephasing [Y.-P. Wang et al., arXiv:2310.03069] in a steady-state boundary-driven setting, confirming superdiffusive magnetization transport in the thermodynamic limit. The emergence of superdiffusion is rather interesting as the Lindblad operators causing it are a coherent sum of two terms, each of which would separately cause diffusion. One therefore has a quantum phenomenon where a coherent sum of two diffusive terms results in superdiffusion. We also study perturbations of the superdiffusive model, finding that breaking the exact form of dissipators, as well as adding interactions to the XX chain, results in superdiffusion changing into diffusion.
We calculate a nonequilibrium steady state of a quantum XX chain in the presence of dephasing and driving due to baths at chain ends. The obtained state is exact in the limit of weak driving while the expressions for one- and two-point correlations are exact for an arbitrary driving strength. In the steady state the magnetization profile and the spin current display diffusive behavior. The spin–spin correlation function, on the other hand, has long-range correlations which, though, decay to zero in either the thermodynamical limit or for equilibrium driving. At zero dephasing a nonequilibrium phase transition occurs from a ballistic transport having short-range correlations to a diffusive transport with long-range correlations.
Quantum transport in nonequilibrium settings plays a fundamental role in understanding the properties of systems ranging from quantum devices to biological systems. Dephasing---a key aspect of out-of-equilibrium systems---arises from interactions with a noisy environment and can profoundly modify transport properties. Here we investigate the impact of dephasing on the nonequilibrium steady-state transport properties of noninteracting fermions on a one-dimensional lattice with long-range hopping (proportional to $1/{r}^{\ensuremath{\alpha}}$), where $\ensuremath{\alpha}>1$. We demonstrate the emergence of distinct transport regimes as the long-range hopping parameter, $\ensuremath{\alpha}$, is tuned. In the short-range limit ($\ensuremath{\alpha}\ensuremath{\gg}1$), transport is diffusive. Conversely, in the long-range limit $[\ensuremath{\alpha}\ensuremath{\sim}\mathcal{O}(1)]$, we observe a superdiffusive transport regime. Using numerical simulations of the Lindblad master equation and corroborating these with an analysis of the current-operator norm, we identify a critical long-range hopping parameter, ${\ensuremath{\alpha}}_{c}\ensuremath{\approx}1.5$, below which superdiffusive transport becomes pronounced and rapidly becomes independent of the dephasing strength. Our results elucidate the intricate balance between dephasing and unitary dynamics, revealing steady-state transport features.
We study transport in the boundary-driven XX spin chain with onsite disorder and randomly positioned onsite dephasing, observing a transition from diffusive to subdiffusive spin transport below a critical density of sites with dephasing. We then present an exactly solvable semiclassical model of conductors and insulators, which exhibits both diffusive and subdiffusive phases, and qualitatively reproduces the results of the quantum system. The subdiffusion in these models is a consequence of rare insulating regions, and they therefore exhibit, in a more controlled setup, the physics of which have been conjectured to cause the subdiffusive transport observed in interacting many-body localizable systems. For the quantum model we show that finite-size effects come from the interplay of three characteristic lengths: one associated with disorder (the localization length), one with dephasing, and the third with the percolation problem defining large, rare, insulating regions. The latter grows logarithmically with system size, and we conjecture that this may be the reason why the heavy-tailed distributions typical of Griffiths effects have not been observed in subdiffusive interacting systems.
In this work we study the heat transport in an XXZ spin-1/2 Heisenberg chain with homogeneous magnetic field, incoherently driven out of equilibrium by reservoirs at the boundaries. We focus on the effect of bulk dephasing (energy-dissipative) processes in different parameter regimes of the system. The non-equilibrium steady state of the chain is obtained by simulating its evolution under the corresponding Lindblad master equation, using the time evolving block decimation method. In the absence of dephasing, the heat transport is ballistic for weak interactions, while being diffusive in the strongly-interacting regime, as evidenced by the heat-current scaling with the system size. When bulk dephasing takes place in the system, diffusive transport is induced in the weakly-interacting regime, with the heat current monotonically decreasing with the dephasing rate. In contrast, in the strongly-interacting regime, the heat current can be significantly enhanced by dephasing for systems of small size.
In order to reveal mechanisms to control and manipulate spin currents, we perform a detailed investigation of the dephasing effects in the open XX model with a Lindblad dynamics involving global dissipators and thermal baths. Specifically, we consider dephasing noise modeled by current-preserving Lindblad dissipators acting on graded versions of these spin systems, that is, systems in which the magnetic field and/or the spin interaction are growing (decreasing) along the chain. In our analysis, we study the nonequilibrium steady state via the covariance matrix using the Jordan-Wigner approach to compute the spin currents. We find that the interplay between dephasing and graded systems gives rise to a nontrivial behavior: When we have homogeneous magnetic field and graded interactions we have rectification enhancement mechanisms, and when we have fully graded systems we can control the spin current in order to keep the direction of the particle and/or spin flow even with inverted baths. We describe our result in detailed numerical analysis and we see that rectification in this simple model indicates that the phenomenon may generally occur in quantum spin systems.
Finite-temperature spin transport in integrable isotropic spin chains (i.e., spin chains with continuous non-Abelian symmetries) is known to be superdiffusive, with anomalous transport properties displaying remarkable robustness to isotropic integrability-breaking perturbations. Using a discrete-time classical model, we numerically study the crossover to conventional diffusion resulting from both noisy and Floquet isotropic perturbations of strength $\ensuremath{\lambda}$. We identify an anomalously-long crossover timescale ${t}_{★}\ensuremath{\sim}{\ensuremath{\lambda}}^{\ensuremath{-}\ensuremath{\alpha}}$ with $\ensuremath{\alpha}\ensuremath{\approx}6$ in both cases. We discuss our results in terms of a kinetic theory of transport that characterizes the lifetimes of large solitons responsible for superdiffusion.