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Register a new userDuality in power-law localization in disordered one-dimensional systems
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The transport of excitations between pinned particles in many physical systems may be mapped to single-particle models with power-law hopping, $1/r^a$. For randomly spaced particles, these models present an effective peculiar disorder that leads to surprising localization properties. We show that in one-dimensional systems almost all eigenstates (except for a few states close to the ground state) are power-law localized for any value of $a>0$. Moreover, we show that our model is an example of a new universality class of models with power-law hopping, characterized by a duality between systems with long-range hops ($a<1$) and short-range hops ($a>1$) in which the wave function amplitude falls off algebraically with the same power $gamma$ from the localization center.
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One-dimensional quasi-periodic systems with power-law hopping, $1/r^a$, differ from both the standard Aubry-Azbel-Harper (AAH) model and from power-law systems with uncorrelated disorder. Whereas in the AAH model all single-particle states undergo a transition from ergodic to localized at a critical quasi-disorder strength, short-range power-law hops with $a>1$ can result in mobility edges. Interestingly, there is no localization for long-range hops with $aleq 1$, in contrast to the case of uncorrelated disorder. Systems with long-range hops are rather characterized by ergodic-to-multifractal edges and a phase transition from ergodic to multifractal (extended but non ergodic) states. We show that both mobility and ergodic-to-multifractal edges may be clearly revealed in experiments on expansion dynamics.
We show that in the regime when strong disorder is more relevant than field quantization the superfluid--to--Bose-glass criticality of one-dimensional bosons is preceded by the prolonged logarithmically slow classical-field renormalization flow of the superfluid stiffness at mesoscopic scales. With the system compressibility remaining constant, the quantum nature of the system manifests itself only in the renormalization of dilute weak links. On the insulating side, the flow ultimately reaches a value of the Luttinger parameter at which the instanton--anti-instanton pairs start to proliferate, in accordance with the universal quantum scenario. This happens first at astronomic system sizes because of the suppressed instanton fugacity. We illustrate our result by first-principles simulations.
We investigate the transition induced by disorder in a periodically-driven one-dimensional model displaying quantized topological transport. We show that, while instantaneous eigenstates are necessarily Anderson localized, the periodic driving plays a fundamental role in delocalizing Floquet states over the whole system, henceforth allowing for a steady state nearly-quantized current. Remarkably, this is linked to a localization/delocalization transition in the Floquet states of a one dimensional driven Anderson insulator, which occurs for periodic driving corresponding to a nontrivial loop in the parameter space. As a consequence, the Floquet spectrum becomes continuous in the delocalized phase, in contrast with a pure-point instantaneous spectrum.
We analyze the effects of disorder on the correlation functions of one-dimensional quantum models of fermions and spins with long-range interactions that decay with distance $ell$ as a power-law $1/ell^alpha$. Using a combination of analytical and numerical results, we demonstrate that power-law interactions imply a long-distance algebraic decay of correlations within disordered-localized phases, for all exponents $alpha$. The exponent of algebraic decay depends only on $alpha$, and not, e.g., on the strength of disorder. We find a similar algebraic localization for wave-functions. These results are in contrast to expectations from short-range models and are of direct relevance for a variety of quantum mechanical systems in atomic, molecular and solid-state physics.
We investigate the phase transition between an ergodic and a many-body localized phase in infinite anisotropic spin-$1/2$ Heisenberg chains with binary disorder. Starting from the Neel state, we analyze the decay of antiferromagnetic order $m_s(t)$ and the growth of entanglement entropy $S_{textrm{ent}}(t)$ during unitary time evolution. Near the phase transition we find that $m_s(t)$ decays exponentially to its asymptotic value $m_s(infty) eq 0$ in the localized phase while the data are consistent with a power-law decay at long times in the ergodic phase. In the localized phase, $m_s(infty)$ shows an exponential sensitivity on disorder with a critical exponent $ usim 0.9$. The entanglement entropy in the ergodic phase grows subballistically, $S_{textrm{ent}}(t)sim t^alpha$, $alphaleq 1$, with $alpha$ varying continuously as a function of disorder. Exact diagonalizations for small systems, on the other hand, do not show a clear scaling with system size and attempts to determine the phase boundary from these data seem to overestimate the extent of the ergodic phase.
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