Interplay of correlations and localization
in long-range disordered systems
Interplay of correlations and localization
in long-range disordered systems
Many-body localization is rather complicated phenomenon for the analytical description due to the exponential growth of the Hilbert space with the system size. But even in the single-particle setting, the Anderson localization transition itself is of crucial importance for the fields of condensed matter theory, quantum information, machine learning, and computer science as well as for statistical physics. The understanding of localization-delocalization principles of conducting electrons and of impurity effects become more and more important also due to the miniaturization of transistors in state-of-art computers, which are already close to the limit of the fully quantum transport. It is especially important because of the fact that the Anderson localization in low-dimensional systems, such as thin films and narrow wires, occurs at any small impurity concentration.
Many-body localization is rather complicated phenomenon for the analytical description due to the exponential growth of the Hilbert space with the system size. But even in the single-particle setting, the Anderson localization transition itself is of crucial importance for the fields of condensed matter theory, quantum information, machine learning, and computer science as well as for statistical physics. The understanding of localization-delocalization principles of conducting electrons and of impurity effects become more and more important also due to the miniaturization of transistors in state-of-art computers, which are already close to the limit of the fully quantum transport. It is especially important because of the fact that the Anderson localization in low-dimensional systems, such as thin films and narrow wires, occurs at any small impurity concentration.
To overcome this problem the effects of electron-photon and electron-phonon scattering causing long-range jumps of electrons have been considered in the literature. The possibility for excitations to jump far away from parent ions, helps to overcome disorder effects and restores the diffusive character of the transport at moderate impurity concentrations.
To overcome this problem the effects of electron-photon and electron-phonon scattering causing long-range jumps of electrons have been considered in the literature. The possibility for excitations to jump far away from parent ions, helps to overcome disorder effects and restores the diffusive character of the transport at moderate impurity concentrations.
However, the presence of long-range excitation jumps is not enough for the delocalization. Indeed, as is shown in our work "Correlation-induced localization" the correlations in long-range hopping, which are present in many (e.g., dipolar) models, change drastically the localization-delocalization phase diagram (see Fig. on the left) and cause even the localization of almost all eigenstates or the formation of multifractal state, depending on the correlation properties.
However, the presence of long-range excitation jumps is not enough for the delocalization. Indeed, as is shown in our work "Correlation-induced localization" the correlations in long-range hopping, which are present in many (e.g., dipolar) models, change drastically the localization-delocalization phase diagram (see Fig. on the left) and cause even the localization of almost all eigenstates or the formation of multifractal state, depending on the correlation properties.
A new paradigm of Anderson localization caused by correlations in the long-range hopping along with onsite disorder is considered which requires an essential revision of the basic localization-delocalization principles.
A new paradigm of Anderson localization caused by correlations in the long-range hopping along with onsite disorder is considered which requires an essential revision of the basic localization-delocalization principles.
The divergence of the spectrum of long-range hopping and the newly developed matrix inversion trick result in the inequivalence of the Anderson's resonance counting and Mott's ergodicity principle, which opens the gap for robust non-ergodic extended phases.
The divergence of the spectrum of long-range hopping and the newly developed matrix inversion trick result in the inequivalence of the Anderson's resonance counting and Mott's ergodicity principle, which opens the gap for robust non-ergodic extended phases.
A new class of random Hamiltonians with translation-invariant hopping (see Fig. in the very top) and the duality with respect to the Floquet transform is suggested and characterized in detail (see the phase diagram in Fig. on the right).
A new class of random Hamiltonians with translation-invariant hopping (see Fig. in the very top) and the duality with respect to the Floquet transform is suggested and characterized in detail (see the phase diagram in Fig. on the right).
Again due to the divergent spectrum of disorder-free hopping model, Euclidean matrices hold a few high-energy delocalized states which accumulate the main hopping weight and lead to the localization in the bulk of the spectrum.
Again due to the divergent spectrum of disorder-free hopping model, Euclidean matrices hold a few high-energy delocalized states which accumulate the main hopping weight and lead to the localization in the bulk of the spectrum.
As the complement method to the matrix inversion trick, one can consider the spatial renormalization group considered in our work "Renormalization to localization without a small parameter". Unlike the matrix inversion trick, it works for smooth enough hopping potentials, but generalize the latter to the case of the only off-diagonal disorder.
As the complement method to the matrix inversion trick, one can consider the spatial renormalization group considered in our work "Renormalization to localization without a small parameter". Unlike the matrix inversion trick, it works for smooth enough hopping potentials, but generalize the latter to the case of the only off-diagonal disorder.
As an example one can consider the whole class of Euclidean random matrices of randomly distributed particles with long-range hopping potentials. Such models arise naturally in various systems, such as the ones with non-crystalline structures like gases, liquids, amorphous materials, and glasses. Long-range Euclidean matrices are applied to the analysis of the systems of particles with Coulomb interactions in two-dimensional irregular confinement, disordered classical Heisenberg magnets with uniform antiferromagnetic interactions, systems with dipole-dipole interactions such as dipolar gases, systems of ultracold Rydberg atoms and even to describe the effects of photon localization in atomic gases.
As an example one can consider the whole class of Euclidean random matrices of randomly distributed particles with long-range hopping potentials. Such models arise naturally in various systems, such as the ones with non-crystalline structures like gases, liquids, amorphous materials, and glasses. Long-range Euclidean matrices are applied to the analysis of the systems of particles with Coulomb interactions in two-dimensional irregular confinement, disordered classical Heisenberg magnets with uniform antiferromagnetic interactions, systems with dipole-dipole interactions such as dipolar gases, systems of ultracold Rydberg atoms and even to describe the effects of photon localization in atomic gases.
Two-dimensional dilute systems more relevant for experimental realizations in ultracold atoms and trapped ions, are usually characterized not only by the long-ranged interactions mentioned above, but also by the embedded anisotropy of a dipolar kind .
Two-dimensional dilute systems more relevant for experimental realizations in ultracold atoms and trapped ions, are usually characterized not only by the long-ranged interactions mentioned above, but also by the embedded anisotropy of a dipolar kind .
In the literature anisotropic terms are usually considered as quasi-disorder and may lead to the localization-delocalization transition. However, as shown in our work "Anisotropy-driven localization transition in quantum dipoles" the situation is more subtle as anisotropy does not always in favor of delocalization and may give rise to the reentrant localization phase diagram (see Fig. on the right).
In the literature anisotropic terms are usually considered as quasi-disorder and may lead to the localization-delocalization transition. However, as shown in our work "Anisotropy-driven localization transition in quantum dipoles" the situation is more subtle as anisotropy does not always in favor of delocalization and may give rise to the reentrant localization phase diagram (see Fig. on the right).
Remarkably, we show both analytically and numerically that in 2d dipolar system with power-law interactions (1- β cos2фr)/ra the localization extends beyond the usual convergence criteria of the locator expansion, a>d, and this extension is more pronounces both at small β<<1 (nearly isotropic) and large β>>1 (strongly anisotropic) tilt of dipoles. This effect is explained with help of the duality of the system with respect to the transformation β → β/(β-1).
Remarkably, we show both analytically and numerically that in 2d dipolar system with power-law interactions (1- β cos2фr)/ra the localization extends beyond the usual convergence criteria of the locator expansion, a>d, and this extension is more pronounces both at small β<<1 (nearly isotropic) and large β>>1 (strongly anisotropic) tilt of dipoles. This effect is explained with help of the duality of the system with respect to the transformation β → β/(β-1).
The above models are just the very peak of an iceberg of the field. Immediate steps in this direction include the simulation of the non-ergodic extended phases of matter (related to the corresponding direction) with Poisson level statistics as well as the effects of time-reversal symmetry breaking and partial correlations (such as the ones considered in the work "Robustness of delocalization to the inclusion of soft constraints in long-range random models").
The above models are just the very peak of an iceberg of the field. Immediate steps in this direction include the simulation of the non-ergodic extended phases of matter (related to the corresponding direction) with Poisson level statistics as well as the effects of time-reversal symmetry breaking and partial correlations (such as the ones considered in the work "Robustness of delocalization to the inclusion of soft constraints in long-range random models").