Non-ergodic phases of matter
Non-ergodic phases of matter
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The unique property of the many-body localized (MBL) systems to avoid thermalization and to keep information about their initial state opens new horizons in applying such systems in the area of quantum memory and optically controlled systems, (such as ultra cold atoms). As the description of MBL systems is hardly accessible analytically, it is of particular concern and high demand to model essential attributes of such systems in less complicated single-particle systems.
The unique property of the many-body localized (MBL) systems to avoid thermalization and to keep information about their initial state opens new horizons in applying such systems in the area of quantum memory and optically controlled systems, (such as ultra cold atoms). As the description of MBL systems is hardly accessible analytically, it is of particular concern and high demand to model essential attributes of such systems in less complicated single-particle systems.
The paper "A random matrix model with localization and ergodic transitions" which initiated this research direction represents the first rigorous example of a Rosenzweig-Porter random matrix ensemble realizing a robust non-ergodic (multifractal) phase, separated both from the localized and ergodic ones by quantum phase transitions (see Fig. on the left). Typically non-ergodic extended states in disordered media present only at phase transitions and live on fractal structures of the underlying Hilbert space. Unlike this, in the Rosenzweig-Porter model (being just the usual Gaussian random ensemble perturbed by the special diagonal) such fractal states are shown to be robust in this model and forming a phase of fractal extended states with a parametrically narrow miniband in the energy domain.
The paper "A random matrix model with localization and ergodic transitions" which initiated this research direction represents the first rigorous example of a Rosenzweig-Porter random matrix ensemble realizing a robust non-ergodic (multifractal) phase, separated both from the localized and ergodic ones by quantum phase transitions (see Fig. on the left). Typically non-ergodic extended states in disordered media present only at phase transitions and live on fractal structures of the underlying Hilbert space. Unlike this, in the Rosenzweig-Porter model (being just the usual Gaussian random ensemble perturbed by the special diagonal) such fractal states are shown to be robust in this model and forming a phase of fractal extended states with a parametrically narrow miniband in the energy domain.
Recently the above model has become relevant and found applications in quantum computing and a so-called quantum random-energy model with further applications to machine learning, quantum annealing, and artificial intelligence in general and even in nonergodic physics and localization of the quantum Sachdev-Ye-Kitaev model.
Recently the above model has become relevant and found applications in quantum computing and a so-called quantum random-energy model with further applications to machine learning, quantum annealing, and artificial intelligence in general and even in nonergodic physics and localization of the quantum Sachdev-Ye-Kitaev model.
Implementation of quantum search algorithms as well as quantum machine learning are the tasks of extreme importance for the future technologies. These processes are characterized by the hierarchy of specification levels: from global blurred contours to fine local details. This hierarchy and self-similarity of different scales finds immediate representation in fractals, describing various things from forms of growing leaves and coastlines to the wave functions representing the probability of quantum particle locations. Thus, the scientists have an essential demand in the quantum models realizing simple and robust fractal wave functions, both in their distributions in spatial and energy domains.
Implementation of quantum search algorithms as well as quantum machine learning are the tasks of extreme importance for the future technologies. These processes are characterized by the hierarchy of specification levels: from global blurred contours to fine local details. This hierarchy and self-similarity of different scales finds immediate representation in fractals, describing various things from forms of growing leaves and coastlines to the wave functions representing the probability of quantum particle locations. Thus, the scientists have an essential demand in the quantum models realizing simple and robust fractal wave functions, both in their distributions in spatial and energy domains.
Some aspects for boosting weak learners based on the above mentioned nonergodic phases are also under a spotlight in the quantum information community.
Some aspects for boosting weak learners based on the above mentioned nonergodic phases are also under a spotlight in the quantum information community.
In particular, the above robust nonergodic extended states provide a unique opportunity for speeding-up quantum annealing, parallel tempering, or reverse annealing optimization algorithms due to efficient population transfer between low-lying energy function minima (forming a miniband) via coherent multi-qubit tunneling. Due to the extensive number of miniband energies these multifractal states are also useful for swift and accurate sampling of the rare local minima in the Hilbert space (see Fig. on the left) crucially needed for accurate and efficient training for machine learning.
In particular, the above robust nonergodic extended states provide a unique opportunity for speeding-up quantum annealing, parallel tempering, or reverse annealing optimization algorithms due to efficient population transfer between low-lying energy function minima (forming a miniband) via coherent multi-qubit tunneling. Due to the extensive number of miniband energies these multifractal states are also useful for swift and accurate sampling of the rare local minima in the Hilbert space (see Fig. on the left) crucially needed for accurate and efficient training for machine learning.
Thus, the search for other models bearing non-ergodic extended phases is of particular interest here. The drawback of the Rosenzweig-Porter model is that its non-ergodic phase shows only fractal properties (all fractal dimensions are the same Dq = 2 – γ for any moment q) due to its simplicity. As shown by us in two recent preprints [arXiv:2002.02979, arXiv:2006.04827] in order to realize a genuine multifractal phase of matter one should make the distribution of off-diagonal matrix elements fat-tailed (e.g., logarithmically normal distributed).
Thus, the search for other models bearing non-ergodic extended phases is of particular interest here. The drawback of the Rosenzweig-Porter model is that its non-ergodic phase shows only fractal properties (all fractal dimensions are the same Dq = 2 – γ for any moment q) due to its simplicity. As shown by us in two recent preprints [arXiv:2002.02979, arXiv:2006.04827] in order to realize a genuine multifractal phase of matter one should make the distribution of off-diagonal matrix elements fat-tailed (e.g., logarithmically normal distributed).
The parameter p controlling the fatness of the tails, deforms the phase diagram of the model and unexpectedly shrinks the multifractal phase and even collapses it at p>1. In order to resolve this paradox one has to distinct between the conventional fully-ergodic phase and a novel weakly-ergodic phase. This new weakly ergodic phase is characterized by the breakdown of the basis rotation invariance and is very likely related to the anomalous diffusion predicted and observed on the ergodic side of the many-body localization transition in various strongly interacting models.
The parameter p controlling the fatness of the tails, deforms the phase diagram of the model and unexpectedly shrinks the multifractal phase and even collapses it at p>1. In order to resolve this paradox one has to distinct between the conventional fully-ergodic phase and a novel weakly-ergodic phase. This new weakly ergodic phase is characterized by the breakdown of the basis rotation invariance and is very likely related to the anomalous diffusion predicted and observed on the ergodic side of the many-body localization transition in various strongly interacting models.
It is this weakly ergodic phase which "eats" the multifractal phase with increasing fatness of the tails. However, as we show in the above works, any cutoff of the tails shows the fragileness of this phase which immediately leaves the stage back to the multifractality.
It is this weakly ergodic phase which "eats" the multifractal phase with increasing fatness of the tails. However, as we show in the above works, any cutoff of the tails shows the fragileness of this phase which immediately leaves the stage back to the multifractality.
Another example of robust multifractality demonstrates a phase separation of ergodic, localized and non-ergodic extended states with an entire multifractal sub-band emerging in a single-particle system. This is an example of robust multifractality in the system with short-range hopping and quasiperiodic "disorder" under a generic out-of-equilibrium periodically driven conditions. In both cases multifractal states are robust to perturbations and are found at a whole range of parameters.
Another example of robust multifractality demonstrates a phase separation of ergodic, localized and non-ergodic extended states with an entire multifractal sub-band emerging in a single-particle system. This is an example of robust multifractality in the system with short-range hopping and quasiperiodic "disorder" under a generic out-of-equilibrium periodically driven conditions. In both cases multifractal states are robust to perturbations and are found at a whole range of parameters.
The question about the existence of a robust non-ergodic extended phase in many-body systems and of two corresponding quantum phase transitions instead of the only many body localization transition encouraged us to investigate the multifractal phenomena in a paradigmatic XXZ Heisenberg spin-1/2 chain model. The state-of-art extensive numerical simulations demonstrate the presence of the nonergodic extended phase before system reaches localization.
The question about the existence of a robust non-ergodic extended phase in many-body systems and of two corresponding quantum phase transitions instead of the only many body localization transition encouraged us to investigate the multifractal phenomena in a paradigmatic XXZ Heisenberg spin-1/2 chain model. The state-of-art extensive numerical simulations demonstrate the presence of the nonergodic extended phase before system reaches localization.
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