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arXiv:2008.08584 · Journal of Physics A Mathematical and Theoretical

The mean-field Bose glass in quasicrystalline systems

Dean Johnstone, Patrik Öhberg, Callum W. Duncan

Published Aug 16, 2021• 21 citations
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Abstract

Abstract We confirm the presence of a mean-field Bose glass (BG) in 2D quasicrystalline Bose–Hubbard models. We focus on two models where the aperiodic component is present in different parts of the problem. First, we consider a 2D generalisation of the Aubry–André (AA) model, where the lattice geometry is that of a square with a quasiperiodic onsite potential. Second, we consider the randomly disordered vertex model, which takes aperiodic tilings with non-crystalline rotational symmetries, and forms lattices from the vertices and lengths of the tiles. For the disordered vertex models, the mean-field BG forms across large ranges of the chemical potential, and we observe no significant differences from the case of a square lattice with uniform random disorder. Small variations in the critical points in the presence of random disorder between quasicrystalline and crystalline lattice geometries can be accounted for by the varying coordination number and the different rotational symmetries present. In the 2D AA model, substantial differences are observed from the usual phase diagrams of crystalline disordered systems. We show that weak modulation lines can be predicted from the underlying potential and may stabilise or suppress the mean-field BG in certain regimes. This results in a lobe-like structure for the mean-field BG in the 2D AA model, which is significantly different from the case of random disorder. Together, the two quasicrystalline models studied in this work show that the mean-field BG phase is present, as expected for 2D quasiperiodic models. However, a quasicrystalline geometry is not sufficient to result in differences from crystalline realisations of the BG, whereas a quasiperiodic form of disorder can result in different physics, as we observe in the 2D AA model.

DOI
10.1088/1751-8121/ac1dc0

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Abstract We confirm the presence of a mean-field Bose glass (BG) in 2D quasicrystalline Bose–Hubbard models. We focus on two models where the aperiodic component is present in different parts of the problem. First, we consider a 2D generalisation of the Aubry–André (AA) model, where the lattice geometry is that of a square with a quasiperiodic onsite potential. Second, we consider the randomly disordered vertex model, which takes aperiodic tilings with non-crystalline rotational symmetries, and forms lattices from the vertices and lengths of the tiles. For the disordered vertex models, the mean-field BG forms across large ranges of the chemical potential, and we observe no significant differences from the case of a square lattice with uniform random disorder. Small variations in the critical points in the presence of random disorder between quasicrystalline and crystalline lattice geometries can be accounted for by the varying coordination number and the different rotational symmetries present. In the 2D AA model, substantial differences are observed from the usual phase diagrams of crystalline disordered systems. We show that weak modulation lines can be predicted from the underlying potential and may stabilise or suppress the mean-field BG in certain regimes. This results in a lobe-like structure for the mean-field BG in the 2D AA model, which is significantly different from the case of random disorder. Together, the two quasicrystalline models studied in this work show that the mean-field BG phase is present, as expected for 2D quasiperiodic models. However, a quasicrystalline geometry is not sufficient to result in differences from crystalline realisations of the BG, whereas a quasiperiodic form of disorder can result in different physics, as we observe in the 2D AA model.
We study the ground state phases of the Bose-Hubbard model with disordered potentials for quasicrystalline systems, with a focus on the Bose-Glass phase. Generally speaking, disorder can lead to the formation of a Bose-Glass, which is characterised by the lack of global phase coherence across the lattice. Here, we will look at two models; the interacting 2D Aubry-Andre model and disordered vertex models from quasicrystalline tiling patterns. Unlike typical disorder in homogeneous, periodic systems, quasicrystalline models possess self-similarity. This leads to a fascinating interplay between correlated, quasiperiodic order and uncorrelated, random disorder. In this work, we combine Gutzwiller mean-field theory with a percolation analysis of superfluid clusters, allowing the critical points and phase regions of these disordered systems to be mapped. When the long-range order is separate to the random disorder, as is the case for the disordered vertex models, then the physics reflects that of periodic lattices with disorder. However, we find that long-range order present in the disorder term of the 2D Aubry-Andre model can result in some peculiarities to the physics of the Bose-Glass. These peculiarities include stabilisation from weak disorder lines and intricate, ordered structures of the phase itself that may provide fruitful areas of future study.
Clarifying similarities and differences in physical properties between crystalline and quasicrystalline systems is one of central issues in studying quasicrystals. To contribute to this, we apply multifractal and hyperuniform analyses to nonuniform spatial patterns in the Bose-Hubbard model on the Penrose and Ammann-Beenker tilings. Based on the mean-field approximation, we obtain real-space distributions of local superfluid amplitude and boson density. In both Mott insulating and superfluid phases, the distributions are hyperuniform. Analyzing the order metric that quantifies the complexity of nonuniform spatial patterns, we find that both quasicrystals show a significant increase of the order metric at a phase boundary between the Mott insulating and superfluid phases, in stark contrast to the case of a periodic square lattice. Our results suggest that hyperuniformity is a useful concept to differentiate between crystalline and quasicrystalline bosonic systems. The order metric clarifies if the distribution of a physical quantity reflects the point distribution or not, and quantifies how complex the distribution is in comparison with the point distribution. Moreover, we introduce on-site random potentials into these quasicrystalline Bose-Hubbard models, leading to a Bose glass phase. Contrary to the Mott insulating and superfluid phases, we find that the Bose glass phase is multifractal. The same multifractality appears on a Bose glass phase in the periodic square lattice. Therefore, multifractality is common in a Bose glass phase irrespective of the periodicity of systems.
Abstract We study the ground state phases of interacting bosons in the presence of a 2D Aubry–André (AA) potential. By using a mean-field percolation analysis, we focus on several superlattice and quasicrystalline regimes of the 2D AA model, including generalisations that account for a tilting or skewing of the potential. We show that barriers to the onset of macroscopic phases naturally arise from weakly modulated domains in the 2D AA model. This leads to the formation of extended crossover domains, in which the macroscopic properties are dominated by a minority of the system. The phase diagrams then exhibit substantially different features when compared against crystalline systems, including a lobe-like or wave-like appearance of the Bose glass, sharp extrusions and thin, percolating clusters. By studying the 2D AA model across multiple regimes, we have shown that these extended crossover domains are not distinct to a small set of parameters.
We study the zero-temperature phase diagram of the disordered spin-111 Bose-Hubbard model in a two-dimensional square lattice. To this aim, we use a mean-field Gutzwiller ansatz and a probabilistic mean-field perturbation theory. The spin interaction induces two different regimes, corresponding to a ferromagnetic and antiferromagnetic order. In the ferromagnetic case, the introduction of disorder reproduces analogous features of the disordered scalar Bose-Hubbard model, consisting in the formation of a Bose glass phase between Mott insulator lobes. In the antiferromagnetic regime, the phase diagram differs more from the scalar case. Disorder in the chemical potential can lead to the disappearance of Mott insulator lobes with an odd-integer filling factor and, for sufficiently strong spin coupling, to Bose glass of singlets between even-filling Mott insulator lobes. Disorder in the spinor coupling parameter results in the appearance of a Bose glass phase only between the nnn and the n+1n+1n+1 lobes for nnn odd. Disorder in the scalar Hubbard interaction inhibits Mott insulator regions for occupation larger than a critical value.
We study the single-particle properties of two-dimensional quasicrystals where the underlying geometry of the tight-binding lattice is crystalline but the on-site potential is quasicrystalline. We will focus on the two-dimensional (2D) generalized Aubry-André model which has a varying form to its quasiperiodic potential, through a deformation parameter and varied irrational periods of cosine terms, which allows a continuous family of on-site quasicrystalline models to be studied. We show that the 2D generalized Aubry-André model exhibits many single-particle mobility edges which we confirm for finite systems and supports critical states across large parameter regions. Critical states are neither fully localized nor extended. We observe that diagonal quasicrystalline models can support many energy intervals of critical states in the spectrum while stabilizing both localized and extended states in other energy intervals; we refer to these as anomalous mobility edges. We show that critical states are present independent of system size through a scaling analysis of the inverse participation ratio and that they are present in spectra that also contain extended and localized states, confirming that at least one anomalous mobility edge is present. Due to this, these models exhibit anomalous diffusion of initially localized states across the majority of parameter regions, including deep in the normally localized regime. The presence of critical states in large parameter regimes and throughout the spectrum will have consequences for the many-body properties of quasicrystals, including the formation of the Bose glass and the potential to host a many-body localized phase. Published by the American Physical Society 2024
We numerically investigate the two-dimensional Bose-Hubbard model with local onsite disorder, where the competition between disorder and short-range interactions leads to the emergence of a Bose glass (BG) phase between the Mott insulator (MI) and superfluid (SF) phases. In order to analyze the inhomogeneous system we employ real-space bosonic dynamical mean-field theory (RBDMFT) and perform an ensemble average over disorder realizations. To distinguish the MI from the BG phase, we compare the Edwards-Anderson order parameter and the compressibility with the energy-gap condition. To identify the insulator to SF transition, we apply a percolation analysis to the condensate order parameter. In qualitative accordance with the theorem of inclusions we always find an intermediate BG phase between the SF and MI. However, the quantitative comparison indicates significant deviations between the MI to BG phase boundary expected in the thermodynamic limit and the one obtained for a finite system size. Additionally, RBMDFT is capable of reliably calculating spectral information throughout the phase diagram. Analyzing the spectral function reveals evidence for analytically predicted damped localized modes in the dispersion relation in the strong-coupling regime.