MATERIALS SCIENCE AND CHEMISTRY

Abstract

Within the project we numerically investigated 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 [1]. To solve the inhomogeneous system, we employed real-space bosonic dynamical mean-field theory [2], which maps the complicated many-body problem to a collection of numerically solvable impurity models. Within our approach we always find an intermediate BG phase between the SF and MI. Analyzing the spectral function in the strong coupling regime reveals evidence for analytically predicted damped localized modes in the dispersion relation [5].

Report

Within our project we worked with a disordered variant of the Bose-Hubbard model on a square lattice. The Hubbard model features three main parameters namely the Hubbard U, which models the short-range repulsion two particles feel when they are close to each other and the Hubbard J, which is connected to the tunneling rate between two adjacent sites. Lastly, there is also the chemical potential μ , which sets the filling of the lattice. The disorder is introduced in the form of a random potential ϵ_i, which changes the chemical potential at each site.

According to previous studies [1], the competition between interaction and disorder leads to the emergence of a new insulating phase called the Bose glass phase. Pictorially, this phase may be explained as many small puddles of superfluidity enclosed by insulating background such that overall no superfluidity can emerge. Yet, due to these puddles the Bose glass phase is distinct from the Mott insulating phase as its spectrum is gapless. In this case this means the spectral function in the Bose glass phase has a non-zero value at the ω = 0 frequency point.

Additionally, the disorder also changes the excitation spectrum in the Mott insulator as shown previously, analytically in [5]. In the following illustration, we depict the k = 0 mode of the spectral function the disordered Bose-Hubbard model on a square lattice for different disorder strengths as calculated using our numerical RBDMFT algorithm. Higher amplitudes indicate that an excitation with energy ℏ ω is favored by the system. The resonant frequency tracks the so-called stable excitation in the system because it is a very narrow peak in the spectral density, which indicates a long lifetime. The resonant peak is dressed by a background constituting damped localized states, which are only present in the disordered model. Following the resonant frequency for larger disorder values for low momentum reveals that its frequency is first to go to zero, which indicates that it drives the transition to the gapless Bose glass phase. These are two key predictions of previous analytical work, which we were able to support using our numerical analysis. Overall, we find very good agreement in the excitation spectrum between our self-consistent numerical calculation using real-space bosonic dynamical mean field theory and the analytic approach via a strong coupling expansion.

On top of the spectral analysis we also investigated possible diagnostics for the different quantum phase transitions. In contrast to classical phase transitions, which are driven by thermal fluctuations, quantum phase transitions are driven by quantum fluctuations and can be accessed by varying physical parameters, such as the energy scales in the Bose–Hubbard Hamiltonian, even at zero temperature. Depending on the parameters U, J, μ and ϵ_i  the particles in the lattice system can be found in either of the three possible phases. The Mott insulator is an insulating phase with a finite integer number of particles per site with no fluctuation of the particle number and thus a vanishing compressibility and no superfluidity. The Bose glass phase only emerges in the disordered system and is another insulating phase with no global superfluidity but with local fluctuation of the particle number leading to a non-zero compressibility. For large enough hopping with respect to the interaction the system enters a superfluid phase, which is characterized by delocalized particles. In this phase excitations are be produced without energy cost and flow frictionless through the system. In order to obtain an appropriate diagnostics for the insulator to superfluid transition of the disordered system, we employ a percolation analysis similar to [4] applied to the local condensate order parameter. To distinguish the Mott from the Bose glass phase the Edwards-Anderson order parameter, which is a measure of the variance in local occupation, performs similar to the compressibility. We elaborate the discussion of the phase diagram and spectral function in [6] and show an evolution of the spectral function from the Mott through the Bose glass to the superfluid phase in the case of unit filling in the provided clip.

References

[1] M. P. A. Fisher et al. en. In: Physical Review B 40.1 (July 1989), p. 546.

[2] M. Snoek et al. en. In: Cold Atoms. Vol. 1. IMPERIAL COLLEGE PRESS, Apr. 2013, p. 355.

[3] V. Gurarie et al. en. In: Physical Review B 80.21 (Dec. 2009), p. 214519.

[4] A. E. Niederle et al. In: New Journal of Physics 15.7 (2013). Publisher: IOP Publishing, p. 075029.

[5] R. S. Souza et al. In: New Journal of Physics 25.6 (2023). Publisher: IOP Publishing, p. 063015.

[6] B. Schindler et al. arXiv preprint: arXiv:2509.01230.