We considered a Bose-Hubbard model on a two-dimensional lattice. We showed that the model parameters (the lattice structure, the hopping constants, and the chemical potential) can be tuned in such a way that the ground state is generated by applying, to the vacuum, N creation operators of locally-defined quasi-particles, where N is proportional to the total number of lattice points. The resulting model has a free parameter β that controls the degree of quantum fluctuation; β = 0 for no fluctuation and β = 1/2 for the maximum fluctuation. We carried out Monte Carlo simulation of this model at zero temperature, and discovered that the model undergoes a quantum phase transition when we change this parameter. While the system is 2+1 dimensional, the new phase transition exhibits the characteristics of 2 dimensions, i.e., the Kosterlitz-Thouless phase transition analogous to the classical XY model in 2 dimensions. Accordingly, the large β phase is the quasi Bose-Einstein condensation phase.

The above-mentioned ground state is expressed exactly as a tensor network with the bond dimension 2. Tracing out this tensor network is equivalent to calculating the partition function of some classical loop gas model. This is why the quantum criticality exhibits the purely 2-dimensional characteristics. It also allows us to construct a Monte Carlo algorithm for efficient sampling from the space of classical occupation-number configurations. In other words, we can stochastically trace out the tensor network to obtain various physical quantities of interest, such as the static structure factors and the helicity modulus.

Fig. 1. The lattice considered (p=1). Within each cluster specified by x or u, a tuned hopping between the black and the white sites is defined.

The lattice is generated from the square lattice by duplicating each bond into p copies and introducing an additional site in the middle of each copy. (Fig. 1) The direct hopping is allowed only within local clusters labelled by either x or u. While bosons interact with each other, the model is constructed in such a way that the dispersion of a single particle has a flat band if influence of the other paricles could be neglected. The result of the Monte Carlo simulation are shown in Fig. 2. The static structure factor diverges around β = 0.2 and, at the same time, the helicity modulus jumps from 0 to 2/π, the universal value characteristic to the KT transition.

Fig. 2. The static structure factor S (top) and the helicity modulus (bottom) as functions of β for p=2. The horizontal line in the bottom panel indicates the universal jump 2/π, the thermodynamic value characteristic to the KT transition point.

References

• [1] H. Katsura, N. Kawashima, S. Morita, A. Tanaka, and H. Tasaki, Phys. Rev. Research, 3 033190 (2021).

Authors

• H. Katsura^a, N. Kawashima, S. Morita, A. Tanaka^b, and H. Tasaki^c
• ^aThe University of Tokyo
• ^bNational Institute of Technology
• ^cGakushuin University