Zero temperature superconductor–edge metal–insulator transition in two-dimensional bosonic systems

Motivated by the recent experimental observation of an intermediate bosonic metallic state in the two-dimensional superconductor-insulator transition at $T=0$, we study an extended Bose-Hubbard model in the limit of a large number of particles per site. Using a representation of this in terms of two coupled $XY$ models, we find, in addition to an insulating phase and a $(2+1)\mathrm{D}$ superfluid phase, two other phases. One phase is a 2D superfluid phase where a crossover from $(2+1)\mathrm{D}$ to 2D has taken place as a result of incipient charge ordering, signaled by $\theta$ disordering, and which is closely related to a supersolid phase. The other new phase is an edge metal state characterized by zero superfluid stiffness, zero charge ordering, and zero bulk compressibility. However, the edge compressibility of the system is nonzero. While we do not find any intermediate state with 2D metallic conductivity, we are able to connect these results to STM experiments on ${\mathrm{MoS}}_2$ showing brims of finite density of states around the entire edge of 2D ${\mathrm{MoS}}_2$ samples.

Figure 1

Zero temperature phase diagrams for the coupled $XY$ model in Eq. (12). (a) $V_1-V_0$ plane with $J=0.5$ fixed. (b) $V_1-J$ plane with $V_0=0.5$ fixed. (a) features three distinct phases: a superconducting (SC) phase for low values of $V_1$ and $V_0$, the edge metal (EM) phase characterized by nonzero edge compressibility for intermediate values of $V_1$, and an insulating (I) phase for high values of $V_1$. The edge compressibility is equal to the susceptibility for creating excess charge of equal sign on two neighboring lattice sites. (b) features four phases, namely the three phases mentioned above and in addition a strictly 2D critical superfluid phase (CSC). The transition from 3D to 2D SC for large $J$ is driven by the proliferation of 2D vortices (instantons) in the phase field $\theta$, where $\mathbf{\nabla }\theta$ is a conjugate variable to the excess charge on two neighboring points. These results are obtained from Monte Carlo simulations, described in more detail in Sec. 3, on a $20\times 20\times 20$ system.

Figure 2

Results from Monte Carlo simulations on the coupled $XY$ model in Eq. (13) with $J=0.5, V_0=0.5$ with $L=L_{\tau }=20$. (a) Superfluid stiffness given by Eq. (22), bulk compressibility given by Eq. (23), and edge compressibility given by Eq. (24). (b) Action susceptibility given by Eq. (32). Shows how the system transitions from a superconductor to an edge metal at $V_1\simeq 0.214$ and then from an edge metal to an insulator at $V_1\simeq 0.47$. Error bars are typically smaller than the data points.

Figure 3

Results for the SC-EM phase transition, using the model in Eq. (13) with $J=0.5, V_0=0.5$ and system sizes $L=L_{\tau }=16,20,24,28$. (a) Order parameter of $\varphi$ given by Eq. (30). (b) Action susceptibility given by Eq. (32). (c) Binder cumulant given by Eq. (31). Shows a 3D $XY$-like phase transition at $V_1=0.214$.

Figure 4

Conductivity (per quantum conductance) as a function of the scaling variable $\alpha /n-n/L$, for system parameters $J=V_0=0.5$ and sizes $L=L_{\tau }=16,20,24,28$. The conductivity at the SC-M transition, ${\sigma }^{*}/{\sigma }_Q$, is given by the value at $\alpha /n-n/L=0$.

Figure 5

Results for the EM-I phase transition, using the model in Eq. (13) with $J=0.5, V_0=0.5$ and system sizes $L=L_{\tau }=16,20,24,28$. (a) Edge compressibility given by Eq. (24). (b) Action susceptibility given by Eq. (32). Shows a BKT transition at $V_1\simeq 0.47$.

Figure 6

Measurements of the correlation function given by Eq. (29), using the model in Eq. (13) with $J=0.5, V_0=0.5$ and $L=L_{\tau }=20$. Top left: Three-parameter curve fit of Monte Carlo data to the ansatz in Eq. (40). Normal plot (top right), log-normal plot (bottom left), and log-log (bottom right) plot of the correlation function for different values of $V_1$, indicated by the color bar on the right. In all figures data have been sampled along the $x$ axis, with one point fixed at the origin. Data have also been averaged over all time slices (before MC average).

Figure 7

Results for the SC-CSC phase transition, using the model in Eq. (13) with $J=1.2, V_0=0.5$ and system sizes $L=L_{\tau }=16,20,24,28$. (a) Edge compressibility given by Eq. (24). (b) Action susceptibility given by Eq. (32). (c) Bulk compressibility and superfluid density given by Eqs. (23), (22). Shows a BKT transition in $\theta$ at $V_1\simeq 0.5$ along with a crossover from 3D to 2D ordering of $\varphi$.

Figure 8

Results for the I-CSC phase transition, using the model in Eq. (13) with $V_1=0.8, V_0=0.5$ and $L=L_{\tau }=20$. (a) Bulk compressibility and superfluid density given by Eqs. (23), (22). (b) Action susceptibility given by Eq. (32). Shows a BKT transition in $\varphi$ at $J\simeq 1.03$.

Figure 9

Results for the SC-I phase transition, using the model in Eq. (13) with $J=0.75, V_0=0.5$ and $L=L_{\tau }=20$. (a) Edge compressibility given by Eq. (24). (b) Action susceptibility given by Eq. (32). (c) Order parameter of $\varphi$ given by Eq. (30). Shows a BKT transition in $\theta$ and a 3D $\mathit{XY}$ transition $\varphi$ at $V_1\simeq 0.48$.