Analysis and resolution of the ground-state degeneracy of the two-component Bose-Hubbard model

We study the degeneracy of the ground-state energy $E$ of the two-component Bose-Hubbard model and of the perturbative correction $E_1$. We show that the degeneracy properties of $E$ and $E_1$ are closely related to the connectivity properties of the lattice. We determine general conditions under which $E$ is nondegenerate. This analysis is then extended to investigate the degeneracy of $E_1$. In this case, in addition to the lattice structure, the degeneracy also depends on the number of particles present in the system. After identifying the cases in which $E_1$ is degenerate and observing that the standard (degenerate) perturbation theory is not applicable, we develop a method to determine the zeroth-order correction to the ground state by exploiting the symmetry properties of the lattice. This method is used to implement the perturbative approach to the two-component Bose-Hubbard model in the case of degenerate $E_1$ and is expected to be a valid tool to perturbatively study the asymmetric character of the Mott insulator to superfluid transition between the particle and hole side.

Figure 1

Panels (a), (c), and (e) are examples of states in $|\sigma ,\lambda \rangle \mathrm{s}$ for the case of a one-dimensional lattice with periodic boundary condition and $M=6$. These states correspond to $(k_a=1,k_b=1,A=0,B=0)$, $(k_a=1,k_b=1,A=3,B=1)$, and $(k_a=1,k_b=1,A=5,B=3)$, respectively. The color (dark) blue refers to $\mathcal{A}$ bosons and (light) red to $\mathcal{B}$ bosons. They are represented pictorially by panels (b), (d), and (f), where blue (dark) sites form the set $\sigma$, red (light) sites form the set $\lambda$, purple (circled in black) sites form the intersection of $\sigma$ and $\lambda$, and white (circled in gray) sites form the set of sites with neither extra $\mathcal{A}$ nor extra $\mathcal{B}$ bosons.

Figure 2

Panels (a)–(i) represent a sequence of states in $O_g$ with $k_a=k_b=A=0$ and $B=3$ for the case of a connected one-dimensional lattice. The sequence connects states (a) and (i).

Figure 3

Panels (a) and (b) are two unconnected states in $O_g$ for the case of a connected one-dimensional lattice with $A=1,B=3$. Panels (c) and (e) are two connected states in $O_g$ for the case of a 2-connected lattice with $A=1,B=3$. Panel (d) is an intermediate state in the sequence connecting (c) to (e). The two crosses in (d) indicate that the removal of site $j$ leaves the remaining lattice still connected.

Figure 4

Panels (a)–(d) represent states in $O_g$ with $A=1,B=3$ for the case of a lattice which is not 2-connected. The absence of 2-connectivity implies that the removal of the site 3 leaves the remaining lattice unconnected. Panels (a), (b), and (c) are connected with each other, but they are not connected with (d).

Figure 5

5(a) and 5(b) are two unconnected states in $O_g$ for the case of a one-dimensional lattice of $M=6$ sites with periodic boundary condition and $A=3,B=2$.

Figure 6

Panel (a) displays a 2-connected lattice constructed by adding path 2 and 3 to the original circle 1. Panels (b) and (g) represent two connected states in $O_g$ for the case of the lattice represented in (a). Here $M=12$ and $A=4,B=6$. Panels (c)–(f) are intermediate states in the sequence connecting (b) to (g). Panels (h) and (i) show states (d) and (e) on sublattices. Panel (i) also shows three different paths (dashed, dot-dashed, and dotted lines) linking the end points of path 3 (see text).

Figure 7

Panels (a) and (b) display two different ways of viewing the same lattice. They can both be viewed as an already constructed lattice plus an added path. In (a), two ends of the added path (dark green) form a bond on the already constructed lattice (light pink). In (b), two ends of the added path (dark green) does not form a bond on the already constructed lattice (light pink).

Figure 8

Panel (a) displays a lattice automorphism $r$ on an hexagon. Panel (b) displays the action on Fock states of the corresponding operator $S_r$: The lattice is rotated while the physical position of particles is unchanged.

Figure 9

Panels (a)–(e) are examples of states in $O_g$ on a 2-connected lattice with $A=3,B=6$. The steps for moving the (dark) blue color from site $i$ to $j$ are explained in the text and displayed pictorially in the sequence (a)–(e). The dashed arrow indicates a path connecting $i$ and $j$. Black arrows indicate the path along which the white (circled in gray) color is moved at each step.

Figure 10

Panels (a)–(i) represent states in $O_g$ for the case of a lattice consisting of a circle with one added path linking to unbonded sites. In this example $A=3,B=5$. States (a) and (i) are connected through the sequence (b)–(h). Black arrows indicate how the white (circled in gray) color is moved at each step. The dashed circle shows the order of the color which needs to be keep fixed (see text).