Rhombi-chain Bose-Hubbard model: Geometric frustration and interactions

We explore the effects of geometric frustration within a one-dimensional Bose-Hubbard model using a chain of rhombi subject to a magnetic flux. The competition of tunneling, self-interaction, and magnetic flux gives rise to the emergence of a pair-superfluid (pair-Luttinger liquid) phase besides the more conventional Mott-insulator and superfluid (Luttinger liquid) phases. We compute the complete phase diagram of the model by identifying characteristic properties of the pair-Luttinger liquid phase such as pair correlation functions and structure factors and find that the pair-Luttinger liquid phase is very sensitive to changes away from perfect frustration (half-flux). We provide some proposals to make the model more resilient to variants away from perfect frustration. We also study the bipartite entanglement properties of the chain. We discover that, while the scaling of the block entropy pair-superfluid and of the single-particle superfluid leads to the same central charge, the properties of the low-lying entanglement spectrum levels reveal their fundamental difference.

Figure 1

One-dimensional lattice of rhombi with $M$ cells and $L$ sites. Each cell contains three sites labeled $A,B,C$. Solid (dashed) line connections indicate a tunneling amplitude $-J(-J{\mathrm{e}}^{i\varphi })$. (a) is the a representation of the model in real space showing the number of sites, whereas (b) is a schematic diagram to illustrate the cells more clearly. (c) is an illustration of a restricted single-particle tunneling for the fully frustrated system.

Figure 2

The single-particle energy band $E\left(k\right)$ as a function of the lattice momentum $k$. The fully frustrated $\varphi =\pi$ bands are the solid lines, whereas the nonfrustrated $\varphi =0$ are shown by dashed lines. An intermediate frustration (at $\varphi =\frac{\pi }{6})$ is shown by the dotted lines.

Figure 3

Illustration of the type of cage terms that control the movement and the interaction of the pairs of particles. $\tilde{U}$ is the on-cell interaction of pairs, $\tilde{J}$ is the hopping of pairs between cells, and $\tilde{V}$ can be interpreted as nearest-neighbor interaction or correlated swapping of two particles across neighboring sites.

Figure 4

The phase-diagram with different phase shifts $\varphi$ against the tunneling coefficient $J/U$. The Luttinger liquid (LL), pair-Luttinger liquid (PLL), and the Mott insulator (MI) regions are labeled. The critical points delimiting the MI region are obtained from the energy gap as in Sec. 4. The critical points separating the LL and PLL phases are obtained by looking at the decay of the single and pair correlation functions. The error bars have been omitted when they are smaller than the marker size. (a) is the full variation of $\varphi$ using a filling=1, while (b) and (c) are regions close to full frustration for filling=1 and filling=2, respectively. The $★$ in (a) denotes $\tilde{G}\left(\pi \right)$, where Vidal and Douçot predicted the LL-PLL transition to be at $\varphi =\pi$. The dashed line represents their MI-LL transition prediction, $G^{*}\left(\varphi \right)$ in Eq. (13), which is only valid for small $\varphi$ values.

Figure 5

Ground-state phase diagram at unit filling $N=L$ and full frustration $\varphi =\pi$ in the $\mu /U-J/U$ plane. Circles and squares represent the numerical data for ${\mu }_{+}$ and ${\mu }_{-}$, respectively. The region ${\mu }_{-}<\mu <{\mu }_{+}$ is the Mott-insulator lobe. The two lines are cubic splines approximations. Their crossing occurs at the BKT point $J_c\approx (0.78\pm 0.03)U$ indicated by the shaded region.

Figure 6

Illustration of the spread of the on-site density $\langle {\widehat{n}}_{\alpha }\rangle$ averaged over the lattice of 226 sites for $\chi =300$ as the tunneling and frustration is varied. Here the nonfrustrated $\left(\varphi =0\right)$ is shown by the red circles $(\circ )$, the intermediate frustration $\left(\varphi =0.9\pi \right)$ is shown by the green dotted pentagrams $\left(★\right)$, and the fully frustrated $\left(\varphi =\pi \right)$ are the blue dashed diamonds $(\diamond )$. The density of the $B$ sites is shown by filled markers and sites $A$ and $C$ (which have equal on-site density) are shown by the empty markers. Again the approximate critical points are highlighted by the shaded regions at $J_c^{[\varphi =0]}=0.14$ (red), $J_c^{[\varphi =0.9\pi ]}=0.33$ (green), and $J_c^{[\varphi =\pi ]}=0.78$ (blue). The inset is a zoomed in version close to the $\varphi =0$ MI-LL transition.

Figure 7

The spatial correlation functions against the intersite distance for a intermediate frustrated case $(\varphi =0.5\pi$ and $J=0.9U)$ in (a) and for the fully frustrated case $\left(\varphi =\pi \right)$ in (b)–(d) for a chain of length $L=226$ with DMRG bond dimension $\chi =300$. In (a)–(c), values below a threshold of ${10}^{-10}$, which constitutes numerical error, have been excluded. In the diagrams, ${\eta }_1$ (dark blue $◯)$ is the largest eigenvalue, ${\eta }_2$ (red $*)$ is the second largest and ${\eta }_3$ (green $\square )$ is the smallest eigenvalue. [(b) and (c)] The decay of the eigenvalues of the pair correlations $\langle {\widehat{b}}_i^{{†}^2}{\widehat{b}}_{i+r}^{{}^2}\rangle$ for (b) $J=0.4U$ in semi-logarithmic scale and for (c) $J=0.9U$ in a double logarithmic scale. The inset is the decay of the eigenvalues of the correlation matrices for the corresponding single correlations $\langle {\widehat{b}}_i^{†}{\widehat{b}}_{i+r}\rangle$ at $J=0.4$ (MI phase) and $J=0.9U$ (PLL phase). (d) The pair correlations for the $B$ sites $\langle {\widehat{b}}_i^{{†}^2}{\widehat{b}}_{i+r}^{{}^2}\rangle$ from $i\approx L/4$ for $J/U=0.9$ and the corresponding ${\eta }_1^{\left[p\right]*}$ values, which are the maximum eigenvalues retaining their sign.

Figure 8

[(a) and (b)] Parameter $\kappa \left[p\right]\left(\kappa \left[s\right]\right)$ obtained from fitting the pair(single)-correlation function for $\varphi =\pi \left(\varphi =0\right)$ with a power law [see Eq. (19)] as a function of $J/U$ for L=226. The shaded region indicates the Mott-PLL(Mott-LL) transition region of uncertainty obtained in Sec. 3.

Figure 9

Eigenvalues ${\zeta }_{\varepsilon }^{\left[p\right]}$ of the structure factor matrix for the pair correlations for $J=0.9U,\chi =300$ for (a) $L=226$ and as a function of the crystal momentum $k$ and (b) different lengths of the peak at $k=\pi$.

Figure 10

Block entanglement entropy for $J=0.9U$ and $L=226$ as a function of the chord length $ln\left(d\left(m\right|M)\right)=ln\left(\left[\frac{M}{\pi }sin\left(\frac{m\pi }{M}\right)\right]\right)$ and compared to the CFT prediction Eq. (21) for (c) the nonfrustrated case $\left(\varphi =0\right)$ and (a) the fully frustrated case $\left(\varphi =\pi \right)$. (b) is an illustration of the different cuts that can be made on the model. The cuts after A, B, and C are distinguished using the symbols $*,+,\text{and}★$, respectively.

Figure 11

(a) The entanglement spectrum as a function of the dispersion from uniform filling $\delta N$ of the number of bosons for the cut after $C$. $\varphi =0$ denotes the unfrustrated case $(\circ )$, $\varphi =0.9\pi$ the intermediate frustration $(\square )$ and $\varphi =\pi$ denotes the fully frustrated case $\left(★\right)$, simulated at $J=0.9U$. In (a), a solid line is used to join the degenerate eigenvalues in all cases. (b) Approximate parabolas for $\varphi =0,\varphi =0.9\pi$ and $\varphi =\pi$ (left to right) based on the length $L=226$. In the third panel, different colors denote the possible curve fitting to even and odd. (c) The unfrustrated ES for a cut after $C$ at $L=226$ with the thermodynamic limit approximation shown by the parabolas. The legend shows the degeneracy of each parabola.

Figure 12

The phase diagram for an infinite estimated length of the Hamiltonian Eq. (A1) for amplitude modulation of $cos\left(\varphi \right)$ against parameters $J/U$. The MI, LL, and PLL regions are as labeled. The MI-LL and the MI-PLL transitions are again obtained from the compressibility of the energy gap (see Sec. 4). The LL and PLL phases are characterized by the decay of the correlation functions as illustrated in Sec. 5.