Exotic Gapless Mott Insulators of Bosons on Multileg Ladders

We present evidence for an exotic gapless insulating phase of hard-core bosons on multileg ladders with a density commensurate with the number of legs. In particular, we study in detail a model of bosons moving with direct hopping and frustrating ring exchange on a 3-leg ladder at $\nu =1/3$ filling. For sufficiently large ring exchange, the system is insulating along the ladder but has two gapless modes and power law transverse density correlations at incommensurate wave vectors. We propose a determinantal wave function for this phase and find excellent comparison between variational Monte Carlo and density matrix renormalization group calculations on the model Hamiltonian, thus providing strong evidence for the existence of this exotic phase. Finally, we discuss extensions of our results to other $N$-leg systems and to $N$-layer two-dimensional structures.

Figure 1

Top: Illustration of the 3-leg ladder geometry with periodic boundary conditions in both the $x$ and $y$ directions. The hopping strength is $J$ in the $x$ direction and $J_{\perp }$ in the $y$ direction; the ring term $K$ moves a pair of hard-core bosons on diagonal corners of a plaquette to the other diagonal corners (if empty). Bottom: The $d_1$ and $d_2$ bands for the 3-leg system at $\nu =1/3$ are shown for the gapless Mott insulator studied here. There are exactly enough bosons so that the $d_2$ fermions fill up one band completely. The bold curves indicate occupied momentum states, and note that there are two degenerate bands at $k_y=\pm 2\pi /3$.

Figure 2

Summary of our numerical results: phase diagram in the $K/J-J_{\perp }/J$ plane of the 3-leg ladder at $\nu =1/3$ filling as determined by DMRG and VMC calculations.

Figure 3

Comparison of DMRG and VMC calculations of the boson momentum distribution $n_b(\mathbf{q})$ (top panel) and density-density structure factor $D_b(\mathbf{q})$ (middle panel; inset is a zoom of $q_y=0$ curves) in the GMI phase of the 3-leg ladder at the point $K/J=2.7$ and $J_{\perp }/J=1.0$. The lack of singular features in $n_b(\mathbf{q})$ and $q_x^2$ dependence of $D_b(q_x,q_y=0)$ around $q_x=0$, both highlight the insulating nature of the GMI phase. However, the density-density structure factor still “sees” the gaplessness of the phase due to the partially filled $d_1$ bands [4], hence the singular features in $D_b(q_x,q_y=\pm 2\pi /3)$. These singularities can be identified with “$2k_F$” wave vectors in the $d_1$ fermion’s band filling configuration (bottom panel) in which the $k_y=0$ band contains 12 fermions and the $k_y=\pm 2\pi /3$ bands each contain 6 fermions. Of course, the $d_2$ fermion configuration (not shown) consists only of one fully filled $k_y=0$ band.

Figure 4

Effective central charge $c$ as inferred from scaling of the entanglement entropy versus subsystem size $S(x)$. For the 3-leg ladder, we fit the von Neumann entanglement entropy of leftmost contiguous blocks of size $3x$ to the scaling form $S(x)=(c/3)\ln [(L_x/\pi )\sin (\pi x/L_x)]+A$ to extract $c$ [15]. In the main plot, $c$ is plotted versus $K/J$ at $J_{\perp }/J=1.0$ for a system of length $L_x=24$; the jump from $c\simeq 1$ to $c\simeq 2$ at the SF-GMI boundary is striking. In the inset, we plot $S(x)$ in both the SF ($K/J=1.0$) and GMI ($K/J=2.7$) phases; the points are the DMRG data, while the curves are the fits to the scaling form.