Strong-coupling phases of frustrated bosons on a two-leg ladder with ring exchange

Developing a theoretical framework to access the quantum phases of itinerant bosons or fermions in two dimensions that exhibit singular structure along surfaces in momentum space but have no quasiparticle description remains a central challenge in the field of strongly correlated physics. In this paper we propose that distinctive signatures of such two-dimensional (2D) strongly correlated phases will be manifest in quasi-one-dimensional “$N$-leg ladder” systems. Characteristic of each parent 2D quantum liquid would be a precise pattern of one-dimensional (1D) gapless modes on the $N$-leg ladder. These signatures could be potentially exploited to approach the 2D phases from controlled numerical and analytical studies in quasi-one-dimension. As a first step we explore itinerant-boson models with a frustrating ring-exchange interaction on the two-leg ladder, searching for signatures of the recently proposed two-dimensional $d$-wave-correlated Bose liquid (DBL) phase. A combination of exact diagonalization, density-matrix renormalization-group, variational Monte Carlo, and bosonization analysis of a quasi-1D gauge theory provide compelling evidence for the existence of an unusual strong-coupling phase of bosons on the two-leg ladder, which can be understood as a descendant of the two-dimensional DBL. We suggest several generalizations to quantum spin and electron Hamiltonians on ladders, which could likewise reveal fingerprints of such 2D non-Fermi-liquid phases.

Figure 1

Schematics of discrete $q_y=2\pi /N$ cuts through Bose surfaces in some anisotropic 2D system placed on an $N$-leg ladder. Here for illustration $N=8$. Such Bose surfaces appear in a DBL construction described in the text. To avoid clutter, we do not show the Fermi surfaces of the $d_1$ and $d_2$ slave particles.

Figure 2

(Color online) $d_1$ and $d_2$ bands for the DBL construction on the two-leg ladder. Top panel: $d_1$ fermions partially occupy both the bonding and the antibonding bands. Bottom left: In the DBL[2,1] phase, $d_2$ fermions occupy only the bonding band. Bottom right: In the DBL[2,2] phase, $d_2$ fermions partially occupy both bands. This state is likely unstable toward a boson-paired phase, as discussed in Appendix . In the wave-function construction, only the occupied orbitals and not the band dispersions are important.

Figure 3

(Color online) Phase diagram for the two-leg boson system at filling $\rho =1/3$. The symbols represent DMRG estimates with shaded gray circles indicating the superfluid phase, the filled blue squares the DBL, and the open red diamonds the $s$-wave-paired state. The schematic line boundaries are obtained from VMC calculations.

Figure 4

(Color online) Phase diagram for the two-leg boson system at filling $\rho =1/9$. The symbols indicate DMRG estimates for the various phases with shaded gray circles for the superfluid phase, the filled blue squares for the DBL, and the open green diamonds for the $d$-wave-paired state. The schematic line boundaries are obtained from VMC calculations.

Figure 5

(Color online) DMRG results for (a) the boson occupation number $n\left(\mathbf{q}\right)$ and (b) the density-density structure factor $D\left(\mathbf{q}\right)$ for the two-leg system at $\rho =1/3$, $J_{\perp }=J$, $K=J$; the system length is $L_x=48$. In (a), the $q_y=\pi$ data were multiplied by a factor of 20. The results are representative for the superfluid phase.

Figure 6

(Color online) DMRG results for the boson pairing correlations in real space for the SF system described in Fig. 5. We show correlations for pairs on the diagonals $d$ and $\overline{d}$, but roughly similar values are obtained for all pair orientations.

Figure 7

(Color online) Exact diagonalization results for a system of size $L_x=12$ at filling $\rho =1/3$ and $J_{\perp }=J$. Shown are the superfluid stiffness (top panel), the average sign of the ground-state wave function (middle panel), and the average double occupancy of rungs (lower panel) as a function of the ring exchange $K$.

Figure 8

(Color online) (a) The boson occupation number $n\left(\mathbf{q}\right)$ and (b) the density-density structure factor $D\left(\mathbf{q}\right)$ for the two-leg system at $\rho =1/3$, $J_{\perp }=J$, $K=3J$; the system length is $L_x=48$. The results are representative of the DBL phase.

Figure 9

(Color online) Positions of the singular momenta in the DBL phase as a function of the ring exchange $K$ for a system with $J_{\perp }=J$ and filling $\rho =1/3$. Data for systems with length $L_x=36$ and $L_x=48$ are given. The momenta are determined from the peak values of $n\left(q_x,q_y\right)$ calculated by DMRG such as shown in the top panel of Fig. 8.

Figure 10

(Color online) The boson pairing correlations in real space for the DBL system described in Fig. 8 calculated by DMRG and VMC. We show correlations for pairs on the diagonals $d$ and $\overline{d}$, while values for other pair orientations are much smaller.

Figure 11

(Color online) Detection of the boson propagation around the loop by applying a phase twist $\theta$ to the periodic boundary connection along the $\widehat{x}$ direction. The energy differences $\Delta E=E\left(\theta =\pi \right)-E\left(\theta =0\right)$, multiplied by $L_x$, for a uniform system with $J_{\perp }=J$ for all bonds (top panel) and for a specially designed system with $J_{\perp }=J$ in one half and $J_{\perp }=-J$ in the other half of the system (bottom panel). The calculations are done by DMRG.

Figure 12

(Color online) DMRG results for (a) the boson occupation number $n\left(\mathbf{q}\right)$ and (b) the density-density structure factor $D\left(\mathbf{q}\right)$ for a two-leg system at $\rho =1/3$, $J_{\perp }=0.1J$, $K=1.4J$; the system length is $L_x=48$. The results are representative for the $s$-wave-paired phase.

Figure 13

(Color online) The boson pairing correlations in real space for the same $s$-wave-paired system described in Fig. 12. We show diagonal pairs but roughly similar correlations are obtained also for pairs on the rungs, while correlations for pairs in the chains are significantly smaller.

Figure 14

(Color online) (a) The boson occupation number $n\left(\mathbf{q}\right)$ and (b) the density-density structure factor $D\left(\mathbf{q}\right)$ for a two-leg system at $\rho =1/9$, $J_{\perp }=0.1J$, $K=3J$; the system length is $L_x=54$. The results are representative of the $d$-wave-paired phase.

Figure 15

(Color online) The boson pairing correlations in real space for the same system described in Fig. 14.