Bose metals and insulators on multileg ladders with ring exchange

We establish compelling evidence for the existence of new quasi-one-dimensional descendants of the $d$-wave Bose liquid (DBL), an exotic two-dimensional quantum phase of uncondensed itinerant bosons characterized by surfaces of gapless excitations in momentum space [O. I. Motrunich and M. P. A. Fisher, Phys. Rev. B 75, 235116 (2007)]. In particular, motivated by a strong-coupling analysis of the gauge theory for the DBL, we study a model of hard-core bosons moving on the $N$-leg square ladder with frustrating four-site ring exchange. Here, we focus on four- and three-leg systems where we have identified two novel phases: a compressible gapless Bose metal on the four-leg ladder and an incompressible gapless Mott insulator on the three-leg ladder. The former is conducting along the ladder and has five gapless modes, one more than the number of legs. This represents a significant step forward in establishing the potential stability of the DBL in two dimensions. The latter, on the other hand, is a fundamentally quasi-one-dimensional phase that is insulating along the ladder but has two gapless modes and incommensurate power-law transverse density-density correlations. While we have already presented results on this latter phase elsewhere [M. S. Block et al., Phys. Rev. Lett. 106, 046402 (2011)], we will expand upon those results in this work. In both cases, we can understand the nature of the phase using slave-particle-inspired variational wave functions consisting of a product of two distinct Slater determinants, the properties of which compare impressively well to a density matrix renormalization group solution of the model Hamiltonian. Stability arguments are made in favor of both quantum phases by accessing the universal low-energy physics with a bosonization analysis of the appropriate quasi-1D gauge theory. We will briefly discuss the potential relevance of these findings to high-temperature superconductors, cold atomic gases, and frustrated quantum magnets.

Figure 1

(Top panel) Band filling for the DBL$[4,2]$ showing the Fermi wave vectors for the right moving $d_1$ and $d_2$ partons. (Bottom panel) An overhead view of the occupied momentum states for the DBL$[4,2]$, highlighting the 2D nature of the phase; it is clear that the $d_1$ Fermi sea is compressed along $\widehat{x}$, while the $d_2$ Fermi sea is compressed along $\widehat{y}$.

Figure 2

Schematic picture of how the anisotropy of the Fermi seas in the parton representation is driven by the ring-exchange term in the model, Eq. (11). If one allows the $d_1$ and $d_2$ partons to virtually hop along their preferred directions, an effective ring exchange for the bosons is generated. Because we are considering partons with fermionic statistics, the sign of the ring-exchange interaction is rendered positive [cf. Eq. (13)]; in contrast, bosonic partons would generate negative ring exchange.[34, 35] The resulting sign can intuitively be understood by noting that the ring-exchange process involves one crossing of the partons, as depicted in the second leftmost plaquette shown above, so that the ring-exchange energy is minimized by having a positive (negative) coefficient for fermionic (bosonic) partons.

Figure 3

Phase diagram of the four-leg system at boson density $\rho =5/12$, using system sizes $12\times 4$ with $N_b=20$ bosons, $18\times 4$ with $N_b=30$ bosons, and (for some points) $24\times 4$ with $N_b=40$ bosons. The colored regions are delineated using VMC data. The gray region indicates phase separation and is delineated schematically by considering the boson density in real space with the DMRG approach. DMRG points for the superfluid, DBL$[4,2]$, and phase separation are indicated by white circles, green squares, and gray diamonds, respectively. Finally, in the region labeled “[4,4],” the VMC finds a DBL state with four equally occupied bands for both the $d_1$ and $d_2$ partons, although this region is likely just a superfluid.

Figure 4

Boson momentum distribution function (top panel) and density-density structure factor (bottom panel) for the superfluid point at $J_{\perp }/J=1$, $K/J=1$ for a $24\times 4$ system size with $N_b=40$ ($\rho =5/12$), as obtained by DMRG. The values of $n_b\left(\mathbf{q}\right)$ at $q_y=\pm \pi /2,\pi$ have been scaled up by a factor of 20. The $\mathbf{q}=0$ condensation is readily apparent in $n_b\left(\mathbf{q}\right)$, as is the $|q_x|$ behavior of $D_b\left(\mathbf{q}\right)$ near $q_x=0$ at $q_y=0$.

Figure 5

Boson momentum distribution function (top panel) and density-density structure factor (middle panel) for a characteristic DBL$[4,2]$ point: $J_{\perp }/J=1$, $K/J=2.5$. The system size is $24\times 4$ with $N_b=40$ ($\rho =5/12$). DMRG data are joined with solid curves while the VMC data are joined with dashed curves. (Bottom panel) Schematic representation of the energy-optimized VMC state for this point. The circles represent momentum orbitals in each of the four bands; filled circles denote occupied orbitals. Each band is centered about $k_x=0$ and we are using antiperiodic boundary conditions in both the $\widehat{x}$ and $\widehat{y}$ directions for the partons.

Figure 6

The same quantities are plotted as in Fig. 5 for a $24\times 4$ system with $N_b=40$ bosons ($\rho =5/12$), except now the measurements are calculated at the point $J_{\perp }/J=0.1$, $K/J=2.5$. The bottom panel depicts the new energy-optimized VMC state, whose $d_1$ configuration differs from that of Fig. 5. Note the feature in the VMC data for $n_b\left(\mathbf{q}\right)$ at $q_x=\pm 3\pi /4$, which is a “suppressed” singularity resulting from creating a boson on the same side of the $d_1$ and $d_2$ coincident bands: $\pm (k_{F1}+k_{F2})=\mp 3\pi /4$. Analogous features are also present in the VMC data of Fig. 5. As well as demonstrating the ability to vary the location of the Bose points within the phase diagram, these results in conjunction with those of Fig. 5 also highlight the general nonuniversality of the amplitudes of the power-law singularities.

Figure 7

Phase diagram of the three-leg system at boson density $\rho =1/3$, using a system of size $24\times 3$ with $N_b=24$ bosons. The colored regions are delineated using VMC data; the white regions indicate where our understanding of the phase diagram is limited. DMRG points for the superfluid, rung Mott insulator, and DBL$[3,0]$ are indicated by blue circles, green squares, and yellow triangles, respectively. This figure has been reproduced from Ref. 19.

Figure 8

Density-density structure factor evaluated at $q_y=\pm 2\pi /3$, i.e., $D_b(q_x,q_y=\pm 2\pi /3)$, for various $K/J$ and fixed $J_{\perp }/J=1$ on a $30\times 3$ system with $N_b=30$ ($\rho =1/3$). The qualitative change in this function is striking as one crosses the first-order phase transition from the superfluid to the DBL$[3,0]$ at around $K/J\approx 2.2$. In particular, $D_b(q_x,q_y=\pm 2\pi /3)$ is a featureless function in the superfluid phase, but has singular structure in the DBL$[3,0]$ at wave vectors originating from the “$2k_F$” wave vectors within the $d_1$ parton band filling configuration. Furthermore, the evolution of the singular features within the DBL$[3,0]$ phase is fully consistent with the three $d_1$ parton bands becoming more equally occupied as $K$ is increased. For very large $K$, e.g., $K/J=20$, the system is in a phase with an approximately conserved number of bosons in each rung and chain, which is well represented by three equally filled $d_1$ bands. These calculations were done with DMRG.

Figure 9

Boson momentum distribution function (top panel) and density-density structure factor (middle panel) for a characteristic DBL$[3,0]$ point: $J_{\perp }/J=1$ and $K/J=2.7$. The system size is $24\times 3$ with $N_b=24$ ($\rho =1/3$). DMRG data are joined with solid curves while the VMC data are joined with dashed curves. (Bottom panel) Schematic representation of the energy-optimized VMC state for this point, shown in an analogous way to the DBL$[4,2]$ state in the bottom panels of Figs. 5 and 6. Here, the $d_1$ configuration resembles a traditional gapless parton Fermi sea, while the $d_2$ configuration is gapped due to the commensurate $\rho =1/3$ density. This figure has been reproduced from Ref. 19

Figure 10

Entanglement entropy scaling for different system sizes on the three-leg system at $\rho =1/3$ in the superfluid ($K/J=1$) and DBL$[3,0]$ ($K/J=2.7$) phases, as obtained by DMRG with fully periodic boundary conditions; $J_{\perp }/J=1$ in both cases. The dashed lines are fits to the scaling form, Eq. (21). For the superfluid point, all the data collapse extremely well onto a curve with $c=1.00$ and $A=1.12$, which is the lower dashed line. The dashed line for the DBL$[3,0]$ data is that used in Fig. 4 of Ref. 19 in which $24\times 3$ data were used and the four smallest/largest $X$ values were discarded in the fit; the obtained fit parameters are $c=1.96$ and $A=1.41$. The collapse of the $L_x=24$ and 30 data is reasonable, although there is a very weak downward shift of the slope in the latter case for the largest $X$. Shell-filling effects or other corrections to the scaling form (which we have not considered) are likely the cause of the less impressive data collapse in the DBL$[3,0]$ as compared to that observed in the superfluid.

Figure 11

Density-density structure factor at $q_y=\pm 2\pi /3$ in the DBL$[3,0]$ phase ($J_{\perp }/J=1,K/J=2.7$) for various system sizes, $L_x\times 3$, using DMRG with cylindrical boundary conditions. The main power-law singularity (see inset) is robust up to systems of size $72\times 3$. This is consistent with the long-distance properties of the gapless DBL$[3,0]$ phase, although we have not pursued a full characterization of the long-distance power-law behavior. There is some small irregular behavior of the main singularity on the scale of $2\pi /L_x$, but this is to be expected with cylindrical boundary conditions and shell-filling effects present in our DBL$[3,0]$ theory.

Figure 12

Ground-state momentum diagram for a $10\times 3$ system with $N_b=10$ ($\rho =1/3$). The colored regions denote the ground-state momentum of the variational wave function that optimizes the trial energy at each point in parameter space. The symbols indicate ED data for the true ground-state momentum of the system. The notation for the quantities in parentheses is $(Q_x,Q_y)$, where $Q_x$ and $Q_y$ are measured in units of $2\pi /L_x$ and $2\pi /L_y$, respectively. For example, $(4,1)\to (4\times 2\pi /10,1\times 2\pi /3)=(4\pi /5,2\pi /3)$. The agreement between the ED and VMC predictions of ground-state momenta at this system size is rather remarkable.

Figure 13

Three quantities are plotted on these axes for a $10\times 3$ system with $N_b=10$ ($\rho =1/3$) with $J=0$ where $K/J_{\perp }$ is the only dimensionless parameter. We plot the independent variable on a logarithmic scale to better interpolate between the two limits. Small $K/J_{\perp }$ corresponds to the rung Mott phase where the rungs of the ladder effectively decouple and there exists a gap to all excitations; large $K/J_{\perp }$ should be the gapless Mott insulator phase or DBL$[3,0]$. The solid and dashed curves are the VMC results, while the points are ED results; the vertical dashed line is the rung Mott-GMI phase boundary as determined by VMC. The blue curve with blue circles is the $x$ component of the ground-state momentum measured in units of $2\pi /10$; the red curve with red squares is the $y$ component of the ground-state momentum measured in units of $2\pi /3$; finally, the black curve with black triangles is the ground-state energy per site. Remarkably, the variational wave functions capture the energetics exceedingly well for the rung Mott and the DBL$[3,0]$ phases, much more so than for the DBL$[4,2]$ and other conducting DBL$[n,m]$ states.

Figure 14

Phase diagram of the three-leg system at boson density $\rho =1/4$, using a $24\times 3$ system with $N_b=18$. The colored regions are delineated using VMC data. Phase separation is determined with VMC methods by means of a Maxwell construction. Commensurate densities tend to be more stable for large $K$ and so phase separation for this system generally consists of a region of zero density and a region at $1/3$ density. DMRG points for the superfluid, DBL$[3,1]$, and phase separation are indicated by white circles, green squares, and gray diamonds, respectively. Our calculations indicate that the DBL$[3,1]$ phase is never stabilized in the system with isotropic hopping ($J_{\perp }/J=1$).

Figure 15

Boson momentum distribution function (top panel) and density-density structure factor (bottom panel) for a typical point in the DBL$[3,1]$ phase. The parameters are $J_{\perp }/J=0.75$ and $K/J=2.5$ on a system of size $24\times 3$ with $N_b=18$ ($\rho =1/4$). DMRG data are joined with solid curves while the VMC data are joined with dashed curves.

Figure 16

Boson momentum distribution function (top panel) and $\widehat{y}$-$\widehat{y}$ current-current structure factor (middle panel) for a typical point in the putative “bond-chiral superfluid” (BCSF) region at $\rho =1/2$ on the three-leg ladder. The parameters are $J_{\perp }/J=1$ and $K/J=3$ on two system sizes: $L_x=24,12$. As explained in the text, the features appearing in these correlations can be qualitatively rationalized by a classical solution, $b_{\mathbf{r}}=\sqrt{\rho }\exp \left(i{\phi }_{\mathbf{r}}\right)$, with phases given by Eq. (B4). In the bottom panel, we display this solution by showing a vector $(\cos {\phi }_{\mathbf{r}},\sin {\phi }_{\mathbf{r}})$ at each site. Since true long-range order of the boson phase is prohibited for our three-leg ladder, this is a potentially appropriate description only on short length scales. Note, however, that the current-current correlations can still be truly long ranged.