Spin Bose-metal phase in a spin-$\frac{1}{2}$ model with ring exchange on a two-leg triangular strip

Recent experiments on triangular lattice organic Mott insulators have found evidence for a two-dimensional (2D) spin liquid in close proximity to the metal-insulator transition. A Gutzwiller wave function study of the triangular lattice Heisenberg model with a four-spin ring exchange term appropriate in this regime has found that the projected spinon Fermi sea state has a low variational energy. This wave function, together with a slave particle-gauge theory analysis, suggests that this putative spin liquid possesses spin correlations that are singular along surfaces in momentum space, i.e., “Bose surfaces.” Signatures of this state, which we will refer to as a “spin Bose metal” (SBM), are expected to manifest in quasi-one-dimensional (quasi-1D) ladder systems: the discrete transverse momenta cut through the 2D Bose surface leading to a distinct pattern of 1D gapless modes. Here, we search for a quasi-1D descendant of the triangular lattice SBM state by exploring the Heisenberg plus ring model on a two-leg triangular strip (zigzag chain). Using density matrix renormalization group (DMRG) supplemented by variational wave functions and a bosonization analysis, we map out the full phase diagram. In the absence of ring exchange the model is equivalent to the $J_1\text{−}{J}_2$ Heisenberg chain, and we find the expected Bethe-chain and dimerized phases. Remarkably, moderate ring exchange reveals a new gapless phase over a large swath of the phase diagram. Spin and dimer correlations possess singular wave vectors at particular “Bose points” (remnants of the 2D Bose surface) and allow us to identify this phase as the hoped for quasi-1D descendant of the triangular lattice SBM state. We use bosonization to derive a low-energy effective theory for the zigzag spin Bose metal and find three gapless modes and one Luttinger parameter controlling all power law correlations. Potential instabilities out of the zigzag SBM give rise to other interesting phases such as a period-3 valence bond solid or a period-4 chirality order, which we discover in the DMRG. Another interesting instability is into a spin Bose-metal phase with partial ferromagnetism (spin polarization of one spinon band), which we also find numerically using the DMRG.

Figure 1

Top: Heisenberg plus ring exchange model on a two-leg triangular strip. Bottom: convenient representation of the model as a $J_1\text{−}{J}_2$ chain with additional four-site terms; the Hamiltonian is written out in Eq. (1).

Figure 2

(Color online) Phase diagram of the ring model [Eq. (1)] determined in the DMRG using system sizes $L=48–96$. Filled squares (red) denote Bethe-chain phase. Open squares (with black outlines) denote valence bond solid with period 2. Open circles (blue) denote spin Bose metal. Open circles with crosses denote where the DMRG has difficulties converging to singlet for the larger sizes, but where we still think this is spin-singlet SBM. Star symbols denote points where the ground state appears to have true nonzero spin (for all points here, the magnetization is smaller than full polarization of the smaller Fermi sea in the SBM interpretation). Filled diamonds (magenta) denote VBS with period 3. Our identifications are ambiguous in the lower VBS-3 region approaching VBS-2. Lines indicate phase boundaries determined in VMC using spin-singlet wave functions described in the text (we also used appropriate dimerized wave functions for the VBS states). A detailed study of a cut $K_{\text{ring}}/J_1=1$ is presented in Sec. 3 (cf. Fig. 8).

Figure 3

(Color online) Spinon dispersion for $t_2>0.5t_1$ showing two occupied Fermi sea segments (here and throughout we use the 1D chain language; see bottom Fig. 1). Gutzwiller projection of this is the zigzag SBM state at the focus of this work.

Figure 4

(Color online) Spin, dimer, and chirality structure factors at a representative point in the spin Bose-metal phase, $K_{\text{ring}}=J_1=1$, $J_2=0$ (close to the Bethe-chain phase), measured in the DMRG for system size $L=144$. We also show the structure factors in the Gutzwiller projection of two Fermi seas $\left(N_1,N_2\right)=\left(61,11\right)$ and in the improved Gutzwiller wave function with parameters $f_0=1$, $f_1=0.65$, and $f_2=-0.5$ (see Appendix ; the parameters are determined by optimizing the trial energy within the given family of states). Vertical lines label important wave vectors expected in the SBM theory for such spinon Fermi sea volumes. We discuss the comparison of the DMRG, VMC, and analytical theory in the text.

Figure 5

(Color online) Spin, dimer, and chirality structure factors at a representative point in the spin Bose-metal phase, $K_{\text{ring}}=J_1=1$, $J_2=3.2$, (between the VBS-3 and VBS-2 phases) measured in the DMRG for system size $L=144$. We also show results in the Gutzwiller projection of two Fermi seas $\left(N_1,N_2\right)=\left(44,28\right)$, and in the improved Gutzwiller wave function with parameters $f_0=1$, $f_1=0.75$, and $f_2=-0.1$ (see Appendix for details). Vertical lines label important wave vectors expected in the SBM theory. We discuss the comparison of the DMRG, VMC, and analytical theory in the text.

Figure 6

(Color online) Evolution of the structure factors in the spin Bose-metal phase between the Bethe-chain and VBS-3 (or partial FM), measured by the DMRG for system size $L=96$. We can track singular wave vectors (Bose surfaces) as spinons are transferred from the first to the second Fermi sea upon increasing $J_2$. In the spin structure factor, the wave vector $q_{\text{high}}$ that starts near $\pi$ is identified as $2k_{F1}$ and the wave vector $q_{\text{low}}$ that starts near 0 is identified as $2k_{F2}$ (see Fig. 4 and text for details). The $q_{\text{high}}$ and $q_{\text{low}}$ are summarized in Fig. 8.

Figure 7

(Color online) Evolution of the structure factors in the spin Bose-metal phase between VBS-3 and VBS-2, measured by the DMRG for system size $L=96$. The $q_{\text{high}}=2k_{F1}$ and $q_{\text{low}}=2k_{F2}$ bracket the $\pi /2$ and approach each other with increasing $J_2$ (they are summarized in Fig. 8). Very notable here is the strong $4k_{F2}$ peak in the dimer structure factors that evolves out of the Bragg peak present in the VBS-3 phase at $2\pi /3$ (see Sec. 5A and Fig. 15); the $2k_{F1}$ moves away from the $2\pi /3$ in the opposite direction.

Figure 8

(Color online) Cut through the phase diagram (Fig. 2) at $K_{\text{ring}}/J_1=1$, showing evolution of the most prominent wave vectors in the spin structure factor. In the Bethe-chain phase we have singular antiferromagnetic $q_{\text{AF}}=\pi$ (cf. Fig. 22). In the spin Bose metal we have singular $q_{\text{high}}=2k_{F1}$ and $q_{\text{low}}=2k_{F2}$ located symmetrically about $\pi /2$ (cf. Figs. 6, 7). In the VBS-3 we have singular $q_{\text{AF}}=\pi /3$ (cf. Fig. 15). In the VBS-2 region for large $J_2$ (cf. Fig. 23), we have dominant correlations at $\pi /2$ corresponding to the decoupled legs fixed point, which is likely to be unstable toward opening a spin gap (Refs. 18, 19). The dotted lines show results for the improved Gutzwiller wave function.

Figure 9

(Color online) Entanglement entropy at representative points in the Bethe-chain $\left(J_2=-1\right)$, SBM ($J_2=0$ and $J_2=3.2$), VBS-3 $\left(J_2=2\right)$, and VBS-2 $\left(J_2=4\right)$ phases, taken from the cut $K_{\text{ring}}=J_1=1$, measured in the DMRG for system size $L=72$ with periodic boundary conditions. We use Eq. (33) to fit data over the range $6\le X\le 66$, which gives the best estimates as $c=1.0$ (Bethe chain), $c=3.1$ (SBM at $J_2=0$), $c=3.2$ (SBM at $J_2=3.2$), $c=1.6$ (VBS-3), and $c=2.1$ (VBS-2).

Figure 10

(Color online) Entanglement entropy for various system sizes for the Bethe-chain $\left(J_2=-1\right)$ and SBM $\left(J_2=0\right)$ points (cf. Fig. 9) plotted versus scaling variable $d=\frac{L}{\pi }\text{sin}\frac{\pi X}{L}$. We also show fits to Eq. (33) done for the larger sizes. The Bethe-chain data collapse well and are fitted with $c=1$. The SBM data are fitted with $c=3.1$ (we show the same fit as in Fig. 9 for this SBM point). The collapse of different $L$ is less good, which is likely due to imprecise scaling of the discrete shell filling numbers with $L$ (see text for details).

Figure 11

Valence bond solid with period 2, where thicker lines indicate stronger bonds. To emphasize the symmetries of the state, we also show second- and third-neighbor bond energies, but details can be different in different regimes. For example, the VBS-2 state in the $K_{\text{ring}}=0$ case has dominant first-neighbor dimerization. On the other hand, in the model with $K_{\text{ring}}=J_1=1$, $J_3=0.5$, the putative VBS-2 region between the Bethe-chain and SBM phases in Fig. 16 has significant third-neighbor modulation but only very small first-neighbor one.

Figure 12

Top: valence bond solid with period 4 suggested as one of the instabilities out of the spin Bose-metal phase [Sec. 4B2(a)]. Thick lines indicate stronger bonds. Bottom: in the two-leg triangular ladder drawing, we see roughly independent spontaneous dimerization in each leg.

Figure 13

Top: chirality order with period 4 suggested as one of the instabilities out of the SBM phase [Sec. 4B2(b)]. In the 1D chain picture, the chirality pattern is given by Eq. (54); $\chi \left(x\right)$ is associated with the $\left[x-1,x,x+1\right]$ loop and the arrows on the links show one “gauge choice” to produce such “fluxes” in the spinon hopping. Bottom: in the two-leg triangular ladder drawing, we see alternating chiralities on the up triangles along the strip and alternating chiralities on the down triangles.

Figure 14

(Color online) Valence bond solid states with period 3. Thick lines indicate stronger bonds. Remaining effective spin-1/2 degrees of freedom are also shown. Coexisting with the translational symmetry breaking, we have $1/x$ power law spin correlations with the antiferromagnetic (dynamic) pattern as shown. The two cases have slightly different microscopics but are qualitatively similar on long length scales.

Figure 15

(Color online) Spin and dimer structure factors at a representative point in the VBS-3 phase, $K_{\text{ring}}=J_1=1$, $J_2=2$, measured in the DMRG for system size $L=96$ (we do not show the chirality as it is not informative). The most notable features are the dimer Bragg peak at $2\pi /3$ corresponding to the static VBS order and also the spin singularity at $\pi /3$ corresponding to the effective spin-1/2 chain formed by the nondimerized spins (see Fig. 14). The trial VBS-3 wave function is constructed as described in the text.

Figure 16

(Color online) Phase diagram of the $J_1\text{−}{J}_2\text{−}{K}_{\text{ring}}$ model with additional antiferromagnetic third-neighbor coupling $J_3=0.5J_1$ introduced to stabilize spin-singlet states. The study is along the same cut $K_{\text{ring}}=J_1$ as in Fig. 8, and the overall features are similar, with the following differences. The ground state is singlet throughout eliminating the difficult partial FM region to the left of the VBS-3 (and the VBS-3 phase is somewhat wider). There is a sizable spin-gapped region between the Bethe-chain and SBM phases (see text for more details). A new spin-gapped phase with period-4 chirality order appears inside the SBM to the left of the VBS-3.

Figure 17

(Color online) Spin, dimer, and chirality structure factors at a point in the tentative chirality-4 phase, $K_{\text{ring}}=J_1=1$, $J_2=1.5$, and $J_3=0.5$, measured in the DMRG for system size $L=96$. Note the absence of sharp features in the spin correlations, which suggests a spin gap, while the dimer correlations have only a feature at $\pi$ corresponding to period-2 modulation. On the other hand, the chirality structure factor shows a Bragg peak at $\pi /2$ that grows with increasing system size. The pattern of chirality correlations in real space is consistent with the order shown in Fig. 13.

Figure 18

(Color online) Spin and dimer structure factors at a tentative point with partial ferromagnetism, $K_{\text{ring}}=J_1=1$, $J_2=1$, and $J_3=-0.5$, measured in the DMRG for system size $L=96$. The calculations are done in the sector $S_{\text{tot}}^z=10$ where the DMRG is well converged and gives $S_{\text{tot}}=10$, which we think is the true ground-state spin. The VMC state has the second Fermi sea fully polarized with $N_{2\uparrow }=20$, $N_{2\downarrow }=0$, while the first Fermi sea is unpolarized with $N_{1\uparrow }=N_{1\downarrow }=38$. Vertical lines label important wave vectors $2k_{F1}$, $2k_{F2}$, and $-k_{F2}-k_{F1}$.

Figure 19

(Color online) Separate $⟨S_q^z{S}_{-q}^z⟩$ and $⟨S_q^x{S}_{-q}^x⟩$ structure factors for the same system as in Fig. 18.

Figure 20

View of the wave function constructed by filling $k$ states of spinons. Here momenta $k=2\pi n/L$, $n=0,1,\dots ,L-1$, form a closed circle. Each filled dot is occupied by both spin up and spin down, producing spin singlet. The projected wave function remains unchanged if all momenta are shifted by the same amount. Only the relative configuration matters, and here we show symmetric configuration of the two Fermi seas separated by $L/4$ unoccupied $k$ states on either side. This is our “bare Gutzwiller” wave function for the spin Bose metal.

Figure 21

(Color online) Spin, bond energy, and chirality structure factors in the bare Gutzwiller wave function with two Fermi seas $\left(N_1,N_2\right)=\left(104,24\right)$ on the 1D chain of length $L=256$. The expected singular wave vectors are marked by vertical lines (here $k_{F1}$ and $k_{F2}$ are defined as in Fig. 3 and all indicated wave vectors are $\text{mod}2\pi$). The structure factors are symmetric with respect to $q\to -q$, and we only show $0\le q\le \pi$. The character of the singularities is consistent with the special case $g=1$ in the SBM theory of Sec. 2.

Figure 22

(Color online) Spin, dimer, and chirality structure factors at a representative point in the Bethe-chain phase, $K_{\text{ring}}=J_1=1$, $J_2=-1$, measured in the DMRG for system size $L=192$. We also show structure factors in the bare Gutzwiller-projected single Fermi sea state, $\left(N_1,N_2\right)=\left(96,0\right)$, and in the improved Gutzwiller wave function with parameters $f_0=1$, $f_1=0$, and $f_2=-1.4$ (see Appendix for wave function details).

Figure 23

(Color online) Spin and dimer structure factors at a point in the VBS-2 phase, $K_{\text{ring}}=J_1=1$, $J_2=4$, measured in the DMRG for system size $L=192$. The exhibited trial wave function is the Gutzwiller projection of two equal Fermi seas, $\left(N_1,N_2\right)=\left(48,48\right)$. This wave function gives decoupled legs expected in the large $J_2$ limit and does not have any VBS-2 order; however, it reproduces the DMRG data quite well, so at this $K_{\text{ring}}=J_1$ cut the system is close to the decoupled-legs limit even just outside the SBM.