Triangular-lattice anisotropic dimerized Heisenberg antiferromagnet: Stability and excitations of the quantum paramagnetic phase

Motivated by experiments on nonmagnetic triangular-lattice Mott insulators, we study one candidate paramagnetic phase, namely the columnar dimer (or valence-bond) phase. We apply variants of the bond-operator theory to a dimerized and spatially anisotropic spin-1/2 Heisenberg model and determine its zero-temperature phase diagram and the spectrum of elementary triplet excitations (triplons). Depending on model parameters, we find that the minimum of the triplon energy is located at either a commensurate or an incommensurate wave vector. Condensation of triplons at this commensurate-incommensurate transition defines a quantum Lifshitz point, with effective dimensional reduction that possibly leads to nontrivial paramagnetic (e.g., spin-liquid) states near the closing of the triplet gap. We also discuss the two-particle decay of high-energy triplons, and we comment on the relevance of our results for the organic Mott insulator EtMe${}_3$P[Pd(dmit)${}_2$]${}_2$.

Figure 1

(a) Lattice structure of the model (1), with exchange couplings $J$, $J^{\prime }$, and $J^{\prime \prime }$ shown as black (thin), red (thick), and dashed black lines, respectively. Solid (open) circles represent spin ${\mathbf{S}}^1$ (${\mathbf{S}}^2$) of each dimer. ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are the primitive vectors of the dimerized lattice. (b) Dimerized ground state in the limit of large $J^{\prime }$. Here, ellipses represents a singlet. (c) Brillouin zone of the dimerized lattice (solid), together with the (hexagonal) Brillouin zone of the original triangular lattice (dashed). The open circles indicate the minimum position of ${\omega }_{\mathbf{q}}$ [Eq. (17)] for parameters $J^{\prime }=1.5$ and $J^{\prime \prime }=1$. Here $\mathbf{K}=(4\pi /3a,0)$, ${\mathbf{K}}^{\prime }=(2\pi /3a,2\pi /\sqrt{3}a)$, $\mathbf{X}=(\pi /a,0)$, $\mathbf{Y}=(0,\pi /\sqrt{3}a)$, and $\mathbf{M}=(\pi /a,\pi /\sqrt{3}a)$, with $a$ being the lattice spacing of the triangular lattice.

Figure 2

Schematic phase diagram of the Heisenberg model (1) as a function of the couplings $J^{\prime }$ and $J^{\prime \prime }$, keeping $J=1$. For large $J^{\prime }$, a gapped dimer phase is realized (shaded). Various other limits are indicated in the figure; see text for details. The multicritical point, where the collinear LRO, noncollinear LRO, and gapped dimer phases meet, is the quantum Lifshitz point. [The classical limit of (1) is discussed in Appendix .]

Figure 3

Quantitative phase diagram for the Heisenberg model (1), obtained using the bond-operator formalism. The dimer phase boundary () has been obtained from the HF-cubic approximation described in the paper; for comparison, we also show the results of the harmonic mean-field[27] () and harmonic spin-wave approximations (dashed blue).[37] The dashed-dotted line separates the commensurate (C) and incommensurate (INC) regions of the dimer phase at large $J^{\prime }$. It continues into the dotted line, which separates the phases with collinear and noncollinear LRO at small $J^{\prime }$—note that this line is an estimate based on series-expansion[9] and classical-limit results (Appendix ). Also shown are the isotropic point $J^{\prime }=J^{\prime \prime }=1$ () and the point $J^{\prime }=1$, $J^{\prime \prime }=1.05$ () for EtMe${}_3$P[Pd(dmit)${}_2$]${}_2$.[22] (Here, longer-range or multiple-spin interactions can also be expected to be relevant.) Finally, on the $J^{\prime \prime }=0$ axis we have indicated the QMC value of $J_c^{\prime }$ () for the square-lattice staggered dimerized Heisenberg model.[33]

Figure 4

Upper row: Triplon dispersion relation at the harmonic level (solid black) along paths in the dimerized Brillouin zone [Fig. 1c], for (a) $J^{\prime }=3.0$, $J^{\prime \prime }=0$ and (b) $J^{\prime }=1.5$, $J^{\prime \prime }=1$. For comparison, the dispersion obtained within the HF approximation for the quartic term (thin green line, see Sec. 4a) is also shown, indicating that the corrections from the quartic term are minor. Finally, the dashed red line represents the bottom of the two-particle continuum at the harmonic level. Lower row: Contour plot of the triplon energy at the harmonic level, for (c) $J^{\prime }=3.0$, $J^{\prime \prime }=0$ and (d) $J^{\prime }=1.5$, $J^{\prime \prime }=1$.

Figure 5

$x$ and $y$ components of the momentum ${\mathbf{Q}}_0$ (see text for definition) as a function of $J^{\prime \prime }$ at the harmonic (dashed), HF (), and HF-cubic () approximations for the lines $J^{\prime }=1.7$ (a) and $3.0$ (b). Nonzero values of ${\mathbf{Q}}_0$ correspond to incommensurate correlations.

Figure 6

$x$ and $y$ components of the momentum ${\mathbf{Q}}_0$ (see text for definition) as a function of $J^{\prime }$ at the harmonic (dashed), HF (), and HF-cubic () approximations for the line $J^{\prime \prime }=0.8$. In the strong-dimer limit of large $J^{\prime }$, the harmonic ${\mathbf{Q}}_0$ is approached for both the HF and HF-cubic results.

Figure 7

Parameter $N_0$ as a function of $J^{\prime }$ for the lines $J^{\prime \prime }=0$ and $J^{\prime \prime }=1$ at the harmonic and Hartree-Fock levels.

Figure 8

(a) Renormalized cubic vertices ${\Gamma }_1(\mathbf{k},\mathbf{p})$, (1), and ${\Gamma }_2(\mathbf{k},\mathbf{p})$, (2). Here $(\alpha ,\beta ,\gamma )$ $=$ $(x,y,z)$, $(z,x,y)$, and $(y,z,x)$. (b)–(e) Lowest-order diagrams resulting from the combination of vertices (1) and (2) that contribute to the normal self-energy ${\Sigma }_3(\mathbf{k},\omega )$.

Figure 9

Triplon dispersion relation along some particular lines of the dimerized Brillouin zone for (a) $J^{\prime }=3$ and $J^{\prime \prime }=0$ and (b) $J^{\prime }=1.5$ and $J^{\prime \prime }=1$ at the HF (dashed black line) and HF-cubic (solid red line) approximations. The triplon decay rate ${\tilde{\Gamma }}_{\mathbf{k}}$ (dot-dashed blue line) and the bottom of the two-particle continuum (dotted green line) both at the HF-cubic approximation are also shown. The $A-B$ line includes the triplon minimum energy of the corresponding approximation.

Figure 10

Triplon gap $\Delta$ as a function of $J^{\prime }$ for (a) $J^{\prime \prime }=0$, (b) $J^{\prime \prime }=0.8$, and (c) $J^{\prime \prime }=1$ at the harmonic (), HF (), and HF-cubic () approximations. The solid line is a fit to the data using Eq. (37).

Figure 11

Contour plot of the triplon energy ${\Omega }_{\mathbf{k}}$ (HF-cubic) for $J^{\prime }=1.5$ and $J^{\prime \prime }=0.8$, i.e., near the CIT.

Figure 12

Classical phase diagram for the Heisenberg model (1), obtained from numerically solving Eq. (A3).