Nonlinear bond-operator theory and $1/d$ expansion for coupled-dimer magnets. I. Paramagnetic phase

For coupled-dimer Heisenberg magnets, a paradigm of magnetic quantum phase transitions, we develop a systematic expansion in $1/d$, the inverse number of space dimensions. The expansion employs a formulation of the bond-operator technique and is based on the observation that a suitably chosen product-state wave function yields exact zero-temperature expectation values of local observables in the $d\to \infty$ limit, with corrections vanishing as $1/d$. We demonstrate the approach for a model of dimers on a hypercubic lattice, which generalizes the square-lattice bilayer Heisenberg model to arbitrary $d$. In this paper, we use the $1/d$ expansion to calculate static and dynamic observables at zero temperature in the paramagnetic singlet phase, up to the quantum phase transition, and compare the results with numerical data available for $d=2$. Contact is also made with previously proposed refinements of bond-operator theory as well as with a perturbative expansion in the interdimer coupling. In a companion paper, the present $1/d$ expansion will be extended to the ordered phase, where it is shown to consistently describe the entire phase diagram including the quantum critical point.

Figure 1

Left: Phase diagram of the coupled-dimer model (2) on the hypercubic lattice as function of the control parameter $q=Kd/J$ and the inverse spatial dimension $1/d$. A QPT separates the paramagnetic singlet phase from the AFM phase. The solid line shows our result (5) for the phase boundary $q_c$ to order $1/d$ for the symmetric case $\kappa =0$ (3); the dashed line represents the solution of the equation $\Delta \left(q\right)=0$ with $\Delta \left(q\right)$ from Eq. (7). The cross marks the numerically exact result for $d=2$ obtained in Ref. [19]. The shaded areas indicate the points of departure for the large-$d$ expansion (red) and the small-$K/J$ expansion (blue, Sec. 5), respectively. Right: Sketch of the model in $d=2$, with solid (hollow) dots denoting the spins 1 (2) of each dimer.

Figure 2

Feynman diagrams for the ground-state energy up to order $1/d^2$.

Figure 3

Feynman diagrams for the normal $\tau$ self-energy up to order $1/d$, with vertices from ${\mathcal{H}}_{2b}^{\prime }$ and ${\mathcal{H}}_4^{\prime }$.

Figure 4

Feynman diagrams for the contribution of cubic (${\mathcal{H}}_3^{\prime }$) terms to the normal $\tau$ self-energy up to order $1/d$.

Figure 5

Triplon dispersion (65) derived from the $1/d$ expansion for the coupled-dimer model (2), showing results for $d=\infty$ (solid line), $d=3$ with $\kappa =0$ (dashed line), and $d=3$ with $\kappa =1$ (dashed-dotted line), where $\kappa$ is the asymmetry parameter (3). Left: results for fixed $q=Kd/J=0.45$. Right: results at criticality $q=q_c$, where here $q_c$ is defined by ${\Omega }_{\vec{Q}}=0$ with ${\Omega }_{\vec{k}}$ from Eq. (65) at fixed $d$; the value of this $q_c$ is distinct from the expansion result (68) evaluated at fixed $d$.

Figure 6

Triplon gap (66) for $d=\infty$ (solid line), $d=3$ with $\kappa =0$ (dashed line), and $d=3$ with $\kappa =1$ (dashed-dotted line).

Figure 7

Feynman diagrams for anomalous self-energies contributing to order $1/d$.

Figure 8

Relevant graphs ${\mathcal{G}}_i$ with $i\in \{1,...,11\}$ to calculate the one-particle dispersion up to order five in $k$ and the ground-state energy per site up to order seven in $k$, respectively. Here, filled circles represent dimers which are connected to their nearest neighbors by links shown as solid lines.

Figure 9

Sample self-energy diagram with two cubic ${\Phi }_{31}$ vertices.