Reduced dimensionality in layered quantum dimer magnets: Frustration vs. inhomogeneous condensates

Motivated by recent experiments on $\mathrm{Ba}\mathrm{Cu}{\mathrm{Si}}_2{\mathrm{O}}_6$, we investigate magnetic excitations and quantum phase transitions, driven either by pressure or magnetic field, of layered dimer magnets with interlayer frustration. We consider two scenarios, (A) a lattice with one dimer per unit cell and perfect interlayer frustration, and (B) an enlarged unit cell with inequivalent layers, with and without perfect frustration. In all situations, the critical behavior at asymptotically low temperatures is three-dimensional, but the corresponding crossover scale may be tiny. Magnetic ordering in case (B) can be discussed in terms of two condensates; remarkably, perfect frustration renders the proximity effect ineffective. Then, the ordering transition will be generically split, with clear signatures in measurable properties. Using a generalized bond-operator method, we calculate the low-temperature magnetic properties in the paramagnetic and antiferromagnetic phases. Based on the available experimental data on $\mathrm{Ba}\mathrm{Cu}{\mathrm{Si}}_2{\mathrm{O}}_6$, we propose that scenario (B) with inequivalent layers and imperfect frustration is realized in this material, likely with an additional modulation of the interlayer coupling along the $c$ axis.

Figure 1

(Color online) (a) bct lattice structure of dimers, with couplings $J$ (intradimer), $J^{\prime }$ (intralayer), and $J_z$ (interlayer). [(b) and (c)] Illustrations of possible symmetry breaking. In (b), the interlayer couplings $J_z^{\Delta }$ are modulated within a unit cell such that the two diagonals are inequivalent. In (c), the interlayer couplings are further modulated along the $c$ axis, enlarging the unit cell.

Figure 2

(Color online) Zero-temperature magnetization vs applied field in the coupled-dimer model near $H_{c1}$ at $T=0$, calculated using bond operators (Sec. 4), for three scenarios: (A) equivalent layers and perfect frustration (dashed line), (B1) inequivalent layers and perfect frustration (dash-dot line), and (B2) inequivalent layers and imperfect frustration (solid). Case (B1) clearly shows the secondary transition as a kink in the magnetization curve, whereas this feature is smeared out due to the proximity effect in case (B2). Details will be discussed in Sec. 7; the parameter values are those in Figs. 9, 16, 17 below. We caution the reader that logarithmic corrections, not captured by our approximation, may somewhat modify the shape of the magnetization curve close to the critical field, but the existence of a sharp kink in (B1) will be unaffected.

Figure 3

(Color online) Diagrams occurring in the perturbation expansion of the order-parameter theory ${\mathcal{S}}_{\varphi }$ (7). The solid lines are $\varphi$ propagators (depending on in-plane momentum ${\vec{k}}_{\parallel }$ and layer index $n$), the full circle is the local four-point vertex $(\propto u_0)$, and the cross is the frustrated interlayer hopping $(\propto \eta k_x{k}_y)$. (a) Interlayer density interaction [open circle; this includes both $u_1$ and $u_2$ terms in ${\mathcal{S}}_{AB}$ (12)], generated from a $u_0^2$ process. For negative $u_1$, the $u_1{({\vec{\varphi }}_n\cdot {\vec{\varphi }}_{n+1})}^2$ term leads to collinear spin correlations in the $z$ direction. (b) Unfrustrated second-neighbor vertical hopping (star, $\propto {\eta }^{\prime }$), generated from interaction processes. This hopping is responsible for 3D behavior at lowest energies, irrespective of the interlayer frustration. (c) Additional diagrams present in the ordered phase; the open square denotes the coupling to the condensate. The first diagram corresponds to nearest-neighbor vertical hopping; the second diagram is the leading “vertical” contribution to the free energy.

Figure 4

Diagrams in the Brueckner bond-operator approach. The solid lines are $t_{\vec{q}\alpha }$ propagators, here depending on the full 3D momentum $\vec{q}$. (a) Self-energy from the hard-core repulsion; the shaded square is the effective four-point vertex $\Gamma$ (A5) obtained from a ladder summation. (b) Leading self-energy from the quartic term ${\mathcal{H}}_4$, with the circle corresponding to $J_4$. (c) Second-order self-energy in ${\mathcal{H}}_4$—its ${\mathcal{H}}_{4z}^2$ portion is required to obtain the leading term in the vertical dispersion on the ideal bct lattice. (d) Second-order self-energy in ${\mathcal{H}}_{3z}$, where the triangle is the three-point vertex of strength $J_3^{\pm }$. The momentum-space structure of the vertex suppresses this diagram at ${\vec{q}}_{\parallel }=(\pi ,\pi )$.

Figure 5

(Color online) Triplon dispersion (solid) and boundary of the two-particle continuum (shaded), calculated using the Brueckner bond-operator approach (Sec. 4B and Appendix ) on an $8^3$ lattice. The parameter values are $J=1$, $J^{\prime }∕J=0.15$, $-J_{2z}=J_{4z}=0.2J^{\prime }$, $J_{3z}=0$, and $H=0$. The inset shows an energy zoom into the almost flat vertical dispersion. For comparison, the dotted line shows a dispersion calculated within the harmonic approximation, with parameter values chosen to match the Brueckner results at wave vectors (0,0,0), $(\pi ,\pi ,0)$, and $(0,0,\pi )$: $J=1.047$, $J^{\prime }∕J=0.140$, and $-J_{2z}∕J^{\prime }=0.196$.

Figure 6

(Color online) Triplon dispersion (solid) and two-triplon continuum (shaded) as in Fig. 5, but with parameter values closer to the zero-field transition: $J=1$, $J^{\prime }∕J=0.25$, $-J_{2z}=J_{4z}=0.2J^{\prime }$, $J_{3z}=0$, and $H=0$. Compared to Fig. 5, the bandwidth along $(\pi ,\pi ,q_z)$ is significantly enhanced.

Figure 7

(Color online) Triplon dispersion as in Fig. 5, but now in a finite field: $J=1$, $J^{\prime }∕J=0.15$, $-J_{2z}=J_{4z}=0.2J^{\prime }$, $J_{3z}=0$, and $H=0.4$.

Figure 8

(Color online) Schematic $T=0$ phase diagrams for the bct lattice coupled-dimer model, for the situation of equivalent layers with perfect frustration. (a) Pressure tuning. Here, generically two scenarios are possible (where the second one applies for the parameters relevant to $\mathrm{Ba}\mathrm{Cu}{\mathrm{Si}}_2{\mathrm{O}}_6$). (b) Field tuning.

Figure 9

(Color online) (a) Triplon energy gaps and (b) uniform (solid) as well as staggered (dashed) magnetization, calculated as function of the external field $H$ using bond operators in the harmonic approximation. Parameter values describe the ideal bct lattice: $J=4.66\mathrm{meV}$, $J^{\prime }=0.5\mathrm{meV}$, and $-J_{2z}=0.1\mathrm{meV}$.

Figure 10

(Color online) Zero-field triplon dispersion for inequivalent layers with perfect frustration, calculated using the harmonic bond-operator approximation. Parameter values are $J_A=4.27\mathrm{meV}$, $J_B=5.04\mathrm{meV}$, $J_A^{\prime }=J_B^{\prime }=0.5\mathrm{meV}$, and $-J_{2z}=J_z=0.1\mathrm{meV}$. Compared to the equivalent-layer case, the vertical dispersion at the unfrustrated point, i.e., along $(0,0,q_z)$, is both quenched and modified in shape, from $\propto J_z\cos q_z$ to $\propto \pm J_z^2\cos 2q_z$.

Figure 11

(Color online) As in Fig. 10, but now for inequivalent layers with imperfect frustration. Parameter values are $J_A=4.27\mathrm{meV}$, $J_B=5.04\mathrm{meV}$, $J_A^{\prime }=J_B^{\prime }=0.5\mathrm{meV}$, $J_{z1}=0.05\mathrm{meV}$, and $J_{z2}=0.15\mathrm{meV}$—the latter two values are the couplings along the two inequivalent $J_z$ diagonals [see Fig. 1b].

Figure 12

(Color online) Bandwidths of the vertical dispersion along $(\pi ,\pi ,q_z)$ and $(0,0,q_z)$ as function of the average ${\overline{J}}_z$, for equivalent layers $(J=4.66\mathrm{meV})$ and perfect frustration (solid, squares), inequivalent layers ($J_A=4.27\mathrm{meV}$ and $J_B=5.04\mathrm{meV}$) with perfect frustration (dotted, triangles), imperfect frustration with $J_{z1}∕J_{z2}=3$ (dashed, circles), and imperfect frustration plus vertical modulation with $J_{zA1}∕J_{zA2}=11∕9$, $J_{zB1}∕J_{zB2}=19$, and ${\overline{J}}_{zA}={\overline{J}}_{zB}$ (dash-dot, no symbols; see also Fig. 18). In all cases, $J^{\prime }=0.5\mathrm{meV}$.

Figure 13

(Color online) Vertical dispersion of the lowest mode, $\omega (q_x,q_y,\pi ∕2)-\omega (q_x,q_y,0)$, as a function of in-plane momentum ${\vec{q}}_{\parallel }$ for inequivalent-layer situations. (a) Perfect frustration; parameter values as in Fig. 10. (b) Imperfect frustration plus vertical modulation; parameter values as in Fig. 18.

Figure 14

(Color online) Schematic $T=0$ phase diagrams for the coupled-dimer model with inequivalent layers and perfect frustration. Due to the absence of a proximity effect, there is no coupling between the condensates on the subsystems $A$ and $B$ (on even and odd layers), leading to two magnetic transitions. (a) Pressure tuning. (b) Field tuning.

Figure 15

(Color online) (a) Schematic overall temperature-field phase diagrams for the coupled-dimer model with inequivalent layers and perfect frustration, for different relations between $J_{A,B}$ and $J_{A,B}^{\prime }$. Solid (dashed) lines show the phase transitions in the $A$ $\left(B\right)$ subsystem. (b) Schematic zero-temperature magnetization curves for the three scenarios in (a). For perfect frustration, the slope will change discontinuously at the transitions.

Figure 16

(Color online) (a) Triplon energy gaps and (b) uniform (solid) and staggered (dashed) magnetizations as in Fig. 9, but for inequivalent layers with perfect frustration—the splitting of the phase transitions is clearly visible. Parameter values are $J_A=4.27\mathrm{meV}$, $J_B=5.04\mathrm{meV}$, $J_A^{\prime }=J_B^{\prime }=0.5\mathrm{meV}$, and $-J_{2z}=J_z=0.1\mathrm{meV}$. The field is displayed in units of the average $\overline{J}=4.66\mathrm{meV}$. The thin lines in (b) show the individual contributions of the $A$ and $B$ subsystems (i.e., even and odd layers) to the magnetizations. The presence of two Goldstone modes in the intermediate field range is an artifact of the harmonic approximation: inclusion of interaction effects would split the two modes.

Figure 17

(Color online) (a) Triplon energy gaps and (b) uniform (solid) and staggered (dashed) magnetizations as in Fig. 16, but now for inequivalent layers with imperfect frustration—here the secondary phase transitions are smeared out due to the proximity effect. Parameter values are $J_A=4.27\mathrm{meV}$, $J_B=5.04\mathrm{meV}$, $J_A^{\prime }=J_B^{\prime }=0.5\mathrm{meV}$, $J_{z1}=0.05\mathrm{meV}$, and $J_{z2}=0.15\mathrm{meV}$—the latter two values are the couplings along the two inequivalent $J_z$ diagonals [see Fig. 1b].

Figure 18

(Color online) Zero-field triplon dispersions for inequivalent layers with partially frustrated and modulated vertical coupling [Fig. 1c]. Parameter values are $J_A=4.27\mathrm{meV}$, $J_B=5.04\mathrm{meV}$, $J_A^{\prime }=J_B^{\prime }=0.5\mathrm{meV}$, $J_{zA1}=0.09\mathrm{meV}$, $J_{zA2}=0.11\mathrm{meV}$, $J_{zB1}=0.01\mathrm{meV}$, and $J_{zB2}=0.19\mathrm{meV}$. The bandwidth along $(\pi ,\pi ,q_z)$ is reduced by a factor of 3 as compared to the situation without modulation (Fig. 11), and can be made arbitrarily small by reducing $\mid J_{zA1}-J_{zA2}\mid$, but the magnetization data in the present case are essentially undistinguishable from the ones in Fig. 17.