Magnetic properties of nanoscale compass-Heisenberg planar clusters

We study a model of spins $1/2$ on a square lattice, generalizing the quantum compass model via the addition of perturbing Heisenberg interactions between nearest neighbors, and investigate its phase diagram and magnetic excitations. This model has motivations both from the field of strongly correlated systems with orbital degeneracy and from that of solid-state based devices proposed for quantum computing. We find that the high degeneracy of ground states of the compass model is fragile and changes into twofold degenerate ground states for any finite amplitude of Heisenberg coupling. By computing the spin structure factors of finite clusters with Lánczos diagonalization, we evidence a rich variety of phases characterized by ${\mathbb{Z}}_2$ symmetry that are either ferromagnetic, $C$-type antiferromagnetic, or of the Néel type and analyze the effects of quantum fluctuations on phase boundaries. In the ordered phases, the anisotropy of compass interactions leads to a finite excitation gap to spin waves. We show that for small nanoscale clusters with large anisotropy gap the lowest excitations are column-flip excitations that emerge due to Heisenberg perturbing interactions from the manifold of degenerate ground states of the compass model. We derive an effective one-dimensional $XYZ$ model that faithfully reproduces the exact structure of these excited states and elucidates their microscopic origin. The low-energy column-flip or compass-type excitations are robust against decoherence processes and are therefore well designed for storing information in quantum computing. We also point out that the dipolar interactions between nitrogen-vacancy centers forming a rectangular lattice in a diamond matrix may permit a solid-state realization of the anisotropic compass-Heisenberg model.

Figure 1

(a) Structure of an NV center in a diamond matrix, which can be described by an effective spin, (b) four NV centers forming a rectangle are controlled by dipolar interactions between the associated spins $\left\{{\vec{S}}_i\right\}$. Dipole interaction $H_{\mathrm{dip}}$ involves compass-type terms of various amplitudes $\{J_x,J_z\}$, depending on the pair considered, and Heisenberg interactions. (c) Ground state of anisotropic compass-Heisenberg model represented by the spins of NV centers and arranged in a rectangular array. (d) Distinct low-energy excitations of this system: a column flip (CF) reverses the spins of a whole column, while a spin flip (SF) reverses a single spin, contributing to spin wave excitations.

Figure 2

Parameterization of the compass-Heisenberg model Eq. (2.1) by two angles $\{\phi ,\theta \}$. A point in parameter space is indicated by the (blue) dot on the sphere; interaction parameters $I$, $J_x$, and $J_z$ correspond to its cartesian coordinates on the respective axes, while angles $\theta$ and $\phi$ are its spherical coordinates.

Figure 3

Two equivalent parameterizations of the compass model by either couplings $J_z$ and $J_x$ or by the radius $J_c$ and angle $\phi$ defined by Eqs. (2.2) and (2.3), respectively. On this circle, isotropic compass points $|J_x|=|J_z|$ are shown by filled (blue) square and diamond for the model with AF $J_z>0$ interactions and by empty (purple) square and diamond for the model with FM $J_z<0$, while Ising points are indicated by dots ($J_x=0$) or vertical bars ($J_z=0$). The points $\phi =3\pi /20$ and $\phi =23\pi /20$, frequently considered hereafter, are also indicated by radial (red) bars.

Figure 4

Structure factors (a) $S^z\left(q\right)$ and (b) $S^y\left(q\right)$ (divided by $N$) obtained for the AF CH model with the $N=36$ cluster (symbols). The lines are guides to the eye. Parameters: $\phi =\pi /10$; the data correspond to four different values of $I/J_c$: $0.3$ ($G_z$ phase), $+{10}^{-6}$, $-0.3$ ($C_z^{\prime }$ phase), and $-0.6$ ($F_y$ phase). High-symmetry points $\Gamma =(0,0)$, $X=(\pi ,0)$, $M=(\pi ,\pi )$, and $Y=(0,\pi )$ in the 2D Brillouin zone are defined in the inset.

Figure 5

Structure factors (a)–(c) $S^z\left(q\right)$ and (d)–(f) $S^y\left(q\right)$ obtained for the isotropic compass model $J_x=J_z$ ($\phi =\pi /4$) and increasing values of the Heisenberg interaction $I$ for clusters of different size: (a) and (d) $N=16$, (b) and (e) $N=24$, and (c) and (f) $N=36$ clusters. The $N=24$ cluster is rectangular with $(L_x,L_z)=(4,6)$, as indicated by the points in (b) and (e). The high-symmetry points $\Gamma$, $X$, $Y$, and $M$ are as in Fig. 4. The compass quasi-1D order is characterized by constant values of $S^z\left(q\right)$ along the $M-Y$ line. The onset of the AF order in the TL is evidenced by the increasing maximum of $S^z\left(q\right)$ at $M=(\pi ,\pi )$, which is visible already for very small values of $I>0$.

Figure 6

Phase diagram of the compass-Heisenberg model in the $(J_x,I)$ plane for fixed AF interaction $J_z=1$. Long-range order is stabilized by any finite $I$. Square ($J_x=J_z$, $I=0$) and diamond ($J_x=-J_z$, $I=0$) at the compass line indicate multicritical points, where in each case four ordered phases meet. The spin order of the different phases $\{G_z,G_x,C_z^{\prime },C_x,C_x^{\prime },F_x,F_y\}$ is depicted in a corresponding inset, and the subscript $\alpha =x,y,z$ indicates the type of symmetry breaking in spin space, see Table . The QPTs between $F_x$ and $C_z^{\prime }$ phases, and between $C_x$ and $G_z$ phases (solid lines) are modified by quantum corrections with respect to the corresponding classical transitions (dashed straight lines).

Figure 7

Evolution of order parameters $S^y(0,0)/N$ and $S^z(0,\pi )/N$ across the $C_z^{\prime }\leftrightarrow F_y$ transition, by varying $I/J_z$; anisotropic compass interactions have the same parameter $\phi =\pi /10$ as in Fig. 4. The cluster $N=24$ has dimensions $L_x=4\text{and}{L}_z=6$. The dashed-dotted lines correspond to perturbative estimations of order parameters given for the $C_z^{\prime }$ phase by Eq. (4.5). The transition coincides here with the classical value $I_c=J_z/2$.

Figure 8

Size scaling of the order parameters of (a) $C_z^{\prime }$ and (b) $F_y$ phases for fixed anisotropy parameter $\phi =\pi /10$ and for values of $I/J_c$ at and close to the value $I=-0.4755J_c$ of the transition between both phases. The data points are obtained with the $N=8,16,20,24,30,32,36$ clusters. Two data points for $N=24$ correspond to two different rectangular clusters: $4\times 6$ and $6\times 4$.

Figure 9

Variation of order parameters $S^x\left(\Gamma \right)/N$ and $S^z\left(Y\right)/N$ across the $C_z^{\prime }\leftrightarrow F_x$ transition as function of $I/J_z$ at fixed $\phi =-3\pi /20$, and for system sizes $N=16,4\times 6,32$. Here the quantum phase transition is shifted by quantum fluctuations from $I_c^0/J_z=-0.245$ to $I_c/J_z=-0.21\left(1\right)$.

Figure 10

Evolution of $1-f$, where $f$ denotes the fidelity Eq. (4.8), as function of $I/J_z$ for parameters (a) $\phi =-3\pi /20$ and (b) $\phi =\pi /10$, and for two cluster sizes $N=16$ and 24 (in both cases $L_z=4$). Smooth peaks reflect phase transitions between the $F_x$ or $F_y$ phase and the $C_z^{\prime }$ phase that are continuous for finite systems. The sharp peaks at $I=0$ indicate the $C_z^{\prime }\leftrightarrow G_z$ transition.

Figure 11

(a) Lowest excitation energies $\Delta E$ with momentum either $k=\Gamma =(0,0)$ or $k=Y=(0,\pi )$, across the $F_x-C_z^{\prime }$ and $C_z^{\prime }-G_z$ transitions as a function of $I/J_z$ at $\phi =-3\pi /20$ and for different cluster sizes. (b) Ground-state energy $E_0$, found with momentum $\Gamma$, reflecting the avoided crossing of the $F_x-C_z^{\prime }$ transition at $I/J_z=-0.21\left(1\right)$.

Figure 12

Lowest excitation energies $\Delta E$ for $N=16$ across the transitions between the $F_y$ and $C_z^{\prime }$ phases (for three distinct values of $J_x/J_z$ between 0 and 1, corresponding to $\phi =\pi /10$, $\pi /5$, and $\pi /4$).

Figure 13

Phase diagram of the compass-Heisenberg model in the $(J_x/J_z,I/J_z)$ plane for fixed FM coupling $J_z=-1$. Similar to Fig. 6, square ($J_x=J_z$) and diamond ($J_x=-J_z$) indicate multicritical points, and the spin order of the different phases is depicted in insets. Phase labels follow the same convention as in Fig. 6. The QPTs between $F_z$ and $C_x^{\prime }$ phases and between $C_z$ and $G_x$ phases (solid lines) are modified by quantum corrections with respect to classical transitions (dashed straight lines).

Figure 14

Spin wave dispersions obtained within the LSW theory (lines) and by exact diagonalization (symbols) of various clusters with size $N\le 32$ for $\phi =23\pi /20$ and for different Heisenberg coupling constants: (a) $I=-0.2J_c$, (b) $I=0.2J_c$, and (c) $I=0.6J_c$. These different parameters correspond to the $F_z$, $C_z$, and $G_y$ phases, respectively. The abscissa $x_k$ follows a path in the Brillouin zone identical as in Fig. 4. For $N=32$, only one of the two lowest spin-wave branches [see second paragraph below Eq. (5.8)] is shown.

Figure 15

Anisotropy gap $E_a/J_c$ of spin waves in the $G_z$ phase as a function of increasing $J_x/J_z=\tan \left(\phi \right)$, for Heisenberg amplitude $I=J_c/5$ and various cluster sizes $N$. (Inset) Size scaling for the gap for $\phi =\pi /4$ (isotropic case) and $\phi =\pi /5$ (finite gap).

Figure 16

Low-energy spectra of columnar excitations $E/J_c$ obtained by exact diagonalization of a $L_x=L_y=4$ periodic cluster, as a function of $I/J_c$, for (a) FM interactions ($\phi =23\pi /20$) and (b) AF interactions ($\phi =3\pi /20$), respectively. Corresponding results obtained from the effective Hamiltonian $H_{\mathrm{col}}^{\mathrm{FM}/\mathrm{AF}}+H_{\mathrm{col}}^{\mathrm{FM}/{\mathrm{AF}}^{\prime }}$ for the same cluster and interaction parameters are shown for the FM (c) and AF (d) cases, respectively. Momenta of various eigenstates are indicated by symbols; continuous and dashed lines indicate even and odd states with respect to time reversal. The lowest line in each panel represents the two quasidegenerate ground states, while the next band (central branch) contains $2^4-4=12$ columnar excitations.

Figure 17

Pseudospins of a CH nanocluster representing $L_x=5$ columnar spin chains in the case $|J_z|\gg |J_x|,|I|$. (a)–(d) show the successive steps in the initialization and utilization of two essentially decoupled protected qubits: (a) the system is in one of the two quasidegenerate ground states, $|{\Phi }_0\rangle$, such that for each pseudospin $j$ one has ${\tau }_j^z={(-1)}^{j-1}$; (b) after flipping the column (or pseudospin) $j=3$, the system is in an excited state $|{\Phi }^{*}\rangle$ belonging to the central branch of columnar excitations; (c) the qubits encoded by pseudospins ${\vec{\tau }}_2$ and ${\vec{\tau }}_4$ are initialized in the state ${\mathrm{H}}_2{\mathrm{H}}_4|{\Phi }^{*}\rangle$, where ${\mathrm{H}}_{\mathrm{j}}$ rotates the pseudospin ${\vec{\tau }}_j$ by $\pi /2$ along the $y$ axis in pseudospin space (which should not be confused with a uniform rotation of all spins in the column $j$); and (d) an example of a state produced from the former by an operation on pseudospin ${\vec{\tau }}_2$.

Figure 18

Lowest single spin-flip excitation energy $E_a$ in the $C_z$ phase in comparison with the maximal ($E_c^{+}$) and minimal ($E_c^{-}$) energies of the central column-flip excitation branch (see text) for open clusters: (a) as a function of $I/J_c$ for fixed $L_x=3$, $L_z=4$ and either $\phi =11\pi /10$ or $\phi =23\pi /20$, here $E_c^{+}$ and $E_c^{-}$ are undistinguishable; and (b) as a function of $L_z$ for fixed $I/J_c=0.1$, $\phi =11\pi /10$ and either $L_x=3$ or 5.