Schwinger boson mean-field theory: Numerics for the energy landscape and gauge excitations in two-dimensional antiferromagnets

We perform some systematic numerical search for Schwinger boson mean-field states on square and triangular clusters. We look for possible inhomogeneous ground states as well as low-energy excited saddle points. The spectrum of the Hessian is also computed for each solution. On the square lattice, we find gapless $U\left(1\right)$ gauge modes in the nonmagnetic phase. In the ${\mathbb{Z}}_2$ liquid phase of the triangular lattice, we identify the topological degeneracy as well as vison states.

Figure 1

Square lattice model. (Top) Spin gap as a function of $\kappa$. In the thermodynamic limit this state is associated to magnetic long-range order (vanishing spin gap) for $\kappa >{\kappa }_c\simeq 0.39$.[7] (Middle) Smallest eigenvalue of the Hessian for the ground state on 36, 64, and 144-site square lattices. (Bottom) Scaling of the lowest Hessian eigenvalue as a function of the system size $N_s$. This shows that the Hessian is gapless for $\kappa =0.2$ and gapped for $\kappa =0.8$. The Hessian gap is finite in the magnetic phase, while it vanishes in the thermodynamic limit for $\kappa <{\kappa }_c$. For $\kappa =0.2$, first eigenvalues decays faster than $1/N_s$ whereas the second eigenvalue goes to zero as $\sim 1/N_s$ (see text).

Figure 2

Hessian eigenvector $d{A}_l$ corresponding to the second smallest eigenvalue (0.26224396339) for a 64-site square lattice with $\kappa =0.2$. The thickness of each bond $l$ is proportional to the modulus $|d{A}_l|$ while the color represents the complex argument of $d{A}_l/A_l$. Red: argument is $+\pi /2$, blue is $-\pi /2$. Here, $dA$ is a gauge excitation, it corresponds to a long wavelength modulation of the flux through each plaquette. Here, the wave vector is parallel to the horizontal bonds. This gauge mode is associated to the smallest nonzero wave vector on the square lattice and is therefore fourfold degenerate.

Figure 3

Hessian eigenvector $d{A}_l$ corresponding to the smallest eigenvalue (2.62955994344) for a 36-site square lattice. Same representation as in Fig. 2. The complex argument of $d{A}_l/A_l$ takes only two values: $\pm \pi /2$ (blue and red), indicating that $dA$ is a gauge excitation. This mode corresponds to an increase of the flux through large horizontal loops, and no change for the flux going through local plaquettes.

Figure 4

First excited mean-field solution on a square lattice with 36 sites and $\kappa =1$. The modulus of the bond parameter (right color scale) is indicated, as well as the chemical potentials (left scale). This solution shows a localized excitation (here around site number 14). The flux decays rapidly with distance from the center: $0.4046133\pi$ on plaquette[8], 9,15,[14], $0.037028\pi$ on plaquette $[1,2,8,7]$, $-0.005291300\pi$ on $[0,1,7,6]$, $0.0001131\pi$ on $[4,5,11,10]$, etc. See Fig. 5 for some the spin-spin correlations in this state.

Figure 5

Spin-Spin correlation in the first excited of a 36-site square lattice with $\kappa =1$ (same state as in Fig. 4). The radius of the circle on site $j$ proportional to the correlation $|\langle {\vec{S}}_{i0}{\vec{S}}_j\rangle |$ with the reference site $i0$ (the sign is available on the color scale). (Top) Reference spin $i_0=14$ is at the center of the localized excitation. (Bottom) Reference spin $i_0=35$ is “far” from the center of the excitation.

Figure 6

Two low-energy mean-field solutions on a square lattice with 36 sites and $\kappa =0.1$. $\left|A\right|$ takes only two values and the bonds with the stronger $\left|A\right|$ are indicated by fat grey lines (for values, see Tab. ). All the sites are equivalent and the chemical potential is uniform in both cases.

Figure 7

A low-energy mean-field solution on a square lattice with 36 sites and $\kappa =0.1$. The modulus of the bond parameters is indicated—the right color scale shows deviations from the minimum value. The left color scale indicates the chemical potentials (dot on each site). This solution shows modulations with the symmetries of a 3 unit cell VBC.

Figure 8

(Top) Smallest eigenvalue of the Hessian for the ground state state on 36-site and 144-site triangular clusters. (Bottom) Spin gap. In the thermodynamic limit, this state is associated to magnetic long-range order (vanishing spin gap) for $\kappa \gtrsim 0.34$.[9]

Figure 9

Hessian eigenvector $dA/A$ corresponding to the smallest (degenerate) eigenvalue (0.1943) for a 144-site triangular lattice with $\kappa =0.1$ (uniform ground state). The complex argument of $dA/A$ take only two values: $\pm \pi /2$ (blue and yellow), indicating that $dA$ is a (gapped U(1)) gauge excitation. (Bottom) Infinitesimal flux variation $dF$ associated to the gauge mode above. For each diamond, the magnitude of the flux variation is indicated by the radius of the dot in its center (see color scale for the sign).

Figure 10

First excited state on a triangular cluster with 36 sites and $\kappa =1$. The bond moduli are invariant by lattice rotations about the site largest chemical potential (grey circle). The flux vanish on all diamonds except for the six diamonds which diagonal bond is marked by a red dot. The later have flux $\pi$. Notice the similarity with Fig. 4.

Figure 11

First excitation on a triangular cluster with 36 sites and $\kappa =0.1$. This state is threefold degenerate. With the ground state, it forms the fourfold topological degeneracy expected for a ${\mathbb{Z}}_2$ liquid on a torus. Notice (scale on the right) that the difference between the largest $|A_{ij}|$ and the smallest one is only $\sim 6.{10}^{-4}$ and should vanish in the thermodynamic limit.

Figure 12

(Top) Modulus $\left|A\right|$ and chemical potential for a saddle point with two visons at distance $d=5\sqrt{3}/3$, ($6\times 6$ triangular lattice, $\kappa =0.2$). (Bottom) Flux vanishes on all diamonds except those with the diagonal tagged with a grey circle. The visons are localized on the triangles with three such circles.

Figure 13

Modulus $\left|A\right|$ and chemical potential for a saddle point with two visons at distance $d=4\sqrt{3}$, ($12\times 12$ triangular lattice, $\kappa =0.2$). A string of reversed $A_{ij}$ (not shown) goes from the first vison core to the second.

Figure 14

(Top) Energy cost of a pair of visons as a function of distance in a 144-site triangular cluster. (Bottom) Difference between the spinon gap $\Delta$ in presence of the two visons, and the gap ${\Delta }_0$ in the absence of visons. This difference has been scaled by $\kappa$ to compare $\kappa =0.1$ and $\kappa =0.3$). Since $\Delta >{\Delta }_0$, there is some repulsion between a spinon and a vison pair.

Figure 15

Highly excited solution on the triangular lattice (144 sites) with $\kappa =0.6$ (ordered phase). (Top) Bond amplitudes. (Bottom) Spin-spin correlations and fluxes. If $\langle S_0·S_i\rangle$ is positive while $\langle S_1·S_i\rangle$ and $\langle S_2·S_i\rangle$ are negative, the site $i$ is said to belong to the “0” sublattice (blue circles). Likewise, the site which are positively correlated with site 1 (respectively 2) are marked in red (respectively green). The other sites are marked in black and the radius of the circle is proportional to the maximum of $|\langle S_0·S_i\rangle |$, $|\langle S_1·S_i\rangle |$ and $|\langle S_2·S_i\rangle |$. The diamonds have vanishing flux except for those marked by a grey circle, which have a flux $\pi$ (see text).