 an alternating phase factor coming from the antiferromagnetic nature of the magnetic order.
For any spatial slice, one can define the Skyrmion number Q(τ ) = 1 4π n · (∂ xn × ∂ yn ) τ, which is a topological invariant. Then, a monopole creation operator is defined as a topological defect at the spacetime point which changes Q(τ ) by +1 across it. By the further duality mapping, this monopole in the NLσM will be mapped into the monopole of the CP 1 theory. As the center of the monopole cannot have a finite spin direction, the monopole is located at a dual lattice. When we have a monopole event in the spacetime, it must give a branch-cut structure on the image of ω(r) because ω(r) must change by 4π around the monopole location in the space. More intuitively, when monopole residing on the dual lattice encircles a single site, the imaginary time trajectories of all spins except the one encircled by the monopole would oscillate just back and forth, while the trajectory for the encircled one would entirely wind its ω by 4π upon the completion of encircling as in Fig. 13(b). Thus, one can view this problem as a monopole hopping around the dual 2D lattice in which each site in the original lattice gives a 4πSη r flux through the plaquettes of the dual lattice. (Monopole is a charged object under this

Gp4g
Gp4m "flux" emanated from a spin-S.) The associated phase factor is independent of the exact imaginary time location of the monopole event. Thus for S = 1/2 case, one can fix the system into a certain gauge and view this as if ±π/2 Aharonov-Bohm phase factor gets accumulated for each hopping process for monopoles. The monopole transformation rule is summarized in Tab. V. R site π/2 is a site (spin) centered rotation. Note that it is important to fix the convention for the rotation center because translation symmetry is projective. Here, we choose a rotation to be defined with respect to the black spin in Fig. 13(a). R plaq π/2 is a plaquette centered rotation, which is defined with respect to the center of four spins in Fig. 13(a). Under the unit translations T x,y, time reversal T, and plaquette centered rotation R plaq π/2, the Néel order changes its sign. It means that the flux pattern is reversed under such transformations, thus we need to transform a monopole into an anti-monopole to compensate for the change. For reflections σ x,y, although n does not flip, the definition of the Skyrmion number tells that we need to change the sign for the number of monopoles. Thus, a monopole transforms into an antimonopole again. After figuring out whether a monopole transforms into a monopole or anti-monopole, we need to multiply it by an additional phase factor α g to account for the Berry phase effect.
Assume the topological term S B is absent momentarily, which is how CP 1 theory in Eq. (4) is derived. This is a usual practice because unlike the first term inside the parenthesis in Fig. A1, the second term, S B, cannot be straightforwardly extended to the continuum field theory description. Under the absence of S B, the monopole insertion operator M † just makes a global adjustment of the Néel order configuration to increase the Skyrmion number by one, without any additional phase factor.
However, we know from the existence of S B in the lattice description that the Berry phase effect is important. In order to take into account the Berry phase effect, we need to examine how the monopole transforms under each symmetry action. Under the active transformation where the coordinate system remains the same, we have where the action of g on M † is determined by the previous argument on whether the monopole transforms into the monopole or anti-monopole upon the symmetry transformation.
To determine α g, we need to fix a gauge first. Fixing a gauge is important because a monopole is always created in a pair with an anti-monopole. Thus, the monopole event always has a reference point (anti-monopole) connected by the branch-cut. The Berry phase factor is independent of the way we draw the branch-cut because going around a single spin-lattice site gives a 2π phase. Let's fix the reference gauge such that the monopole created at (0,0) gives a Berry phase β. Then, for a generic coordinate r, inserting a monopole would give the Berry phase factor η(r)β where η(r) = 1, i, −1, or − i depending on whether the coordinates (r x, r y ) are (even, even), (odd, even), (odd, odd), (even, odd) [81]. This is shown explicitly in Fig. 13(c) as moving the monopole along the arrow gives an additional phase factor i. Inserting an anti-monopole at r gives a phase factor η * (r)β * since it would give an exactly opposite contribution to the Berry phase term by having ω → −ω. Now, by fixing β = 1, the insertion of the monopole and anti-monopole at each dual lattice site simply gives a phase η(r) and η * (r).
To illustrate the further procedure, let us consider two examples, R site π/2 and T x. Under R site π/2, a monopole remains monopole, but its location changes as r → R site π/2 r. By calculating the relative phase factor between the monopole created at r and R site π/2 r, we obtain its transformation rule in the continuum description: In the case of T x, a monopole transforms into an antimonopole under the symmetry action. At the same time, its location changes as r → r +x. Following the similar procedure, we obtain the following rule: Following the similar analysis, we can obtain α g for all symmetry transformations summarized in Tab. V. In the case of time-reversal symmetry, we do not need an extra phase factor because time-reversal already complexconjugates the phase factor of a monopole operator, which matches with the phase factor given by the antimonopole at the same site. Moreover, any monopole condensation amplitude would be time-reversal symmetric because T : M → M † * = M. So far, we assumed β = 1. However, a different choice of β can be made, which would affect to the transformation rule for the symmetries that filps monopole to anti-monopole. In fact, we can show that this is related to the identification rule between monopole M † and the VBS order parameters v x and v y. For β = 1, we get the relation Eq. (6); for example, T x : M † → −iM implies that the T x -invariant monopole condensation corresponds to the condition that Re M + Im M = 0. However, if β = β = e iπ/4, we would get T x : M † → −M, implying that the T x -invariant condition is Re M = 0. In such a case, we can deduce that M † ∼ v x + iv y.
Once we fix the gauge choice and determine how the monopole transforms under the square lattice symmetries (p4m), how the monopole transforms under the Shastry-Sutherland lattice symmetries (p4g) can be deduced easily. This is because the Shastry-Sutherland lattice symmetries can be expressed in terms of the square lattice symmetries if we take the Shastry-Sutherland lattice in In this section, we present our analysis on the square lattice with spin-1/2. Using the iDMRG simulation, we studied these models on an infinite cylinder with a circumference size up to L = 10 lattice sites. Here, we focused on the correlation length spectra. The simulation is explicitly U (1) z symmetric, and we can plot the correlation spectra for each U (1) z quantum number, S z. Since following models are SO(3) symmetric in the microscopic Hamiltonian, there must exist some degeneracy between different S z sectors, which can be interpreted as the spectrum for a higher spin.
First, let us consider the case where the square lattice symmetry is broken. The following model realizes the phase transition between Néel order and dimerized phase: where red and blue bonds are shown in Fig. 14(a). Here, the dimerized phase does not break any symmetry be-cause the square lattice symmetry is already broken in the model. Therefore, the transition should be described by the Landau-Ginzburg theory. Indeed, it is known from the quantum Monte Carlo simulation [33] that the system realizes O(3) Wilson-Fisher critical point at J 1 /J 1 = 0.523. In the iDMRG simulation, we also observed that the Néel order parameter develops at J 1 /J 1 ∼ 0.52. At the transition, a single monopole event is not suppressed because different configurations for single monopole event cannot exactly cancel each other due to the absence of symmetries. As a result, the gauge fluctuation (VBS order parameter fluctuation) becomes confining, and the CP 1 theory is no longer valid. Instead, the critical theory is described by the classical NLsM with O(3) Néel vector. The correlation spectra in Fig. 14(c) shows that the spin-triplet correlation length is the largest across the phase transition while the spin-singlet correlation