We have realized that the transition between the up-up-down and 0-coplanar states near the Ising limit is actually of the first order, although the hysteresis region is very narrow. In Fig. 1(a), we present the corrected quantum phase diagram, in which the boundary of the corresponding transition for 0 < J=J z < 0.437 is replaced by a thick blue line. This minor correction does not affect the rest of the phase diagram and the main conclusions of our Letter, including the novel degeneracy-lifting mechanism that gives rise to the new π-coplanar state.A recent theoretical work based on the density matrix renormalization group method has suggested the existence of the first-order transition for 0 < J=J z ≲ 0.4 [3]. Therefore, we reexamined the magnetization curve m z ðHÞ ¼ P i hŜ z i i=M for small positive values of the anisotropy J=J z . As shown in Fig. 1(b), the magnetization curve is three valued in a finite range of H=J z near the end point of the plateau (H ¼ H c2 ), which implies that the transition is of the first order. When J=J z increases, the sign of the susceptibility χ c2 ≡ dm z =dHj H¼H c2 just above the plateau changes from negative to positive; i.e., there is a tricritical point (TCP) where the transition nature changes from first order to second order. In Fig. 1(c), we show the extrapolation of the inverse of χ c2 with respect to the scaling parameter λ for different values of J=J z . The location of the TCP is estimated to be ðJ=J z Þ TCP ≈ 0.437, for which the extrapolated value of χ −1 c2 is 0. This result is consistent with Ref.[3].[1] S. Wessel and M. Troyer, Phys. Rev. Lett. 95, 127205 (2005).[2] L. Bonnes and S. Wessel, Phys. Rev. B 84, 054510 (2011).[3] D. Sellmann, X.-F. Zhang, and S. Eggert, arXiv:1403.0008.FIG . 1 (color online). (a) Ground-state phase diagram of the spin-1=2 triangular-lattice XXZ model obtained by the cluster mean-field method combined with a scaling scheme (CMF þ S) (J z > 0). The thick blue (thin black) solid curves correspond to first-(second-)order transitions. The dot marks the tricritical point. The latest quantum Monte Carlo data [1,2] are shown by the red dashed (first-order) and dotted (second-order) curves. The symbol (×) is the value from the dilute Bose gas expansion. (b) An example of the magnetization curve that exhibits a first-order transition. We show the magnetization m z divided by the saturation value S ¼ 1=2 as a function of the magnetic field H=J z . The vertical dashed line marks the first-order transition point. (c) Cluster-size scalings of the CMF data for the inverse susceptibility χ −1 c2 just above the plateau. The lines indicate linear fits to the data obtained from the three largest clusters for each J=J z .
We make a detailed study of the moduli space of winding number two (k = 2) axially symmetric vortices (or equivalently, of co-axial composite of two fundamental vortices), occurring in U(2) gauge theory with two flavors in the Higgs phase, recently discussed by Hashimoto-Tong and Auzzi-Shifman-Yung. We find that it is a weighted projective space W CP 2 (2,1,1) ≃ CP 2 /Z 2 . This manifold contains an A 1 -type (Z 2 ) orbifold singularity even though the full moduli space including the relative position moduli is smooth. The SU(2) transformation properties of such vortices are studied. Our results are then generalized to U(N) gauge theory with N flavors, where the internal moduli space of k = 2 axially symmetric vortices is found to be a weighted Grassmannian manifold. It contains singularities along a submanifold.
We show that local/semilocal strings in Abelian/non-Abelian gauge theories with critical couplings always reconnect classically in collision, by using moduli space approximation. The moduli matrix formalism explicitly identifies a well-defined set of the vortex moduli parameters. Our analysis of generic geodesic motion in terms of those shows right-angle scattering in head-on collision of two vortices, which is known to give the reconnection of the strings.Introduction. -The issue of reconnection (intercommutation, recombination) of colliding cosmic strings attracts much interest recently (see [1,2,3]), owing to the fact that the reconnection probability is related to the number density of the cosmic strings, which is strongly correlated with possible observation of them. However, solitonic strings may appear in numerous varieties of field theories, which certainly makes any prediction complicated. In this Letter, we employ the moduli matrix formalism [4] to show that, in a wide variety of field theories admitting supersymmetric generalization, inevitable reconnection of colliding solitonic strings (i.e. reconnection probability is unity) is universal. The inevitable reconnection of local strings in Abelian Higgs model [5] (see also [6]) has been known for decades, and for non-Abelian local strings in U (N C ) gauge theories with N F (= N C ) flavors, this universality was found in [7] by a topological argument. Here, via a different logic and explicit computations, we show the concrete dynamics of the inevitable reconnection (note that [7] does not describe dynamics). Furthermore, our results extend the universality to semilocal strings [8]
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.