Deconfined quantum criticality in the two-dimensional antiferromagnetic Heisenberg model with next-nearest-neighbor Ising exchange

We have considered the $S=1∕2$ antiferromagnetic Heisenberg model in two dimensions, with an additional Ising next-nearest-neighbor interaction. Antiferromagnetic next-nearest-neighbor interactions will lead to frustration, and the system responds with flipping the spins down in the $xy$ plane. For large next-nearest-neighbor coupling the system will order in a striped phase along the $z$ axis, this phase is reached through a first-order transition. We have considered two generalizations of this model, one with random next-nearest-neighbor interactions, and one with an enlarged unit cell, where only half of the atoms have next-nearest-neighbor interactions. In both cases the transition is softened to a second-order transition separating two ordered states. In the latter case we have estimated the quantum critical exponent $\beta \approx 0.25$. These two cases then represent candidate examples of deconfined quantum criticality.

Figure 1

A chain of six spins, depicted with an operator sequence of length $n=8$ in the $\beta$ direction. The filled bars denote diagonal interactions $S_i^z{S}_j^z$ and the open bars are flip operators $S_i^{+}{S}_j^{-}+S_i^{-}{S}_j^{+}$. The dashed region shows spins and /or $p$ slices which have been frozen by the next-nearest-neighbor bond in the middle.

Figure 2

(Color online) The staggered magnetization along the $z$ axis in the ground state $(\beta =10)$, as a function of $\kappa$.

Figure 3

(Color online) The striped magnetization in the ground state $(\beta =10)$, as a function of $\kappa$. There is a first-order transition at $\kappa \approx 1.205$.

Figure 4

Phase diagram for the model in the $\kappa ,T$ plane. For $\kappa >{\kappa }_c$ and $\kappa <0$, the model has magnetic order, striped and staggered, respectively. In the intermediate $\kappa$ range values there is no finite $T$ magnetic order, however there is a superfluid order which persists into the finite $T$ region. The line separating topological 2D $XY$ order from the normal phase is determined with considerably less precision than the two other lines.

Figure 5

(Color online) Disorder averaged value of $M^z(\pi ,0)$ as a function of ${\kappa }_0$ for two different system sizes. The indicated error bars are standard error estimates from the independent disorder realizations. The temperature coupling is $\beta =10$.

Figure 6

The binary system consisting of two different atomic species. Only type $A$ $(●)$ has next-nearest-neighbor interaction, illustrated with diagonal lines.

Figure 7

(Color online) The staggered magnetization of the $A$ sites in the ground state of the $AB$ model, as a function of $\kappa$. The solid line is a least-squares fit to a power law.

Figure 8

(Color online) The distribution of NNN operators at the phase transition for one instance of the systems shown in Fig. 3 and Fig. 7; clearly the uniform system shows bimodal behavior found at a first-order transition, whereas the $AB$ system has a unimodal distribution expected from a continuous phase transition. (Observe that the curves have been shifted horizontally to fit on the same plot, we have therefore not retained any tick marks on the axes.)