Topological terms on topological defects: A quantum Monte Carlo study

Dirac fermions in (2 + 1) dimensions with dynamically generated anticommuting SO(3) antiferromagnetic and ${\mathrm{Z}}_2$ Kekulé valence-bond solid (KVBS) masses map onto a field theory with a topological $\theta$ term. This term provides a mechanism for continuous phase transitions between different symmetry-broken states: topological defects of one phase carry the charge of the other and proliferate at the transition. The $\theta$ term implies that a domain wall of the ${\mathrm{Z}}_2$ KVBS order parameter harbors a spin-$1/2$ Heisenberg chain, as described by a (1 + 1)-dimensional SO(3) nonlinear sigma model with $\theta$ term at $\theta =\pi$. Using pinning fields to stabilize the domain wall, we show that our auxiliary-field quantum Monte Carlo simulations indeed support the emergence of a spin-$1/2$ chain at the ${\mathrm{Z}}_2$ topological defect. Surprisingly, the consequences of the topological term are seen far from the critical point such that the physics of apparently unrelated model systems are naturally understood by invoking them. This concept can be generalized to higher dimensions where (2 + 1)-dimensional SO(4) or SO(5) theories with topological terms are realized at a domain wall.

Figure 1

(a) Schematic ground-state phase diagram [6]. Dashed line indicates scans considered here. (b) In our model, fermions acquire a $\pi$-flux when circulating around a plaquette and are coupled to Ising spins on lattice bonds with magnitude $\pm \xi$. We consider periodic (open) boundary conditions in the ${\mathit{a}}_1({\mathit{a}}_2$) directions and freeze the Ising spins on the open boundary to impose the domain wall. (c) Real-space bond energy change, $\mathrm{\Delta }{\widehat{B}}_{\mathit{i},{\mathit{a}}_l}$. Here, $L_1=30$ and $L_2=17$.

Figure 2

Real-space correlation functions for (a) spin, (b) dimer, and (c) bond at the domain wall of ${\mathrm{Z}}_2$ KVBS (solid symbols). Here, $L_2=17$ and $x=L_1\sin (\pi i_1/L_1)$ is the conformal distance [31]. Open symbols indicate QMC results for the one-dimensional Hubbard model at $U/t=4$ and at half filling. The solid line corresponds to $x^{-1}$.

Figure 3

Real-space bond energy change at $\mathit{i}=(i_1=0,i_2)$ (left panels) and spin-spin correlation functions along the domain wall of ${\mathrm{Z}}_2$ KVBS (right panels) for $L_2=17$ and $x=L_1\sin (\pi i_1/L_1)$ with $L_1=30$. (a), (b) $1/h=0.333$ (${\mathrm{Z}}_2$ KVBS phase) and (c), (d) $1/h=0.275$ (${\mathrm{Z}}_2$ KVBS phase close to the critical point) at $U=7$. At $h=\infty$, our model maps onto the pure Hubbard model on the $\pi$-flux lattice and $U>5.71\left(1\right)$ [32] places us in the AFM phase. In (e), (f) [(g), (h)], we consider this limit with [without] pinning fields at $U=7$, and in (i), (j), the same limit with pinning fields at $U=12$.