A Novel Non-Perturbative Lattice Regularization of an Anomaly-Free $1 + 1d$ Chiral $SU(2)$ Gauge Theory

Abstract

We present a numerical treatment of a novel non-perturbative lattice regularization of a $1+1d$ $SU(2)$ Chiral Gauge Theory. Our approach follows recent proposals that exploit the newly discovered connection between anomalies and topological (or entangled) states to show how to create a lattice regularization of any anomaly-free chiral gauge theory. In comparison to other methods, our regularization enjoys on-site fermions and gauge action, as well as a physically transparent fermion Hilbert space. We follow the `mirror fermion' approach, in which we first create a lattice regularization of both the chiral theory and its mirror conjugate and then introduce interactions that gap out only the mirror theory. The connection between topological states and anomalies shows that such interactions exist if the chiral theory is free of all quantum anomalies. Instead of numerically intractable fermion-fermion interactions, we couple the mirror theory to a Higgs field driven into a symmetry-preserving, disordered, gapped phase.

Figures (5)

• Figure 1: (color online) (a) Schematic of an IQH state. Each IQH state has a single low-energy mode on each boundary but is otherwise gapped. (b) General schematic of our model. Each black line represents a $2+1$ dimensional lattice structure. Each $1+1d$ edge gives rise to a single left- or right-moving mode. We stack $8$$\nu=1$ IQH states (blue) with $8$$\nu=1$ parity-reversed IQH states (red). Next, we organize these modes into $SU(2)$ representations. Each representation is illustrated by a white square with the spin label except the trivial spin-0 representations which we omit. This results in a $1_{R}\oplus (0_{R})^{5}\oplus (1/2_{L})^{4}$ theory on the left edge and its mirror $(L\leftrightarrow R)$ theory on the right edge. We couple the fermions on the right edge to a Higgs field that gaps out the right edge, leaving only the chiral theory on the left edge.
• Figure 2: (color online) Integrated Density of States (IDOS) for various choices of the coupling strength $g$ and the correlation length of the Higgs field $\xi$ with $L=80$. We first find the low-magnitude eigenvalues and order them in as $|\lambda_{n}|\leq |\lambda_{n+1}|$. Each plot shows $n$ vs. $|\lambda_{n}|$. States localized on the chiral edge are denoted by green circles, all others are denoted by blue squares. A black dashed line indicates the magnitude of the lowest mirror-edge eigenvalue. In the upper panels, we use the same configuration of the Higgs field with $\xi\approx 10.4$ and slowly turn on the interaction. For $g=0$, the IDOS is just that for two copies of our chiral edge theory. As the interaction strength increases, states not localized to the chiral edge are gapped out until, at $g=1$ only the chiral theory remains (below $|\lambda|\lesssim.45$). The resulting IDOS is just that for the chiral theory. The slight momentum renormalization can be mitigated by increasing $\ell_{w}$, though at high computational cost. In the lower panels, we fix $g=1$ and progressively smooth the Higgs field, increasing the Higgs correlation length $\xi$ from $\xi\approx .6$ until $\xi\approx 11.4$. For $\xi\approx .6$, there are still many low-energy mirror edge states, though their momentum structure is wiped out by the strongly disordered $\phi$. As we increase $\xi$, the mirror edge gap increases until at $\xi\approx 11.4$ no mirror-edge states remain (below $|\lambda|\lesssim.45$).
• Figure 3: (a) Mirror theory gap as a function of correlation length $\xi$ for various choices of $L$. We couple the fermions on the mirror theory edge to a Higgs field fluctuating with correlation length $\xi$ and look for the smallest magnitude eigenvalue for states not strongly localized on the chiral theory edge. $\xi=L$ would correspond to the usual symmetry-breaking Higgs mechanism. We choose a finite $\xi$, independent of $L$. For $\xi \gtrsim 8$, the mirror edge is gapped with $\Delta\approx .35$, and we see that this gap is independent of $L$ for $L\gtrsim 40$, indicating thermodynamic behavior. Note that, in the $\xi=L\to\infty$ 'symmetry-breaking' limit, $\Delta=g=1$.
• Figure 4: (color online) (a): Spatial IQH state hopping model. Fermion sites are shown as spheres, with hopping terms as links. A yellow link indicates a hopping of $+1$, a red link a hopping of $-1$, and a green link a hopping of $+i$ in the direction of the arrow and $-i$ in the opposite direction. Hopping around any plaquette generates a phase of $\pi$, hence with $1$ fermion per site this is a $\nu=1$ IQH state. (b): Spacetime lattice with $L_{w}=2$. Each spatial component is just a slice of a IQH state shown in (a), while the purple links represent a Hopping of $ti$ in only the direction of the double arrows. In addition, each site is given the onsite chemical potential $-t\psi^{\dagger}_{i}\psi_{i}$, where we later set $t=3$.The Hopping matrix corresponding to this model is precisely our spacetime Lagrangian. (c) Dispersion relation for the spatial lattice with $L_{w}=2$. (d) Complex Eigenvalues of the spacetime hopping model.
• Figure 5: Total gap as a function of correlation length $\xi$ for various choices of $L$. We introduce a second, independently fluctuating Higgs field to the chiral theory edge, in addition to the fluctuating Higgs field introduced on the Mirror Theory edge ($\xi$ is the minimum correlation length of the two fields). We then measure the total gap, and plot it as a function of the minimum correlation length over the two edges. This calculation allows us to extend our analysis to larger $L$ and confirm the previous results. For $\xi\gtrsim8$, the system is again gapped with $\Delta\approx .35$, and we reach thermodynamic, $L$ independent behavior for $L\gtrsim40$. Note that, in the $\xi=L\to\infty$ 'symmetry-breaking' limit, $\Delta=g=1$.