Chiral Lattice Gauge Theories from Symmetry Disentanglers

Abstract

We propose a Hamiltonian framework for constructing chiral gauge theories on the lattice based on symmetry disentanglers: constant-depth circuits of local unitaries that transform not-on-site symmetries into on-site ones. When chiral symmetry can be realized not-on-site and such a disentangler exists, the symmetry can be implemented in a strictly local Hamiltonian and gauged by standard lattice methods. Using lattice rotor models, we realize this idea in 1+1 and 3+1 spacetime dimensions for $U(1)$ symmetries with mixed 't Hooft anomalies, and show that symmetry disentanglers can be constructed when anomalies cancel. As an example, we present an exactly solvable Hamiltonian lattice model of the (1+1)-dimensional "3450" chiral gauge theory, and we argue that a related construction applies to the $U(1)$ hypercharge symmetry of the Standard Model fermions in 3+1 dimensions. Our results open a new route toward fully local, nonperturbative formulations of chiral gauge theories.

Figures (4)

• Figure 1: Using a symmetry disentangler to construct a chiral theory. Within a stack of free-fermion $G$-SPTs (blue), the lower boundary of each such SPT is $G$-symmetrically gapped (blue-yellow region) against a corresponding $D$ dimensional model (yellow) with the same anomaly but with a not-on-site realization of $G$, resulting in a topologically trivial lower boundary. Then the only remaining low-energy degrees of freedom are those on the upper boundaries of the free-fermion SPTs (blue), yielding a chiral model. The final step is to make an anomaly free subgroup $G' \subset G \times \ldots \times G$ on-site by applying a symmetry disentangler that acts across the stack of $D$-dimensional Hilbert spaces.
• Figure 2: Lattice model of rotors and fermions, with the fermionic decoration adapted from Wang_2018Wang_2020. The black tetrahedron shown is part of a simplicial decomposition of the spatial $3$-manifold. This simplicial decomposition has a branching structure, which we may take to be induced from a global ordering of the black vertices: orientations on the black edges point from smaller to larger numbers. The rotors are located at the black vertices. The red dots represent Majorana fermions. Each physical fermionic degree of freedom is represented by a pair of such Majorana fermions straddling a triangular face of the black tetrahedron. In the above figure we show $8$ Majorana fermions, corresponding to $4$ physical fermions associated to the $4$ faces of the black tetrahedron. The dotted red lines form the 'resolved dual lattice' in the terminology of Wang_2018Wang_2020, who show how to define a so-called Kasteleyn orientation on it. The fermionic part of the $U(1)_A$ generator (eq. \ref{['eq:defK']}) is defined using a fermionic bilinear where the Majorana fermions are paired according to a pairing $p_\phi$ controlled by the rotor configuration $\{\phi\}$. This pairing just decorates Kitaev chains on odd vorticity loops. More precisely, the Poincare duals of the black triangular faces carrying odd vorticity $d\lfloor d\phi \rceil$ form closed loops (blue) in which the Majorana fermions are paired within the black tetrahedra; the remaining Majorana fermions are paired across the black triangular faces. In the figure above, the vortex penetrates the faces $124$ and $234$.
• Figure 3: Short loop of resolved dual edges (dotted red lines) surrounding the edge $e$ on the resolved dual lattice. For clarity we have projected from $3$ dimensions to $2$ dimensions, so the central point represents an edge $e$ oriented perpendicular to the page and the black lines represent faces $f$ that contain $e$. The short loop is Kasteleyn oriented, because an odd number of its edges are oriented clockwise, or counter-clockwise (since there are always an even number of resolved dual edges in any such short loop, this property is independent of the choice of clockwise or counter-clockwise orientation along the loop). The red dots represent Majorana fermions, and the gray ovals represent the 'trivial' pairing $p_0$, with each such pair defining a complex fermion. The Kasteleyn property implies that if we instead chose the complementary pairing $p_1$ of the Majorana fermions, i.e. formed a Kitaev chain along the loop, then $\prod_{\langle \gamma \gamma' \rangle \in p_0} i\gamma\gamma' = \prod_{\langle \gamma \gamma' \rangle \in p_1} i\gamma\gamma'$, where by convention the orientations always point from $\gamma$ to $\gamma'$.
• Figure 4: Variant of the $D>2$ construction from the anomaly in-flow perspective. A stack of $N$$(D{+}1)$-dimensional free-fermion $G$-SPTs (blue slab) is coupled across an interface (blue-yellow region) to a stack of $(D{+}1)$-dimensional commuting-projector $G$-SPTs (yellow slab). This interface is assumed to be fully gapped, topologically trivial, and preserves the $G$ symmetry, while the top surface of the blue slab remains gapless. A symmetry disentanger (grey) supported near the bottom of the yellow slab, renders on-site an anomaly-free subgroup $G'\subset G^N$, enabling the bottom boundary to be trivially gapped by an on-site Hamiltonian without breaking $G'$. This construction provides a Hamiltonian realization of the SMG scenario for constructing chiral gauge theories.

Theorems & Definitions (2)

• proof
• proof