Abstract
Symmetry protected topological (SPT) states have boundary 't Hooft anomalies that obstruct an effective boundary theory realized in its own dimension with UV completion and an on-site $G$-symmetry. In this work, yet we show that a certain anomalous non-on-site $G$ symmetry along the boundary becomes on-site when viewed as an extended $H$ symmetry, via a suitable group extension $1\to K\to H\to G\to1$. Namely, a non-perturbative global (gauge/gravitational) anomaly in $G$ becomes anomaly-free in $H$. This guides us to construct exactly soluble lattice path integral and Hamiltonian of symmetric gapped boundaries, always existent for any SPT state in any spacetime dimension $d \geq 2$ of any finite symmetry group, including on-site unitary and anti-unitary time-reversal symmetries. The resulting symmetric gapped boundary can be described either by an $H$-symmetry extended boundary of bulk $d \geq 2$, or more naturally by a topological emergent $K$-gauge theory with a global symmetry $G$ on a 3+1D bulk or above. The excitations on such a symmetric topologically ordered boundary can carry fractional quantum numbers of the symmetry $G$, described by representations of $H$. (Apply our approach to a 1+1D boundary of 2+1D bulk, we find that a deconfined gauge boundary indeed has spontaneous symmetry breaking with long-range order. The deconfined symmetry-breaking phase crosses over smoothly to a confined phase without a phase transition.) In contrast to known gapped interfaces obtained via symmetry breaking (either global symmetry breaking or Anderson-Higgs mechanism for gauge theory), our approach is based on symmetry extension. More generally, applying our approach to SPT, topologically ordered gauge theories and symmetry enriched topologically ordered (SET) states, leads to generic boundaries/interfaces constructed with a mixture of symmetry breaking, symmetry extension, and dynamical gauging.
Highlights
After the realization that a spin-1=2 antiferromagneticHeisenberg chain in 1 þ 1 dimensions (1 þ 1D) admits a gapless state [1,2] that “nearly” breaks the spin rotation symmetry, many physicists expected that spin chains with higher spin, having fewer quantum fluctuations, might be gapless with algebraic long-range spin order
Some concluding and additional remarks follow: (1) We provide a UV complete lattice regularization of the Hamiltonian and path integral definition of gapped interfaces based on the symmetry-extension mechanism, partly rooted in Ref. [51]
(2) The anomalous non-on-site G-symmetry at the boundary indicates that, if we couple the G-symmetric boundary to the weakly fluctuating background probed gauge field of G, there is an anomaly in G
Summary
Heisenberg chain in 1 þ 1 dimensions (1 þ 1D) admits a gapless state [1,2] that “nearly” breaks the spin rotation symmetry (i.e., it has “symmetry-breaking” spin correlation functions that decay algebraically), many physicists expected that spin chains with higher spin, having fewer quantum fluctuations, might be gapless with algebraic long-range spin order. Haldane phases are trivial gapped states, just like the disordered product state of spin-0’s [13] Soon after their classification in 1 þ 1D, bosonic SPT states in higher dimensions were classified based on group cohomology. Symmetry breaking gives a straightforward way to construct gapped boundary states or interfaces, since SPT phases are completely trivial if one ignores the symmetry. We further expand our approach to construct anomalous gapped symmetry-preserving interfaces (i.e., domain walls) between bulk SPT states, topological orders (TO), and SETs [53]. Each Hp acts on the spins contained in an octagon [Fig. 3], flipping the spins in a plaquette if all adjacent pairs of spins are equal This Hamiltonian is U CZX invariant, 1⁄2UCZX ; H 1⁄4 0; ð2:10Þ in the case of a system without boundary (an infinite system or a finite system with periodic boundary conditions). It has been shown in Ref. [14] that the non-on-site nature of the effective Z2 -symmetry gives an obstruction to making the boundary gapped and symmetry preserving
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