Lieb-Schultz-Mattis, Luttinger, and 't Hooft - anomaly matching in lattice systems

Abstract

We analyze lattice Hamiltonian systems whose global symmetries have ’t Hooft anomalies. As is common in the study of anomalies, they are probed by coupling the system to classical background gauge fields. For flat fields (vanishing field strength), the nonzero spatial components of the gauge fields can be thought of as twisted boundary conditions, or equivalently, as topological defects. The symmetries of the twisted Hilbert space and their representations capture the anomalies. We demonstrate this approach with a number of examples. In some of them, the anomalous symmetries are internal symmetries of the lattice system, but they do not act on-site. (We clarify the notion of “on-site action.”) In other cases, the anomalous symmetries involve lattice translations. Using this approach we frame many known and new results in a unified fashion. In this work, we limit ourselves to 1+1d systems with a spatial lattice. In particular, we present a lattice system that flows to the c=1c=1 compact boson system with any radius (no BKT transition) with the full internal symmetry of the continuum theory, with its anomalies and its T-duality. As another application, we analyze various spin chain models and phrase their Lieb-Shultz-Mattis theorem as an ’t Hooft anomaly matching condition. We also show in what sense filling constraints like Luttinger theorem can and cannot be viewed as reflecting an anomaly. As a by-product, our understanding allows us to use information from the continuum theory to derive some exact results in lattice model of interest, such as the lattice momenta of the low-energy states.

Solvable models for 2+1D quantum critical points: Loop soups of 1+1D conformal field theories

“…Our construction generates a new kind of non-local frustration by combining two primary ingredients: (1) a non-trivial dependence of the Casimir energy of the decoration CFT on the chain length modulo some integer, and (2) a geometrical constraint which forces all loops on a given lattice to have a certain length modulo some integer. The first ingredient was recently understood as arising from effectively twisted boundary conditions for certain chain lengths [43]. More generally, our work demonstrates the importance of the coefficients c r (which encode the Casimir energy dependence on the chain length modulo some integer) as an additional property of a CFT Hamiltonian with crucial physical consequences.…”

“…However, if c was negative, the energy per site would be minimized by taking L → ∞, leading to a loop which is macroscopically long and which can thus host a 1+1D theory with a vanishing gap. Interestingly, this scenario can be realized by choosing CFTs which are "frustrated" for certain chain lengths [43] and which can effectively behave as if c < 0, as we now show. A key insight is that, for certain CFTs, the value of c which appears in the Casimir energy is not always equal to the actual central charge and may depend on the chain length modulo some integer.…”

“…Another generalization is to consider other decoration Hamiltonians beyond the ones we have proposed here. This motivates further work on classifying the possible c r sequences for known CFTs, especially beyond c = 1 theories [43].…”

“…From a spin-liquid perspective, we will see that our construction starts with a familiar route, in which local frustration (in our case Ising antiferromagnetism on a triangular lattice) leads to an exponentially large number of classical ground states which is described by a familiar ensemble of fully packed loops (FPL) [40][41][42]. However, the source of the exotic criticality we will describe below -fully packed self-avoiding walks described by a non-unitary c = −1 theory -lies elsewhere: It is due to an additional kind of non-local frustration whereby all allowed loops on the honeycomb lattice realize an effectively twisted boundary condition [43] for the CFT which lives on it.…”

Majorana chain and Ising model - (non-invertible) translations, anomalies, and emanant symmetries

“…A characteristic example of such a constraint is the famous Lieb-Schultz-Mattis (LSM) theorem [3] and its generalizations . Various authors [14,16,17,19,20,29,32] have suggested that these results should be phrased as 't Hooft anomalies. In particular, [32] has presented a framework to couple the lattice system to background gauge fields and to probe for anomalies as a failure of their gauge symmetry.…”