Generalized Boundary Condition Applied to Lieb-Schultz-Mattis-Type Ingappabilities and Many-Body Chern Numbers

We introduce a new boundary condition which renders the flux-insertion argument for the Lieb-Schultz-Mattis-type theorems in two or higher dimensions free from the specific choice of system sizes. It also enables a formulation of the Lieb-Schultz-Mattis-type theorems in arbitrary dimensions in terms of the anomaly in field theories in $1+1$ dimensions with a bulk correspondence as a BF theory in $2+1$ dimensions. Furthermore, we apply the anomaly-based formulation to the constraints on a half-filled spinless fermion on a square lattice with $\pi$ flux, utilizing a time-reversal, magnetic translations and an on-site internal $\mathrm{U}(N)$ symmetries. This demonstrates the role of the time-reversal anomaly on the ingappabilities of a lattice model. Moreover, by our new boundary condition, we show that the many-body Chern number of this lattice model is nonvanishing as $N$ mod $2N$ in the presence of $\mathrm{U}(N)$ and magnetic translations. This can be a general mechanism of anomaly-based constraints on quantized Hall conductance, which generally depends on high-energy physics, from field theory.

Popular Summary

Ringlike paths and spiral-rise structures are common in our daily life: Circular tracks encircle parks and spiral staircases grace theaters. In closed quantum systems, however, the former takes precedence. So-called periodic boundary conditions direct a particle to periodically return to its initial position after completing a cycle along one direction. In such systems, researchers have paid little attention to spiral-rise trajectories. We find that a tilted boundary condition with a spiral-rise nature—where a particle is raised up after going around one cycle—can reveal in complex electronic crystals low-energy properties that are otherwise missed when periodic boundary conditions are enforced.

Quantifying the behavior of interacting electronic systems is a complex problem and our study refines and expands the well-known Lieb-Schultz-Mattis theorem, which states that certain conditions on symmetries and the number of particles per unit volume require either gapless excitations or a ground-state degeneracy. The tilted boundary condition improves a proof of the theorem in higher dimensions by eliminating an artificial restriction on system sizes. Furthermore, it generalizes the theorem to a series of strongly correlated spin systems in arbitrary dimensions.

One of the new perspectives is a systematic way to set physical boundary conditions so that they can give sensible constraints on low-energy behaviors of systems. The new boundary condition can also help us make a precise connection between the system and the surface of a generalized topological insulator. Our work might help researchers search for new quantum materials and understand their properties.

Figure 1

TLBC when $d=2$.

Figure 2

${\mathbb{Z}}_N\times {\mathbb{Z}}_N$ bundle on $\tilde{X}=T^2$ where the domain walls are inserted beyond $\partial X=S^1$ and a corresponding transformation is acted across a domain wall according to the domain-wall orientation indicated above.

Figure 3

$X_{1,2}^{\prime }$ are glued along their common boundary $T^3$. There is a unit $\mathrm{U}(N)$ instanton labelled by “*” inside $D_{X_1^{\prime }}\cup -D_{X_2^{\prime }}$ inducing a flux through $D_{X_1^{\prime }}$.

Figure 4

The bulk-interface-bulk equivalence of the partition function (110) where the PV regulator mass is $-m_0$ in Eqs. (a1) and (a2).