Twisted boundary condition and Lieb-Schultz-Mattis ingappability for discrete symmetries

TL;DR

It is shown that, if the system is gapped, the ground-state degeneracy under the twisted boundary condition also implies a ground- state (quasi-degeneracy) under the periodic boundary conditions, giving a compelling evidence for the recently proposed Lieb-Schultz-Mattis-type ingappability due to the on-site discrete symmetry in two and higher dimensions.

Abstract

We discuss quantum many-body systems with lattice translation and discrete onsite symmetries. We point out that, under a boundary condition twisted by a symmetry operation, there is an exact degeneracy of ground states if the unit cell forms a projective representation of the onsite discrete symmetry. Based on the quantum transfer matrix formalism, we show that, if the system is gapped, the ground-state degeneracy under the twisted boundary condition also implies a ground-state (quasi-)degeneracy under the periodic boundary conditions. This gives a compelling evidence for the recently proposed Lieb-Schultz-Mattis type ingappability due to the onsite discrete symmetry in two and higher dimensions.

Figures (6)

• Figure 1: $W$-twisted tilted boundary condition sketched for $d=2$ with $W=R_z^\pi$ for SU$(2)$ spins and $W=W_N$ for SU$(N)$.
• Figure 2: The traditional lattice translation $T_1$ is not a symmetry for the twisted Hamiltonian while $\tilde{T}_1\equiv\hat{r}_z^\pi(\vec{C})T_1$ is the proper symmetry to consider.
• Figure 3: The QTM picture of the gauge transformation $r_z^\pi(\vec{E})$ on the original Hamiltonian: the fact that the gauge transformation does not change the partition function is reflected by the commutation $[\tilde{U}_g,\tilde{H}]=0$ in the QTM paradigm.
• Figure 4: The tensor which is an element of the MPO. The vertical lines act on the local physical Hilbert space, while the horizontal lines do on the virtual space.
• Figure 5: The MPO construction of the QTM.