Foliated-Exotic Duality and Anomaly Inflow in Fracton Quantum Field Theories

Abstract

Fracton phases are new types of phases of matter characterized by subsystem global symmetry, which is a generalized global symmetry whose symmetry operator is partially topological. Their continuum low-energy effective descriptions admit two different formulations: an exotic quantum field theory (QFT) using exotic tensor gauge fields, and a foliated QFT constructed from a foliation structure and foliated gauge fields. For certain fracton QFTs, these two descriptions are equivalent, which is called the foliated-exotic duality. In this dissertation, we extend the foliated-exotic duality by combining it with the anomaly inflow mechanism for 't Hooft anomalies of subsystem symmetries. This dissertation has two main results. First, we discuss the exotic and foliated $BF$ theories in 2+1 dimensions, which exhibit the mixed 't Hooft anomaly of $\mathbb{Z}_N \times \mathbb{Z}_N$ subsystem symmetry. This anomaly is captured by a subsystem symmetry-protected topological (SSPT) phase for $\mathbb{Z}_N \times \mathbb{Z}_N$ subsystem symmetry in one dimension higher. By extending the foliated-exotic duality in the fractonic $BF$ theory to the SSPT phase, we establish the field correspondences in the SSPT phase and construct the foliated description of the SSPT phase. Second, we discuss the exotic $φ$-theory in 2+1 dimensions -- a fractonic gapless scalar field theory, which has the 't Hooft anomaly of $U(1) \times U(1)$ subsystem symmetry. The anomaly is captured by an SSPT phase for $U(1) \times U(1)$ subsystem symmetry in 3+1 dimensions via the anomaly inflow mechanism. Extending the foliated-exotic duality to the $φ$-theory, we establish field correspondences in the $φ$-theory and construct the foliated $φ$-theory that is equivalent to the exotic $φ$-theory. This provides the first example of the foliated-exotic duality in gapless theories.

Figures (10)

• Figure 1: The cube operator $C_c$ (left) and the cross operator $X_s^k$ (right) in the X-cube model. The red and blue links represent the operators $A_e$ and $B_e$, respectively.
• Figure 2: The orange dashed loop is the zigzagging closed loop $\hat{\mathcal{C}}^{ij}$ in the $x^i x^j$-plane. The red links represent the operator $A_e$.
• Figure 3: The blue line is the closed line $\mathcal{C}^{k}$ in the $x^k$ direction. The blue links also represent the operator $B_e$.
• Figure 4: The fracton excitation created by acting with $A_e$. The red links represent $A_e$ and the colored cubes correspond to the fractons.
• Figure 5: The lineon excitations created by acting with $B_e$. The blue links represent $B_e$ and the colored sites correspond to the lineons.