Fracton phases of matter are gapped phases of matter that, by dint of their sensitivity to UV data, demand non-standard quantum field theories to describe them in the IR. Two such approaches are foliated quantum theory and exotic field theory. In this paper, we explicitly construct a map from one to the other and work out several examples. In particular, we recover the equivalence between the foliated and exotic fractonic BF theories recently demonstrated at the level of operator correspondence. We also demonstrate the equivalence of toric code layers and the anisotropic model with lineons and planons to the foliated BF theory with one and two foliations, respectively. Finally, we derive new exotic field theories that provide simple descriptions of hybrid fracton phases from foliated field theries known to do so. Our results both provide new examples of exotic field theories and pave the way toward their systematic construction from foliated field theories.
Abstract Fracton phases of matter are gapped phases of matter that, by dint of their sensitivity to UV data, demand non-standard quantum field theories to describe them in the IR. Two such approaches are foliated quantum theory and exotic field theory. In this paper, we explicitly construct a map from one to the other and work out several examples. In particular, we recover the equivalence between the foliated and exotic fractonic BF theories recently demonstrated at the level of operator correspondence. We also demonstrate the equivalence of toric code layers and the anisotropic model with lineons and planons to the foliated BF theory with one and two foliations, respectively. Finally, we derive new exotic field theories that provide simple descriptions of hybrid fracton phases from foliated field theries known to do so. Our results both provide new examples of exotic field theories and pave the way toward their systematic construction from foliated field theories.
There has been proposed two continuum descriptions of fracton systems: foliated quantum field theories (FQFTs) and exotic quantum field theories. Certain fracton systems are believed to admit descriptions by both, and hence a duality is expected between such a class of FQFTs and exotic QFTs. In this paper we study this duality in detail for concrete examples in 2+1 2 + 1 and 3+1 3 + 1 dimensions. In the examples, both sides of the continuum theories are of BF B F -type, and we find the explicit correspondences of gauge-invariant operators, gauge fields, parameters, and allowed singularities and discontinuities. This deepens the understanding of dualities in fractonic quantum field theories.
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
ZN×ZN subsystem symmetry. This anomaly is captured by a subsystem symmetry-protected topological (SSPT) phase for
ZN×ZN 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)×U(1) subsystem symmetry. The anomaly is captured by an SSPT phase for
U(1)×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.
Due to the recent studies of the fracton topological phases, the host of which deconfined quasiparticle excitations with mobility restrictions, the concept of symmetries has been updated. Focusing on one of these new symmetries, multipole symmetries, including global, dipole, and quadruple symmetries, and gauge fields associated with them, we construct new sets of
ZN(2+1)-dimensional foliated background-field (BF) theories, where BF theories of conventional topological phases are stacked in layers with couplings between them. By investigating gauge-invariant nonlocal operators, we show that our foliated BF theories exhibit unusual ground-state degeneracy depending on the system size; it depends on the greatest common divisor between
N and the system size. Our result provides an important connection between UV lattice models of the fracton topological phases and other unconventional ones in view of foliated field theories.
We introduce a new kind of foliated quantum field theory (FQFT) of gapped fracton orders in the continuum. FQFT is defined on a manifold with a layered structure given by one or more foliations, which each decompose spacetime into a stack of layers. FQFT involves a new kind of gauge field, a foliated gauge field, which behaves similar to a collection of independent gauge fields on this stack of layers. Gauge invariant operators (and their analogous particle mobilities) are constrained to the intersection of one or more layers from different foliations. The level coefficients are quantized and exhibit a duality that spatially transforms the coefficients. This duality occurs because the FQFT is a foliated fracton order. That is, the duality can decouple 2+1D gauge theories from the FQFT through a process we dub exfoliation.
Abstract This work investigates the gapped interfaces of 3+1d fracton phases of matter using foliated gauge theories and lattice models. We analyze the gapped boundaries and gapped interfaces in X cube model, and the gapped interfaces between the X-cube model and the toric code. The gapped interfaces are either “undecorated” or “decorated”, where the “decorated” interfaces have additional Chern-Simons like actions for foliated gauge fields. We discover many new gapped boundaries and interfaces, such as (1) a gapped boundary for X-cube model where the electric lineons orthogonal to the interface become the magnetic lineons, the latter are the composite of magnetic planons; (2) a Kramers-Wannier-duality type gapped interface between the X-cube model and the toric code model from gauging planar subsystem one-form symmetry; and (3) an electromagnetic duality interface in the X-cube model that exchanges the electric and magnetic lineons.