String and membrane condensation on three-dimensional lattices

In this paper, we investigate the general properties of lattice spin models that have string and/or membrane condensed ground states. We discuss the properties needed to define a string or membrane operator. We study three three-dimensional spin models which lead to $Z_2$ gauge theory at low energies. All the three models are exactly soluble and produce topologically ordered ground states. The first model contains both closed-string and closed-membrane condensations. The second model contains closed-string condensation only. The ends of open strings behave like fermionic particles. The third model also has condensations of closed membranes and closed strings. The ends of open strings are bosonic while the edges of open membranes are fermionic. The third model contains a different type of topological order.

Figure 1

A three-dimensional model with Majorana fermions on the sites. The six Majorana fermions label the six vectors from a site in the following way: $x\mapsto +\widehat{x},\overline{x}\mapsto -\widehat{x},y\mapsto \widehat{x},\overline{y}\mapsto -\widehat{y},z\mapsto \widehat{z},\overline{z}\mapsto -\widehat{z}$. The plaquettes in the three planes are shown. ${\widehat{F}}_{p_{xy}},{\widehat{F}}_{p_{yz}},{\widehat{F}}_{p_{zx}}$ are, respectively, the plaquettes ${\widehat{F}}_{{\mathbf{i}}_1{\mathbf{i}}_2{\mathbf{i}}_3{\mathbf{i}}_4},{\widehat{F}}_{{\mathbf{i}}_2{\mathbf{i}}_3{\mathbf{i}}_6{\mathbf{i}}_7},{\widehat{F}}_{{\mathbf{i}}_4{\mathbf{i}}_3{\mathbf{i}}_6{\mathbf{i}}_5}$, where ${\mathbf{i}}_1=\mathbf{i}$; ${\mathbf{i}}_2=\mathbf{i}+\widehat{\mathbf{x}}$; ${\mathbf{i}}_3=\mathbf{i}+\widehat{\mathbf{x}}+\widehat{\mathbf{y}}$; ${\mathbf{i}}_4=\mathbf{i}+\widehat{\mathbf{y}}$; ${\mathbf{i}}_5=\mathbf{i}+\widehat{\mathbf{y}}+\widehat{\mathbf{z}}$; ${\mathbf{i}}_6=\mathbf{i}+\widehat{\mathbf{x}}+\widehat{\mathbf{y}}+\widehat{\mathbf{z}}$; ${\mathbf{i}}_7=\mathbf{i}+\widehat{\mathbf{x}}+\widehat{\mathbf{z}}$.

Figure 2

The dashed lines represent the cube in the cubic lattice. The solid lines represent the octahedron.

Figure 3

(Color online) The cube operator and the corner loop operator with the nomenclature of the Majorana fermions forming a corner loop.

Figure 4

A triple of noncommuting corner loops. The corner operator above does not commute with the two below. Each corner term affects three cubes.

Figure 5

The cube operator and four corner loop operators. The cube is made of the product of only four such loops. These four operators all commute with each other because they have no Majorana fermion in common. The four corner loops shown in the figure are chosen on the odd sites.

Figure 6

A section of the cubic lattice is shown. The picture shows the set of commuting corner operators selected for the Hamiltonian. At even sites, we choose the upper corners (marked with a 엯 in the $xy$ and $\overline{xy}$ quadrants and the lower corners (marked with an $\times$) in the other two quadrants. We make the complementary choice on odd sites. This is equivalent to alternately taking all the lower corners or all the upper corners in a square. This means that in even cubes all the corners are taken, but in the odd ones none.

Figure 7

(Color online) The string featured in the model.