Fracton (subdimensional particle)

The X-cube model is constructed on a cubic lattice, with qubits on each edge of the lattice.

The Hamiltonian is given by

${\displaystyle H=-\sum _{c}A_{c}-\sum _{{\textrm {}}v}(B_{v,x}+B_{v,y}+B_{v,z})}$

Here, the sums run over cubic unit cells and over vertices. For any cubic unit cell ${\displaystyle c}$, the operator ${\displaystyle A_{c}}$ is equal to the product of the Pauli ${\displaystyle X}$ operator on all 12 edges of that unit cube. For any vertex of the lattice ${\displaystyle v}$, operator ${\displaystyle B_{v,\mu }}$ is equal to the product of the Pauli ${\displaystyle Z}$ operator on all four edges adjacent to vertex ${\displaystyle v}$ and perpendicular to the ${\displaystyle \mu }$ axis. Other notation conventions in the literature may interchange ${\displaystyle X}$ and ${\displaystyle Z}$.

In addition to obeying an overall ${\displaystyle Z_{2}\times Z_{2}}$ symmetry defined by global symmetry generators ${\displaystyle \prod _{\ell }X_{\ell }}$ and ${\displaystyle \prod _{\ell }Z_{\ell }}$ where the product runs over all edges in the lattice, this Hamiltonian obeys subsystem symmetries acting on individual planes.

All of the terms in this Hamiltonian commute and belong to the Pauli algebra. This makes the Hamiltonian exactly solvable. One can simultaneously diagonalise all the terms in the Hamiltonian, and the simultaneous eigenstates are the Hamiltonian's energy eigenstates. A ground state of this Hamiltonian is a state ${\displaystyle |GS\rangle }$ that satisfies ${\displaystyle A_{c}|GS\rangle =1}$ and ${\displaystyle B_{v,\mu }|GS\rangle =1}$ for all ${\displaystyle c,v,\mu }$. One can explicitly write down a ground state using projection operators ${\displaystyle {\frac {1+A_{c}}{2}}}$ and ${\displaystyle {\frac {1+B_{v,\mu }}{2}}}$.

It is important to note that the constraints posed by ${\displaystyle A_{c}=1}$ and ${\displaystyle B_{v,\mu }=1}$ are not all linearly independent when the X cube model is embedded on a compact manifold. This leads to a large ground state degeneracy that increases with system size. On a torus with dimensions ${\displaystyle L_{x},L_{y},L_{z}}$, the ground state degenearcy is exactly ${\displaystyle 2^{2L_{x}+2L_{y}+2L_{z}-3}}$ .[4] A similar degeneracy scaling, ${\displaystyle \log GSD\propto L}$, is seen on other manifolds as well in the thermodynamic limit.

The X cube model hosts two types of elementary excitations, the fracton and lineon (also known as the one-dimensional particle).

If a quantum state is such that the eigenvalue of ${\displaystyle A_{c}=-1}$ for some unit cube ${\displaystyle c}$, then we say that, in this quantum state, there is a fracton located at the position ${\displaystyle c}$. For example, if ${\displaystyle |GS\rangle }$ is a ground state of the Hamiltonian, then for any edge ${\displaystyle \ell }$, the state ${\displaystyle X_{\ell }|GS\rangle }$ features four fractons, one each on the cubes adjacent to ${\displaystyle \ell }$.

Given a rectangle ${\displaystyle R}$ in a plane, one can define a "membrane" operator as ${\displaystyle Z_{R}\equiv \prod _{\ell \in R}Z_{\ell }}$ where the product runs over all edges ${\displaystyle \ell }$ perpendicular to the rectangle that pass through this rectangle. Then the state ${\displaystyle Z_{R}|GS\rangle }$ features four fractons each located at the cubes next to the corners of the rectangle. Thus, an isolated fracton can appear in the limit of taking the length and width of the rectangle ${\displaystyle R}$ to infinity. The fact that a nonlocal membrane operator acts on the ground state to produce an isolated fracton is analogous to how, in smaller dimensional systems, nonlocal string operators can produce isolated ${\displaystyle \pi }$ flux particles and domain walls.

This construction shows that an isolated fracton cannot be mobile in any direction. In other words, there is no local operator that can be acted on an isolated fracton to move it to a different location. In order to move an individual isolated fracton, one would need to apply a highly nonlocal operator to move the entire membrane associated with it.

If a quantum state is such that the eigenvalue of ${\displaystyle B_{v,x}=B_{v,y}=-1}$ for some vertex ${\displaystyle v}$, then we say that, in this quantum state, there is a lineon located at the position ${\displaystyle v}$ that is mobile in the ${\displaystyle z}$ direction. A similar definition holds for lineons that are mobile in the ${\displaystyle x}$ direction and lineons that are mobile in the ${\displaystyle y}$ direction. In order to create an isolated ${\displaystyle z}$ lineon at a vertex ${\displaystyle v}$, one must act on the ground state with a long string of Pauli ${\displaystyle X}$ operators acting on all the edges along the ${\displaystyle z}$ axis that are below the lineon. Lineon excitations are mobile in one direction only; the Pauli ${\displaystyle X}$ operator can act on lineons to translate tham along that direction.

An ${\displaystyle x,y}$ and ${\displaystyle z}$ lineon can all fuse into the vacuum, if the lines on which each of them move concur. That is, there is a sequence of local operators that can make this fusion happen. The opposite process can also happen. For a similar reason, an isolated lineon can change direction of motion from ${\displaystyle x}$ to ${\displaystyle y}$, creating a new lineon moving in the ${\displaystyle z}$ direction in the process. The new lineon is created at the point in space where the original lineon changes direction.

It is also possible to make bound states of these elementary excitations that have higher mobility. For example, consider the bound state of two fractons with the same ${\displaystyle x}$ and ${\displaystyle y}$ coordinates separated by a finite distance ${\displaystyle w}$ along the ${\displaystyle z}$ axis. This bound state, called a planeon, is mobile in all directions in the ${\displaystyle xy}$ plane. One can construct a membrane operator with width ${\displaystyle w}$ in the ${\displaystyle z}$ axis and arbitrary length in either the ${\displaystyle x}$ or ${\displaystyle y}$ direction that can act on the planeon state to move it within the ${\displaystyle xy}$ plane.

It is possible to remotely detect the presence of an isolated elementary excitation in a region by moving the opposite type of elementary excitation around it. Here, as usual, "moving" refers to the repeated action of local unitary operators that translate the particles. This process is known as interferometry. It can be considered analogous to the idea of braiding anyons in two dimensions.

For example, suppose a lineon (either an ${\displaystyle x}$ lineon or a ${\displaystyle y}$ lineon) is located in the ${\displaystyle xy}$ plane, and there is also a planeon that can move in the ${\displaystyle xy}$ plane. Then we can move the planeon in a full rotation that happens to encompass the position of the lineon. Such a planeon movement would be implemented by a membrane operator. If this membrane operator intersects with the Pauli-${\displaystyle X}$ string operator attached to the lineon exactly one time, then at the end of the rotation of the planeon the wave function will pick up a factor of ${\displaystyle -1}$, which indicates the presence of the lineon.[5]

It is possible to construct the X cube model by taking three stacks of toric code sheets, on along each of the three axes, superimposing them, and adding couplings to the edges where they intersect.[3] This construction explains some of the connections that can be seen between the toric code topological order and the X cube model. For example, each additional toric code sheet can be understood to contribute a topological degeneracy of 4 to the overall ground state degeneracy of the X cube model when it is placed on a three dimensional torus; this is consistent with the formula for the ground state degeneracy of the X cube model.

Another example of a type I fracton model is the checkerboard model.[6]

This model also lives on a cubic lattice, but with one qubit on each vertex. First, one colours the cubic unit cells with the colours ${\displaystyle A}$ and ${\displaystyle B}$ in a checkerboard pattern, i.e. such that no two adjacent cubic cells are the same colour. Then the Hamiltonian is

${\displaystyle H=-\sum _{c\in A}\prod _{v\in c}X_{v}-\sum _{c\in B}\prod _{v\in c}Z_{v}}$

This model is also exactly solvable with commuting terms. The topological ground state degeneracy on a torus is given by ${\displaystyle log_{2}GSD=4L_{x}+4L_{y}+4L_{z}-6}$ for lattice of size ${\displaystyle 2L_{x},2L_{y},2L_{z}}$ (as a rule the dimensions of the lattice must be even for periodic boundary conditions to make sense).

Like the X Cube model, the checkerboard model features excitations in the form of fractons, lineons, and planeons.

Haah's code is defined on a cubic lattice with two qubits on each vertex. We can refer to these qubits using Pauli matrices ${\displaystyle {\vec {\sigma }}_{v}}$ and ${\displaystyle {\vec {\mu }}_{v}}$, each acting on a separate qubit. The Hamiltonian is

${\displaystyle H=-\sum _{c}(A_{c}+B_{c})}$.

Here, for any unit cube ${\displaystyle c}$ whose eight vertices are labeled as ${\displaystyle p}$, ${\displaystyle q_{1}=p+(1,0,0)}$, ${\displaystyle q_{2}=p+(0,1,0)}$, ${\displaystyle q_{3}=p+(0,0,1)}$, ${\displaystyle r_{1}=p+(0,1,1)}$, ${\displaystyle r_{2}=p+(1,0,1)}$, ${\displaystyle r_{3}=p+(1,1,0)}$ , and ${\displaystyle s=p+(1,1,1)}$, the operators ${\displaystyle A_{c}}$ and ${\displaystyle B_{c}}$ are defined as

${\displaystyle A_{c}=\sigma _{s}^{z}\mu _{s}^{z}\prod _{j=1}^{3}\mu _{q_{j}}^{z}\sigma _{r_{j}}^{z}}$

${\displaystyle B_{c}=\sigma _{p}^{x}\mu _{p}^{x}\prod _{j=1}^{3}\mu _{q_{j}}^{x}\sigma _{r_{j}}^{x}}$

This is also an exactly solvable model, as all terms of the Hamiltonian commute with each other.

The ground state degeneracy for an ${\displaystyle L\times L\times L}$ torus is given by

${\displaystyle \log _{2}GSD=4{\textrm {deg}}({\textrm {gcd}}(1+(1+x)^{L},1+(1+\omega x)^{L},1+(1+\omega ^{2}x)^{L})_{\mathbb {F} _{4}})-2}$,[8][9]

Here, gcd denotes the greatest common divisor of the three polynomials shown, and deg refers to the degree of this common divisor. The coefficients of the polynomials belong to the finite field ${\displaystyle \mathbb {F} _{4}}$, consisting of the four elements ${\displaystyle \lbrace 0,1,\omega ,\omega ^{2}\rbrace }$ of characteristic 2 (i.e. ${\displaystyle 1+1=0}$). ${\displaystyle \omega }$ is a cube root of 1 that is distinct from 1. The greatest common divisor can be defined through Euclid's algorithm. This degeneracy fluctuates wildly as a function of ${\displaystyle L}$. If ${\displaystyle L}$ is a power of 2, then according to Lucas's theorem the three polynomials take the simple forms ${\displaystyle x^{L},(\omega x)^{L},(\omega ^{2}x)^{L}}$, indicating a ground sate degeneracy of ${\displaystyle 2^{4L-2}}$. More generally, if ${\displaystyle 2^{m}}$ is the largest power of 2 that divides ${\displaystyle L}$, then the ground state degeneracy is at least ${\displaystyle 2^{2^{m+2}-2}}$ and at most ${\displaystyle 2^{4L-2}}$.

Thus the Haah's code fracton model also in some sense exhibits the property that the logarithm of the ground state degeneracy tends to scale in direct proportion to the linear dimension of the system. This appears to be a general property of gapped fracton models. Just like in type I models and in topologically ordered systems, different ground states of Haah's code cannot be distinguished by local operators.

Haah's code also features immobile elementary excitations called fractons. A quantum state is said to have a fracton located at a cube ${\displaystyle c}$ if the eigenvalue of ${\displaystyle A_{c}}$ is ${\displaystyle -1}$ for this quantum state (an excitation of the ${\displaystyle B_{c}}$ operator is also a fracton. Such a fracton is physically equivalent to an excitation of ${\displaystyle A_{c}}$ because there is a unitary map exchanging ${\displaystyle A_{c}}$ and ${\displaystyle B_{c}}$, so it suffices to consider excitations of ${\displaystyle A_{c}}$ only for this discussion).

If ${\displaystyle |GS\rangle }$ is a ground state of the Hamiltonian, then for any vertex ${\displaystyle v}$, the state ${\displaystyle \mu _{v}^{x}|GS\rangle }$ features four fractons in a tetrahedral arrangement, occupying four of the eight cubes adjacent to vertex ${\displaystyle v}$ (the same is true for the state ${\displaystyle \sigma _{v}^{x}|GS\rangle }$, although the exact shape of the tetrahedron is different).

In an attempt to isolate just one of these four fractons, one may try to apply additional ${\displaystyle \mu _{v'}^{x}|GS\rangle }$ spin flips at different nearby vertices to try annihilate the three other fractons. Doing so simply results in three new fractons appearing further away. Motivated by this process, one can then identify a set ${\displaystyle S}$ of vertices in space that together form some arbitrary iteration of the three-dimensional Sierpiński fractal. Then the state ${\displaystyle \prod _{v\in S}\mu _{v}^{z}|GS\rangle }$ features four fractons, one each at a cube adjacent to a corner vertex of the Sierpinski tetrahedron. Thus we see that an infinitely large fractal-shaped operator is required to generate an isolated fracton out of the ground state in the Haah's code model. The fractal-shaped operator in Haah's code plays an analogous role to the membrane operators in the X-cube model.

Unlike in type I models, there are no stable bound states of a finite number of fractons that are mobile. The only mobile bound states are those such as the completely mobile four-fracton states like ${\displaystyle \mu _{v}^{x}|GS\rangle }$ that are unstable (i.e. can transform into the ground state by the action of a local operator).

Many of the known type I fracton model are in fact in the same foliated fracton order as the X Cube Model, or in the same foliated fracton order as multiple copies of the X cube model. However, not all are. A notable known example of a distinct foliated fracton order is the twisted foliated fracton model.[10]

Explicit local unitary maps have been constructed that demonstrate the equivalence of the X cube model with various other models, such as the Majorana checkerboard model and the semionic X cube model. The checkerboard model belongs to the same foliated fracton order as two copies of the X cube model.[6]

Just like how topological orders tend to have various invariant quantities that represent topological signatures, one can also attempt to identify invariants of foliated fracton orders.

Conventional topological orders often exhibit ground state degeneracy which is dependent only on the topology of the manifold on which the system is embedded. Fracton models do not have this property, because the ground state degeneracy also depends on system size. Furthermore, in foliated fracton models the ground state degeneracy can also depend on the intricacies of the foliation structure used to construct it. In other words, the same type of model on the same manifold with the same system size may have different ground state degeneracies depending on the underlying choice of foliation.[4]

By definition, the number of superselection sectors in a fracton model is infinite (i.e. scales with system size). For example, each individual fracton belongs to its own superselection sector, as there is no local operator that can transform it to any other fracton at a different position.

However, a loosening of the concept of superselection sector, known as the quotient superselection sector, effectively ignores two-dimensional particles (e.g. planeon bound states) which are presumed to come from two-dimensional foliating layers.[5] Foliated fracton models then tend to have a finite list of quotient superselection sectors describing the types of fractional excitations present in the model. This is analogous to how topological orders tend to have a finite list of ordinary superselection sectors.

Generally for fracton models in the ground state, when considering the entanglement entropy of a subregion of space with large linear size ${\displaystyle R}$, the leading order contribution to the entropy is proportional to ${\displaystyle R^{2}}$, as expected for a gapped three dimensional system obeying an area law. However, the entanglement entropy also has subleading terms as a function of ${\displaystyle R}$ that reflect hidden nonlocal contributions. For example, the ${\displaystyle \propto R}$ subleading correction represents a contribution from the constant topological entanglement entropy of each of the 2D topologically ordered layers present in the foliation structure of the system.

Since foliated fracton order is invariant even when disentangling such 2D gapped layers, an entanglement signature of a foliated fracton order must be able to ignore of the entropy contributions both from local details and from 2D topologically ordered layers.

It is possible to use a mutual information calculation to extract a contribution to entanglement entropy that is unique to the foliated fracton order. Effectively, this is done by adding and subtracting entanglement entropies of different regions in such a way as to get rid of local contributions as well as contributions from 2D gapped layers.[12][11]

For example, in the U(1) scalar charge model, the fracton charge density (${\displaystyle \rho }$) is related to a symmetric electric field tensor (${\displaystyle E_{ij}}$, a theoretical generalization of the usual electric vector field) via ${\displaystyle \rho =\partial _{i}\partial _{j}E_{ij}}$, where the repeated spatial indices ${\displaystyle i,j=1,2,3}$ are implicitly summed over. Both the fracton charge (${\displaystyle q}$) and dipole moment (${\displaystyle p_{i}}$) can be shown to be conserved:

${\displaystyle {\begin{aligned}q&=\int \rho \;d^{3}x=\int \partial _{i}(\partial _{j}E_{ij})\;d^{3}x=0\\p_{i}&=\int x_{i}\rho \;d^{3}x=\int x_{i}\partial _{j}\partial _{k}E_{jk}\;d^{3}x=-\int \partial _{k}E_{ik}\;d^{3}x=0\end{aligned}}}$

When integrating by parts, we have assumed that there is no electric field at spatial infinity. Since the total fracton charge and dipole moment is zero under this assumption, this implies that the charge and dipole moment is conserved. Because moving an isolated charge changes the total dipole moment, this implies that isolated charges are immobile in this theory. However, two oppositely charged fractons, which forms a fracton dipole, can move freely since this does not change the dipole moment. [13]

One approach to constructing an explicit action for scalar fractonic matter fields and their coupling to the symmetric tensor gauge theory is the following.[3] Suppose the scalar fractonic matter field is ${\displaystyle \Phi }$. A global charge conservation symmetry would imply that the action is symmetric under the transformation ${\displaystyle \Phi \to e^{i\alpha }\Phi }$ for some spatially uniform real ${\displaystyle \alpha }$, as is the case in usual ${\displaystyle U(1)}$ charged theories. A global dipole moment conservation symmetry would imply that the action is symmetric under the transformation ${\displaystyle \Phi ({\vec {r}})\to e^{i{\vec {\lambda }}\cdot {\vec {r}}}\Phi ({\vec {r}})}$ for an arbitrary real spatially uniform vector ${\displaystyle {\vec {\lambda }}}$. The simplest kinetic terms (i.e. terms featuring the spatial derivative) that are symmetric under these transformations are quartic in ${\displaystyle \Phi }$.

${\displaystyle {\mathcal {L}}=|\partial _{t}\Phi |^{2}-m^{2}|\Phi |^{2}-g|\Phi \partial _{i}\partial _{j}\Phi -\partial _{i}\Phi \partial _{j}\Phi |^{2}-g'\Phi ^{*2}(\Phi \partial _{i}\partial _{j}\Phi -\partial _{i}\Phi \partial _{j}\Phi )+\ldots }$

Now when gauging this symmetry, the kinetic expression ${\displaystyle \Phi \partial _{i}\partial _{j}\Phi -\partial _{i}\Phi \partial _{j}\Phi }$ gets replaced with ${\displaystyle \Phi \partial _{i}\partial _{j}\Phi -\partial _{i}\Phi \partial _{j}\Phi -iA_{ij}\Phi ^{2}}$, where ${\displaystyle A_{ij}}$ is a symmetric tensor that transforms under arbitrary gauge transformations as ${\displaystyle A_{ij}\to A_{ij}+\partial _{i}\partial _{j}\alpha }$. This shows how a symmetric tensor field couples to scalar fractonic matter fields.

The U(1) scalar charge theory is not the only symmetric tensor gauge theory that is gives rise to limited mobility particles. Another example is the U(1) vector charge theory.

In this theory, the fractonic charge is a vector quantity ${\displaystyle {\vec {\rho }}}$. The symmetric tensor gauge field transforms under gauge transformations ${\displaystyle {\vec {\alpha }}}$ as ${\displaystyle A_{ij}\to A_{ij}+\partial _{i}\alpha _{j}+\partial _{j}\alpha _{i}}$. The Gauss law for this theory takes the form ${\displaystyle \partial _{i}E_{ij}=\rho _{j}}$, which implies both a total charge conservation and a conservation of total angular charge moment ${\displaystyle \int d^{3}x{\vec {\rho }}\otimes {\vec {x}}}$. The latter conservation law implies that isolated charges are restricted to move parallel to their corresponding charge vectors. Thus these particles appear to be similar to the lineons in Type I fractons, except here they are in a gapless theory.