A fracton is an emergent topological quasiparticle excitation which is immobile when in isolation.[1] [2] Many theoretical systems have been proposed in which fractons exist as elementary excitations. Such systems are known as fracton models. Fractons have been identified in various CSS codes as well as in symmetric tensor gauge theories.
Gapped fracton models often feature a topological ground state degeneracy that grows exponentially and sub-extensively with system size. Among the gapped phases of fracton models, there is a non-rigorous phenomenological classification into "type I" and "type II". Type I fracton models generally have fracton excitations that are completely immobile, as well as other excitations, including bound states, with restricted mobility. Type II fracton models generally have fracton excitations and no mobile particles of any form. Furthermore, isolated fracton particles in type II models are associated with nonlocal operators with intricate fractal structure.[3]
The paradigmatic example of a type I fracton model is the X-cube model. Other examples of type I fracton models include the semionic X-cube model, the checkerboard model, the Majorana checkerboard model, the stacked Kagome X-cube model, the hyperkagome X-cube model, and more.
The X-cube model is constructed on a cubic lattice, with qubits on each edge of the lattice.
The Hamiltonian is given by
H=-\sumcAc-\sumrm{v}(Bv,x+Bv,y+Bv,z)
Here, the sums run over cubic unit cells and over vertices. For any cubic unit cell
c
Ac
X
v
Bv,\mu
Z
v
\mu
X
Z
In addition to obeying an overall
Z2 x Z2
\prod\ellX\ell
\prod\ellZ\ell
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
|GS\rangle
Ac|GS\rangle=1
Bv,|GS\rangle=1
c,v,\mu
1+Ac | |
2 |
1+Bv,\mu | |
2 |
The constraints posed by
Ac=1
Bv,\mu=1
Lx,Ly,Lz
2Lx+2Ly+2Lz-3 | |
2 |
logGSD\proptoL
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
Ac=-1
c
c
|GS\rangle
\ell
X\ell|GS\rangle
\ell
Given a rectangle
R
ZR\equiv\prod\ellZ\ell
\ell
ZR|GS\rangle
R
\pi
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
Bv,x=Bv,y=-1
v
v
z
x
y
z
v
X
z
X
An
x,y
z
x
y
z
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
x
y
w
z
xy
w
z
x
y
xy
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
x
y
xy
xy
X
-1
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. 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.[5]
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
A
B
H=-\sumc\in\prodvXv-\sumc\in\prodvZv
This model is also exactly solvable with commuting terms. The topological ground state degeneracy on a torus is given by
log2GSD=4Lx+4Ly+4Lz-6
2Lx,2Ly,2Lz
Like the X cube model, the checkerboard model features excitations in the form of fractons, lineons, and planeons.
The paradigmatic example of a type II fracton model is Haah's code. Due to the more complicated nature of Haah's code, the generalisations to other type II models are poorly understood compared to type I models.[6]
Haah's code is defined on a cubic lattice with two qubits on each vertex. We can refer to these qubits using Pauli matrices
\vec{\sigma}v
\vec{\mu}v
H=-\sumc(Ac+Bc)
Here, for any unit cube
c
p
q1=p+(1,0,0)
q2=p+(0,1,0)
q3=p+(0,0,1)
r1=p+(0,1,1)
r2=p+(1,0,1)
r3=p+(1,1,0)
s=p+(1,1,1)
Ac
Bc
Ac=
z | |
\sigma | |
s |
z | |
\mu | |
s |
3 | |
\prod | |
j=1 |
z | |
\mu | |
qj |
z | |
\sigma | |
rj |
Bc=
x | |
\sigma | |
p |
x | |
\mu | |
p |
3 | |
\prod | |
j=1 |
x | |
\mu | |
qj |
x | |
\sigma | |
rj |
This is also an exactly solvable model, as all terms of the Hamiltonian commute with each other.
The ground state degeneracy for an
L x L x L
log2GSD=4\deg(\gcd(1+(1+x)L,1+(1+\omegax)L,1+(1+\omega2
L) | |
x) | |
F4 |
)-2.
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
F4
\lbrace0,1,\omega,\omega2\rbrace
1+1=0
\omega
L
L
xL,(\omegax)L,(\omega2x)L
24L-2
2m
L
2m+2-2 | |
2 |
24L-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
c
Ac
-1
Bc
Ac
Ac
Bc
Ac
If
|GS\rangle
v
x | |
\mu | |
v |
|GS\rangle
v
x | |
\sigma | |
v |
|GS\rangle
In an attempt to isolate just one of these four fractons, one may try to apply additional
x | |
\mu | |
v' |
|GS\rangle
S
\psi=\prodv
z | |
\mu | |
v |
|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
x | |
\mu | |
v |
|GS\rangle
One formalism used to understand the universal properties of type I fracton phases is called foliated fracton order.
Foliated fracton order establishes an equivalence relation between two systems, system
A
B
HA
HB
HA
HB
H1
H2
It is important in this definition that the local unitary map remains at finite depth as the sizes of systems 1 and 2 are taken to the thermodynamic limit. However, the number of gapped systems being added or removed can be infinite. The fact that two-dimensional topologically ordered gapped systems can be freely added or removed in the transformation process is what distinguishes foliated fracton order form more conventional notions of phases.To state the definition more precisely, suppose one can find two (possibly empty or infinite) collections of two-dimensional gapped phases (with arbitrary topological order),
2D,A | |
H | |
j |
2D,B | |
H | |
j |
U
U
HA ⊗ otimesj
2D,A | |
H | |
j |
HB ⊗ otimesj
2D,B | |
H | |
j |
HA
HB
More conventional notions of phase equivalence fail to give sensible results when directly applied to fracton models, because they are based on the notion that two models in the same phase should have the same topological ground state degeneracy. Since the ground state degeneracy of fracton models scales with system size, these conventional definitions would imply that simply changing the system size slightly would alter the entire phase. This would make it impossible to study the phases of fracton matter in the thermodynamic limit where system size
L\toinfty
H
H(Lx,Ly,Lz)
H(L'x,L'y,L'z)
Foliated fracton order is not a suitable formalism for type II fracton models.
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.[9]
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.
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.
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. 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
R
R2
R
\proptoR
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.[10] [11]
The immobility of fractons in symmetric tensor gauge theory can be understood as a generalization of electric charge conservation resulting from a modified Gauss's law. Various formulations and constraints of symmetric tensor gauge theory tend to result in conservation laws that imply the existence of restricted-mobility particles.
For example, in the U(1) scalar charge model, the fracton charge density (
\rho
Eij
\rho=\partiali\partialjEij
i,j=1,2,3
q
pi
\begin{align} q&=\int\rho d3x=\int\partiali(\partialjEij) d3x=0\\ pi&=\intxi\rho d3x=\intxi\partialj\partialkEjk d3x=-\int\partialkEik d3x=0 \end{align}
One approach to constructing an explicit action for scalar fractonic matter fields and their coupling to the symmetric tensor gauge theory is the following. Suppose the scalar fractonic matter field is
\Phi
\Phi\toei\alpha\Phi
\alpha
U(1)
\Phi(\vecr)\toei\Phi(\vec{r})
\vecλ
\Phi
l{L}=|\partialt\Phi|2-m2|\Phi|2-g|\Phi\partiali\partialj\Phi-\partiali\Phi\partialj\Phi|2-g'\Phi*2(\Phi\partiali\partialj\Phi-\partiali\Phi\partialj\Phi)+\ldots
Now when gauging this symmetry, the kinetic expression
\Phi\partiali\partialj\Phi-\partiali\Phi\partialj\Phi
\Phi\partiali\partialj\Phi-\partiali\Phi\partialj\Phi-iAij\Phi2
Aij
Aij\toAij+\partiali\partialj\alpha
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
\vec{\rho}
\vec{\alpha}
Aij\toAij+\partiali\alphaj+\partialj\alphai
\partialiEij=\rhoj
\intd3x\vec{\rho} ⊗ \vec{x}
Fractons were originally studied as an analytically tractable realization of quantum glassiness where the immobility of isolated fractons results in a slow relaxation rate.[12] [13] This immobility has also been shown to be capable of producing a partially self-correcting quantum memory, which could be useful for making an analog of a hard drive for a quantum computer.[14] [15] Fractons have also been shown to appear in quantum linearized gravity models[16] and (via a duality) as disclination crystal defects.[17] However, aside from the duality to crystal defects, and although it has been shown to be possible in principle,[18] [19] other experimental realizations of gapped fracton models have not yet been realized. On the other hand, there has been progress in studying the dynamics of dipole-conserving systems, both theoretically[20] [21] [22] and experimentally,[23] [24] which exhibit the characteristic slow dynamics expected of systems with fractonic behavior.
U(1) symmetric tensor gauge theory | type-I | type-II | ||
---|---|---|---|---|
energy spectrum | gapless | gapped | gapped | |
example models | scalar charge | X-cube [25] | Haah's cubic code [26] | |
example characteristics | conserved dipole moment | conserved charge on stacks of two-dimensional surfaces | fractal conservation laws, no mobile particles |
It has been conjectured [27] that many type-I models are examples of foliated fracton phases; however, it remains unclear whether non-Abelian fracton models[28] [29] [30] can be understood within the foliated framework.