Quantum Field Theory of X-Cube Fracton Topological Order
and Robust Degeneracy from Geometry
We propose a quantum field theory description of the X-cube model of fracton topological order. The field theory is not (and cannot be) a topological quantum field theory (TQFT), since unlike the X-cube model, TQFTs are invariant (i.e. symmetric) under continuous spacetime transformations. However, the theory is instead invariant under a certain subgroup of the conformal group. We describe how braiding statistics and ground state degeneracy are reproduced by the field theory, and how the the X-cube Hamiltonian and field theory can be minimally coupled to matter fields. We also show that even on a manifold with trivial topology, spatial curvature can induce a ground state degeneracy that is stable to arbitrary local perturbations! Our formalism may allow for the description of other fracton field theories, where the only necessary input is an equation of motion for a charge density.
Just as the initial theoretical discovery of (liquid Zeng and Wen (2015)) topologically ordered phases of matter Wen (2016); Lan et al. () led to incredible discoveries, the same is now occurring in the context of non-liquid topological order, particularly fracton topological order Bravyi and Haah (2013); Vijay et al. (2015); Vijay and Fu (); Brown et al. (2016); Bravyi et al. (2011); Haah (2011); Yoshida (2013); Ma et al. (2017); Prem et al. (2017); Chamon (2005); Williamson (2016); Slagle and Kim (2017); Hsieh and Halász (2017); Halász et al. (); Prem et al. (); Devakul et al. (); Petrova and Regnault (). Both kinds of topological order have a finite energy gap to all excitations (although gapless versions of liquid and non-liquid Pretko (2017a, b); Rasmussen et al. (); Xu (2006); Xu and Wu (2008); Pretko () phases also exist), and both host degenerate ground states which are stable to arbitrary perturbations and can only be distinguished by nonlocal operators. (This is in contrast to spontaneous symmetry breaking states where the degenerate ground states are protected by symmetry and can be distinguished by local order parameters.) Both liquid and non-liquid topological orders also host topological excitations, which can only be annihilated via contact with the appropriate antiparticle(s).
The low energy physics of liquid topologically ordered phases Zeng and Wen (2015) is topologically invariant: i.e. symmetric under any continuous (and bijective) spacetime transformation which preserves the topology of the spacetime manifold. For example, the multiplicity of the ground state degeneracy depends only on the topology of the spatial manifold. Additionally, the braiding statistics of the topological excitations depend only on the topology of the paths that they take. Exactly solvable lattice models exist for many of these phases Levin and Wen (2005). However, the topological nature of these phases lends to a more minimal and universal description in the form of a topological quantum field theory (TQFT) Chern and Simons (1974); Bartlett (); Schwarz (); Witten (1988); Atiyah (1988); Dijkgraaf and Witten (1990) which makes the topological nature of these phases explicit via an explicit topological invariance, which is not possible in a lattice model. For example, Kitaev’s toric code lattice model and the BF theory TQFT in 2+1D (or equivalently Chern-Simons theory with -matrix Wen and Zee (1992) ) both describe topological order Kitaev (2003); Blau and Thompson (1991); Putrov et al. (2017).
Non-liquid topologically ordered phases retain many of the exotic properties of the liquid topological phases, except that the long distance physics is not topologically invariant. The simplest example of a non-liquid phase is a decoupled (or weakly coupled) stack of 2D toric codes in three dimensions. A less trivial example is the X-cube model Vijay et al. (2016) with fracton topological order which we study in this work. Both models have a ground state degeneracy which is stable to arbitrary perturbations; however, on an torus their degeneracy is exponentially large with , and thus depends on more than just the topology of the spatial manifold. Both phases also have constrained dynamics which are stable to perturbations. In the stack of toric codes, the topological charge and flux excitations can not move between different toric code stacks, which allows the point-like charge and flux particles to have non-trivial braiding statistics in a 3D phase. And the X-cube model has dimension-1 particles Vijay et al. (2015) which can only move in the , , or directions. These movement constraints are not invariant under spatial deformations, and thus these phases aren’t topologically invariant and therefore can’t be described by a TQFT.
Nevertheless, it seems important to ask if it is possible to describe these non-liquid topological phases with a field theory which captures as much of the spacetime symmetry as possible (i.e. some subgroup of the group of continuous spacetime transformations). However, as is often the case in quantum field theories, an obstruction presents itself in the form of an infinite quantity. In the case of the stacked toric code example, a natural field theory is BF theory with an extra coordinate to index the different stacks:
However, this field theory appears to have an infinite degeneracy on a torus: a factor of 4 for each value of , for which there are infinitely many. Nevertheless, we argue that this dilemma can be solved by applying the standard weapon against infinities in field theories: a short distance cutoff. That is, if we impose a short distance cutoff , then the degeneracy on an torus is finite and equal to . Thus, we can view the field theory as describing a periodic cubic lattice with degeneracy where . We propose that the same trick can be applied to our field theory for the X-cube model, which also has a degeneracy which is exponentially large in system size.
A more practical concern is: how do we write down the correct field theory to describe a given non-liquid phase? This task was manageable for TQFTs because the topological invariance greatly limited the possible Lagrangians that could be written down. For the case of non-topologically invariant (i.e. non-liquid) topological phases, we do not have this luxury.
However, for the case of toric code and its BF theory TQFT description, the terms in the toric code Hamiltonian can be precisely related to the terms in the BF Lagrangian and also to the gauge invariance of the TQFT. That is, the time components of the gauge fields act as Lagrange multipliers which impose zero charge and flux constraints, where the terms in the toric code Hamiltonian are lattice discretizations of the charge and flux densities in BF theory. And the gauge invariance is related to the fact that all of the terms in the toric code Hamiltonian commute with each other. We review this relationship in Appendix A (and Appendix B for 3+1D BF theory).
In Sec. I we use this intuition to systematically derive a field theory (Eq. (9)) for the X-cube model Vijay et al. (2016) of fracton topological order. The precise relations discussed in the previous paragraph continue to hold. The field theory is not topologically invariant, but is instead invariant under a certain subgroup of the conformal group of spacetime transformations which transforms all coordinates independently (Sec. I.4). However, there are new surprises which challenged our previous intuition. For example, we will see that the parallel movement of a pair of fractons requires a fracton dipole current in the region between the pair of fractons (Fig. 2), which is related to the fact that in the lattice model a membrane operator can be used to move a pair of fractons. This peculiarity is necessitated by the exotic charge conservation constraints (Eq. (16)) which enforce e.g. the immobility of isolated fractons.
In Sec. I.2 we explain the generic braiding processes of the X-cube model and how they are described by our field theory. In particular, the motion of dimension-1 particles around the edges of a cube results in a phase factor which depends on the number of fractons within the cube (modulo for the model) foo (). And the motion of a pair of oppositely charged fractons around the top and bottom edge of a cylinder oriented along the -axis generates a phase which depends on the difference in the number of -axis and -axis dimension-1 particles Ma et al. (2017).
In Sec. I.3 we show how the X-cube Hamiltonian and field theory can be coupled to matter fields with subdimensional symmetries. Before the matter fields are coupled to the gauge fields, the matter excitations have the same mobility constraints as the fractons and dimension-1 particles of the X-cube model. However, while the mobility constraints of the X-cube model are robust (i.e. stable under arbitrary local perturbations), the mobility constraints of the matter fields in the absence of the gauge fields is instead protected by subdimensional symmetry.
In Sec. I.5 we explain how the ground state degeneracy of the X-cube model can be calculated from either the lattice model or the field theory. As is well known, the degeneracy is not topologically invariant, but is instead exponentially large with system size (on a torus) Vijay et al. (2016); Vijay (); Ma et al. (2017). However, when the log of the degeneracy is expressed as an integral over space (Eq. (63)), it is invariant under the spacetime transformation discussed in Sec. I.4 when the cutoff is transformed appropriately. We then ask: since the degeneracy isn’t topologically invariant, is a nontrivial topology of the spatial manifold actually necessary for a stable ground state degeneracy in the X-cube model? In Sec. II we show that the answer is no: a cubic lattice with curvature defects can host a stable ground state degeneracy. As the size of the curved portion of the lattice increases, the degeneracy can be made exponentially large while the energy splitting of the degeneracy due to perturbations becomes exponentially small.
In Appendix C we attempt to use our field theory generating formalism to derive field theories for new non-liquid topological phases. With our current formalism, the only necessary input is an equation for a charge density. Unfortunately, in this work we only rule out certain simple possibilities. For example, we find that when the scalar charge fracton phase Rasmussen et al. (); Pretko (2017a) is “Higgsed” down to , that the fractons in the theory become mobile (and thus not fractons) in the theory. Other possibilities are left to future work.
Before we begin, we will briefly explain some of the nonstandard notation that we use. Roman letters denote spatial indices. Greek letters denote spacetime indices. ( and for the 2+1D theories in Appendix A). is the time index. Hats are placed above operators. A semicolon (e.g. in in Sec. I) is used to indicate that the indices following the semicolon do not transform under spacetime transformations (Sec. I.4).
We use the convention that all spatial and spacetime indices are implicitly summed unless they appear on both sides of the equation or the right hand side is zero. For example,
could be written more explicitly as
where means for all . The semicolon does not denote a covariant derivative. Instead, in the is used to emphasize that transforms like a time-component of the current , while “” mereley indexes the different time-components of ; the semicolon is used to seperate these different kinds of indices.
I X-Cube Quantum Field Theory
We will begin by systematically deriving a quantum field theory (QFT) for the X-cube model of fracton topological order Vijay et al. (2016). See Appendix A.1 and Appendix B, for analogous derivations for BF theory in 2+1D and 3+1D.
The X-cube model is defined by the following Hamiltonian Vijay et al. (2016):
where denotes the spatial coordinates. and are defined in Fig. 1 in terms of and , which are generalizations of Pauli operators:
If , then and reduce to the usual Pauli operators and . is the fracton operator, where is the number of fracton excitations module . is a dimension-1 particle operator; if and , then there is a z-axis dimension-1 particle at . (Fig. 9)
In order to connect the lattice model to the field theory, we will rewrite the lattice operators as exponents of fields and :
where is a spatial index (roman letters are used to denote spatial indices). and are fracton and dimension-1 particle densities, respectively. For the purposes of this work, we will only interpret Eq. (3) as a rough correspondence. The integrals integrate over small regions near . Specifically: is an integral across the link that lives on; integrates over the dual plaquette that is orthogonal to the link that lives on; integrates over the cube that is centered at; and integrates over the a plaquette in the place of the operator.
We will usually view and as real-valued fields, which are distinguished from their corresponding operators and by hats. However, when and are viewed as operators, they have the following equal time commutation relation:
where “” is used to emphasized that these will be equations of motion and not strict equalities. and (Fig. 1) can be viewed as lattice discretizations of and . Note that and commute (i.e. via the bracket in Eq. (4)); this occurs because and commute (i.e. ).
Regarding the notation, is just the absolute value of the Levi-Civita symbol, and merely forces , , and to be different spacetime indices. We will use the convention that all spatial and spacetime indices are implicitly summed unless they appear on both sides of the equation or the right hand side is zero. Thus, in the equation for , is not summed, but both and are implicitly summed over even though only appears once and appears three times. The semicolon is used to indicate that the indices following the semicolon do not transform under spacetime transformations (Sec. I.4).
The Lagrangian description of the degenerate ground state manifold can now be written down:
The first term describes the equal-time commutation relation (Eq. (4)), while the second and third terms enforce a zero charge constraint (Eq. (5)) via the Lagrange multipliers and . The final four terms are generic couplings of the fields ( and ) to the current sources ( and ). Eq. (7) is a local Hilbert space constraint, which results from the fact that (Eq. (5)) and .
In order to transition to a more standard notation, we will redefine , , , and in term of , , , and , respectively 111In case the reader is interested, a previous version of this work Slagle and Kim () was written up using and fields instead, which are more closely connected to the Pauli operators and .:
In terms of the and fields, the Lagrangian (Eq. (6)) becomes:
The equations of motion for the currents are
The gauge invariance can be derived as follows:
where the brackets are evaluated using Eq. (4) written in terms of and fields:
A constraint on (Eq. (13)) is imposed since it does not reduce the generality of the gauge transformation, and because it will be needed to fulfill the constraint on (Eq. (10)) under its gauge transformation (Eq. (15)). The transformation of the fields ( and ) corresponds to conjugating the lattice operators ( and ) by the terms in the Hamiltonian ( and ) at the positions where and are nonzero: e.g. . The gauge invariance is a direct result of the fact that the terms in (Eq. (1)) commute with each other. For example, and are invariant under the above transformation because and commute (i.e. ), and and commute because and commute.
To derive how and transform, the above gauge transformations can be inserted into (Eq. (9)), and and can be solved for to find:
Finally, in order for the coupling of the gauge fields ( and ) to the currents ( and ) in (Eq. (9)) to be gauge invariant, the currents must obey the following constraint:
where the “” means that we don’t sum over in the last equation, which specifies three separate constraints. These are generalized charge conservation constraints (analogous to Eq. (80) for BF theory), which encode the movement restrictions of the fracton current and dimension-1 particle current . 222Eq. (16) is similar to the generalized continuity equation in Ref. Pretko (2017c).
A single stationary fracton at the origin is simply described by
where is the Dirac delta function. However, the current source () describing the creation of fractons is more exotic. Eq. (16) allows four fractons to be created at and , forming a fracton quadrupole, via the following current configuration
where is the Heaviside step function:
is nonzero on a square at time , which then creates four fractons () at the corners for . This is analogous to the X-cube lattice model where fractons are created at the corners of membrane operators. The double derivative in in Eq. (16) is the reason why fractons are created at corners of membrane operators in the field theory (instead of at the ends of string operators as is typically the case). Physically, is can be regarded as a fracton dipole current; see Fig. 2.
A -axis dimension-1 particle at the origin is represented by
This is similar to the lattice model where a -axis particle excites the and operators. The -axis particle can only move in the direction. This motion is described by
More generally, an -axis particle at the origin is given by
If both an -axis and -axis dimension-1 particle are at the origin, then this is equivalent to a -axis antiparticle:
The first line in Eq. (22) is Eq. (21) summed over ; this corresponds to the presence of both an -axis and a -axis particle. The second line shows that this is equivalent to the negation of Eq. (21) with , which corresponds to just a single -axis antiparticle. (This fusion rule can also be understood from the lattice operators.)
i.2 “Braiding” Statistics
If no additional excitations are created, isolated fractons are immobile and isolated dimension-1 particles can only move along straight lines. However, when we consider braiding statistics of topological excitations, we are allowed to create additional excitations. For example, to measure a flux in toric code we imagine 1) creating two charges, 2) moving one of the charges around the flux, and then 3) annihilating the two charges. We will find a slightly more exotic scenario for the “braiding” of X-cube excitations.
Dimension-1 Particle “Braiding”
As a first example, we will use our field theory description to demonstrate that in order to count the number of fractons (modulo ) within a cube, we can create dimension-1 particles and move them around the edges of the cube foo ().
The fracton current that describes the presence of a single fracton at is
A solution to (11) to describe this (motionless) current is
Using Eq. (3), this can be interpreted as (at the mean field level 333 For example, a mean field wavefunction can be defined by (e.g. with given in Eq. (23)). The physical wavefunction is the mean field wavefunction projected onto the desired dimension-1 particle charge configuration: where the projection operator (if ) projects onto the charge configuration . In Eq. (24), the charge configuration is zero (), which corresponds to . If we consider (instead of Eq. (23)) and keep , then is the exact ground state of the X-cube Hamiltonian (Eq. (1)). ) on a square membrane with a corner at . Such a wavefunction can be obtained by acting on the ground state of the lattice model (Eq. (1)) by a product of operators on the membrane.
In order to obtain a nonzero braiding statistic with the fracton, we will consider a dimension-1 particle current at time around the corners of a cube of length (Fig. 3) which can be written as
This current describes a dimension-1 particle which can only move in straight without creating any additional excitations. In this current configuration, twelve different dimension-1 particles are created so that the edges of a cube are traced out. (And indeed, these currents satisfy the constraints in Eq. (16).)
Although we won’t need them, there are a couple nice solutions to (11) to describe the current :
which is trivial to integrate since the integral just replaces the in by . Another gauge equivalent solution is
where is nonzero only inside the cube.
We can now evaluate the action (Eq. (9)) for this configuration. Since the equations of motion for and are satisfied, the Lagrangian simplifies:
The second line results because the first and third terms in the first line are zero since (Eq. (23)) and (Eq. (17)). Plugging in the expressions for and gives the third line. The integrand is nonzero only where the red current and blue membrane intersect in Fig. 3. Thus, the presence of a fracton in the cube is detected by “braiding” dimension-1 particles around the edges of a cube.
As a second example, we show how the presence of a dimension-1 particle can be detected by moving fractons around it. Ma et al. (2017)
The current describing an -axis dimension-1 particle at is
which has the following field solution:
To detect the -axis dimension-1 particle within a cube of length , we can simultaneously move two oppositely charged fractons around the top and bottom edges of the cube (Fig. 4a). The fractons are capable of moving by exchanging fracton dipoles (see Fig. 2). However, the current that describes this process might seem surprising, so we’ll instead begin with the lattice membrane operator that generates this current and fracton motion (Fig. 4a):
The membrane consists of four squares, which each create fractons at the corners which cancel out with the fractons generated by neighboring squares. From the and commutation relations (Eq. (2)), this operator will rotate the expectation value of . Using Eq. (3) and (I.1) to relate to , and the equations of motion for (Eq. (11)) to shift via the term in Eq. (11), we find the following fracton current:
is nonzero in the blue regions of Fig. 4a.
It may seem surprising that the fracton current (Eq. (31)) is nonzero on a membrane, even though we are trying to describe the movement of two point-like fractons. However, as explained in Fig. 3, this is due to the fact that fractons are created at corners of membrane operators, which implies that a membrane operator (or membrane current) is required to move a pair of fractons. This is manifested in the conservation law (copied below) by the second derivative in the second term:
Thus, is best regarded as a fracton dipole current.
In order to understand why we must consider a fracton dipole current in the field theory, let us define a new quantity , which we’ll call the fracton flow, in terms of the fracton current :
Unlike the fracton current (), the fracton flow equations of motion obey the usual current conservation law:
The fracton flow (green in Fig. 4) corresponds to the more intuitive notion of net movement of fractons. However, on a closed manifold the fracton flow does not uniquely specify the fracton current (), nor is it sufficient to calculate the resulting phase from a braiding process. For example, Fig. 4b shows a different current configuration, which results in the same fracton flow but a different phase factor; on a 3D torus, there is no reason to prefer one current configuration over the other. Thus, the current carries some addition information: fractons can only move by exchanging fracton dipoles (Fig. 2), and specifies where the dipole current occurs.
Although we won’t need them, (Eq. (31)) has a couple of nice field configuration solutions:
Another gauge equivalent solution is
which is nonzero inside the cube.
We can now evaluate the action (Eq. (9)) for this configuration. Making use of the equations of motion for and and the fact that for our current configuration, we find:
Thus, the presence of a dimension-1 particle in the cube is detected by moving a pair of fractons around the top and bottom edges of the cube. A more detailed analysis would show that this motion of fractons actually counts the difference in the number of -axis and -axis dimension-1 particles inside the cube.
i.3 Minimal Coupling to Matter
In this section we will show how the X-cube Hamiltonian (Eq. (1)) and field theory (Eq. (9)) can be coupled to matter, which is related to the gauging procedures introduced in Ref. Vijay et al. (2015); Williamson (2016). We will leave further study of these models to future work. See Appendix A.2, for an analogous treatment for toric code and BF theory in 2+1D.
In the lattice model, matter can be introduced by introducing fracton matter operators at the centers of the cubes and three dimension-1 matter operators () on the sites of the cubic lattice. The fracton operator and and dimension-1 particle operator are multiplied by and , respectively. We also introduce hopping terms and for the fracton and dimension-1 matter, respectively (Fig. 5). The Hamiltonian with this matter coupling is
and are fracton and dimension-1 matter number operators, and “h.c.” denotes the addition of the Hermitian conjugate of the preceding operators.
If and/or is small, then the and/or gauge fields are “Higgsed”, and is in a trivial phase with no topological order. This occurs because the Wilson and ’t Hooft loop operators (Fig. 6), which describe the ground state degeneracy, don’t commute with the and operators, respectively. When and are large, the matter has a large mass gap and has no effect on the phase.
The fracton matter hopping operator () hops fracton matter (i.e. excitations of ) with the same mobility constraints as the fracton excitations in the original X-cube model (Eq. (1)). For example, it can create four fractons from the vacuum, or it can move a fracton dipole along a plane as fracton dipoles are dimension-2 particles. Similarly, the dimension-1 matter hopping operator () hops dimension-1 matter (i.e. excitations of ) along straight lines. Thus, and are analogous to the and operators, respectively, in the X-cube model without explicit matter coupling (Eq. (1)). In the X-cube model, the mobility constraints of the fracton and dimension-1 particle excitations was robust; i.e. stable to arbitrary local perturbations. If we were to consider the matter content of (Eq. (37)) in the absence of the gauge fields and , then the mobility constraints of the matter would instead be enforced by subdimensional symmetries.
We can describe the same physics in the field theory by introducing -periodic (i.e., vortices are allowed) matter fields and . The simplest way to systematically construct a gauge invariant Lagrangian is to first define currents and (not to be confused with and in Eq. (6)) equal to and , respectively, and then apply a gauge transformation (Eq. (12) and (15)) with and :
where is off-diagonal and symmetric, and and are coupling constants. We can now define a Lagrangian for the matter fields:
where we have imposed a local constraint on , analogous to the local constraint placed on (Eq. (10)). (Without the constraint, would have a trivial local symmetry .) 444Note, in this work we are writing all Lagrangians in real time for consistency; in imaginary time two signs will flip in Eq. (39) so that is positive definite.
The advantage of this construction is that the Lagrangian is gauge invariant as long as and transform as
This construction also guarantees that the equations of motion for and imply that the matter currents and obey the mobility (or generalized charge conservation) constraints in Eq. (16).
Note that before is coupled to the gauge field (e.g. set in ), has a subdimensional symmetry