Corner Modes and Ground-State Degeneracy in Models with Gauge-Like Subsystem Symmetries
Subsystem symmetries are intermediate between global and gauge symmetries. One can treat these symmetries either like global symmetries that act on subregions of a system, or gauge symmetries that act on the regions transverse to the regions acted upon by the symmetry. We show that this latter interpretation can lead to an understanding of global, topology-dependent features in systems with subsystem symmetries. We demonstrate this with an exactly-solvable lattice model constructed from a 2D system of bosons coupled to a vector field with a 1D subsystem symmetry. The model is shown to host a robust ground state degeneracy that depends on the spatial topology of the underlying manifold, and localized zero energy modes on corners of the system. A continuum field theory description of these phenomena is derived in terms of an anisotropic, modified version of the Abelian K-matrix Chern-Simons field theory. We show that this continuum description can lead to geometric-type effects such as corner states and edge states whose character depends on the orientation of the edge.
It is widely known that discrete gauge symmetries can give rise to topological order in 2+1D Kogut (1979); Lüscher (1982); Woit (1983); Kogut et al. (1983); Wen (1990); Levin and Wen (2003). This began with work on 2+1D lattice gauge theory descriptions of quantum dimer models and resonating valence Bond states Fradkin and Shenker (1979); Fradkin and Kivelson (1990); Wen (1991a); Moessner et al. (2001); Ardonne et al. (2004); Levin and Wen (2006); Fradkin (2013). Since then, there has been intense theoretical effort studying the properties of topological ordered lattice gauge theories Jersák et al. (1983); Brown et al. (1988); Stack et al. (1994); Engels et al. (1995); Greensite (2003); Wang et al. (2011). Key features of these systems include, a robust ground state degeneracy which depends on the topology of the underlying spatial manifold/lattice Wen (1989); Misguich et al. (2002); Levin and Wen (2006), fractionalized quasiparticles with unusual statistics Wen and Zee (1991); Wen (1991b, c); Kitaev (2006); Nayak et al. (2008); Levin and Gu (2012), and long-range entangled ground states Kitaev and Preskill (2006); Chen et al. (2010); Jiang et al. (2012).
A quintessential example of emergent topological order is Kitaev’s toric code model, which realizes the deconfined phase of a lattice gauge theory Kitaev (1997); Bravyi and Kitaev (1998); Kitaev (2003); Castelnovo and Chamon (2008); Tupitsyn et al. (2010). The model consists of a square lattice with spin- degrees of freedom defined on the links of lattice. The gauge transformation consists of flipping all spins around a single elementary plaquette. When defined on a manifold of genus , the toric code system has a ground state degeneracy of , which corresponds to the number of ways gauge fluxes can be threaded through non-contractible loops in the system.
Recently, there has also been significant work in understanding the role of subsystem symmetries in topological phases of matter. For a dimensional system, subsystem symmetries (also refereed to as Gauge-Like symmetries) are sets of symmetries that act independently on dimensional subregions, with . Subsystem symmetries can be viewed as intermediate between gauge symmetries ( dimensional subregions) and global symmetries ( dimensional subregions).
In connection to topology, it has been shown that subsystem symmetries can lead to unique topological phases of matter known as subsystem symmetry protected topological (SSPT) phases You et al. (2018a). SSPT phases have edge degrees of freedom that transform projectively under the subsystem symmetry. For open boundaries, SSPT’s have a subextensive ground state degeneracy protected by the subsystem symmetries. In this way SSPT’s are a subsystem generalization of (global) symmetry protected topological phases Devakul et al. (2018).
Subsystem symmetries have also been studied in connection to fractonic phases of matter Chamon (2005); Haah (2011); Nandkishore and Hermele (2018). Fracton systems are 3+1D phases of matter, characterized by immobile excitations, or excitations which are confined to sub-dimensional regions. It has been found that gauging a subsystem symmetry can lead to a fractonic phaseBravyi et al. (2011); Yoshida (2013); Williamson (2016); Vijay et al. (2016); You et al. (2018b); Vijay et al. (2016). Since fracton systems are believed to be described by rank 2 symmetric gauge theories, this field has also gained attention due to possible connections to elasticity and gravity theories Pretko (2017); Pretko and Radzihovsky (2018).
Currently, the study of subsystem symmetries has been largely based on viewing a -dimensional subsystem symmetry as a global symmetry acting on -dimensional subregions. However, there is also a complimentary view of a -dimensional subsystem symmetry as a gauge symmetry acting on a -dimensional subregion. For example, consider a plane with coordinates , where a subsystem symmetry acts along (const.) lines. Restricted to lines, the subsystem symmetry is a global symmetry. However, for (const.) lines the subsystem symmetry is a local/gauge symmetry, since it only acts at the point .
Since subsystem symmetries behave like gauge symmetries in certain subregions, we believe that salient features of lattice gauge theories may occur in systems where the low energy physics is invariant under a subsystem symmetry. In particular we ask if subsystem symmetries can lead to interesting global phenomena in the same way that gauge symmetries do in topologically ordered phases. We answer this question in the affirmative by using a model of bosons with a subsystem symmetry. Using two complimentary descriptions, we show that this model has multiple ground states on a torus, which cannot be locally distinguished. Furthermore, we show that for a rectangular system with open boundaries, there are gapless degrees of freedom that are localized to the system’s corners.
This paper is organized as follows. In Section II, we construct the subsystem symmetry invariant model by using a coupled wire construction. In Section III we construct an effective projector Hamiltonian and use it to study the system. In Section IV we construct and analyze a continuum description of the subsystem symmetry invariant model. In Section V we generalize the continuum description and discuss its features. Finally, we discuss and conclude these results in Section VI.
Ii Subsystem Symmetry Invariant Model
To construct our subsystem symmetry invariant model, we consider an array of complex bosonic wires on a square lattice with unit directions and . The Hamiltonian for the wire array with wires aligned parallel to the -direction is given by
where is a complex valued boson, and is a chemical potential. For a lattice, this model has symmetries which correspond to rotating the phase of a given wire. Formally, this symmetry operation is given by , where , and is a real function that is constant along the direction (). The factor of included in this definition is necessary for this system to have non-trivial features.
We now want to couple these wires in such a way that the subsystem symmetries are preserved. To do this, we will introduce a new set of fields defined on the links that connect sites r and . These fields transform as under the subsystem symmetries. Introducing these fields, the Hamiltonian becomes
This model now has subsystem symmetries given by and , where is a real function that is constant along the direction. The coupling in Eq. 2 can be viewed as a subsystem generalization of a gauge connection, i.e., a way of coupling the bosons such that the subsystem symmetry is preserved. This coupling has also introduced vortex configurations where the value of jumps by . The term proportional to adds an energy cost to creating these vortices. Since the terms only couple fields that are neighbors in the -direction, these vortex excitations can only propagate along the -direction.
To gain more insight into this Hamiltonian, let us restrict our attention to a line along the direction defined as where is a constant. Let us extract the section of the Hamiltonian that acts only on . The resulting Hamiltonian for this subregion is
where . This is exactly the Hamiltonian for charge bosons coupled to a gauge field . The gauge transformations are given by and . This is exactly the subsystem transformation of the full system restricted to the line. So, along the subregion, the subsystem symmetry corresponds to a gauge symmetry.
Motivated by this, we can consider the expectation value of the Wilson loops of the dimensionally reduced system . For periodic boundary conditions, the expectation value of can be changed by a factor of by threading a unit of flux through the system. In terms of the fields, the flux threading sends , for each . In the full system, becomes the operator . This operator is invariant under the subsystem symmetries of Eq. 1. For periodic boundaries in the direction, we can also define a ”flux insertion” operation that sends for each r. This will change the expectation value of by a factor of .
It is clear that is similar to the Wilson loops of a lattice gauge theory. To illustrate the similarities and differences between lattice gauge theories and Eq. 2, let us consider these systems on a torus. For a lattice gauge theory there are two distinct non-contactable Wilson loops: one oriented in the direction, and one oriented in the direction. The expectation value of these loops can be changed by threading flux through the or directions respectively. However, for Eq. 2, the Wilson loop-like operator is fixed to be oriented in the direction. As a result, the system only responds to threading flux through the direction. Motivated by this, it will prove useful to think of Eq. 2 as a gauge theory where the Wilson loops are restricted to be oriented in the direction, or equivalently where flux can only be inserted in the direction.
Now let us tune such that there is a large boson occupancy per site. can then be replaced with the rotor variable , where corresponds to the phase of the complex boson Phillips (2012). The Hamiltonian then becomes
The subsystem symmetry is now given by , and where is constant along the direction. This model is the main result of this section.
It is worth noting that due to the generalized Elitzur’s theorem Batista and Nussinov (2005), the continuous subsystem symmetry of Eq. 4 cannot be spontaneously broken. So the ground state of Eq. 4 must be invariant under under all subsystem symmetry transformations, as must all local observables. This is similar to gauge theories, where the ground state and local observables must also be invariant under all local gauge transformations.
Iii Effective Projector Hamiltonian
To better study Eq. 4, it will be useful to construct an effective description in terms of an exactly solvable model of commuting projectors. The resulting model will be non-local, however it will be useful to determine key features of Eq. 4 such as ground state degeneracy, and edge physics. In Section IV, we will rederive these results using a local continuum description of Eq. 4.
We will consider the case where while remains finite. The low energy excitations will thereby be violations of the term proportional to (vortices of ) in Eq. 4. To be explicit, let us consider an effective description for . The vortices of will therefore be -vortices, where . In the large limit we can rewrite as
where is a -valued variable ( only takes on values of or ) that corresponds to the vortices of the field. Let us now examine how these fields transform under a subsystem symmetry transformation given by satisfying . It will be useful to decompose , where takes on values in and is a -valued function. Under such a transformation
where we have used the fact that is periodic. Comparing Eq. 5 and 6, we see that the transformation law for is . So is only acted on by transformation generated by . Since , the transformations generated by form a subgroup of the full group of subsystem symmetry transformations.
The aforementioned subsystem symmetry generated by flips the spins on an even number of columns. In terms of the spin variables, this symmetry transformation is generated by , where (see Fig. 1).
The full Hilbert space of Eq. 7 is spanned by . These are eigenstates with eigenvalues () and (). In the limit, we will only consider states that satisfy . Using this the Hamiltonian becomes
In this limit the phase fluctuations are frozen out energetically and the effective model acts on the restricted Hilbert space spanned only by the spin operators . Formally this is a mapping that takes a state
Additionally, due to the generalized Elitzur’s theorem, all observables must be invariant under the subsystem symmetries. Because of this, we should focus on just the ”physical subspace” of this reduced Hilbert space, which consists of states that are invariant under the subsystem symmetries generated by . Under the aforementioned mapping, the physical subspace of the full Hilbert space maps to a subspace of the restricted Hilbert space that is invariant under the subsystem symmetry subgroup that acts on . To project the restricted Hilbert space onto the corresponding physical subspace, we note that a subsystem symmetry invariant state will satisfy for all columns . This condition can be enforced in the low-energy subspace by adding the term (with ) to the Hamiltonian Eq. 8. The resulting effective projector Hamiltonian is
The low energy sector will now be invariant under the subsystem symmetry. The second term in this Hamiltonian is notably non-local. This is an artifact of projecting to the physical Hilbert space. Nevertheless, this effective model provides a simple and useful description that we can use to study the low energy features of the full system Eq. 4.
It will now be useful to simplify the lattice on which we have defined this effective spin model. Let us define a new lattice such that the sites of the new lattice are the links connecting the sites r and of the original lattice. This means that the fields now live on sites instead of links. The new lattice is shown Fig. 2. After switching to the new lattice the Hamiltonian simplifies to
where r are the sites on the new lattice, and and are now the unit directions of the new lattice. is now the set of spins along a given straight line in the direction.
This spin model is the main result of this section. All terms in the Hamiltonian commute, and so the spin model is exactly solvable. The subsystem symmetry here is generated by
This operation is shown in Fig. 3. As we can see, the non-local second term in Eq. 11 guarantees that the ground state of the system is invariant under this transformation. Eq. 11 also has a second subsystem symmetry generated by
where is a line of spins in the direction. Due to the first term in Eq. 11, the ground state will be invariant under this second subsystem symmetry as well.
Indeed, the quantum compass model and the spin model Eq. 11 share the same subsystem symmetries, and Eq. 11 can also arise as the effective description of the phase of Eq. 14 in finite sized systems. In this case, the effective will be proportional to . However, despite the apparent similarities, these models have different ground state properties in the thermodynamic limit. It is known that the quantum compass model has 2 phases corresponding to and Dorier et al. (2005). In both phases, the number of ground states scales as for an system. The point marks a first order phase transition that connects these two phasesChen et al. (2007). In contrast, the spin model Eq. 11 has a gapped phase with a finite number of ground states, even in the thermodynamic limit. This will be shown in the following sections.
iii.1 Ground States and Excitations
The ground state of the effective spin model Eq. 11 can be found by minimizing each of the commuting terms. We can intuitively understand the nature of the ground state in the following way. The terms proportional to in Eq. 11 describe an array of decoupled Ising chains. Thus, for , the spin model is simply an array of Ising chains in the ferromagnetic phase. In the low-energy subspace, each chain can then be characterized by a single magnetization variable .
The terms proportional to in Eq. Eq. 11 flip all spins on a pair of the neighboring Ising chains (see Fig. 3), i.e., each term flips a pair of magnetizations, e.g., and . Let us define the operator . Since , . In terms of , the Hamiltonian Eq. 11 becomes
This Hamiltonian is just another ferromagnetic Ising chain, with the ferromagnetism oriented in the -direction. So the effect of the term proportional to in Eq. 11 is to orient the magnetization of the original Ising chains. In particular, if we start with a ground state for , we can determine the ground state for by acting on the ground state with the operator
To see this, consider a state that minimizes Eq. 11 with . Then for all r. Since , minimizes all terms in Eq. 11. It is also true that , for all and by extension, . So also minimizes all terms in Eq. 11. thereby minimizes the entire Hamiltonian with and is the ground state.
We note here that is in fact exactly the projection operator that projects the restricted Hilbert space of Eq. 8 to the subsystem symmetry invariant physical subspace of the restricted Hilbert space. As we shall demonstrate below, the number of ground states will depend on the topology of the lattice. The excited states of the spin model are characterized by having either or , which have an excitation energy of and respectively.
iii.2 Ground State Degeneracy
A key feature of the subsystem symmetry invariant model Eq. 4 is the existence of multiple ground states that cannot be locally distinguished. We will demonstrate this by considering the effective spin model Eq. 11 on a torus. To find the number of ground states, we will identify operators that commute with the Hamiltonian, and use them to label the degenerate ground states. The non-trivial operators that commute with the Hamiltonian Eq. 11 are
where is a closed loop in the direction, and is a closed loop in the direction. On a torus, the and loops will be the two cycles that define the torus. These loops are shown in Fig. 4. For an torus, the total number of operators is and the number of operators is . Since , both loop operators are operators. We can identify the symmetry operator as the product of the neighboring loop operators and , and similarly identify as the product of and .
All loops and on the torus intersect once, and so all operators anti-commute with all operators. The minimum dimension needed to represent this anti-commuting algebra is 2, leading to 2 distinct ground states. If we were to diagonalize the ground state subspace to label them by their eigenvalue, then the 2 ground states would be related by the acting on a given ground state with an operator . Since and are both non-local operators, this degeneracy is robust to local perturbations.
The degeneracy can also be found by counting the number of constraints for spins on a torus. Let us first consider the terms proportional to in Eq. 11. These terms describe a system of Ising chains with periodic boundaries. Each chain contributes unique constraints, leading to unique constraints from the terms in Eq. 11. The terms proportional to in Eq. 11 then give unique constraints. Since all terms in Eq. 11 commute, all these constraints can be simultaneously satisfied, leading to constraints in total. There is thereby net free spin degree of freedom which corresponds to the ground states that were previously identified.
It is useful to compare these results to the case of a lattice gauge theory on a torus. In lattice gauge theory models, there are 2 additional ground states on a torus (for a total of 4 ground states)Kitaev (2006). These 2 additional ground states occur since non-contractible loops of operators oriented in the direction, and non-contractible loops of oriented in the direction also commute with the lattice gauge theory Hamiltonian, and anti-commute with each other. These operators do not commute with the spin model Eq. 11, and so the number of ground states is reduced to 2.
On a sphere all string operators and commute, and so the ground state of Eq. 11 on a sphere is unique. We also show this explicitly in Appendix A by counting constraints. This topology-dependent degeneracy is reminiscent of the topological ground state degeneracy found in topological ordered systems, though it is important to note that our spin model has a non-local constraint. The non-locality will be removed when we discuss the continuum limit.
iii.3 Open Boundaries and Corner Modes
We shall now consider the system with open boundaries. For simplicity, we shall take the lattice to be an rectangle with open boundaries. For this geometry, the terms proportional to in Eq. 11 give constraints, and the terms proportional to give constraints, leading to constraints which can be simultaneously satisfied. There is then a single free spin 1/2 degree of freedom, leading to 2 ground states.
In the string picture, this can be seen by the anti-commutation between the zero energy operators and from Eq. 17 where (resp. ) is now a string in the (resp. ) direction stretching from one boundary to the other. Since the system has open boundaries, the string operators do not have to form closed loops to commute with the Hamiltonian and be invariant under the subsystem symmetries of the model. Since all operators anti-commute with all operators there are degenerate 2 ground states.
Furthermore, these 2 ground states correspond to anti-commuting corner degrees of freedom. To show this, we will switch to a Majorana representation of the spin-1/2 degrees of freedomKitaev (2003). This is done by introducing 4 Majorana degrees of freedom at each site , , , and . The spin degrees of freedom then become , , and , with the local constraint that . Setting , the spin model can be described by the mean field Hamiltonian
where corresponds to the Majorana dimerization pattern given in Fig 5. The construction of this mean field Hamiltonian is outlined in Appendix B. The ground state of the spin model can then be found by projecting the mean field ground state of this Majorana Hamiltonian onto the physical states using the projector , and then using the above identification between the spin operators and Majorana fermions.
For the rectangular geometry, there are 4 free Majorana degrees of freedom located at the corners of the lattice (see Fig. 5). This leads to 4 ground states, but only 2 are physical after projecting onto the physical states. The reduction in ground states can be viewed as a consequence of the full model being bosonic, i.e., fermion parity even. For the rectangular geometry, the degrees of freedom for the spin model are thereby zero energy corner operators leading to a robust ground state degeneracy. We note that this particular dimerization pattern and corner mode configuration is similar to an insulator model presented in Ref. Benalcazar et al., 2017a that has corner charge, but vanishing quadrupole moment. Indeed, the corner modes considered here are similar to what is found in higher order topological insulatorsBenalcazar et al. (2017b); Langbehn et al. (2017); Schindler et al. (2018); Khalaf (2018). However, for the spin model Eq. 11, the corner operators are non-local. This is because individual Majorana corner operators do not commute with the projector operator . Only pairs of Majorana corner operators commute with and are physical.
Iv Continuum Theory
where fields are now defined along both and oriented links, and are neighboring sites. The sum over is over plaquettes with corners and . In the low energy () limit, is pinned to be by the cosine term. Upon substituting this into Eq. 19, the Hamiltonian reduces to Eq. 4 with . This model has the subsystem symmetry given by , and where . This is the same symmetry as in Eq. 4. We note that is invariant under these transformations. Eq. 19 is the Hamiltonian for bosons minimally coupled to a vector field with an additional mass term for the fields oriented along the direction. It is worth explicitly stating that this model is not a gauge theory due to the additional mass term for .
The continuum description of Eq. 19 in Euclidean space is
where is a constant, and we have included a current that couples to the fluctuations of the field. To study the dynamics of the phase , we will introduce the variable , and shift . After this Eq. 20 becomes
where we have introduced . Since the field is now massive, it can be integrated out, leaving a theory just in terms of . The field is now also coupled to , causing the excitations of the field to have a corresponding current . As desired, this model has a subsystem symmetry where and is function of and only.
From the equations of motion for , we have that . As a result, in the low energy limit where , vanishes. In this limit, (the current in the direction) is removed from the theory, and the only current is in the direction (). This means that the excitations of the field will only move in the direction. This is a form of sub-dimensional dynamics, where the excitations are only able to move in certain lower dimensional subregions.
We will also need to consider the vortex dynamics of the field. This is done by introducing a vortex current , and setting . To enforce this constraint, we will introduce the field as a Lagrange multiplier for Eq. 20
In this construction, there are vortex currents in both the and directions ( and ). However, in the original lattice model Eq. 4, the vortices were only able to move in the direction. To remedy this, we will add the term . The equation of motion for then gives that , and in the low energy limit () . In this limit, the vortex current is removed from the theory, and there is only a vortex current in the -direction (). As a result, the vortices of the field are confined to move in the -direction as in the lattice model.
After adding the field, integrating out the massive field, and keeping only the long wavelength contributions, the Lagrangian density becomes
This model is the main result of this section. It is worth noting that this theory has the form of a mutual Chern-Simons theory with additional mass terms for and Bardeen and Stephen (1965); Hansson et al. (2004); Diamantini et al. (2006). This observation will be allow us to generalize this model in Section V.
In Eq. 23, it is also apparent that there is a second subsystem symmetry where and is only a function of and . This is the same as the subsystem symmetry generated by (Eq. 13) in the effective projector Hamiltonian.
iv.1 Ground State Degeneracy
We will now calculate the ground state degeneracy of the continuum model on a torus. To do this, we will first rotate back to Minkowski space, and set the currents ,
From the equations of motion for and , we have that . At low energies, and , and the action becomes
If we minimize the action with respect to and we find the equations of motion and . On a torus, these equations of motion are solved by
Here is a function of and only and is periodic on the torus, is a function of and and is periodic on the torus, and and are functions of only. are the length dimensions of the torus.
After substituting these terms into Eq. 25 and integrating over the and coordinates, the action reduces to
Using canonical commutation relations, we have that . Since and are periodic variables, the observables are and , which obey the commutation relationship,
In order to satisfy this operator algebra, there must be ground states. This is consistent with what was found using the effective projector model with . We note that for a conventional mutual Chern-Simons theory the ground state degeneracy would be
iv.2 Corner Modes
To find the edge degrees of freedom for a system with open boundaries we will use the low energy description with no external currents in Minkowski space (Eq. 25). For a rectangular system with open boundaries, the equations of motion for and are solved by
Using this, the action becomes
which is a total derivative for both and . If the system is defined on the rectangle and , the action becomes
This action describes localized operators , which are defined at the corners of the system . Since the Hamiltonian corresponding to the action Eq. LABEL:eq:CornerAct vanishes, the are zero energy operators. It should be noted that there is a redundancy in the corner mode description, since .
Using the canonical commutation relationships from Eq. LABEL:eq:CornerAct, the operators satisfy the algebra
Naively this would lead to ground states. However, if the constraint is taken in account there are actually only ground states. This agrees with what was found in using the effective model for . As opposed to Abelian Chern-Simons field theories, where the edge theory is a CFT Balachandran et al. (1991); Dunne (1999); Wen (2004); Fujita et al. (2009); Lu and Vishwanath (2012), the edge theory of the subsystem symmetry invariant model is given by corner modes.
V Generalized Continuum Theory
To generalize the continuum description to include more vector fields, we note that Eq. 23 has the form of a Chern-Simons field theory with -matrix