Supersymmetric quantum field theory with exotic symmetry in 3+1 dimensions and fermionic fracton phases

We propose a supersymmetric quantum field theory with exotic symmetry related to fracton phases. We use superfield formalism and write down the action of a supersymmetric version of the $\varphi$ theory in 3+1 dimensions. It contains a large number of ground states due to the fermionic higher pole subsystem symmetry. Its residual entropy is proportional to the area instead of the volume. This theory has a self-duality similar to that of the $\varphi$ theory. We also write down the action of a supersymmetric version of a tensor gauge theory, and discuss BPS fractons.


Introduction and summary
Recently, fracton phases are attracting a lot of attention. For review, refer to [1,2] and references therein. Such a system shows exotic properties, such as sub-extensive entropy, local particle-like excitation with restricted mobility, and so on. There are two approaches to fracton phases: solvable lattice models and continuum description. In this paper, we investigate continuum description.
One of the most important developments in this topic is the discovery of higher pole subsystem symmetries. It has been pointed out by [3,4] that higher pole symmetries play a key role in the exotic properties of fracton phases. Higher pole symmetries have been further investigated by a lot of papers including [5,6,7,8,9,10,11,12,13]. Higher pole symmetries are also important in this paper.
One open question raised in [2] is whether there are intrinsically fermionic fracton phases. We want to approach this question from the perspective of continuum description. However, we do not have nice guiding principles, since fracton systems do not have the Lorentz symmetry nor continuous spacial rotational symmetry. Therefore, in this paper, we propose supersymmetry that relates bosons and fermions. We employ this supersymmetry as a guiding principle to introduce fermions to the system. We expect that the fermionic part of this supersymmetric theory is a natural fermionic fracton system.
In this paper, we first propose a supersymmetrization of the theory [7,13] in 3 + 1 dimensions. Let us explain this theory in our notation. Let and ì = ( , , ) be time and space coordinates, respectively. We impose a periodic boundary conditions in the , , directions for all fields throughout this paper. It is convenient to introduce the differential operators where is a real positive constant. The theory includes a single scalar field with the periodicity ∼ + 2 . The Lagrangian density of the theory is given by where 0 is a real positive constant. This theory has momentum and winding quadrupole symmetry and exhibits a self-duality as shown in [13].
The strategy for supersymmetrization of the theory in this paper is as follows. The Lagrangian density (1.2) resembles to that of (1 + 1)-dimensional massless free scalar theory in which ± are replaced by the derivatives in light-cone directions. Therefore we try to supersymmetrize the theory in the same way as 1 + 1 dimensions as if ± were the derivatives in the light-cone directions. We employ the superfield formalism. In this paper, we only consider an analogue of N = (1, 1) supersymmetry in 1 + 1 dimensions.
Here is the short summary of the results of this paper. Our strategy works for some cases including the supersymmetric version of the theory. We find that the supersymmetric theory is obtained by adding "the theory" ( ± : real fermionic fields.) (1.3) to the theory (1.2). Therefore we claim that this theory is a natural intrinsically fermionic fracton system. In the supersymmetric theory, ± are the superpartners of the quadrupole symmetry currents. Therefore they satisfy the conservation laws which are nothing but the equations of motion derived from (1.3). We find that fermionic quadrupole charges ∮ are conserved by the same argument as usual higher pole symmetries. Due to these fermionic charges, the residual entropy or log of the ground state degeneracy is proportional to the area of the system instead of the volume. We also show that our supersymmetric theory exhibits the self-duality, as the theory does [13]. We use our superfield formalism to show this self-duality.
We also formulate the supersymmetric tensor gauge theory that appears in gauging the global part of quadrupole symmetry of the supersymmetric theory. This is the supersymmetric version of the tensor gauge theory considered in [7,8,13]. The multiplet of the gauge field includes two real fermions ± and a real scalar in addition to the tensor gauge field 0 , . There are many future issues that may lead to some intriguing results. One issue is supersymmetry as a subsystem symmetry. Unfortunately, our supersymmetry in this paper is only global supersymmetry. It would be very interesting if one finds subsystem supersymmetry by improving our result.
Another issue is a lattice fermionic system described by the theory (1.3) in the low energy limit. Lattice supersymmetry is a very difficult problem, but it may be feasible to find a lattice realization of the fermionic part. We make an attempt in appendix A. There have been a few studies of fermionic fracton phases from lattice models [14,15,16].
There are several other issues. One is constructing interacting supersymmetric theory, which is not possible in the formulation in this paper. It will be also interesting to construct theories with extended supersymmetry, for example, N = (2, 2) and N = (2, 0). Finally, the analysis of fractons in our supersymmetric system is also a big issue.
The construction of this paper is as follows. In section 2, we use superfields to formulate the supersymmetric theory. We discuss how to write down the supersymmetric action. We also show that this supersymmetric theory has a self-duality. In section 3, we consider supersymmetric tensor gauge theory. We write down the action by using the superfield formalism. We discuss BPS defects as fractons. In appendix A, we give an fermionic lattice model. We count the ground state degeneracy and show the residual entropy is proportional to the area instead of the volume.

Superfields and supersymmetric theory
In this section, we introduce superspace and superfields in order to write down the supersymmetric action. In particular, we write down the supersymmetric theory. This theory has fermionic quadrupole symmetry in addition to bosonic quadrupole symmetry of the theory. We show this supersymmetric theory has a self-duality.
We introduce real fermionic coordinates + , − in addition to the spacetime coordinates , ì = ( , , ). Then we define the following derivatives in order to describe the supersymmetry transformation.
where ± = 1 2 ( ± ) are differential operators of (1.1). Then the anti-commutation relation between them are given by The last anti-commutation relation is important when we write down the action. A real superfield is written as , ± , are fields in the spacetime called "components." If Φ is bosonic, and are real bosonic fields and ± are real fermionic fields. On the other hand, if Φ is fermionic, and are real fermionic fields and ± are real bosonic fields.
We define the supersymmetry transformation by where ± are infinitesimal fermionic parameters of the transformation. In terms of components, the supersymmetry transformation is written as (2.5) Let us call Φ a "superfield" if it is transformed as (2.4) by the supersymmetry transformation.
If Φ is a superfield, the derivatives D ± Φ are also superfields, since Q ± and D ± anti-commute to each other, in the same way as usual supersymmetry. On the other hand, if Φ 1 and Φ 2 are superfields, the product Φ 3 := Φ 1 Φ 2 is not a superfield. In other words, Φ 3 does not follow the transformation law (2.4) due to the third-order derivatives in Q ± . This property is quite different from usual supersymmetry and the main obstacle in our formulation. Before constructing the action, let us look at the supersymmetry algebra. Suppose we have a theory invariant under the transformation (2.4). This theory includes supersymmetry generators ± that satisfy the relation (2.6) Then, we find anti-commutation relations of ± given by 1 Here is the Hamiltonian, and is the generator of the transformation where is a infinitesimal bosonic parameter. Notice that is different from the product of momenta . We conclude that this supersymmetry does not exist unless the theory is invariant under the transformation (2.8).

The action of the supersymmetric theory
The difficulty in our formulation is that, unlike the ordinary supersymmetry, the product of superfields does not become a superfield. In spite of this difficulty, the following theorem allows us to write the action of a free field theory. Theorem 1. Let Φ 1 , Φ 2 be superfields which are transformed by the transformation law (2.4). Then is a total derivative. Here integral ∫ 2 is defined by Proof. Because of an identity = (total derivative), (2.11) the differential operator Q in (2.4) satisfies the relation Therefore the left-hand side of (2.9) becomes = (total derivative in spacetime). (2.13) Let us write down the action of the supersymmetric theory with the help of theorem 1. The field in this theory is a real bosonic superfield Φ with periodicity Φ ∼ Φ + 2 . Φ is expressed by components as 14) The periodicity of Φ implies the periodicity of the component as ∼ + 2 . The Lagrangian density is written as where 0 is a real parameter. The supersymmetry transformation of this Lagrangian density becomes a total derivative according to theorem 1. Therefore this theory has the supersymmetry. The Lagrangian density is written by components as One can integrate out the auxiliary field . Then, the theory is the sum of the theory (1.2) and the theory (1.3).

Quadrupole symmetry current and the ground state degeneracy
In our supersymmetric theory, we have momentum and winding quadrupole symmetry currents as superfields. Let us consider superfields where we use equations of motion in the expression by components. Since the equations of motion can be expressed as D + D − Φ = 0, these currents satisfy the conservation law By the same argument as the -theory [13], we find that ∮ ± , are all conserved. Actually, the bosonic charges of this theory are nothing but the momentum and winding quadrupole charges of the theory [13]. The momentum and winding quadrupole charge density in [13] are expressed, respectively, as Besides these bosonic charges, we also have the same number of fermionic quadrupole charges in our theory. Let be the number of bosonic charges. By the argument of [13], is proportional to the area if we regularize the theory by the lattice. Let us assume that we have a regularization that preserves the supersymmetry. Then we have the same number of fermionic charges. Let , ( = 1, . . . , ) be these fermionic charges. We can choose the basis so that the canonical anticommutation relations become (2.21) This is nothing but the Clifford algebra. Since these 's commute with the Hamiltonian, the space of ground states must be a representation space of this Clifford algebra. Therefore the ground state degeneracy is 2 [ /2] . We conclude that the residual entropy [ /2] log 2 shows a sub-extensive behavior; it is proportional to the area instead of the volume.
In appendix A, we consider a fermionic lattice model that may be a regularization of the theory. We find that the number of fermionic charges and the residual entropy are proportional to the area.

Duality
Here we will derive the supersymmetric version of self-duality given in [13]. As a warm up, we rederive the self-duality of the theory in our notation. We start from the Lagrangian density where , ± ,˜ ± are real bosonic fields. has periodicity ∼ + 2 .
If we integrate out˜ ± in (2.22), we obtain constraints˜ ± = ± . Then we integrate out ± and obtain the theory (1.2).
On the other hand, if we first integrate out ± , we obtain Then we integrate out and obtain the constraint as a necessary condition. This constraint can be solved by introducing a real bosonic field˜ as where is a constant that is determined so that˜ is 2 periodic. Let us determine . Besides the constraint (2.24), we obtain some additional constraints from the 2 periodicity of . Consider the Fourier modes of the fields The 2 periodicity of implies only the periodicity of the zero mode ì 0 ∼ ì 0 + 2 . Let us focus on this zero mode. The part of the action including this ì 0 is where is the volume of the space = ∫ 3 ì 1. The canonically conjugate momentum of ì 0 is given by Because of the periodicity of ì 0 , the eigenvalues of ì 0 must be integers. From the relation (2.25), we obtain˜ Finally we obtain the dual Lagrangian This is just the same result as [13]. Notice that in [13] is expressed as = 1 0 2 . Next let us turn to the duality of supersymmetric theory. Starting from the action where ± , ± are real fermionic superfields and Φ is a real bosonic superfield with the periodicity Φ ∼ Φ + 2 .
If we first integrate out ± , we obtain the constraint ± = D ± Φ. Then we integrate out ± and obtain the supersymmetric -theory (2.16).
On the other hand, if we first integrate out ± , we obtain Then we integrate out Φ and obtain the constraint which is solved in terms of a bosonic superfield Φ as The coefficient is determined so that Φ has the periodicity Φ ∼ Φ + 2 . Finally we obtain the dual theory as Notice that we only use the quadratic actions during this procedure, and therefore the supersymmetry is manifest.

Supersymmetric tensor gauge theory
In this section, we supersymmetrize the tensor gauge theory. We write down the supersymmetric tensor gauge theory action. We also discuss BPS Wilson line defects and fractons.

Gauge superfields
Let us consider gauging the shift symmetry of Φ in the supersymmetric theory(2.16). The parameter of the shift symmetry is promoted to a real bosonic superfield with periodicity ∼ + 2 . The gauge transformation of Φ is given by Let us introduce real fermionic superfields Γ ± and the super covariant derivatives ∇ ± by The gauge transformation law of Γ ± are determined so that ∇ ± Φ are gauge invariant, and given by Let us look at the components of Γ ± and their gauge transformation. First, let us denote the components of as where , are real bosonic fields and ± are real fermionic fields. We also denote the components of Γ ± as Γ + = + − 2 + + + − ( + ) − 2 + − ( + + + − ), where , , ± are real bosonic fields, and ± , ± are real fermionic fields. Notice that these are the most generic form of two real fermionic superfields. The gauge transformations (3.3) of the components read We can construct the gauge invariant superfield Σ from Γ ± as In order to find the component expression of Σ, it is convenient to choose Wess-Zumino (WZ) gauge In this gauge, Γ ± are expressed as We obtain the expression of Σ in terms of components as Here let us mention the relation to the tensor gauge field in [7,8,13]. The gauge fields ± and the field strength +− in this paper are related to the gauge fields 0 , and the field strength in [13] as Thus the superfield Γ ± are supersymmetry completions of the tensor gauge fields 0 , .

Gauge theory action
Here we construct the supersymmetric gauge theory action. The gauge field part is written by the gauge invariant superfield Σ and consists of the kinetic term and the potential term. The potential term must be quadratic polynomial or linear function in order to preserve the supersymmetry in our formalism. The kinetic term in the Lagrangian density is written as (4 + − + 2 + − + + 2 − + − + (2 +− ) 2 ), (3.12) where is a real coupling constant. We find that this theory includes a boson with the same kinetic term as the theory(1.2), and ± that are a copy of the theory (1.3) in addition to the gauge theory kinetic term. The quadratic term is written as where is a real constant. This term includes axion-like interaction between and ± , as well as a fermion mass term. The linear term is the theta term where is a real constant with the periodicity ∼ + 2 .
We can add the supersymmetric theory as a matter to this gauge theory. The matter part of the action is obtained by replacing the derivatives D ± with covariant the derivatives ∇ ± as One can expand this action by components and obtain the fully gauge invariant action by components. Instead of writing down the gauge unfixed action in components, we choose WZ gauge (3.8) and write down the action in components. In WZ gauge the matter part of the action is given by In this action, ± couple to the fermionic quadrupole currents ∓ , and therefore one can interpret that ± are fermionic analogs of the tensor gauge fields. Finally, let us make some comments on the supersymmetry of the action (3.16) in WZ gauge. Since the supersymmetry transformation (2.4) breaks WZ gauge condition (3.8), the matter action (3.18) One can explicitly check that ′ transformation of L matter is a total derivative.

BPS defects as fractons
We have a BPS defects in the supersymmetric tensor gauge theory constructed above. Let us consider the Wilson line exp ∫ ( + + − + ) . This Wilson line is one in [13] dressed by . This Wilson line is invariant under the gauge transformation (3.3). It is also invariant under a half of the supersymmetry transformation with parameters − = + , and therefore we call it as a BPS Wilson line. This Wilson line cannot move because of the gauge symmetry. Therefore this Wilson line describes the probe limit of a fracton.
In [13], it has been shown that once four of these defects form a quadrupole, it can move collectively. This is also true in our defect in our supersymmetric theory. However, all the supersymmetry is broken if the quadrupole moves even if the velocity is constant. It is an interesting future problem to find a nice dressing to make the moving quadrupole BPS, or some no-go theorem. We conclude that the residual entropy is proportional to the area instead of the volume of the system.