Spontaneously broken supersymmetric fracton phases with fermionic subsystem symmetries

We construct a purely fermionic system with spontaneously broken supersymmetry that shares the common feature with a fracton phase of matter. Our model is gapless due to the Nambu-Goldstone mechanism. It shows a ground-state degeneracy with the"Area-law"entropy due to fermionic subsystem symmetries. In the strongly coupled limit, it becomes a variant of the Nicolai model, and the ground-state degeneracy shows the"Volume-law"entropy. Gauging the fermionic subsystem symmetry has an t'Hooft anomaly by itself, but the would-be gauged theory may possess a fermionic defect that is immobile in certain spatial directions.


Introduction
Ordinary matter has extensive entropy at finite temperature, which is proportional to the whole volume of the system, while it has zero entropy at zero temperature, satisfying the third law of the thermodynamics. Certain peculiar matter (e.g. in the glassy phase) shows non-zero entropy at zero temperature that is proportional to the volume. There exists, however, even more peculiar matter which shows the "Area-law" entropy at zero temperature. One of the most well-known, yet mysterious, such examples is a black hole [1]. 1 Another example is a ground-state entropy of the fracton phase of matter [2,3,4,5].
A fracton is an extraordinarily peculiar excitation and has attracted a lot of attention these days. See e.g. [6,7,8] for reviews.
In the fracton phase of matter, the origin of the "Area-law" entropy can be attributed to the so-called subsystem symmetry [9,10,11,12,13,14,15,16,17,18,19]. The Noether charge for the ordinary symmetry (in d dimensional space) is obtained by integrating the charge density over the whole volume: Q 0 = d d x ρ while that for the subsystem symmetry is obtained by integrating the charge density only over codimension q subsystems Q q = d d−q x ρ. A subsystem symmetry can be regarded as infinitely many copies of ordinary symmetries and it results in a huge degeneracy in the energy spectrum of the system. Although many realizations of subsystem symmetries have been studied (both in lattice models and in continuum field theories) in the context of fracton physics, a construction of a fermionic fracton has been a challenging issue. In [24], they proposed a supersymmetric field theory that has a fermionic subsystem symmetry. In this paper, we construct a purely fermionic system with spontaneously broken supersymmetry that shares the common feature with a fracton phase of matter 2 .
Our model has one parameter, and the weak coupling limit gives a tight-binding model of free fermions with a peculiar dispersion relation E = |k x k y | with respect to the wavenumber k x and k y . In the strongly coupled limit, our model becomes a two-1 Uncharged black holes have zero entropy at zero temperature satisfying the third law of thermodynamics, but charged black holes have zero-temperature entropy that is proportional to the area of the horizon. 2 Note that our effective field theory description, similar to those in [20,21,22,24,26,25], will not include a "fracton" as a dynamical degree of freedom, but it appears as a probe defect.
dimensional variant of the Nicolai model [27]. The Nicolai model is a supersymmetric lattice model whose Hamiltonian is constructed purely out of fermionic creation and annihilation operators. The ground states on the chain are exponentially degenerate with respect to the system size, and the structure has been extensively studied in the literature [28,29,30]. When the coupling constants are randomly generated, it may also be regarded as a supersymmetric analogue of the Sachdev-Ye-Kitaev model [33,34,35,36]. Our model therefore interpolates a fracton phase of matter and a higher-dimensional variant of the Nicolai model.
The organization of the paper is as follows. In section 2, we construct a purely fermionic system with spontaneously broken supersymmetry that shares the common feature with a fracton phase of matter as a continuum field theory. In section 3, we present a lattice regularization and study the ground state degeneracy. In section 4, we discuss some related models. In section 5, we study gauging of the fermionic subsystem symmetry in relation to the defect operators. In section 6, we conclude with some discussions. The paper has two appendices. In Appendix A, we review the superfield formalism, and in Appendix B, we present some general aspects of gauging shift symmetries.

Continuum Model
Let us consider a continuum field theory in d = 1 + 2 dimensions (t, x) = (t, x, y) with a complex supercharge where χ and ψ are complex fermions with the canonical (equal-time) anti-commutation We define the supersymmetric Hamiltonian as the anti-commutator of the complex supercharges: where H int the four-fermi interaction: By construction, this model is supersymmetric although we have only fermionic canonical fields. It turns out that the supersymmetry Q (as well as Q † ) is spontaneously broken when g = 0. There, a superpartner of a fermionic particle is a pair of the fermionic particle itself and a Numbu-Goldstnoe fermion.
This model has a fermionic subsystem symmetry (or kinematic supersymmetry): where the fermionic parameter ξ(x) depends only on the x coordinate but not on the y coordinate. The corresponding charges are given by where the integration is over only the y coordinate. These charges are parametrized by the coordinate x, and they are all conserved because ∂ t ψ = ∂ y (...) from the equations of motion. In the infinite g limit, our model has another fermionic subsystem symmetry: where the fermionic parameter ζ(y) depends on the y coordinate but not on the x coordinate. This subsystem symmetry, however, is explicitly broken by the four-fermi interaction This model also possesses the U(1) fermion number symmetry: It is generated by the Noether charge Other discrete symmetries include x-parity symmetry: x → −x, y → y with χ → χ and ψ → −ψ, y-parity symmetry: x → x, y → −y with χ → χ, ψ → −ψ as well as charge conjugation (or particle-hole) symmetry: χ → χ † , ψ → ψ † . Note that in this model x and y coordinates are treated differently as we see in H int , and there is no π/2 rotation symmetry (which is sometimes common in a fracton phase of matter [21]). Our supercharge and hence Hamiltonian is stable in the sense that there is no other term consistent with the above symmetry without adding more derivatives.
Our model may be regarded as a two-dimensional generalization of the extended Nicolai model [31] equipped with the fermionic subsystem symmetry. In the g = 0 limit, it may be regarded as a two-dimensional generalization of the original Nicolai model.
We would like to note here that g −1 plays the role of the coupling constant of the model. To see this, we rescale the time t → t g and observe that the action is equipped with the canonical kinetic terms with the four-fermi interaction proportional to g −1 : In the infinite g limit (i.e., free limit), the dispersion relation of ψ and χ is E = |k x k y | in terms of the wave nunmber k x and k y , and we have a ground-state degeneracy with the "Area-law" entropy (i.e., S ∝ Area). 3 Around infinite g, naive dimension counting suggests that g −1 is irrelevant in the renormalization group sense.
Supersymmetry (Q and Q † ) is spontaneously broken when g = 0. For large enough g, the Hamiltonian is dominated by the constant term d 2 x g 2 and the ground-state energy cannot be zero. From the anti-commutation relation {Q † , Q} = H, if the vacuum energy is non-zero, the supersymmetry must be spontaneously broken [37]. By using the holomorphy argument, we expect that the phase transition only occurs at g = 0 if any.
Indeed, we can formally argue for the spontaneously breaking of the supercharge Q in the following way.
Let us introduce a local operator O † : such that {Q, O † } = g . This implies that the supercharge Q is spontaneously broken for non-zero g because otherwise 0|{Q, O † }|0 = 0. The argument is valid as long as the regularization and the renormalization do not spoil the supersymmetry transformation (10). We will discuss a lattice regularization in the next section.
The zero-energy excitation (from the dispersion relation E = |k x k y | in the infinite g limit) can be understood as a Nambu-Goldstone fermion χ (or more precisely O introduced above) with k x = k y = 0. On the other hand, the zero-energy mode of ψ with k y = 0, k x = 0 corresponds to Goldstone fermions for the fermionic subsystem symmetry (4). The two distinct Nambu-Goldstone fermions coincide at k x = k y = 0 due to the kinetic mixing between χ and ψ. The zero-energy mode of ψ with k x = 0, k y = 0 is a consequence of the emergent fermionic subsystem symmetry (6) in the free limit and does not survive once we introduce the interaction with finite g.
For finite g, the interaction will gap some of the zero-energy excitations that are not protected by the fermionic subsystem symmetry. The zero-energy mode protected by the fermionic subsystem symmetry is the k y = 0 mode. The ground-state entropy is smaller but it is still proportional to L x but is insensitive to L y (once we put the theory on a finite box with length L x and L y ).
When g = 0, the model may be regarded as a variant of the Nicolai model [27] and we conjecture that the ground states are much more degenerate: the ground-state entropy shows the "Volume law": S ∝ Volume ∼ L x L y . One supersymmetric ground state is the Fock vacuum: ψ( x)|Ω = χ( x)|Ω = 0, but there are many others.
It is trivial to generalize our model to three dimensions (t, x) = (t, x, y, z): with As in two dimensions, the supersymmetry Q is spontaneously broken when g = 0.
The corresponding action is and it has the fermionic subsystem symmetries generated by Morally speaking, the free limit of the resulting theory is a fermionic part of the supersymmetric model studied in [24] with the dispersion relation E = |k x k y k z | (except that our fermions are complex rather than real). We can argue that for finite g, the ground-state entropy shows the "Area law": S ∼ L 2 . When g = 0, we conjecture that the ground-state entropy shows the "Volume law": S ∼ L 3 .

Lattice model and ground-state degeneracy
One lattice realization of our continuum model is to consider the following supercharge on the square lattice: Here, i labels the sites and c i and d i are complex fermion operators with canonical anti- In the continuum limit, c i and d i , respectively, correspond to χ( x) and ψ( x) in the previous section. Here and hereafter, i + nx + mŷ denotes the lattice site that is translated by n lattice units in x direction and m lattice units in y direction from site i.
The Hamiltonian is given by the anti-commutator: where the interaction Hamiltonian H int is A similar supercharge calculation can be found in [31], where the extended Nicolai model was studied.
As in the continuum field theory, we can argue for the spontaneous symmetry breaking of Q by the trick used in [31,32]. Let us introduce the local operator O † i : This immediately implies that the supersymmetry Q is spontaneously broken unless g = 0 on the finite lattice. A careful analysis of the infinite volume limit may be completed as in [32].
In the free limit g → ∞, the dispersion relation becomes E = 4| sin kx 2 sin ky 2 | (after the rescaling of time t → g −1 t). The gapless mode at k x = k y = 0 corresponds to a Nambu-Goldstone fermion for the spontaneously broken supersymmetry Q (when g = 0).
Let us now study the fermionic subsystem charges and ground-state degeneracy on the finite lattice. We consider a square lattice with the lattice size L x × L y (i.e. i = i + L xx = i + L yŷ ), imposing periodic boundary conditions on fermions.
We can clearly see that the fermionic subsystem charges commute with the Hamiltonian, and they satisfy the standard anti-commutation relations of L x complex fermions: The ground states form a non-trivial representation of this algebra so that the groundstate entropy shows S ∼ L x at least. When g = ∞ it is enhanced to S ∼ L y + L x due to the extra zero modes from m d mx+nŷ . When g = 0, our model becomes a variant of the Nicolai model and we conjecture that the ground-state entropy is given by S ∼ L x L y .
We can estimate a lower bound of the ground-state entropy at g = 0 as follows. First note that if we take any of the (classical) supersymmetric ground states of the Nicolai chain (see e.g. [28]) copied in y directions are the supersymmetric ground states of our two-dimensional model (leading to S ∼ L x ground-state entropy). Now we may still obtain the supersymmetric ground states by flipping some of the occupancies of c i while keeping the d j states. If this happens to be the case, we can repeat the flipping arbitrarily many times along the c i in the y directions. This procedure is local in the x direction so we obtain S ∼ L x L y entropy.
We offer numerical studies of the ground-state degeneracy with small lattices. With When g = 0, we may give a more precise conjecture about the ground-state degeneracy.
For L x > 2, it is conjectured to be 4 · 2 Lx . The number can be explained as follows. We have L x zero modes from n d mx+nŷ as well as one zero mode from the spontaneously broken supercharge Q, leading to the 2 Lx+1 -fold degeneracy. We further observe that the charge conjugation Q → Q † acts non-trivially on them so that we have the 2 Lx+2 -fold degeneracy in total. 4 For L x = 2, the situation is slightly different. We have L y zero modes from m d mx+nŷ and two zero modes from n d mx+nŷ , but there is one trivial relation n ( m d mx+nŷ ) = m ( n d mx+nŷ ) among them. In addition, the supercharge Q gives another zero mode, leading to the 2 Ly+2 -fold degeneracy. We observe that the charge conjugation Q → Q † acts non-trivially (at least when L y = 4, 5) so that we have the 2 Ly+3 -fold degeneracy in total.
We claim that there are no other zero modes that are not associated with any symmetry. We have less evidence for L x = 2 and L y = 2, but for small numbers of L x and L y we have explicitly checked with a computer that there are no other zero modes. The above counting does not depend on g (as long as g = 0), and we do not expect any phase 4 The extra degeneracy from the charge conjugation had already appeared in the one-dimensional extended Nicolai chain [38]. Suppose that the charge conjugation acts on a ground state |Ω trivially (otherwise the extra degeneracy is obvious), then we can always find a quartet |Ω , Q|Ω , Q † |Ω and QQ † |Ω (= (E −Q † Q)|Ω ) as long as the supersymmetry is spontaneously broken. One can then rearrange the quartet into a pair Q † |Ω and QQ † |Ω and their charge conjugates. The argument is generic and it applies to any spontaneously broken supersymmetric models with the charge conjugation symmetry (in finite systems).
transition at finite g.
Similarly, in three dimensions, we consider the supercharge on the cubic lattice: and define the Hamiltonian by the anti-commutator H = {Q † , Q}. We can study the degeneracy of the ground states explicitly and verify the "Area-law" entropy when g = 0.
For example, when L x = L y = L z = 2 the degeneracy is 256 both at g = 0 and g = 1. 5 When g = 0, we conjecture that the ground-state degeneracy gives the "Volume law" on a larger lattice.

Other models
Here, we discuss other models that have similar features to our models studied above.
The first model is to construct a supercharge out of one fermion χ per one lattice unit (instead of two fermions χ and ψ per one lattice unit). The supercharge is given by with the Hamiltonian H = {Q † , Q}. Note that this model does not possess the U(1) symmetry when g = 0. In the infinite g limit, the dispersion relation is E = |k x k y | and the ground-state degeneracy shows the "Area-law" entropy. On the other hand, with finite but non-zero g, while the supersymmetry Q is still spontaneously broken, the ground-state degeneracy is lifted because the interaction lacks the subsystem symmetry. One lattice realization of this model on the triangular lattice was studied in Appendix F of [33].
The second model is to restore the asymmetry between x and y directions. For this purpose, we may introduce two ψ fields ψ 1 and ψ 2 . Then the supercharge with the Hamiltonian H = {Q † , Q} gives the model that retains the π/2 rotation symmetry (under which ψ 1 and ψ 2 are non-trivially exchanged). In the infinite g limit, the dispersion relation is E = |k x k y | (from ψ 1 and ψ 2 modes) together with a flat band at zero energy (from χ modes). With finite g, the flat band is lifted and we conjecture that the ground-state degeneracy shows the "Area-law" entropy because of the fermionic The third model is to achieve the full fermionic subsystem symmetry in both x and y directions: ψ( x) → ψ( x) + ξ(x) + ζ(y). For this purpose, we introduce two χ fields χ 1 and χ 2 and consider the supercharge with the Hamiltonian H = {Q † , Q}. It has the fermionic subsystem symmetry ψ( x) → ψ( x) + ξ(x) + ζ(y). In the infinite g limit, it shows the dispersion relation E = |k x k y | (from χ 2 and ψ modes) with an additional flat band at zero energy (from χ 1 modes).
With finite but non-zero g, we conjecture that the ground-state degeneracy shows the "Area-law" entropy.

(Anomalous) Gauging and defect operators
Our models discussed so far do not possess a fracton as a dynamical degree of freedom, but we may consider the analogue of a probe fracton by considering a defect in the gauged theory. Our gauge theory will be a fermionic analogue of the tensor gauge theory studied in the context of a fracton phase of matter [39,40] (see also [41,42]).
We would like to gauge the shift symmetry of ψ fields by replacing ordinary derivatives ∂ t and ∂ y with covariant derivatives in the action. In the spatial direction, it is not difficult.
We simply replace ∂ y ψ with the covariant derivative D y ψ = ∂ y ψ + Φ y by assuming the fermionic gauge transformation law: Here the gauge fields Φ t and Φ y as well as the gauge parameter λ are fermionic.
On the other hand, gauging of the shift symmetry with the first order time derivative requires extra care. We will see that gauging of our model itself is anomalous in the t'Hooft sense and cancellation of the anomaly is necessary to make the gauge field dynamical. The difficulty comes from the fact that the action is invariant under the shift only up to total derivatives in time. 6 Let us postulate the action of the form to describe a gauged kinetic term. The variation of the gauge fields Φ t cancel the first order variation of ψ fields, but we see that the action fails to be invariant because of the remainder: which cannot vanish for non-zero Φ t , which we call "anomaly" (because it only depends on the gauge fields rather than the matter field ψ). The counter-term that could be used to cancel the anomaly would be iΦ † t ∂ −1 t Φ t , but it is non-local. Therefore, in order to gauge the fermionic shift symmetry, we need a certain cancellation mechanism by adding other degrees of freedom. In this discussion, we did not assume the supersymmetry, but we will see the same difficulty in the full supersymmetric action in Appendix A. The only difference is that the gauge field must have a reality condition Under the assumption of the anomaly cancellation, we may study the defect line operator that would correspond to a fermionic "fracton" or "lineon" [7]. We can construct a gauge invariant fermionic defect line operator The integration is over the worldline trajectory of the probe. Since we do not have the gauge field Φ x , the defect line operator cannot "move" in x direction (i.e., the probe trajectory cannot possess the non-trivial dx component). This defect operator becomes an analogue of "fracton" or "lineon" in our model.

Discussions
In this paper, we have constructed a purely fermionic system with spontaneously broken supersymmetry that shares the common feature with a fracton phase of matter. Our model is gapless due to the Nambu-Goldstone mechanism. It shows a ground-state degeneracy with the "Area-law" entropy due to the fermionic subsystem symmetries. In the strongly coupled limit, it becomes a variant of the Nicolai model, and the ground-state degeneracy shows the "Volume-law" entropy.
Our model is characterized by a coupling constant g. The phase boundary of the spontaneous supersymmetry breaking is located at g = 0. This is expected from the holomorphic nature of the supersymmetric model with a complex supercharge. It may be interesting to consider the Majorana version of our model so that the phase transition at finite g is possible. On the one dimensional chain, we find that there exists a phase transition at finite g [43].
From a formal perspective, one might have noticed that the subsystem symmetry in our model is rather trivially realized as a linear combination of fundamental fermionic operators d i (or field ψ). One may even try to remove these zero-mode states from the model and the exponential degeneracy of the ground states would be gone. We note that the apparent simplicity of removing the zero mode associated with the subsystem symmetry is almost always the case in the effective field theory description [20,21,22,24,26,25]. In the continuum description, however, removing the zero modes leads to a non-locality and break down of the effective field theory.
For future directions, it seems important to understand the anomaly associated with gauging of the fermiomic shift symmetry. We have demonstrated it from the continuum field theory perspective, but understanding of it from the lattice viewpoint may be of interest. It may be related to the difficulty to construct a model of fermionic fracton phase of matter.
The study of the fracton effective field theories is still in its infancy. Non-trivial UV/IR mixture may make the study of the interaction and the renormalization group non-trivial [25,44]. Realization in string theory and holography may help better understand the strongly interacting phase [45] and reveal a possible connection to the black hole physics through the "Area-law" entropy formula.
The supersymmetric action can be constructed as superspace integration by noting θθ component of a general superfield and θ component of a chiral superfield is supersymmetric (up to total derivatives in time). In our model, it is given by Our superspace formalism is much simpler than the one used in [24], where the anti-commutator of the supercharge also involves the spatial derivatives.
We can integrate out f ψ and f χ to obtain the Hamiltonian In order to gauge the shift symmetry in a manifestly supersymmetric fashion, we introduce a fermionic real superfield V and a fermionic chiral superfield Ξ y and impose invariance under the superfield gauge transformation with respect to a fermionic chiral superfield Ω, which plays a role of the gauge parameter.
Then the supersymmetric matter action is superfield gauge invariant.
We would like to discuss the supersymmetric analogue of the t'Hooft anomaly mentioned in section 5. Consider the candidate supersymmetric action which would give a kinetic term. This is superfield gauge invariant up to the anomaly term V Ω +ΩV , but without extra degrees of freedom, there is nothing we can add to cancel it. Note in particular that V 2 = 0 since V is fermionic. 8 In order to partially fix the gauge symmetry induced by Ω, while assuming the anomaly cancellation, we may use the Wess-Zumino gauge with the component This fixes the real part of the top component of Ω (i.e. θ independent component of Ω) and θ component of Ω. We still have the residual gauge transformation Here we note that Φ t is pure imaginary while Φ y is complex. The fermionic gauge parameter λ is pure imaginary. In component, (36) gives the supersymmetric version of the anomalous action that we studied in section 5.
In the superspace language, the supersymmetric defect line operator is given by Note that our superspace does not involve spatial coordinates, so it preserves the same amount of supersymmetry as the bulk action.
B Gauging bosonic/fermionic shift symmetry in quan-

tum mechanics
In this appendix, we will demonstrate the anomaly of gauging a fermionic shift symmetry in the simple quantum mechanics in comparison with a bosonic shift symmetry. Let us begin with a bosonic case. Consider a trivial particle quantum mechanics (in zero spatial dimension) with the action that gives the zero Hamiltonian. The equal time commutation relation is [x, p] = i. It has a shift symmetry of p → p + const and x → x + const.
If we tried to gauge the shift of both x and p, we would obtain the gauged action which is anomalous. This is because the shift symmetry is a symmetry of the Lagrangian Instead, if we only gauge the shift of x, the action is not anomalous: The Hamiltonian is H = −pA x and in this case we see that the gauge constraint p = 0 is consistent with the Hamiltonian dynamics. In this way, we may be able to gauge the shift symmetry of the bosonic system, but only one of the shift of x or p can be gauged.
The fermionic system has a drastic difference. Consider a complex fermion with the action which implies the zero Hamiltonian. The equal-time anti-commutation relation (in zero spatial dimension) is {ψ † , ψ} = 1. It has a shift symmetry of ψ → ψ + const. We will see that gauging the shift symmetry of ψ is anomalous.
Without loss of generality, let us try to gauge the shift of ψ + ψ † . Would-be gauged action becomes but the gauge transformation is anomalous. The Hamiltonian is H = −iΨ(ψ + ψ † ), but the gauge constraint is inconsistent with the Hamiltonian dynamics i[H, ψ + ψ † ] = 2Ψ unless Ψ = 0.
Our discussion is generic, but there is a possible loophole: one may add extra degree of freedom and/or modify the gauge transformation of Ψ. One successful example of gauging a fermionic shift symmetry is the super-Higgs mechanism in supergravity, where the space-like component of the gravitino and the gravity play the role of the extra degrees of freedom and the modification of the gauge transformation is accompanied [46].