The Fracton Gauge Principle

Michael Pretko Department of Physics and Center for Theory of Quantum Matter University of Colorado, Boulder, CO 80309 (Dated: May 30, 2019) A powerful mechanism for constructing gauge theories is to start from a theory with a global symmetry, then apply the “gauge principle,” which demands that this symmetry hold locally. For example, the global phase rotation of a system of conserved charges can be promoted to a local phase rotation by coupling to an ordinary U(1) vector gauge field. More recently, a class of particles has been studied featuring not only charge conservation, but also conservation of higher moments, such as dipole moment, which leads to severe restrictions on the mobility of charges. These particles, called fractons, are known to be intimately connected to symmetric tensor gauge fields. In this work, we show how to derive such tensor gauge theories by applying the gauge principle to a theory of ungauged fractons. We begin by formulating a field theory for ungauged fractons exhibiting global conservation of charge and dipole moment. We show that such fracton field theories have a characteristic non-Gaussian form, reflecting the fact that fractons intrinsically interact with each other even in the absence of a mediating gauge field. We then promote the global higher moment conservation laws to local ones, which requires the introduction of a symmetric tensor gauge field. Finally, we extend these arguments to other types of subdimensional particles besides fractons. This work offers a possible route to the formulation of non-abelian fracton theories.

Introduction. The concept of a gauge theory is one to absorb the non-covariant piece of this transformation. of the most important elements of the toolbox of mod- We do this by defining a gauge-covariant derivative as: ern theoretical physics, describing phenomena ranging iα from the fundamental particles of the Standard Model DµΦ = (∂µ − iAµ)Φ → e DµΦ (3) to topological phases of matter in solid state systems. Gauge theories are different from more conventional field where the gauge field Aµ must transform as Aµ → Aµ + theories, in that they have an enormously large degree ∂µα. We can then easily write down a gauge-invariant of symmetry, partially due to a significant redundancy Lagrangian for the theory in terms of the gauge covariant in the description of physical states. Specifically, gauge derivative and field invariants of the gauge field itself:

theories have local symmetries, involving independent 2 µν symmetry transformations at each point in spacetime, as L = Lm[Φ,DΦ,D Φ, ...] + Fµν F (4) opposed to the global symmetry transformations encoun- where the first term describes matter and its interaction tered in ungauged field theories. A particularly elegant with the gauge field, while F = ∂ A − ∂ A is the and powerful approach to constructing gauge theories is µν µ ν ν µ standard field strength tensor of a U(1) gauge field, de- to start from an ungauged theory, with some global sym- scribing the dynamics of a photon mode. In this way, metry, then make the demand that this symmetry should promoting the phase rotation of a set of conserved par- continue to hold locally. Philosophically, the notion that ticles from a global to a local symmetry has led to the a global symmetry should continue to hold at the local familiar Maxwell gauge theory. level is known as the “gauge principle.”1 Taking other types of global symmetries and “gaug- As a concrete example, consider a system of conserved ing” them will result in other types of gauge theories. For particles, described by a field Φ. Particle number con- example, gauging a global symmetry described by a non- servation will be encoded in the theory in the form of abelian Lie group will lead to a Yang-Mills theory.1 As invariance of the action under a global phase rotation: another example, gauging a symmetry protected topo- Φ → eiαΦ (1) logical (SPT) phase protected by group G will result in a topologically ordered phase with gauge group G.2 These for constant α. In order to construct a gauge theory, we

arXiv:1807.11479v3 [cond-mat.str-el] 29 May 2019 types of gauging procedures are now well-established. In now demand that the theory should be invariant under recent years, however, a new set of unfamiliar symme- independent phase rotations at each point in spacetime, tries and conservation laws has come to light, pertaining iα(x,t) i.e. under Φ → e Φ for a function α(x, t) with ar- to higher charge moments. These new conservation laws bitrary spacetime dependence. Adding such spacetime are manifested in systems of particles known as fractons, dependence creates problems for derivative operators in which have severely restricted mobility tracing directly the theory, which no longer transform covariantly: to conservation of quantities such as dipole moment.3 4 ∂ Φ → eiα(∂ + i∂ α)Φ (2) (We refer the reader to a recent review for a broad per- µ µ µ spective on fractons, and to selected literature3,5–26 for In order to restore gauge-covariance and thereby obtain further details.) While global charge conservation is as- a gauge-invariant action, we must introduce a gauge field sociated with a global phase rotation, Φ → eiαΦ, global 2 dipole conservation is associated with a linearly varying ciple. This work opens a possible door to investigations phase rotation, Φ → ei~λ·~xΦ. This symmetry transforma- of non-abelian fracton theories, via gauging non-abelian tion is neither a gauge symmetry nor one of the recently analogues of higher moment symmetries. discussed subsystem symmetries.8,27,28 (Note that a sub- Ungauged Fracton Field Theory. We begin by con- system symmetry is equivalent to an infinite number of structing a field theory for ungauged fractons, focusing higher moment symmetries, which provide the more gen- on the simplest type: U(1) fractons exhibiting conserva- eral framework.) It is not immediately clear what sort of tion of both charge and dipole moment. We describe the theory will result from “gauging” such a set of symme- fractons by a matter field Φ along with its corresponding tries and conservation laws. A significant clue is provided charge density operator, ρ = Φ†Φ. We now wish to write by fracton phases in certain spin, Majorana, and quan- down an action for this theory which is consistent with tum rotor systems3,6–11, which are often described in the global conservation of charge and dipole moment. Since language of tensor gauge theories, as opposed to the more the charge density ρ generates rotations of the phase of familiar vector gauge fields obtained from gauging con- Φ, the demand that the action respect charge conser- R d ventional symmetries. In this work, we will demonstrate vation ( d x ρ = constant) requires invariance of the explicitly how tensor gauge fields arise via application of action under the global transformation: the gauge principle to a system of fractons, promoting Φ → eiαΦ (5) global higher moment conservation laws and symmetries to local ones. This work serves as a useful complement to for constant α. We also demand that the theory obeys R d the existing literature on gauging subsystem symmetries conservation of dipole moment, d x (ρ~x) = constant, to obtain discrete fracton models.8,10–12,25,27,29,30 which requires an additional invariance under: ~ In order to study gauging a system of fractons, it is Φ → eiλ·~xΦ (6) first necessary to have a field theory for ungauged U(1) fractons, which has not yet been studied in the fracton for constant vector ~λ. In other words, the phase of the literature. (Earlier work29 has studied a field theory for field can now change by a linear function, instead of sim- a condensate of ungauged fractons, though as we will ply by a global constant, as in the case of ordinary con- discuss, such field theories cannot describe the mobil- served charges. For convenience, we combine both types ity restrictions of uncondensed fractons.) We therefore of transformations into the form: begin by formulating a field-theoretic description of un- Φ → eiα(x)Φ (7) gauged fractons exhibiting global conservation of both charge and dipole moment. We show that there is gener- where, in the present context, α(x) is restricted to be a ically no nontrivial Gaussian action (i.e. quadratic in the linear function. fracton fields) with these properties. Rather, the natural In order to construct an action with the desired sym- field theory for the simplest fracton system is quartic in metries, we look for operators O which transform covari- the fracton fields. Similarly, field theories for fracton sys- antly under the phase rotation, O → einαO for integer n. tems conserving even higher moments will feature even For ordinary globally conserved charges, where α(x) is a higher powers of the fields in the action. This type of constant, the field Φ and all of its derivatives and pow- non-Gaussian behavior has already been found by Slagle ers transform covariantly. In the present case, however, and Kim in the context of a gauge theory for the “X- the linear behavior of α(x) restricts the set of covariant cube” fracton model.12 This non-Gaussian nature of the operators involving spatial derivatives. While the field field theory reflects the fact that fractons necessarily in- Φ itself is still covariant, it can readily be checked that teract with each other even in the absence of a mediating any number of derivatives acting on a single power of Φ gauge field, as we will review. In this sense, there is no will not transform covariantly. Rather, the lowest-order true “non-interacting” theory of fractons. covariant derivative operator contains two factors of Φ, With the ungauged theory in hand, we then proceed taking the form: to apply the gauge principle, demanding that the theory Φ∂i∂jΦ − ∂iΦ∂jΦ (8) be invariant under local symmetry transformations. We show that, for a theory conserving charge and dipole mo- Under a generic transformation Φ → eiα(x)Φ, this oper- ment, the gauge principle demands coupling to a rank- ator transforms into: two symmetric tensor gauge field, consistent with pre-  i2α viously studied fracton phases.3 Similarly, local conser- e Φ∂i∂jΦ−∂iΦ∂jΦ vation of higher charge moments will require the intro- 2 duction of even higher rank tensor gauge fields. In this +i∂iαΦ∂jΦ + i∂jαΦ∂iΦ − (∂iα∂jα − i∂i∂jα)Φ  way, the theory of symmetric tensor gauge fields can be 2 derived directly from a gauge principle, in close anal- −i∂iαΦ∂jΦ − i∂jαΦ∂iΦ + ∂iα∂jαΦ ogy with more conventional gauge theories. Finally, we   i2α 2 consider extensions of this logic to other types of sub- = e Φ∂i∂jΦ − ∂iΦ∂jΦ + (i∂i∂jα)Φ dimensional particles besides fractons, which also yield tensor gauge theories upon application of the gauge prin- (9) 3

FIG. 2. Fractons can interact via the exchange of virtual dipoles, allowing two fractons to “push off” of each other. FIG. 1. Theories with only charge conservation have local dipole creation operators, which lead to quadratic terms in the continuum field theory limit. In contrast, a theory with dipole moment conservation has no such dipole creation op- interact with each other, even in the absence of a medi- erators. In this case, the smallest creation operators are ating gauge field. While a fracton cannot move by itself, quadrupolar, leading to quartic terms in the field theory. a fracton is capable of limited mobility by “pushing off” other fractons in the system via the following process: A fracton moves in one direction by emitting a dipole in For the phase rotations under consideration, where α(x) the opposite direction, which then propagates to and is is a linear function, we have ∂i∂jα = 0, such that: absorbed by a second fracton, as depicted in Figure 2. Such processes, which lead to a “gravitational” attrac- Φ∂ ∂ Φ − ∂ Φ∂ Φ → ei2α(Φ∂ ∂ Φ − ∂ Φ∂ Φ) (10) i j i j i j i j tion between fractons14, are neatly captured by the two It is then straightforward to construct a Lagrangian re- quartic terms in the fracton action. specting the charge and dipole conservation laws. To We mention in passing that, to describe a symme- lowest order, this Lagrangian takes the form: iθ(x,t) try broken system, such that Φ(x, t) = Φ0e for 2 2 2 2 constant Φ0 and dynamical phase θ, the Lagrangian of L = |∂tΦ| − m |Φ| − λ|Φ∂i∂jΦ − ∂iΦ∂jΦ| Equation 11 will simplify to an ordinary Gaussian field 0 ∗2 2 i (11) −λ Φ (Φ∂ Φ − ∂iΦ∂ Φ) theory on the phase θ, which transforms as θ → θ +α(x) where we have freely added a term with a single time under the symmetries. The resulting action takes the derivative and a mass term, which have no interplay with form: the spatially dependent phase rotation. The constants λ 0 0 1 K and λ are arbitrary couplings. Note that, while the λ L = (∂ θ)2 − (∂ ∂ θ)2 (12) term contains fewer derivatives than the λ term, it only 2 t 2 i j 2 contains diagonal second derivatives (e.g. ∂x, but not ∂x∂y), indicating that this term only describes longitudi- for some parameter K. For such symmetry broken nal motion of dipoles. As such, it is necessary to keep the phases, where the mobility restrictions have been re- λ term in order to correctly describe transverse motion laxed, the non-Gaussian nature of the original action of dipoles. Also note that we have here assumed rota- is unimportant. However, for describing a symmetric tional invariance, for simplicity. Additional anisotropic system with immobile gapped fractons, one must retain terms may arise for certain lattice symmetries. the full quartic structure of the field theory. Note that, 2 Equation 11 represents a field theory for ungauged if (∂i∂jθ) is replaced by cos(∂i∂jθ) in the above La- fractons, respecting global conservation of both charge grangian, then accessing the uncondensed phase becomes and dipole moment. Importantly, this lowest-order non- possible. However, such cosine terms inherently require trivial action is already non-Gaussian before coupling to a choice of underlying lattice and do not represent a any gauge field, containing fourth powers of Φ in the ac- true continuum field theory. Rather, the quartic action tion. This non-Gaussian behavior could have been antic- of Equation 11 must be used for a completely continuum ipated based on microscopic considerations. For a theory description of the fracton phase. of ordinary conserved charges, local charge creation oper- Application of the Gauge Principle. In the previous ators take the form of microscopic dipoles, Φ†(x+1)Φ(x), section, we derived a fracton field theory which is invari- which lead to a quadratic term in the action upon Tay- ant under Φ → eiα(x)Φ for linear functions α(x), thereby lor expansion. Meanwhile, in a theory with conserved respecting global conservation of charge and dipole mo- dipole moment, the local charge creation operators take ment. We now wish to apply the gauge principle and the form of microscopic quadrupoles, as depicted in Fig- demand that the theory have a local symmetry, such ure 1, leading to quartic terms upon Taylor expansion. that α(x) is an arbitrary function of x. From the form Similarly, a theory with conserved quadrupole moment of the transformation seen in Equation 9, it is easy to would only have microscopic octupole operators, leading see that we can construct a gauge-covariant object of the to an action which is octic in the fracton field. This logic form: will extend to conservation laws of any higher moment. The non-Gaussian nature of the fracton action is also 2 to be expected, since fractons have an intrinsic ability to Φ∂i∂jΦ − ∂iΦ∂jΦ − iAijΦ (13) 4 if we have the transformation rules: one-dimensional particles, which carry a vector-valued

iα charge ~ρ. We take these vector charges to exhibit global Φ → e Φ (14) conservation of both charge (R ddx ~ρ = constant) and R d Aij → Aij + ∂i∂jα (15) angular charge moment ( d x (~ρ × ~x) = constant). We describe these vector particles via a field Φ correspond- For notational simplicity, we define a gauge-covariant i 2 ing to each component of the charge vector, such that second derivative operator Dij acting on Φ as follows: † ρi = Φi Φi, where no summation over i is implied. In this 2 2 DijΦ = Φ∂i∂jΦ − ∂iΦ∂jΦ − iAijΦ (16) section alone, all summations will be indicated explicitly. Conservation of vector charge implies that the action More properly, we should define Dij as a bilinear op- for the theory should be invariant under independent erator acting on two functions Φ and Ψ as Dij[Φ, Ψ] = phase rotations on each component of the field: Φ∂i∂jΨ−∂iΦ∂jΨ−iAijΦΨ, where both fields must have iαi the same charge under the gauge transformation in or- Φi → e Φi (20) der for the operator to be covariant. (Note that, in the presence of multiple charged fields, only the total for constant vector αi, where this equation is to be inter- charge and dipole moment of the system will be con- preted component-wise in some particular basis. Mean- served, not for each field separately.) For present pur- while, the angular moment conservation law implies in- poses, however, we will only need the “diagonal” ele- variance under a second set of transformations: 2 ment, DijΦ ≡ Dij[Φ, Φ]. We can also allow α to have P i ijkλj xk arbitrary time dependence if we introduce the gauge- Φi → e jk Φi (21) covariant time derivative: for constant vector λj. For convenience, we combine DtΦ = (∂t − iφ)Φ (17) these transformations as:

iαi(x) where the field φ transforms as: Φi → e Φi (22) φ → φ + ∂ α (18) t where for present purposes we have αi(x) = α0,i + P  λ x for constants α and λ. Note that the behaving analogously to a “timelike” component of the jk ijk j k 0,i gauge field. We can also construct gauge-invariant quan- field Φi does not transform nicely under rotations (i.e. tities involving only the gauge fields, analogous to the it is not a valid vector). Nevertheless, at the end of the electric and magnetic fields of ordinary U(1) gauge the- day, we will obtain a true tensor gauge field upon apply- ory. For example, in three spatial dimensions these take ing the gauge principle. The loss of manifest rotational k ` invariance introduced by Φi may simply be a mathemat- the form Eij = ∂tAij −∂i∂jφ and Bij = ik`∂ A j. Mak- ing use of these quantities, the Lagrangian for the gauged ical artifact. It remains an open problem whether this theory, to lowest order in derivatives, must take the form: theory can be rewritten in a manifestly rotationally in- variant way. 2 2 2 2 2 0 ∗2 i 2 L = |DtΦ| − m |Φ| − λ|DijΦ | − λ (Φ DiΦ + c.c.) We now wish to construct an invariant action by iden- −λ0 + EijE − BijB tifying the covariant operators in this theory. As usual, ij ij operators without derivatives are automatically covari- (19) ant. Meanwhile, a derivative on Φi transforms as: Note the need for the complex conjugate (c.c.) in the λ0 iαj iαj X term, which is not automatically real-valued. The gauge ∂iΦj → e (∂i + i∂iαj)Φj = e (∂i + i ikjλk)Φj field sector of this Lagrangian is exactly of the form stud- k ied in previous works on fracton tensor gauge theories, (23) specifically that of the scalar charge theory studied in We see that, at the level of single-field derivative opera- Reference 3. However, we now have an explicit coupling tors, we have covariance only when i = j, giving us the to matter fields which accounts for the higher moment covariant longitudinal derivative operators: conservation laws of fractons. Through similar applica- ∂ Φ → eiαi ∂ Φ (24) tion of the gauge principle to theories with even higher i i i i conserved moments, we could also derive gauge theories for each component i. To include transverse derivatives featuring tensor gauge fields of rank higher than two, in the action, however, we must once again proceed to which we do not carry through here. two-field operators. We can easily construct a covariant Extensions to Other Subdimensional Particles. We operator as follows: now wish to extend these ideas to other types of subdi- mensional particles, which have restricted mobility only Φi∂iΦj + Φj∂jΦi → in certain directions. As a concrete example, we will   i(αi+αj ) focus on one-dimensional particles, restricted to mo- e Φi∂iΦj + Φj∂jΦi + i(∂iαj + ∂jαi)ΦiΦj tion along a line. A similar analysis will hold for two- dimensional particles. We focus on the simplest type of (25) 5

P Taking αi(x) = α0,i + jk ijkλjxk, the last term above of subdimensional particles can be derived from a gauge vanishes, leaving us with: principle, just like those for fractons. Conclusions. In this work, we applied the “gauge prin-   ciple” (i.e. promotion of a global symmetry to a local i(αi+αj ) Φi∂iΦj + Φj∂jΦi → e Φi∂iΦj + Φj∂jΦi (26) symmetry) to a system of fractons. We have shown how to formulate field theories of ungauged fractons, obey- We can now write down a Lagrangian respecting vector ing global conservation of higher charge moments. These charge and angular moment conservation as a function field theories are generically non-Gaussian, reflecting the of these covariant operators: intrinsic ability of fractons to interact without the need for a mediating gauge field. We then promoted the global

L[Φi, ∂iΦi, Φi∂iΦj + Φj∂jΦi] (27) higher moment conservation laws to local ones, which we have shown requires coupling the theory to a symmetric where the lowest order term involving transverse spatial tensor gauge field. In this way, the theory of symmetric derivatives is quartic in the fields. tensor gauge fields arises from a gauge principle in much Starting from this theory with global conservation the same way as an ordinary vector gauge field. We also laws, we can now apply the gauge principle, giving discussed extensions of this logic to other types of subdi- mensional particles besides fractons, which obey similar αi(x, t) arbitrary spacetime dependence, to promote the conservation laws to local ones. In this case, the gauge- gauge principles. covariant operators become: This work has the potential to open several new direc- tions in fracton physics. For example, by considering a fracton field transforming under a nonabelian Lie group, Φ ∂ Φ + Φ ∂ Φ − iA Φ Φ (28) i i j j j i ij i j is it possible to construct a nonabelian tensor gauge the- ory? And what would the properties be of such a system? Also, with an explicit field theory for fractons, is it now (∂ − iA )Φ (29) i ii i possible to develop more powerful technical tools to an- alyze fracton theories, such as an analogue of Feynman diagrams? By discretizing the derivatives, can we write DtΦi = (∂t − iφi)Φi (30) fracton theories on arbitrary lattices? How do these field theories need to be modified to apply to “type-II” U(1) where we have introduced a tensor gauge field which fractons, featuring no mobile bound states?31 Does this transforms as: field theory shed any light on the theory of elasticity and its associated phase transitions?22,23,32,33 There are Aij → Aij + ∂iαj + ∂jαi (31) many exciting questions remaining to be answered. Acknowledgments. I acknowledge useful conversations with Wojciech de Roeck, Mike Hermele, Shriya Pai, φi → φi + ∂tαi (32) Leo Radzihovsky, Rahul Nandkishore, Han Ma, Abhi- nav Prem, and Kevin Slagle. I also thank Juven Wang The corresponding action can be written down directly for pointing out some typos in a previous version of this from Equation 27 by replacing all derivative operators manuscript. This work is supported partially by NSF with their gauge-covariant versions, plus adding the ap- Grant 1734006, by a Simons Investigator Award to Leo propriate field invariants. The resulting gauge theory Radzihovsky, and by the Foundational Questions Insti- has exactly the structure of the vector charge theory dis- tute (fqxi.org; grant no. FQXi-RFP-1617) through their cussed in Reference 3. In this way, tensor gauge theories fund at the Silicon Valley Community Foundation.

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