Fracton topological order from Higgs and partial confinement mechanisms of rank-two gauge theory
Han Ma Michael Hermele Xie Chen





Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA


Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA


Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA



Fractons are gapped point-like excitations in d=3  topological ordered phases whose motion is constrained. They have been discovered in several gapped models but a unifying physical mechanism for generating them is still missing. It has been noticed that in symmetric-tensor U(1) gauge theories, charges are fractons and cannot move freely due to, for example, the conservation of not only the  charge but also the  dipole moment. To connect these theories with fully gapped fracton models, we study Higgs and partial confinement mechanisms in rank-2 symmetric-tensor gauge theories, where charges or magnetic excitations, respectively, are condensed. Specifically, we describe two different routes from the rank-2 U(1) scalar charge theory to the X-cube fracton topological order, finding that a combination of Higgs and partial confinement mechanisms is necessary to obtain the fully gapped fracton model. On the other hand, the rank-2 ℤ_2 scalar charge theory, which is obtained from the former theory upon condensing charge-2 matter, is equivalent to four copies of the d=3 toric code and does not support fracton excitations. We also explain how the checkerboard fracton model can be viewed as a rank-2 ℤ_2 gauge theory with two different Gauss' law constraints on different lattice sites.


Fracton topological order from Higgs and partial confinement mechanisms 
 of rank-two gauge theory
    Xie Chen
    December 30, 2023
==================================================================================================





§ INTRODUCTION 


Topological order in three dimensional systems can exhibit completely new features that are not present in lower dimensions. For example, following earlier work of Chamon,<cit.> Haah proposed a three dimensional exactly solvable model – the cubic code – with topological properties that are very different from all previously known examples.<cit.> The cubic code model is topological in the sense that, when defined on a three torus, the model is gapped and has a ground state degeneracy that is robust against any local perturbation. The ground state degeneracy, in sharp contrast to usual three dimensional topological models, changes with system size and is upper bounded by an exponential of the linear system size<cit.>. Similar to usual topological states, the cubic code supports fractional point-like excitations. However, the point-like excitations cannot move freely in three dimensional space. Instead, sets of four point excitations move in a coordinated way, and they can only be separated from each other by a fractal-shaped operator. Similar fractal structures have been found in a class of fractal quantum codes.<cit.>

Beginning with Chamon, and more recently, a class of gapped three dimensional topological models have been discovered which also host point-like excitations that cannot move freely, but without fractal structure.<cit.> In general, whether or not fractal structure is present, such point-like excitations are dubbed `fractons' or `sub-dimensional particles', and the corresponding models are said to have fracton topological order. The motion of the excitations differs from model to model: some are sub-dimensional particles that move only along a line or in a plane; some are fractons that can only move in coordination with others. This leads to a range of fundamental questions that need to be addressed: What are the universal physical properties characterizing a fracton topological phase? What kinds of fracton topological order are possible? What are physical mechanisms leading to fracton topological order?

Fracton topological order has been studied from a number of different perspectives,<cit.> and several physical mechanisms leading to it have been proposed.
Some fracton phases can be obtained via coupled-layer constructions, where the appearance of fractons is driven by  `particle loop condensation,' starting from a system of decoupled two-dimensional topological states, <cit.> obtained by coupling one-dimensional chains.<cit.>  Ref. hsieh2017fractons introduced parton theories of fracton states, providing a route to construct variational wave functions in more physically realistic models.



In a closely related development, it was pointed out by Pretko that fractons appear in higher rank U(1) gauge theories, i.e. those where the electric field and vector potential are symmetric tensors of rank two or higher.<cit.> In a vector gauge theory, Gauss' law leads to the conservation of total charge; once a positive-negative charge pair is created, each of of the charges can move freely in space without violating charge conservation. For a higher rank gauge theory, the situation can be very different. A modified Gauss's law can lead to the conservation of not only the total charge, but also the conservation of dipole moment, quadrupole moment, etc. Because of the extra conservation laws, the charge excitations cannot move freely any more – they become fractons (or sub-dimensional particles). We also note several earlier studies of higher-rank U(1) gauge theories, although the restricted mobility of charged excitations was not pointed out.<cit.>

Higher rank U(1) gauge theories, being studied extensively recently<cit.>, are different from fracton topological phases as they exhibit gapless photon modes. Is it possible to remove the gapless modes and make a connection with the gapped fracton models mentioned above. A natural way to remove gapless modes in a continuous gauge theory is to `Higgs' the gauge field and reduce the gauge group to a discrete one.<cit.> For example, when the normal vector U(1) gauge theory is Higgsed down to Z_2, we get a Z_2 gauge theory – the U(1) gauge charge reduces to the Z_2 gauge charge; the U(1) flux loop reduces to the Z_2 flux loop. Similarly, if we Higgs a gapless higher rank U(1) gauge theory, we get a gapped higher rank Z_N gauge theory, which seems to give a natural way to generate gapped fracton topological models. 

Surprisingly, as we show in this paper, the fracton nature of the charge excitations may be lost via the Higgs mechanism. This conclusion was also addressed in a recent study.<cit.> Here, we show in detail that while the scalar charge rank-2 U(1) gauge theory contains fractons, its Higgsed version does not and is equivalent to several copies of a discrete vector gauge theory. To arrive at a gapped fracton phase via the Higgs mechanism, something more is needed.

In particular, we discuss two paths that lead from the scalar charge rank-2 U(1) gauge theory to the fracton topological order of the X-cube model, a gapped fracton model introduced in Ref. vijay2016fracton.[The ground state degeneracy on a 3-torus of the exactly solvable X-cube model is stable under local perturbation. This can be verified using degenerate perturbation theory. If we add a local perturbation V= λ∑𝒪_loc to the Hamiltonian, the matrix elements in the resulting effective Hamiltonian for the degenerate ground state space are proportional to λ̃^L, where λ̃∝λ is a constant, and L is the linear size of the system.  This holds because only logical operators supported on a region of size L or larger have non-vanishing matrix elements within the ground state subspace. Since the degenerate subspace has dimension ∼ c^L for some constant c, the matrix Frobenius norm is bounded by (c λ̃)^L, which in turn bounds all the eigenvalues. Therefore, as long as λ is below some finite threshold, the splitting of the ground state subspace is exponentially small and approaches to zero in the thermodynamic limit. A very similar argument applies to many other gapped fracton models.]  As shown in Fig.<ref>, in one of the paths we can first Higgs the rank-2 U(1) gauge theory by condensing a charge-two matter field, arriving at four copies of Z_2 vector gauge theory (the toric code). Then we can condense certain flux loops to partially confine the gauge fields and arrive at the X-cube topological order. 

The second path to the X-cube topological order begins by first condensing certain monopole excitations of the rank-2 U(1) gauge theory, to arrive at a distinct `hollow' rank-2 U(1) theory, whose field tensors only have the off-diagonal components. This theory was studied previously in Ref. xu2008resonating, where it was shown that it is unstable to confinement arising from proliferation of instantons. Nonetheless, upon condensing charge-two matter in this theory, we again obtain the X-cube fracton topological order.

Similar to the X-cube model, the checkerboard fracton model<cit.> can also be interpreted as a rank-2 ℤ_2 scalar charge theory. An important difference is that two different forms of Gauss' law are alternately enforced on the even/odd layers of the system.



We would like to emphasize that in this paper we are not concerned with the critical properties of transitions between phases, or even whether continuous transitions exist.  Instead, we are interested in physical mechanisms by which one phase can be driven into another phase, by condensation of some excitations. Such mechanisms can be considered independent of critical phenomena, and indeed are still relevant even in cases where a transition is driven first order by fluctuations.  All transitions between phase in this paper are discussed in this spirit, leaving the study of critical phenomena for future work.

The paper is organized as follows. We begin our discussion by reviewing rank-2 U(1) gauge theory with scalar charge in Sec. <ref>, including a “hollow” version of this theory that has only off-diagonal elements in the electric field and gauge field tensors.
Sec. <ref> studies the effect of Higgsing the conservation laws of higher-rank gauge theories, and the resulting predictions for the mobility of charges in the corresponding gapped states. Sec. <ref> begins the left-hand branch of Fig. <ref>; we couple the rank-2 U(1) scalar charge theory to charge-2 matter, and condense the charge-2 matter to obtain the rank-2 ℤ_2 scalar charge theory. This theory does not support fractons and is found in Sec. <ref> to be equivalent to four copies of the d=3 toric code, with some further details given in Appendix <ref>. We complete the left-hand branch of Fig. <ref> in Sec. <ref>, where we show that condensing flux loops in a selective manner results in the X-cube model. Appendix <ref> studies a two-dimensional version of the rank-2 ℤ_2 scalar charge theory on the square lattice, showing it is equivalent to three copies of the d=2 toric code.

The right-hand branch of Fig. <ref> is taken in Sec. <ref>. First, in Sec. <ref>, we implement an electric-magnetic duality transformation to describe the gapped magnetic excitations of the rank-2 U(1) scalar charge theory. These excitations are point-like magnetic monopoles that move in two-dimensional planes, and we show that condensing these excitations in their planes of motion leads to the hollow U(1) gauge theory. Sec. <ref> then shows that condensing charge-2 matter in the hollow U(1) gauge theory leads to the X-cube model. Finally, in Sec. <ref> we show that the checkerboard fracton model can be interpreted as a rank-2 ℤ_2 gauge theory, with two different forms of Gauss' law on different lattice sites. The paper concludes with a discussion in Sec. <ref>.




§ REVIEW: RANK-2 U(1) GAUGE THEORY 


Let us start with a brief review of higher rank U(1) gauge theories studied in Ref. pretko2016generalized,pretko2016subdimensional. Unlike the higher form gauge theories with antisymmetric tensors as their gauge fields, which do not give us new topological orders in three dimensions, the higher rank gauge fields are symmetric tensors.  This difference leads to deconfined phases with sub-dimensional charges in d ≥ 3.

In this paper, we focus on rank-2 U(1) gauge theories and related phases<cit.>. While we will be interested in studying these theories defined on the lattice, it is useful to first describe them without dynamical matter in the continuum. We note that lattice regularization plays a more important role for these theories as compared to conventional gauge theories, and it is not clear whether these theories make sense with both dynamical matter and the full symmetry of Euclidean three-dimensional space; see Ref. fractons_review for a brief discussion of this issue.

The electric field E_μν and vector potential A_μν are both symmetric tensors, with the Greek indices running over spatial directions, i.e. μ, ν = x,y,z.   Eigenvalues of E_μν and A_μν are real numbers, and we have the commutation relations

    [ A_μν (), E_λσ(') ] = -i ( δ_μλδ_νσ + δ_μσδ_νλ ) δ( - ')  ,

where , ' are positions in d=3 space.

Such rank-2 gauge theories can be classified according to their Gauss laws and the nature of the corresponding charges. For example, if the Gauss law contracts all the indices of the tensor electric field and gives scalar charge, we have a scalar charge theory. On the other hand, if the Gauss law gives a charge transforming as a vector, then we have a vector charge theory.

We consider the scalar charge theory, where the Gauss law is ∂_μ∂_ν E_μν = ρ, where  repeated indices are summed over. This Gauss law leads to invariance under  gauge transformations A_μν→ A_μν + ∂_μ∂_ν f, where f is an arbitrary function of spatial position.  From A_μν we can construct the gauge-invariant magnetic field tensor B_μν = ϵ_μλσ∂_λ A_σν, which is traceless but not symmetric, and satisfies ∂_μ B_μν = 0 in the non-compact theory.  Deferring until later a discussion of dynamical matter degrees of freedom, the Hamiltonian density is H = 1/2 E_μν E_μν + 1/2 B_μν B_μν.

The unconventional Gauss law leads to a conservation of both electric charge and dipole moment.  Consider some bounded spatial region V,  with boundary ∂ V.  The total charge in V is given by

    Q = ∫_V ρ  d^3  = ∫_∂ V∂_ν E_μν  n_μ dS ,

where n_μ is a unit vector field normal to ∂ V and dS is the surface area element.  Because the right-hand side is a boundary term, this implies that it is impossible to locally create electric charges; of course, this is familiar from vector gauge theory.  Here it is also true that the dipole moment d = ∫_V ρ  d^3 can be written as an integral of the electric field over ∂ V, so it is also impossible to locally create dipole moments.  One dramatic consequence of this dipole conservation is that single electric charges are immobile, because moving an electric charge changes the dipole moment.




Here, we discuss how to put the rank-2 scalar charge theory on the simple cubic lattice, starting from the continuum theory.  Previous works  considered the related rank-2 vector charge theory, also on the simple cubic lattice <cit.>.  As shown in Fig. <ref>, the off-diagonal elements E_μν=E_νμ (also A_μν=A_νμ) with μ≠ν are defined on the plaquettes in the μ - ν plane, while each site hosts all three diagonal elements E_μμ (also A_μμ). Each conjugate pair A_μν, E_μν is an O(2) quantum rotor, with A_μν a 2π-periodic phase variable; this makes the theory compact.  The continuum form of the commutation relations implies that [A_μν, E_μν ] = -i, with  μ≠ν, when the two variables lie on the same plaquette.  For the diagonal components we have [ A_μμ, E_μμ ] = - 2i (no sum on μ).  This implies that off-diagonal elements of E have integer eigenvalues, while diagonal elements have even integer eigenvalues.  This distinction is somewhat undesirable; we will see how to correct it below.

On the lattice, the Gauss law is given by

    _μ_ν E_μν = n_ r

where Δ_μ is a finite-difference operator, and n_ r is the charge at site r.  In more detail, Gauss law can be written as

    2 (Δ_x  Δ_y E_xy + Δ_y  Δ_z E_yz +Δ_x  Δ_z E_xz   ) 
       + ( Δ_x Δ_x E_xx +Δ_y Δ_y E_yy + Δ_z Δ_z E_zz) = n_ r.

Here, similar to the difference in commutation relations, there is a factor of two difference in how the diagonal and off-diagonal components of E appear in Gauss' law.  We address these two undesirable features by defining E'_μμ = E_μμ / 2, so that E'_μμ takes integer eigenvalues and [ A_μμ, E_μμ ] = - i.  Putting this into Gauss' law, we see that n_ r is now restricted to be an even integer, so we define the integer-valued charge n'_ r = n_ r / 2.  Dropping the primes, we then have the commutation relations [A_μν, E_μν ] = -i, and the Gauss law

    (Δ_x  Δ_y E_xy + Δ_y  Δ_z E_yz +Δ_x  Δ_z E_xz   ) 
       + ( Δ_x Δ_x E_xx +Δ_y Δ_y E_yy + Δ_z Δ_z E_zz) = n_ r.

The Gauss' law directly determines those charge configurations that can be created locally, which are illustrated in Fig. (<ref>).  All these “locally-creatable” charge configurations have vanishing dipole moment. 





We now include dynamical electrically charged matter degrees of freedom, and at the same time describe the lattice Hamiltonian. The matter fields are O(2) quantum rotors placed on the cubic lattice sites r, with number n_ r and phase θ_ r, satisfying [θ_ r, n_ r] = i δ_ r,  r'.  The Hamiltonian is

    H    =     U ∑_ r,μ≤ν E^2_μν - K ∑_ r,μ,νcos ( B_μν ) 
       +    u ∑_ r n^2_ r  - J ∑_ r,μ≤νcos[ Δ_μΔ_νθ -A_μν] .

Here the lattice magnetic field is B_μν = ϵ_μλσΔ_λ A_σν, which can be viewed as a traceless but not symmetric tensor field defined on the dual cubic lattice, where diagonal components reside on dual sites (cube centers of the original lattice), and off-diagonal components reside on dual plaquettes.  The Gauss law is given by Eq. (<ref>), which leads to gauge transformations

    A_μν   →    A_μν + Δ_μΔ_ν f 
    θ_ r   →   θ_ r + f_ r.

When u and K are the largest energy scales, the charged matter is gapped, and there is a stable deconfined phase that can be described by expanding the -K cos(B_μν) terms to leading (quadratic) order<cit.>.  This phase has a gapless, linearly dispersing photon mode with five polarizations and gapped charge excitations. There are also gapped magnetic monopole particle excitations, described in Sec. <ref>.

We also consider a variant of the rank-2 scalar charge theory, where E_μν and A_μν remain symmetric but have only off-diagonal elements.  We refer to this theory as the “hollow” rank-2 scalar charge theory, because if we write E and A as 3 × 3 matrices, the diagonal elements are zero.  The Gauss law constraint is

    Δ_x Δ_y E_xy + Δ_y Δ_z E_yz + Δ_xΔ_z E_x z = n_ r,

which leads to the same form of gauge transformations as in Eqs. (<ref>, <ref>).  The Gauss law can also be written

    1/2Δ_μΔ_ν E_μν = n_ r,

where we take the diagonal components of E to be zero.

This Gauss law implies that the charge on every {1 0 0} lattice plane is conserved.  Let R be a bounded subset of some lattice plane p.   Without loss of generality, we take p to be an xy plane.  We have

    Q_p = ∑_ r∈ R n_ r = 1/2∑_ r∈ RΔ_μΔ_ν E_μν.

Each term in the summand contains at least one of Δ_x or Δ_y, so the sum reduces to a boundary term on ∂ R, which implies that Q_p is conserved.


If we try to define a magnetic field tensor using the same expression B_μν = ϵ_μλσΔ_λ A_σν, but setting terms with diagonal components of A to zero, we find that off-diagonal elements of B are no longer gauge-invariant.  Therefore we have only the diagonal elements B_μμ (no sum on μ), which satisfy a tracelessness constraint  ∑_μ B_μμ = 0, where the sum is over the three diagonal elements residing on the same dual lattice site.  This gives us two independent elements of the magnetic field tensor, which is the correct number of degrees of freedom, as there are three independent elements of A_μν and one unphysical gauge degree of freedom.

The Hamiltonian for the hollow gauge theory is

    H_ hollow   =     U ∑_ r,μ < ν E^2_μν - K ∑_ r, μcos ( B_μμ ) 
       +    u ∑_ r n^2_ r  - J ∑_ r,μ < νcos[ Δ_μΔ_νθ -A_μν] .

This theory was studied previously in Ref. xu2008resonating, where it was shown that a putative deconfined phase with gapped matter is unstable to confinement via proliferation of instantons.  Even though this theory does not have a deconfined phase (at least with gapped matter), it will still be of interest to us, because it can be Higgsed to obtain the same quantum phase of matter as the X-cube model.

The hollow gauge theory is not just a variant of the ordinary rank-2 scalar charge theory, but it can be obtained from it by a partial confinement transition, as we show in Sec. <ref>. Starting from the scalar charge theory, we can make U different for the diagonal and off-diagonal elements of E, so that the E^2 part of the Hamiltonian becomes

    H_ electric = U_d ∑_ r, μ E_μμ^2 + U_od∑_ r, μ < ν E_μν^2 .

Making U_d large results in a state where E_μμ≈ 0.  This is precisely the hollow gauge theory, which is clear because the Gauss law of the scalar charge theory becomes that of the hollow gauge theory in this limit.  We expect there should be a phase transition as U_d is increased, from the deconfined phase of the scalar charge theory to the (necessarily) confined hollow gauge theory.  Indeed, in Sec. <ref>, we show that condensing certain monopoles in the scalar charge theory leads to the hollow gauge theory.






§ HIGGSING THE CONSERVATION LAWS 



As we mentioned before, the motion of U(1) charge is subject to extra conservation laws in  higher rank gauge theories, such as dipole moment conservation. When the U(1) gauge theory is Higgsed, we can understand the effects on mobility of charges by directly studying how the conservation laws are modified, which can then be confirmed by a more detailed analysis starting from a Hamiltonian or functional integral formulation.  We thus refer to Higgsing of the conservation laws themselves.  In this section, we study the effect of a condensate of charge-N matter fields, which breaks the gauge group down to ℤ_N. We discuss charge conservation, dipolar conservation, and the planar conservation law obtained by Higgsing the hollow U(1) gauge theory.

Below, all the gauge theories studied have scalar charges living on the cubic lattice sites r=(x,y,z), with lattice spacing a=1.




 §.§ Charge conservation


We begin this section with the charge conservation law as a warm up. In a U(1) gauge theory, the U(1) charge is conserved, meaning, as discussed in Sec. <ref>, that the total charge Q in a region cannot be changed by acting with local operators within that region.  Upon Higgsing the theory to ℤ_N, charge-N objects can appear from and be absorbed into the condensate.  Therefore Q is now well-defined only modulo N, and becomes a conserved ℤ_N charge.  This of course is familiar from vector gauge theory.  The conservation of ℤ_N charge puts no constraints on the mobility of charges.



 §.§ Dipole conservation 


Now we consider the conservation of dipole moment that arises in the rank-2 scalar charge theory.  On the lattice, the dipole moment d of some region V is given by d = ∑_ r∈ V r  n_ r, and is conserved in the sense that it cannot be changed locally.

We now suppose we have a condensate of N-charge objects, which results in ℤ_N charges. By analogy with charge conservation, we can write down the Higgsed form of dipole conservation, which is simply that d N is conserved, i.e. each component of d is separately conserved modulo N.  Here it is important that we set the lattice constant to one; then charge ± N objects appearing from the condensate can change each component of d by integer multiples of N.

To understand the effects of the Higgsed dipole conservation law, it is useful first to consider configurations of charges lying on the x-axis, i.e. with r_y = r_z = 0.  We describe such a charge configuration by a set of ordered pairs { (q_1, r_x1), (q_2, r_x2), …}, where q_i is the value of the charge at r_x = r_xi, and the charge is zero for points not listed explicitly.

Dipole conservation tells us that {(+1,0), (-1,r_x)} can be locally created only when r_x is a multiple of N. Moreover, it is instructive to note that the locally creatable configuration {(+1,0),(-1,N)} can  be obtained starting from a charge configuration with vanishing U(1) charge and dipole moment, and then exploiting the condensate to add and remove charge-N objects.  We consider the configuration { (+1,0), (-N, i), (N,i+1), (-1, N) }, which is easily seen to have vanishing charge and dipole moment.  The charges at site i and i+1 can be absorbed into the condensate, and this configuration is thus equivalent to {(+1,0),(-1,N)}.

On the other hand, the configuration {(+1,0),(-1,k)} with 0 < k < N has non-vanishing d N and is not locally creatable.  Moreover, a configuration with a +1 charge at the origin and a -1 charge at r = (r_x, r_y, r_z) is easily seen to be have vanishing dipole moment modulo N, and thus be locally creatable, only when each of r_x, r_y and r_z are integer multiplies of N.  From these observations we see that, although the charge in the rank-2 U(1) gauge theory is immobile, charges in the Higgsed rank-2 ℤ_N gauge theories can hop freely in any direction, but can only hop N lattice spacings at a time along the x, y and z directions. This suggests that the rank-2 ℤ_N gauge theories obtained in this way are equivalent to the usual vector ℤ_N gauge theories. Indeed, we will show in Sec. <ref> that the rank-2 ℤ_2 scalar charge theory on the cubic lattice is equivalent to four copies of vector ℤ_2 gauge theories (toric codes). With increasing N, the number of distinct particle species is expected to increase. In the limit of N →∞, the charge can only hop an infinite number of lattice spacings at a time, and we go back to the U(1) case where charge excitations are  immobile.






 §.§ Planar conservation law



In the U(1) scalar charge theory, the charge and dipole conservation laws impose strong constraints on the motion of excitations. However, after Higgsing, the ℤ_N charges turn out to be fully mobile. In order to further restrict the motion of ℤ_N charges to be sub-dimensional, we obviously need stronger constraints resulting from more powerful conservation laws.

If the charge on every lattice plane is separately conserved, then clearly single charges will not be able to move.  This is true even for a discrete gauge theory with ℤ_N charges, and indeed occurs in the X-cube model (see Sec. <ref> and Sec. <ref>).   In other words, this strong conservation law would lead to fractons and sub-dimensional excitations in discrete gauge theory.

In Sec. <ref>, we showed that the hollow U(1) gauge theory indeed has conservation of charge on each { 100 } lattice plane.  Upon Higgsing this conservation law by condensing charge-N objects, we obtain ℤ_N charges that are conserved separately in each plane, which are thus immobile fractons.  At N=2, this is exactly the same conservation law as in the X-cube model, and we expect that condensing charge-2 objects in the hollow U(1) gauge theory will lead to the X-cube topological order.  This expectation is verified in Sec. <ref>.





§ X-CUBE MODEL VIA ℤ_2 SCALAR CHARGE THEORY 




 §.§ Charge-2 condensation 


The results on the mobility of ℤ_N charges predicted above by Higgsing the conservation laws can be obtained explicitly  by coupling the lattice model  to a charge-N matter field.  We do this here for the rank-2 U(1) scalar charge theory.  Starting with the Hamiltonian in Eq. <ref>, we add a charge-2 matter field on sites r with number N_ r and phase Θ_ r. We add the following terms to the Hamiltonian in Eq. <ref>

    H_2e   = u_2 ∑_ r N_ r^2 - Δ∑_ rcos[ Θ_ r -2 θ_ r] 
        - J_2 ∑_ r,μ≤νcos[ Δ_μΔ_νΘ
    -2 A_μν( r)] .

The Gauss law in Eq. (<ref>) is modified to become

    (Δ_x  Δ_y E_xy + ⋯ )  + ( Δ_x Δ_x E_xx + ⋯) = n_ r + 2 N_ r.


Tuning J_2 to be large results in a condensation of the charge-2 field e^iΘ.  We also take Δ large for convenience, and we treat the Δ and J_2 cosine terms as constraints.  To solve the constraint imposed by Δ→∞, we  define a ℤ_2 charge creation operator τ^z_ r≡ e^i ζ_ r, where

    ζ_ r = 1/2Θ_ r - θ_ r =0,π.

The operator τ^z_ r anticommutes with  τ^x_ r≡exp[ i π n_ r], which justifies the notation suggestive of Pauli matrices.


The second cosine leads to a rank-2 ℤ_2 gauge field Z_μν≡exp (i η_μν) on plaquettes (μ≠ν) and sites (μ=ν), where

    η_μν = 1/2Δ_μΔ_νΘ - A_μν( r)=0,π.

The  operator Z_μν anticommutes with the rank-2 ℤ_2 electric field X_μν≡exp (i π E_μν).  The ℤ_2 magnetic flux is given by F_μν = exp(i B_μν ) = exp(i ϵ_μλσΔ_λη_λν ). Two elements of F_μν are shown in Fig. (<ref>). 


We can now write down the effective Hamiltonian describing the system upon condensing e^iΘ, in terms of the new  ℤ_2 variables, which is a rank-2 ℤ_2 scalar charge gauge theory. We have

    H_ℤ_2   = -J ∑_ r,μ≤νexp(i Δ_μΔ_νζ ) Z_μν
       - K ∑_ r, μ,ν F_μν
    - u ∑_ rτ^x_ r - U ∑_ r,μ≤ν X_μν .

Here we have replaced the n_ r^2 and E_μν^2 terms with -τ^x_ r and - X_μν, respectively.  These are the simplest terms in the ℤ_2 variables that penalize non-zero charge and electric field, respectively.

The Gauss' law is obtained by exponentiating that of the U(1) theory, and is

    exp( i π ( Δ_xΔ_x E_x x + ⋯ ) + (Δ_xΔ_y E_xy + ⋯) ) = τ^x_ r.

The left-hand side, which we denote G_ r,  is a product of 18 X_μν operators on the edges and vertices of an octahedron, as illustrated in Fig. <ref>.

We are interested in the deconfined phase of this theory, where magnetic flux is suppressed and where the matter fields are gapped.  An extreme limit of the deconfined phase obtains for J = U = 0, where the model becomes exactly solvable.  The U term gives a tension to non-trivial electric field configurations, and, equivalently, leads to fluctuations of the magnetic field.  For large enough U, we expect the theory to become confining.  The J term is a kinetic energy for the ℤ_2 charges, which we expect to condense for large enough J.

We will analyze the deconfined phase at its exactly solvable point.  To do this, it will be convenient to exploit a mapping to a local bosonic model with a tensor product Hilbert space (i.e., not a gauge theory).  This mapping works in the same way as the familiar mapping between  vector ℤ_2 gauge theory and the ℤ_2 toric code.<cit.>  The degrees of freedom in the bosonic model are Pauli operators Z̃_μν and X̃_μν.  These are related to the gauge theory degrees of freedom by

    X̃_μν   =    X_μν
    Z̃_μν   =   exp(i Δ_μΔ_νζ ) Z_μν.

Using Gauss' law to express τ^x_ r in terms of G_ r, the Hamiltonian for the bosonic model is

    H̃_ℤ_2   =    - K ∑_ r, μ,νF̃_μν  - u ∑_ rG̃_ r
       -    J ∑_ r,μ≤νZ̃_μν  -  U ∑_ r,μ≤νX̃_μν,

where F̃_μν and G̃_ r are products of Z̃_μν and X̃_μν, respectively, given by the same expressions as F_μν and G_ r.







 §.§ Lattice ℤ_2 scalar charge theory 



We now study the deconfined phase of the ℤ_2 scalar charge theory at its exactly solvable point.  We work with the local bosonic model of Eq. (<ref>), set U = J = 0, and drop the tildes that distinguish between operators acting in the gauge theory and tensor product Hilbert spaces. We will show that this model is  equivalent to four copies of the three dimensional toric code (equivalently, four copies of  ℤ_2 vector gauge theory).  In the next section, we start from this model and describe how to obtain the X-cube model via a confinement transition.


First, in Appendix <ref>, we establish that the ground state degeneracy GSD on a L × L × L torus is a constant: log_2(GSD)=12, independent of system size.  Moreover, the degenerate ground states cannot be locally distinguished, and the model is topologically ordered.  This is consistent with the model being equivalent to four copies of the d=3 toric code.


Here, we describe the excited states of the model, which can be labeled by the eigenvalues of the commuting operators G_ r and F_μν.  First, we consider “electric” excitations where G_ r = -1 for some sites r contained in a bounded region, and where F_μν = 1.  We will see that any such excitation can be built up from four independent types of fully mobile ℤ_2 point charges.

Acting with a Z_μν operator on a ground state creates point-like charges at sites by flipping the sign of some G_ r eigenvalues.  Depending on whether μ = ν or μ≠ν, two charge configurations are possible, and are the same as in Fig. <ref>, but with ℤ_2 charges. In the linear charge configuration of Fig. <ref>b, obtained by acting with Z_μμ, there are now only two nontrivial charges, as the middle one vanishes into the condensate.  Because of this, single charges are able to hop from site to site by an even number of lattice spacings. Therefore, by acting repeatedly with Z_μμ operators, we can transform a general electric excitation into one supported on a cube of eight sites r = (r_x, r_y, r_z), where r_x, r_y, r_z = 0,1, and where we take the origin to be one corner of the cube.

At this point it might appear that there are eight types of independent charges, corresponding to the eight vertices of the cube.  However, acting with off-diagonal Z_μν operators on the faces of the cube, any charge configuration can be brought to one with excitations at only four of the eight vertices, as shown in Fig. <ref>.  There are thus four types of ℤ_2 charges, corresponding to G_ r = -1 at only a single one of the four vertices of Fig. <ref>.  We label the charge types by τ^z_1,2,3,4, as shown in the figure.



The existence of four types of ℤ_2 charge is consistent with equivalence to four copies of the d=3 toric code.  To go further, we should also establish the existence of magnetic loop excitations that have non-trivial braiding with the charges.  For a single copy of the d=3 toric code, there is a statistical phase θ = π when a point charge braids around a loop.  Here, upon braiding any composite of the elementary charges τ^z_1,2,3,4, we expect a statistical phase of θ = 0 or θ = π.  We need to  show that there exist four different types of magnetic loops, such that if we braid a single point-like excitation with each of the four loops, the pattern of statistical phases uniquely determines the charge type.   A set of such loop excitations is shown in Fig. <ref>; in each case the loop and the membrane operator creating it lies in a yz plane.  The statistics of each of the four elementary charges with a given loop is then easily determined by noting whether the string operator transporting the charge in the x-direction, which is a product of Z_xx operators, commutes or anti-commutes with the loop's membrane operator.  This information is shown in Table <ref>, and it is straightforward to verify that the charge type is fully resolved by braiding with the four different loops.





Based on the ground state degeneracy and properties of excitations and logical operators, we conclude that the  ℤ_2 scalar charge model has the same topological order as four copies of the d=3 toric code.  There are no sub-dimensional particle excitations in this system, which is consistent with our prediction from Higgsing the conservation law.

We note that a similar analysis can be carried out in two dimensions, starting with the rank-2 U(1) scalar charge theory on the square lattice, and Higgsing it by coupling to charge-2 matter.  There, the resulting ℤ_2 scalar charge theory is equivalent to three copies of the d=2 toric code.  This analysis is presented in Appendix <ref>.



 §.§ X-cube model from selective flux loop condensation 
 

Here we will describe how to get the X-cube fracton topological order from the rank-2 ℤ_2 scalar charge theory, by condensing certain flux loops. Before doing this, we discuss a simpler but similar transition from the cubic-lattice d=3 ℤ_2 toric code to a stack of decoupled d=2 toric code layers.





We consider a perturbed d=3 ℤ_2 toric code, which has a single qubit on each nearest-neighbor link ℓ of the simple cubic lattice.  
The Hamiltonian is

    H_3dTC = - ∑_p B_p - ∑_ r A_ r - h ∑_ℓ∥ z X_ℓ

where p labels square plaquettes, r labels cubic lattice sites, and we take h ≥ 0.  The sum in the last term is over links parallel to the z-axis.  The plaquette term is given by  B_p = ∏_ℓ∈ p Z_ℓ, while the vertex term is A_ r = ∏_ℓ∼ r X_ℓ, where latter product is over the six links touching the site r.  When h = 0, we have the usual exactly solvable d=3 toric code, which has point charge excitations where A_ r = -1, and flux excitations that are loops along which B_p = -1.

Now we consider the effect of the perturbation, which tends to freeze the degrees of freedom between xy plane layers.  In the strong coupling h →∞ limit, we set X_ℓ = 1 for all ℓ∥ z.  The remaining terms can be treated in first-order degenerate perturbation theory.  Projecting the Hamiltonian into the degenerate subspace, each  A_ r term becomes a product of X_ℓ over the four touching r and lying in an xy plane.  Plaquette terms B_p where p lies in an xy plane survive the projection unchanged, while other plaquette terms have vanishing projection into the degenerate subspace.  The resulting Hamiltonian is simply that of a stack of decoupled d=2 toric code layers.

The transition from d=3 toric code to d=2 toric code layers can be interpreted in terms of a condensation of certain flux loops.  Considering first small h, we see that acting with X_ℓ for ℓ∥ z creates a small flux loop lying within an xy plane between two square lattice layers.  For intermediate h, many such xy-plane flux loops are created, and eventually we expect these loops will condense at infinite h.  The point charge excitations of the d=3 toric code can no longer move in the z-direction, because they have a statistical interaction with the loop condensate.  However, the charges can still move freely within xy-plane square lattice layers, where the loop condensate does not penetrate.  In addition, a flux loop that intersects a single square lattice layer at two points, as shown in Fig. <ref>, reduces to two point-like flux excitations at the intersection points, in the presence of the condensate. 


Now we return to the rank-2 ℤ_2 scalar charge theory discussed in the previous section, from which we obtain the X-cube topological order via a similar selective flux loop condensation.  We add the following term to the Hamiltonian

    H' = - h∑_ r∑_μ = x,y,z X_μμ( r) ,

where we recall that the diagonal components X_μμ reside at lattice sites r.
In the strong coupling limit, h →∞, X_μμ ( r) is restricted to be 1.  As in the d=3 toric code case, we treat the remaining terms in first-order degenerate perturbation theory.  Projecting the G_ r term into the degenerate subspace with the projector P results in  

    P G̃_ r P = ∏_p ∼ r X_μν(p) ,

where the product is over the 12 plaquettes p with a corner at the site r.  The off-diagonal F_μν terms have vanishing projection into the degenerate subspace, while the diagonal terms F_μμ survive unchanged.  The resulting model has only off-diagonal (μ≠ν) Pauli operators X_μν and Z_μν residing on plaquettes.  Replacing the cubic lattice with its dual, the plaquette variables become link variables X_ℓ and Z_ℓ, where ℓ labels dual lattice links.  It is straightforward to see that 

    P G_ r P = ∏_ℓ∈ c X_ℓ,

where c is the dual lattice cube centered at r, and the product is over the 12 edges of the cube.  This is precisely the cube term of the X-cube model.  Moreover,

    F_μμ = ∏_ℓ∼ s ,  ℓ⊥μ Z_ℓ,

where the product is over the four links touching the dual lattice site s that are perpendicular to the μ-direction; this is a vertex term of the X-cube model.  Therefore, the Hamiltonian projected to the degenerate subspace is precisely the X-cube model.



Again, we can develop a physical picture based on flux loop condensation by considering small and intermediate h.  Acting with the perturbation creates certain flux loops within xy, yz and xz planes, as shown in Fig. <ref>.  When these loops condense, the charge excitations of the rank-2 ℤ_2 scalar charge theory are confined, as they have a statistical interaction with the loops.  However, bound states of two charges separated along the x, y or z axis can propagate freely (see Fig. <ref>); string operators transporting such bound states are a product of off-diagonal Z_μν operators, which commute with the perturbation.  We thus arrive at the conclusion that individual charges are now created at the corners of a rectangular membrane operator, and become fractons.


We can also obtain the one-dimensional particles of the X-cube model from flux loops in this picture.  Acting on a plaquette p with an off-diagonal operator such as X_xz(p) in the rank-2 ℤ_2 scalar charge theory creates an excitation where eight different F_μν operators (crosses in Fig. <ref>) flip sign.  Four of these are off-diagonal operators with μ≠ν, and upon condensing loops these excitations disappear into the condensate.  The other four operators are the F_xx and F_zz terms adjacent to the plaquette p, which remain as gapped excitations.  By acting with a product of X_xz over a stack of xz-plane plaquettes (see Fig. <ref>), these two excitations can be separated along a line.  These are the one-dimensional particle excitations of the X-cube model.







§ X-CUBE MODEL VIA HOLLOW U(1) SCALAR CHARGE THEORY



Now we study the right-hand path of Fig. <ref>, which is a different route to the X-cube model from the rank-2 U(1) scalar charge theory.  In Sec. <ref>, we already noted that adding U_d ∑_ r, μ E^2_μμ to the Hamiltonian of the scalar charge theory, and making U_d large, freezes out the diagonal degrees of freedom, resulting precisely in the hollow U(1) gauge theory.  Evidently this is some kind of partial confinement transition, but by what mechanism does it proceed?  Here, in Sec. <ref>, we carry out an electric-magnetic duality transformation and show that increasing U_d drives the condensation of certain point-like magnetic monopoles, and that the hollow U(1) gauge theory describes the system in the presence of the monopole condensate. The same hollow U(1) scalar charge theory was studied before<cit.>, and it was found that it does not have a stable deconfined phase.  Nonetheless, the effect of condensing charge-2 matter can still be studied as a mechanism to drive the system into another phase, and in Sec. <ref> we show that this results in the X-cube model.  Because of the confinement in the hollow U(1) gauge theory, we can interpret the condensation described in Sec. <ref>  as a mechanism for a transition between a confined phase and a phase with the topological order of the X-cube model.




 §.§ Magnetic monopole condensation in the rank-2 U(1) scalar charge theory 



In order to understand the magnetic sector of the rank-2 U(1) scalar charge theory, we go to its dual theory. We will work with a discrete-time Euclidean action that can be derived from  the lattice Hamiltonian of Eq. (<ref>).  Since we are interested only in the magnetic sector, we drop the coupling to matter fields and consider a pure gauge theory.  The Euclidean action takes the form

    S    =    i ∑_τ,  r∑_μ≤ν E_μνΔ_τ A_μν - i ∑_τ,  rϕ(τ, r) ∑_μ≤νΔ_μΔ_ν E_μν
       +   ∑_τ,  r[ u_d ∑_μ E_μμ^2 +  u_od∑_μ < ν E_μν^2 ]  
    - k ∑_τ,  r∑_μ, νcos( B_μν ) .

Here, pairs τ,  r label space-time points, E_μν is an integer-valued field residing on sites (μ = ν) and plaquettes (μ≠ν) of the spatial lattice, and A_μν is a 2π-periodic variable residing in the same locations.  The magnetic field tensor B_μν was defined in terms of A_μν in Sec. <ref>, is not symmetric but is traceless, and resides on sites and plaquettes of the spatial dual cubic lattice.  The field ϕ(τ, r) is 2π-periodic  Lagrange multiplier that imposes the Gauss' law constraint.

Integrating out ϕ( r, τ), the Gauss' law constraint is solved by writing E_μν in terms of a dual gauge field α_μν taking values in 2πℤ:

    E_μμ   =   1/2 πϵ_μλσΔ_λα_σμ  (no sum over μ)
    
    E_μν   =   1/2π( ϵ_μλσΔ_λα_σν + ϵ_νλσΔ_λα_σμ)    ( μ≠ν ) .

Here, α_μν lives on  dual lattice sites if μ=ν and on dual plaquettes if μ≠ν.  The field α_μν transforms under dual gauge transformations by

    α_μν→α_μν + Δ_μλ_ν + δ_μν f ,

where λ_μ lives on the dual lattice links and f lives on dual sites, and both λ_μ and f take values in 2πℤ. 

Next, we proceed to the Villain representation for the cosine terms in the action, i.e. we replace

    -kcos( B_μν ) →1/2 k D_μν^2+ i D_μν B_μν

where D_μν is a new integer valued field that is summed over.  Similarly to α_μν,  D_μν is a tensor field living on the dual lattice sites and plaquettes.

Now we can integrate out A_μν field, which results in a constraint on D_μν, that is solved by

    D_μν = 1/2π ( Δ_τα_μν - Δ_μψ_ν  - δ_μνθ) .

Here,  invariance of D_μν under dual gauge transformations requires introducing 2 πℤ-valued fields ψ_μ and θ living on the links and sites of the dual lattice, respectively.  These fields transform under dual gauge transformations by

    ψ_μ   →ψ_μ + Δ_τλ_μ,
    θ   →θ + Δ_τ f .
 

We thus obtain the dual action

    S_ dual   =   1/4 π^2∑_τ, r[ u_d ∑_μ (ϵ_μλσΔ_λα_σμ)^2 
       +    u_od∑_μ < ν (ϵ_μλσΔ_λα_σν + ϵ_νλσΔ_λα_σμ )^2 ] 
       +   1/8 π^2 k∑_τ, r∑_μ, ν (  Δ_τα_μν - Δ_μψ_ν - δ_μνθ)^2 .


To proceed, we promote the discrete fields to real-valued fields and introduce cosine terms to softly restore the discreteness constraint.  It is convenient to allow for real-valued dual gauge transformations; in order to do this and keep the cosine terms gauge invariant, we need to introduce new compact fields  ϕ_μ on  dual  links and  γ  on  dual  sites.  These fields transform as ϕ_μ→ϕ_μ +λ_μ and γ→γ+f under dual gauge transformations. We add the cosine terms

    S'    =    
    - t_m ∑_τ, r∑_μ, νcos(Δ_μϕ_ν  - α_μν -γδ_μν) 
       -    t”_m ∑_τ, r∑_μcos(Δ_τϕ_μ - ψ_μ ) 
       -    t_θ∑_τ, rcos(Δ_τγ - θ) 
    .

The discreteness of θ will not play an important role in our discussion, so we take t_θ = 0 and integrate out γ to obtain the action

    S'_ dual   =    S_ dual
    - t_m ∑_τ, r∑_μ≠νcos(Δ_μϕ_ν  - α_μν ) 
       -    t'_m ∑_τ, r∑_μ,νcos (Δ_μϕ_μ  - Δ_νϕ_ν -α_μμ+ α_νν) 
       -    t”_m ∑_τ, r∑_μcos(Δ_τϕ_μ - ψ_μ ) .

Here, we see that ϕ_μ is a matter field coupled to the dual gauge field.  We therefore expect it to represent gapped magnetic excitations, as we shall see below.

Next, we decouple the second term in Eq. (<ref>) with a Hubbard-Stratonovich transformation,

    1/8 π^2 k   (   Δ_τα_μν - Δ_μψ_ν - δ_μνθ)^2 
       →    2 π^2 k   β_μν^2 - i β_μν ( Δ_τα_μν - Δ_μψ_ν - δ_μνθ)   .

Here, β_μν should be interpreted as the magnetic field; that is, β_μν∼ B_μν.  We note that integrating out θ gives the constraint ∑_μβ_μμ = 0.
Next, we go to a Villain representation for the last term in Eq. (<ref>), i.e. 

    t”_m cos   (   Δ_τϕ_μ - ψ_μ ) →1/t”_m  n_μ^2
    + i  n_μ ( Δ_τϕ_μ - ψ_μ )

This introduces the “number operator” n_μ conjugate to the phase ϕ_μ of the dual matter field.

Integrating out ψ_μ gives the dual Gauss' law,

    Δ_μβ_μν = n_ν,

which shows that n_μ is a point-like (in space) vector magnetic charge, or magnetic monopole, created by e^i ϕ_μ.  From this constraint, it is straightforward to show that the quantity ∑_μ r_μ n_μ is conserved.  Conservation of this quantity allows a n_μ monopole to hop freely in a d=2 plane normal to μ, but forbids the monopole to move independently in the μ-direction.  Therefore, these monopoles are two-dimensional particles.  This is born out by inspecting the the first two terms in Eq. (<ref>), which describe two different dynamical processes of monopoles.  The t_m term is a hopping of n_μ monopoles in a plane perpendicular to μ.  The t'_m term is a cooperative two-monopole hopping process.

We now consider the effect of condensing monopoles within their d=2 planes of motion, by making t_m and t”_m large, while keeping t'_m fixed.  We will see that, because we are keeping t'_m fixed, this does not fully Higgs out the dual gauge field.  To understand why this is the partial confinement transition we expect when U_d (and hence u_d) becomes large, we note that the t_m cosine originates from the discreteness constraint on off-diagonal elements of α_μν.  Now, diagonal elements of E_μν only depend on off-diagonal elements of a_μν, in contrast to off-diagonal elements of E_μν, which depend on both diagonal and off-diagonal elements of a_μν [see Eqs. <ref> and <ref>].  Therefore, as U_d is increased in the original Hamiltonian, the discreteness of diagonal elements E_μμ becomes more important, and we expect t_m to increase in this effective dual description.  

To analyze the large t_m and t”_m limit, we make the changes of variables

    α_μν   →   α_μν + Δ_μϕ_ν
    ψ_μ   →   ψ_μ + Δ_τϕ_μ.

We thus obtain constraints α_μν = 0 (for μ≠ν), and ψ_μ = 0.  The first of these constraints implies E_μμ = 0, as expected in the large U_d limit.   We define h_μ≡α_μμ, and write the resulting action as

    S^ hollow_ dual   =   u_od/4 π^2∑_τ, r[ (Δ_z h_x - Δ_z h_y)^2
    + (Δ_x h_y - Δ_x h_z)^2 
       +    (Δ_y h_z - Δ_y h_x)^2 ]
    + 1/8 π^2 k∑_τ, r∑_μ ( Δ_τ h_μ - θ )^2  
       -    t'_m ∑_τ, r∑_μ < νcos( h_μ - h_ν  )  .

which is invariant under the gauge transformations

    h_μ   →    h_μ + f 
    θ   →   θ + Δ_τ f ,


It was shown previously by Xu and Wu that this theory is a dual description of the hollow U(1) scalar charge theory,<cit.> so that this theory indeed describes the system upon condensing monopoles within their d=2 planes of motion.  Moreover, Xu and Wu also showed that the cosine term in the above action is relevant, corresponding to a proliferation of instantons in space-time, that results in a gapped confined phase.<cit.>  




 §.§ X-cube model from lattice hollow U(1) scalar charge theory by Higgs mechanism




Here, we couple the hollow U(1) rank-2 scalar charge theory to charge-2 matter, and show that condensing this matter results in the X-cube model.  The details of the analysis are very similar to that carried out in Sec. <ref>, so we only summarize the key points.  Starting with the Hamiltonian Eq. (<ref>), we add a charge-2 matter field with number N_ r and conjugate phase Θ_ r.  We add the following terms to the Hamiltonian

    H_2e   = u_2 ∑_ r N_ r^2 - Δ∑_ rcos[ Θ_ r -2 θ_ r] 
        - J_2 ∑_ r,μ < νcos[ Δ_μΔ_νΘ
    -2 A_μν( r)] .

The charge-2 matter condenses for sufficiently large J_2.  As in Sec. <ref>, we take both J_2 and Δ to be large, which allows us to treat the corresponding cosine terms as constraints.

We obtain a model defined in terms of symmetric-tensor Z_μν, X_μν Pauli operators with μ≠ν living on plaquettes, and τ^x_ r, τ^z_ r Pauli operators living on sites.  As in Sec. <ref>, X_μν is a rank-2 ℤ_2 electric field, Z_μν the conjugate rank-2 ℤ_2 gauge field, and τ^x_ r is the ℤ_2 gauge charge of the un-condensed charge-1 matter.  The Hamiltonian is

    H^ hollow_ℤ_2   =    - J ∑_p, μ < ντ^z_ r_1τ^z_ r_2τ^z_ r_3τ^z_ r_4  Z_μν(p) 
       -    K ∑_ r, μ F_μμ - u ∑_ rτ^x_ r - U ∑_ r, μ < ν X_μν.

Here, the sum in the first term is over plaquettes p, and the sites r_1, …,  r_4 lie at the corners of p.  F_μν is defined in Sec. <ref>; only the diagonal elements play a role here, because only the diagonal elements B_μμ appear in the U(1) hollow gauge theory.  The Gauss' law is

    G^h_ r≡∏_p ∼ r X_μν(p) =  τ^x_ r,

where the product is over the 12 plaquettes sharing a corner with r.

This theory can be viewed as a hollow rank-2 ℤ_2 scalar charge theory on the cubic lattice.  To see it is equivalent to the X-cube model, we exploit the same mapping to a local bosonic model discussed in Sec. <ref>.  The resulting Hamiltonian is 

    H̃^ hollow_ℤ_2   =    - J ∑_p, μ < νZ̃_μν(p) - K ∑_ r, μF̃_μμ
       -    u ∑_ rG̃^h_ r - U ∑_ r, μ < νX̃_μν.

Passing to the dual lattice so that the plaquette variables become link variables, the F̃_μμ term becomes precisely the vertex term of the X-cube model, while the G̃^h term becomes the cube term (see Fig. <ref>).  Therefore, we have obtained precisely the X-cube model, perturbed by the J and U terms, which give dynamics to the fractons and one-dimensional particle excitations, respectively.









§ CHECKERBOARD MODEL AS A ℤ_2 SCALAR CHARGE MODEL 



All the rank-2 gauge theories considered thus far have a Gauss' law of the same form at every point in space.  Here, we construct a rank-2 ℤ_2 gauge theory with two different Gauss' laws at different vertices of the cubic lattice, and show that this theory is equivalent to the checkerboard fracton model.<cit.>.


We work on the cubic lattice, and place qubits on xz and yz plaquettes, with no degrees of freedom residing on xy plaquettes.  We denote the corresponding Pauli operators by X_xz (yz) and Z_xz (yz).  It is convenient to work with electric field E_μν and gauge field A_μν variables introduced by defining X_μν = exp(i π E_μν ), and Z_μν = exp(i π A_μν), where E_μν and A_μν take values in { 0, 1}.  We write

    E_μν = [ [    0 E_xz E_yz; E_zx    0    0; E_yz    0    0 ]]      A_μν = [ [    0 A_xz A_yz; A_zx    0    0; A_yz    0    0 ]]

Taken together, the centers of the xz and yz plaquettes form a finer simple cubic lattice.


We impose two  Gauss' laws.  On layers with even z coordinate, we impose Δ_x Δ_z E_xz + Δ_y Δ_z E_yz = n_ r.  The corresponding charge resides at even-z vertices of the cubic lattice, as shown in Fig. <ref>a.  On odd-z layers, we impose Δ_y Δ_z E_xz + Δ_x Δ_z E_yz = n_ r', where r' is the center of an xy plaquette, so that charge resides on xy plaquettes (see Fig. <ref>b).  Viewing these Gauss' laws in terms of the finer cubic lattice, we see that they are defined on a “checkerboard” of edge-sharing cubes whose centers (where charges reside) form a FCC lattice.



From the Gauss law, we can obtain the gauge transformation of A_μν, e.g. A_yz ( r) → A_yz( r) + Δ_y f( r- 1/2ẑ) + Δ_x f( r+ 1/2ẑ) as shown in Fig. <ref>(d). The simplest gauge-invariant combinations of the A_μν are sums of eight variables that take precisely the same form as the sums in the two Gauss' laws.  Passing to a local bosonic model as above, the Hamiltonian consists of a term imposing the  Gauss' laws, and the gauge-invariant terms built from A_μν.  In terms of Pauli matrices, 
we have

    H_ckb = -∑_C∏_i ∈ C  X_i -∑_C∏_i ∈ C  Z_i

where C denotes the edge-sharing cubes of the checkerboard lattice.  This is precisely the Hamiltonian of the checkerboard fracton model.








§ DISCUSSION



In this work, we showed that Higgs and partial confinement mechanisms provide relationships among gapped fracton phases and U(1) symmetric-tensor gauge theories.  These relationships also encompass more conventional topological orders, e.g. the rank-2 ℤ_2 scalar charge theory that is obtained from the rank-2 U(1) scalar charge theory by condensing charge-2 matter, and which becomes the X-cube fracton topological order upon selective loop condensation. Our results are a starting point to investigate quantum critical phenomena at transitions between different fracton states.  At a more basic level, the mechanisms we discuss give insight into fracton phases, by allowing us to understand the degrees of freedom in terms of another proximate phase.

In this work, we focused on the scalar charge theory, but there are other rank-2 U(1) gauge theories<cit.> with different forms of Gauss' law, e.g. the vector charge theory and traceless scalar charge theories. It will be interesting to investigate what types of fracton topological order can arise from more general higher-rank gauge theories, via Higgs mechanisms and partial confinement transitions. In addition, it may be fruitful to explore whether new types of fracton topological order can arise as gauge theories with different forms of Gauss' law on different lattice sites, along the lines of our presentation of the checkerboard fracton model as a rank-2 gauge theory.

Another open question is whether any “type II” fracton phases like that in Haah's code, where all non-trivial excitations are fractons created at the corners of fractal operators, are  related to U(1) symmetric-tensor gauge theory in a manner similar to the examples discussed here.  In a certain sense, it is obvious that Haah's code can be viewed as a gauge theory, where one type of term in the Hamiltonian (either the X or Z term) is viewed as implementing a Gauss' law constraint energetically.  The question then becomes whether this Gauss' law can emerge from simpler or more familiar theories by some sequence of Higgs and/or partial confinement transitions, which could shed light on type II fracton topological order beyond the realm of exactly solvable models.






§ ACKNOWLEDGEMENT


H.M thanks Michael Pretko for insightful discussion. M.H. and H.M. are supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences (BES) under Award number DE-SC0014415. X.C. is supported by the Caltech Institute for Quantum Information and Matter, the Walter Burke Institute for Theoretical Physics, the Alfred P. Sloan research fellowship, and National Science Foundation under award number DMR-1654340.

Note added: We became aware of a parallel investigation <cit.>
in which the Higgs mechanism is applied to several rank-2 gauge theories and the resulting phases are studied. Where our results overlap, they agree. 









§ GROUND STATE DEGENERACY AND LOGICAL OPERATORS OF THE ℤ_2 SCALAR CHARGE THEORY


In this section, we calculate the ground state degeneracy in the deconfined phase of the rank-2 ℤ_2 gauge theory on the cubic lattice.  We also discuss the logical operators and argue that the degenerate ground states are locally indistinguishable, i.e. the ground state degeneracy is topological in nature.  We work at the exactly solvable point and use the description in terms of a bosonic model (i.e. one with a tensor product Hilbert space), whose Hamiltonian is given by Eq. (<ref>) with U = J = 0.

In a L× L × L system, there are L^3 unit cells with four distinct sites within each unit cell. Three sites lie on the plaquettes, with one qubit on each plaquette. The other site lies on the vertex and has three qubits. In total, the number of qubits is 6L^3.  We always take L to be even.

We count the number of independent stabilizers. There is one octahedron term G_ r per vertex, thus we have L^3 of these terms in total.  However, these terms are not all independent and satisfy some constraints.  We separate the vertices into eight groups, according to the parity p_μ = r_μ 2 of the coordinates (μ = x,y,z), labeling the groups by (p_x, p_y, p_z).  The product G_(p_x, p_y, p_z) = ∏_ r ∈ (p_x, p_y, p_z) G_ r is a product of X_xy over xy planes with z  2 = p_z, X_yz over yz planes with x  2 = p_x, and X_zx over zx planes with y  2 = p_y. We can think of the operators G_(p_x, p_y, p_z) as lying at the vertices of a cube, and it is easy to check that the product of these operators over any face of the cube is unity.  This gives four independent constraints, and the number of independent vertex terms is L^3-4.


The magnetic field terms are more complicated, and to count them we resort to a numerical method employed previously in Ref. ma2017fracton, which we also use to check the counting of X stabilizers above.  A product of X or Z Pauli operators is viewed as an element of the 𝔽_2 vector space V_X(Z)≃ (𝔽_2)^6 L^3, with vector addition corresponding to operator multiplication.  The set of all X (Z) stabiliers is a subspace S_X (Z)⊂ V_X(Z), and the number of independent stabilizers is the dimension of this subspace.  Any X stabilizer is a product of the L^3 operators G_ r; in linear algebra language, we can say that G_ r gives a spanning set for S_X.  This spanning set can be represented as a L^3 × 6 L^3 matrix, and its rank, which can be determined via row reduction, gives the dimension of S_X.  The same method can be applied to Z stabilizers, where F_μν constitute a generating set.  We used this method for even L = 2,…,12 to confirm that the number of independent X-stabilizers is L^3 - 4, and to determine that the number of independent Z-stabilizers is 5L^3 - 8.

This implies that the ground state degeneracy is log_2 (GSD)=6L^3- (L^3-4)-(5L^3 - 8) = 12, which is equal to the degeneracy of four copies of the toric code on a 3-torus.  To check that the degeneracy is topological in nature, using the same numerical method as above, we return to the space S_Z of Z-stabilizers described by its generating set.  For each type of charge τ^z_1,…, τ^z_4, we add to the generating set three large string operators that transport the charge around the three cycles of the 3-torus.  These string operators are products of Z_μμ, and we choose them to run along straight lines, with their transverse position arbitrary.  Upon adding these operators to the generating set, we now find 5 L^3 + 4 independent Z-operators, including both Z-stabilizers and string logical operators.  Adding the string operators thus fully resolves the space of degenerate ground states.  Moreover, because the transverse position of the string operators is arbitrary, they can be chosen to avoid the position of any local operator of interest, and it follows that the ground states are locally indistinguishable, as expected for four copies of the d=3 toric code.




§ ℤ_2 SCALAR CHARGE THEORY IN TWO DIMENSIONS 


In this section, we construct the rank-2 ℤ_2 scalar charge theory on the square lattice in two dimensions. The ℤ_2 variables are defined in a similar way as in d=3: each site has two diagonal components while the off-diagonal components are defined on the plaquettes. The Hamiltonian is

    H_ℤ_2^2d = -K ∑_ r,μ,ν F^2d_μν - u ∑_ r G^2d_ r

where F_μν is a product of 4 Z_μν operators around each link and G_ r is a product of 8 X_μν operators around a site, as shown in Fig. <ref>. 


We now show that this model is equivalent to three copies of the d=2 toric code. Numerically, using the method we reviewed in Appendix <ref>, we obtained the ground state degeneracy of this model on a L × L 2-torus, with even L=2,…,20, finding log_2 ( GSD)= 6. This is the expected result for three copies of the d=2 toric code.



Point-like charge excitations are created at the sites by acting with Z_μν. Acting with Z_μμ( r), we create two charge excitations at the two neighbors of r in the ±μ direction. Charges can thus hop two lattice spacings in the x and y directions (Fig. <ref>a). Naively, it might seem that the four charges at corners of a plaquette are distinct types of excitations.  But since these four charges can be created together by acting with Z_xy, there are actually three different types of charges.


Acting  with X_μν  creates magnetic flux excitations living on the links, denoted as m_1,2,3 (Fig. <ref>(b)). They are also mobile excitations. For example, m_1 can be transported along the y direction by acting with a string operator that is a product of X_xx operators along the y direction. The same excitation can be transported along the x direction by applying a “thick” string operator, as shown in Fig. <ref>(b). m_1 can hop one lattice spacing in the y-direction, and two lattice spacings in the x-direction. The other flux excitations have similar mobility. Similar to the charge excitations, four flux excitations on the edges of a plaquette are created together by acting with X_xy, which means that there are three rather than four distinct types of fluxes.




Next, we can check the braiding between magnetic fluxes and electric charges by simply considering the commutation relation between the corresponding string operators at a crossing point. The result is listed in Table <ref>. It is straightforward to verify that the flux type is fully resolved by braiding with the three different electric charges.





All the analysis above, including the ground state degeneracy, logical operators and properties of the excitations, shows that the rank-2 ℤ_2 scalar charge theory is three copies of d=2 toric code, which is of course a conventional topological order without any fracton or sub-dimensional excitations. This is consistent with our prediction from Higgsing the conservation law in Sec. <ref>. In three dimensions, we can gap the diagonal components of the tensor fields to get a hollow tensor gauge theory describing fracton topological order. However, in two dimensions, the corresponding hollow gauge theory gives a classical state, because it lacks a gauge-invariant flux term.


