Summary of fracton stuff so far

Transcription

1 Summary of fracton stuff so far Ethan Lake (Dated: August 5, 2016) Contents I. Introduction 1 II. Reconstructing the X-cube model from coupled toric codes 2 A. Doing the coupling 2 B. Excitation spectrum and braiding statistics 3 C. Other ways to couple the planes together 5 1. ZZ coupling 5 2. Y Y coupling 5 D. Ground state degeneracy 6 1. From the excitation spectrum 6 2. From counting degrees of freedom 7 3. GSD with other coupling terms 8 E. Z N generalization of X-cube model 8 F. Reducing the dimensionality of the e excitations 9 III. Reconstructing the checkerboard model from coupled toric codes 10 IV. The X-cube and checkerboard models from coupled Ising layers 12 A. The X-cube model 12 B. The checkerboard model 13 V. List of approaches that haven t worked 14 A. Obtaining the toric code by gauging the 2d Ising model 14 References 15 I. INTRODUCTION The following notes are a summary of the fracton-related things I ve worked out over the last month or so. We ll primarily be looking at how to construct fracton models from TQFT building blocks. The most ambitious question we might like to address is To what extent can fracton models be described by TQFT? Of course, we (so far) can only begin to answer this question. To take the first steps, we will primarily focus on reproducing the fracton topological phases that have previously appeared in the literature rather than attempting to produce new examples of fracton phases. In particular, we will primarily be concerned with reproducing the X-cube model 1 and the checkerboard model 1,2. Before we begin, some comments on notation. There are essentially two types of spins we will be interested in throughout this paper: those associated with gauge fields (like the operators appearing in the toric code) and those associated with physical spins (like the operators appearing in the Ising Hamiltonian, which we will use less frequently). We will write the Pauli operators for gauge field spins (which normally are written as σ x, σ y, σ z ) as X, Y, Z, and the Pauli operators for the physical spins (normally written as τ x, τ y, τ z ) as X, Y, Z. Additionally, we will use fraktur font to label subdimensional particles and fractons. Fracton models generically have electric and magnetic excitations, which we will write as e and m, respectively. This is done to avoid confusion with the particles in the toric code, which we denote as usual by e and m.

2 2 II. RECONSTRUCTING THE X-CUBE MODEL FROM COUPLED TORIC CODES A. Doing the coupling First, we will show how to reproduce the X-cube model by coupling together layers of toric codes, which was the idea that kicked off this whole project. The building blocks of our model are orthogonal copies of 2d toric codes, aligned in the xy, yz, and xz planes which are copied along each unit vector in 3d to form a giant stack of planes. We denote Pauli operators for a link l on the plane p as Z p l and X p l. We couple the 2d planes together along their links, so that two spins live on each link of the lattice. Explicitly, we couple the layers together by adding a term that favors the alignment of the x component of the two spins present on each link. With this choice, the Hamiltonian looks like H = H p T C h X p l Xq l, (1) p p q l p q where p and q run over all planes in the stack and H p T C is the normal toric code Hamiltonian on plane p: H p T C = + p A p + p B p, (2) where +s denote vertices, s denote plaquettes, and we have choosen the convention B p = A p + = X l. (3) l Z l The second term of Eq. (1) is the coupling between layers. As h, the x-components of the spins on each link align with one another, but the z-components remain decoupled, giving a single effective Pauli algebra on each site, which is consistent with the size of the Hilbert space used in the original formulation of the X-cube model. Since the XX coupling does not commute with the plaquette terms for original toric code, our model is not exactly soluble as it stands. To find a better formulation of the plaquette terms, we treat them as a perturbation, writing the Hamiltonian as H = H 0 + H 1, where H 0 = A p + h X p l Xq l, p + p p q l p q H 1 = t (4) B p p p for t 1 and h 1. The plaquette operator in H 1 flips the x-components of four spins a plaquette, which increases the energy of the system by 8ht, and hence takes us out of the low-energy manifold, forming a coupling between ground states and excited states. If we treat this B coupling perturbatively, we can focus only on operators that stay within the low-energy manifold, which means that we must get rid of the original B plaquette term. To find what sort of operator we can replace the plaquette term with, we construct different combinations of the plaquette operator, and find which combinations of plaquettes act to stabilize the low-energy manifold. We choose the operator from this list with the smallest power of t, and put that operator in the Hamiltonian. The quickest way to get back to the low-energy manifold after the action of a B operator is to flip all the 24 spins on a cube which includes as a face. This requires five additional applications of B, taking us to sixth order in perturbation theory. Let us write this combination of six B operators as a new cube operator, defining for any cube c in the lattice the operator C c = B p, (5) c where the product is over all plaquettes on the boundary of the cube c. This cube operator enters the Hamiltonian at O(t 6 ), and so our effective low-energy theory is (with a schematic t 6 coefficient) H = t 6 C c A p + h X p l Xq l. (6) c p + p p q l p q l + We then obtain the X-cube Hamiltonian by passing to the h limit.

3 How do we interpret this construction from a physical perspective? One can begin by noting that [H, X l,p X l,q ] = 0 if l p q, and so objects created by the action of X l,p X l,q have no energy, and so the ground state will consist of a superposition of all possible things that can be created with X p l Xq l operators. Remembering the original B plaquette term, we see that X p l Xq l creates a cross-shaped bunch of four m-particles. We can imagine connecting these m-particles with magnetic strings free to live in the continuum. We label these loops of magnetic string m-chains, which look like 3 In the picture, the s mark the location of m particles, the dashed grey lines are m-strings, the s are the effective spins that lives on the links after we pass to the h limit, and the blue string represents an m-chain loop (created by acting with X p l Xq l on the center link) that forms the building blocks of the condensate. So, we see that the X-cube model is secretly a theory of condensed m-chains, which are magnetic string loops built out of two m-strings on a pair of orthogonal planes. (7) B. Excitation spectrum and braiding statistics We now examine the types of excitations produced in this model, an analysis which is made more physically transparent by thinking in terms of the condensed m-chains picture. There are two types of excitations, both of which are subdimensional: one of the types is completely immobile on its own and is associated with m particles, being essentially half of a single m particle. More explicitly, we can see from the Hamiltonian that two excitations are created at the ends of broken m-chains. That is, the ground states of the theory are superpositions of closed m-chains, with open m-chains appearing as excitations (this is in contrast to 3d toric code, where magnetic loops cannot break). The m-chain end fractons are just the 1 eigenstates of the cube operator C c, and so we will also call them cube excitations, which we denote by m. It s straightforward to check that the m excitations can only appear in multiples of four, and are located at the corners of a membrane operator created by acting with X on a rectangular array of links (which creates a sheet of m-chain lines). While they are immobile on their own, pairs of cube excitations are mobile in two dimensions. More specifically, a pair of cube excitations lying at the ends of one edge of a membrane of X operators can move freely in the plane normal to the edge, but are not allowed to move along the edge direction (as they create additional cube excitations if they attempt to do so). The m cube excitations are associated with m particles, and so we should expect the existence of another type of fracton excitation associated with the e particles. We will denote this other excitation (the electric fracton) as e, and we will show that it is only mobile along 1d submanifolds. Additionally, we will show that an e excitation has π mutual statistics with an m cube excitation, provided that the m cube is not free to move in the plane normal to the movement of the line excitation (in which case the mutual statistics is trivial). In order for the e excitations to be deconfined, they must have trivial braiding with the m-chain condensate. Figuring out what kinds of things braid trivially with the m-chain condensate will tell us what the e excitations have to be. Thinking about the braiding of prospective e particles with m-chains is a bit tricky if we work at the lattice level, and so we will think of things as being in the continuum limit in what follows to aid in visualization. The kind of braiding process we need to look at is one in which an m-chain loop winds around an e particle. This braiding procedure is illustrated in the following figure, where we have marked the e exicitation as a black dot and

4 the m-chain loop as a solid loop, with the small circles marking the intersection points of the m-chain loop with the xy plane (which marks the location of a pair of m xy particles): 4 e z y x (n.b. this figure is not on the same scale as Figure 7, and the black dot at the center is an e excitation, not a spin!) The curvy loop with the arrow in the figure illustrates the path that the intersection of the loop with the xy plane takes during the braiding procedure. Since the m-chain loops are just stacks of a bunch of m particles, the intersection of the loop with each plane defines the location of an m particle in that plane. Thus, moving the left small black dot around the e particle is equivalent to braiding e with an m xy, giving a braiding phase of θ e,mxy. We don t need to worry about the intersection of the m-chain loop with any other xy plane in the stack of xy planes, since only the xy plane on which the e excitation sits will be relevant for computing the braiding statistics (the same applies for the yz and zx planes). However, the θ e,mxy braiding is not the only braiding process that occurs. If we track the intersections of the m-chain loop with the xz plane, we see that we get a trajectory like (8) z x (9) where the big black dot is the e excitation. This is equivalent to braiding an m xz around the e, and so we get another phase factor of θ e,mxz. Finally, we need to pay attention to the intersection of the m-chain loop with the yz plane. This one requires a bit more imagination, but once we visualize the procedure correctly, we see that we get a picture like z y (10) which is equivalent to braiding an m yz about the e excitation. Putting everything together, we see that θ m chain loop,e = θ e,mxy θ e,myz θ e,mzx. (11)

5 Of course we are free to wiggle the loop around before we begin the braiding procedure, and so this braiding phase is the same with all m-chain loops in the condensate. We thus see that if e is to built from e particles and is to avoid confinement, it must be a composite of two distinct e-particles which live on different planes. This in turn implies that it must be restricted to move along one dimension. This argument tells us that the only e s that survive the m-chain condensation must exist in pairs. However, we note that we could have seen this from the Hamiltonian without having to draw pictures. If we take the h limit, it s impossible to create a single pair of e s that live on the same plane, since the relevant Z operator would anti-commute with the hxx term. However, the ZZ operator that creates two pairs of e s does commute with the hxx term, and so creating pairs of es is allowed. 5 C. Other ways to couple the planes together For completeness, we now briefly mention two other ways to couple the toric codes together, namely with the coupling terms hxx and hy Y. Something like h(xx + ZZ) is boring, since taking h over-constrains the dimensions of the Hilbert spaces on each site and leads to a trivial theory. 1. ZZ coupling First we consider hzz coupling, and treat the vertex term A + perturbatively. The effective Hamiltonian is H eff = Z p l Zq l, (12) vertices v D v p p B p h p q where the D v operator appears at third order in perturbation theory and is a product of X s over all 12 spins adjacent to the vertex v (i.e. a product of all three distinct A p + operators at v). In this case, an e-chain consisting of four e-particles located at two adjacent vertices (created by applying Z p l Zq l at a link l) has zero energy cost, and so these e-chains will condense. What happens to the m-particles? It is instructive to take the strong coupling h limit, in which the only way we can create m s is by acting with XX on a link, which creates the mini chain loop of four ms familiar from the XX coupling case. This is sort of the electromagnetic dual of the hxx coupling situation here, e chain loops condense, and m s must appear in pairs. There is a key difference, however. In the hxx coupling case, the pairs of e s we could create were deconfined (as long as they moved along a straight line). Here though, m strings have a constant energy density, since at each segment in the m string, little transverse m strings must appear, which cost finite energy since the B term is still present in the Hamiltonian. So while pairs of m s can survive the e chain loop condensation, they are completely immobile, and cannot move without creating additional excitations. However, they are not fractons in the usual sense, since operators which create them do not have their energy densities supported entirely within finite regions localized at the ends of the operators. What about the excitations created by the D v term? We can create them in pairs by acting with Z on one spin of a link. Notice that applying Z operators in a string creates a string for these D v excitations, and so they behave like e particles in the 3d toric code, in the sense that they are completely deconfined, and free to move in any direction. Thus, we end up with a theory with no fractons, a single confined magnetic excitation, and a completely deconfined electric charge. l p q 2. Y Y coupling Now for an hy Y coupling term. If we treat the coefficients of both the vertex and plaquette terms as small compared to h, then the effective (And schematic) Hamiltonian is H eff = cubes c C c vertices v D v h p q l p q Y p l Y q l, (13) where the D v operator is the same operator as in the ZZ coupling case. What kind of things condense in this case? Acting on XX or ZZ on a link is allowed, which creates no excitations and commutes with the interlayer coupling. This means that chain loops of m particles and chain loops of e particles should both be condensed.

6 We cannot act with just X or Z on a spin without violating the coupling term. So if we take the h strong coupling limit, C c excitations and D v excitations cannot appear on their own. Even in the h < case, the strings for both types of excitations have a constant and nonzero energy density. The only types of excitations that we can create, then, are created by hitting a spin with Y, which creates a cluster of two D v excitations and the usual set of four C c excitations. However, it is straightforward to check that these excitations are completely immobile, since their strings cannot be created without also creating other excitations. Again, they are not fractons in our sense of the word, since membrane operators which create them have a constant energy density. So, a Y Y coupling term leads to a pretty boring phase. Finally, we may wonder if anything new happens if we couple the toric code planes together along their faces, rather than along their links. As one may expect, this essentially just implements electromagnetic duality, and gives us nothing new. 6 D. Ground state degeneracy 1. From the excitation spectrum In Fu & Co. s paper 1, the GSD for the X-cube and checkerboard models is obtained by using powerful but rather opaque algebraic geometry stuff. It d be a bit nicer if we could get reproduce their results just using toric code technology, which we will do first for the X-cube model. There are a few ways to get the ground state degeneracy. We ll start with the one that involves the spectrum of quasiparticle excitations. From what we know from our knowledge of regular topological order, we can compute the GSD by choosing from either the electric or magnetic sectors, and determining the number of different ways to thread electric or magnetic fluxes through the non-contractible cycles of whatever manifold we re working over (which in our case will be the 3-torus). First, let s look at the magnetic sector. One might think that we just need to count the different ways to thread m-strings through the different cycles of the 3-torus. This isn t quite true, however we actually need to count the different ways to thread m-strings. There are two ways to make an m-string. One is act with X on a series of parallel links, arranged in a pattern like the rungs of a ladder (like the m-string operators in the regular toric code), which act on a single spin on each link. This creates two cube excitations at each end of the string. The other way is to act with X on a series of parallel links in the same pattern, but acting on both spins on each link. This creates an m-string along the ladder direction, as well as a bunch of mini m-strings transverse to the ladder direction, creating a closed m-chain loop with no cube excitations. The difference between these two operators is that the latter can be applied while staying within the ground state subspace of the system, since such an operator creates an m-chain loop that is part of the condensate. On the other hand, the first method defines an m-string operator that is not part of the condensate, and creates cube excitations at the string ends. The reason for bringing up this point is that it is not enough to compute the GSD by counting the number of m strings that run along each noncontractible cycle of the 3-torus, since both procedures outlined in the last paragraph can do this. The GSD is only affected by m strings that loop around a noncontractible cycle and are of the first type: i.e. are such that if we cut them, we produce cube excitations (as opposed to the m-strings in the condensate, which create no excitations when cut). By cutting, we mean inserting an extra empty lattice plane in the middle of the string, which is oriented orthogonally to the string direction. How do we classify such m-string loops? Consider an m-string that forms a loop around a given noncontractible cycle of the 3-torus, which we will take to be of size L L L, where L is the number of lattice sites along a side of the 3-torus. Now consider cutting the m-string by inserting an extra plane in the lattice, on which no X operators have been applied and which is oriented so as to be normal to the m-string in question. This cutting procedure creates four cube excitations, which must fuse in pairs to the vacuum if we collapse the inserted plane. This means that in order to find the GSD, it is enough to count the number of inequivalent ways to place two cube excitations in a plane, with the total GSD computed by raising this number to the third power (one power for every noncontractible cycle in the 3-torus). To complete the classification of ways to arrange pairs of cubes on a plane, we observe that pairs of cube excitations are always free to move within a plane normal to the line which connects them. For the case at hand, one of these directions must be along the direction of the noncontractible m string in question, while the other can be chosen from the two remaining directions. Let p denote the L L plane in question, which is normal to the m-string. The m-string ends can form two types of cube excitations in p those that are free to move along the rows of p, and those that are free to move along the columns of p (the terminology is appropriate since p is a square lattice). Since each column (row) can be either occupied or unoccupied by a column- (row)-moving cube, m-strings passing through p

7 contribute 2 L 2 L ground states. This isn t quite correct though, since cubes must always appear in pairs, and so we need to restrict ourselves to those states with an even number of cube fluxes passing through p. This means that p actually contributes 2 2L 1 degenerate ground states. Since there are three in-equivalent choices for the plane p (three noncontractible cycles on the torus), we obtain log 2 (GSD X cube ) = 6L 3, (14) which agrees with the algebraic geometry calculation perfrormed in Ref 1. Thinking in terms of the e excitations rather than the m excitations also works, but is a bit more complicated to think about due to the more restricted mobilities of the e particles From counting degrees of freedom We now overview an alternate, but perhaps simpler way of computing the GSD, done by directly counting degrees of freedom in the Hilbert space. Each plane in our system contains 2L 2 spins, giving 2L 3 spins in a stack of all of the planes of a given orientation, and thus giving 6L 3 spins in the entire system, as there are three orientations of stacked planes in our construction. Thus, our unconstrained Hilbert space has dimension 2 6L3. Our effective Hamiltonian of course imposes constraints on the configurations of these spins in order to remain in the ground state. We have one constraint for each vertex term A + in H. There are 3L 3 such vertex terms, since acting with all vertex terms acts on each spin twice. We also have one constraint for each cube term C c (L 3 of them), and one constraint for each hxx term (3L 3 of them), and so we have 7L 3 constraints. Uh oh is our system over-determined? Not quite, since we also have relations between these constraints. One such relation involves the product of the three types of A + terms at each vertex. We said that the only excitations allowed at vertices were pairs of e excitations. Thus, an even number of A + terms at a given vertex must have eigenvalues of 1. This means that if we know the eigenvalues of two A + s at the vertex +, we automatically know the the third, due to the hxx constraints which constrain the product of the A + terms at each vertex. This gives us one relation for each of the L 3 vertices in the system. Additionally, we see that the 3-torus geometry means that the product of A + terms over any plane must be equal to 1, giving us one constraint per plane. There are 3L planes in the system, giving extra 3L relations. We can say the same thing for products of cube operators C c throughout planes, producing an additional 3L relations. However, the products of the C c cube operators don t actually give us 3L relations between the constraints. They actually only give us 3L 3 relations, since the product of C c s over all but one plane of the 3-torus tells us the product of C c s over the last remaining plane. In summary, we have #(spins) = 6L 3 #(constraints) = 3L 3 + 3L 3 + L 3 #(relations) = L 3 + 3L + 3L 3 (15) implying a ground state degeneracy of log 2 (GSD X cube ) = 6L 3, (16) confirming our earlier result. It is interesting to compare this result to what we would get from a stack of decoupled toric codes. Since each plane in the stack is its own torus with GSD = 4 and since there are 3L such planes, we obtain log 2 (GSD decoupled ) = 6L, (17) which is different from the result established earlier by only the universal subleading correction of 3. Evidently the effect of introducing coupling between the planes leads to a very slight increase in information (8 bits), which is independent of how many planes we couple together! Pretty cool. This universal contribution of 3 to log 2 (GSD) can also be understood from a slightly different perspective, a la Pretko s approach in Ref 3. To turn our coupled toric codes model into a theory of weird electromagnetism, we can associate the X Pauli operators with e ie and the Z operators with e ia, where E and A are schematically the electromagnetic field and vector potential. We can write E as a 3 3 matrix, where the rows index the orientation of a plane in the stack (i.e. the xy, yz, or zx planes) and the columns index the direction (x, y, or z) of the electric field.

8 Since each plane in the stack is two-dimensional, it can only have two nonzero electric field components. This means that we can set three of the entries in E to be zero the z component of E on the xy plane, the x component of E on the yz plane, and the y component of E on the zx plane. Thus, before we add the δh coupling, the E matrix has six independent Z 2 degrees of freedom (since the E field is Z 2 -valued). When we add the δh hxx coupling and work at energy scales below h, we can further constrain the E matrix. Namely, we can set the x components of E to be equal for the xz and xy planes, the y components of E to be equal for the xy and yz planes, and the z components of E to be equal for the yz and zx planes. This reduces the number of Z 2 degrees of freedom in the E matrix to three. Thus, we see that the effect of the δh coupling on the GSD is to simply reduce it by three Z 2 degrees of freedom, leading into a contribution to log 2 (GSD) of GSD with other coupling terms For completeness, we will also compute the GSD for the other ways of coupling the planes together, although it is easy to guess the answers. First let s look at the hzz coupling. As we saw earlier, the electric sector of the excitation spectrum consists of a single completely deconfined excitation, that is free to move however it likes. Thus, the GSD is simply log 2 (GSD ZZ ) = 3, (18) because each of the three noncontractible loops of the 3-torus can either be threaded or not threaded by a flux of the electric sector excitation. Can this also be understood in terms of the magnetic sector? The magnetic excitation is a zero-dimensional nonfracton excitation, and so at first it seems like we can t say much in terms of flux threading. However, we note that we can still act with a membrane XX operator that spans an entire plane, since such an operator creates no m-particle or D v excitations. Such a membrane is mobile in the direction normal to the plane defined by the membrane (by acting with D v operators), and so such membranes are specified by the two noncontractible loops they span (the membrane is equivalent to a 2-torus, and so are classified by which orientation of 2-torus they correspond to). There are three ways to choose two noncontractible loops, and since a membrane can either span or not span these two loops, we confirm that the GSD is 2 3 = 8. As for the Y Y coupling, we only have one immobile non-fracton excitation, and so we can t compute the GSD by threading fracton flux through the cycles of the 3-torus. However, we observe that a plane of parallel lines built out of X p l Zq l operators stretching across two noncontractible loops has zero energy and yet cannot be created smoothly out of the vacuum without producing additional excitations. Crucially, this membrane is completely immobile in the direction normal the membrane. Thus there are 2 L possible membrane configurations for each membrane orientation, giving log 2 (GSD Y Y ) = 3L, (19) which is purely subextensive and has no subleading constant term, in contrast with both other types of coupling. This tells us that there is nothing really topological about this theory, which agrees with our assessment of its excitation spectrum. We see that just by changing the ways in which the planes are coupled, we change the GSD from being purely sub-extensive, to sub-extensive with a topological correction of 8, and finally to a completely topological value of 8. Thus, the amount of information carried by the system is extremely sensitive to the way we couple the planes together. E. Z N generalization of X-cube model Obtaining a Z N generalization of the X-cube model is pretty straightforward. Figuring out how to modify the X and Z operators to the Z N case is straightforward: we use the Z N shift and clock matrices, which are defined explicitly through N N X = k k + 1, Z = e 2πik/N k k, XZ = e 2πi/N ZX. (20) k=1 k=1 Figuring out how to modify the Hamiltonian is not as straightforward, since the algebra defined by X and Z is no longer Hermitian. A proper Hamiltonian can be constructed by giving both the faces and vertices of the square lattice alternating colors, but it s kind of a pain (see 4 for the basic idea for the 2d case).

9 Rather than writing out the full Hamiltonian, we will appeal to the what else can it be? tactic to find the excitation spectrum. To create the Z N X-cube model, we take three stacks of Z N toric codes and couple them together. We then perform m-chain loop condensation by allowing magnetic strings in each layer to lift up and propagate in the full 3-dimensional stacked system. The ground state, as in the N = 2 case, consists of a superposition of closed m-chain loops parametrized by integers n Z N and a choice of orientation. Of course, the regular 2-dimensional B plaquette operators can no longer appear in H, since we have relaxed the condition that the net flux through any plaquette vanish. As in the N = 2 case, doing perturbation theory means finding an operator for the magnetic fluxes that has an eigenvalue of 1 in the ground state. This operator is a cube operator C c, and is similar to the one in the N = 2 case it measures the amount of magnetic flux coming out of a cube. The only difference in the Z N generalization is that now it needs to keep track of wether the flux on a cube face is passing into the cube or out of the cube. In the ground state, the net flux through any cube is zero, corresponding to a C c = 1 eigenstate of the cube operators. The magnetic excitations in the excitation spectrum are simply Z N generalizations of the cube excitations in the N = 2 version. They have the same movement as before, but can now fuse amongst themselves to other nontrivial cube excitations, with the fusion rules of course given by addition in Z N. The electric excitations are also qualitatively the same, although there are more of them than in the Z 2 case. To avoid confinement, the total electric charge of any subdimensional e excitation must vanish (see the arguments in Sec II B). In the Z 2 case this meant that the e excitations had to be one of e = (1, 1, 0), (1, 0, 1), (0, 1, 1), where we have written (a, b, c) = e a xye b yze c zx. Since one of a, b, c is always had to be 0, this meant that the e excitations were always free to move along one direction (the direction along which they carry no electric charge). However, in the more general Z N case the e excitations can look like (a, b, N a b), and so we can have completely immobile e excitations if a, b 0 and a + b N. Finally, while counting degrees of freedom is a little tricky, one can use the properties of the excitations to show that the ground state degeneracy for the Z N theory is just log N (GSD) = 6L 3. (21) As before, the basic idea for doing the computation is to imagine splitting the system along a plane, studying the patterns that excitations can arrange themselves on the split plane, and imposing the constraint that the total charge of the excitations on the plane vanish modulo N. 9 F. Reducing the dimensionality of the e excitations Let s now return to the regular Z 2 X-cube model. In order to explore new phases and obtain something more checkerboard-like, we can consider trying to turn the 1-dimensional e particles into 0-dimensional particles by restricting their movement through additional terms in the Hamiltonian. Typically, these terms will consist of products of X operators, which are designed so as to restrict what sort of e fracton strings we can act with (recall that in order to commute with δh, e fracton strings must look like a series of ZZ operators applied along a line of links). One way we could imagine making the e fractons more 0-dimensional would be to make it very energetically costly for e fracton strings to terminate, while still allowing them to turn for a small amount of energy. If we could impose this condition fully, the electric sector of the excitation spectrum would consist of 0-dimensional e fractons that could only appear at the corners of rectangular ZZ membrane operators, with their movement being identical to the movement of the m cube fractons. Unfortunately we can t make the e excitations fully 0-dimensional, as it is straightforward to show that this would over-constrain the Hilbert space and imply a trivial ground state degeneracy. How ever, we can go part of the way by changing the coupling term δh in the following way: δh δh + h O v, (22) vertices v where h is as big as h and where O v is the product of X operators over the links emanating from v which hits the z

10 10 links twice and all the other links once. Graphically, we can schematically write O v as O v = = X Notice that O v commutes with all the other terms in the Hamiltonian, and so the exact solubility of the model is preserved. Adding this term prevents us from adding open e fracton strings that point in the x or y directions, while still allowing for closed e strings with corners in the xy plane. The closed e strings support e fractons at their corners, which behave in the same way as the m cube fractons. However, we are still allowed to create open e fracton strings which point along the z direction, which create regular 1-dimensional e excitations. As mentioned earlier, trying to restrict the behavior of the e strings further results in a trivial theory. (23) III. RECONSTRUCTING THE CHECKERBOARD MODEL FROM COUPLED TORIC CODES Now we turn to the more onerous task of reconstructing the checkerboard model Hamiltonian. Currently, this can be done only by using a single stack of 2d toric code layers with a non-onsite δh coupling term. This tactic allows us to reproduce the Hamiltonian and excitation spectrum, but not the Hilbert space of the original checkerboard model, due to the non-onsite nature of the coupling term. That said, I don t think that the Hilbert space issue is all that important for now. Our checkerboard model consists of a stack of 2d planes, each of which is built from two interpenetrating toric codes. Each layer looks like When we stack these layers vertically to form the full 3d system, we alternate the colors on the lattices on successive layers in order to produce the checkerboard pattern. That is, if the lattice in the above picture is for the plane p, then the lattice for planes p 1 and p + 1 (which are below and above the plane p in the stack) is the above picture but with the red and blue sublattices interchanged. The toric code spins live on the intersection of the blue and red links, which are associated with the vertices of the checkerboard lattice (meaning that we have two toric code spins at each vertex of the checkerboard lattice). To see how this gives a checkerboard lattice structure, we can let ( ) lattice points denote sites which lie on faces of checkerboard cubes with unit normal in the ẑ ( ẑ) direction, which in the above figure points out of (into) the (24)

11 page. We will denote the toric code Pauli operators by Z and Z, depending on which sublattice (blue or red) they are affiliated with. This means that the base Hamiltonian is H = (A + + A + + B + B ), (25) p,{+ } p where p labels a 2D plane in the stack and the A l + X l and B l Z l operators are defined as usual. Figuring out what the appropriate coupling term is turns out to be kind of a pain. The least complicated choice seems to be δh = h (Z r zẑ Z r Z r Z r+zẑ + X r X r+zẑ X r+zẑ X r+2zẑ ), (26) r p z 2Z where h is big, p is a plane in the stack, r runs over all the spin sites in p, and the sum over z runs over all even integers. Pictorially, the coupling looks like 11 Z X (27) None of the A and B operators commute with δh, and so we must treat them all perturbatively. Staring at the lattice a while, we see that the first term we end up getting is H = (A +,pa +,p 1 + B,p B,p+1 ) + δh, (28) p,{+, } p where A +,p 1 is the vertex operator immediately below A +,p in the stack of planes, and likewise for the B operators. With a little more staring at the lattice, we see that this is actually equivalent to the regular checkerboard model Hamiltonian, and that as long as we work at energy scales below h, any combination of Pauli operators that we act with is either a term in δh (and thus creates something in the condensate), or creates cube excitations of the form dictated by the checkerboard model. The objects that condense in this model are of course determined by the terms in δh. We see that the first term in δh condenses a vertical stack of an e pair on one layer, a combination of an e pair with an e pair on the next layer up, and an e pair yet another layer up. Due to the summation over 2Z, these objects can only condense when the middle layer occurs on a plane with even z-coordinate. The second term in δh condenses a similar object, this time made out of m-particles with and arranged such that the middle layer of the stack always occurs on a plane with odd z-coordinate. This staggered way of condensing the stacks is needed to ensure that the two different objects (conglomerates of es and conglomerates of ms) don t confine one another. While this approach reproduces the Hamiltonian and excitation spectrum of the checkerboard model (and does it with only a single stack of 2d layers, which is potentially more relevant for experiments), it doesn t really reproduce the checkerboard model, since the Hilbert spaces in this model and the real checkerboard model are different. This is due to the nonlocal δh coupling term that we had to add, which prevents us from having a single Pauli algebra on each site of the lattice in our model, we definitely don t have the right Hilbert space, since our effective Pauli algebras are smeared out between multiple sites. Additionally, a tentative calculation of the GSD for this approach gives log 2 (GSD) = 2L 2, a factor of 3 smaller than the GSD in the actual checkerboard model, although the comparison may not be particularly meaningful at this stage.

12 12 IV. THE X-CUBE AND CHECKERBOARD MODELS FROM COUPLED ISING LAYERS There are actually two ways of examining the models that we have been focusing on. The first, which we have already talked about, is to notice that they can be realized as coupled stacks 2d topological orders. The second is to build them out of gauged versions of models built from coupled layers of 2d Ising theories (by Ising theories we mean those with regular rr interactions, not those with multi-body interactions considered in Ref 1 ). Since the toric code is just a gauged version of the 2d Ising theory (see Appendix A), these two ways of understanding fracton phases can be summarized by the following commutative diagram: 2D Ising-like model couple coupled Ising-like theory gauge toric code-like theory couple gauge fracton topological order (29) Combined with the (standard) results of Appendix A, we have already shown how to proceed along the lower path in the diagram. In what follows, we will show how to follow the top path for both X-cube and checkerboard models. As a reminder, we will use X, Y, Z to denote gauge field Pauli operators and X, Y, Z to denote physical spin Pauli operators. A. The X-cube model We now examine how to get to the X-cube model from an Ising spin model starting point. Unlike the approach taken in the fracton topological phases paper, we will use regular nearest-neighbor Ising interactions, rather than assuming the plaquette-type Ising interactions, although the distinction isn t incredibly significant. To get things to work out correctly, we will couple the Ising layers along their links, just as we coupled the toric codes, so that there are three Ising spins at each site of the lattice. The naive approach would be to couple the Ising layers along their faces, since we might guess that if the toric code is the dual of the 2D Ising model we should couple the Ising layers in the dual of the way that coupled the toric codes. This ends up giving us weird octahedral interactions though, so it s better to couple the layers along links. Anyway, what should the coupling look like? We know that when we coupled the toric codes together we used a XX term. Since the duality between the toric code and the 2D Ising model involves the relation Z r Z r = X rr, we see that a XX coupling corresponds to a coupling of Ising interaction terms Z r,p Z r,pz r,q Z r,q, where p and q are the two planes that contain the link rr. Adding in the transverse fields to give the spins dynamics, the coupled Ising Hamiltonian is H = X r,p. (30) rr,p q Z r,p Z r,pz r,q Z r,q r,p We still have a global spin-flip symmetry along any plane, which we now attempt to gauge in the regular way, by coupling a single Z operator to each interaction term. There are three types of generators of gauge transformations, corresponding to the three different orientations of planes that possess the spin-flip symmetry. As usual we work in the restricted Hilbert space where we can replace the X operators by the corresponding gauge transformation generators. After making this replacement we see that [H, Z r,p ] = 0, and so we can set Z r,p = 1, giving H = p + p A p + l Z l, (31) where A p + is the standard vertex operator (built from Xs) at the site + and operating in the plane p. The lone Z l doesn t commute with the A p + terms, and so we go to perturbation theory as usual. The first term in perturbation theory that commutes with all types of the A p + operators is at 12th order, and is precisely the product of Z l s over the links of a cube. Thus, we recover the X-cube model.

13 13 B. The checkerboard model The idea is to construct a checkerboard lattice out of a single stack of 2D square lattices. We decompose each 2D layer into a superposition of two square lattices as before, with a generic layer as before looking like We stack layers like this in an alternating fashion, so that spins lie directly above and below spins and vice versa. In terms of the 3D checkerboard lattice, the lattice represents spins that live on the faces of the checkerboard cubes whose unit normal is ẑ and the lattice represents spins that live on the faces with unit normal ẑ. We take each layer to have generic nearest-neighbor Ising interactions, and couple Ising interactions on and lattices that overlap with one another. We do this by introducing the gauge field spins with Pauli operators X, Y, Z at the points on the lattice where the red an blue lines cross. Note that we are only introducing one type of gauge field it carries no / flavor index. We also introduce an Ising coupling between spins on one layer and spins on the layer below them (but not between spins and spins on the layer above them!). The gauge-coupled interaction term is then (note that we are not coupling the / inter-layer interaction to a gauge field) (32) H int = rr Z r Z r Z r Z r Z rr r Z r Z r ẑ (33) The subscripts are a bit imprecise, but hopefully the schematic notation does the job for now. Note that this interaction doesn t have a subextensive spin-flip symmetry r p X / r where p is an arbitrary plane in the stack, due to the inter-layer coupling term. However, it does have the symmetry generated by the operator G which acts with simultaneous spin flips along two parallel sheets: G = r p X r X r+ẑ, (34) where again p is a plane in the stack. To get to the checkerboard model we will gauge this symmetry. We first add the transverse field term δh = X r X r+ẑ to the Hamiltonian. This doesn t commute with H int as it stands, and so we need to figure out what the generator of gauge transformations is. After staring at the picture for a bit, we see that the generator of gauge transformations is C X c = l c X l, (35) where c is a cube of the checkerboard lattice. We then go over to the restricted Hilbert space and replace the XX transverse field term in the Hamiltonian by an appropriate Cc X term. As usual, this replacement means that [H, Z / ] = 0, and so we can set all the Z / s to +1. The remaining Z term doesn t commute with the A operators, and so we go to perturbation theory to figure out what we should replace it with. This gives us an operator which is the product of Z operators over the cube, and we can write H = c (C X c + C Z c ), (36)

14 where the sum is over all cubes in the checkerboard lattice and Cc Z is the same as Cc X, but with X Z. This is precisely the checkerboard model, which we got simply by coupling layers of plain nearest-neighbor Ising models. We should point out that the approaches outlined above for both models are different from the approaches used in 1, since we never construct our models from theories with multi-body interactions (although we do have to do a more complicated multi-interaction coupling procedure). Also, at a technical level, we determine the cube operator(s) perturbatively, rather than the approach in Ref 1, which is to determine the vertex operators perturbatively. We feel that this is a bit more natural since from an Ising point of view, the vertex operators are obtained as energetic ways to enforce gauge invariance, and thus should always be large (and hence should not be treated perturbatively). 14 For the X-cube model: V. LIST OF APPROACHES THAT HAVEN T WORKED Getting to H X cube through coupled 2d layers, which seems to be pretty difficult (if not impossible?). For the checkerboard model: Using three orthogonal coupled layers of interpenetrating square lattice toric codes with onsite coupling, which at first seemed to be the most natural thing to do. The difficulty here was reducing the degrees of freedom on each site to a single Ising spin while maintaining the checkerboard pattern. Using coupled layers of honeycomb lattices, with the Hamiltonian on each site given by a product of Xs and Zs over each hexagon (aka the color code). The method that almost worked was to use four hexagonal layers with three spins per site, with coupling terms like XX + ZZ added to every site. The only problem here was extending the coupling terms throughout the entire lattice in a consistent way. Using coupled layers of square lattice toric codes, with each layer folded into a snake pattern like. While I could get this to work for a small group of cubes, the difficulty once again was assigning coupling terms to the entire lattice in a consistent way. I m actually not convinced that this can t work, but visualizing the geometry was complicated enough to make me pursue other ideas. Haah s code: This approach almost works, and it might be worthwhile to look at further. The idea is to notice that Haah s code 3 (the one with three XX and three ZZ operators) is essentially just built from Wen plaquette models. Explicitly, the front, right, and bottom faces are Wen plaquette models for the first spin index, and the back, top, and left faces are Wen plquette models for the second spin index. The hard part is figuring out how to do the coupling in a consistent way throughout the whole lattice. Appendix A: Obtaining the toric code by gauging the 2d Ising model We already know how to do this, but to connect better with the approach used in the main text, we will obtain the toric code in a slightly different way than usual. The Hamiltonian for a single layer of 2d Ising spins is given by H = rr Z r Z r r X r, (A1) where r and r are lattice sites sharing a link. We then introduce a Z 2 gauge field with Pauli operators X, Y, Z to promote the global r X r spin-flip symmetry to a local one: H = rr Z r Z rr Z r r X r, (A2) where the X, Y, Z gauge fields live on the links of the lattice, not on the links of the dual lattice (which is the choice often chosen in the literature). Note that we are not writing a l X l term since we want to work in a region of parameter space where the X, Y, Z gauge fields have no dynamics.

15 15 We want the X r spin-flip symmetry to be local, but [H, X r ] 0 as it stands. This problem is fixed if we can get the gauge field to transform appropriately under the spin-flip. We see that the Hamiltonian does commute with G(r) = X r X l, (A3) l + where + means all the links emanating from the site r = +. Thus, the operator A + = l + X l (A4) is the generator of Ising gauge transformations. We can then impose the constraint A + = X r on the Hilbert space of the problem to get H = + A + rr Z r Z rr Z r. (A5) Now Z commutes with the present form of H, and so we can work with Z = +1 eignenstates without loss of generality. However, Z l doesn t commute with A +, and so we are led to perform a perturbative expansion in the Zs. The first term that appears in perturbation theory which acts within the ground state manifold is the plaquette operator B = l Z l, and so we recover H = + A + B (A6) for the familiar toric code Hamiltonian. 1 S. Vijay, J. Haah, and L. Fu, ArXiv e-prints (2016), S. Vijay, J. Haah, and L. Fu, PRB 92, (2015), M. Pretko, ArXiv e-prints (2016), M. D. Schulz, S. Dusuel, R. Orús, J. Vidal, and K. P. Schmidt, New Journal of Physics 14, (2012),