Hybrid Fracton Phases: Parent Orders for Liquid and Non-Liquid Quantum Phases
HHybrid Fracton Phases: Parent Orders for Liquid and Non-Liquid Quantum Phases
Nathanan Tantivasadakarn, ∗ Wenjie Ji,
2, 3, † and Sagar Vijay ‡ Department of Physics, Harvard University, Cambridge, MA 02138, USA Department of Physics, University of California, Santa Barbara, CA 93106, USA Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
We introduce hybrid fracton orders: three-dimensional gapped quantum phases that exhibit the phenomenol-ogy of both conventional three-dimensional topological orders and fracton orders. Hybrid fracton orders hostboth (i) mobile topological quasiparticles and loop excitations, as well as (ii) point-like topological excitationswith restricted mobility, with non-trivial fusion rules and mutual braiding statistics between the two sets of exci-tations. Furthermore, hybrid fracton phases can realize either conventional three-dimensional topological ordersor fracton orders after undergoing a phase transition driven by the condensation of certain gapped excitations.Therefore, they serve as parent orders for both long-range-entangled quantum liquid and non-liquid phases. Westudy the detailed properties of hybrid fracton phases through exactly solvable models in which the resultingorders hybridize a three-dimensional Z topological order with (i) the X-Cube fracton order, or (ii) Haah’s code.The hybrid orders presented here can also be understood as the deconfined phase of a gauge theory whose gaugegroup is given by an Abelian global symmetry G and subsystem symmetries of a normal subgroup N alonglower-dimensional sub-regions. A further generalization of this construction to non-Abelian gauge groups ispresented in forthcoming work [1]. CONTENTS
I. Introduction 1II. Hybrid Toric Code Layers 4III. Fractonic Hybrid X-Cube Order 6IV. Lineonic Hybrid X-Cube Order 12V. Hybrid Haah’s Code 14VI. Hybrid Phases as Parent Orders for Toric Code andFracton Orders 16VII. Discussion 18Acknowledgments 19References 19A. Abelian 2-subsystem symmetries 20B. Derivation of the fractonic hybrid X-Cube model 21C. Derivation of the lineonic hybrid X-Cube model 23D. Ground State Degeneracy 24E. A Hybrid of X-Cube and Two Toric Codes 26F. Hybrid Haah’s Code as a Parent Order 29 ∗ [email protected] † [email protected] ‡ [email protected] I. INTRODUCTION
Gapped fracton phases of matter [2–6] are quantum phasescharacterized by the presence of fractionalized, point-like ex-citations with highly restricted mobility, and a robust ground-state degeneracy that can grow sub-extensively with systemsize, due to an intricate pattern of long-range entanglement.This phenomenology is in stark contrast to that of more famil-iar, topologically-ordered phases that can host mobile point-like excitations with non-trivial self- and mutual-statistics intwo spatial dimensions, along with loop excitations in threedimensions. Gapped fracton orders appear in two broad cat-egories: Type I orders, such as the X-Cube fracton order [6],host both immobile quasiparticles (fractons) as well as frac-tionalized excitations with restricted mobility, while in TypeII orders, such as Haah’s code [3], all fractionalized excita-tions are immobile, and cannot be separated without incurringan energy cost. More exotic non-Abelian fracton orders havebeen recently explored, in which certain excitations with re-stricted mobility have a protected internal degeneracy [7–10],analogous to the quantum dimension of non-Abelian quasi-particles in two-dimensional topological phases.Progress has been made towards understanding some uni-versal properties of fracton orders that are characteristic ofthe phase, such as fusion and braiding processes for fraction-alized excitations [11], topological entanglement entropy inthe ground-state [12–14], and the foliated structure of certainType I fracton orders which allows these phases to easily “ab-sorb” two-dimensional topological orders through the actionof a finite-depth quantum circuit [15–17], in contrast to a con-ventional quantum liquid , which can similarly absorb short-range-entangled degrees of freedom [18]. Quantum field the-ories that capture the low-energy properties of Type I fractonorders have been recently proposed [19–24].Though fracton orders have attracted intense study, theyhave so far been treated as exotic non-liquid phases thatstand alone from conventional, three-dimensional topologi-cal orders, which can be described at low energies by topo- a r X i v : . [ c ond - m a t . s t r- e l ] F e b logical quantum field theories in (3 + 1) spacetime dimen-sions ( TQFT ). Some indirect relations between fractonorders and conventional quantum liquids have been iden-tified. First, strongly-coupled stacks of lower-dimensionaltopological phases can realize certain fracton orders or three-dimensional topological orders, independently [9, 25–36].Second, lattice models in which the gapped excitations con-tain non-Abelian fractons as well as mobile particles whichbehave similarly to the charges in a three-dimensional D gauge theory, have been recently proposed [37–40]. However,key properties of these models including (i) their relationshipto conventional quantum liquid orders such as the D gaugetheory and (ii) the braiding and fusion of excitations, are notfully understood. Whether properties of certain long-range-entangled quantum liquid and non-liquid states can coexist,or be possibly unified into a “parent” order, has remained anopen question.In this work, we answer this question directly, by proposinga family of hybrid fracton orders, which host both the exoticexcitations of a fracton phase, as well as the point- and loop-like excitations that appear in a TQFT . Within these hybridphases, the two kinds of excitations have non-trivial mutualstatistics and fusion rules – i.e. collections of excitations na-tive to the fracton order can fuse into excitations of the quan-tum liquid order and vice versa – so that the hybrid phase istruly distinct from a tensor product of the two orders. Thesehybrid orders further serve as clear parent phases for both con-ventional three-dimensional topological orders and Type I orType II fracton orders, since they can realize either order af-ter undergoing a phase transition in which an appropriate setof gapped excitations condense. After introducing a frame-work for understanding the emergence of these orders, weconcretely characterize certain hybrid ordered phases througha series of exactly solvable models, which provide a theoret-ical toolbox to determine the topological data – including fu-sion, braiding, and the mobilities of excitations – in full detail.A number of outstanding questions about these phases thatwe introduce, and their generalizations, remain to be ad-dressed. First, it remains to be understood whether hybridfracton phases can fit into the existing framework of foliatedfracton orders. For example, can a two-dimensional topolog-ical order be “exfoliated” from Type I hybrid fracton mod-els? This would also clarify their entanglement renormaliza-tion group flow, which exhibits dramatically different behav-iors between liquid and non-liquid phases [41, 42]. Whetherhybrid fracton phases can quantitatively improve upon the per-formance of existing quantum memories based on Type IIfracton orders [43] also remains to be studied. Finally, a field-theoretic understanding of such orders could shed light onthe universal properties of these states at low energies, otherproximate phases, and other hybridizations of liquid and non-liquid orders that are possible. We hope to address this lastquestion in future work [1]. Summary of Main Results:
We now provide a detailedsummary of our main results, and an outline of this work.To illustrate the properties of hybrid fracton orders, we in-troduce four exactly solvable models of these hybrid phases,in increasing levels of complexity. All of the models intro- (a) (b)(c) (d)(e) (f)FIG. 1.
Hybrid Haah’s Code:
In the hybrid Haah’s code, the twospecies of fracton excitations ( e and m ) that are native to Haah’scode [3] are created in the geometric arrangements shown in (a) and(b). Pairs of e fractons fuse into a mobile Z charge ( e ) as in (c).The hybrid order also hosts a flux loop m . Two identical, rectan-gular flux loops fuse to generate an arrangement of the m fractonexcitations, shown schematically in (d). The precise geometry of thegenerated fracton excitations is presented in Fig. 13 in Sec. V. The e charge has identical mutual statistics with a flux loop m as in the3d toric code as shown in (e). In (f), braiding the flux loop m aroundthe e fracton gives a non-trivial phase that is consistent with the fu-sion and braiding in (c) and (e). duced can be thought of as a hybrid of a (liquid) Z toric codein three dimensions, and a (non-liquid) Z fracton model, dueto the fact that the hybrid order hosts both the gapped excita-tions of the toric code, as well as the exotic excitations of thefracton order. In fact, the hybrid orders we introduce have thesame ground state degeneracy as the tensor product of the twoorders on the three-torus. Nevertheless, the hybrid phases dif-fer from a trivial tensor product due to the non-trivial fusionand braiding of the gapped excitations.The excitations in the hybrid orders that we consider, alongwith some of their braiding and fusion rules, may be summa-rized succinctly. All of these orders host a mobile Z charge(labeled e ) and a Z flux loop (labeled m ), which have thesame mobility and mutual statistics as the Z charge and fluxin the 3d toric code. Furthermore, the hybrid order hosts anexcitation with restricted mobility (labeled e ) and its conju-gate excitation (labeled m ) which are in correspondence withthe excitations in a particular Type I or Type II fracton order.For example, e can correspond to the fracton excitation in theX-Cube model, with m then corresponding to the conjugate TABLE I.
Fusion in Hybrid Fracton Orders:
In the hybridorders studied in this work, the excitations e ( m ) resemble thecharge (flux) in the 3d Z toric code, while e and m resemble thetwo species of excitations with reduced mobility in a Z fractonorder, respectively. These excitations and their composites formall of the gapped excitations in the hybrid order, and some of theircharacteristic fusion rules are shown below. The inverses of e and m – labeled ¯ e and ¯ m , and defined by the relation ¯ e × e = ¯ m × m = 1 – have identical mobilities as e and m , respectively. Excitations & Fusion Rules
Generating Set = { , e, e , m, m } e ≡ e × e = mobile Z charge m = flux loop m × m = e × e = 1 Hybrid Toric Code Layers (Sec. II) e ≡ planon m ≡ planon m × m = planons ( m ) along loop Fractonic Hybrid X-Cube (Sec. III) e ≡ fracton m ≡ lineon m × m = lineons ( m ) at corners Lineonic Hybrid X-Cube (Sec. IV) e ≡ lineon m ≡ fracton m × m = fractons ( m ) at corners Hybrid Haah’s Code (Sec. V) e ≡ fracton m ≡ fracton m × m = fractons ( m ) along loop aa The precise geometric arrangement of the fractons generated by the loopfusion is presented in Sec. V. excitation in the X-Cube phase which is only mobile alonglines (the lineon). The resulting hybrid phase is then termedthe fractonic hybrid X-Cube order, where “fractonic” refers tothe mobility of the e excitation. In select situations, this label-ing is unnecessary as the hybrid order can be defined unam-biguously. In Haah’s code, for example, both species of exci-tations are fractons, which are further exchanged by a dualitytransformation [41]. Therefore, in constructing a “hybridiza-tion” of Haah’s code with the 3d toric code, choosing e to beeither of the fracton excitations in Haah’s code yields the samehybrid order, whose properties are summarized schematicallyin Fig. 1.Our labeling of the gapped excitations in the hybrid frac-ton orders is suggestive of their fusion rules, which are sum-marized in Table I. These fusion rules yield new phenomenathat are not separately possible in either the fracton or three-dimensional toric code orders. For example, in both the hybridHaah’s code and in the fractonic hybrid X-Cube order, twofractons ( e × e ) fuse into a completely mobile quasiparticle ( e ) . This is particularly striking, as a single fracton is com-pletely immobile, and collections of these fractons can onlyform excitations with significantly reduced mobility in a TypeI fracton order. Additionally, the fusion of a pair of loop exci-tations ( m × m ) yields a geometric pattern of m excitations, which are fractons in the hybrid Haah’s code.Apart from the fusion rules, we obtain a universal braidingphase for two excitations e a and m b , when at least one of thetwo excitations exhibits enough mobility to remotely detectthe other. This braiding process leads to the accumulation ofa universal phase e iθ ab where θ ab = iπ ab, (1)and with a , b ∈ { , , , } , in all of the hybrid orders thatwe present. Other braiding processes that are specific to eachhybrid order are also studied, which are not summarized here.The emergence of hybrid fracton orders may be more gen-erally understood in two complementary ways. First, our hy-brid orders can be obtained by starting with a Type I or TypeII fracton order which is enriched by an on-site Abelian globalsymmetry (e.g. Z ), so that certain excitations of the fractonorder carry fractional quantum numbers under the symmetry.Gauging this global symmetry then yields a hybrid fracton or-der, in which certain excitations of the original fracton ordercan fuse into the gauge charge in a conventional topologicalorder (e.g. the gapped charge in a Z gauge theory).Equivalently, the hybrid order can be thought of as the de-confined phase of a gauge theory. We may start with a short-range-entangled (SRE) quantum system with global symme-try G and subsystem symmetries N , where N is a normal sub-group of G ; the subsystem symmetries are defined as symme-try transformations along extensive sub-regions of the lattice(e.g. planes). Importantly, the subsystem and global sym-metries are not independent of each other, and their interplayis such that the gapped, symmetric excitations of the SREphase can be (i) charged under the global symmetry or (ii)charged under a combination of planar symmetries and theglobal symmetry, so that gauging these symmetries yields ahybrid fracton order. As a consequence of this construction,we also refer to the hybrid order as a ( G, N ) gauge theory,and in this work we restrict our attention to Abelian groups G and N , where G/N = Z . Because the hybrid order is thedeconfined phase of a gauge theory, we will often refer to itsgapped excitations as charges or fluxes depending on whetherthe excitation is related to a (i) gapped, symmetric excitationin the ungauged, SRE phase which transforms under the sym-metry group (charge) or (ii) a defect of the symmetry group(flux). ( G, N ) gauge theories for more general groups G and N are studied extensively in a forthcoming work [1], where itis found that gauging the abelian global symmetry G and sub-system symmetries N of a SRE state yields a hybrid order thathybridizes a 3d G/N toric code and a fracton model based onthe subsystem symmetry gauge group N .We now provide an outline of this work. In Sec. II, weintroduce the simplest example of a hybrid phase, which hy-bridizes the order in a stack of two-dimensional (2d) Z toriccodes and the three-dimensional (3d) Z toric code. This or-der – termed the hybrid toric code layers – can be obtainedeither as a generalized gauge theory, or by condensing a set ofgapped excitations in a stack of 2d Z toric codes. These twocomplementary ways of obtaining the hybrid order provide animportant understanding about the fusion and braiding statis-tics of the gapped excitations. The intuition obtained from thisexample extends to the hybrid fracton orders that we considersubsequently.In the remaining sections, we introduce more complex hy-brid phases that hybridize a fracton order with a 3d toric codetopological order. For the hybrid Type I fracton models thatwe present, we choose the X-Cube order [6] as our input. Inthis case, there are two possible hybrid orders that can be ob-tained, if the gapped excitation e is chosen to be the fracton inthe X-Cube model or the lineon excitation. The former yieldsthe fractonic hybrid X-Cube order, in which a pair of frac-tons fuse to the mobile charge e , and is introduced in Sec.III. The latter case, where a pair of lineons fuse to the mobilecharge, is presented in Sec. IV. The equivalance between theground-state degeneracy of the fractonic hybrid X-Cube orderand of the tensor product of the X-Cube and toric code orderson the three-torus is related to an isomorphism between thealgebra of closed Wilson loop and membrane operators in theground-state subspace of these orders, which we identify. Hy-brid Type II orders can also exist and we introduce a hybridof the 3d Z toric code and Haah’s code [3] in Sec. V andstudy its properties in detail.Lastly, in Sec. VI, we study the proximate phases of thehybrid fracton orders, which establishes that these models areparent states for both conventional topological orders, as wellas fracton orders. We explicitly demonstrate that for eitherthe Type I or Type II hybrid fracton orders that we introduce,condensing an appropriate set of gapped excitations can drivea phase transition into either a Z topologically ordered phaseor a Z fracton phase. We show that the phase transition be-tween one of the hybrid orders and an X-Cube fracton ordercan be direct and continuous , and related to the Higgs tran-sition in a three-dimensional Z gauge theory in a particularlimit, though the generic nature of this phase transition re-mains to be understood.Interestingly, we find that a common feature of the hy-bridized model is that they can be thought of as promotingcertain Z degrees of freedom in the tensor product of a liquidand non-liquid order into Z degrees of freedom. More con-cretely, starting from a product of the Z toric code and a Z fracton model, the hybridization can be viewed as pairing upqubits of the toric code with qubits of the fracton model, andpromoting these pairs to a Z qudit. For cases where the de-grees of freedom of the two models both live on edges, suchas the models in Secs. II and IV, we are able to rewrite theHamiltonian as a mix of Z qubits and Z qudits. For those inSecs. III and V, the positions of the degrees of freedom of thetoric code and fracton model do not match, and the algebra ofoperators in the hybrid model is more involved (see Eqs. (22)and (59)).In Appendix A, we give a self-contained discussion of thedefinition of an abelian ( G, N ) symmetry, and a qualitative Unlike a type II fracton order, in which all topological excitations arestrictly immobile, a hybrid
Type II order hosts the excitations of both aliquid order and a type II fracton order, and can have mobile topologicalexcitations.
Toric Code ℤ e e e e e e m “ loop” m Condensed m Con fi ned ¯ m FIG. 2.
Hybrid Toric Code Layers from a stack of Z ToricCodes:
An alternate construction of the hybrid toric code layers,which clarifies the nature of the flux excitations is shown schemati-cally. Starting with a stack of Z toric codes, we condense pairs of e anyons in adjacent layers. The anyons m and ¯ m in each layer isconfined, but a composite excitation composed of m and ¯ m pairs ineach layer – the “ m loop”– braids trivially with the condensate andtherefore remains as a topological excitation. description of the process of gauging such a symmetry. Amore general construction of hybrid fracton models involvinga general finite group G will be presented in a forthcomingwork [1]. II. HYBRID TORIC CODE LAYERS
In this section, we begin by describing the simplest con-struction of a hybrid order, which hybridizes a stack of 2d toriccode layers with a 3d toric code, as described by the Hamil-tonian (3). The resulting hybrid order contains the excitationsof both the 2d toric code, which are restricted to move withinplanes, as well as those of the 3d toric code, with non-trivialbraiding and fusion rules.We first obtain this hybrid order by gauging the symme-tries of a short-range-entangled (SRE) phase. We start with L independent copies of a two-dimensional, SRE states, eachwith a global Z symmetry, and in a trivial gapped, symmetry-preserving (paramagnetic) state. The full symmetry group ofthe stacked layers is Z L . The excitations in each layer that arecharged under this symmetry group (charges) can be labeledby an integer modulo 4, and cannot move across layers.Next, we may break the Z L symmetry by adding a cou-pling between adjacent layers that allows charge-2 excitationsto tunnel between layers. We observe that since the charge ineach layer is only now conserved modulo 2, the Z symmetryin each layer is now broken down to Z . Nevertheless, theglobal Z symmetry defined as the diagonal Z symmetries ofall layers is still preserved.The charge excitations in the SRE state still transform as a Excitation Creation operator Charge Local Wilson operatorplanon e End points of Z on x, y edges A v = i, A dv = − B p xy = closed e loop around p xy mobile charge e End points of Z on x, y edges A v = − B dp = closed e loop around p End points of Z on z edgesloop m Boundary of X membrane in xy , plane B dp = − A v = closed m membrane around v Boundary of X membrane in xz , yz plane B dp = − , B p xy = ± i planon m End points of X in xy plane B p xy = − A dv = closed m loop around v TABLE II.
Excitations in the hybrid toric code layers:
A summary of the pure charge and flux excitations in the hybrid toric code layers isprovided above, along with the local operators that measure these excitations in the lattice model Eq. (3). Z representation under the global symmetry, so we they canstill be labeled by an integer modulo 4. However, they ex-hibit mobility constraints due to the fact that they transformunder the remaining Z planar symmetry in each layer. Theeven charges transform trivially under the Z planar symme-try, meaning they are fully mobile, while the odd charges arealso charged under the planar symmetry, meaning they canonly move within their respective planes. From this, it is alsoapparent that a fusion of two identical odd charges results in afully mobile even charge.We now gauge all of the symmetries of this model. Theproperties of the charge excitations of the SRE state carry overto the gauge charges of the resulting hybrid order. Qualita-tively, we can first gauge the Z planar symmetries, whichcreates stacks of Z toric codes. The global symmetry is nowreduced from Z to Z because we have also gauged a Z sub-group of the global symmetry, which is the product of the Z planar symmetry in every layer. As a consequence, the stackof toric codes are each enriched by the remaining global Z symmetry. In particular, the global symmetry fractionalizeson the toric code anyon e in every layer. Finally, we may alsogauge the global Z symmetry to obtain the desired hybridmodel.While the above construction is well-defined, it sheds lesslight on the nature of the flux excitations in the final hybridmodel. To study the flux excitations, we find it more insight-ful to consider the following alternate route, which will resultin the same hybrid order. We start with the SRE state and tem-porarily neglect the interlayer couplings, so that each layer hasthe full Z symmetry. By gauging the Z symmetry in eachlayer, we obtain a stack of Z toric codes. Then, to restore theinterlayer couplings, we condense pairs of e anyons of the Z toric code between every adjacent layer, as shown in Figure 2.As a result, the e anyons in each layer are all now in the samesuperselection sector in the condensate phase, making the e particle mobile in the z direction. In addition, the unit flux m ,which braids non-trivially with the e pairs is confined, buta composite loop excitation composed of m - ¯ m pairs in eachlayer remains deconfined. We will refer to this loop excitationas the “ m -loop”, a gauge flux of the hybrid model. The anyon m , however braids trivially and survives as a well-definedexcitation in the condensed phase. It therefore remains as apoint particle confined to each layer.The considered condensation has interesting consequences in terms of the mobility of the particles under fusion. Iden-tically to the charge excitations before gauging, the gaugecharge e is a planon, but fusion with another gauge chargegives e , a fully mobile excitation. In addition, obtaining thehybrid phase by condensing excitations in a stack of Z toriccodes allow us to determine the fusion of the flux excitations.The m -loop is fully mobile, but upon fusion with itself, it de-composes into pairs of m planons in each layer. The types ofexcitations and their mobilities in the hybrid toric code layersare summarized in Table II.We note that other than the unusual mobilities, the statistics of the excitations are the same as those of a Z toric codemodel. That is, the mutual statistics of the excitations e a and m b for a = 0 , , , is just i ab . While it could be suggestiveto think that the final hybrid model simply decouples into a 3dtoric code (with mobile excitations e and m ) tensored witha stack of 2d toric codes (formed by planar excitations e and m ) because each of the pairs above has a mutual − braidingstatistics, this can be refuted by noticing the mutual statisticsof i between the m loop and the e planon, which cannot occurin the stacked model.A simple exactly-solvable lattice model for the hybrid toriccode layers can be explicitly constructed. The model is a hy-brid of the 2d and 3d toric codes. On a square lattice, we placea Z qubit on the z links with the usual Z and X Pauli oper-ators, and a Z qudit on the x and y links with the followingclock and shift operators Z = (cid:88) n =0 i n | n (cid:105) (cid:104) n | , X = (cid:88) n =0 | n + 1 (cid:105) (cid:104) n | , (2)which satisfy ZX = i X Z . The Hamiltonian is given by H Hybrid = H (cid:48) T C + H (cid:48) T C ,H (cid:48) T C = − (cid:88) v A v + A † v − (cid:88) p B dp ,H (cid:48) T C = − (cid:88) v A dv − (cid:88) p (cid:107) B p xy + B † p xy . (3)where p xy refer to plaquettes that are in the xy plane only.The explicit form of the operators are A v = X † X † XX XX , (4) B dp = Z Z Z Z , Z Z Z Z , Z Z Z Z (5) A dv = X X X X , (6) B p xy = Z ZZ † Z † . (7)The hybridization of the 2d and 3d Z toric codes can be seenfrom the fact that the edges in the xy plane have been modifiedfrom Z to Z qudits. The two models are coupled in such away that the vertex term of the 3d toric code A v squares to thevertex term of the 2d toric code A v = A dv . Furthermore, forplaquettes in the xy plane, the plaquette term of the 2d toriccode B p xy squares to the xy plaquette of the 3d toric code B dp xy .The phenomenology of the hybrid model can be readilychecked from this lattice model as illustrated in Figure 3 andsummarized in Table II. The planon e corresponds to A v = i ,and can only be excited at the end points of a string of Z in the xy plane. Squaring this string operator creates the excitation e at its end points, which satisfy A v = − . However, e isa mobile particle because it can also hop in the z direction byacting with Z on z edges.To create the flux loop m , we apply X on every x or y link,and X on every z link that intersects a surface S (cid:48) on the duallattice. The eigenvalues of the plaquette terms violated at theboundary of S (cid:48) are given by B dp = − and B p xy = ± i .Squaring the operator that creates the loop, we find that theonly terms that are violated are B p xy = − , which corre-spond to m excitations created at the positions where theoriginal m loop pierces each xy plane. The m excitationsare planons because there are no local operators that can movethem out of the plane.Finally, moving the e planon around the m loop. we seethat there is a single overlap of the Z and X operators, whichresults in a braiding phase of i .In Appendix D 1, we calculate the ground state degeneracyof the model on a torus to be L + 3 and explicitly constructthe logical operators. III. FRACTONIC HYBRID X-CUBE ORDER
We now present a hybrid order that combines the phe-nomenology of a (three-foliated) fracton order with that of the
Z Z ZZ e (planon) ¯ e (planon) Z e (mobile) Ze (mobile) X m (planon) m (planon) m (loop) X XXX
X XXX XZ Z Z X X FIG. 3.
Geometry of the Excitations in the Hybrid Toric CodeLayers:
Excitations of the hybrid toric code layers from the Hamil-tonian (3) are shown. For charge excitations, the colors magenta, red,and orange correspond to A v = i, − , − i , respectively. For flux ex-citations, blue refers to B dp , B p,xy = − , while cyan and purplerefers to B p xy = i, − i , respectively.
3d toric code. Here, we will find that the charges and fluxesthat were originally planons in the hybrid toric code layerswill become fractons and lineons that behave similarly to theexcitations of the X-Cube model. Furthermore, the fractonwill square to a mobile topological charge, and thus we willrefer to the resulting order as the fractonic hybrid X-Cube or-der. This hybrid order can be intuitively understood as the de-confined phase of a gauge theory, which is obtained by gaug-ing a collection of symmetries – including both global sym-metries, as well as symmetries along three intersecting planes(planar subsystem symmetries) – of a short-range-entangledphase.
A. Paramagnet with Global and Subsystem Symmetries
To illustrate the gauging procedure, we consider a four-dimensional Hilbert space at each vertex of a cubic lattice,with the Z clock and shift operators defined at each latticesite, as in Eq. (2). We may consider a product state with X v = +1 at all lattice sites, which is trivially the ground-stateof a Hamiltonian H = − (cid:88) v [ X v + X v + X v ] . (8)We will consider the gapped symmetric excitations of thisparamagnet which are invariant under a global Z symme-try transformation (cid:81) v X v , along with a planar Z symmetryalong any plane p in the xy , yz or xz directions (cid:81) v ∈ p X v .After gauging these symmetries, these excitations are in one-to-one correspondence with the gapped, fractionalized chargeexcitations of the resulting hybrid phase.The elementary excitations of the paramagnet are createdas follows. First, for a plaquette p , the operator ∆ p = Z i Z † j Z k Z † l excites four charge excitations at the corners i, j, k, l . These excitatons are charged ± i under the global Z symmetry. In addition, they are also charged − under the Z planar symmetry, which renders them immobile. Theseexcitations will correspond to the fracton e after the gaug-ing procedure. However, applying this operator twice createsfour particles which are charged − under the global Z , butcharge neutral under the planar Z . Therefore, these chargesare mobile, and can be hopped using ∆ e = Z i Z f , where i and f are the endpoints of the edge e . Explicitly, ∆ ijkl ) = ∆ ( ij ) ∆ ( ik ) ∆ ( il ) . (9) B. Hybrid Order
We will now gauge the aforementioned symmetry, the de-tails of which we will relegate to Appendix B. Qualitatively,we separate the gauging into two steps. First, we gauge the Z planar symmetries. This results in a Z X-Cube model wherethe remaining Z global symmetry fractionalizes on the frac-ton excitation. Further gauging this Z global symmetry willgive the hybrid model we will now present. Instead, we opt tomotivate the resulting Hamiltonian as a hybridization betweenthe 3d toric code and the X-Cube model.We consider a cubic lattice with an additional diagonal edgeadded to each plaquette on the cubic lattice as shown in Figure4, and place a Z gauge field (qubit) on each edge and each(square) plaquette of this lattice. In addition, we assign a localordering of the vertices to each edge e = ( if ) and each squareplaquette p = ( ijkl ) as shown in Fig. 4.The Hamiltonian can be thought of as first starting with a Z toric code defined with Z gauge fields on each edge ofthe lattice tensored with a Z X-Cube model defined with Z gauge fields on each square plaquette. Then, we couple thetwo gauge fields by modifying the vertex term in the toric codeand the cube term in the X-Cube model. The Hamiltonian isgiven by H Hybrid = H (cid:48) T C + H (cid:48) XC ,H (cid:48) T C = − (cid:88) v A v + A † v − (cid:88) B ,H (cid:48) XC = − (cid:88) v A XCv − (cid:88) c (cid:88) r = x,y,z B c,r + B † c,r , (10) iii jkk kll ljiff f fff j FIG. 4.
Description of the Lattice for the Hybrid X-Cube Model:
Diagonal edges are added to each plaquette in the cubic lattice.
Left :each edge e = ( if ) is oriented, pointing outward from an “initial”vertex i towards a “final” vertex f . Right : ordering of vertices foreach square plaquette p = ( ijkl ) where A v = (cid:89) e → v ξ † e (cid:89) e ← v ξ e , (11) B = (cid:89) e ∈ Z e , (12) A XCv = (cid:89) p ⊃ v X p , (13) B c,r = (cid:89) p ∈ c (cid:48) r ζ † p (cid:89) p ∈ c r ζ p . (14)Here, Z e is the Pauli- Z operator on each edge, and X p is thePauli- X operator on each plaquette. Visually, the operatorsabove are shown in Fig. 5. To clarify the notation above, e → v ( e ← v ) in the vertex term A v denotes the incoming(outgoing) edges towards (from) the vertex v as defined in Fig.4, and shown in orange (magenta) in Fig. 5. For the plaquetteterm B , the sum is over all triangular plaquettes . The cubeterm B c,r , as in the X-Cube model, depends on an orientation r . In particular, c r and c (cid:48) r for r = x, y, z , are each a set of twoplaquettes surrounding the cube c shown in cyan and purplerespectively in Fig. 5. As in the X-Cube model, they satisfy B c,x B c,y B c,z = 1 .Now, we notice that if ξ e and ζ p were Pauli- X operators oneach edge and Pauli- Z operators on each plaquette, then thisHamiltonian is indeed just a stack of the toric code and the X-Cube models. However, in the hybrid model, the operators ξ e and ζ p are Pauli operators decorated with additional Cliffordgates X e → ξ e = X e (cid:89) p ∈ n ( e ) CNOT e,p , (15) Z p → ζ p = Z p S ( ij ) S † ( ik ) S ( il ) , (16)where S = (cid:18) i (cid:19) , CNOT = . (17) A v = A v = A XCv = B = B c,x = B c,y = B c,z = B c,x = B c,y = B c,z = ξ e = ≡ ξ e = ζ p = ≡ ζ p = FIG. 5.
Lattice Model for the Fractonic Hybrid X-Cube Order:
Visualization of the operators in the fractonic hybrid X-Cube model. Thecolor coding used is red= X , blue= Z , green = S , yellow = S † , magenta = ξ , orange = ξ † , cyan = ζ , purple = ζ † , → = CNOT. Here, for the CNOT gate in Eq. (15), the qubit on the edge e is the control and the plaquette p is the target. Furthermore, n ( e ) is the set of plaquettes p such that the edge e appears as ( ij ) , ( ik ) , or ( il ) of p as defined in Fig. 4 (See also Eq. (B6))Because the operators ξ e and ζ p act on both the gauge fieldson edges and on plaquettes, we can anticipate that the excita-tions created by them will display features pertinent to boththe toric code and and X-Cube models. For example, as wewill see, a fracton excitation can have non-trivial statisticswith the flux loop. Let us analyze the algebra of these op-erators.First, when restricted to terms only on plaquettes or onedges, the operators act the same as Z Pauli operators, ζ p X p (cid:48) = ( − δ p,p (cid:48) X p (cid:48) ζ p , (18) Z e ξ e (cid:48) = ( − δ e,e (cid:48) ξ e (cid:48) Z e , (19) [ ξ e , ξ e (cid:48) ] = [ ζ p , ζ p (cid:48) ] = [ ξ e , X p ] = [ ζ p , Z e ] = 0 . (20)Second, the square of the modified operators are also Paulioperators, ξ e = (cid:89) p ∈ n ( e ) X p , ζ p = Z ( ij ) Z ( ik ) Z ( il ) . (21)Third, the operators ζ p and ξ e either commute, or act as theclock and shift operators of a Z qudit, depending on if e is a certain edge of the plaquette p shown in Fig. 4, ζ p ξ e = + i ξ e ζ p ; e = i p j p , i p l p , − i ξ e ζ p ; e = i p k p , ξ e ζ p ; otherwise . (22)The first and second properties implies that the ζ p is still a Z gauge field on plaquettes with respect to the electric field X p .Similarly, ξ e is still the Z electric field on each edge with re-spect to the gauge field Z e . The replacement only modifies thestar term A v and the cage term B c,r . Because of the secondproperty, the vertex term of the toric code A v term squaresto the vertex term of the X-Cube model A XCv , and the cubeterm of the X-Cube model B c,r squares to a product of fourtriangular plaquette terms B of the toric code as shown inFig. 5.
1. Excitations and fusion
Since the model is a commuting projector Hamiltonian, itis exactly solvable. Therefore, we can explicitly write downthe excitations and compare the similarities to the hybrid toriccode layers in the previous section. Using the commutationrelations Eq. (22), we see that ζ p commutes with B c,r and B , but violates the projector containing A v at the four ver-tices at the corners of p . In particular, this implies that the fourcorners of ζ p are charged ± i under the operator A v . Further- Excitation Creation operator Charges Local Wilson operatorfracton e Corners of ζ p membrane A v = i , A XCv = − B c = Closed loop of e − ¯ e dipole around c mobile charge e Corners of ζ p membrane A v = − B = Closed loop of e around .End points of Z e stringloop m Boundary of ξ e membrane B p = ± i (at corners), B = − A v = Closed membrane of m around v lineon m Corners of ξ e membrane B p = − A XCv = Closed cage of m around v End point of X p stringTABLE III. Excitations in the fractonic hybrid X-Cube model:
A summary of the pure charge and flux excitations in the fractonic hybridX-Cube model is provided above, along with the local operators that measure these excitations in the lattice model. more, since A v = A XCv these excitations are also charged − under A XCv and are therefore fractons. We will call theexcitations e, ¯ e for the excitation A v = i, − i , respectively. Ingeneral, a product of ζ p over a surface S creates such fractonsat the corners of S fracton e, ¯ e : (cid:89) p ∈S ζ p . (23)The other type of charge excitation is created by a product of Z e on an open open string L ,mobile e : (cid:89) e ∈ L Z e . (24)The end points of the string operator above are charged − under A v , and commute with other local terms in the Hamil-tonian. Now, the operator ζ p also creates such excitations onthe corners of p , as it is charged − under A v at the corners.Thus, we will call A v = − the point excitation e . This canbe seen from the fact that ζ p can be written as a product of Z e operators. More generally, the product of ζ p on a surface S creates the point charges at the corners of S ,mobile e on corners : (cid:89) p ∈S ζ p , ζ p = Z ( ij ) Z ( ik ) Z ( il ) , (25)which is just the dualized form of Eq. (9).The e excitations are fully mobile, since a string of Z e operators can hop individual e excitations. This can also beseen from the fact that it is not charged under A XCv , whichdetects the fracton.Next, we define operators that violate B p and B , butcommute with A v and A XCv . Excitations created from suchoperators are flux excitations. First, acting with ξ e on alledges intersecting a given surface S (cid:48) on the dual lattice createsa loop excitation at the boundary of that surface.mobile loop m : (cid:89) e ⊥S (cid:48) ξ e , (26)More precisely, the operators B along the boundary of S (cid:48) are charged − . Interestingly, we find that the corners of theloop operators are moreover charged ± i under two of the three B c operators. This is shown in cyan and purple in Fig, 6. Lastly, applying X p violates B c,r on the two cubes adjacentto p , creating two lineon excitations. In general, the lineonoperator is the product of X p along a rigid string L (cid:48) on thedual square lattice,lineon m : (cid:89) p ⊥ L (cid:48) X p , (27)and are charged − under two of the three B c,r operators.In particular, a lineon mobile in the direction r (cid:48) = x, y, z ischarged under the B c,r for r (cid:48) (cid:54) = r .The excitations of this model are summarized in Table IIIand shown in Figure 6.These excitations also have interesting fusion rules. Twofractons fuse into a mobile charge, as we can see from com-paring Eq. (23) with Eq. (25). More surprisingly, two identi-cal loop excitations fuse into a number of lineons. To see this,we consider a fusion of the loop m with itself by applying ξ e = (cid:89) p ∈ n ( p ) X p , (28)to all edges in a dual surface S (cid:48) . One can verify that (cid:89) e ∈S (cid:48) ξ e = (cid:89) p ∈ ∂ S (cid:48) X p . (29)That is, the product is equal to applying X p to all plaquettesalong the boundary of S (cid:48) , which is just the lineon string alongthe original loop. Therefore, a lineon excitation is created atevery corner of the original loop excitation, and the lineon ismobile along the direction normal to both edges of the loopmeeting at the given corner. This is illustrated in Figure 7.To summarize our results, the excitations in this exactlysolvable model can be created by cutting open “closed” Wil-son operators. The excitations are “topological” in the sensethat once created, they can fluctuate as far as their mobilitiesallow without an energy cost. To see this, we point out that thefollowing Wilson operators are just products of the stabilizers,0 e (fracton)¯ e (fracton) ¯ e (fracton) m (loop) m (lineon) e (mobile) e (mobile) e (fracton) m (lineon) FIG. 6.
Geometry of the Excitations in the Hybrid X-Cube Or-der:
Excitations of the fractonic hybrid X-Cube model and their cor-responding creation operators are shown. The excitations e , e , m and m are created using a membrane of ζ p (cyan), a flexible stringof Z e (blue), a membrane of ξ e (magenta), and a rigid string of X p (red), respectively. and therefore commute with the Hamiltonian W e ( L ) = (cid:89) e ∈ L Z e = (cid:89) ∈ S B , (30) W m (Σ) = (cid:89) e ⊥ Σ ξ e = (cid:89) v ∈V A v , (31) W e − ¯ er ( S r ) = (cid:89) p ∈ S r ζ p = (cid:89) c ∈ Ω r B c,r , (32) W m cage ( C ) = (cid:89) p ∈ C X p = (cid:89) v ∈ V A XCv . (33)Here, L is a loop which encloses a surface S , Σ is a surfacewhich encloses a volume V , S r is a closed ribbon within aplane perpendicular to ˆ r which encircles the region Ω r , and C is a rigid cage configuration which encloses a block volume V . This means that open Wilson operators that act on differ-ent sub-manifolds but share the same boundary will create thesame excitations. For example, the m loop excitation, thoughcreated by a surface operator, does not depend on the choiceof surface in which we choose to fill the loop.
2. Braiding
The hybrid model has non-trivial braiding processes be-tween the excitations of different mobilities. Similar to usualtopological phases, we can prepare an excitation using anopen Wilson operator, then act with a different closed Wilsonoperator to perform the braiding. The closed Wilson opera-tor describes the limit of a braiding process in the space-timepicture, performed in an infinitesimally small amount of time.As in a Z gauge theory, there is a braiding phase of − between the mobile charge e and the flux loop m , reminis-cent of the braiding in a Z toric code. We show this in Fig.1e. Similarly, there is also a braiding phase of − betweena fracton dipole pointing in direction ˆ r with a lineon mobilealong the ˆ r (cid:48) direction if r (cid:54) = r (cid:48) as shown in Fig. 8(a). Thiscan be compared to the fracton-lineon braiding process in theX-Cube model[11].The more interesting braiding that makes this model dif-ferent from a stack of the toric code and X-Cube models isthat the fracton e and the flux loop m has an Aharonov-Bohmphase of i . There are two ways to see this. One way is torealize that acting with the closed Wilson operator W m (Σ) around a fracton braids a flux loop around that vertex. Since W m (Σ) is just a product of A v operators enclosed within thesurface Σ , and the fracton is charged i under A v , this impliesthat the braiding process gives a statistical phase of i .Alternatively, we propose an exotic braiding process be-tween a fracton dipole and a corner of an m loop, as shownin Figure 8(b). We consider a fracton dipole and use theclosed Wilson operator W e − ¯ er ( S r ) to hop the fracton dipolein a closed trajectory perpendicular to the direction r . Since W e − ¯ er ( S r ) is a product of B c,r operators, and the corner ofthe m loop is charged ± i under B c,r in two of the three di-rections, we find that if the trajectory of the fracton dipole en-closes a corner of the m -loop within the same plane, then theprocess can detect a phase of ± i . Specifically, there is a statis-tical phase if the m -loop pierces the W e − ¯ er ( S r ) membrane. Itis interesting to note that although the fracton is immobile, itis allowed to move when paired up as a dipole. Furthermore,it is only when the dipole braids with a corner of the m loop Closed membraneof m ( W m (Σ)) fracton e i charge e − Closed string of e − ¯ e dipole ( W e − ¯ er ( S r )) Closed string of e charge ( W e ( C )) loop m ± i (at corners of the loop) − lineon m − TABLE IV.
Braiding Data:
Summary of the braiding phases in thefractonic hybrid X-Cube model. The braiding process is obtainedby applying the closed Wilson operator over the excitations of thefractonic hybrid X-Cube model, as shown in Figs. 1e, and 8. m (lineon) = m (loop)corner of FIG. 7.
Loop fusion in the fractonic hybrid X-Cube order:
Fusion of two identical m excitations results in lineon excitations ( m ) at thecorners of the membrane. The direction of mobility (blue double arrow) for each lineon is perpendicular to the two segments of the loopmeeting at that corner. that only one of the fractons winds up forming a link with the m -loop .To conclude, the loop excitation in this hybrid model hasexotic braiding properties which makes it distinct from a loopexcitation in a pure TQFT . Though the loop is fully mobile,its corners can be detected with a phase i by fracton dipolesdefined in the same plane (in two of the three directions). Notethat this braiding is also consistent with fusion, since the m loop corners square to lineons, which can be detected with anidentical process with braiding phase − .Finally, it is important to point out certain braiding pro-cesses with trivial statistics. The first is the trivial braidingbetween the mobile charge e and the lineon m . This canbe seen from the fact that the operators that excite each par-ticle do not overlap (one acts on edges, while the other actson plaquettes). Furthermore, since e is mobile and both arepoint particles, any possible braiding is homotopic to a triv-ial braiding process. The second is trivial three-loop braid-ing statistics, which is an important topological invariant for(liquid) 3d topological orders [26, 44–50]. To show this, weuse the fact that the m -loop can be excited by a membrane of ξ e , which satisfies [ ξ e , ξ e (cid:48) ] = 0 . Therefore, any braiding ofloops cannot produce a phase, including any three-loop braid-ing processes.A summary of the braiding phases are given in Table IV. One might be concerned that the notion of a dipole detecting a corner ofan m -loop might not be well-defined away from the exactly solvable limit.In particular, whether the notion of a m loop corner is well defined pointin space if the membrane operator that creates the loop excitation has alarger support. However, we know that in the exactly solvable limit, theHamiltonian has a conservation law that the product of B c,r on all cubesin a given plane perpendicular to the direction r is the identity. Since B c,r detects the corners of the flux loop, this conservation law guarantees thateach plane always has an even number of flux loop corners. Therefore,the notion of a flux loop corner is well defined for every plane. It followsthat away from the exactly solvable limit, there is an equivalent conserva-tion law (adiabatically connected to the B c,r operator) that pins the fluxloop corners to specific planes. Hence, the braiding process is well-definedthroughout the hybrid phase.
3. Ground state degeneracy and logical operators on a torus
The ground state degeneracy of the hybrid model can becalculated via similar methods used for the toric code and X-Cube models. In Appendix D, we count the number of inde-pendent stabilizers and compare it to the total dimension ofthe Hilbert space. We find that the ground state degeneracy ofthe model on a torus of size L x × L y × L z , is log GSD = 2( L x + L y + L z ) . (34)To distinguish the different ground states, we restrict our-selves to the ground state subspace and explicitly construct thelogical (non-local Wilson) operators in this subspace by tun-neling excitations around the torus. In the following, we arguethat the logical operators can be factored to a “toric code” sub-space, consisting of operators that tunnel e and m -loops, andan “X-Cube” subspace, consisting of operators that tunnel e -dipoles and m -lineons. This will allow us to conclude thatthe ground state degeneracy is log GSD = log GSD
T C + log GSD XC (35) = 3 + 2( L x + L y + L z ) − L x + L y + L z ) . Therefore, they form a complete set of logical operators.First, consider tunneling an e − ¯ e dipole around the torususing the operator W e − ¯ e ( R ) = (cid:89) p ∈ R ζ p , (36)where R is a cyan ribbon shown in Fig. 9. Now, althoughthe corners of ζ p are fractons with Z fusion rules, the ribbonof ζ p is actually a Z operator in the ground state subspace,since it squares to a product of B operators, which is set toone. This operator anticommutes with the following operatorthat tunnels the m lineon W m ( L (cid:48) ) = (cid:89) p ⊥ L (cid:48) X p , (37)for some rigid string L (cid:48) that intersects the ribbon R (shown inred). These set of operators form L x + L y + L z ) − pairsof independent Z logical operators identically to those in theX-Cube model.2 e (fracton) m (lineon)¯ e (fracton) corner of m (loop)¯ e (fracton) e (fracton) (a)(b) FIG. 8.
Braiding Processes for an e − ¯ e Dipole: (a)
The braidingof an e − ¯ e dipole (pointing in the x direction) with an m lineonmobile along the y direction. As the dipole moves around a closedloop in the yz plane, if the dipole winds around the lineon, it picksup a phase of − . (b) An analogous braiding of an e − ¯ e dipole witha corner of the m loop in xz plane. If the path of the e fracton formsa link with the m loop, it picks up a phase of i . The two braidingprocesses are consistent with the fusion of two m loops in Fig. 7. Next, consider tunneling the m -loop around a non-trivial2-cycle Σ (cid:48) of the torus (shown in magenta), which can be im-plemented by applying W m (Σ (cid:48) ) = (cid:89) e ⊥ Σ (cid:48) ξ e . (38)Similarly, this operator is a Z operator in the ground statesubspace, since squaring this operator gives at most a productof A XCv operators. This operator anticommutes with W e ( C ) = (cid:89) e ∈ C Z e , (39)for some 1-cycle C (shown in blue) that intersects transver-sally with Σ (cid:48) . On a torus, there are such pairs.Lastly, the pairs W e − ¯ e and W m commute, which can beargued from the fact that an e -dipole braids trivially with an m -loop when there are no corners . This ensures that the log-ical operators factor into the two subsets as claimed. More explicitly, for each plaquette p on which the two Wilson operatorsoverlap, W m contains ζ p and W e − ¯ e contains either ξ ik ξ ij or ξ ik ξ il .These two sets of operators always commutes using Eq. (22). FIG. 9.
Logical (non-local Wilson loop) operators of the Hamil-tonian (B31) on a three-torus:
Left: W e − ¯ e ( R ) and W m ( L (cid:48) ) logi-cal operators of the “X-Cube” subspace. They describe the tunnelingof e − ¯ e fracton dipole (cyan) and m lineons (red), respectively.Right: W e ( C ) and W m (Σ (cid:48) ) logical operators of the “toric code”subspace. They describe the tunneling of e mobile charge (blue)and m mobile loop (magenta), respectively. IV. LINEONIC HYBRID X-CUBE ORDER
In the previous model, the fractons and lineons were treatedas charge and flux excitations respectively. We will now con-sider the opposite scenario, where the lineons are charges andthe fractons are fluxes. The model in this section is thereforean example of a different type of hybridization between thetoric code and the X-Cube model. To distinguish it from theformer, we will refer to this hybridization as the Lineonic hy-brid X-Cube model.We remark that although this is the simplest model toconstruct in the case that lineons are charges, the model isanisotropic. As we will see, only lineons mobile in the x or y direction will square to a mobile particle, while the lineonmobile in the z direction will square to the vacuum superselec-tion sector. This is because the fusion rule of the three lineons e x × e y × e z = 1 forbids all three lineons from squaring tothe same mobile Z particle. Nevertheless, it is possible toconstruct a different hybrid model where the lineons square totwo different mobile particles. Such a model would instead bea hybrid between the X-Cube model and two 3d toric codes.We construct such a model explicitly in Appendix E.Following the structure of the previous section, the Isingmodel and its hybrid model are described in Secs. IV A andIV B, respectively. In Appendix C, we show that this modelcan be obtained by a similar p -string condensation [27, 28] tothe X-Cube model by replacing the stacks of toric codes in the xy planes with the hybrid toric code layers of Sec. II. A. Paramagnet with Global and Subsystem Symmetries
To obtain the previous model, the paramagnet had an onsiteplanar symmetry in three directions, where each onsite termis generated by the same normal subgroup of the global Z symmetry. Therefore, an excitation is charged under planarsymmetries along all three directions, resulting in an immo-bile charge in the gauged modelTo start off differently, our paramagnet now has a global G = Z × Z = (cid:104) a = b = 1 (cid:105) symmetry. However, the3 Excitation Creation operator Charges Local Wilson operatorlineon e x End point of Z on x edges A v = i , A XCv,y = A XCv,z = − B c = Cage of e x , e y , e z around c lineon e y End point of Z on y edges A v = i , A XCv,x = A XCv,z = − lineon e z End point of ZI on z edges A XCv,x = A XCv,y = − mobile charge e End point of Z on x, y edge, A v = − B p = Closed loop of e around p .End points of IZ on z edgeloop m Boundary of IX membrane in xy plane B p = − A v = Closed membrane of m around v Boundary of X membrane in xz, yz plane B p = − , B c = ± i (at corners)fracton m Corners of XI membrane in xy plane B c = − A XCv,r = Closed loop of m dipole around v Corners of X membrane in xz, yz planeTABLE V. Excitations in the lineonic hybrid X-Cube model:
A summary of the pure charge and flux excitations in the lineonic hybridX-Cube model is provided above, along with the local operators that measure these excitations in the lattice model. planar symmetry N for each direction of planes is generatedby a different subgroup of G . In particular, the xz , yz and xy planar symmetries are generated by the Z subgroups a , a b and b , respectively. As a result, excitations of this paramagnetare charged under only two of the three planar symmetries andare therefore lineons.To obtain the model we are to present, we first gauge theplanar symmetries of the model to obtain the X-Cube model.The remaining Z global symmetry fractionalizes on the li-neon mobile in the x and y directions. We can then gauge theglobal Z symmetry to obtain the hybrid model. B. Hybrid Order
The model is defined on a cubic lattice with a Z qudit oneach x and y edge, and two qubits on each z edge. The Hamil-tonian of the model is H Hybrid = H (cid:48) T C + H (cid:48) XC ,H (cid:48) T C = − (cid:88) v A v + A † v − (cid:88) p B p ,H (cid:48) XC = − (cid:88) v (cid:88) r = x,y,z A XCv,r − (cid:88) c B c + B † c , (40)where A v = IX X IX X X † X † , (41) B p = Z IZ IZ Z , IZIZ Z Z , Z Z Z Z , (42) A XCv,x = X XIXI X , A XCv,y = X X X X , (43) A XCv,z = XIXI X X , B c = Z ZI ZZ † ZI ZZ † Z † Z † Z ZIZI . (44)Note that the X-Cube model is defined on a dual square latticecompared to that of Sec. III. The vertex terms of the X-Cubemodel satisfy, A XCv,x A XCv,y A XCv,z = 1 . Furthermore, because ofthe hybridization, the vertex term of the toric code A v squaresto A XCv,z of the X-Cube model, and the cube term B c of theX-Cube model squares to a product of two B p plaquettes.
1. Excitations, fusion, and braiding
The excitations in this model are shown in Fig. 10. First wediscuss the charges, which are violations of the vertex terms.Because this model is anisotropic, the lineons e x and e y inthe xy plane are excited with Z on a rigid string in the xy plane. On the other hand, the lineon e z is excited with ZI ona rigid string in the z direction. The lineon e r correspond to A XCv,r (cid:48) = − for r (cid:54) = r (cid:48) . Note that like the X-Cube model, thethree lineons fuse to the vacuum.lineon e x , ¯ e x : (cid:81) e ∈ L x Z e e y , ¯ e y : (cid:81) e ∈ L y Z e e z , ¯ e z : (cid:81) e ∈ L z ZI e . (45)The fusion rules for each lineon species, however, is different. e z × e z = vacuum e x × e x = e y × e y ≡ e . (46)4 Z Z ZZ e x (lineon) e y (lineon) Z e (mobile) IZe (mobile) m (loop) X IX X XXX XZ Z Z ZI Z e z (lineon) ZIm (fracton) X XI X X X X X XI XIIX IXm (fracton) FIG. 10.
Excitations and their Creation Operators in the Li-neonic X-Cube Model
The excitation e is a mobile particle, and can move in the xy plane using Z , as well as in the z direction using IZ .mobile e : (cid:89) e ∈ L x ,L y Z e , (cid:89) e ∈ L z IZ e . (47)This mobile excitation is charged − under A v .Next, we discuss the flux excitations. The first is the m -loop, which is a violation of plaquettes. To excite an m loop,we apply IX on a z -edge , or X , X † on an x or y edge. How-ever, we also notice that when the m -loop is oriented in the xz or yz plane, the corners of the m -loop are charged ± i under = m (loop) m (fracton) FIG. 11.
Loop fusion in the lineonic hybrid X-Cube order:
Afterfusing two identical m loops in the lineonic hybrid X-Cube model,the corners of the m loops fuse into fractons if the loops are orientedin the xz or yz planes. Otherwise, they fuse into the vacuum. the B c operator as shown in cyan and purple in Fig. 10 .loop m : (cid:89) e ⊥S xy IX e , (cid:89) e ⊥S xz , S yz X e . (48)Finally, the fracton is the excitation B c = − . Four fractonscan be created on the four cubes adjacent to an edge using X acting on an x or y edge, or using XI on a z edge.fracton m : (cid:89) e ⊥S xy XI e , (cid:89) e ⊥S xz , S yz X e . (49)Similarly to the previous model, the fusion of flux excita-tions gives immobile point excitations at its corners. Interest-ingly, if the m -loop is oriented in the xy plane, then two m -loops fuse to the vaccuum. However, if the m -loop is orientedin the xz or yz planes, then there will be fracton excitationsleft at the corners after the fusion. This is shown in Figure 11.The local Wilson operators are products of stabilizers ofthe Hamiltonian, and correspond to closed trajectories of theexcitations in the Hamiltonian as summarized in Table V.In addition to the usual − phase between e and m sim-ulating the toric code, and between e and m simulating X-Cube, we also have a braiding phase of i when an m -loopmoves around the lineons. By realizing that a product of A v is a closed configuration of the m -loop, we see that the e x , e y and e z obtains a phase of i , − i , and − when an m -loop isbraided around each particle respectively. V. HYBRID HAAH’S CODE
In this section, we construct a type-II hybrid fracton model.Here we choose the subsystem symmetry to be the fractalsymmetry corresponding to Haah’s code [3]. Therefore, wewill call this model the hybrid Haah’s code. The symmetric,short-range-entangled state we start with has Z d.o.f. on ver-tices of a cubic lattice. The SRE state is again the ground-stateof the Hamiltonian H = − (cid:88) v [ X v + X v + X v ] . (50)5 A v = B = A HCv = B c = ξ e = ≡ ξ e = ζ (1) v = ≡ ζ p = ζ (2) v = ≡ ζ p = FIG. 12.
Lattice Model for the Hybrid Haah’s code:
Visualization of the operators in the hybrid Haah’s code. The color coding used isred= X , blue= Z , green = S , magenta = ξ , orange = ξ † , cyan = ζ , purple = ζ † , → = CNOT. We enforce a global Z symmetry generated by (cid:81) v X v anda Z fractal symmetry (cid:81) v ∈ fractal X , the latter of which isprecisely the symmetry that is gauged to obtain Haah’s code[3]. The fractal symmetry replaces the planar symmetry pre-viously considered when constructing the hybrid X-Cube or-ders.Excitations above the paramagnet ground state are obtainedby applying the following operators ∆ e = Z i Z f , (51) ∆ (1) v = Z v Z v +ˆ x Z v +ˆ y Z v +ˆ z , (52) ∆ (2) v = Z v Z v +ˆ x +ˆ y Z v +ˆ y +ˆ z Z v +ˆ x +ˆ z . (53)where the unit vectors ˆ x , ˆ y and ˆ z denotes translation by a unitcell in the x , y , and z directions respectively.Next, we follow through the process of gauging the symme-try. We first gauge the Z fractal symmetry to obtain Haah’scode where the gauge charge is fractionalized by the remain-ing global Z symmetry. Then, we gauge the global Z sym-metry to obtain the hybrid model.The hybrid model lives on the same lattice as in Figure 4,and we adopt the same convention of ordering of vertices oneach edge. The Hilbert space of this model, however, is dif-ferent. We place a qubit on each edge and two qubits on eachvertex of this lattice. Furthermore, it is helpful to define anindex α = 1 , . On edges, e (1) and e (2) denotes upright ordiagonal edges of this lattice, respectively, while on vertices, v (1) and v (2) denote the first and second qubit of that vertex,respectively.The algebra of this model is generated by ζ ( α ) v , ξ ( α ) e , Z e , X ( α ) v . ζ p and ξ e are modified operators given by ξ ( α ) e = X e CNOT e,i ( α ) e , (54) ζ ( α ) v = Z ( α ) v (cid:89) e ( α ) ← v S e . (55)Here, e ← v denotes edges incoming to v as defined in Fig 4.The operators satisfy the following algebra ζ ( α ) v X ( α (cid:48) ) v (cid:48) = ( − δ v,v (cid:48) δ α,α (cid:48) X ( α (cid:48) ) v (cid:48) ζ ( α ) v , (56) Z e ξ ( α ) e (cid:48) = ( − δ e,e (cid:48) ξ ( α ) e (cid:48) Z e , (57) [ ξ ( α ) e , ξ ( α (cid:48) ) e (cid:48) ] = [ ζ ( α ) p , ζ ( α (cid:48) ) p (cid:48) ] = [ ξ ( α ) e , X p ] = [ ζ ( α ) p , Z e ] = 0 . (58)which makes them look like Z Pauli operators when definedsolely within the vertex or edge subspace, except that we alsohave the following commutation relations between terms be-tween vertices and edges ζ ( α ) v ξ ( α (cid:48) ) e = (cid:40) i ξ ( α (cid:48) ) e ζ ( α ) v ; if e ← v and α = α (cid:48) , ξ ( α (cid:48) ) e ζ ( α ) v ; otherwise . (59)The Hamiltonian of the hybrid Haah’s code is then ex-pressly H Hybrid = H (cid:48) T C + H (cid:48) HC ,H (cid:48) T C = − (cid:88) v A v + A † v − (cid:88) B ,H (cid:48) HC = − (cid:88) v A HCv − (cid:88) v B v + B † v , (60)6where A v = (cid:89) e → v ξ † e (cid:89) e ← v ξ e , , (61) B = (cid:89) e ∈ Z e , (62) A HCv = X (1) v X (1) v − ˆ x X (1) v − ˆ y X (1) v − ˆ z X (2) v X (2) v − ˆ x − ˆ y X (2) v − ˆ y − ˆ z X (2) v − ˆ z − ˆ x , (63) B v = ζ (1) v ζ (1) v +ˆ x +ˆ y ζ (1) v +ˆ y +ˆ z ζ (1) v +ˆ z +ˆ x ζ (2) v † ζ (2) v +ˆ x † ζ (2) v +ˆ y † ζ (2) v +ˆ z † . (64)The operators are visualized in Figure 12.Similarly to previous examples, this model is a hybridmodel of a Z toric code and a Z Haah’s code. If we hadreplaced ξ ( α ) e and ζ ( α ) v in the definitions of A v and B v withPauli matrices X e and Z ( α ) v , respectively, then the modelwould be a tensor product of the 3d toric code defined onthe edges and Haah’s code defined on the vertices. However,by replacing X e → ξ ( α ) e , Z ( α ) v → ζ ( α ) v , the degrees of free-dom on the edges and vertices are now coupled. The vertexterm A v now squares to A HCv in Haah’s code. At the sametime, the B v term of Haah’s code now squares to a product oftwelve B terms of the toric code. A. Summary of excitations, fusion, and braiding
The fracton e is created via the operators ζ ( α ) v , which com-mutes with all B v and B , but violates four A v projectors.in particular they are charged i under the operator A v . Asin Haah’s code, a general charge configuration can be createdat the corners of a Sierpinski pyramid by applying ζ ( α ) v in afractal pattern fracton e : (cid:89) v ∈ fractal ζ ( α ) v . (65)The mobile charge can be created at the end points of a stringof Z e , which is charged − under A v . Furthermore, they canalso be created by applying (cid:16) ζ ( α ) v (cid:17) in a fractal patternmobile e : (cid:89) v ∈ fractal (cid:16) ζ ( α ) v (cid:17) , (cid:16) ζ ( α ) v (cid:17) = (cid:89) e ( α ) ← v Z e . (66)Therefore, e is a mobile charge.Next, we define operators that create the flux excitations.First, to create a loop excitation, we apply ξ e on edges withintersect a surface S (cid:48) in the dual lattice.mobile loop m : (cid:89) e ⊥S (cid:48) ξ p . (67)The boundary of this surface is charged − under B . Fur-thermore, we find that depending on the shape of S , the loopis also sporadically charged i, − , or − i under various B v operators at the boundary. FIG. 13.
Loop fusion in the hybrid Haah’s code:
Fusing two iden-tical m loops positioned at the dashed square in the xz plane resultsin fractons ( m ) at the positions shown in blue. Finally, applying X ( α ) v for α = 1 , violates four B c terms,and creates flux fracton excitations m .fracton m : (cid:89) v ∈ fractal X ( α ) v . (68)The fusion follows from the definition of the operatorsabove. A fusion of two charge fractons e results in a mobileparticle e . Furthermore, a fusion of two m -loops results ina number of m flux fractons sporadically placed around theloop depending on its shape. For example, the fusion of two m -loops on a square in the xz plane is shown in Fig. 13.Braiding is very similar to the hybrid X-Cube models. Inaddition to the usual e and m particle-loop braiding as inthe toric code, an m -loop can also braid around the fracton e to give a phase of i . Unfortunately, we are unaware of well-defined braiding processes between e fracton and m fractonin Haah’s code. Such types of braiding, if they exist, couldgive further braiding processes in this model as well. VI. HYBRID PHASES AS PARENT ORDERS FOR TORICCODE AND FRACTON ORDERS
In this section, we argue that the hybrid orders introducedin this paper are natural parent states for both liquid and frac-ton topological orders. We tailor the discussion in this sectiontowards the fractonic hybrid X-Cube model in Sec. III, anddemonstrate that driving a phase transition that condenses anappropriate set of excitations in this hybrid order can yield ei-ther the Z toric code or the Z X-Cube model. While wefocus on this particular example here, similar arguments canbe drawn for all the other models presented in this work. Allof the models described in this paper are proximate to the Z toric code or a Z fracton order through a similar phase tran-sition.We may study phases proximate to the the fractonic hy-brid X-Cube model by adding perturbations to the Hamilto-nian H Hybrid in Eq. (10). We first add longitudinal and trans-7verse fields which act as hopping terms for the mobile charge e and the lineon m in the hybrid order. H = H Hybrid − t e (cid:88) e Z e − t m (cid:88) p X p . (69)First, let t e = 0 and consider the limit of large t m . Sincethe operator X p hops a lineon ( m ), the lineons “condense”in the limit of large t m . As a consequence, the fractons e , ¯ e , which braid non-trivially with the lineons are confined .The only remaining topological excitations are e created by Z e , which is charged − under A v , and the m -loop, createdby (26). Therefore, the m -lineon condensate is in the sametopological phase as the d toric code.Indeed, we can derive the effective Hamiltonian in this limitby imposing the constraint that X p = 1 on H Hybrid . The op-erator B c,r does not commute with this constraint and, as aresult, does not contribute to the effective Hamiltonian at lead-ing order in perturbation theory in /t m . The other terms inthe Hamiltonian reduce to A v → (cid:89) e ⊃ v X e , (70) B → B = (cid:89) e ∈ Z e , (71) A XCv → . (72)The simplification of the first term follows from the fact thatCNOT e,p = X − Ze p → − Ze = 1 . (73)The effective Hamiltonian in this subspace, obtained by thereplacements in Eq. (70), exactly yields the 3d toric code.Next, we consider the limit that t m = 0 and t e → + ∞ .In this limit, the mobile charge e is condensed, as the op-erator Z e hops the mobile e particle. As a result, the loopexcitation m , which braids non-trivially with e is no longer atopological excitation, and we are left with the fracton e andthe lineon m . They are exactly the excitations in the X-Cubefracton order. More explicitly, we may again obtain the ef-fective Hamiltonian by projecting into the subspace in which Z e = 1 . The operator A v brings us out of this constrainedsubspace and does not contribute to the effective Hamiltonianat leading order. The remaining operators reduce to B → , (74) A XCv → A XCv = (cid:89) p ⊃ v X p , (75) B c,r → (cid:89) p ∈ c r ,c (cid:48) r Z p . (76) The energy cost to separate a set of fractons now grows linearly in theirseparation. This is in contrast to the constant (logarithmic) energy barrierto separate these excitations in a Type I (II) hybrid order.
FIG. 14.
Schematic Phase Diagram:
For the fractonic hybrid X-Cube model, condensing the mobile charge ( e ) or the lineon ( m )drives a phase transition into the X-Cube fracton order or a 3d toriccode topological order, respectively, as obtained by studying theHamiltonian (69). The phase transition between the hybrid order andthe X-Cube order in this phase diagram is continuous and in the sameuniversality class as the Higgs transition in a three-dimensional Z gauge theory, near the line t m = 0 , as described in the text. The na-ture of the phase transition between the hybrid and toric code ordersis not known. We note that the geometry of the phase boundariesshown here is not meant to be exact, and we are unaware if otherintermediate phases can arise in the “interior” of the phase diagram.A similar phase diagram is obtained for the hybrid Haah’s code inAppendix F. The simplification of the last line follows from S e = i − Ze → i − = 1 . (77)Therefore, the remaining stabilizers are exactly those of the X-Cube model, and the ground-state exhibits the X-Cube fractonorder.Finally, we can consider t e , t m → ∞ . In this case both e and m are condensed, since e and m have non-trivial braid-ing with the above set of excitations, there are no topologi-cal excitations left. The phase is a trivial confined phase. Aschematic phase diagram is shown in Figure 14.We may derive some properties of the phase transition be-tween the fractonic hybrid X-Cube and X-Cube orders by ob-serving that the operator Z e hops the mobile e excitation andcommutes with all of the terms in the fractonic hybrid X-CubeHamiltonian, except for A v . Furthermore, the X p term, whichhops a lineon, commutes with all terms in the fractonic hy-brid X-Cube Hamiltonian, except for B c,r . As a result, wemay study the Hamiltonian in Eq. (69) within a constrainedHilbert space, within which there are no m flux loop excita-tions, or e fracton excitations, as enforced by B = 1 and A XCv = A v = 1 . Projecting the Hamiltonian into this con-8strained subspace yields P HP = − (cid:88) v A v − t e (cid:88) e Z e − (cid:88) c,r B c,r − t m (cid:88) p X p , (78)with P denoting the projection. Here, we have also used thefact that P A † v P = A v and P B c,r P = B c,r .When t m = 0 , the lineons are non-dynamical, and wemay further set B c,r = 1 . The algebra satisfied by A v and Z e is precisely the algebra between the “star” operator in a3d toric code, which measures the Z charge, and a trans-verse field, which has the effect of hopping the charge exci-tation. As a result, the phase transition between the fractonichybrid X-Cube order and the X-Cube order is precisely relatedto the Higgs transition in a three-dimensional Z gauge the-ory, which is driven by the condensation of the Z charge.This transition is known to be direct and continuous, anddual to an Ising symmetry-breaking phase transition in (3+1)-dimensions [51], if the dynamical Z flux excitations are for-bidden.The generic nature of the transition between the fractonichybrid X-Cube and X-Cube orders, when other excitations(fractons or lineons) are allowed is not known. We may show,however, that the phase transition remains continuous when t m is turned on. In this case, lineons may be created, thoughthey are highly “massive” excitations when t m is small, andwe may integrate out the lineon excitations to obtain an ef-fective description of the critical point. To leading order inperturbation theory, the effective Hamiltonian describing thesystem when t m (cid:28) may be obtained from Eq. (78) as H eff = − (cid:88) v A v − t e (cid:88) e Z e − (cid:88) c,r B c,r − K (cid:88) v A v + · · · (79)to leading order in perturbation theory in t m , where K ∼ O ( t m ) . Since A v and B c,r commute with the effectiveHamiltonian, the critical point separating the fractonic hybridX-Cube order and the X-Cube model is unchanged at this or-der in perturbation theory, and the transition remains continu-ous, and admits a description as a Higgs transition in a three-dimensional Z gauge theory.In Appendix F, we demonstrate through a similar deriva-tion that the Hybrid Haah’s code is a parent state for both thetoric code and Haah’s code. Namely, a condensation of thefracton m drives the system into the toric code phase, and acondensation of the mobile charge e drives the system intothe Haah’s code phase. VII. DISCUSSION
In this work, we introduced exactly solvable models of hy-brid fracton phases, which consists of excitations with bothmobile and immobile excitations. As phases of matter, they are distinct from a stack of a fracton phase with a liquid topo-logical ordered phase because of its unusual fusion and braid-ing properties. Our work raises a challenge to classify gappedquantum phases in terms of liquids and non-liquids.Although in this paper, we focused on abelian hybrid mod-els, an exactly solvable model can in fact be constructed foran arbitrary finite gauge group, which will be presented in up-coming work [1].In the following, we present some interesting directions forfurther studies of abelian hybrid fracton models.
Twisted hybrid fracton models : In this paper, our hybridfracton models are obtained by starting with a product stateand gauging the ( G, N ) symmetry. If we instead start witha Subsystem Symmetry Protected Topological (SSPT) stateprotected by G subsystem symmetry[31, 52, 53], and breakthe symmetry explicitly to ( G, N ) it should be possible to ob-tain more exotic twisted hybrid models after gauging. Onemight also be interested in finding phases protected directlyby the ( G, N ) symmetries. This can be broadly searched bystudying consistency conditions of the symmetry defects [54]. Fermionic ( G, N ) symmetries : The current constructioncan be straightforwardly generalized to fermionic symmetries.One can start with a charge- n superconductor, which is in aatomic insulating phase and impose an extra fermion paritysymmetry on subregions. Then, one can gauge the fermionicsubsystem symmetry [55, 56] to obtain a Z fracton model en-riched by a Z n global symmetry and study whether the sym-metry enrichment is different from its bosonic counterpart. Ifso, further gauging the Z n global symmetry will give rise to adifferent hybrid model. Subsymmetry-Enriched Topological (SSET) phases : Ourhybrid fracton models can be used to construct SSET phases[39] by condensing the particle m along with charges of anIsing model with the same restricted mobility (or equivalentlyby gauging the higher-rank symmetry associated to the Wilsonoperator of m ). In particular, using the models introduced inthis paper, we expect the resulting model to be a Z toric codeenriched by Z subsystem symmetries. It would be particu-larly interesting to investigate the properties a Z toric codeenriched by the fractal symmetry of Haah’s code. Error Correcting Codes : The exotic mix of immobile andmobile excitations in these models might be useful for quan-tum error correction. Indeed, the absence of string-like logi-cal operators (topologically non-trivial Wilson loops) in TypeII fracton orders is intimately related to their improved abil-ity to act as a self-correcting quantum memory [43]. In thehybrid Haah’s code, the presence of a loop excitation as wellas a conjugate fracton excitation that precisely resembles thefractons in Haah’s code may lead to improved performance asa quantum memory, which deserves further study.
Emergent symmetries.
The excitations of the fractonic hy-brid X-Cube model in Sec. III seem to have a cubic symmetry,even though the Hamiltonian does not. It would be interestingto see whether the Hamiltonian can be written in a form thatpreserves the cubic symmetry as well, or whether the cubicsymmetry can only emerge at low energies.
Non-liquid orders beyond hybrid models.
As a third prox-imate phase to the hybrid models, as discussed in Sec. VI,9it would be interesting to consider condensing the compositeexcitation e m in the hybrid model. Such a phase may poten-tially realize a non-liquid phase that is not a hybrid between atopological order and a fracton order. ACKNOWLEDGMENTS
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In this Appendix, we give a proper definition of the ( G, N ) symmetry which act as the gauge group for the hybrid fractonmodels. Here, we focus on abelian ( G, N ) symmetries andgive a proper definition for a general group G in Ref. [1].
1. Definition
The hybrid fracton models derived in this paper can bethought of as the result of gauging a paramagnet with a partic-ular mix of global and subsystem symmetries, which we terma . This name is an homage to 2-groupsymmetries[57–59], which can be thought of as a particularextension of global (0-form) symmetries by 1-form symme-tries. Likewise, the relevant 2-subsystem symmetries in thispaper can be thought of as a subsystem symmetry correspond-ing to the underlying fracton order extended by a global Z symmetry. The hybrid fracton orders can therefore be under-stood as a “2-subsystem gauge theory”.An abelian 2-subsystem symmetry can be defined given thefollowing data1. An abelian global symmetry group G
2. A normal subgroup
N (cid:47) G
3. The type of region on which N acts as a subsystem sym-metry (e.g. 1-foliated, 3-foliated, fractal,...)To realize the 2-subsystem symmetry in a lattice model, wefirst define R gv an onsite representation of g ∈ G at each ver-tex v . The global G symmetry is defined as (cid:81) v R gv . Further-more, for a group element n in the normal subgroup N the N subsystem symmetry is defined as (cid:81) v ∈ sub R nv , where “sub” isa subregion specified by the type of subsystem symmetry.Let us first emphasize two important points. First, since N is a subgroup of G , the N subsystem symmetry is not an in-dependent symmetry from the global symmetry. For example,1if the subsystem symmetry is 1-foliated, then a product of all N planar symmetries is a global N symmetry, a subgroup ofthe global G symmetry.Second, a 2-subsystem symmetry contains both global sym-metries and subsystem symmetries as limiting cases. For ex-ample, a pure global G symmetry can be written as ( G, Z ) since it only has a trivial subgroup acting as a subsystem sym-metry. Furthermore, a pure N subsystem symmetry can bedenoted ( N, N ) . It is worth nothing that gauging the symme-tries in these two examples will lead to 3d topological ordersand fracton orders, respectively, which are not hybrid models.Therefore, the 2-subsystem symmetries we are interested in,are those for which N is a strict but non-trivial normal sub-group of G . For example, the gauge group corresponding tothe Hybrid toric code layers in Sec. II is a 1-foliated ( Z , Z ) subsystem symmetry.Mathematically, if we neglect the input geometry of thesubsystem symmetry (which specifies the region on which thesubsymmetry acts), then ( G, N ) has a structure of an abeliangroup. Starting with abelian groups G and N , the central ex-tension → N −→ G −→ Q → (A1)is given uniquely by specifying a representative cocycle ω of H ( Q, N ) , where Q = G/N is the quotient group.Now, let L be the number of independent N subsystemsymmetries. The group ( G, N ) is then formally given by thecentral extension → N L −→ ( G, N ) −→ Q → (A2)where N L ≡ N × · · · × N L is a product of L identical copiesof N . The extension can be specified by a representative co-cycle ω (cid:48) of H ( Q, N L ) chosen to be ω (cid:48) = (cid:81) Ll =1 ι l ◦ ω where ι l : N → N L is the embedding of N to the l th copy of N L .It is important to note that even though ( G, N ) is an abeliangroup, and therefore can be written as a product of cyclicgroups, specifying the group extension (A2) encodes infor-mation about which subgroup of ( G, N ) acts as subsystemsymmetries.
2. Gauging 2-subsystem symmetries
The process of gauging a 2-subsystem symmetry to obtaina hybrid fracton model can be broken into two steps, as shownin Figure 15. We start by first gauging the subsystem symme-try N , to obtain a fracton model[6, 60–65] (with gauge group N ). After gauging, the global symmetry G is reduced to thequotient group Q = G/N . Nevertheless, because of the in-terdependence of the global and subsystem symmetries, theremaining global symmetry Q enriches the fracton phase. Inparticular, for abelian 2-subsystem symmetries, Q fractional-izes on the gauge charge (which has restricted mobility) of thefracton model. The fractionalization implies that these gaugecharges can fuse into charges of Q . Now, since gauging Q promotes the charges of Q into gauge charges, the result isa gauge theory where gauge charges with restricted mobility can fuse into fully mobile gauge charges. This is precisely ourhybrid fracton model. Furthermore, consistency with braid-ing implies that the flux loops should also square into variousfluxes with restricted mobility. These properties are demon-strated explicitly in our exactly solvable lattice models.The explicit process of gauging 2-subsystem symmetriesis omitted from the main text. Instead, we give a detailedcalculation of gauging a 2-subsystem symmetry that gives the(fractonic) hybrid X-Cube model of Sec. III in Appendix B. Appendix B: Derivation of the fractonic hybrid X-Cube model
In this appendix, we present details of the derivation of thefractonic hybrid X-Cube model. We first present the form ofthe corresponding Ising model with ( Z , Z ) symmetry. Wethen gauge the symmetry to obtain the corresponding gaugetheory. We note that the final model we obtain given in Eq.(B31) is slightly different from Eq. (10) in the main text, butonly differs in the energetics. That is, the stabilizers of the twomodels are the same. A prescription for gauging a general 2-subsystem symmetry is presented in Ref. 1.
1. Ising model
We begin with a cubic lattice with an additional diagonaledge added to each plaquette as shown in Figure 4. The ad-ditional edge, though not essential to the construction, is forconvenience. Namely, it makes the minimal coupling to theglobal symmetry unambiguous. In addition, it will also makethe resulting terms after gauging are non-Pauli stabilizers ofthe ground state subspace. Lastly it will make the form of theWilson operators apparent.On each vertex v , we place a Z qudit associated to clockand shift operators Eq. (2) reproduced here Z = i − − i , X = . (B1)The Hamiltonian is the following transverse-field Potts model H = − (cid:88) v X v + X v + X v − h E (cid:88) e =( if ) e − h P (cid:88) p =( ijkl ) p + ∆ p + ∆ p (B2)where there are two type of Ising terms: one on each edge,and one on each square plaquette ∆ e = Z i Z f , (B3) ∆ p = Z i Z † j Z k Z † l . (B4)The Hamiltonian has a global Z symmetry generated by (cid:81) v X v , and a planar Z symmetry (cid:81) v ∈ plane X v . We note that2 ℤ Symmetry-Enriched Fracton Model symmetry fractionalizes on the gauge charge ℤ FIG. 15. Summary of the gauging process to obtain the abelian hybrid fracton models in this paper. without ∆ e , the model would instead have a larger symme-try, namely a Z symmetry generated by (cid:81) v ∈ plane X for eachplane.When h E , h P (cid:28) , the model is in the paramagnetic phase.For simplicity, we will now analyze the model in the limitwhere h E = h P = 0 . In this limit, gapped charged excitationsto the ground state are created by applying ∆ e , and ∆ p , thenature of which has been described in the main text.
2. Gauging Z planar symmetries To gauge the symmetry, it is helpful to first rewrite each ver-tex as a tensor product of two qubits . From the expressionsin Eq. (B1), notice that Z = Z ⊗ S, X = ( ⊗ X ) × CNOT , , (B5) Z = ⊗ Z, X = X ⊗ . (B6)where CNOT , = (B7)is the CNOT gate controlled by the second qubit and targettedat the first qubit. Therefore, we see that the planar symmetry (cid:81) v ∈ plane X v in this basis is just a product of Pauli X operatorsin the first qubit, and we can perform a generalized Kramers-Wannier duality [6, 64, 65] on the first qubit to gauge the Z planar symmetries and obtain the X-Cube model. Under theduality, the first qubit is mapped to qubits on plaquettes of thecubic lattice, while the second qubit remains on each vertex.At the level of operators, it is defined as ( Z ⊗ I ) i ( Z ⊗ I ) j ( Z ⊗ I ) k ( Z ⊗ I ) l → Z p , (B8) ( X ⊗ I ) v → (cid:89) p ⊃ v X p . (B9) A similar mapping which differs by a Hadamard on the first qubit waspointed out in Ref. 31. In Ref. 1 we generalize the mapping presented toany group extension described by a factor system.
Note that only operators that commute with the planar sym-metries will be mapped to local operators. The result of thismapping on the symmetric operators is ∆ e → ˜ ζ e = Z i Z f , (B10) ∆ p → ˜ ζ p = Z p S i S † j S k S † l , (B11) X → ˜ A v = X v × (cid:89) p ⊃ v CNOT v,p , (B12) X → ˜ A v = (cid:89) p ⊃ v X p . (B13)Here, we note the operators with tildes because the remainingglobal symmetry action is not written in an onsite manner. Wewill momentarily fix this via a basis transformation and denoteoperators in the new basis without tildes.Now, because a product of four ∆ p operators around a cubeis the identity, we also must enforce constraints ˜ B c,r = (cid:89) p ∈ c (cid:48) r ˜ ζ † p (cid:89) p ∈ c r ˜ ζ p = (cid:89) p ∈ c r ,c (cid:48) r Z p = 1 (B14)where c (cid:48) r and c r are the set of plaquettes surrounding a cubedefined in Figure 5. This can be enforced energetically via aprojector. Therefore, the Hamiltonian after gauging the planarsubsystem symmetries is H SEF = − (cid:88) v A v + ˜ A v + ˜ A v − (cid:88) p (cid:88) r = x,y,z B c,r + ˜ B c,r + ˜ B c,r . (B15)Looking at the terms defined only on plaquettes, given by ˜ A v and ˜ B c,r one notices that these are indeed just the terms thatappear in the X-Cube model. However, the model is coupledto a Z matter field living on vertices in a non-trivial manner. Nevertheless, the model has a global symmetry that en-riches the X-Cube fracton order, thus realizing a Symmetry-Enriched Fracton (SEF) model. Since we have gauged the Z To elaborate, the term ˜ A v is not a minimally coupled term between theX-cube model and a Z matter field since ˜ A v has order four. Z global symmetry is now reducedto a Z global symmetry, generated by (cid:81) v ˜ A v (the reason ithas order two is because (cid:81) v ˜ A v is just the identity). However,written in the current basis, the symmetry is not onsite because ˜ A v acts both on the vertex v as well as nearby plaquettes, andtherefore is not a local symmetry action. Indeed, this propertyallows us to demonstrate that this Z symmetry fractionalizeson the fracton excitation, which we will now prove. Considera fracton excitation at a vertex v , which is a charge ˜ A v = − of the vertex term of the X-Cube model. If we consider a localaction of the Z symmetry in a region R , denoted (cid:81) v ∈R ˜ A v such that only this fracton excitation is in the region R , thenthe local symmetry action squared is (cid:81) v ∈R ˜ A v = − . Thisshows that the fracton is indeed fractionalized under the global Z symmetry.
3. Gauging the global Z symmetry Our next goal is to gauge the remaining Z global symme-try. However, in its current form, the global symmetry (cid:81) v ˜ A v is not onsite. Therefore, we have to perform a basis transfor-mation to make the symmetry onsite. That is, a basis wherethe local symmetry action is X v instead of A v . Our basistransformation is the following unitary U = (cid:89) p X Zi (cid:16) − Zj + − Zk + − Zl (cid:17) p . (B16)Using this unitary to conjugate all operators, we find thatthe operators in the new basis (now written without tildes) are A v = (cid:89) p | v = j,k,l X − ZiZv p × X v × (cid:89) p | v = i X − ZvZj + − ZvZk + − ZvZl p , (B17) A v = (cid:89) p ⊃ v X p , (B18) B c,r = (cid:89) p ∈ c r ζ p , (B19) ζ e = Z i Z f , (B20) ζ p = Z p ( S i CZ ij S j )( S i CZ ik S k ) † ( S i CZ il S l ) . (B21)In particular, the global Z symmetry now acts as (cid:81) v A v = (cid:81) v X v , which is onsite, since all the X p factors cancel in theproduct.We can now perform the Kramers-Wannier duality to gaugethe global Z symmetry. This maps the remaining qubit oneach vertex, to qubits on each edge. At the level of operators,the map is given by Z i Z f → Z e , (B22) S i CZ if S f → S e , (B23) X v → (cid:89) e ⊃ v X e . (B24) The gauged operators are now denoted in bold, and are A v = (cid:89) p | v = j,k,l X − Ziv p × (cid:89) e ⊃ v X e × (cid:89) p | v = i X − Zvj + − Zvk + − Zvl p , (B25) A v = (cid:89) p ⊃ v X p , (B26) B c,r = (cid:89) p ∈ c r ζ p , (B27) ζ e = Z e , (B28) ζ p = Z p S ij S † ik S il . (B29)We remark that X − Ze p = CNOT e,p . Therefore, defining ξ e as in Eq. (15), we realize that A v can be rewritten as in Eq.(11).Enforcing a fluxless condition using for each triangleon the lattice using B energetically, we finally obtain theHamiltonian for the ( Z , Z ) gauge theory H = − (cid:88) v A v + A v + A v − (cid:88) B (B30) − (cid:88) c (cid:88) r = x,y,z B c,r + B c,r + B c,r , (B31)This Hamiltonian has the same ground state and types of ex-citations as that of the fractonic hybrid X-Cube model in Eq.(10), and only differs in the energetics. In particular, we canidentify A v with A T Cv in the main text.
Appendix C: Derivation of the lineonic hybrid X-Cube model
In this Appendix, we derive the lineonic hybrid X-Cubemodel. However, instead of deriving this model from gaug-ing a SRE state with particular symmetry like in Appendix B,we will show that Lineonic hybrid X-Cube model can in factbe alternatively be derived via p -string condensation[28]. Westart with the hybrid toric code layers of Sec. II (where the lay-ers are in the xy planes) tensored with stacks of 2d toric codesalong the xz and yz planes. The stabilizers of this stacked4model are A (1) v = XIXIIXIIXI B (1) p xz = ZIIZIZIZII A (2) v = XIIXIXIIXI B (2) p yz = ZIIZI ZI IZI A (3) v = I X † I X † I X I X IIXIIX B (3) p xz = IIZI Z I Z IIZ B (3) p yz = I Z IIZ IIZI Z B (3) p xy = I Z I Z † I Z I Z † (C1)Next we perform a p -string condensation, by enforcing theconstraint ( X X ) e = 1 (for x, y edges) (C2) ( XXI ) e = 1 (for z edges) (C3)This condenses the particle-string composed of the m anyonsof the toric code stacks in the xz and yz planes, along withthe m planon of the hybrid toric code layers.The constraint above reduces the size of the Hilbert space.We can define effective Pauli operators which commutes withthe above constraints and satisfy the same algebra as the orig-inal ones. For x and y edges, we choose I X ≡ X , I X = XI, ≡ X Z Z ≡ Z , (C4)and on z edges, we choose XII = IXI ≡ XI, IIX ≡ IX, ZZI ≡ ZI, IIZ ≡ IZ. (C5)We can now derive the effective stabilizers in this subspaceby restricting to only product of the stabilizers in Eq. (C1)that commutes with the constraints. The stabilizers restrict tothe following stabilizers of the lineonic hybrid X-Cube modelunder the substitution A (1) v → A XCv,x , (C6) A (2) v → A XCv,y , (C7) (cid:16) A (3) v (cid:17) → A XCv,z , (C8) A (3) v → A v, , (C9) B (3) p xz → B p xz , (C10) B (3) p yz → B p yz , (C11) (cid:16) B (3) p yz (cid:17) → B p xy . (C12) Now, although B (1) p xz , B (2) p yz , and B (3) p xy do not commute withthe constraints, the product of such operators around the sixplaquettes surrounding a cube does commute. The restrictionof such cube operators is exactly B c in the lineonic model.Therefore, we have shown that the effective Hamiltonian afterthe condensation realizes the lineonic hybrid X-Cube modelas desired.We remark that the lineonic ( Z , Z ) hybrid model pre-sented in Appendix E can similarly be derived by startingwith a product three copies of the hybrid 1-foliated model pre-sented in Sec. II, oriented in the three directions. The conden-sate is a product of the three m -loops of each model. Interest-ingly, since m -loops of the 1-foliated model square to a prod-uct of m planons in each layer, the square of the term thatcondenses the three m -loops is exactly the term that inducesa p -string condensate of the m planons. A further study ofthe relation between flux loops and p -strings could shed fur-ther light on the relation between higher-form symmetries andtheir foliated versions[66]. Appendix D: Ground State Degeneracy
In this Appendix, we calculate the ground state degeneracyof the hybrid fracton models introduced in the paper. We firstperform explicit calculations for the hybrid toric code layersand the fractonic hybrid X-Cube model on a torus, and thenprove generally that the ground state degeneracy of the hybridmodel is equal to that of the tensor product model.
1. Hybrid Toric Code Layers
The Hamiltonian (3) has a Z qubit and two Z qubits perunit cell. Therefore, log (dim H ) = 5 L x L y L z (D1)Since the Hamiltonian is a commuting projector, the groundstate subspace are eigenstates for which we set all the stabiliz-ers to one A v = A dv = B dp = B p xy = 1 . (D2)Let us count how many constraints this imposes on the Hilbertspace.First, there is one A v operator per unit cell. Since A v haseigenvalues ± and ± i , restricting to A v = 1 divides thesize of the Hilbert space by four, meaning it produces twoconstraints per vertex, and so it imposes L x L y L z constraintsin total. Not all such constraints are independent. For each xy plane, we have the identity (cid:81) v ∈ xy A v = 1 . There are L z such identities. Furthermore, we also have the constraint (cid:81) v A v = 1 . Note that (cid:81) v A v = 1 is already accounted forby the previous identities. Therefore we have L x L y L z − L z − (D3)constraints from A v . We note that since A v = A dv , there areno further constraints from setting A dv = 1 .5Next, there are three B dp operators per unit cell. The eigen-values of B dp are ± , so this produces L x L y L z constraints.We subtract by L x L y L z − for each cube where a productof six B dp operators around the cube is the identity; the extraone is because of overcounting all the cubes, and we subtractby three for the product of B dp around the three 2-cycles ofthe torus being identity. The number of independent B dp con-straints is therefore L x L y L z − ( L x L y L z − − L x L y L z − . (D4)Finally, there is one B p xy operator per unit cell, which haseigenvalues ± and ± i . However, B p xy = B dp xy , so this onlyimposes L x L y L z further constraints. For each plane, we havethe identity (cid:81) v ∈ xy B p xy = 1 , so subtracting the redundan-cies, there are only L x L y L z − L z (D5)additional constraintsPutting everything together, the ground state degeneracy isgiven by log GSD =5 L x L y L z − (2 L x L y L z − L z − − (2 L x L y L z − − ( L x L y L z − L z )= 2 L z + 3 (D6)which is the same as the GSD of L z layers of 2d toric codestensored with a 3d toric code.Let us construct these logical operators explicitly. First, ineach layer, we can tunnel the e planon around the x cycle ofthe 2d torus using W ex,l = (cid:89) e x ,y = y ,z = l Z e (D7)for some fixed y . There are L z such operators for each layer l = 1 . . . L z . This anticommutes with W m y,l = (cid:89) e x ,x = x ,z = l X e (D8)for some fixed x . This operators that tunnels the m planonaround the y cycle of the 2d torus in the layer l . Each pairspans a Hilbert space of dimension two, therefore so far theyfit in a Hilbert space of dimension L z .In addition, we can also consider tunneling the m looparound the yz W myz = (cid:89) e x ,x = x X e (D9)This commutes up to a phase i with W el for each l . Note that (cid:0) W myz (cid:1) = (cid:89) l W m y,l . (D10)It turns out that we can minimally extend the size of theHilbert space by a single qubit to accomodate this operator. That is, the algebra of these operators fit in a Hilbert space ofdimension L z +1 . Explicitly, we can express the above oper-ators using L z + 1 qubits as W ex,l = Z l S L z +1 , (D11) W myz = X L z +1 (cid:89) l CNOT L z +1 ,l , (D12) W m y,l = X l , (D13)which satisfies the same algebra. Note that there is anotherset of operators constructed identically by swapping x and y ,independent of this set of operators.Finally, we can tunnel the e mobile particle in the z directionusing W e z = (cid:89) e z ,x = x ,y = y Z e (D14)This anticommutes with W mxy = (cid:89) e z ,z = z X e (D15)which tunnels m around the 2-cycle in the xy plane. Thispair generates a Hilbert space of dimension two independentof the two aforementioned sets. Putting everything together,the Hilbert space dimension of the logical subspace is log (dim H logical ) = 2( L z + 1) + 1 = 2 L z + 3 (D16)in agreement with our ground state degeneracy.
2. Fractonic Hybrid X-Cube
The Hamiltonian has nine qubits per unit cell, and thereforelives in a Hilbert space with log (dim H ) = 9 L x L y L z . Sincethe Hamiltonian is a commuting projector, the ground statesubspace are eigenstates for which we set all the stabilizers toone A v = A T Cv = B = B c,r = 1 . (D17)First, there is one A v per unit cell, but since it has eigenval-ues ± and ± i , restricting to A v = 1 divides the size of theHilbert space by four, meaning it produces two constraints pervertex, and so it imposes L x L y L z constraints in total. Notall such constraints are independent. Similarly to X-Cube, wesubtract by L x + L y + L z − because (cid:81) plane A v = 1 for each xy , yz , and xz plane; the overcount of two is because theproduct over all parallel planes is (cid:81) v A v . Then, analogouslyto the toric code, we subtract by one because (cid:81) v A v = 1 ( (cid:81) v A v = 1 already being accounted just earlier). To con-clude there are L x L y L z − ( L x + L y + L z − (D18)independent A v constraints. Since A v = A T Cv , there are nofurther constraints from setting A T Cv = 1 B operators per unit cell. The eigen-values of B are ± , so this produces L x L y L z constraints.We subtract by L x L y L z − for each cube where a productof twelve B ’s around the cube is the identity; the extra oneis because of overcounting all the cubes, and we subtract bythree for the product of B around the three 2-cycles of thetorus being identity. The number of independent B con-straints is therefore L x L y L z − ( L x L y L z − − . (D19)Finally there are two independent B c,r operators per unitcell, each of which has eigenvalues ± and ± i . Thus, thereare L x L y L z constraints. We need to subtract by L x + L y + L z − because (cid:81) plane B c,r = 1 for each r perpendicular tothe plane, the extra one for overcounting the product over allpossible planes being identity. Furthermore, each B c squaresto a product of four B operators, meaning we have to furthersubtract by L x L y L z . Therefore, the independent constraintsof B c,r up to B operators is L x L y L z − ( L x + L y + L z − (D20)Putting everything together, the ground state degeneracy isgiven by log GSD =9 L x L y L z − [2 L x L y L z − ( L x + L y + L z − − [6 L x L y L z − ( L x L y L z − − − [2 L x L y L z − ( L x + L y + L z − L x + L y + L z ) (D21)which is consistent with the number of logical operatorscounted in the main text.
3. General calculation
Let us now show in general that a hybrid model betweena Z toric code and a Z fracton model will have the sameground state degeneracy on any manifold. To warm up, let uscalculate the ground state degeneracy in the stacked model.We have stabilizers (cid:16) A T C (cid:17) = (cid:16) B T C (cid:17) = (cid:16) A frac (cid:17) = (cid:16) B frac (cid:17) = 1 . (D22)We omit the cell which each stabilizers is defined on for sim-plicity. The dimension of the Hilbert space is given by log dim H stack = log dim H TC + log dim H frac . (D23)The stabilizers of the toric code A T C and B T C imposes log dim H TC − independent constraints while those of thefracton model A frac and B frac will impose log dim H frac − log GSD frac constraints. Therefore the ground state degener-acy of the stacked model is log GSD stack = log GSD frac + 3 . (D24)Let us now argue that the ground state degeneracy in the hy-brid model must be the same. The dimension of the Hilbert space in the hybrid model is equal to that of the stacked model,while the stabilizers satisfy (cid:16) A T C (cid:17) = A frac , (cid:16) B frac (cid:17) = (cid:89) B T C , (cid:16) A frac (cid:17) = (cid:16) B T C (cid:17) = 1 (D25)where (cid:81) is an appropriate product of toric code plaquetteterms, depending on the hybrid model. Because of this, thenumber of constraints that A frac and B T C impose are un-changed, while the additional constraints that A T C and B frac impose (up to A frac and B T C terms) are also the same as be-fore. This implies that the ground state degeneracy of the hy-brid model is the same as that of stacked model.It is important to note that the above result does not implythat the logical operators of the hybrid model satisfy the samealgebra to those in the stacked model. Though this is indeedthe case in the hybrid X-Cube model, we have demonstratedthat the algebra is different in the case of the hybrid toric codelayers; some logical operators in the hybrid model have orderfour, while all logical operators have order two in the stackedmodel.
Appendix E: A Hybrid of X-Cube and Two Toric Codes
In this Appendix, we introduce an isotropic version of theLineonic Hybrid X-cube model presented in Sec. IV. Thismodel is more involved, because it is a hybrid between theX-Cube model and two 3d toric codes. Nevertheless, all thelineons in this model will square to mobile charges. Descrip-tively, let e x , e y , e z be the lineons in the X-Cube model and e , e be the mobile charges of the toric code. Then the fu-sion rules are given by e x = e e y = e , e z = e e (E1)Because of the gauge group underlying this model, we willcall it a Lineonic ( Z , Z ) gauge theory.
1. Paramagnet
First, we describe the paramagnet with a ( Z , Z ) Z qudits per vertex H = − (cid:88) v X I v + X I v + X I v I X v + I X v + I X v (E2)The global Z symmetry has three Z subgroups. Theyare generated by (cid:81) v X I v , (cid:81) v I X † v , and the diagonal (cid:81) v X † X v . Furthermore, there are three Z planar symmetries (cid:81) v ∈ xz X I v , (cid:81) v ∈ yz I X v , (cid:81) v ∈ xy X X v . The product of all7 A v, =
11 232 3 A v, = B p, = , , A v, =
33 121 2 A v, = B p, =
112 2 , ,
113 3 A v, =
22 313 1 A v, = B p, =
331 1 ,
221 1 ,
22 33 B c = , B c = =
221 1 113 313 3 121 21 =
112 23 311 3 311112 2 = FIG. 16.
Lattice Model for the Lineonic ( Z , Z ) gauge theory: Visualization of the operators in the Lineonic ( Z , Z ) gauge theory. Here, , , IX, X I, X † X (magenta), , , IX, X † I, X X (orange), red = X I , , , Z I, IZ, Z Z (blue), cyan = Z I , purple = Z † I . planar symmetries in a particular direction is respectively the Z subgroup of the three Z global symmetries.The operators that create charged excitations that commutewith the above symmetry are defined on each edge e ρ , where ρ = x, y, z and depends on the direction of the edge. ∆ e x = ( Z I ) i ( Z † I ) f (E3) ∆ e y = ( I Z ) i ( I Z † ) f (E4) ∆ e z = ( Z † Z † ) i ( ZZ ) f (E5)We note that the convention of the generators and hoppingoperators above have been carefully chosen such that the endpoint i of ∆ e x is charged , i, − i under the three Z genera-tors, and charged , − , − under the planar Z symmetries.This makes the charges at the endpoint a lineon excitation.Furthermore, the product ∆ e x ∆ e y ∆ e z which shares the sameendpoint i has no charge at site i . In the corresponding gaugedmodel, this means that the three lineons mobile in the three di-rections fuse to the vacuum.
2. Hybrid Order
The hybrid model is defined on a 3d cubic lattice. For eachedge, we place a Z qudit and a Z qubit. The Hamiltonian ofthe hybrid model is H Hybrid = − (cid:88) r = x,y,z (cid:34)(cid:88) v A v,r + A v,r + A v,r (cid:88) p B p,r (cid:35) − (cid:88) c B c + B c + B c , (E6) where A v,r = (cid:89) e → v ξ † e,r (cid:89) e ← v ξ e,r , (E7) A v,r = (cid:89) e ⊃ v r ( X I ) e , (E8) B p,r = (cid:89) e ∈ p ζ e,r (E9) B c = (cid:89) e ∈ c (cid:48) ( Z I ) e (cid:89) e ∈ c ( Z I ) e . (E10)Here, e → v and e ← v denote edges entering and exitingthe vertex v as in the main text. This is shown in Figure 16as orange and magenta respectively. Furthermore, e ∈ c (cid:48) and e ∈ c refer to the purple and cyan edges of a cube c in theFigure.To define ξ e,r and ζ e,r , we note that its definition dependson the orientation of the link e . Therefore, we have to defineit separately for e x , e y , and e z . For simplicity in definingthese operators, we substitute x, y, z with , , so that wecan define ξ e ρ ,r = IX ; r − ρ ≡ mod X I ; r − ρ ≡ mod X † X ; r − ρ ≡ mod (E11) ζ e ρ ,r = Z I ; r − ρ ≡ mod IZ ; r − ρ ≡ mod Z Z ; r − ρ ≡ mod (E12)Note that for a fixed edge e , ξ e,x ξ e,y ξ e,z = 1 , and ζ e,x ζ e,y ζ e,z = 1 . In particular this implies that A v,x A v,y A v,z = 1 . Furthermore, we would like to point8out that ξ e,r and ζ e,r (cid:48) commutes for r = r (cid:48) and otherwiseanticommutes.We can see that the above model is a hybrid between twotoric codes and the X-Cube model in the following way. Sup-pose the terms ξ e,r and ζ e,r were instead Pauli matrices ξ e ρ ,r = IX ; r − ρ ≡ mod XI ; r − ρ ≡ mod XX ; r − ρ ≡ mod (E13) ζ e ρ ,r = ZI ; r − ρ ≡ mod IZ ; r − ρ ≡ mod ZZ ; r − ρ ≡ mod (E14)Then the pair A v,r and B p,r forms two copies of the toriccode (note that there are three terms for r = x, y, z but onlytwo are independent). Furthermore, the pair A v,r and B c arestabilizers for the X-Cube model. However, by promoting thefirst variable into a Z variable, the models are coupled in sucha way that the vertex term of each toric code now squares tothe vertex term (that detects the lineon) in the X-Cube model,and the cube term which detects the fracton squares to a prod-uct of plaquette terms of the two toric codes. This is illustratedin Figure 16. a. Summary of excitations, fusion and braiding Because the model is exactly solvable, we can explicitlywrite down the excitations . The operator Z I on a rigid stringin the direction ρ violates the vertex terms A v,r for r (cid:54) = ρ .In particular, the end points are charged ± i under the oper-ator A c,r , and − under A v,r , and are therefore lineons. Itcommutes with all B c r and B ’s. We will label the corre-sponding lineon e ρ , where ρ = x, y, z is the direction of therigid string L ρ . lineon e ρ , ¯ e ρ : (cid:89) e ρ ∈ L ρ Z I e ρ . (E15)Similarly, the end points of Z I creates excitations whichare charged − under two of the three A v,r operators. sothe point excitation e ρ is created. However, this excitation ismobile. To hop e ρ in the direction r we use a flexible string L of ζ e,r in Eq. (E12). Note that in for r = ρ , ζ e,r is just equalto Z I as expected.mobile e ρ : (cid:89) e ∈ L ζ e,r , (E16)Next, we define operators that violate B p,r and B c , butcommute with A v,r . The first kind are loop excitations la-beled m r for r = 1 , , , and satisfy m × m × m = 1 .The loop m r is created by acting with ξ e,r on all edges inter-secting a given surface S (cid:48) on the dual lattice. This creates aloop excitation at the boundary of that surface. Without lossof generality, let us choose the S (cid:48) to intersect the edges in asingle direction ρ . Because of the commutation relations be-tween ξ e,r and ζ e,r (cid:48) , the loop excitation are charged − under = m m (fracton)= m (loop) m (fracton)= m (loop) m (fracton)(loop) FIG. 17.
Loop fusion in the Lineonic ( Z , Z ) gauge theory: Fu-sion of two identical m r loops results in fracton excitations ( m ) atthe corners if at least one of the two loop segments meeting at thecorner points in the r direction. B p,r (cid:48) at the boundary of S if r (cid:54) = r (cid:48) . Furthermore, ξ e ρ ,r com-mutes with ( Z I ) e ρ if r = ρ , otherwise they commute up toa phase ± i . The result of this is that the corners of the loopexcitation are charged ± i under B c only for r (cid:54) = ρ . In otherwords, for a given r , the loop excitation m r are charged under B c in two of the three directions.mobile loop m r : (cid:89) e ⊥S ξ e,r , (E17)Lastly, the fracton excitation is a violation of B c . Fourfractons can be created at the corners of the operator ( X I ) e .fracton m : (cid:89) e ⊥S ( X I ) e (E18)The fusion rules can be seen from the explicit form of theoperators. The lineons e ρ mobile in the ρ direction fuse into e ρ which are mobile particles.On the other hand, the loop excitation m r is mobile. How-ever, we notice that ξ e ρ ,r is equal to ( X I ) e for r (cid:54) = ρ . There-fore, the loop excitation fuses with itself to fractons at thecorner in two of the three directions as shown in Figure 17.The braidings that differ from a stack of two toric codesand an X-Cube model is a braiding between the lineon e ρ andthe loop m r . Using the commutation relations of Z I , whichcreates the lineon and ξ which creates the loop, one finds that9the braiding phase is r − ρ ≡ mod i ; r − ρ ≡ mod − i ; r − ρ ≡ mod (E19) Appendix F: Hybrid Haah’s Code as a Parent Order
In this Appendix, we give an identical argument to SectionVI that the hybrid Haah’s code is a parent state for both thetoric code and Haah’s code.We add the following perturbations to the Hybrid Haah’scode Hamiltonian H Hybrid in Eq. (60). H = H Hybrid − t e (cid:88) e Z e − t m (cid:88) α =1 , (cid:88) v X ( α ) v , (F1)First, we derive the effective Hamiltonian for t e = 0 and t m → ∞ which is the condensate of the fractonic fluxes m by setting X ( α ) v = 1 for α = 1 , , . We discard B v since itbrings us out of the subspace. The other terms in the Hamil-tonian reduce to A v → (cid:89) e ⊃ v X e (F2) B → B = (cid:89) e ∈ Z e A XCv → Therefore the effective Hamiltonian in this subspace has sta-bilizers of the 3d toric code.Next, we consider t m = 0 and t e → ∞ , which is thecondensate limit of the mobile charge e . By restricting to thesubspace where Z e = 1 , the operator A v is discarded, sinceit brings us out of the subspace. The remaining stabilizersreduce to B → (F3) A HCv → X (1) v X (1) v − ˆ x X (1) v − ˆ y X (1) v − ˆ z X (2) v X (2) v − ˆ x − ˆ y X (2) v − ˆ y − ˆ z X (2) v − ˆ z − ˆ x B c,r → Z (1) v Z (1) v +ˆ x +ˆ y Z (1) v +ˆ y +ˆ z Z (1) v +ˆ z +ˆ x Z (2) v Z (2) v +ˆ x Z (2) v +ˆ y Z (2) v +ˆ zz