TSLB_Talks_files/Lee SB.pdf

Space Group Fractionalization in two dimensional nonsymmorphic crystals

SungBin Lee IBS workshop

June 2016

ArXiv 1605.08042

Outline

✴ Motivation

✴ Space Group Symmetries (Symmorphic vs Nonsymmorphic)

✴ Review : Original vs Extended HOLSM Theorem

✴ Glide Fractionalization in Shastry-Sutherland lattice

MotivationTopological Phases : gapped to all excitations, protected gapless edge states, non-trivial quasiparticle excitations

SPT (Symmetry Protected Topological) phases ex) Topological Insulators : protected by U(1) and Time-Reversal symmetries, connected to trivial product states if the symmetry is broken, short-range entangled states. HgTe,Bi1-xSbx etc

SPT vs SET ??

SET (Symmetry Enriched Topological) phases ex) (Fractional) Quantum Hall, Spin Liquids : topologically ordered with fractional quasiparticle excitations, long-range entangled states, robust to arbitrary perturbations/ symmetry breaking as long as the gap stays open. GaAs, ZnCu3(OH)6Cl26C(Herbertsmithite) + numerics

Motivation

In the presence of symmetry, interplay between symmetry and topological order ? ex) In FQHE, observation of quantized Hall conductance due to the interplay between U(1) charge symmetry and topological order

Q) Many focuses related to on-site symmetries. Role of spatial symmetries ?

157 of 230 space groups are non-symmorphic

Space Group symmetries

symmorphic : translations ⊗ point group symmetries ex) square, honeycomb, cubic

Q) How do the spatial symmetries affect to topologically ordered phases?

nonsymmorphic : include glide or screw symmetries ex) Shastry-Sutherland lattice, diamond, pyrochlore

Symmorphic vs Nonsymmorphic

8

FIG. 2. (color online) Z4 electric field configuration at ⌫ = 2 onthe SSL. Here, the Gauss law requires

Qr02hrr0i

err0 = �1 on each

site. Solid (Black) and dashed (magenta) links indicate electric fielderr0 = 1 and -1 respectively. Note that all space-group symmetriesare preserved by this pattern.

FIG. 3. (color online) Vison creation operators at a square plaquette⌦: products of electric field operator err0 and e

†rr0 along arrow lines.

visons28 that are constructed by inserting Z4 fluxQrr

0arr

0 on

a single plaquette (of either shape). Since the electric fieldoperator shifts the value of a

rr

0 , and we wish to only changethe flux through a single plaquette, it follows that in order tocreate a vison we must apply electric field operators along a‘string’ of bonds on the lattice. Fig. 3 shows an exampleof such a flux insertion operator ˆF⌦ at the square plaquettelabeled ⌦:

ˆF⌦ =

Y

rr

0!1e(†)rr

0 = e12e†23e34e

†45 · · · , (25)

where the indices 1, 2, 3 · · · label each site along the ‘string’identified in Fig. 3. In order to further examine the visonproperties, it is once again convenient to move to a dual rep-resentation of the Z4 gauge theory.

B. Dual Z4 clock model

As in the Ising case, the dual theory is a convenient lan-guage to study the vison, as the nonlocal duality mapping ren-ders the vison creation operator a local object. To that end, weintroduce a a new set of Z4 operators E

r

and Ar

that resideon each site ¯

r of the dual pentagon lattice. These variables

have similar Hilbert space structure as in (20) and are relatedto a, e via

err

0= ⌘

rr

0A†r

Ar

0 (26)Y

rr

022,Marr

0= E

r

, (27)

The Gauss law constraint in the original lattice site maps tothe product of ⌘

rr

0 values for every pentagon:Q

r

02hrr0ierr

0=

Q

rr

02D⌘rr

0 . The dual theory then takes the form of a Z4 clock

model on the pentagonal lattice,

Hg = �hX

rr

0

⌘rr

0A†r

Ar

0 �KX

r

Er

+ h.c., (28)

where the bond strengths ⌘rr

0 satisfyY

rr

02D⌘rr

0= i⌫ (29)

at boson filling ⌫. The non-trivial product of bond variablesaround a plaquette for ⌫ = 1 indicate that the clock model isfrustrated29. Note that we can readily construct a bond con-figuration satisfying (29) by examining the electric field con-figurations in Figs. 1, 2, and associating the value of ⌘

rr

0 ona bond of a dual lattice with the value of the electric field onthe direct lattice bond bisected by ¯

r

¯

r

0. Once this assignmentis made, the couplings ⌘

rr

0 are held fixed, i.e. they are notdynamical objects.

In the dual theory, the vison creation operator ˆF⌦ definedby (25) is represented via

ˆF⌦ =

⇣ Y

rr

0!1⌘⇤rr

0

⌘A1. (30)

Although this includes a non-local string product of the bondstrengths ⌘

rr

0 , as we have already noted, these are fixed andnon-dynamical. Thus, as promised, in the dual theory ˆF⌦ is alocal operator; we now study its symmetry properties.

C. Vison symmetry analysis

In order to study the fractionalization of space group sym-metries, we now consider the transformation of the A

r

underlattice symmetries.

The symmetries of the group p4g (shared by both the SSLand the dual pentagonal lattice) are generated by the follow-ing operations: translations along orthogonal lattice primitivevectors (a1 ⌘ (2, 0) and a2 ⌘ (0, 2)):

Ta1 : (x, y) 7! (x, y) + a1

Ta2 : (x, y) 7! (x, y) + a2, (31)

mirror reflections along planes oriented at ⇡/4 with respect tothe lattice vectors:

�xy : (x, y) 7! (y, x)

�xy : (x, y) 7! (�y,�x), (32)

honeycomb Shastry-Sutherland: nonsymmorphic rank S=2

Q) Can we find any single point that is invariant under all lattice symmetries? Yes if symmorphic & No if nonsymmorphic

glide symmetry : reflection + fractional (1/2) translations

Review : HOLSM Theorem

Hastings-Oshikawa-Lieb-Schulz-Mattis (HOLSM) Theorem :

At fractional ν, unique and translationally invariant gapped ground state is forbidden in the system with U(1) charge.

(I) The system remains gapless (ex) carry spin but no charge no spectral gap and no topological order

(II) The system is gapped, breaks translational symmetry effective filling of enlarged unit cell is integer

(III)The system is gapped and preserves translational symmetries degenerate ground states that are topologically distinct


Simple proof

Torus in p.b.cxy

𝚽= ∫ A∙dr

Begin with a ground state |ψ> that satisfies Tx|ψ> = eiP0 |ψ>.

After inserting 2𝝅 flux (via uniform vector potential A=2𝝅/Nxa x), a new state |ψʹ> also satisfies Tx|ψʹ> = eiP0 |ψʹ>.

Hamiltonian with 2𝝅 flux insertion requires Gauge transformation : a final state |ψ> = Û |ψʹ>

3

thermodynamic limit. We impose periodic boundary condi-tions that identify r and r+Niai. Note that in the case when⌫ is a fraction, we choose Nc so to ensure that Q = ⌫Nc is aninteger, in accord with the quantization of charge. We work inunits where ~ = e = 1, so that the quantum of flux is 2⇡.

The original HOLSM theorem states that at fractional ⌫, itis impossible for the system to be insulating with respect tothe U(1) charge and have a unique, translationally-invariantgapped ground state. This leaves open the following possibil-ities for an insulating ground state:

(i) The system remains gapless. This possibility emergesin the case when the ‘bare’ constituent particles carrymultiple global quantum numbers, e.g. separately con-served spin and charge for electronic systems. In thiscase, the system is in a sense fractionalized: the emer-gent low energy degrees of freedom carry only somestrict subset of the global quantum numbers: e.g. theycarry spin, but not charge. We do not have a sharpground state degeneracy, as there is no spectral gap, andhence there is no topological order.

(ii) The system is gapped, and breaks translational sym-metry, thereby enlarging the unit cell. The effective fill-ing in the new unit cell is then an integer. In this case,we can adiabatically deform the ground state into that ofa band insulator, precluding fractionalization.

(iii) The system is gapped, and preserves translationalsymmetries. In this case, we have a ground state de-generacy that cannot be associated with a broken sym-metry. One route to this is for the low-energy excitationsto be fractionalized, so that the emergent low-energy de-scription is as the deconfined phase of a lattice gaugetheory19,20; the degenerate ground states may then be as-sociated with processes that create a quasiparticle-holepairs from the vacuum, thread them around a noncon-tractible loop (here we assume periodic boundary con-ditions) before fusing them back into the vacuum. Theresulting states are topologically distinct from the orig-inal ground state, but no local observable — includingthose associated with spontaneously broken symmetries— can tell them apart.

We exclusively focus on case (iii) in this paper.We now sketch the argument that leads to these conclu-

sions; we give an intuitive proof, and refer the reader toRefs.4–7, 21, and 22 for a more formal treatment. We be-gin with a ground state | i and thread a flux quantum througha periodic direction, which, by gauge invariance, returns us tothe original Hamiltonian. This procedure produces an eigen-state |˜ i. Earlier work has argued that for an insulator, |˜ imust be a ‘low energy’ state, i.e. its energy approaches that ofthe ground state in the thermodynamic limit.6–8,2324 The keystep is to show that |˜ i is distinct from | i, which would thenestablish ground state degeneracy. In the case of fractional fill-ing, these states differ in crystal momentum6–8,23, as we nowdemonstrate.

For specificity25, let us take the example of a square latticein d = 2 with lattice spacing a. Now, imagine adiabatically

threading 2⇡ flux through a handle of the torus (recall that oursystem is defined with periodic boundary conditions), say theone enclosed by a noncontractible loop parallel to the x-axis.The particles are assumed to couple minimally to this gaugeflux, with unit charge. Suppose we began with a ground state| i; as translation in the x-direction is a symmetry, we mayassume that this is a state of fixed momentum: in other words,we have

ˆTx| i = eiP0 | i (1)

for some crystal momentum P0. After inserting 2⇡ flux, weassume that we are in a new state | 0i; however, it is clearthat the flux insertion can be implemented via a uniform vec-tor potential A =

2⇡N

x

aˆ

x and that this preserves Tx as a sym-metry. Thus, we may conclude that ˆTx| 0i = eiP0 | 0i, i.e.the crystal momentum is unchanged during the adiabatic fluxthreading. However, under the flux insertion, the Hamiltonianalso changes, from ˆH(0) to ˆH(2⇡) — it now describes a sys-tem with an inserted flux. We complete the adiabatic cycle byperforming a large gauge transformation, implemented by theoperator

Û = exp

⇢i2⇡

Nx

Zd2r ˆx · r⇢(r)

�(2)

where we have employed second-quantized notation and ⇢(r)is the density operator corresponding to the conserved chargeˆQ. It is straightforward to show that

ˆTxÛ ˆT�1

xÛ�1

= ei2⇡N

x

R⇢(r)

= ei2⇡Q/Nx

= ei2⇡⌫Ny . (3)

Using this, we see that the final result of the adiabatic cycle isa state |˜ i = Û | 0i, whose crystal momentum P1 is given by

eiP1 |˜ i ⌘ ˆTx|˜ i = ˆTxÛ | 0i (4)

= ei2⇡⌫Ny

Û ˆTx| 0i = ei(P0+2⇡⌫Ny

)|˜ i.Clearly, if Ny is chosen relatively prime to q (note that wemay still choose Nx so that ⌫NxNy is an integer, so there isno inconsistency) then | i, |˜ i differ in their crystal momen-tum, i.e. the ground state after flux insertion is distinct fromthe one we began with. One resolution is of course that thesystem is gapless (case (i) above), in that there is a finite den-sity of low-lying excited states arbitrarily close to the groundstate in the thermodynamic limit. In that case, inserting fluxincreases the energy and so |˜ i is no longer degenerate with| i in the thermodynamic limit. Another option is that thesystem breaks translational symmetry (here, along x); this isconsistent with a gapped phase adiabatically connected to aband insulator, case (ii) above; the broken symmetry can bemeasured via an appropriate local order parameter (note alsothat the breaking of translational symmetry will lead to a re-definition of the filling to account for the enlarged unit cell). Afinal possibility is case (iii): there is a finite number of groundstates that are degenerate, but apart from these there is a spec-tral gap. This is consistent with topological order. We remarkthat on a finite system one can distinguish case (i) and case(iii) by studying the scaling of the gap between the low-lyingmany-body eigenstates: it will scale roughly ⇠ L�1 in case(i), while it vanishes exponentially in case (iii).


xy

𝚽= ∫ A∙dr

If Ny is chosen relatively prime to q (ν = p/q), then |ψ>, |ψ> are differ in their crystal momentum. The ground state after flux insertion is distinct with the initial ground state.

3

thermodynamic limit. We impose periodic boundary condi-tions that identify r and r+Niai. Note that in the case when⌫ is a fraction, we choose Nc so to ensure that Q = ⌫Nc is aninteger, in accord with the quantization of charge. We work inunits where ~ = e = 1, so that the quantum of flux is 2⇡.

The original HOLSM theorem states that at fractional ⌫, itis impossible for the system to be insulating with respect tothe U(1) charge and have a unique, translationally-invariantgapped ground state. This leaves open the following possibil-ities for an insulating ground state:

(i) The system remains gapless. This possibility emergesin the case when the ‘bare’ constituent particles carrymultiple global quantum numbers, e.g. separately con-served spin and charge for electronic systems. In thiscase, the system is in a sense fractionalized: the emer-gent low energy degrees of freedom carry only somestrict subset of the global quantum numbers: e.g. theycarry spin, but not charge. We do not have a sharpground state degeneracy, as there is no spectral gap, andhence there is no topological order.

(ii) The system is gapped, and breaks translational sym-metry, thereby enlarging the unit cell. The effective fill-ing in the new unit cell is then an integer. In this case,we can adiabatically deform the ground state into that ofa band insulator, precluding fractionalization.

(iii) The system is gapped, and preserves translationalsymmetries. In this case, we have a ground state de-generacy that cannot be associated with a broken sym-metry. One route to this is for the low-energy excitationsto be fractionalized, so that the emergent low-energy de-scription is as the deconfined phase of a lattice gaugetheory19,20; the degenerate ground states may then be as-sociated with processes that create a quasiparticle-holepairs from the vacuum, thread them around a noncon-tractible loop (here we assume periodic boundary con-ditions) before fusing them back into the vacuum. Theresulting states are topologically distinct from the orig-inal ground state, but no local observable — includingthose associated with spontaneously broken symmetries— can tell them apart.

We exclusively focus on case (iii) in this paper.We now sketch the argument that leads to these conclu-

sions; we give an intuitive proof, and refer the reader toRefs.4–7, 21, and 22 for a more formal treatment. We be-gin with a ground state | i and thread a flux quantum througha periodic direction, which, by gauge invariance, returns us tothe original Hamiltonian. This procedure produces an eigen-state |˜ i. Earlier work has argued that for an insulator, |˜ imust be a ‘low energy’ state, i.e. its energy approaches that ofthe ground state in the thermodynamic limit.6–8,2324 The keystep is to show that |˜ i is distinct from | i, which would thenestablish ground state degeneracy. In the case of fractional fill-ing, these states differ in crystal momentum6–8,23, as we nowdemonstrate.

For specificity25, let us take the example of a square latticein d = 2 with lattice spacing a. Now, imagine adiabatically

threading 2⇡ flux through a handle of the torus (recall that oursystem is defined with periodic boundary conditions), say theone enclosed by a noncontractible loop parallel to the x-axis.The particles are assumed to couple minimally to this gaugeflux, with unit charge. Suppose we began with a ground state| i; as translation in the x-direction is a symmetry, we mayassume that this is a state of fixed momentum: in other words,we have

ˆTx| i = eiP0 | i (1)

for some crystal momentum P0. After inserting 2⇡ flux, weassume that we are in a new state | 0i; however, it is clearthat the flux insertion can be implemented via a uniform vec-tor potential A =

2⇡N

x

aˆ

x and that this preserves Tx as a sym-metry. Thus, we may conclude that ˆTx| 0i = eiP0 | 0i, i.e.the crystal momentum is unchanged during the adiabatic fluxthreading. However, under the flux insertion, the Hamiltonianalso changes, from ˆH(0) to ˆH(2⇡) — it now describes a sys-tem with an inserted flux. We complete the adiabatic cycle byperforming a large gauge transformation, implemented by theoperator

Û = exp

⇢i2⇡

Nx

Zd2r ˆx · r⇢(r)

�(2)

where we have employed second-quantized notation and ⇢(r)is the density operator corresponding to the conserved chargeˆQ. It is straightforward to show that

ˆTxÛ ˆT�1

xÛ�1

= ei2⇡N

x

R⇢(r)

= ei2⇡Q/Nx

= ei2⇡⌫Ny . (3)

Using this, we see that the final result of the adiabatic cycle isa state |˜ i = Û | 0i, whose crystal momentum P1 is given by

eiP1 |˜ i ⌘ ˆTx|˜ i = ˆTxÛ | 0i (4)

= ei2⇡⌫Ny

Û ˆTx| 0i = ei(P0+2⇡⌫Ny

)|˜ i.Clearly, if Ny is chosen relatively prime to q (note that wemay still choose Nx so that ⌫NxNy is an integer, so there isno inconsistency) then | i, |˜ i differ in their crystal momen-tum, i.e. the ground state after flux insertion is distinct fromthe one we began with. One resolution is of course that thesystem is gapless (case (i) above), in that there is a finite den-sity of low-lying excited states arbitrarily close to the groundstate in the thermodynamic limit. In that case, inserting fluxincreases the energy and so |˜ i is no longer degenerate with| i in the thermodynamic limit. Another option is that thesystem breaks translational symmetry (here, along x); this isconsistent with a gapped phase adiabatically connected to aband insulator, case (ii) above; the broken symmetry can bemeasured via an appropriate local order parameter (note alsothat the breaking of translational symmetry will lead to a re-definition of the filling to account for the enlarged unit cell). Afinal possibility is case (iii): there is a finite number of groundstates that are degenerate, but apart from these there is a spec-tral gap. This is consistent with topological order. We remarkthat on a finite system one can distinguish case (i) and case(iii) by studying the scaling of the gap between the low-lyingmany-body eigenstates: it will scale roughly ⇠ L�1 in case(i), while it vanishes exponentially in case (iii).

using

eiP1 |ψ>≣ Tx|ψ> = TxÛ |ψʹ> = ei2𝝅νNy ÛTx |ψʹ> = ei(P0+2𝝅νNy)|ψ>

Tx|ψ> = eiP0 |ψ>.

2𝝅 flux insertion + gauge transformation Û

finite number of degenerate ground states topologically ordered

Review : Extended HOLSM Theorem

4

1. Extending HOLSM to Integer Filling

If we attempt to apply the above arguments at integer filling(⌫ 2 Z), it is clear that the change in momentum upon fluxinsertion is always a reciprocal lattice vector: in other words,we cannot use crystal momentum to differentiate between | iand |˜ i. However, on non-symmorphic lattices, one can stilldistinguish these states using the quantum numbers of the non-symmorphic operations. Let us review how this argument pro-ceeds. For simplicity, since we are working at integer ⌫, wemay take Ni = N . Now, consider a non-symmorphic symme-try ˆG that involves a point-group transformation g followed bya translation through a fraction of a lattice vector ⌧ in a direc-tion left invariant by g: in other words, we have

G : r ! gr + ⌧ . (5)

In this paper, we will be concerned with the case when g is amirror reflection, in which case ⌧ is always one-half a recip-rocal lattice vector, and ˆG is termed a glide reflection. This isthe only possible non-symmorphic symmetry in d = 2.

As before, we begin with a ground state | i, and assume itis an eigenstate of all the crystal symmetries, including ˆG, i.e.

ˆG| i = ei✓| i (6)

We consider the smallest reciprocal lattice vector k left in-variant by g, so that gk = k and k generates the invariantsublattice along ˆ

k. We now thread flux by introducing a vec-tor potential A = k/N (Note that as k is in the reciprocallattice, k · ai is always an integer multiple of 2⇡, so this isalways a pure gauge flux; the case studied above is simplya specific instance of this.) In the process of flux insertion| i evolves to a state | 0i that is degenerate with it. Onceagain, to compare | 0i to | i, we must return to the originalgauge, which can be accomplished by the unitary transforma-tion | 0i ! Û

k

| 0i ⌘ |˜ i, where

Ûk

= exp

⇢i

N

Zddr k · r⇢(r)

�(7)

removes the inserted flux. Since A is left invariant by ˆG,threading flux does not alter ˆG eigenvalues, so | i and | 0ihave the same quantum number under ˆG; however, on actingwith Û

k

, the eigenvalue changes, as can be computed from theequation:

ˆG Ûk

ˆG�1=

Ûk

e2⇡i�g

(k)Q/N (8)

where we have defined the phase factor �g(k) = ⌧ · k/2⇡,and Q = ⌫N3 is the total charge. It may be readily ver-ified that since gk = k, �g(k) is unchanged by a shift inreal-space origin. For a non-symmorphic symmetry opera-tion ˆG, this phase �g(k) must be a fraction. This followssince ⌧ is a fractional translation. (If a lattice translation hadthe same projection onto k as ⌧ , this would yield an integerphase factor.26 However, this would render the screw/glide re-movable i.e. reduced to point group element⇥translation bychange of origin and hence not truly non-symmorphic.) Thus,

for ˆG non-symmorphic, �g(k) = p/SG, with p,SG relativelyprime. From (8) we conclude that | i and |˜ i have distinctˆG eigenvalues whenever �g(k)Q/N = pN2⌫/SG is a frac-tion. Since we may always choose N relatively prime to theSG, the result of flux insertion is a state distinguished fromthe original state by its ˆG eigenvalue, unless the filling is amultiple of SG. For a glide SG = 2.

From this argument, we see that in any 2D crystal witha glide reflection plane, we can extend the applicability ofthe HOLSM theorem to odd integer fillings, by consideringground states that are invariant under the glide symmetry (inaddition to translations). Similar arguments can be made alsofor screw rotations in d = 3, but we focus on the d = 2 casein this paper.

B. Topological Order, Gauge Theories and CrystalMomentum Fractionalization

As we have discussed, assuming the absence of symmetrybreaking and the presence of a gap, the HOLSM theoremsrequire a ground-state degeneracy on the torus, and that theground states differ by crystal momenta or other point groupsymmetry quantum numbers. For the square lattice at half-filling, an effective low-energy description that is consistentwith this picture is that the ground state exhibits Z2 topologi-cal order. This is a fractionalized, translationally invariant in-sulating phase, whose ground state is not unique in a multiplyconnected geometry (e.g., the periodic boundary condition-torus considered here) owing to the presence of a gapped Z2

vortex or vison excitation in the spectrum. The degeneracy isthen associated with the presence or absence of a vison thread-ing a non-contractible loop of the torus and hence topological.The splitting between the vison/no vison states vanishes expo-nentially with system size in the thermodynamic limit, sincethe tunneling of a vison ‘into’ or ‘out of’ the torus costs anenergy that scales with L, as the vison is a gapped bulk exci-tation.

With these preliminaries, we are ready to study the frac-tionalization of symmetries in our ⌫ = 1/2 square lattice ex-ample. We introduce the effective low-energy theory for thetopological phase: introducing Ising degrees of freedom ⌧µ

rr

0

(µ = x, y, z) on each link (rr

0) on the square lattice, we have

the Ising gauge theory Hamiltonian

HIGT = �hX

hrr0i⌧xrr

0 �KX

p

Y

rr

02p

⌧zrr

0 , (9)

with a Gauss law constraint for every site r

Y

r

02hrr0i⌧xrr

0 = (�1)

2⌫ , (10)

where h· · ·i labels nearest-neighbor sites and and p labels pla-quettes. As the microscopic origins of HIGT are detailed inseveral excellent references, and since we also give a detailedaccount of similar constructions in the non-symmorphic casebelow, we do not repeat them here. Note that HIGT is in anordered phase for K ⌧ h and is in a deconfined phase for

nonsymmorphic symmetry G

Glide reflection : 𝞽 = a/2

4



G : r ! gr + ⌧ . (5)



ˆG| i = ei✓| i (6)



k

| 0i ⌘ |˜ i, where

Ûk

= exp

⇢i

N

Zddr k · r⇢(r)

�(7)


k


ˆG Ûk

ˆG�1=

Ûk

e2⇡i�g

(k)Q/N (8)








rr

0




HIGT = �hX

hrr0i⌧xrr

0 �KX

p

Y

rr

02p

⌧zrr

0 , (9)


Y

r

02hrr0i⌧xrr

0 = (�1)

2⌫ , (10)


4



G : r ! gr + ⌧ . (5)



ˆG| i = ei✓| i (6)



k

| 0i ⌘ |˜ i, where

Ûk

= exp

⇢i

N

Zddr k · r⇢(r)

�(7)


k


ˆG Ûk

ˆG�1=

Ûk

e2⇡i�g

(k)Q/N (8)








rr

0




HIGT = �hX

hrr0i⌧xrr

0 �KX

p

Y

rr

02p

⌧zrr

0 , (9)


Y

r

02hrr0i⌧xrr

0 = (�1)

2⌫ , (10)


4



G : r ! gr + ⌧ . (5)



ˆG| i = ei✓| i (6)



k

| 0i ⌘ |˜ i, where

Ûk

= exp

⇢i

N

Zddr k · r⇢(r)

�(7)


k


ˆG Ûk

ˆG�1=

Ûk

e2⇡i�g

(k)Q/N (8)








rr

0




HIGT = �hX

hrr0i⌧xrr

0 �KX

p

Y

rr

02p

⌧zrr

0 , (9)


Y

r

02hrr0i⌧xrr

0 = (�1)

2⌫ , (10)


2𝝅 Flux threading A = k/N

Unitary transformation

Return to original gauge

4



G : r ! gr + ⌧ . (5)



ˆG| i = ei✓| i (6)



k

| 0i ⌘ |˜ i, where

Ûk

= exp

⇢i

N

Zddr k · r⇢(r)

�(7)


k


ˆG Ûk

ˆG�1=

Ûk

e2⇡i�g

(k)Q/N (8)








rr

0




HIGT = �hX

hrr0i⌧xrr

0 �KX

p

Y

rr

02p

⌧zrr

0 , (9)


Y

r

02hrr0i⌧xrr

0 = (�1)

2⌫ , (10)


4



G : r ! gr + ⌧ . (5)



ˆG| i = ei✓| i (6)



k

| 0i ⌘ |˜ i, where

Ûk

= exp

⇢i

N

Zddr k · r⇢(r)

�(7)


k


ˆG Ûk

ˆG�1=

Ûk

e2⇡i�g

(k)Q/N (8)








rr

0




HIGT = �hX

hrr0i⌧xrr

0 �KX

p

Y

rr

02p

⌧zrr

0 , (9)


Y

r

02hrr0i⌧xrr

0 = (�1)

2⌫ , (10)


4



G : r ! gr + ⌧ . (5)



ˆG| i = ei✓| i (6)



k

| 0i ⌘ |˜ i, where

Ûk

= exp

⇢i

N

Zddr k · r⇢(r)

�(7)


k


ˆG Ûk

ˆG�1=

Ûk

e2⇡i�g

(k)Q/N (8)








rr

0




HIGT = �hX

hrr0i⌧xrr

0 �KX

p

Y

rr

02p

⌧zrr

0 , (9)


Y

r

02hrr0i⌧xrr

0 = (�1)

2⌫ , (10)


4



G : r ! gr + ⌧ . (5)



ˆG| i = ei✓| i (6)



k

| 0i ⌘ |˜ i, where

Ûk

= exp

⇢i

N

Zddr k · r⇢(r)

�(7)


k


ˆG Ûk

ˆG�1=

Ûk

e2⇡i�g

(k)Q/N (8)








rr

0




HIGT = �hX

hrr0i⌧xrr

0 �KX

p

Y

rr

02p

⌧zrr

0 , (9)


Y

r

02hrr0i⌧xrr

0 = (�1)

2⌫ , (10)


where k.ai = 2𝝅

Thus, for nonsymmorphic symmetry G, phase is a fraction

4



G : r ! gr + ⌧ . (5)



ˆG| i = ei✓| i (6)



k

| 0i ⌘ |˜ i, where

Ûk

= exp

⇢i

N

Zddr k · r⇢(r)

�(7)


k


ˆG Ûk

ˆG�1=

Ûk

e2⇡i�g

(k)Q/N (8)








rr

0




HIGT = �hX

hrr0i⌧xrr

0 �KX

p

Y

rr

02p

⌧zrr

0 , (9)


Y

r

02hrr0i⌧xrr

0 = (�1)

2⌫ , (10)


4



G : r ! gr + ⌧ . (5)



ˆG| i = ei✓| i (6)



k

| 0i ⌘ |˜ i, where

Ûk

= exp

⇢i

N

Zddr k · r⇢(r)

�(7)


k


ˆG Ûk

ˆG�1=

Ûk

e2⇡i�g

(k)Q/N (8)








rr

0




HIGT = �hX

hrr0i⌧xrr

0 �KX

p

Y

rr

02p

⌧zrr

0 , (9)


Y

r

02hrr0i⌧xrr

0 = (�1)

2⌫ , (10)


SG is nonsymmorphic rank : For glide reflection, SG=2

Review : Extended HOLSM Theorem

For 2D crystals with glide reflection (SG=2), the ground states with odd integer fillings are topologically ordered.

4



G : r ! gr + ⌧ . (5)



ˆG| i = ei✓| i (6)



k

| 0i ⌘ |˜ i, where

Ûk

= exp

⇢i

N

Zddr k · r⇢(r)

�(7)


k


ˆG Ûk

ˆG�1=

Ûk

e2⇡i�g

(k)Q/N (8)








rr

0




HIGT = �hX

hrr0i⌧xrr

0 �KX

p

Y

rr

02p

⌧zrr

0 , (9)


Y

r

02hrr0i⌧xrr

0 = (�1)

2⌫ , (10)


4



G : r ! gr + ⌧ . (5)



ˆG| i = ei✓| i (6)



k

| 0i ⌘ |˜ i, where

Ûk

= exp

⇢i

N

Zddr k · r⇢(r)

�(7)


k


ˆG Ûk

ˆG�1=

Ûk

e2⇡i�g

(k)Q/N (8)








rr

0




HIGT = �hX

hrr0i⌧xrr

0 �KX

p

Y

rr

02p

⌧zrr

0 , (9)


Y

r

02hrr0i⌧xrr

0 = (�1)

2⌫ , (10)


, have different crystal momentum if ν/SG is not integer.

Topologically ordered ground states with multiple degeneracies.

Q) What are the signatures of topologically ordered states?

Signature of topological order

original HOLSM theorem extended HOLSM theorem

topological order due to fractional filling ν

topological order even for certain integer filling ν (ν/SG ∉Z) due to nonsymmorphicity with rank SG

Q) How are they differ in terms of low energy excitations?

Related to original HOLSM theorem, the system has vison (flux creation) excitations that break translational symmetry (Odd Ising gauge theory). Then what about extended HOLSM theorem with nonsymmorphicity?

A) vison excitations that break glide reflections

Z4 gauge theory in Shastry-Sutherland lattice

Model construction

6

— for instance, it is possible to show that any gauge choiceconsisting with half-odd integer filling on the square latticealso violates fourfold rotation symmetry, and thus this oper-ation is also implemented projectively on single-vison states.A more refined question is to ask, are there situations wherevisons transform regularly under translations, so that the pro-jective symmetry arises purely a property of the other crystalsymmetries?

The answer to this is in the affirmative, and perhaps un-surprisingly, is intimately connected to the extension of theHOLSM theorem to integer filling discussed above. In thebalance of this paper, we detail how to generalize the ‘odd fill-ing $ odd Ising gauge theory’ connection to integer fillings innon-symmorphic 2D crystals by constructing low-energy ef-fective descriptions of fractionalized phases in these crystals.We then examine the signatures of fractionalization of glidereflection symmetry in numerics.

III. MODEL

We now begin the central analysis of this paper, where weexplore space group symmetry fractionalization at integer fill-ing. We will consider the case of the 2D non-symmorphicspace group p4g, which has S = 2, at filling ⌫ = 1. This isprecisely a situation where the only commensurability condi-tion is the extended HOLSM theorem that relies on the glidereflection symmetries in p4g. We note that among the 17 2Dspace groups (also termed wallpaper groups) there are threeother non-symmorphic groups: pg, p2mg and p2gg. How-ever, these have reduced symmetry compared to p4g, andare apparently of less relevance to materials. While we donot consider these in detail, we expect that the broad fea-tures of glide symmetry fractionalization should apply to theseas well. We restrict our attention to a (generalized) Bose-Hubbard model. Denoting the boson creation and annihilationoperators at site r by B†

r

and Br

, we have

H = �tX

hrr0i(B†

r

Br

0+ h.c) + V [{N

r

}], (17)

where the first term hops bosons between nearest-neighborshrr0i and the second term is an interaction that depends onthe boson number on site r, N

r

= B†r

Br

. For t � V , the bo-son kinetic energy dominates and we enter a superfluid phasewith broken U(1) symmetry. We are primarily interested inthe opposite limit, t ⌧ V , where the system is in a Mott insu-lating phase. In particular, we focus on the case ⌫ = 1, wherethe extended HOLSM theorem requires that any symmetry-preserving insulator exhibit fractionalization.

Our central example consists of a model of s-orbitals ar-ranged on the sites of the so-called Shastry-Sutherland lattice,familiar in the context of frustrated magnetism; here, the glidesymmetry operates by interchanging the spatial coordinates ofthe four orbitals in a unit cell (Fig. 1), but the orbitals them-selves transform trivially. At ⌫ = 1, as there are four orbitalsin each unit cell, we have an average site filling of 1/4.

While this s-orbital boson model could be realized andstudied in cold atom systems, we note that it also has signifi-

cant relevance to frustrated magnetism of spins on the Shastry-Sutherland lattice. This follows from taking the hard corelimit of (17); in this limit, by associating the presence (ab-sence) of a boson on a site with spin up (down) with respectto a reference axis, we arrive at a Hamiltonian for U(1) spins,with the total charge related to the total magnetization alongthe specified axis. Such a spin Hamiltonian can describe themagnetization process of spins in a field, with the Mott insu-lating phases corresponding to magnetization plateaus.

We note that there are a variety of Shastry-Sutherland lat-tice materials that exhibit magnetization plateaus, such asSrCu2(BO3)2, Yb2Pt2Pb, TmB4 and ErB4. Of these, theplateau at 1/2 the saturation magnetization has been observedin SrCu2(BO3)2, TmB4 and ErB4.15? –17 We will present a de-tailed numerical study of the structure of the ⌫ = 1/2 phasediagram elsewhere.

In Appendix A we also examine a different model, contain-ing two orthogonally oriented d-orbitals in each unit cell. Thisshares the same p4g space group as the Shastry-Sutherlandlattice, but encodes the symmetries in its nontrivial orbitalcontent, rather than in terms of the spatial coordinates of theorbitals within a unit cell. While we relegate detailed dis-cussion of this model to the appendix, the symmetry frac-tionalization is essentially identical to that for the Shastry-Sutherland example; the sole technical difference is that thereis a straightforward parton construction that directly leads toa Z2 gauge theory, rather than accessing this via the inter-miedate step of a Z4 theory. Nevertheless, the d-orbital modelserves as a check that our results depend only on the spacegroup rather than a particular implementation of the symme-try in our model.

IV. GLIDE FRACTIONALIZATION

As noted above, we will focus on the richer example of theShastry-Sutherland s-orbital model; readers interested in see-ing how the story plays out in the d-orbital model are referredto Appendix A.

A. Z4 Gauge Theory

We begin our study of the s-orbital SSL model by con-structing a parton mean-field theory that appropriately cap-tures the fractionalization of the degrees of freedom. Recallthat we are working at ⌫ = 1 and there are four sites in eachunit cell; therefore, an intuitive choice for a translationally-invariant parton mean-field solution is to split each of the‘bare’ charge-1 bosons into four partons each with 1/4 charge,and engineer a translationally-invariant parton Mott insulatorin which each parton is frozen on a single site. Operationally,this may be implemented by introducing a rotor representationof these 1/4-charges: we define a parton number operator n

r

,with conjugate phase variable �

r

. In terms of these operators,we may rewrite our original theory in terms of parton variables

Interested in V>>t limit and focus on ν=1

8


Qr02hrr0i

err0 = �1 on each





0arr

0 on


rr


ˆF⌦ =

Y

rr

0!1e(†)rr

0 = e12e†23e34e

†45 · · · , (25)




r

and Ar




err

0= ⌘

rr

0A†r

Ar

0 (26)Y

rr

022,Marr

0= E

r

, (27)


rr


r

02hrr0ierr

0=

Q

rr

02D⌘rr



Hg = �hX

rr

0

⌘rr

0A†r

Ar

0 �KX

r

Er

+ h.c., (28)


0 satisfyY

rr

02D⌘rr

0= i⌫ (29)


rr


r

¯

r


rr



ˆF⌦ =

⇣ Y

rr

0!1⌘⇤rr

0

⌘A1. (30)


rr




r



Ta1 : (x, y) 7! (x, y) + a1

Ta2 : (x, y) 7! (x, y) + a2, (31)


�xy : (x, y) 7! (y, x)

�xy : (x, y) 7! (�y,�x), (32)

Model is realizable in cold atom but is also related to frustrated magnetism that shows 1/2 magnetization plateau seen in SrCu2(BO3)2, TmB4 and ErB4.

Let’s construct Z4 gauge theory. Why Z4? one ‘bare’ charge 1 boson split into translationally invariant parton with charge 1/4 (one parton per site (4 site/ unit cell) can form a trivial Mott insulator )

7

via the mapping

B†r

= (b†r

)

4, (18)nr

= 4Nr

(19)

where b†r

= ei�r is the parton creation operator. Eq. (19)should be viewed as a constraint on the parton Hilbert spacethat reduces it to the physical Hilbert space of the originalbosons.

To proceed, we follow the standard procedure of soften-ing the constraint: we enlarge the Hilbert space to includeparton configurations not satisfying (19), but compensate forthis by introducing a gauge field whose role is to implementthe constraint by dynamically projecting out unphysical de-grees of freedom; as we now show, a minimal choice for thegauge group is Z4. Recall that gauge fields are associated withlinks of the lattice. We take the Hilbert space on a single link` = (r, r0) to consist of four states, h` = {|0i, |1i, |2i, |3i},and introduce a Z4 vector potential operator a` and the asso-ciated electric field operator e` via their action on h`:

arr

0 |ki = e2⇡ik/4|ki and err

0 |ki = |k + 1i, (20)

where we identify |4i ⌘ |0i As links may be traversed in ei-ther direction, given a fiducial orientation used to define (20),for the reversed orientation we have

ar

0r

= a†rr

0 and er

0r

= e†rr

0 . (21)

In terms of these variables, the constraint (19) appears as aGauss law for the Z4 electric field sourced by the partons:

Y

r

02hrr0ierr

0= e2⇡inr/4, (22)

where the product is over the links connecting r to its neigh-boring sites r0.

The above considerations allow us to rewrite (17) as an ef-fective gauge theory,

H = Hg +Hm

Hm = �tX

hrr0i(b†

r

arr

0br

0+ h.c.) + ˜V [{n

r

}], (23)

Hg = �hX

hrr0i(e

rr

0+ h.c.)�K

X

p22,M(

Y

rr

02p

arr

0+ h.c.)

where h, t,K > 0 and the sum over p ranges over the two dif-ferent types of lattice plaquettes (triangular and square plaque-ttes) as shown in Fig. 1, and we have assumed K2 = KM =

K. The link product for each plaquette assumes a fixed ori-entation, here taken to be anticlockwise. Hg represents thedynamics of the gauge field, required in order to implement(19). Hm represents the parton degrees of freedom that inter-act with the gauge field via the usual minimal coupling on thelattice, and ˜V is a parton interaction term written as a functionof the parton density n

r

.For h � K, the low-energy configurations of (23) have

err

0= 1, and from the Gauss law (22) we conclude that in

this parameter regime nr

⌘ 0 (mod 4). This is the confiningphase of the gauge theory: the partons remain tied together

FIG. 1. (color online) SSL structure and Z4 electric field configu-ration at ⌫ = 1 on the SSL. Lattice vectors a1 and a2. The Gausslaw constraint

Qr02hrr0i

err0 = i must be satisfied at each site. Solid

(Black) and dashed (magenta) links indicate electric field err0 = 1and -1 respectively. Blue arrows on the diagonal links representerr0 = i when the link is traversed in the direction of the arrow.The dotted square shows a single unit cell that includes four sites;note that the e-field pattern is the same in every unit cell, but breakspoint-group symmetries.

into the original bosons, and therefore there are no states inthe Hilbert space where free charges can be asymptoticallyseparated. From the HOLSM arguments, it follows that thisphase necessarily breaks symmetry, as it is gapped but notfractionalized.

We are more interested in the limit of K � h, where thepartons may move independently of each other and the elec-tric field is no longer confined. We may consider the partonfields to be gapped, and assume that the interaction term ˜Vis such that the partons form a strong Mott insulating groundstate with n

r

= 1; since we have a single parton on each site,it is evident that this phase is translationally invariant. As thepartons are gapped, we may integrate them out; the result-ing theory is a pure Z4 gauge theory (that will have the sameform as Hg , possibly with renormalized parameters). For low-energy configurations below the parton gap, we have n

r

= 1

and therefore we may rewrite the Gauss law (22) asY

r

02hrr0ierr

0= i. (24)

Observe that (24) requires a non-trivial Z4 gauge flux throughevery plaquette; this is the Z4 analog of the nontrivial fluxin the odd Ising gauge theory. From our discussion above,it is straightforward to see that for filling ⌫ of the originalbosons, the RHS is e2⇡i⌫/4. We show an electric field config-uration satisfying (24) in Fig. 1. Note that, unlike in the caseof the odd Ising gauge theory, the Z4 electric field configu-ration does not enlarge the unit cell — reflecting the preser-vation of translational symmetries — but breaks point groupsymmetries, specifically the glide symmetry. We will explorethe consequences of this shortly. In contrast, for boson fill-ing ⌫ = 2, it is straightforward to see that all the space groupsymmetries are preserved by field configurations that satisfythe Gauss law, as shown in Fig. 2.

The lowest-energy excitations of the pure gauge theory are

7

via the mapping

B†r

= (b†r

)

4, (18)nr

= 4Nr

(19)

where b†r



arr


0 |ki = |k + 1i, (20)


ar

0r

= a†rr

0 and er

0r

= e†rr

0 . (21)


Y

r

02hrr0ierr

0= e2⇡inr/4, (22)



H = Hg +Hm

Hm = �tX

hrr0i(b†

r

arr

0br

0+ h.c.) + ˜V [{n

r

}], (23)

Hg = �hX

hrr0i(e

rr

0+ h.c.)�K

X

p22,M(

Y

rr

02p

arr

0+ h.c.)



r


err





Qr02hrr0i





r


r

= 1


r

02hrr0ierr

0= i. (24)



7

via the mapping

B†r

= (b†r

)

4, (18)nr

= 4Nr

(19)

where b†r



arr


0 |ki = |k + 1i, (20)


ar

0r

= a†rr

0 and er

0r

= e†rr

0 . (21)


Y

r

02hrr0ierr

0= e2⇡inr/4, (22)



H = Hg +Hm

Hm = �tX

hrr0i(b†

r

arr

0br

0+ h.c.) + ˜V [{n

r

}], (23)

Hg = �hX

hrr0i(e

rr

0+ h.c.)�K

X

p22,M(

Y

rr

02p

arr

0+ h.c.)



r


err





Qr02hrr0i





r


r

= 1


r

02hrr0ierr

0= i. (24)



four states |k> = {|0>,|1>,|2>,|3>}

Gauss law for Z4 electric field

7

via the mapping

B†r

= (b†r

)

4, (18)nr

= 4Nr

(19)

where b†r



arr


0 |ki = |k + 1i, (20)


ar

0r

= a†rr

0 and er

0r

= e†rr

0 . (21)


Y

r

02hrr0ierr

0= e2⇡inr/4, (22)



H = Hg +Hm

Hm = �tX

hrr0i(b†

r

arr

0br

0+ h.c.) + ˜V [{n

r

}], (23)

Hg = �hX

hrr0i(e

rr

0+ h.c.)�K

X

p22,M(

Y

rr

02p

arr

0+ h.c.)



r


err





Qr02hrr0i





r


r

= 1


r

02hrr0ierr

0= i. (24)



7

via the mapping

B†r

= (b†r

)

4, (18)nr

= 4Nr

(19)

where b†r



arr


0 |ki = |k + 1i, (20)


ar

0r

= a†rr

0 and er

0r

= e†rr

0 . (21)


Y

r

02hrr0ierr

0= e2⇡inr/4, (22)



H = Hg +Hm

Hm = �tX

hrr0i(b†

r

arr

0br

0+ h.c.) + ˜V [{n

r

}], (23)

Hg = �hX

hrr0i(e

rr

0+ h.c.)�K

X

p22,M(

Y

rr

02p

arr

0+ h.c.)



r


err





Qr02hrr0i





r


r

= 1


r

02hrr0ierr

0= i. (24)



7

via the mapping

B†r

= (b†r

)

4, (18)nr

= 4Nr

(19)

where b†r



arr


0 |ki = |k + 1i, (20)


ar

0r

= a†rr

0 and er

0r

= e†rr

0 . (21)


Y

r

02hrr0ierr

0= e2⇡inr/4, (22)



H = Hg +Hm

Hm = �tX

hrr0i(b†

r

arr

0br

0+ h.c.) + ˜V [{n

r

}], (23)

Hg = �hX

hrr0i(e

rr

0+ h.c.)�K

X

p22,M(

Y

rr

02p

arr

0+ h.c.)



r


err





Qr02hrr0i





r


r

= 1


r

02hrr0ierr

0= i. (24)



7

via the mapping

B†r

= (b†r

)

4, (18)nr

= 4Nr

(19)

where b†r



arr


0 |ki = |k + 1i, (20)


ar

0r

= a†rr

0 and er

0r

= e†rr

0 . (21)


Y

r

02hrr0ierr

0= e2⇡inr/4, (22)



H = Hg +Hm

Hm = �tX

hrr0i(b†

r

arr

0br

0+ h.c.) + ˜V [{n

r

}], (23)

Hg = �hX

hrr0i(e

rr

0+ h.c.)�K

X

p22,M(

Y

rr

02p

arr

0+ h.c.)



r


err





Qr02hrr0i





r


r

= 1


r

02hrr0ierr

0= i. (24)



err’=-1err’= i


7

via the mapping

B†r

= (b†r

)

4, (18)nr

= 4Nr

(19)

where b†r



arr


0 |ki = |k + 1i, (20)


ar

0r

= a†rr

0 and er

0r

= e†rr

0 . (21)


Y

r

02hrr0ierr

0= e2⇡inr/4, (22)



H = Hg +Hm

Hm = �tX

hrr0i(b†

r

arr

0br

0+ h.c.) + ˜V [{n

r

}], (23)

Hg = �hX

hrr0i(e

rr

0+ h.c.)�K

X

p22,M(

Y

rr

02p

arr

0+ h.c.)



r


err





Qr02hrr0i





r


r

= 1


r

02hrr0ierr

0= i. (24)



Interested in deconfined phase where K>>h limit and focus on ν=1

8


Qr02hrr0i

err0 = �1 on each





0arr

0 on


rr


ˆF⌦ =

Y

rr

0!1e(†)rr

0 = e12e†23e34e

†45 · · · , (25)




r

and Ar




err

0= ⌘

rr

0A†r

Ar

0 (26)Y

rr

022,Marr

0= E

r

, (27)


rr


r

02hrr0ierr

0=

Q

rr

02D⌘rr



Hg = �hX

rr

0

⌘rr

0A†r

Ar

0 �KX

r

Er

+ h.c., (28)


0 satisfyY

rr

02D⌘rr

0= i⌫ (29)


rr


r

¯

r


rr



ˆF⌦ =

⇣ Y

rr

0!1⌘⇤rr

0

⌘A1. (30)


rr




r



Ta1 : (x, y) 7! (x, y) + a1

Ta2 : (x, y) 7! (x, y) + a2, (31)


�xy : (x, y) 7! (y, x)

�xy : (x, y) 7! (�y,�x), (32)


vison (flux) creation operator

8


Qr02hrr0i

err0 = �1 on each





0arr

0 on


rr


ˆF⌦ =

Y

rr

0!1e(†)rr

0 = e12e†23e34e

†45 · · · , (25)




r

and Ar




err

0= ⌘

rr

0A†r

Ar

0 (26)Y

rr

022,Marr

0= E

r

, (27)


rr


r

02hrr0ierr

0=

Q

rr

02D⌘rr



Hg = �hX

rr

0

⌘rr

0A†r

Ar

0 �KX

r

Er

+ h.c., (28)


0 satisfyY

rr

02D⌘rr

0= i⌫ (29)


rr


r

¯

r


rr



ˆF⌦ =

⇣ Y

rr

0!1⌘⇤rr

0

⌘A1. (30)


rr




r



Ta1 : (x, y) 7! (x, y) + a1

Ta2 : (x, y) 7! (x, y) + a2, (31)


�xy : (x, y) 7! (y, x)

�xy : (x, y) 7! (�y,�x), (32)

To understand vison (string) operator, it is better to work with it’s dual pentagonal lattice.

9

FIG. 4. (color online) Configuration of bond signs ⌘rr0 on thedual pentagonal lattice for ⌫ = 1. At this filling,

Q

rr02D⌘rr0 = i

(link products are taken in the anti-clockwise sense). Solid (Black)and dashed (magenta) links indicate electric field ⌘rr0 = 1 and -1respectively. Blue arrows on each diagonal link represents ⌘rr0 = i.The dotted square shows a single six-site unit cell. Note that the pat-tern breaks point group symmetry but not translations, as expected.

and glide reflections about axes parallel to the lattice vectors:

Gx : (x, y) 7! (x,�y) +1

2

(a1 � a2)

Gy : (x, y) 7! (�x, y)� 1

2

(a1 � a2) , (33)

(See Fig. 1 for the lattice structure and lattice vectors.)Note that we have chosen a center of symmetry that renders�xy,�xy very simple and underscores that they do not involveany translations, at the cost of making the glide operationslightly more involved. The crucial point is that the associatedtranslations are not projections of lattice vector onto the glideplanes, fact that guarantees that the glide can not be removedby a suitable change of origin30.

These transformation properties map the values of field op-erators at different lattice sites into each other. The transfor-mation properties of the A

r

depends crucially on the set of⌘rr

0 and hence implicitly on the original boson filling. For⌫ = 2, the ⌘

rr

0 configuration does not break any of spacegroup symmetries, and therefore it is a straightforward exer-cise to show that A

r

transforms trivially under lattice sym-metries. In contrast, for ⌫ = 1 the assignment of ⌘

rr

0 sat-isfying

Q

rr

02D⌘rr

0= i necessarily breaks point-group sym-

metries and therefore Ar

transforms projectively. In order todetermine the transformation laws of the A

r

under symmetry,it suffices to consider how the transformations (31-33) act onA

r

while keeping the combinationP

rr

0 ⌘rr

0A†r

Ar

0+h.c. in-

variant. This amounts to constructing the projective symmetrygroup in the standard terminology of the parton constructionof fractionalized phases. [SP: Is this true?]

First, note that it is straightforward to see that the Ar

transform trivially under translations Ta1 and T

a2 since ⌘rr

0

phases do not enlarge the unit-cell. We may therefore consideronly the point-group symmetries. It is useful to introducesome notation: let us label the unit cells by integers (x, y)


Q

rr02D⌘rr0 =

�1. Solid (Black) and dashed (magenta) links indicate electric field⌘rr0 = 1 and -1 respectively. Note that the pattern breaks no sym-metries.

such that ¯r(m,n) = ma1 + na2 and label the six dual latticesites within a single unit cell as shown in Fig. . By exam-ining how the action of the four symmetries (�xy , �xy , Gx,Gy) relates these six sublattice indices while simultaneouslytransforming the unit cell coordinates we arrive at Table I. Asan example of how to construct the entries in Table I, let usconsider the reflection �xy . From Fig. 4, we see that underthis symmetry, the sublattices transform via 1 $ 3, 2 $ 6,4 ! 4 and 5 ! 5. Furthermore, note that owing to the phasedifference ⌘14 = �⌘34 = 1, we must require that �xy inducea sign change only on sublattice 4 so that ⌘

rr

0A†r

Ar

0 remainsinvariant. Finally, introducing such a sign change only forsublattice 4 requires that A

r

be transformed into its conjugateA†

r

0 in order to leave the hopping between sublattices 4 and5 unchanged. Proceeding in this fashion, we may constructthe other entries in Table I. The transformation of A

r

! A†r

0

can be also understood as a flux-antiflux transformation un-der reflection; this immediately allows us to conclude that allpoint-group operations that incorporate a reflection must alsoconjugate the flux creation operator.

Table I allows us to compute relations between differ-ent symmetries when acting on single-vison states. Opera-tionally, we may obtain these relations by constructing thestate |v

r

i ⌘ A†r

|0i and acting upon it with the different sym-metry operators in turn. First, we find that a subset of thespace group symmetries satisfy a ‘trivial’ algebra, in that theydo not exhibit any difference when acting on single visonscompared to their multiplication table computed within thespace group (without reference to the vison states):

T va1T va2

= T va2T va1

(34a)

(�vxy)

2= 1 (34b)

(�vxy)

2= 1 (34c)

Gvy�

vxy = �v

xyGvx (34d)

T vx (G

vy)

�1= �v

xyGx�vxy (34e)

In the dual theory, vison creation operator

8


Qr02hrr0i

err0 = �1 on each





0arr

0 on


rr


ˆF⌦ =

Y

rr

0!1e(†)rr

0 = e12e†23e34e

†45 · · · , (25)




r

and Ar




err

0= ⌘

rr

0A†r

Ar

0 (26)Y

rr

022,Marr

0= E

r

, (27)


rr


r

02hrr0ierr

0=

Q

rr

02D⌘rr



Hg = �hX

rr

0

⌘rr

0A†r

Ar

0 �KX

r

Er

+ h.c., (28)


0 satisfyY

rr

02D⌘rr

0= i⌫ (29)


rr


r

¯

r


rr



ˆF⌦ =

⇣ Y

rr

0!1⌘⇤rr

0

⌘A1. (30)


rr




r



Ta1 : (x, y) 7! (x, y) + a1

Ta2 : (x, y) 7! (x, y) + a2, (31)


�xy : (x, y) 7! (y, x)

�xy : (x, y) 7! (�y,�x), (32)

non local static string product of

8


Qr02hrr0i

err0 = �1 on each





0arr

0 on


rr


ˆF⌦ =

Y

rr

0!1e(†)rr

0 = e12e†23e34e

†45 · · · , (25)




r

and Ar




err

0= ⌘

rr

0A†r

Ar

0 (26)Y

rr

022,Marr

0= E

r

, (27)


rr


r

02hrr0ierr

0=

Q

rr

02D⌘rr



Hg = �hX

rr

0

⌘rr

0A†r

Ar

0 �KX

r

Er

+ h.c., (28)


0 satisfyY

rr

02D⌘rr

0= i⌫ (29)


rr


r

¯

r


rr



ˆF⌦ =

⇣ Y

rr

0!1⌘⇤rr

0

⌘A1. (30)


rr




r



Ta1 : (x, y) 7! (x, y) + a1

Ta2 : (x, y) 7! (x, y) + a2, (31)


�xy : (x, y) 7! (y, x)

�xy : (x, y) 7! (�y,�x), (32)

local operator with

Z4 gauge theory in Shastry-Sutherland lattice8


Qr02hrr0i

err0 = �1 on each





0arr

0 on


rr


ˆF⌦ =

Y

rr

0!1e(†)rr

0 = e12e†23e34e

†45 · · · , (25)




r

and Ar




err

0= ⌘

rr

0A†r

Ar

0 (26)Y

rr

022,Marr

0= E

r

, (27)


rr


r

02hrr0ierr

0=

Q

rr

02D⌘rr



Hg = �hX

rr

0

⌘rr

0A†r

Ar

0 �KX

r

Er

+ h.c., (28)


0 satisfyY

rr

02D⌘rr

0= i⌫ (29)


rr


r

¯

r


rr



ˆF⌦ =

⇣ Y

rr

0!1⌘⇤rr

0

⌘A1. (30)


rr




r



Ta1 : (x, y) 7! (x, y) + a1

Ta2 : (x, y) 7! (x, y) + a2, (31)


�xy : (x, y) 7! (y, x)

�xy : (x, y) 7! (�y,�x), (32)

8


Qr02hrr0i

err0 = �1 on each





0arr

0 on


rr


ˆF⌦ =

Y

rr

0!1e(†)rr

0 = e12e†23e34e

†45 · · · , (25)




r

and Ar




err

0= ⌘

rr

0A†r

Ar

0 (26)Y

rr

022,Marr

0= E

r

, (27)


rr


r

02hrr0ierr

0=

Q

rr

02D⌘rr



Hg = �hX

rr

0

⌘rr

0A†r

Ar

0 �KX

r

Er

+ h.c., (28)


0 satisfyY

rr

02D⌘rr

0= i⌫ (29)


rr


r

¯

r


rr



ˆF⌦ =

⇣ Y

rr

0!1⌘⇤rr

0

⌘A1. (30)


rr




r



Ta1 : (x, y) 7! (x, y) + a1

Ta2 : (x, y) 7! (x, y) + a2, (31)


�xy : (x, y) 7! (y, x)

�xy : (x, y) 7! (�y,�x), (32)

9


Q

rr02D⌘rr0 = i



Gx : (x, y) 7! (x,�y) +1

2

(a1 � a2)

Gy : (x, y) 7! (�x, y)� 1

2

(a1 � a2) , (33)



r



rr


r


rr

0 sat-isfying

Q

rr

02D⌘rr




r


r


rr

0 ⌘rr

0A†r

Ar

0+h.c. in-




a2 since ⌘rr

0



Q

rr02D⌘rr0 =



rr

0A†r

Ar


r


r


r

! A†r

0



r

i ⌘ A†r


T va1T va2

= T va2T va1

(34a)

(�vxy)

2= 1 (34b)

(�vxy)

2= 1 (34c)

Gvy�

vxy = �v

xyGvx (34d)

T vx (G

vy)

�1= �v

xyGx�vxy (34e)

9


Q

rr02D⌘rr0 = i



Gx : (x, y) 7! (x,�y) +1

2

(a1 � a2)

Gy : (x, y) 7! (�x, y)� 1

2

(a1 � a2) , (33)



r



rr


r


rr

0 sat-isfying

Q

rr

02D⌘rr




r


r


rr

0 ⌘rr

0A†r

Ar

0+h.c. in-




a2 since ⌘rr

0



Q

rr02D⌘rr0 =



rr

0A†r

Ar


r


r


r

! A†r

0



r

i ⌘ A†r


T va1T va2

= T va2T va1

(34a)

(�vxy)

2= 1 (34b)

(�vxy)

2= 1 (34c)

Gvy�

vxy = �v

xyGvx (34d)

T vx (G

vy)

�1= �v

xyGx�vxy (34e)

7

via the mapping

B†r

= (b†r

)

4, (18)nr

= 4Nr

(19)

where b†r



arr


0 |ki = |k + 1i, (20)


ar

0r

= a†rr

0 and er

0r

= e†rr

0 . (21)


Y

r

02hrr0ierr

0= e2⇡inr/4, (22)



H = Hg +Hm

Hm = �tX

hrr0i(b†

r

arr

0br

0+ h.c.) + ˜V [{n

r

}], (23)

Hg = �hX

hrr0i(e

rr

0+ h.c.)�K

X

p22,M(

Y

rr

02p

arr

0+ h.c.)



r


err





Qr02hrr0i





r


r

= 1


r

02hrr0ierr

0= i. (24)



7

via the mapping

B†r

= (b†r

)

4, (18)nr

= 4Nr

(19)

where b†r



arr


0 |ki = |k + 1i, (20)


ar

0r

= a†rr

0 and er

0r

= e†rr

0 . (21)


Y

r

02hrr0ierr

0= e2⇡inr/4, (22)



H = Hg +Hm

Hm = �tX

hrr0i(b†

r

arr

0br

0+ h.c.) + ˜V [{n

r

}], (23)

Hg = �hX

hrr0i(e

rr

0+ h.c.)�K

X

p22,M(

Y

rr

02p

arr

0+ h.c.)



r


err





Qr02hrr0i





r


r

= 1


r

02hrr0ierr

0= i. (24)



Shastry-Sutherland Dual-Hexagonal7

via the mapping

B†r

= (b†r

)

4, (18)nr

= 4Nr

(19)

where b†r



arr


0 |ki = |k + 1i, (20)


ar

0r

= a†rr

0 and er

0r

= e†rr

0 . (21)


Y

r

02hrr0ierr

0= e2⇡inr/4, (22)



H = Hg +Hm

Hm = �tX

hrr0i(b†

r

arr

0br

0+ h.c.) + ˜V [{n

r

}], (23)

Hg = �hX

hrr0i(e

rr

0+ h.c.)�K

X

p22,M(

Y

rr

02p

arr

0+ h.c.)



r


err





Qr02hrr0i





r


r

= 1


r

02hrr0ierr

0= i. (24)



8


Qr02hrr0i

err0 = �1 on each





0arr

0 on


rr


ˆF⌦ =

Y

rr

0!1e(†)rr

0 = e12e†23e34e

†45 · · · , (25)




r

and Ar




err

0= ⌘

rr

0A†r

Ar

0 (26)Y

rr

022,Marr

0= E

r

, (27)


rr


r

02hrr0ierr

0=

Q

rr

02D⌘rr



Hg = �hX

rr

0

⌘rr

0A†r

Ar

0 �KX

r

Er

+ h.c., (28)


0 satisfyY

rr

02D⌘rr

0= i⌫ (29)


rr


r

¯

r


rr



ˆF⌦ =

⇣ Y

rr

0!1⌘⇤rr

0

⌘A1. (30)


rr




r



Ta1 : (x, y) 7! (x, y) + a1

Ta2 : (x, y) 7! (x, y) + a2, (31)


�xy : (x, y) 7! (y, x)

�xy : (x, y) 7! (�y,�x), (32)

vs

⌘rr0⌘A†

1

8


Qr02hrr0i

err0 = �1 on each





0arr

0 on


rr


ˆF⌦ =

Y

rr

0!1e(†)rr

0 = e12e†23e34e

†45 · · · , (25)




r

and Ar




err

0= ⌘

rr

0A†r

Ar

0 (26)Y

rr

022,Marr

0= E

r

, (27)


rr


r

02hrr0ierr

0=

Q

rr

02D⌘rr



Hg = �hX

rr

0

⌘rr

0A†r

Ar

0 �KX

r

Er

+ h.c., (28)


0 satisfyY

rr

02D⌘rr

0= i⌫ (29)


rr


r

¯

r


rr



ˆF⌦ =

⇣ Y

rr

0!1⌘⇤rr

0

⌘A1. (30)


rr




r



Ta1 : (x, y) 7! (x, y) + a1

Ta2 : (x, y) 7! (x, y) + a2, (31)


�xy : (x, y) 7! (y, x)

�xy : (x, y) 7! (�y,�x), (32)


8


Qr02hrr0i

err0 = �1 on each





0arr

0 on


rr


ˆF⌦ =

Y

rr

0!1e(†)rr

0 = e12e†23e34e

†45 · · · , (25)




r

and Ar




err

0= ⌘

rr

0A†r

Ar

0 (26)Y

rr

022,Marr

0= E

r

, (27)


rr


r

02hrr0ierr

0=

Q

rr

02D⌘rr



Hg = �hX

rr

0

⌘rr

0A†r

Ar

0 �KX

r

Er

+ h.c., (28)


0 satisfyY

rr

02D⌘rr

0= i⌫ (29)


rr


r

¯

r


rr



ˆF⌦ =

⇣ Y

rr

0!1⌘⇤rr

0

⌘A1. (30)


rr




r



Ta1 : (x, y) 7! (x, y) + a1

Ta2 : (x, y) 7! (x, y) + a2, (31)


�xy : (x, y) 7! (y, x)

�xy : (x, y) 7! (�y,�x), (32)

8


Qr02hrr0i

err0 = �1 on each





0arr

0 on


rr


ˆF⌦ =

Y

rr

0!1e(†)rr

0 = e12e†23e34e

†45 · · · , (25)




r

and Ar




err

0= ⌘

rr

0A†r

Ar

0 (26)Y

rr

022,Marr

0= E

r

, (27)


rr


r

02hrr0ierr

0=

Q

rr

02D⌘rr



Hg = �hX

rr

0

⌘rr

0A†r

Ar

0 �KX

r

Er

+ h.c., (28)


0 satisfyY

rr

02D⌘rr

0= i⌫ (29)


rr


r

¯

r


rr



ˆF⌦ =

⇣ Y

rr

0!1⌘⇤rr

0

⌘A1. (30)


rr




r



Ta1 : (x, y) 7! (x, y) + a1

Ta2 : (x, y) 7! (x, y) + a2, (31)


�xy : (x, y) 7! (y, x)

�xy : (x, y) 7! (�y,�x), (32)

a1

a2

8


Qr02hrr0i

err0 = �1 on each





0arr

0 on


rr


ˆF⌦ =

Y

rr

0!1e(†)rr

0 = e12e†23e34e

†45 · · · , (25)




r

and Ar




err

0= ⌘

rr

0A†r

Ar

0 (26)Y

rr

022,Marr

0= E

r

, (27)


rr


r

02hrr0ierr

0=

Q

rr

02D⌘rr



Hg = �hX

rr

0

⌘rr

0A†r

Ar

0 �KX

r

Er

+ h.c., (28)


0 satisfyY

rr

02D⌘rr

0= i⌫ (29)


rr


r

¯

r


rr



ˆF⌦ =

⇣ Y

rr

0!1⌘⇤rr

0

⌘A1. (30)


rr




r



Ta1 : (x, y) 7! (x, y) + a1

Ta2 : (x, y) 7! (x, y) + a2, (31)


�xy : (x, y) 7! (y, x)

�xy : (x, y) 7! (�y,�x), (32)

p4g symmetry includes

9


Q

rr02D⌘rr0 = i



Gx : (x, y) 7! (x,�y) +1

2

(a1 � a2)

Gy : (x, y) 7! (�x, y)� 1

2

(a1 � a2) , (33)



r



rr


r


rr

0 sat-isfying

Q

rr

02D⌘rr




r


r


rr

0 ⌘rr

0A†r

Ar

0+h.c. in-




a2 since ⌘rr

0



Q

rr02D⌘rr0 =



rr

0A†r

Ar


r


r


r

! A†r

0



r

i ⌘ A†r


T va1T va2

= T va2T va1

(34a)

(�vxy)

2= 1 (34b)

(�vxy)

2= 1 (34c)

Gvy�

vxy = �v

xyGvx (34d)

T vx (G

vy)

�1= �v

xyGx�vxy (34e)

9


Q

rr02D⌘rr0 = i



Gx : (x, y) 7! (x,�y) +1

2

(a1 � a2)

Gy : (x, y) 7! (�x, y)� 1

2

(a1 � a2) , (33)



r



rr


r


rr

0 sat-isfying

Q

rr

02D⌘rr




r


r


rr

0 ⌘rr

0A†r

Ar

0+h.c. in-




a2 since ⌘rr

0



Q

rr02D⌘rr0 =



rr

0A†r

Ar


r


r


r

! A†r

0



r

i ⌘ A†r


T va1T va2

= T va2T va1

(34a)

(�vxy)

2= 1 (34b)

(�vxy)

2= 1 (34c)

Gvy�

vxy = �v

xyGvx (34d)

T vx (G

vy)

�1= �v

xyGx�vxy (34e)

9


Q

rr02D⌘rr0 = i



Gx : (x, y) 7! (x,�y) +1

2

(a1 � a2)

Gy : (x, y) 7! (�x, y)� 1

2

(a1 � a2) , (33)



r



rr


r


rr

0 sat-isfying

Q

rr

02D⌘rr




r


r


rr

0 ⌘rr

0A†r

Ar

0+h.c. in-




a2 since ⌘rr

0



Q

rr02D⌘rr0 =



rr

0A†r

Ar


r


r


r

! A†r

0



r

i ⌘ A†r


T va1T va2

= T va2T va1

(34a)

(�vxy)

2= 1 (34b)

(�vxy)

2= 1 (34c)

Gvy�

vxy = �v

xyGvx (34d)

T vx (G

vy)

�1= �v

xyGx�vxy (34e)

Q) How do A and A+ transform under these symmetries?

ex) 𝜎xy : 1 <-> 3, 2 <-> 6, 4->4, 5->5 and A <-> A+ due to sign change

9


Q

rr02D⌘rr0 = i



Gx : (x, y) 7! (x,�y) +1

2

(a1 � a2)

Gy : (x, y) 7! (�x, y)� 1

2

(a1 � a2) , (33)



r



rr


r


rr

0 sat-isfying

Q

rr

02D⌘rr




r


r


rr

0 ⌘rr

0A†r

Ar

0+h.c. in-




a2 since ⌘rr

0



Q

rr02D⌘rr0 =



rr

0A†r

Ar


r


r


r

! A†r

0



r

i ⌘ A†r


T va1T va2

= T va2T va1

(34a)

(�vxy)

2= 1 (34b)

(�vxy)

2= 1 (34c)

Gvy�

vxy = �v

xyGvx (34d)

T vx (G

vy)

�1= �v

xyGx�vxy (34e)

A <-> A+ : flux-antiflux transformation under reflection

Z4 gauge theory in Shastry-Sutherland lattice10

�

xy

�

xy

G

x

G

y

A(m,n)1 A

†(n,m�1)3 (�1)m+n

A

†(�n,�m)3 i

2n+1A

†(m,�n�1)3 i

2m+1A

†(�m,n)3

A(m,n)2 A

†(n,m)6 (�1)m+n

A

†(�n,�m)2 i

2n+3A

†(m,�n�1)5 i

2m+1A

†(�m,n+1)4

A(m,n)3 A

†(n+1,m)1 (�1)m+n

A

†(�n,�m)1 i

2n+1A

†(m+1,�n�1)1 i

2m+3A

†(�m,n+1)1

A(m,n)4 �A

†(n,m)4 (�1)m+n

A

†(�n,�m)5 i

2n+3A

†(m,�n)6 i

2m+3A

†(�m,n)2

A(m,n)5 A

†(n,m)5 (�1)m+n

A

†(�n,�m)4 i

2n+3A

†(m+1,�n�1)2 i

2m+1A

†(�m�1,n+1)6

A(m,n)6 A

†(n,m)2 �(�1)m+n

A

†(�n,�m)6 i

2n+1A

†(m+1,�n)4 i

2m+1A

†(�m�1,n)5

TABLE I. Vison symmetries. We list the transformation of the single-vison operator Ar under four lattice symmetries, for a given phaseconfiguration ⌘rr0 that satisfies

Q

rr02D⌘rr0 = i (see Fig. 4)

where the ‘v’ denotes the fact that we are considering the ac-tion on single-vison states. In contrast, the remaining set ofrelations between the space group symmetry generators in-cludes a projective phase factor of (-1) relative to their ex-pected forms:

(Gvx)

2= �T v

a1(35a)

(Gvy)

2= �T v

a2(35b)

�vxyG

vx = �Gv

x�vxy (35c)

�vxy�

vxy = ��v

xy�vxy (35d)

GvxT

va2

= �(T va2)

�1Gvx. (35e)

The non-trivial (-1) phase factor that appears in the above al-gebraic relations is once again an indication that the visonsfractionalize symmetry: in this case, the fractionalized sym-metry corresponds to the glide planes (and the remaining non-trivial relations should be viewed as consequences of this.)We may readily confirm that for ⌫ = 2, such a phase factoris absent: there is no point group symmetry fractionalization.Indeed, our arguments may be straightforwardly extended toall fillings, and we find (perhaps unsurprisingly!) that the rel-evant phase factor is (�1)

⌫ (mod S), so that point-group quan-tum number fractionalization only occurs for fillings that arenot a multiple of the non-symmorphic rank.

D. Condensing Z4 fluxes: confined phases and Z2 gaugetheories

In the previous section, we have studied the symmetries ofsingle vison excitations in the deconfined phase of Hg thatemerges in the limit K � h. We have demonstrated that atodd integer filling the visons fractionalize point-group sym-metries while preserving translational symmetries. We nowfocus on the case of ⌫ = 1, and analyze the proximate phasesthat can be accessed from our Z4 theory by condensing visons.In a Z4 theory, we may choose to condense either one, two,or three visons; each of these leads to distinct possibilities.Note that we do not construct the specific microscopic Hamil-tonians needed to drive the system into these vison-condensedphases; we simply use the preceding symmetry analysis todraw universal conclusions about the symmetry and topologi-cal properties of the vison condensates.

A crucial fact is that condensing particles in the deconfinedphase of a gauge theory confines all particles that have non-trivial mutual statistics with the condensate, but leaves par-ticles with trivial mutual statistics as deconfined excitations.The charges (denoted e) and fluxes (denoted m) in the Z4

gauge theory take values qe, qm 2 {0, 1, 2, 3}, with the mu-tual statistics phase factor for taking an e-particle around anm-particle (or vice-versa) given by e2⇡iqeqm/4. We will nowstudy the phases obtained by the different possibilities for vi-son condensation.Single Vison Condensation: Imagine we exit the deconfinedphase of Hg by condensing a single vison. The resulting phasewill have hA(x,y)µi 6= 0 for some µ 2 {1, 2, . . . , 6}. Since thesingle-vison state corresponding to A has qm = 1, it followsthat in a single-vison condensate, all fluxes are identified withthe vacuum (since it is a condensate of fluxes), and all the e-particles are confined, since they all have nontrivial statisticswith a qe = 1 object. Thus, condensing the vison results inconfinement of the Z4 gauge field, and the symmetry relations(35a-35e) reveal that the system breaks point-group symme-tries, owing to the nonzero value of hAµ

(x,y)i. We note thatcondensing visons provides a convenient unified formalismfor examining broken-symmetry states on the SSL.Vison Pair Condensation: A more interesting situationarises if energetics favor the condensation of paired visonsover the single-vison condensate. We may understand thenature of the resulting phase as follows. As A2 carries twounits of magnetic flux (qm = 2), creating a two-vison con-densate identifies qm = 2 with qm = 0 and hence qm = 1

with qm = 3: in other words, the fluxes now take values inthe group Z2. Now, we see that condensing a qm = 2 ob-ject must confine the qe = 1 and the qe = 3 charges, as theyhave nontrivial mutual statistics with it; however, the qe = 2

charge remains deconfined. Thus, the charges also take val-ues in Z2, and we are left with a Z2 gauge theory. As theqe = 2 charge must be equivalent a two-parton bound state,we conclude that it also carries 1/2 charge of the global U(1)

symmetry. If we can construct a vison-paired state withoutbreaking symmetry, then we will arrive at a simpler fraction-alized description of a symmetry-preserving phase of ⌫ = 1

bosons on the SSL; this would be a phase with deconfinedquasiparticles with 1/2-charge under the global U(1) symme-try, and emergent Z2 gauge flux.

In order to construct such a state, it suffices to consider

Relation between different symmetries

9


Q

rr02D⌘rr0 = i



Gx : (x, y) 7! (x,�y) +1

2

(a1 � a2)

Gy : (x, y) 7! (�x, y)� 1

2

(a1 � a2) , (33)



r



rr


r


rr

0 sat-isfying

Q

rr

02D⌘rr




r


r


rr

0 ⌘rr

0A†r

Ar

0+h.c. in-




a2 since ⌘rr

0



Q

rr02D⌘rr0 =



rr

0A†r

Ar


r


r


r

! A†r

0



r

i ⌘ A†r


T va1T va2

= T va2T va1

(34a)

(�vxy)

2= 1 (34b)

(�vxy)

2= 1 (34c)

Gvy�

vxy = �v

xyGvx (34d)

T vx (G

vy)

�1= �v

xyGx�vxy (34e)

‘trivial’ algebra algebra with phase factor -1

10

�

xy

�

xy

G

x

G

y

A(m,n)1 A

†(n,m�1)3 (�1)m+n

A

†(�n,�m)3 i

2n+1A

†(m,�n�1)3 i

2m+1A

†(�m,n)3

A(m,n)2 A

†(n,m)6 (�1)m+n

A

†(�n,�m)2 i

2n+3A

†(m,�n�1)5 i

2m+1A

†(�m,n+1)4

A(m,n)3 A

†(n+1,m)1 (�1)m+n

A

†(�n,�m)1 i

2n+1A

†(m+1,�n�1)1 i

2m+3A

†(�m,n+1)1

A(m,n)4 �A

†(n,m)4 (�1)m+n

A

†(�n,�m)5 i

2n+3A

†(m,�n)6 i

2m+3A

†(�m,n)2

A(m,n)5 A

†(n,m)5 (�1)m+n

A

†(�n,�m)4 i

2n+3A

†(m+1,�n�1)2 i

2m+1A

†(�m�1,n+1)6

A(m,n)6 A

†(n,m)2 �(�1)m+n

A

†(�n,�m)6 i

2n+1A

†(m+1,�n)4 i

2m+1A

†(�m�1,n)5


Q



(Gvx)

2= �T v

a1(35a)

(Gvy)

2= �T v

a2(35b)

�vxyG

vx = �Gv

x�vxy (35c)

�vxy�

vxy = ��v

xy�vxy (35d)

GvxT

va2

= �(T va2)

�1Gvx. (35e)













visons fractionalize point group symmetries


For ν=2 case, 8


Qr02hrr0i

err0 = �1 on each





0arr

0 on


rr


ˆF⌦ =

Y

rr

0!1e(†)rr

0 = e12e†23e34e

†45 · · · , (25)




r

and Ar




err

0= ⌘

rr

0A†r

Ar

0 (26)Y

rr

022,Marr

0= E

r

, (27)


rr


r

02hrr0ierr

0=

Q

rr

02D⌘rr



Hg = �hX

rr

0

⌘rr

0A†r

Ar

0 �KX

r

Er

+ h.c., (28)


0 satisfyY

rr

02D⌘rr

0= i⌫ (29)


rr


r

¯

r


rr



ˆF⌦ =

⇣ Y

rr

0!1⌘⇤rr

0

⌘A1. (30)


rr




r



Ta1 : (x, y) 7! (x, y) + a1

Ta2 : (x, y) 7! (x, y) + a2, (31)


�xy : (x, y) 7! (y, x)

�xy : (x, y) 7! (�y,�x), (32)

8


Qr02hrr0i

err0 = �1 on each





0arr

0 on


rr


ˆF⌦ =

Y

rr

0!1e(†)rr

0 = e12e†23e34e

†45 · · · , (25)




r

and Ar




err

0= ⌘

rr

0A†r

Ar

0 (26)Y

rr

022,Marr

0= E

r

, (27)


rr


r

02hrr0ierr

0=

Q

rr

02D⌘rr



Hg = �hX

rr

0

⌘rr

0A†r

Ar

0 �KX

r

Er

+ h.c., (28)


0 satisfyY

rr

02D⌘rr

0= i⌫ (29)


rr


r

¯

r


rr



ˆF⌦ =

⇣ Y

rr

0!1⌘⇤rr

0

⌘A1. (30)


rr




r



Ta1 : (x, y) 7! (x, y) + a1

Ta2 : (x, y) 7! (x, y) + a2, (31)


�xy : (x, y) 7! (y, x)

�xy : (x, y) 7! (�y,�x), (32)

9


Q

rr02D⌘rr0 = i



Gx : (x, y) 7! (x,�y) +1

2

(a1 � a2)

Gy : (x, y) 7! (�x, y)� 1

2

(a1 � a2) , (33)



r



rr


r


rr

0 sat-isfying

Q

rr

02D⌘rr




r


r


rr

0 ⌘rr

0A†r

Ar

0+h.c. in-




a2 since ⌘rr

0



Q

rr02D⌘rr0 =



rr

0A†r

Ar


r


r


r

! A†r

0



r

i ⌘ A†r


T va1T va2

= T va2T va1

(34a)

(�vxy)

2= 1 (34b)

(�vxy)

2= 1 (34c)

Gvy�

vxy = �v

xyGvx (34d)

T vx (G

vy)

�1= �v

xyGx�vxy (34e)

9


Q

rr02D⌘rr0 = i



Gx : (x, y) 7! (x,�y) +1

2

(a1 � a2)

Gy : (x, y) 7! (�x, y)� 1

2

(a1 � a2) , (33)



r



rr


r


rr

0 sat-isfying

Q

rr

02D⌘rr




r


r


rr

0 ⌘rr

0A†r

Ar

0+h.c. in-




a2 since ⌘rr

0



Q

rr02D⌘rr0 =



rr

0A†r

Ar


r


r


r

! A†r

0



r

i ⌘ A†r


T va1T va2

= T va2T va1

(34a)

(�vxy)

2= 1 (34b)

(�vxy)

2= 1 (34c)

Gvy�

vxy = �v

xyGvx (34d)

T vx (G

vy)

�1= �v

xyGx�vxy (34e)

-1

all algebra are ‘trivial’, no phase factor -1 unlike the case of ν=1.

visons do not fractionalize symmetries for ν=2, since ν/SG ∈Z

Z4 gauge theory in Shastry-Sutherland latticeAt odd integer filling, the visons fractionalize point-group symmetries while preserving translational symmetries.

4



G : r ! gr + ⌧ . (5)



ˆG| i = ei✓| i (6)



k

| 0i ⌘ |˜ i, where

Ûk

= exp

⇢i

N

Zddr k · r⇢(r)

�(7)


k


ˆG Ûk

ˆG�1=

Ûk

e2⇡i�g

(k)Q/N (8)








rr

0




HIGT = �hX

hrr0i⌧xrr

0 �KX

p

Y

rr

02p

⌧zrr

0 , (9)


Y

r

02hrr0i⌧xrr

0 = (�1)

2⌫ , (10)


9


Q

rr02D⌘rr0 = i



Gx : (x, y) 7! (x,�y) +1

2

(a1 � a2)

Gy : (x, y) 7! (�x, y)� 1

2

(a1 � a2) , (33)



r



rr


r


rr

0 sat-isfying

Q

rr

02D⌘rr




r


r


rr

0 ⌘rr

0A†r

Ar

0+h.c. in-




a2 since ⌘rr

0



Q

rr02D⌘rr0 =



rr

0A†r

Ar


r


r


r

! A†r

0



r

i ⌘ A†r


T va1T va2

= T va2T va1

(34a)

(�vxy)

2= 1 (34b)

(�vxy)

2= 1 (34c)

Gvy�

vxy = �v

xyGvx (34d)

T vx (G

vy)

�1= �v

xyGx�vxy (34e)

comparison: Odd Ising gauge theory for half-filling ν=1/2

Vison creation operator

4



G : r ! gr + ⌧ . (5)



ˆG| i = ei✓| i (6)



k

| 0i ⌘ |˜ i, where

Ûk

= exp

⇢i

N

Zddr k · r⇢(r)

�(7)


k


ˆG Ûk

ˆG�1=

Ûk

e2⇡i�g

(k)Q/N (8)








rr

0




HIGT = �hX

hrr0i⌧xrr

0 �KX

p

Y

rr

02p

⌧zrr

0 , (9)


Y

r

02hrr0i⌧xrr

0 = (�1)

2⌫ , (10)


⊗𝝉rr’

x = -1

Visons fractionalize translational symmetries.

5

K � h. It is the latter that has a finite-energy gapped vi-son excitation and hence has topological order. As shown inRef. 8, linking the Gauss law constraint to the filling through(10) is required in order to satisfy the commensurability con-straints of the HOLSM theorem: it guarantees that the topo-logically degenerate ground states differ by the appropriatecrystal momentum computed via the flux insertion argument.

In the K � h deconfined phase HIGT supports a gappedchargon (or spinon, in the case when the U(1) is a componentof spin) that carries fractional electromagnetic charge (or frac-tional spin quantum number) ⌫e as well as a Z2 Ising charge.It also supports the dual excitation to a chargon, a gapped Isingvortex or “vison” that carries Z2 gauge flux; the vison is neu-tral under the global U(1) charge. Note that we have madea specific choice of Z2 fractionalization in taking the char-gon/spinon to carry fractional charge and the vison to be neu-tral; we will focus exclusively on this choice in throughoutthis paper. We have also assumed that both chargon/spinonand the vison are bosons; as one carries Z2 charge and theother Z2 flux, they acquire a mutual statistics phase of ⇡, ren-dering their bound state a fermion.

For the purposes of understanding crystal momentum frac-tionalization, the central point to note is that when ⌫ is a half-odd integer, the Gauss law (10) requires a non-trivial staticbackground Ising gauge charge on each site — in other words,there is a chargon frozen at each site. In this manner, theHOLSM commensurability constraints force us to work withan odd Ising gauge theory in the terminology of Ref.27. As wenow show, this requires that the vison carry a non-zero crystalmomentum.

A single-vison state is one where a single plaquette p has anon-zero Z2 magnetic flux, i.e.

Y

rr

02p

⌧zrr

0 = �1. (11)

This is non-local in the ⌧z-basis of the gauge theory: to satisfy(11), we must flip ⌧zs on a string of bonds out to infinity. Inorder to further elucidate symmetry properties of the vison, itis convenient to move to a description in which the vison cre-ation operator is local. This is accomplished by mapping ourZ2 gauge theory into its transverse-field Ising model (TFIM)dual, as we now detail.

Let us define dual lattice sites ¯

r that reside at the center ofeach square lattice plaquette. We define a new set of Isingoperators �µ

r

for the TFIM; these are related to the originalIsing variables via

⌧xrr

0 = ⌘rr

0�zr

�zr

0 (12)Y

rr

02pr

⌧zrr

0 = �xr

, (13)

where ¯

r and ¯

r

0 are the dual lattice sites on either side of thethe link (rr

0) in the original square lattice, and p

r

representsthe direct lattice plaquette centered on dual lattice site r. TheZ2 phase ⌘

rr

0 is defined on each link (

¯

r

¯

r

0) of the dual lattice.

Under this mapping, the IGT Hamiltonian (9) is transformedinto the Hamiltonian for the TFIM

HTFIM = �hX

rr

0

⌘rr

0�zr

�zr

0 �KX

r

�xr

(14)

while the Gauss law (10) appears as a phase constraintY

rr

02p

⌘rr

0= (�1)

2⌫ . (15)

As a final step to obtain a well-defined TFIM, we must picka prescription for the phases ⌘

rr

0 that satisfies (15); while forinteger ⌫ we may simply choose ⌘

rr

0= 1, for half-odd integer

⌫ this ‘gauge fixing’ amounts to choosing one frustrated bondon each plaquette, where the Ising coupling flips sign. Thedistinction between even and odd Ising gauge theory (inte-ger and half-odd integer ⌫) is thus mapped into the distinctionbetween ordinary and (fully) frustrated TFIMs. Upon gauge-fixing, the vison becomes a local operator — essentially �z

r

with an appropriate (gauge-fixed) phase — and therefore wecan consider the symmetry properties of vison states by con-sidering the transformation properties of �z

r

. For the case ofodd Ising gauge theory, any gauge fixing satisfying (15) nec-essarily breaks translational symmetry (in other words, thereis no assignment of frustrated bonds that respects the transla-tional symmetry of the lattice). It is straightforward to showthat in any consistent gauge choice, the unit cell is doubled,and consequently a single vison state satisfies

T vx

T vy

(T vx

)

�1 �T vy

��1= �1, (16)

where T vx

and T vy

are the translations of visons by lattice vec-tors x and y on a square lattice.

In other words, visons transform projectively under thetranslations when ⌫ is a half-odd-integer. (Such non-commuting translations should be familiar from the case ofa charged particle in a magnetic field, but that situation is dis-tinct from the example considered here.)

An intuitive understanding of why HOLSM requires the vi-son to carry nontrivial crystal momentum is as follows. Imag-ine starting from the deconfined phase of HIGT and tuning pa-rameters so that we condense one of its gapped excitations.Recall that the vison and spinon are both bosons, but havenontrivial mutual statistics — so either can be condensed, butthis will confine the other. Now, as the chargon carries a globalU(1) charge, it follows that condensing this will break theU(1) symmetry, leading to a superfluid with gapless excita-tions. In contrast, condensing the vison leads to a gapped, con-fined phase. Were the vison to carry trivial quantum numbers,then the resulting vison-condensed phase would be a gappedphase without any fractionalization or translational symmetrybreaking, which is impossible for fractional ⌫. Thus, consis-tency demands that the vison carry non-zero crystal momen-tum, as this will trigger broken symmetry when it condenses.

The discussion above has all the basic ingredients that willpermeate the remainder of this paper: a commensurabilitycondition (HOLSM theorem) requiring a certain non-trivialbackground gauge charge in the low-energy effective theory,in turn forcing a fractionalization of crystal symmetry quan-tum numbers. However, the alert reader will note that we haveonly considered the fractionalization of translational quan-tum numbers, leading to a period-quadrupling structure. Arethere distinct signatures of the fractionalization of other crys-tal symmetries? Of course, there may be other symmetriesthat trivially inherit their projective nature from translations

Condensation of Visons and Z2 gauge theory

What happen if visons condense?charge e & flux m : qe, qm ∈ {0,1,2,3} : mutual statistics phase factor for taking e particle around m particle or vice versa

10

�

xy

�

xy

G

x

G

y

A(m,n)1 A

†(n,m�1)3 (�1)m+n

A

†(�n,�m)3 i

2n+1A

†(m,�n�1)3 i

2m+1A

†(�m,n)3

A(m,n)2 A

†(n,m)6 (�1)m+n

A

†(�n,�m)2 i

2n+3A

†(m,�n�1)5 i

2m+1A

†(�m,n+1)4

A(m,n)3 A

†(n+1,m)1 (�1)m+n

A

†(�n,�m)1 i

2n+1A

†(m+1,�n�1)1 i

2m+3A

†(�m,n+1)1

A(m,n)4 �A

†(n,m)4 (�1)m+n

A

†(�n,�m)5 i

2n+3A

†(m,�n)6 i

2m+3A

†(�m,n)2

A(m,n)5 A

†(n,m)5 (�1)m+n

A

†(�n,�m)4 i

2n+3A

†(m+1,�n�1)2 i

2m+1A

†(�m�1,n+1)6

A(m,n)6 A

†(n,m)2 �(�1)m+n

A

†(�n,�m)6 i

2n+1A

†(m+1,�n)4 i

2m+1A

†(�m�1,n)5


Q



(Gvx)

2= �T v

a1(35a)

(Gvy)

2= �T v

a2(35b)

�vxyG

vx = �Gv

x�vxy (35c)

�vxy�

vxy = ��v

xy�vxy (35d)

GvxT

va2

= �(T va2)

�1Gvx. (35e)













(1) Single vison condensation (qm=1) : all fluxes are identified to vacuum, <A(x,y)μ>≠0 breaks point group symmetries, all e particles are confined with non-trivial mutual statistics with qe=1, thus confining Z4 gauge field,

(3) Vison Pair condensation (qm=2) : qm=2 equivalent to qm=0 & qm=1 equivalent to qm=3 -> Z2 gauge group, just confine qe=1 and 3 charges but qe=2 charges remain deconfined

(2) Vison triplet condensation (qm=3) : qm =3 equivalent to qm=0 thus qm=1 and 2 are also equivalent to vacuum, all e particles are confined, similar to single vison condensate

Condensation of Visons and Z2 gauge theory

Vison Pair condensation (qm=2)

11

nearest-neighbor vison pair operators to identify appropriatecombination(s) of visons that preserve all symmetries. Intu-itively, we wish to identify a vison pair such that acting withany symmetry operator leads to a squaring of phase factors in(35a-35e) so that the (-1) factors are absent: in other words,we wish to identify a two-vison condensate that transformstrivially under symmetries.

To this end, we consider the action of space group symme-tries on linear combinations of vison pairs:

ˆG :

X

hrr0icrr

0Ar

Ar

0 ! ei✓GX

hrr0ic⇤rr

0A†r

A†r

0 , (36)

where crr

0 ⌘ cei✓ss0 depends only on the sublattice index ofsites r, r0 and the ✓ss0 are chosen so as to make the linear com-bination invariant up to the conjugation (that simply encodesthe flux-antiflux transformation under reflection) and overallphase ei✓G . More precisely, we wish to find a choice of c

rr

0

that makes ✓G = 0 for all the four reflection symmetries —two ordinary reflections and two glides. Note that the fact thata flux-2 operator is mapped to a flux-(�2) operator under re-flection is innocuous, since we may add fluxes modulo 4 in aZ4 gauge theory. [SP: Right?]

We first observe that such vison-pair operators have theproperty that pairs built from sublattices (4,5) and (2,6) mixwith each other but not with those built from the other pairsunder the four reflection operaotrs. In other words, the op-erators c45A4A5 + c26A2A6 transforms independtly fromc12A1A2 + c36A3A6 + c14A1A4 + c34A3A4 + c23A2A3 +

c16A1A6 + c35A3A5 + c15A1A5 (we have suppressed spatialindices except for those that denote the sublattice, for sim-plicity.) Let us denote the quantum numbers of the four pointgroup symmetries listed in (32) and (33) as follows:

~✓G ⌘�✓�

xy

, ✓�xy

, ✓Gx

, ✓Gy

�⌘ (✓⌘, ✓� , ✓⇠, ✓) . (37)

In this notation we find four possible solutions that transformin the manner of (36); labelling these as (i) to (iv) (with therelevant ✓ss0 listed in Table II) we find that the phase factorsfor the different reflections satisfy the following nontrivial re-lations:

(i) ✓⌘ = ✓� + ⇡ = ✓⇠ + ⇡/2 = ✓ + 3⇡/2 ⌘ ✓ (38a)(ii) ✓⌘ = ✓� + ⇡ = ✓⇠ + 3⇡/2 = ✓ + ⇡/2 ⌘ ✓ (38b)(iii) ✓⌘ = ✓� = ✓⇠ = ✓ ⌘ ✓ (38c)(iv) ✓⌘ = ✓� = ✓⇠ + ⇡ = ✓ + ⇡ ⌘ ✓. (38d)

Crucially, we identify case (iii) as one where there is a con-sistent choice ✓ = 0 under which the vison pair transformstrivially under all the symmetry operations. [SP: I am a bitconfused: do we need this condition? Mike— what do youthink?] Thus, as we have argued, it is indeed possible to con-dense a vison pair without breaking any of the point groupsymmetries.

In summary: upon condensing any of the vison pair statesidentified above we arrive at a Z2 gauge theory, where the sin-gle vison continues to transform projectively under the point-group symmetries, and the single chargon carries 1/2 unit ofthe global U(1) charge. In Appendix A we show that a simi-lar structure arises in a different model with the same p4g and

U(1) symmetries and filling of ⌫ = 1, but with two sites perunit cell, underlining the fact that the structure is universal tothe symmetry group rather than any particular tight-bindinglattice model.Vison Triplet condensation: Condensing a three-visonbound state once again leads to a confined phase with brokensymmetry, by an argument analogous to the one-vison case.First, observe that all the e particles have nontrivial mutualstatistics and are hence confined. Now, we have identifiedqm = 3 and the vacuum, qm ⌘ 0; but since we began with aZ4 gauge theory, qm = 4 is also the vacuum. We can checkthat this guarantees that the particles with qm = 1, 2 are alsoidentified with the vacuum31 so there are no magnetic flux ex-citations. Finally, we observe that a triplet of fluxes will alsocarry a (-1) factor for the symmetry operations (35a-35e), andhence this condensate breaks symmetry.

V. DETECTING GLIDE FRACTIONALIZATION

[SP: Mike, I think SungBin and I would appreciate a carefulreading of this as you are far more expert on these issues thanwe are!]

A. Numerical Signatures

We now turn to a discussion of how to detect glide sym-metry fractionalization in numerics. As in the rest of the pa-per, we will focus on the case of Z2 topological order; notethat while our initial construction for a topologically orderedphase for the s-orbital Shastry-Sutherland model invoked a Z4

gauge structure with three distinct vison excitations, by con-densing paired visons we were able to access a theory with Z2

topological order with no broken symmetry, where the singleremaining vison excitation continues to transform nontriviallyunder glides. Note that if we also require a time-reversal in-variant ground state, then the simplest possibility consistentwith HOLSM theorems quite generally is a Z2 ‘toric code’type spin liquid, which is the Z � 2 topological order in-voked in the preceding discussion (similar arguments to thoseof Ref. 32 may be used to rule out the alternative ‘doubledsemion’ theory.)

We follow a line of reasoning originally developed by Za-letel, Lu and Vishwanath33 (ZLV), and use similar notationwhere possible. Recall that a Z2 topologically ordered phasehas a four-fold degeneracy on the torus or the (infinite cylin-der). In line with ZLV, we assume the existence of a numericalprocedure that can generate the ground-state manifold in thebasis of ‘minimally entangled states’ (MESs). The MES basisis the unique basis for the ground-state manifold in which theunitary operation of ‘threading anyonic flux a’, denoted Fa

x , isrealized as a permutation of basis states. Formally, Fa

x relatesground states to ground states by creating a pair of anyons(a, a) from vacuum and dragging in opposite directions ±x toinfinity; each MES thus has a unique topological flux thread-ing the cylinder. Let us denote the four anyon types of the toriccode as follows: ‘1’ (the vacuum and any local excitations,

11

nearest-neighbor vison pair operators to identify appropriatecombination(s) of visons that preserve all symmetries. Intu-itively, we wish to identify a vison pair such that acting withany symmetry operator leads to a squaring of phase factors in(35a-35e) so that the (-1) factors are absent: in other words,we wish to identify a two-vison condensate that transformstrivially under symmetries.

To this end, we consider the action of space group symme-tries on linear combinations of vison pairs:

ˆG :

X

hrr0icrr

0Ar

Ar

0 ! ei✓GX

hrr0ic⇤rr

0A†r

A†r

0 , (36)

where crr

0 ⌘ cei✓ss0 depends only on the sublattice index ofsites r, r0 and the ✓ss0 are chosen so as to make the linear com-bination invariant up to the conjugation (that simply encodesthe flux-antiflux transformation under reflection) and overallphase ei✓G . More precisely, we wish to find a choice of c

rr

0

that makes ✓G = 0 for all the four reflection symmetries —two ordinary reflections and two glides. Note that the fact thata flux-2 operator is mapped to a flux-(�2) operator under re-flection is innocuous, since we may add fluxes modulo 4 in aZ4 gauge theory. [SP: Right?]

We first observe that such vison-pair operators have theproperty that pairs built from sublattices (4,5) and (2,6) mixwith each other but not with those built from the other pairsunder the four reflection operaotrs. In other words, the op-erators c45A4A5 + c26A2A6 transforms independtly fromc12A1A2 + c36A3A6 + c14A1A4 + c34A3A4 + c23A2A3 +

c16A1A6 + c35A3A5 + c15A1A5 (we have suppressed spatialindices except for those that denote the sublattice, for sim-plicity.) Let us denote the quantum numbers of the four pointgroup symmetries listed in (32) and (33) as follows:

~✓G ⌘�✓�

xy

, ✓�xy

, ✓Gx

, ✓Gy

�⌘ (✓⌘, ✓� , ✓⇠, ✓) . (37)

In this notation we find four possible solutions that transformin the manner of (36); labelling these as (i) to (iv) (with therelevant ✓ss0 listed in Table II) we find that the phase factorsfor the different reflections satisfy the following nontrivial re-lations:

(i) ✓⌘ = ✓� + ⇡ = ✓⇠ + ⇡/2 = ✓ + 3⇡/2 ⌘ ✓ (38a)(ii) ✓⌘ = ✓� + ⇡ = ✓⇠ + 3⇡/2 = ✓ + ⇡/2 ⌘ ✓ (38b)(iii) ✓⌘ = ✓� = ✓⇠ = ✓ ⌘ ✓ (38c)(iv) ✓⌘ = ✓� = ✓⇠ + ⇡ = ✓ + ⇡ ⌘ ✓. (38d)

Crucially, we identify case (iii) as one where there is a con-sistent choice ✓ = 0 under which the vison pair transformstrivially under all the symmetry operations. [SP: I am a bitconfused: do we need this condition? Mike— what do youthink?] Thus, as we have argued, it is indeed possible to con-dense a vison pair without breaking any of the point groupsymmetries.

In summary: upon condensing any of the vison pair statesidentified above we arrive at a Z2 gauge theory, where the sin-gle vison continues to transform projectively under the point-group symmetries, and the single chargon carries 1/2 unit ofthe global U(1) charge. In Appendix A we show that a simi-lar structure arises in a different model with the same p4g and

U(1) symmetries and filling of ⌫ = 1, but with two sites perunit cell, underlining the fact that the structure is universal tothe symmetry group rather than any particular tight-bindinglattice model.Vison Triplet condensation: Condensing a three-visonbound state once again leads to a confined phase with brokensymmetry, by an argument analogous to the one-vison case.First, observe that all the e particles have nontrivial mutualstatistics and are hence confined. Now, we have identifiedqm = 3 and the vacuum, qm ⌘ 0; but since we began with aZ4 gauge theory, qm = 4 is also the vacuum. We can checkthat this guarantees that the particles with qm = 1, 2 are alsoidentified with the vacuum31 so there are no magnetic flux ex-citations. Finally, we observe that a triplet of fluxes will alsocarry a (-1) factor for the symmetry operations (35a-35e), andhence this condensate breaks symmetry.

V. DETECTING GLIDE FRACTIONALIZATION

[SP: Mike, I think SungBin and I would appreciate a carefulreading of this as you are far more expert on these issues thanwe are!]

A. Numerical Signatures

We now turn to a discussion of how to detect glide sym-metry fractionalization in numerics. As in the rest of the pa-per, we will focus on the case of Z2 topological order; notethat while our initial construction for a topologically orderedphase for the s-orbital Shastry-Sutherland model invoked a Z4

gauge structure with three distinct vison excitations, by con-densing paired visons we were able to access a theory with Z2

topological order with no broken symmetry, where the singleremaining vison excitation continues to transform nontriviallyunder glides. Note that if we also require a time-reversal in-variant ground state, then the simplest possibility consistentwith HOLSM theorems quite generally is a Z2 ‘toric code’type spin liquid, which is the Z � 2 topological order in-voked in the preceding discussion (similar arguments to thoseof Ref. 32 may be used to rule out the alternative ‘doubledsemion’ theory.)

We follow a line of reasoning originally developed by Za-letel, Lu and Vishwanath33 (ZLV), and use similar notationwhere possible. Recall that a Z2 topologically ordered phasehas a four-fold degeneracy on the torus or the (infinite cylin-der). In line with ZLV, we assume the existence of a numericalprocedure that can generate the ground-state manifold in thebasis of ‘minimally entangled states’ (MESs). The MES basisis the unique basis for the ground-state manifold in which theunitary operation of ‘threading anyonic flux a’, denoted Fa

x , isrealized as a permutation of basis states. Formally, Fa

x relatesground states to ground states by creating a pair of anyons(a, a) from vacuum and dragging in opposite directions ±x toinfinity; each MES thus has a unique topological flux thread-ing the cylinder. Let us denote the four anyon types of the toriccode as follows: ‘1’ (the vacuum and any local excitations,

Find pair vison operators that preserve all the symmetries.

-> Z2 gauge theory

15

Following the standard procedure of softening the con-straint, we introduce a gauge field on each link of the lattice.In this case, the minimal choice for the gauge group is Z2

and the Hilbert space on each link ` = (r, r0) consists of twostates h` = {|0i, |1i}. We introduce a Z2 vector potentialoperator al and the corresponding electric field operator e`,which satisfy

arr


0 |ki = |k + 1i. (A3)

Note that as we work in a Z2 theory we may identify |2i and|0i; furthermore, though their conjugates take reversed orien-tations, in a Z2 theory links are non-directional, and hence wedo not require an independent condition such as (21).

In terms of gauge fields, the constraint (A2) can be repre-sented as a Gauss law for the Z2 electric field,

Y

r

02hrr0ierr

0= ei⇡nr , (A4)


In parallel with the analysis in the main text, we rewrite themicroscopic Bose-Hubbard model (17) by placing the partonsin a gapped phase and integrating them out, leaving an effec-tive gauge theory defined on square plaquettes,

Hg = �hX

hrr0i(e

rr

0+ h.c.)�K

X

p22

(

Y

rr

02p

arr

0+ h.c.),

(A5)

where h,K > 0.For h � K, the the ground state of (A5) has e

rr

0= 1and

we have nr

⌘ 0 (mod 2) from the Gauss law (A4). This isthe electric-field-confined phase, where the partons are boundinto the original bosons, and (following the HOLSM theorem)the ground state must break symmetry if it is gapped but notfractionalized.

For K � h, the partons can propagate independently. Thiscorresponds to the deconfined phase, where the Gauss law(A4) is represented as

Y

r

02hrr0ierr

0= �1. (A6)

Fig. 6 shows an electric field satisfying (A6). We emphasizethat the Z2 electric field configuration does not enlarge theunit cell, so it does not break translation symmetry. However,it does breaks point group symmetry, in particular, the glidesymmetry. We now turn to an analysis of the low-energy visonexcitations of this gauge theory.

2. Dual Ising model

In order to investigate the properties of visons in the Z2

gauge theory, it is once again convenient to study the dualtheory, here a quantum Ising model. The dual lattice is also acheckerboard lattice with two sites in each unit cell. At eachdual lattice site ¯

r, we introduce a new set of Ising operators

Er

and Ar

that satisfy (26-27) in terms of which the dual Isingtheory then can be written exactly as (28), as for the dual Z4

clock model in the main text. Under the duality, the Gauss lawconstraint is mapped to a constraint on the product of bonds⌘rr

0 for every dual checkerboard lattice plaquette: we haveY

rr

022

⌘rr

0= (�1)

⌫ . (A7)

at boson filling ⌫. For ⌫ = 1, the product of bond variablessatisfies -1 and the dual theory is the fully frustrated Isingmodel on the checkerboard lattice: each plaquette has exactlyone frustrated bond with ⌘

rr

0= �1. As a reference configu-

ration (Fig. 6), we may assign the links ` = (

¯

r

¯

r

0) on the dual

checkerboard lattice that bisect the dashed magenta links ofthe direct lattice to have ⌘

rr

0= �1, while the ones that bisect

the solid black links have ⌘rr

0= 1.

In the dual theory, the non-local vison creation operator isdescribed by a local operator A

r

, along with a (static) non-local string product of bond strengths ⌘

rr

0 , fixed by the refer-ence configuration above. We may now examine its transfor-mation properties under various lattice symmetries.

3. Vison symmetry analysis

Before we consider the transformation of Ar

, we first listthe symmetries of the checkerboard d-orbital model. There

FIG. 6. (color online) Schematic picture of d-orbital ordering pat-tern between d(x+y)2 and d(x�y)2 on x-y plane. Dashed line in-dicates the enlarged unit cell in the presence of orbital ordering.Solid (Black colored) and dashed (magenta colored) links have elec-tric field configuration err0 = 1 and �1 respectively, satisfyingQr02hrr0i

err0 = �1. (See the main text for details.)

p4g symmetry with orbital ordering : 2site /unit-cell -> Start with Z2 gauge theory and obtain similar symmetry algebra

Symmetry fractionalization in Nonsymmorphic Crystals

Start with HOLSM theorem and focus on topologically ordered state

fractional filling vs nonsymmorphic with integer filling

Ising(Z2) gauge theory

vison fractionalizes translational symmetry

ν ∉ Z

Z4 /Z2 gauge theories

vison fractionalizes glide reflection symmetries

vison condensation : translational symmetry broken state

single/ triple vison condensation : glide reflection broken state vison pair condensation : Z2 gauge group

ν ∈ Z

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