Isotropic Layer Construction and Phase Diagram for Fracton Topological Phases

2 MB·

by  Sagar Vijay

Isotropic Layer Construction and Phase Diagram for Fracton Topological Phases
Sagar Vijay1,2
1Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
2Kavli Institute for Theoretical Physics, Santa Barbara, CA 93106, USA
Startingfromanisotropicconfigurationofintersecting,two-dimensionaltoriccodes,weconstruct
a fracton topological phase introduced in Ref. [26], which is characterized by immobile, point-
liketopologicalexcitations(“fractons”), anddegenerateground-statesonthetorusthatarelocally
indistinguishable. Our proposal leads to a simple description of the fracton excitations and of the
ground-state as a “loop” condensate, and provides a basis for building new 3D topological orders
suchasanatural,Z generalizationofthisfractonphase,whichweintroduce. Wedescribetherich
N
phasestructureofourlayeredZ system. Byinvokingalatticeduality,wedemonstratethatwhen
N
N ≥5, thereisanintermediatephasethatappearsbetweenthedecoupled, layeredsystemandthe
7
fracton topologically-ordered state, which opens the possibility of a continuous transition into the
1
fracton topological phase. We conclude by presenting a solvable model, that interpolates between
0
2 the fracton phase and a confined phase in which the phase transition is first-order.

n
a
Topologicalphasesofmatterhavebeenasubjectofac- the quantum dual of these interacting systems. This
J
tiveinterestincondensedmatterphysics. Theemergence providesageneralizationofthewell-knownWegnerdual-
3
oflocalconservationlawsincertainquantummany-body ity in (2+1)-dimensions that relates symmetry-breaking

systems leads to a convenient, low-energy description of phase transitions to the confinement transition of con-
]
l the system as a gauge theory, which captures the behav- ventional lattice gauge theories [1]. Furthermore, this
e
- iorofawiderangeoftopologicalphasesrangingquantum generalized lattice gauge theory has been used to find
r Hall states to quantum spin liquids [1–13]. The long- new fracton topological phases of bosons and fermions
t
s ranged entanglement in the degenerate, locally indistin- [24,26],andtofindexotic,interactingsystemswithfrac-
.
t guishable ground-states of a topological phases have led tal symmetries such as the dual of Haah’s code [26, 27].
a
m toproposalsfortheiruseasplatformsforuniversalquan- Recently,certaingaplessphasesthathavegapped,charge
tum computation [14–18]. excitations with reduced mobility have also been found
-
d Inrecentyears,newkindsofgappedtopologicalphases in “higher-rank” U(1) gauge theories [28].
n have been discovered in three spatial dimensions by In this work, we introduce a new construction of a
o studying exactly solvable models [19–24]. A general fea- fracton topological phase introduced in Ref. [26] which
c
ture of these topological phases is the presence of cer- yields an intriguing connection between two-dimensional
[
tain point-like topological excitations which are created topological orders and exotic three-dimensional topolog-

1 at the corners of operators with support on fractal- or ical phases that have excitations with reduced mobility.
v membrane-likeregions. Theseexcitationsarestrictlyim- The starting point for our construction is an isotropic
2
mobile at zero temperature, as no local operator can configuration of inter-penetrating, two-dimensional (2D)
6
move a single excitations without creating additional toriccodes[14]. Wedemonstratethatcondensingappro-
7
gapped excitations in the system. This is in contrast to priate excitations in the layered system – termed “com-
0
0 the behavior of point-like charges in discrete gauge the- positeexcitations”–canleadto(i)three-dimensionalZ2
. ories, which may be created at the ends of Wilson lines topological order, (ii) a fracton topological order known
1
and behave like mobile particles. As the point-like exci- as the “X-cube” phase [26], whose phenomenology we
0
7 tationsintheseexotictopologicalphasesappeareitheras thoroughly review, or (iii) a topologically trivial para-
1 fractionsoftopologicalexcitationswithrestrictedmobil- magnetic state. Condensing composite excitations effec-
: ity or at the corners of operators with support on a frac- tively binds the charges or fluxes of the two-dimensional
v
talregionofthelattice,theyarereferredtoas“fractons” toric codes, resulting in an emergent topological excita-
i
X [26]. ApartfromprovidinganexoticalternativetoFermi tion with reduced mobility. Our proposal also gives rise
r or Bose statistics in three dimensions, fracton topolog- to a “loop-gas” picture for the ground-state wavefunc-
a
ical orders are of interest for studying clean quantum tion of fracton topological phases, as well as a simple
systems with “glassy” dynamical behavior [19, 29], and understanding of the origin of the immobile excitations,
for building robust quantum memories such as Haah’s which opens a possible route for constructing other ex-
code [22, 25]. The phenomenology of fracton topological otic, three-dimensional topological orders.
phases is thoroughly reviewed in Ref. [24, 26]. We present a natural generalization of this procedure
Recently, ithasbeenshownthatgappedfractontopo- thatyieldsanew,Z analogoftheX-cubephase,before
N
logical phases may be understood through a generaliza- turning our attention to the rich phase structure of this
tion of conventional lattice gauge theory for interacting microscopic model. Our results are summarized in the
quantum systems that have an extensive set of global schematic phase diagrams in Fig. 8a and b. Of impor-
symmetries (e.g. planar symmetries) [26]. As a conse- tance is a lattice duality between certain phase transi-
quence, fracton topological phases may be obtained as tion(s) into the fracton topological phase and a confine-
2
(xz)
As A(syz) Bs(yz)
s A(sxy) Bs(xz)
s Bs(xy)
2DToricCodeLayer
z
� (a) (b)
y
µ
FIG. 2. Z charge & flux operators for the decoupled
x 2
layers: The locations of the (a) Z charge and (b) Z flux
2 2
operators at each site of the three-dimensional cubic lattice
Intersecting 2D Toric Codes
forthedecoupled,intersectinglayersoftwo-dimensionaltoric
codes; each charge or flux operator is oriented along the xy
(green), yz (red), or xz (blue) planes.
FIG. 1. Intersecting Layers of 2D Toric Codes: A
stack of two-dimensional, square-lattice toric codes in the xy
(green), yz (red) and xz (blue) directions, which intersect at
“plaquette” operators), as shown in the inset in Fig. 1.
sites. The resulting three-dimensional cubic lattice has two
spins per link (σ, µ) as shown. A single layer of the square- We take as our convention that a star operator is given
lattice toric code is shown as well, with the “star” and “pla- as the product of the x-component of the Pauli spins,
quette” operators defined as shown. while a plaquette operator involves a product of the z-
component.
We now place L copies of the square-lattice toric code
menttransitioninaconventional(3+1)-dZ gaugethe- in the xy, yz, and xz planes, respectively; any three mu-
N
ory driven by the condensation of Z flux loops. From tually orthogonal planes intersect at a single site of the
N
this duality and from the well-known phase structure of 2D square lattice. As shown schematically in Fig. 1,
Z gaugetheories,wedeterminethatwhenN 5,there this intersecting, three-dimensional arrangement of the
N
≥
must be an intermediate phase between the decoupled toric codes forms a three-dimensional (3D) cubic lattice
layers and the fracton topological phase, with an emer- with L3 sites and two spins per link. The 2D Z charge
2
gent U(1) gauge symmetry. While we believe this phase operators lie at the sites of the cubic lattice while the
is generically gapped and topologically trivial, we deter- Z flux operators lie at plaquettes. Initially, different
2
minethatthisphaseremainsgaplessalongaspecificline copies of the 2D toric codes are completely decoupled,
of the phase diagram, along which the phase transition and the Hamiltonian for the full system in terms of the
into the fracton topological phase is believed to be con- twospeciesofspins(σandµ)ateachlinkmaybewritten
tinuous. This opens the possibility of a continuum field in the form
theory description of this particular fracton topological
phase, which we leave to future work [62] to explore.
H = J A(j)+ B(j) (1)
We conclude by presenting a solvable projector model in 0 − r r
which the transition from the fracton topological phase (cid:88)r j=x(cid:88)y,yz,xz(cid:104) (cid:105)
to a trivial confined phase is first order, though this may
where the sum is over the sites r on the cubic lattice.
not capture the generic behavior of this transition. Here, A(j) is the four-spin star operator at site r which
r
measures the Z charge along the copy of the square-
2
lattice toric code that is oriented along the jth plane
I. ISOTROPIC LAYER CONSTRUCTION (with j = xy, yz, or xz). We associate three plaque-
tte operators B(j) with each site of the cubic lattice,
r
We begin by introducing the microscopic model for which measure the elementary Z flux through an ele-
2
our layer construction, before analyzing its rich phase mentary plaquette oriented in the jth plane. The star
structureandtheemergenceoffractontopologicalorder. andplaquetteoperatorsinthe3Dcubiclatticeareshown
Consider a single layer of the two-dimensional (2D) toric schematically in Fig. 2. All of the operators in H
toric
code on the square lattice [14], which describes the zero- mutually commute, and the ground-state may be writ-
correlation length limit of the deconfined phase of two- tenexactlyasanequal-amplitudesuperpositionofclosed
dimensional Z gauge theory. The Z gauge field lies on electric-charge loops within each plane. The topological
2 2
thelinksofthesquarelattice,andateachlatticesiteand degeneracy D is simply log D =6L after imposing peri-
2
plaquette,therearefour-spinoperatorsthatmeasurethe odic boundary conditions. The topologically-degenerate
Z chargeandflux,respectively(theso-called“star”and ground-states are locally indistinguishable, so that the
2
3
s z CompositeCharge
0 e Condensation
e
s �szs0µzss0 e e x y e e
e e
�z µz
ss0 ss0| decoupledi | decoupledi | 3DZ2i
FIG.3. Composite Charge Condensation: Theoperator FIG.4. “Gluing”Loops: Condensingthecompositecharge
σszs(cid:48)µzss(cid:48), when acting on the ground-state |Ψdecoupled(cid:105) of the excitation has the effect of “gluing” electric charge loops in
decoupled, two-dimensional toric codes, creates four electric adjacent, orthogonal layers. The resulting wavefunction is
charge excitations, two in each of the orthogonal planes that that of the 3D Z topological phase, as described in Sec. I.
2
meetatthelink,asshown. Condensingthiscompositecharge
excitation leads to a three-dimensional Z topological phase.
2
4, adding a composite electric charge has the effect of
“cutting open” two electric charge loops in orthogonal
2D Z topological order of the decoupled toric codes is
2 layers that appear in a loop configuration in the state
stable in the presence of weak perturbations [30].
Ψ . This“cut”loopconfigurationistobeinter-
decoupled
We now add an interaction |preted as a(cid:105) single, three-dimensional loop of the emer-
gent electricchargeA(xy)A(yz)A(xz),whichremainswell-
H = h σz µz t σx µx (2) s s s
1 − rr(cid:48) rr(cid:48) − rr(cid:48) rr(cid:48) defined in the condensed phase, and the resulting loop
(cid:104)(cid:88)r,r(cid:48)(cid:105) (cid:104)(cid:88)r,r(cid:48)(cid:105) superposition is precisely the wavefunction for the 3D
Z topological phase. As we will soon show, composite
which couples the two spins on each link of the cubic 2
chargecondensationalsohastheeffectofbindingthe2D
lattice. We begin by analyzing the phase diagram of the
Z fluxes into flux loops. In this way, the 3D electric
Hamiltonian 2
charge inherits its bosonic self-statistics and π-mutual
statistics with flux loops from the 2D Z phase.
H =H +H . (3) 2
0 1
WenowderivethisresultfromthemicroscopicHamil-
The ground-state of (3) will have the same topological tonian (3). When h (cid:29) J, t, we let σrzr(cid:48)µzrr(cid:48) = +1 and
order as a decoupled set of 3L toric codes when t, h define a single spin degree of freedom for every link of
J. We refer to this as the “decoupled phase” for th(cid:28)e the cubic lattice as
remainder of this work. In addition, however, we will
τz σz τx σx µx (4)
demonstrate that this Hamiltonian realizes (i) a (3+1)-d rr(cid:48) ≡ rr(cid:48) rr(cid:48) ≡ rr(cid:48) rr(cid:48)
Z topologicalphaseand(ii)afractontopologicalphase,
2 In terms of this spin, we obtain an effective Hamiltonian
along with (iii) a trivial paramagnet which corresponds
in perturbation theory
to a confined limit of the two topological phases.
H(1) = J A J B t τx (5)
eff − r− p− rr(cid:48)
A. (3+1)-d Z2 Topological Order from Composite (cid:88)r (cid:88)p (cid:104)(cid:88)r,r(cid:48)(cid:105)
(cid:101) (cid:101)
Charge Condensation
where the coupling J O(J3/h2). Here, the operator
∼
whSiltearkteinepginfrgomJ tahneddtecfioxuepdl,edwhpihcahsee,vewnetunaolwlyinlecardesasteoha, Ar ≡Ar(xy)(cid:101)A(ryz)A(rxz) = τrxr(cid:48) (6)
(3+1)-dimensionalZ2 topologicalphase. Beforedemon- (cid:101) r(cid:48)∈(cid:89)star(r)
stratingthisexplicitlywiththemicroscopicHamiltonian is precisely the six-spin operator that measures the Z
2
(3), we obtain intuition for this result by studying how charge in a (3+1)-dimensional Z topological phase,
2
the ground-state wavefunction changes as h is increased. while
When acting on the decoupled toric codes, the operator
σz µz createsfourelectricchargeexcitationsintheor- B τz (7)
thsos(cid:48)gosnsa(cid:48)l layers that meet at the link connecting sites s p ≡ rr(cid:48)
and s, as shown in Fig. 3. Increasing h leads to the r,r(cid:89)(cid:48)∈∂p
(cid:48)
condensation of this excitation, which we refer to as the isthefour-spinoperatormeasuringthefluxthroughpla-
composite electric charge. quette p on the cubic lattice. When t = 0, the effec-
Condensing the composite electric charge “glues” the tive Hamiltonian H(1) is precisely that of the 3D toric
eff
planar electric charge loops in the wavefunction for the code [31]. Furthermore, increasing t eventually leads to
decoupled toric codes (Ψ ). As shown in Fig. condensation of the Z flux loops, resulting in a trivial,
decoupled 2
| (cid:105)
4
confined phase. As advertised, the 2D Z fluxes are con-
2
fined, while a bound-state of the Z fluxes survives as
2
a topological excitation, which becomes the flux loop of m
the three-dimensional Z2 topological phase. Condensing s0
acompositeexcitationhasledtotheemergenceofanew µx m
topological excitation whose mobility is restricted. s �sxs ss0 m
0
m
B. “X-Cube” Fracton Topological Order from
Composite Flux Loop Condensation �sxs0µxss0| decoupledi
While composite charge condensation leads to three- FIG. 5. Composite Flux Loop Condensation: The op-
erator σx µx , when acting on the decoupled layers of the
dimensional Z2 topological order, condensing a bound- ss(cid:48) ss(cid:48)
two-dimensionalZ phase,createsfourfluxexcitationsinor-
state of the Z fluxes in adjacent layers – a composite 2
2 thogonal planes, as shown. Condensing this composite flux
flux loop –yieldstheso-called“X-cube”fractontopolog-
leadstothe“X-cube”fractontopologicalphase,withfracton
ical phase, as originally introduced and studied in Ref.
operator O as given in the main text, and shown in Fig. 7.
c
[26] using an exactly solvable, commuting Hamiltonian.
WefirstreviewthephenomenologyoftheX-cubefracton
topological phase before demonstrating that this phase ditional excitations. Pairs of fractons which are mobile
emerges within our layer construction. A detailed de- within planes exhibit π mutual statistics with dim.-1
scription of this fracton phase is provided in Ref. [26]. quasiparticles contained within their plane of motion.
1. Phenomenology of the X-cube Phase 2. Isotropic Layer Construction
The X-cube phase is a gapped topological phase that To observe the emergence of the X-cube phase in our
was introduced in Ref. [26], by studying the dual de- layer construction, we start from the decoupled phase of
scriptionoftheplaquetteIsingmodel,aninteractingspin theHamiltonian(3)andincreaset,whilekeepingJ andh
system with an extensive set of planar spin-flip symme- fixed. Acting with the operator σrxr(cid:48)µxrr(cid:48) on the ground-
tries; the finite-temperature behavior of this system has state of the decoupled planes of toric codes creates four
been studied previously [53–56]. While the plaquette Z2 fluxes in orthogonal layers, which form a closed loop
Ising model exhibits only symmetry-breaking or param- on the dual lattice as shown in Fig. 5a. We refer to this
agneticphasesatzerotemperature,itsdualdescriptionis excitation as a composite flux loop.
far more exotic, describing the “confinement” transition The X-cube fracton phase emerges after condensation
of a fracton topological phase. The solvable Hamilto- of this composite flux loop. When t J, h, we let
(cid:29)
ndwiuaaesnsthfooortwhtnehtigsoeohfrmaavceettorayntooppfhotalhsoeeg,imctaeulrlmgtir-eosdupnitndh-eisnttXaet-recaudcbteiegoenmnseor[d2a6ecl]y,, σdτrzorxrrm(cid:48)(cid:48)µ≡fxrorr(cid:48)σr=zerv(cid:48)e+µrzr1yra(cid:48)l.nindkWdoeeffitnthheeenancoubebfftiaecicntliavatentiscepeffineacsdteiτvgrxerre(cid:48)He≡aomfσfirlrxteroe(cid:48)--,
log D =6L 3 on the length-L three-torus, along with nian in perturbation theory as
2 −
exotic topological excitations whose mobility is severely H(2) =H h τz (8)
restricted. Sincethedegenerateground-statesarelocally eff X-cube− rr(cid:48)
indistinguishable, this degeneracy is stable to perturba- (cid:104)r(cid:88),r(cid:48)(cid:105)
tions, so that the X-cube model [26] describes a stable, where
gapped phase of matter. H = J A(j) K (9)
The X-cube phase has two types of gapped, topolog- X-cube − r − Or
r,j r
ical excitations. The first are point-like, immobile exci- (cid:88) (cid:88)
tations (the “fractons”) which are created by acting on with K O(J6/t5). Here, A(j) is a four-spin operator
r
∼
the ground-state with an operator supported on a flat, for the spins along the four links emanating from site r
rectangular region. The fracton excitations are created that lie in the jth plane:
at the corners of this membrane and may only be cre-
ated in groups of four. No local operator can move a A(rj) = τrxs (10)
single fracton without creating other excitations in the s∈pl(cid:89)anej(r)
system. While individual fracton excitations are immo-
while the operator , given by
bile, a pair of fractons are mobile within a plane and Or
have bosonic self-statistics. In the X-cube fracton phase, B(xy)B(yz)B(xz)B(yz)B(xz)B(xy)
Or ≡ r r r r+xˆ r+yˆ r+zˆ
there is a second type of point-like topological excita-
= τz (11)
tion, referred to as a “dimension-1 quasiparticle”, which rr(cid:48)
may only move along straight lines without creating ad- r,r(cid:48)∈(cid:89)cube(r)
5
can only be broken at pairs of points along xy, yz or xz
planes. As an example, acting on the ground-state with
W = τx , a string operator along a line in the
z r,r(cid:48) rr(cid:48)
z-direction,hastheeffectofcuttingcompositefluxloops
(cid:81)
attwopoints,asshowninFig. 6a,andtheresultingpair
Wz Wz of 2D Z2 fluxes may move within a plane. This operator
createsthemobileanyonexcitationsintheX-cubephase,
z z
which clearly inherit their “topological” properties (i.e.
self-statistics and mutual statistics with the dimension-1
x xy y M⌃ ⌃ quasiparticle), from the 2D Z2 fluxes in the toric code.
M
Anisolatedstringendpointispreciselythefractonex-
(a) (b)
citation of the X-cube model, and may be obtained by
FIG. 6. Loop-Gas Representation: Actingon theX-cube spatially separating the broken endpoints of composite
ground-state with (a) a straight, line-like operator W cre- flux loops. An isolated string endpoint cannot be moved
z
ates a pair of Z flux excitations at the ends. This operator on its own without creating additional excitations, for
2
“cuts”openaclosedloopconfigurationinthefractonground- if such an endpoint were mobile, then it would be pos-
state;alternatively,thisoperatoraddstwoopenstringsatits sible to smoothly deform a configuration of open strings
endpoints, when acting on an empty region. For the indi-
intoonewhichviolatesthepreviously-derivedconstraint.
cated loop configuration, acting with (b) a membrane opera-
The immobile, isolated endpoints then describe the frac-
tor M yields two separated, open strings of composite flux.
Σ ton excitations of the X-cube phase. In practice, act-
An isolated string endpoint cannot be moved on its own, or
ing on the ground-state with a flat membrane operator
else it would be possible to generate a configuration of bro-
= τx will localize these excitations at the
kIIeBn.lToohpessethisaotlavtioedlateexctihteatgiloonbsaalrceonthsteraiminmtsoebxiplelafirnaecdtoinnsS.ec. McorΣnersofrΣ,r,(cid:48)∈aΣssrhro(cid:48)wninFig. 6b. Inthedecoupledtoric
(cid:81)
codes, this operator creates a sequence of 2D Z fluxes
2
in parallel layers. After condensing the composite flux
ispreciselytheproductofthetwelveτz spinssurrounding loops, however, only the ends of this layered excitation
anelementarycubeasshowninFig. 7. TheHamiltonian carry an energy cost, since the bulk of this excitation is
HX cubeispreciselythecommutingHamiltonianthatde- locally indistinguishable from a composite flux loop.
scri−besthe“fixed-point”propertiesoftheX-cubefracton
ThetopologicaldegeneracyoftheX-cubefractontopo-
phase, as introduced in Ref. [26]. The ground-state at
logical phase – which was previously obtained [26] by
tahllers,ojl.vaEbxlecitpaotiinotnssamtiasyfiebseOcrrea|Ψte(cid:105)d=byAa(rcjt)i|nΨg(cid:105)w=ith|Ψst(cid:105)rifnogr cAo(uj)ntoinngthiendthepreeen-dtoernutscuosnisntgraainntaslgoenbrtahiec roepperreasteonrtsatOiorn,
r
or membrane operators to violate these constraints. The
of the solvable Hamiltonian [21, 22] – may now be un-
operators r and A(rj) measure the Z2 “charge” of the derstood by counting the number of topological sectors
O
fracton and dimension-1 excitations (i.e. the presence or of the decoupled, two-dimensional toric codes that are
absence of an excitation), respectively. madeequivalentaftercondensationofthecompositeflux
loop. Amoreextensivediscussionofthiscountingispre-
sented in the Supplemental Material [63].
3. Loop Gas Wavefunction, Excitations and Degeneracy
Our layer construction provides insight into the origin
C. Z X-cube Topological Phase
of the exotic topological excitations and sub-extensive N
ground-state degeneracy of the X-cube phase. First, the
ground-state wavefunction of the X-cube fracton phase The spirit of our proposal motivates the construction
maybewrittenasasuperpositionofcompositefluxloop of new, three-dimensional topological orders, by appro-
excitations of the layered 2D toric codes; heuristically priately condensing “loop” objects in an array of two-
dimensional topological phases. An extended discussion
ΨX cube (12) of new topological phases that may be built in this man-
| − (cid:105)∼ |C(cid:105)
ner is the subject of forthcoming work [62]. Here, we
(cid:88)C
with , a configuration of loops on the dual lattice. Ex- discuss the simplest generalization of our construction,
citatioCns may be obtained at the end-points of “broken” involvinganarrayoftwo-dimensionalZN toriccodes[14]
composite flux loops. We note, however, that while any inthesameconfigurationasshowninFig. 1,whichgives
closed loop configuration is permissible in the ground- rise to a ZN generalization of the X-cube model. After
state, the manner in which these loops may be broken arranging the ZN toric codes in a three-dimensional ar-
is highly constrained. First, a composite flux loop that ray, each link of the three-dimensional cubic lattice now
has broken at a single point is precisely a 2D Z2 flux hastwoZN quditdegreesoffreedom,denotedXrr(cid:48),Zrr(cid:48)
excitation of the underlying toric codes. Since these ex- and Xrr(cid:48), Zrr(cid:48) respectively, and satisfying the algebra
citations must appear in pairs, the composite flux loops XZ = ωZX, XZ = ωZX with ω = e2πi/N. We now
(cid:101) (cid:101)
(cid:101)(cid:101) (cid:101)(cid:101)
6
Z Z†
Z
p Z Z† Z† Sincetheoperators c andA(rj) commute,theground-
v p q Oc⇠ Z† Z†Z ZZ Z ZZ†† satnadtegaopfp(e1d7)exacgitaeidnsstaaOttiessfimesayObce|Ψc(cid:105)rea=tedA(rbjy)|aΨc(cid:105)tin=gw|Ψit(cid:105)h,
s u Z straight Wilson line or flat membrane operators, in a
v rq Oc⇠ Z† Z† ZZ Z† straightforward generalization of the operators in the Z2
AAAs(((xxyyzz)))⇠⇠⇠rXXX†q†q†pXXXupuuXXXssrXXX†v†r†v ZZ†† ZZ Xtmtihc-aeclgeuusnbneeatdinmecdrolcydfhirenaalrgc.gteNo2,nDowreeZxshpcoietwctatoetivrivoieecnrl,ysc,tohcdwaeerhsrdi.ycimhAaeisnsZsaiNionrnhee-es1lureiclqttteu,rdaiacsifprpaoaanmridr-
N
AF(IxGy).7⇠. TXh†qeXZuXNsXX-†vcubeModel: Shownarethecommuting offractonswhicharemobileinaplanehavebosonicself-
AAToX(p(h-yxceezzru))aZb⇠t⇠e2ormcsXXaots†q†phdeXXaedltruiXHsaXcpusr=pXsXes†tea−†vdrKiinn(cid:80)tShcee(cO.socIlB+va,Obislc†eo)bH−taaJmin(cid:80)ieltdro,nbjiy(aAnrr(ejfp)ol+ractAihner(gjZ)Z†N)., dsintiamitteissntpsicilosan,n-eb1uoqtfunmaosonitp-itaorrnitv.iciaHlelemorefu,ctthuhaaerlgssettaaqttiiss∈ttiicc{as1l,ea.in.θ.pgq,leNwθi−ptqh1=}a
Z† →τz and X, X† →τx. 2πpq/N, where p is the magnetic charge carried by the
fracton excitation.
consider the Hamiltonian
II. PHASE DIAGRAM
H =H +H (13)
(cid:48) 0(cid:48) 1(cid:48)
Wenowturntoadiscussionoftherichphasediagram
where H describes the de-coupled layers of Z toric
0 N ofthecoupledZ toriccodes,asdescribedbytheHamil-
N
codes
tonian(13). Ourresultsaresummarizedinthediagrams
shown in Fig. 8a and b when N < 5 and N 5, re-
H = J A(j)+B(j) +h.c. (14) ≥
0(cid:48) − r r spectively. Both figures include the four distinct phases
(cid:88)r j=x(cid:88)y,yz,xz(cid:104) (cid:105) discussed previously, i.e.
with A(rj) and Br(j) the ZN charge and flux operators at I. J (cid:29)h, t: Decoupled 2D ZN Topological Phase
site r, for the toric code oriented along plane j, respec- II. h J, t: 3D Z Topological Phase
N
tively. The layers are coupled through the interactions (cid:29)
III. t J, h: Z X-Cube Fracton Phase
N
(cid:29)
H1(cid:48) =−h Zr†r(cid:48)Zrr(cid:48) +h.c. (15) IV. h, t(cid:29)J: Trivial, Confined Phase
(cid:104)(cid:88)r,r(cid:48)(cid:105)(cid:104) (cid:105) Theremainingdetailsinthephasediagramarisefroman
(cid:101)
t X X +h.c. interesting duality between the Hamiltonian (13) when
− rr(cid:48) rr(cid:48) h=0,andthe3DZ toriccodeinthepresenceofafield;
(cid:104)(cid:88)r,r(cid:48)(cid:105)(cid:104) (cid:105) increasing the strenNgth of this field eventually leads to
(cid:101)
Following our previous analysis, we consider the limit condensationoftheZN fluxloopsandresultsinatrivial,
h J, t, where we find that the ground-state exhibits confined phase. From this duality, and from knowledge
3D(cid:29)ZN topological order. In this limit, we may identify of the phase structure of ZN lattice gauge theory, we
a single ZN degree of freedom on each link of the cubic deduce that when N < 5, the ZN X-cube model has a
lattice (with logical operators X, Z), and obtain the direct, first-order transition to a phase where the layers
effective Hamiltonian H t [X +h.c.] where areeffectivelydecoupled. WhenN 5,however,thereis
H is now the 3D Zt(cid:48)oricto−ric corrd(cid:48)e Hrarm(cid:48) iltonian [61]. an intermediate phase which is dua≥l to a Coulomb phase
t(cid:48)oric N
Alternatively, when t J, h so(cid:80)that we have condensed that is known to appear in the phase diagram of (3+1)-
the “ZN composite fl(cid:29)ux loop”, we obtain the effective d ZN lattice gauge theory [40, 41], along the line h =
Hamiltonian 0. We argue, however, that this phase is gapped and
topologically trivial when h > 0. Interestingly, one of
He(cid:48)ff =HX-cubeN −h Zrr(cid:48) +Z†rr(cid:48) (16) tchonettirnaunosuitsioinnsnouumteorfictahlestguadpileesss[4p3h,a4s4e],iswbheilciehvepdretsoenbtes
(cid:104)r(cid:88),r(cid:48)(cid:105)(cid:104) (cid:105) the intriguing possibility that there exists a continuum
where field theory description for the Z X-cube phase.
N
Several features of the phase diagrams in Fig. 8, in-
HX-cubeN =−K [Oc+Oc†]−J [A(rj)+(A(rj))†] cluding the manner in which these phases meet at the
c r,j center of the phase diagram, as well as the generic be-
(cid:88) (cid:88)
(17) havior of the transitions between the Z X-cube and
N
trivial confined phases, are unknown. We conclude by
is the solvable Hamiltonian for the ZN generalization of presenting a solvable projector model in which the tran-
theX-cubemodel. Themicroscopicformoftheoperators sition is believed to be first-order, and by speculating on
and A(j) are explicitly given in Fig. 7. directions for future work.
c r
O
7
1 ZCNonFdleunxsaLtoioonp (J⇤/t⇤) 1 (J2/t2) (J1/t1)
(3+1)-d
(3+1)-d Confined (3+1)-d CoulombPhase Confined
ZZNnTopologicalOrder [Paramagnet] ZZNnTop. Order [Paramagnet]
h h
tanh ? tanh ? ?
J J
 �  �
DecoupledPhase Z X-CubeFracton DecoupledPhase Confined ZN X-Cube
N
(2+1)-dZNTopologicalOrder TopologicalOrder (2+1)-d [CrystallineOrder] Fracton
ZZNnTop. Order Top. Order
0 0
((tt⇤//JJ)⇤⇤) t/J (t1/J1) Dual (t2/J2) t/J
CoulombPhase
(a) N <5 (b) N ≥5
FIG. 8. Schematic Phase Diagram of the Coupled System: The duality derived in Sec. II relates the points (x,y) =
(t/J,0) with (J/t,1) that lie along the black and blue arrows, respectively, in the schematic phase diagrams above. From
knowledge of the phase diagram of (3+1)-d Z lattice gauge theory, we argue that when N <5, there is a direct, first-order
N
transition between the X-cube phase and the decoupled phase. Alternatively, when N ≥ 5, there must be an intermediate
phase along the line h = 0 which is dual to a Coulomb phase that appears in the phase diagram of the (3+1)-d Z lattice
N
gauge system. We argue that this phase is unstable [50] and becomes a gapped, topologically trivial phase when h>0. From
MonteCarlostudiesofthetransitionalongthebluearrow,betweenthedeconfinedphaseofZ gaugetheoryandtheCoulomb
N
phase [43], we believe that the dual transition along the h=0 line, into the fracton topological phase in (b), is continuous.
A. Confinement Transition from the Z spin models when k is sufficiently large. In addition to
N
Topological Phase a low-temperature symmetry-breaking phase and a dis-
ordered phase at high temperatures, there is an inter-
mediate phase with algebraic correlations that resembles
We begin from the top left corner of the phase dia-
the low-temperature behavior of the XY model. The ex-
gram (h J, t), where our system is in a 3D Z topo-
N
(cid:29) istence of this XY-like phase with an emergent global
logical phase. Increasing the strength of the coupling
U(1) symmetry may be motivated by recalling that in
t eventually leads to a condensation of Z flux loops,
N
theclassical2DXYmodel,Z anisotropyisamarginally
and the associated phase transition is captured by the k
irrelevantperturbationwhenk 4[37]. Amorerigorous
finite-temperaturebehaviorofthe4DclassicalZ lattice
N ≥
argument for the existence of this phase, even when the
gauge theory with a “Wilsonian” imaginary time action
anisotropy is infinitely strong, is provided in Ref. [40].
S [θ] = βK cos[(∆ θ) ], where θ is a Z variable
W p × p N
definedonthelinksofthe4Dcubiclattice,while(∆ θ) Asimilarargumentmaybeusedforthepresenceofan
(cid:80) × p
isthelattic(cid:101)ecurlaroundplaquettep,andthesumisover intermediate phase with emergent U(1) gauge symmetry
all plaquettes on the 4D lattice. inthephasediagramof4DZN latticegaugetheoryinthe
large-N limit [40–42]. Ref. [41] studied a Villain form of
When N < 5, classical Monte Carlo studies [43, 44]
the 4D Z gauge theory, which exhibits a discrete ana-
have observed a direct, first-order phase transition be- N
log of the electric-magnetic self-duality of compact U(1)
tween a disordered phase where the magnetic flux loops
gaugetheory[39],andwhoseactionmaybere-writtenas
have condensed at high temperatures, to one where they
that of Maxwell electrodynamics in the presence of both
remainenergeticallycostly,withtheexpectationvalueof
electric and magnetic charges. While the action exhibits
classicalWilsonloopoperatorsdecayingastheexponen-
only a Z gauge symmetry, it was argued that both the
tial of the area of an enclosed region (“area-law”) and as N
electricandmagneticexcitationsarestronglysuppressed
their total length (“perimeter-law”), respectively, in the
near the self-dual point in the large-N limit, so that the
twophases. Inourproblem,thesetwophasescorrespond
effectiveactionisthatofpureMaxwellelectromagnetism.
to a trivial loop condensate and a phase with Z topo-
N
Asaresult,whenN 5,thereisa“Coulombphase”that
logicalorder,respectively,asindicatedinthetopportion
≥
is encountered in the top region of the phase diagram in
of the phase diagram in Fig. 8a.
Fig. 8b, with the Z variable θ effectively behaving as
WhenN islarge(N 5),three phasesareobservedin N
≥ the emergent U(1) gauge field.
Monte Carlo studies of the classical 4D Z lattice gauge
N
theory [42–44]. This rich phase structure may be un- These three phases may be distinguished by the be-
derstood in a variety of ways. To begin, we recall that a havior of the gauge-invariant Wilson loop operator W ,
C
similarphenomenonoccursin2Dclassical,Z -symmetric whichcreatesaclosedtubeofelectricflux,alongwithits
k
8
(electric-magnetic) dual Γ , which is a membrane-like
C(cid:101) �x µx
operator that creates a closed magnetic flux loop along rs rs ⌘z Brs
loop C on the dual lattice. In the deconfined phase of
the ZN gauge theory, a large Wilson loop WC exhibits �xµx s ⌘z s
ipoerr,imw(cid:101)ehtieler-ltahwe obpephoasviitoerbwehhailveioΓrC(cid:101)ochcausrsairneat-hlaewcobnefihnaevd- r r ⌘z
phase. In the intermediate phase, however, both charge
⌘z
and loop excitations are suppressed [41] and neither ex-
citation has condensed. As a result, the Wilson loop
operator W must exhibit perimeter-law behavior [41].
C
Since this phase is self-dual under the electric-magnetic FIG. 9. Bond-Plaquette Duality: We provide a dual rep-
duality transformation, the same behavior holds for Γ . resentation of the Hamiltonian (3) by introducing spins (η)
As argued in Ref. [38], due to the non-trivial statistiCc(cid:101)s on the links of the dual cubic lattice, which measure the
flux through an elementary plaquette in the decoupled lay-
ofZ chargesandfluxloops, perimeter-lawbehaviorfor
N
ers. This dual representation allows us to demonstrate that
both operators can only occur in the presence of a gap-
the transition from the decoupled layers to the Z X-cube
less gauge field, which is consistent with the emergence N
phaseisdualtoaconfinementtransitioninZ gaugetheory.
N
of a Coulomb phase.
It remains unclear whether the transition between the
Coulomb and confined phases is continuous [45, 46] or ourfollowingdiscussion. RecallthattheHamiltonian(3)
weakly first-order [47, 48]. However, the transition be- for the coupled toric codes when h=0 is given by
tween the deconfined phase of the Z gauge theory and
N
the Coulomb phase appears continuous in Monte Carlo H = J A(j)+B(j) t σx µx (18)
studies [42–44]. Finally, as N increases, the region of − r r − rr(cid:48) rr(cid:48)
stability for the Z topological phase shrinks, while the (cid:88)r,j (cid:104) (cid:105) (cid:104)(cid:88)r,r(cid:48)(cid:105)
N
second transition from the Coulomb phase to the con- Werecognizethatthe2DZ chargeoperatorA(j)andthe
2 r
fined phase remains robust [43]. This is consistent with operator =B(xy)B(yz)B(xz)B(yz)B(xz)B(xy),whichis
theexpectationthatthelimit“N + ”,shouldsome- the produOcrt of sirx fluxr operratorsr+arxˆourn+dyˆanr+ezˆlementary
→ ∞
how reproduce the phase diagram of the pure (3+1)-d cube,asinEq. (11),commutewiththeHamiltonian(18)
compact U(1) gauge theory. andwitheachother[ ,A(j)]=[A(j),H]=[ ,H]=0.
Or r(cid:48) r Or
Therefore, we work in a restricted Hilbert space where
B. Transition(s) from the decoupled phase to the A(j) Ψ = Ψ and Ψ = Ψ (19)
r | (cid:105) | (cid:105) Or| (cid:105) | (cid:105)
Z X-cube phase
N
without loss of generality.
We now introduce a dual description of the transition
We now study the bottom half of the phase diagrams
from the decoupled theory (t J) to the X-cube phase
showninFig. 8a&b. Wefirstdemonstratethatthetran- (cid:28)
(J t) that solves the first of these constraints, by in-
sition from the decoupled layers of 2D toric codes to the (cid:29)
troducing spins (η) on the links of the dual cubic lattice.
X-cube fracton phase is dual to the confinement transi-
These spins are to be interpreted as measuring the flux
tionforthe(3+1)-dZ gaugetheory,whichisdrivenby
N through a plaquette in the decoupled toric code layers.
the condensation of flux loops. This result is intuitively
Weimplementthedualrepresentationbyperformingthe
apparent from our “loop-gas” representation of the X-
replacements
cube phase, which suggests that the transition from the
decoupled layers must be driven by the condensation of B(j) ηx (20)
a loop excitation. Physically, our duality transformation r −→ r,j
σx µx ηz (21)
mapstheclosedcompositefluxloopsintheground-state rs rs −→ Brs ≡ r(cid:48),j
of the X-cube phase into the closed electric flux loops r(cid:48),j∈(cid:89)plaqrs
in the ground-state of the Z topological phase. We
N where is a four-spin operator on a plaquette of the
rs
emphasize that this duality relates the bulk spectrum of B
duallatticepiercedbythelink r, s asshowninFig. 9.
twoseeminglydistinctphysicalsystems,andprovidesre- (cid:104) (cid:105)
Thedualitytransformationpreservesthealgebraoflo-
lations between certain local operators on both sides.
cal operators in (18), and naturally solves the constraint
We begin by explicitly implementing our lattice dual-
A(j) Ψ = Ψ . Furthermore, the dual description of the
ity transformation on the Hamiltonian for the coupled r | (cid:105) | (cid:105)
operator is given by
Z2 toric codes (3) when h=0, before discussing its con- Or
sequences. In practice, our transformation is identical to = ηx (22)
the (2+1)-d Wegner duality that maps the transverse- Or −→Ar s,j
field Ising model to the (2+1)-d Z gauge theory [1], in s,j∈(cid:89)starr
2
each layer of our coupled system; as a result, the gener- where is precisely the six-spin Z charge operator
r 2
A
alization of this duality for the Z case is apparent from which acts along a “star” configuration of the η spins at
N
9
eachsiteofthedualcubiclattice. Wethereforefindthat The nature of the transitions between this trivial phase,
the dual Hamiltonian is precisely the X-cube phase and the decoupled layers are unknown
when h>0. When h=0, however, our duality mapping
H = J ηx t (23) suggeststhatthephasetransitionfromtheX-cubephase
dual − r,j − Brr(cid:48)
to the dual Coulomb phase is dual to the transition be-
(cid:88)r,j (cid:104)(cid:88)r,r(cid:48)(cid:105) tween the deconfined phase of (3+1)-d Z gauge theory
N
supplemented by the Z Gauss’s law constraint on the and the ordinary Coulomb phase, which may be contin-
2
dual cubic lattice uous [42, 43], though we are unaware of the continuum
fieldtheorythatgovernsthepropertiesofthistransition.
r Ψ = Ψ (24) We conclude this section by identifying the following
A | (cid:105) | (cid:105)
order parameters
which describes the confinement transition for a 3D Z
2
topological phase driven by the condensation of flux W X X Γ B(j) (26)
Σ ≡ rs rs Σ ≡ p
loops. Asadvertised,thelatticedualityrelatestheZ X-
cube phase to the deconfined phase of (3+1)-d Z2 ga2uge (cid:104)r,(cid:89)s(cid:105)∈Σ (cid:101) (p,(cid:89)j)∈Σ
theory, while the decoupled toric code layers are dual to that distinguish the X-cube phase, the decoupled phase
the trivial, confined phase. and the dual Coulomb phase along the line h=0. Here,
Our duality transformation establishes that the bulk Σisatwo-dimensionalregionwithareaAΣandperimeter
transition from the decoupled toric codes to the X-cube PΣ. The product appearing in the definition of WΣ is
fractonphaseisdualtofluxloopcondensationinthe3D takenalongthebondsperpendiculartotheregionΣ,and
Z gaugetheory,whichisknowntobeadirect,first-order thisoperatormaybethoughtofasinsertingacomposite
2
transition. The natural ZN generalization of this dual- flux loop along the boundary of Σ. Furthermore, ΓΣ
ity transformation leads us to conclude that along the is the product of the plaquette operators that measure
h = 0 line of the phase diagram (i) when N < 5, there the 2D ZN flux through the region Σ. Our notation is
is a direct, first-order transition between the decoupled meant to emphasize that these operators are dual to the
layers and the X-cube phase and (ii) when N 5, there Wilson loop and membrane operators that distinguish
is an intermediate gapless phase for the layere≥d system, the various phases of the ZN gauge theory. From our
which is dual to the Coulomb phase that emerges in the discussion in Sec. IIA, we determine that the operators
phase diagram of the (3+1)-d ZN gauge theory. From WΣ, ΓΣ exhibit the following behavior along the h = 0
our duality transformation, it appears that this gapless line, when the region Σ is sufficiently large:
“dual Coulomb” phase along the h=0 line of the phase • Decoupled Phase: WΣ e−σAΣ, ΓΣ e−cPΣ
diagramischaracterizedbyanemergentU(1)gaugesym- (cid:104) (cid:105)∼ (cid:104) (cid:105)∼
metry, which should be generated by the natural gener-
alization of the Gauss’s law condition Or = +1 for the • Dual Coulomb Phase: (cid:104)WΣ(cid:105), (cid:104)ΓΣ(cid:105)∼e−cPΣ
X-cube fracton phase. The U(1) generalization of this
condition may be read off from Fig. 7; if we consider
an integer-valued “electric field” tensor Eij with vanish- • X-cube Phase: (cid:104)WΣ(cid:105)∼e−cPΣ, (cid:104)ΓΣ(cid:105)∼e−σAΣ
ing diagonal components E =0, and a conjugate gauge
ii
field A , then this Gauss’s law takes the simple form
ij C. Confinement of the X-cube fracton phase
∆ ∆ E =0 (25)
i j ij
The generic behavior of the confinement transition for
with∆i,thelatticederivativealongdirectioni,andsum- the ZN X-cube fracton topological phase – by condens-
mation performed over repeated indices. ing the dimension-1 quasiparticle excitations – are not
The“higher-rank”compactU(1)gaugetheory[28,49] known. This transition occurs in the far right region of
defined by this Gauss’s law condition is equivalent to a the phase diagrams in Fig. 8. Instead of studying this
generalized dimer model studied in Ref. [50], which was transitiondirectly,however,weproposeasolvableHamil-
showntobeinagappedphasewithcrystallineorderdue tonian that interpolates between the Z2 X-cube fracton
to the proliferation of topological defects in the gauge topologicalorderandaconfinedphaseinthissectionand
fieldconfigurations(i.e. “monopole”events). Duetothis determine that the transition is first-order within this
result, we believe that the dual Coulomb phase is unsta- model. It is unknown whether this captures the generic
ble,andgiveswaytoagapped,topologicallytrivialphase behaviorofthistransition,outsideofthesolvablemodel.
with conventional long-range order when h > 0. The We construct a solvable projector model by placing
h=0linethencorrespondstothefine-tunedlimitwhere spins(τ)onthelinksofathree-dimensionalcubiclattice.
monopole events in the gauge field are suppressed. This Now, however, we consider the Hamiltonian
conclusion is consistent with the fact that the perturba-
tion σz µz corresponds to a highly non-local operator
intermrrs(cid:48)ofrrth(cid:48)edualη spins,andwithourexpectationfor H=−J A(rj)+J  e−hτrzs/J (27)
thebehaviorofthephasediagramintheN + limit. (cid:88)r,j (cid:88)r,j s∈pl(cid:89)anej(r)
→ ∞  
10
supplemented by the constraint It is convenient to re-cast the ground-state wavefunc-
tion in an alternate form. Observe that configuration of
Ψ = Ψ . (28) spins appearing in the state ψ may equivalently
Or| (cid:105) | (cid:105) | fracton(cid:105)
be specified by the action of the A(j) operators on the
r
where A(j) = τx is the four-spin operator referencestate τz =+1 . Asaresult,theground-state
in the Xr-cube mso∈dpellantehj(art) isrsoriented in the jth plane, wavefunction Ψ| gsss(cid:48) may (cid:105)be re-written in terms of dual
(cid:81) | (cid:105)
andmeasuresthepresenceofadimension-1quasiparticle Ising variables µ, σ at the sites of a three-dimensional
excitation at site r. As before, the operator is the cubic lattice, which label the presence or absence of a
r
12-spin τz operator as defined in Eq. (11) thatOmeasures particular operator A(j) acting on the reference config-
r
the fracton “charge” at a cube = τz . uration. In this dual representation, the ground-state
We observe that the HamiltoniOanr exsh,irb(cid:48)i∈tcsubbeo(rt)h rt(cid:48)hse takes the form
H (cid:81)
Z X-cube fracton phase (h J) as well as a trivial
co2nfined phase (h J). Fu(cid:28)rthermore, when h J, Ψgs = 1 e−βH/2 σ , µ (32)
(cid:29) (cid:28) | (cid:105) √ |{ } { }(cid:105)
the Hamiltonian reduces to the effective Hamiltonian
H Z (cid:88){σ}(cid:88){µ}
considered previously for the confinement transition of
the X-cube phase where the classical Hamiltonian βH is given by
h
Heff =−J A(rj)−2h τrzr(cid:48) +O(h2/J) (29) βH = J [σrσr+xˆ+µrµr+yˆ+σrµrσr+zˆµr+zˆ] (33)
(cid:88)r,j (cid:104)r(cid:88),r(cid:48)(cid:105) (cid:88)r
eraTthoresfuΠlrl,Hj a≡m−iltAon(rji)a+nHs(∈27pl)aniseja(rs)uem−hoτfrxsp/rJojaencdtiotnheorpe-- athniPdsrZcellaiimssstiinchaaelrpymaMrotdoitenilot.enCfuanrlcotisotnudZie=so(cid:80)fth{σe}c(cid:80)las{sµi}cael−HβHamfoilr-
fore has a positive semi-definite spectrum. As a result,
(cid:81) tonian βH reveal a first-order phase transition when
a ground-state of the Hamiltonian is found by explic-
h/J 1.13 [51, 52], which implies the presence of a
itly constructing a zero-energy wavefunction. Projec- ≈
first-order phase transition in our model. For exam-
tor Hamiltonians similar to (27) have been extensively
ple, the expectation value of the magnetization M =
studied by considering generalizations of the Rokhsar-
τz ispreciselytheenergyoftheclassicalsystem
Kivelson (RK) point in the two-dimensional quantum (cid:104)r,r(cid:48)(cid:105) rr(cid:48)
E = Ψ M Ψ , which exhibits a discontinuity at the
dimer model [32–36], where the ground-state may be (cid:80) (cid:104) | | (cid:105)
putative phase transition point [51, 52]. Classical corre-
written as an equal-amplitude superposition of dimer
lationfunctionsthatarenotinvariantundertheZ trans-
configurations. For these generalized RK models, the 2
formationσ σ,µ µalonganyplaneofthecubic
Hamiltonian is related to the Markovian transition ma-
→− →−
lattice must vanish, as this is a symmetry of the classi-
trixforaclassicalsystemwhichsatisfiesdetailedbalance
cal Hamiltonian (33). To our knowledge, however, the
[34]. Spatial correlation functions of certain “diagonal”
behavior of higher-point correlation functions near the
operators in the ground-state of the quantum model are
classical phase transition have not yet been studied. An
identicaltoclassicalcorrelationfunctionsinequilibrium.
intriguing possibility is that the dynamical behavior of
Let ψ be a ground-state of the Hamiltonian
fracton
| (cid:105) H thisclassicalsystemis“glassy”inacertainrangeofcou-
when h=0, i.e. one of the ground-states of the solvable
plings h/J [51, 52], which would imply in the quantum-
X-cube fracton Hamiltonian. We may write ψ
fracton
| (cid:105) mechanical problem that the Hamiltonian has a gap-
explicitly as H
less spectrum, and that certain correlation functions ex-
1+A(j) hibit power-law decay in time [34, 35], while remaining
ψ = r τz =+1 (30) spatially short-ranged. We leave an exploration of this
| fracton(cid:105) (cid:32) 2 (cid:33)| ss(cid:48) (cid:105) exotic possibility to future work.
r,j
(cid:89)
where τz =+1 is the state with all spins polarized in
the z-d|irsesc(cid:48)tion. T(cid:105)he ground-state of is then given by ACKNOWLEDGMENTS
H
Ψgs ehτrzs/2J ψfracton (31) We thank Liang Fu for helpful discussions and com-
| (cid:105)∼ | (cid:105)
ments. This work is supported in part by DoE Office of
(cid:104)(cid:89)r,s(cid:105)
BasicEnergySciences,DivisionofMaterialsSciencesand
It is straightforward to check that Ψ is indeed a zero- Engineering under Award de-sc0010526, and by the Na-
gs
| (cid:105)
energyeigenstateoftheHamiltonian, asitisannihilated tionalScienceFoundationunderGrantNo. NSFPHY11-
by all of the projectors Π . 25915.
r,j