Wire constructions of Abelian topological phases in three or more dimensions

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by  Thomas Iadecola

Wire constructions of Abelian topological phases in three or more dimensions
Thomas Iadecola,1 Titus Neupert,2 Claudio Chamon,1 and Christopher Mudry3
1Physics Department, Boston University, Boston, Massachusetts 02215, USA
2Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA
3Condensed Matter Theory Group, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland
(Dated: May 11, 2016)
Coupled-wireconstructionshaveproventobeusefultoolstocharacterizeAbelianandnon-Abelian
topological states of matter in two spatial dimensions. In many cases, their success has been com-
6 plemented by the vast arsenal of other theoretical tools available to study such systems. In three
1 dimensions, however, much less is known about topological phases. Since the theoretical arse-
0 nal in this case is smaller, it stands to reason that wire constructions, which are based on one-
2 dimensional physics, could play a useful role in developing a greater microscopic understanding of
three-dimensionaltopologicalphases. Inthispaper,weprovideacomprehensivestrategy,basedon
y
the geometric arrangement of commuting projectors in the toric code, to generate and character-
a
M ize coupled-wire realizations of strongly-interacting three-dimensional topological phases. We show
how this method can be used to construct pointlike and linelike excitations, and to determine the
topological degeneracy. We also point out how, with minor modifications, the machinery already
0
developed in two dimensions can be naturally applied to study the surface states of these systems,
1
a fact that has implications for the study of surface topological order. Finally, we show that the

strategydevelopedfortheconstructionofthree-dimensionaltopologicalphasesgeneralizesreadilyto
]
l arbitrarydimensions,vastlyexpandingtheexistinglandscapeofcoupled-wiretheories. Throughout
e
the paper, we discuss Z topological order in three and four dimensions as a concrete example of
- m
r this approach, but the approach itself is not limited to this type of topological order.
t
s
.
t
a CONTENTS IV. Conclusion 32
m
- I. Introduction 1 Acknowledgments 33
d
n
A. Deconfinement of defects along the direction of
o II. Three-dimensional wire constructions 3
a wire 33
c A. Decoupled wires 3
[
B. Interwire couplings and criteria for B. Discrete gauge symmetry and ground state in
2 producing gapped states of matter 5
the limit of vanishing kinetic energy 34
v C. Fractionalization 7
1 1. Change of basis 7
References 37
8
2. Compactification, vertex operators, and
9
fractional charges 8
0
0 3. Pointlike and linelike excitations 9 I. INTRODUCTION
. 4. Energetics of pointlike and linelike
1
defects 12
0 The experimental discovery of the integer and frac-
6 5. Statistics of pointlike and linelike defects 14
tional quantum Hall effects excited enormous interest
1 6. Topological ground-state degeneracy 15
in the study of topological states of matter in two di-
v: 7. Topological field theory 18 mensional space. Strongly interacting states of matter
Xi D. Example: Zm topological order in distinguished by the presence of excitations with frac-
three-dimensional space from coupled wires 19
tional quantum numbers or nontrivial boundary modes
ar 1. Definitions and interwire couplings 19 haveattractedparticularattentionfromtheorists. Over
2. Excitations 20 time, a vast arsenal of theoretical tools has been devel-
3. Ground state degeneracy on the oped to study such systems, from the microscopic (e.g.,
three-torus 22 numericaltechniquestostudylatticemodelswithtopo-
4. Surface states 22 logicallyorderedgroundstates)tothemacroscopic(e.g.,
topological quantum field theories).
III. Higher-dimensional wire constructions 25 Wire constructions, which were first undertaken for
A. Hypercubic arrays of quantum wires 25 the integer [1–3], and later the fractional [4–15], quan-
B. Generalizing the results of Sec. IIB 27 tum Hall effect, are conveniently poised midway be-
C. Example: Z topological order in tween these two extremes. The approach in this case
m
four-dimensional space from coupled wires 28 is to model a topological phase by starting from an
2
anisotropic theory of decoupled gapless quantum wires, alization of reversal of time, which acts on-site. It is
and then introducing local couplings between the wires possiblethatafullythree-dimensionalwireconstruction
to produce a gapped state of matter with an isotropic could remedy this peculiarity, although such a descrip-
low-energydescription. Thisapproachhasthevirtueof tion is still lacking.
yielding the edge theory, itself that of a Luttinger liq-
Layer constructions, in which planes of two-
uid, directly, and of providing means to construct the
dimensional topological liquids are stacked on top of
low-lying quasiparticle excitations of the bulk quantum
one another and coupled, were used to construct the
liquid. Furthermore, because wire constructions make single surface Dirac cone of the three-dimensional Z
2
use of well-understood techniques in one-dimensional
TI[49]andtostudysurfacetopologicalorder[50]. Wire
physics, such as Abelian (or non-Abelian) bosoniza-
constructions of three-dimensional topological states of
tion, one can construct analytically tractable theories
matter have also recently been undertaken, yielding
of states of matter that might not otherwise admit a
Weyl semimetals [51, 52] and a class of fractional topo-
controlled analytical description.
logical insulators [53]. However, in all three cases, dif-
Inrecentyears,wireconstructionshavealsobeenused ferent methods are used to develop the wire construc-
to study fractional topological insulators (FTIs) [10, tions themselves, and little effort has been made to ex-
11, 14] and spin liquids [16, 17], and also to develop tendtheseconstructionsbeyondthespecificproblemat
an extension [11] of the ten-fold way for noninteracting hand in each example. In order to attack the most dis-
fermions [18–21] to strongly-correlated systems. tinctive aspects of topological states of matter in three
dimensions,suchassurfacetopologicalorder,itisthere-
Since the prediction [22] and discovery [23–25] of
three-dimensional Z topological insulators (TIs), there fore necessary to develop a framework that lends itself
2
readily to a variety of approaches with minimal modifi-
has been a growing interest in understanding topo-
cations.
logical states of matter in three spatial dimensions.
In addition to generalizing these time-reversal invari- In this paper, we provide a comprehensive strat-
ant Z topological insulator to the strongly-interacting egy to design wire constructions of strongly-interacting
2
regime [26–28], there has been an effort to derive ef- Abeliantopologicalstatesofmatterinthreedimensions.
fective field theories describing the bulk of such TIs, The strategy that we present is to start with decoupled
and to determine the bulk-boundary correspondence quantumwiresplacedonthelinksofatwo-dimensional
in such theories that yields the hallmark single Dirac square lattice, and then to couple the wires with many-
cone on the two-dimensional surface [29–33]. Further bodyinteractionsassociatedwitheachstarandplaque-
work has undertaken efforts to understand broader fea- tteofthelattice. Inthisway,eachinteractiontermthat
tures of three-dimensional topological states of matter, couples neighboring wires can be viewed as correspond-
such as the statistics of pointlike and linelike excita- ing to one of the commuting projectors that enters Ki-
tions [34, 35]. For example, it has been shown that taev’storiccodeHamiltonian[54]. Thiscorrespondence
certain three-dimensional topological phases can only simplifiestheapplicationofacriterion,firstproposedby
be distinguished by the mutual statistics among three Haldane, to ensure that these interaction terms do not
linelike excitations [35]. compete, and are sufficient in number to gap out all
gapless modes in the array of quantum wires when pe-
Another major direction of work concerns three-
riodic boundary conditions are imposed along all three
dimensionalsystemswhosesurfacesarethemselvestwo-
spatial directions.
dimensional topological states of matter. The simplest
example of this phenomenon occurs on the surface of When all interaction terms satisfy this criterion, the
a Z TI when time-reversal symmetry is locally bro- Hamiltonian is frustration-free, and taking the strong-
2
ken by a magnetic field on the surface, in which case coupling limit produces a gapped three-dimensional
a half-integer surface quantum Hall effect develops [36– state of matter. With this done, one can proceed to
39]. Further theoretical work has shown that generic characterize this state of matter in terms of its point-
three-dimensional topological phases, including but not like and linelike excitations, as well as their statistics,
limited to the fermionic Z TI, can exhibit more ex- and calculate the topological degeneracy, if any, of the
2
otic surface topological phases that cannot exist with ground-state manifold. The class of three-dimensional
the same realization of symmetries for local Hamilto- modelsstudiedinthisworkfeaturesatopologicaldegen-
nians in purely two-dimensional space. This family eracygivenby det κ 3,wheretheinteger-valuedmatrix
of surface phenomena is known as surface topological κ contains info|rmati|on about the mutual statistics of
order [40–47]. Several recent works [47, 48] have ap- pointlike and linelike excitations in the theory. This
proached the question of surface topological order by is in close analogy with the K-matrix formalism de-
applyingthequasi-one-dimensionalphysicsofwirecon- veloped for two-dimensional topological states of mat-
structions,althoughitappearsthatthisapproachneces- ter [55]. When periodic boundary conditions are re-
sitates the use of an unusual “antiferromagnetic” time- laxed by the presence of two-dimensional terminating
reversal symmetry rather than the usual (physical) re- surfaces, we further show that gapless surface states
3
result. One can apply the coupled-wire techniques al- propriate interwire couplings and their compatibility in
ready developed in two dimensions to study the various the strong-coupling limit. Finally, in Sec. IIIC, we pro-
gapped surface states that can be produced by intro- vide an example of this generalization by constructing
ducinginterwirehoppingsorinteractionsonthesurface, Z -topologically-ordered states of matter in four di-
m
provided that the added terms are compatible with the mensions, andconstructingtheirpointlike, linelike, and
interactions in the bulk. membranelike excitations, before concluding in Sec. IV.
In addition, we show that the above strategy for con-
structing three-dimensional Abelian topological states
of matter can be readily extended to arbitrary di- II. THREE-DIMENSIONAL WIRE
mensions, vastly expanding the existing scope of the CONSTRUCTIONS
coupled-wire approach. Indeed, much as it is possi-
ble to define higher-dimensional versions of the toric In this section, a method to construct arrays of cou-
code on hypercubic lattices (see, e.g., Ref. [56]), one pled wires realizing topological phases of matter in
can arrange a set of decoupled quantum wires on a d- three-dimensional space is presented. We begin by
dimensional hypercubic lattice and couple them with defining a class of gapless theories describing decoupled
interactions defined on stars and plaquettes of this lat- wires,beforemovingontoadiscussionofinterwirecou-
tice. Applying Haldane’s compatibility criterion, one plings. Inparticular,weprovideasetofalgebraiccrite-
can show that these interactions produce a gapped ria that are sufficient to determine whether the theory
(d+1)-dimensional state of matter, whose excitations is gapped when periodic boundary conditions are im-
and topological properties can be investigated much as posed.
in the three-dimensional case.
The structure of this paper is as follows. In Sec. II,
we develop in detail the strategy discussed above for A. Decoupled wires
constructing three-dimensional topological phases from
coupled wires. In Sec. IIA, we establish the basic no- We consider a two-dimensional array of 2N quantum
tation used to describe the array of decoupled quantum wires, labeled by Latin indices j =1,...,2N, placed on
wires. In Sec. IIB, we present Haldane’s compatibility the links of a two-dimensional square lattice embedded
criterionandaclassofmany-bodyinteractionsbetween in three-dimensional Euclidean space. Each quantum
wiresthatsatisfyit. (Thisclassismainlychosenforan- wire is assumed to be gapless and nonchiral, and there-
alytical expedience, and is not the only class of interac- fore to contain 2M gapless degrees of freedom, labeled
tionsthatcanbeconstructedaccordingtoourstrategy.) by Greek indices α=1,...,2M. We take the wires (of
In Sec. IIC, we show how to use the interacting arrays length L) to lie along the z-direction, and the square
ofquantumwiresdefinedinSecs.IIAandIIBtostudy lattice to lie in the x-y plane. We will impose periodic
states of matter with fractionalized excitations. In par- boundary conditions in all directions (x, y, and z) until
ticular, we show how to construct pointlike and linelike further notice. The set of decoupled quantum wires is
excitations,anddeterminetheirstatistics,aswellasthe described by the quadratic Lagrangian
topological ground state degeneracy. Next, in Sec. IID
we exemplify our strategy with perhaps the simplest L
1 T T
type of topological order in three dimensions, namely Lˆ = dz ∂ Φˆ ∂ Φˆ ∂ Φˆ ∂ Φˆ
Zsumrfatcoeposltoagtiecsalofortdheers.eFZurt-htoeprmoloorgei,cawlley-ionrvdeesrteigdatsetatthees 0 4π (cid:90)0 (cid:34)(cid:16) t (cid:17) K(cid:16) z (cid:17)−(cid:16) z (cid:17) V(cid:16) z (cid:17)(cid:35)
m (2.1a)
of matter, and find that they are unstable to interwire
hoppings. Additionally, a surface fractional quantum
where
HalleffectwithHallconductivity[(2e)2/h] (1/2m)can
×
develop at the expense of breaking time-reversal sym- Φˆ(t,z)..= φˆ (t,z) ... φˆ (t,z)
metry on the surface. We also discuss how these obser- 1,1 1,2M |
(2.1b)
vations regarding surface states can be extended to the (cid:16) T
φˆ (t,z) ... φˆ (t,z)
more general class of interwire interactions introduced ···| 2N,1 2N,2M
in Sec. IIB. (cid:17)
is a vector that collects the 2M scalar fields φˆ (t,z)
Next, in Sec. III, we outline the generalization of our j,α
defined in each of the j =1,...,2N wires. We use ver-
results to arbitrary dimensions. In Sec. IIIA, we dis-
tical bars as a visual aid to separate degrees of freedom
cuss how to define d-dimensional hypercubic arrays of
defined in different wires. The block-diagonal 4MN-
quantum wires that are analogous to the square array
dimensional matrix
of quantum wires used to construct three-dimensional
topological states. Then, in Sec. IIIB, we generalize
..= 1 K, (2.1c)
the results of Sec. IIB regarding the definitions of ap- K 2N ⊗
4
where 1 is the unit matrix of dimension 2N and K as we do here, further requires the use of Klein factors,
2N
is a 2M 2M symmetric matrix with integer entries, which are needed in order to assure that fermionic ver-
×
yields the equal-time commutation relations tex operators (defined below) defined in different wires
anticommutewithoneanother. TheseKleinfactorscan
∂ φˆ (z),φˆ (z(cid:48)) =i2πδ K−1 δ(z z(cid:48)). (2.1d) besubsumedintotheequal-timecommutationrelations
z j,α j(cid:48),α(cid:48) jj(cid:48) αα(cid:48) − for the scalar fields φˆ and φˆ . This can be done
(cid:104) (cid:105) j,α j,α
R L
We will omit the explicit time dependence of the fields by integrating both sides of Eqs. (2.2a) over all z and
from now on. Finally, the block-diagonal 4MN 4MN fixing the arbitrary constant of integration to be the
×
matrix Klein factor necessary to ensure the appropriate anti-
commutation of vertex operators. We refer the reader
..= 1 V, (2.1e)
V 2N ⊗ to the Appendix of Ref. [57] for more details on this
procedure.
where the 2M 2M matrix V is real, symmetric, and
× For bosons, each wire j = 1,...,2N instead con-
positive-definite. The matrix V is set by microscop-
tainsM flavorsofnonchiral scalarfieldsφˆ andφˆ ,
ics within each wire, and will usually be taken to be j,α j,α
1 2
a diagonal matrix in this work. However, the matrix where α1,2 = 1,...,M label “charge” and “spin” de-
K, which enters the commutation relations (2.1d), con- grees of freedom, respectively. These fields obey the
tains crucial data that define the fundamental degrees equal-time commutation relations
of freedom in a wire. The final data necessary to com-
plete the definition of the theory describing the two- ∂zφˆj,α (z),φˆj(cid:48),α(cid:48)(z(cid:48)) =0,
1 1
dimensional array of decoupled quantum wires is the (cid:104) (cid:105)
∂ φˆ (z),φˆ (z(cid:48)) =0, (2.3a)
4MN-dimensional “charge-vector” z j,α j(cid:48),α(cid:48)
2 2
(cid:104) (cid:105)
Q..= Q | Q | ... | Q T. (2.1f) ∂zφˆj,α1(z),φˆj(cid:48),α2(z(cid:48)) =i2πδjj(cid:48)δα1α2δ(z−z(cid:48)),
(cid:104) (cid:105)
The 2M-dimension(cid:0)al integer vector Q (cid:1)collects the U(1) so that the K-matrix for bosons is
electric charges associated with the scalar fields φˆ ,
j,α M
0 1
α=1,...,2M. K ..= . (2.3b)
b 1 0
The theory defined by Eqs. (2.1) can be viewed as
α=1 (cid:18) (cid:19)
aneffectivelow-energydescriptionofatwo-dimensional (cid:77)
array of decoupled physical quantum wires containing We take the bosonic charge vector to be
fermionic or bosonic degrees of freedom.
For fermions, each wire j = 1,...,2N contains M Qb..= 2 1 0 ... 1 0 T, (2.3c)
flavors of chiral scalar fields φˆ and φˆ , where
j,αR j,αL in units where the elec(cid:0)tron charge e(cid:1)is set to unity, so
α = 1,...,M label right- and left-moving degrees
R,L that the fields φˆ carry a U(1) electric charge, while
of freedom, respectively. These fields obey the chiral j,α1
equal-time commutation relations φˆ is neutral. (Of course, one could define a “spin
j,α
2
vector” analogous to Q that encodes the coupling to
∂ φˆ (z),φˆ (z(cid:48)) =+i2πδ δ δ(z z(cid:48)), anotherU(1)gaugefieldforspin,but,forsimplicity,we
z j,αR j(cid:48),α(cid:48)R jj(cid:48) αRα(cid:48)R − will work exclusively with electric charges here.)
(cid:104) (cid:105)
∂ φˆ (z),φˆ (z(cid:48)) = i2πδ δ δ(z z(cid:48)), Thefundamentalexcitationsofafermionicorbosonic
z j,αL j(cid:48),α(cid:48)L − jj(cid:48) αLα(cid:48)L − wire can be built out of the vertex operators
(cid:104) (cid:105)
∂ φˆ (z),φˆ (z(cid:48)) =0, (2.2a)
z j,α j(cid:48),α
(cid:104) R L (cid:105) ψˆf†,b;j,α(z)..= exp −i(Kf,b)αα(cid:48)φˆj,α(cid:48)(z) , (2.4)
and therefore, for fermions, the 2M 2M matrix K
× (cid:16) (cid:17)
entering Eq. (2.1d) is given by for any j = 1,...,2N and α = 1,...,2M, where we
have adopted the convention of summing over repeated
M
+1 0 indices. Any local operator acting within a single wire
K ..= . (2.2b)
f 0 1 can be built from these vertex operators. Similarly, op-
α(cid:77)=1 (cid:18) − (cid:19) erators spanning multiple wires can be built by taking
We further adopt the conventionthat the charge-vector productsofvertexoperatorsfromeachconstituentwire.
The charges of the excitations created by these vertex
Q ..= 1 ... 1 T, (2.2c) operators are measured by the charge operator
f
inunitswheretheelectro(cid:0)ncharge(cid:1)eissettounity,fora L
Q
fermionic wire with 2M channels. Treating an array of Qˆ ..= α δ dz ∂ φˆ (z), (2.5)
j,α 2π αα(cid:48) z j,α(cid:48)
fermionic quantum wires within Abelian bosonization, (cid:90)
0
5
for any j = 1,...,2N and α = 1,...,2M, where L is assume that either charge or number-parity conserva-
the length of a wire. The normalization of the charge tion holds. The former is imposed by demanding that
operator is taken to be such that
[Qˆ ,ψˆ† (z)]=Q δ δ ψˆ† (z) (2.6) T =0 , (2.10a)
j,α f,b;j(cid:48),α(cid:48) α jj(cid:48) αα(cid:48) f,b;j(cid:48),α(cid:48) Q T ∀ T
while the latter is imposed by relaxing the above re-
at equal times, indicating that the vertex operator
ψˆ† carries the charge Q . quirement to
f,b;j,α α
T =0 mod 2 . (2.10b)
Q T ∀ T
B. Interwire couplings and criteria for producing For a detailed discussion of how further symmetry
gapped states of matter requirements constrain the tunneling vectors , see
T
Ref. [11].
Given the two-dimensional array of decoupled and We are now prepared to discuss the strategy we em-
gaplessquantumwiresdefinedinSec.IIA,wewouldlike ploytoproducegappedstatesofmatterfromtheabove
to devise a systematic way of introducing strong single- construction. We first recall that the array of decou-
particleormany-bodycouplingsbetweenadjacentwires pled quantum wires consists of 4MN gapless degrees
in order to yield a variety of gapped topologically- offreedom. AsnotedinRef.[58],andlateremployedin
nontrivial three-dimensional phases of matter. Our Refs.[11,57],asinglecosineterminthesuminEq.(2.7)
strategy will be to extend the approach taken in is capable of removing (i.e., gapping out) at most two
Ref. [11], which considered one-dimensional chains of of these gapless degrees of freedom from the low-energy
wires, to two dimensions. We begin by adding to the sector of the theory. This occurs in the limit UT →∞,
quadratic Lagrangian Lˆ defined in Eq. (2.1) a set of where the argument of the cosine term becomes pinned
0
cosine potentials toitsclassicalminimum. Therefore,inprincipleittakes
only2MN cosinetermstogapoutall4MN degreesof
L freedominthebulkofthearrayofquantumwireswhen
Lˆ := dz U (z) cos T Φˆ(t,z)+α (z) . periodic boundary conditions are imposed. Matters are
{T} T T K T complicated somewhat by the nontrivial commutation
(cid:90)0 (cid:88)T (cid:16) (cid:17) relations (2.1d), which ensure that cosine terms corre-
(2.7)
sponding to distinct tunneling vectors and (cid:48) do not
T T
commute in general. Consequently, it is possible that
Here, the 4MN-dimensional integer vectors encode
T quantum fluctuations may lead to competition between
tunneling processes between adjacent wires. This inter-
the various cosine terms that frustrates the optimiza-
pretation becomes transparent upon recognizing that,
tion problem of simultaneously minimizing all of these
up to an overall phase,
terms. However, in Ref. [58], Haldane observed that if
the criterion
2N 2M
T
e−iTTKΦˆ(z) = ψˆ† (z) j,α, (2.8)
f,b;j,α T (cid:48) =0 (2.11)
j(cid:89)=1α(cid:89)=1(cid:104) (cid:105) T KT
holds, then the cosine terms associated with the tun-
where ψˆf†,b;j,α are the vertex operators defined in neling vectors and (cid:48) can be minimized indepen-
Eq. (2.4). [We follow Ref. [11] in using the shorthand dently, and therTefore doTnot compete with one another.
notation (ψˆ† )−1 ψˆ and in employing an ap- [Note that each tunneling vector must also satisfy
propriate pof,ibn;jt,-αsplitt≡ingfp,br;ejs,αcription when multiplying Eq. (2.11), i.e., we require that TT = 0 for all .]
fermionic operators.] For generic tunneling vectors , Therefore, if one can find a “HaTldaKneTset” H of 2MTN
T
Eq.(2.8)describesamany-bodyorcorrelatedtunneling linearly-independenttunnelingvectors,allofwhichsat-
that amounts to an interaction term in the Lagrangian isfy Eq. (2.11), then it is possible to gap out all degrees
offreedominthearrayofquantumwiresbyaddingsuf-
Lˆ..= Lˆ +Lˆ . (2.9) ficiently strong interactions of the form (2.7). If such
0 {T}
a set H is found, then it suffices to restrict the sum in
The real-valued functions U (z) 0 and α (z) in Eq. (2.7) to H, and to posit that all couplings U
T ≥ T T ∈ T
Eq. (2.7) encode the effects of disorder on the ampli- are sufficiently large in magnitude to gap out all modes
tude and phase of these interwire couplings. in the array of quantum wires.
Distinctstatesofmattercanberealizedbyrestricting Wenowpresentasimplegeometricprescriptiontoaid
the sum over tunneling vectors in Eq. (2.7) to ensure in the determination of the existence (or lack thereof)
that the interaction terms (2.8)Tsatisfy certain symme- ofaHaldanesetHforatwo-dimensionalarrayofquan-
tries. For all examples considered in this work, we will tum wires with some set of desired symmetries. This
6
N (a)
(j)
v
2
W E
(j)
(j) s (j)
v T v
1 1
S �
p
(j)
v
2
�
s
(b)
(j)
w
2
FIG. 1. A single unit cell of the square array of wires,
consisting of a single star s and plaquette p. The dashed (j)
(j) (j)
nmeiadrpeostin-nteoigfbeoarchlinliknskbheolsotnsgatqounaenitguhmbowriinrge,urnepitrecseelnlst.edTbhye w1 Tp �w1
thesymbol×,alignedalongthez-direction(outofthepage).
Anyplaquettepissurroundedbyfourquantumwireslocated
at the four cardinal points p , p , p , and p , repectively.
N W S E
Similarly, any star s is surrounded by four quantum wires
(j)
located at the four cardinal points s , s , s , and s . w
N W S E 2
�
FIG. 2. Pictorial representation of the tunneling vectors
prescription capitalizes on the fact that we have chosen (2.13). The 2M-dimensional integer-valued vectors v1(j,2) and
all2N quantumwirestolieonthelinksofasquarelat- w(j) determine the linear combinations of bosonic fields in
1,2
tice. (Inprinciple,thisisnottheonlypossiblechoiceof each wire that enter the cosine term associated with each
lattice geometry, but it provides a simple way of count- star or plaquette, respectively.
ing degrees of freedom in any dimension, as we will see
belowandinSec.III.)Onasquarelatticewith2N sites,
there are N “stars” (centered on the vertices of the lat-
tice) and N “plaquettes” (centered on the vertices of (j)T (j(cid:48)) =0 s,p,j,j(cid:48). (2.12c)
Ts K Tp ∀
the dual lattice), assuming that periodic boundary con-
ditionsareimposedasinFig.1. Ifweassociatethetun-
Iftheaboveequationsaresatisfied,thenthesetof2MN
neling vectors and with each star s and plaquette
Ts Tp tunnelingvectorsisaHaldaneset,andthereforecapable
p, respectively, then we have a set of 2N tunneling vec-
of yielding a gapped phase in the strong-coupling limit.
tors. Since there are 4MN gapless degrees of freedom
We now turn to the problem of building 2MN tun-
inthearrayofdecoupledquantumwires, wecanobtain
the necessary number 2MN of tunneling vectors by ex- neling vectors s(j) and p(j). Enumerating all solutions
T T
panding this set to include M “flavors” of tunneling tothisproblemforallmatrices isbeyondthescopeof
K
vectors (j) and (j) for each star and plaquette, re- the present work. However, we will present below one
s p
T T way of constructing these tunneling vectors that builds
spectively. Welabeltheseflavorsusingateletypeindex
intheminimalsymmetriesofchargeand/orparitycon-
j = 1,...,M. Imposing the Haldane criterion (2.11)
servation [Eqs. (2.10)] and greatly reduces the number
on this set of tunneling vectors then yields the set of
ofequationsthatmustbesolved[relativetoEqs.(2.12),
equations
which contain an infinite number of linear equations in
Ts(j)T K Ts((cid:48)j(cid:48)) =0 ∀ s,s(cid:48),j,j(cid:48), (2.12a) tmhaettiohnermisopdryonvaidmedic].limInitpNart→icu∞lari,fnifowaedddietisoirnealchinafrogre-
(j)T (j(cid:48)) =0 p,p(cid:48),j,j(cid:48), (2.12b) conservation [Eq. (2.10a)] to hold, we may define the
Tp K Tp(cid:48) ∀
7
tunneling vectors by their nonvanishing components matrix K may in principle describe a distinct gapped
phase of matter.
(Ts(j))j,α..= v1(j,α) (cid:16)+δj,vsE2(j,α)−δδj,js,sWN(cid:17)−δj,sS , (2.13a) a5MnOd(bjMs↔e+rvj1e(cid:48)).t/h2TalhtineEreeqafsro.re(eq2,u.t1ah5tei)osenarscerinistye8mrMiam2aevmtaroircuianubtnledtsoe.raT1she↔itsoi2sf
( (j)) ..= w(j) δ (cid:16)δ (cid:17) important for two reasons. First, the number of equa-
Tp j,α 1,α j,pW − j,pE (2.13b) tions does not scale with the number 2N of quantum
(cid:16)+w(j) δ (cid:17) δ , wiresinthearray. Thisensuresthatasinglesolutionto
2,α j,pN − j,pS theseequationsholdsforanysystemsizewhenperiodic
(cid:16) (cid:17) boundary conditions are imposed. Second, this set of
where we recall that j = 1,...,2N labels the quantum
equations is underconstrained for any M (i.e., there are
wires and α = 1,...,2M labels the degrees of freedom
alwaysmorevariablesthanequations). Thismeansthat
within a wire. Here, v(j), v(j), w(j), and w(j) are arbi-
1 2 1 2 for generic matrices K of fixed dimension 2M, there is
trary 2M-dimensional integer vectors. The Kronecker
in principle more than one solution to Eqs. (2.15).
deltasinthetunnelingvector ensurethatitsnonzero
Ts We aim to construct gapped states of matter that
entriesaredefinedwithinthequantumwiress ,...,s
N W haveanisotropiclow-energydescription. Consequently,
to the north, ..., west of the vertex on which star s is
it is natural to demand that the tunneling vectors de-
centered. The Kronecker deltas in select the quan-
Tp fined in Eqs. (2.13) and depicted in Fig. 2 are indepen-
tum wires p ,...,p , which are defined similarly for
N W dentofdirection. Thiscanbeachievedbyimposingthe
theplaquettep(seeFig.1). Withthesedefinitions, one
additional constraints
verifiesthatEq.(2.10a)holdsindependentlyoftheform
of v1(j,2), w1(j,2), and the charge-vector Q for a single wire. v1(j) =v2(j) =..v(j) (2.16a)
Similarly, when we wish to impose number-parity
conservation [Eq. (2.10b)], we may define for any j = and
1,...,2N and any α=1,...,2M
w(j) =w(j) =..w(j). (2.16b)
( (j)) ..= v(j) δ +δ 1 2
Ts j,α 1,α j,sE j,sW (2.14a) Note that Eq. (2.15c) is solved independently of the
(cid:16)+v2(j,α) δj,sN(cid:17)+δj,sS , form of the 2M-dimensional vectors vµ(j) and wµ(j) if
( (j)) ..= w(j) δ +(cid:16)δ (cid:17) Eqs. (2.16) hold. These constraints reduce the total
Tp j,α 1,α j,pE j,pW (2.14b) number of variables contained in the tunneling vectors
(cid:16)+w(j) δ (cid:17)+δ , Ts and Tp from 8M2 to 4M2, and the number of non-
2,α j,pN j,pS trivial equations to 2M(M +1)/2, i.e.,
(cid:16) (cid:17)
and verify that Eq. (2.10b) holds independently of the
form of v(j), w(j), and Q. v(j)TKv(j(cid:48)) =0, (2.17a)
1,2 1,2
Henceforth, we will focus on the charge-conserving w(j)TKw(j(cid:48)) =0, (2.17b)
tunneling vectors defined in Eqs. (2.13), as all general
criteria discussed below have analogues for the parity- whicharemerelyrewritingsofEqs.(2.15a)and(2.15b).
conserving tunneling vectors defined in Eqs. (2.14). With this, we have arrived at the simplest incarnation
The charge-conserving tunneling vectors defined in of our construction. We will henceforth assume that
Eqs. (2.13) are expressed in a convenient pictorial form Eqs.(2.17)holdforappropriatechoicesofthe2M,2M-
in Fig. 2. From this pictorial representation, it is clear dimensional vectors v(j) and w(j). However, note that
that any two distinct, adjacent stars (be they of the Eqs. (2.16) are sufficient but not necessary in order to
same flavor or different flavors) share a single wire be- produce a state of matter that has an isotropic low-
tween them. The same statement holds for plaquettes. energy description. We will therefore comment, as ap-
However, adjacent stars and plaquettes share two wires propriate, on how our results below generalize to cases
between them, regardless of the flavor. Therefore, one where v(j) =v(j) and w(j) =w(j).
1 (cid:54) 2 1 (cid:54) 2
can show that Eqs. (2.12) are satisfied if and only if
v(j)TKv(j(cid:48)) =0, (2.15a)
µ µ C. Fractionalization
w(j)TKw(j(cid:48)) =0, (2.15b)
µ µ
v(j)TKw(j(cid:48)) v(j)TKw(j(cid:48)) =0, (2.15c) 1. Change of basis
1 2 − 2 1
for all j and j(cid:48) = 1,...,M and µ = 1,2. Equations Inthissection,weoutlinehowtousetwo-dimensional
(2.15)arefundamentaltoourconstruction,aseachsolu- arrays of coupled quantum wires, like those described
tiontotheseequationsforagivendimension2M ofthe in the previous two sections, to study phases of matter
8
with fractionalized excitations. To this end, let us as- ofscalarfieldsobeyingthenewequal-timecommutation
sume that we have a Haldane set H containing 2MN relations
tunneling vectors Ts(j) and Tp(j) with j = 1,...,M de- ∂ φ˜ˆ (z),φ˜ˆ (z(cid:48)) =i2πδ K˜−1 δ(z z(cid:48)), (2.21)
fined by Eqs. (2.13) that satisfy (2.16) and the Hal- z j,α j(cid:48),α(cid:48) jj(cid:48) αα(cid:48) −
danecriterion(2.17). Withtheseassumptions,thetwo- (cid:104) (cid:105)
for j,j(cid:48) = 1,...,2N and α,α(cid:48) = 1, ,2M, which are
dimensional array of coupled quantum wires acquires
neither fermionic nor bosonic in natu··r·e. We allow K˜ to
a gap in the strong-coupling limit, yielding a three-
be any symmetric, invertible, 2M 2M integer matrix,
dimensional gapped state of matter.
×
as long as it is related to K or K by a transforma-
As discussed in the previous section, the phase of f b
tion of the form (2.18). Interactions between wires that
matter obtained in this way is a system of strongly-
yield a gapped state of matter can be constructed by
interacting fermions or bosons. However, for the pur-
following the procedures of the previous section. The
poses of studying fractionalization, it is convenient to
work in a basis where the “fundamental” constituents 2MN integer tunneling vectors ˜s(j) and ˜p(j) obtained
T T
of each wire are not fermions or bosons, but (possibly in this way form a Haldane set H˜ related to the Hal-
fractionalized) quasiparticles. This is achieved by mak- dane set H by the transformation (2.18). For reasons of
ing the change of basis simplicity that will become clear momentarily, we will
concern ourselves in this paper primarily with the tun-
neling vectors ˜(j) and ˜(j) whose nonzero entries are
s p
Φ˜ˆ(z)..= −1Φˆ(z), (2.18a) equal to 1. (OTf course,Tnothing prevents us from also
W ±
considering cases where this does not hold.) The coun-
˜ := T , (2.18b)
V W VW terparts (j) and (j) of these tunneling vectors under
K˜ ..= WTKW, (2.18c) the transTfosrmationTp(2.18) generically have entries with
Q˜..= WTQ, (2.18d) magnitudelargerthan1. Thisfactwillbeofimportance
to us now, as we turn to the issue of compactification.
˜..= −1 , (2.18e)
T W T
where 2. Compactification, vertex operators, and fractional
charges
..= 1 W, (2.18f)
W 2N ⊗
Although the transformation (2.18) might appear in-
for some invertible 2M 2M integer-valued matrix W.
Trehmisaicnhsasnygmemofetvraicriaabnlde×sinhtaesgesrevvearlaulevdi.rtSueecs.onFd,irsQ˜t,rKe˜- nthoecouroyuws,itthhetrheeisLaagfruanndgaiamneLn˜ˆtadlefidniffeedreinncEeqb.e(t2w.2e0e)natnhde
the original fermionic or bosonic theory with the La-
mains integer valued. Third, this change of variables
leaves the quantity T Φˆ(z), which enters the argu- grangianLˆ definedinEq.(2.9)whenperiodicboundary
T K conditions are imposed in the z-direction (as we have
ment of the cosine terms in Eq. (2.7), invariant, i.e.,
assumed from the outset). In the latter theory, which
˜T ˜Φ˜ˆ(z)= T Φˆ(z). (2.19) is a theory of interacting electrons or bosons treated
T K T K within bosonization, the traditional choice of compact-
ification for the scalar fields φˆ (z) with j =1,...,2N
Thus, the linear transformation (2.18) does not change j,α
and α=1, ,2M is
the character of the interaction itself, although it al- ···
ters the tunneling vector and the 4MN-dimensional φˆ (z+L) φˆ (z)+2πn , (2.22)
vector Φˆ of bosonic fieldTs. Furthermore, one verifies j,α ≡ j,α α
that the linear transformation (2.18) does not alter the forn Z2M. Thischoiceensuresthesingle-valuedness
compatibility criteria (2.15) or the quantity T that of thαe∈fermionic or bosonic vertex operators (2.4) un-
Q T
determinesthepresenceorabsenceofchargeornumber- der z z +L, and, in turn, that of the Lagrangian
parity conservation. Lˆ =Lˆ→+Lˆ ,asonecanre-writeLˆ intermsofthe
0 {T} {T}
Given the possibility of performing a change of basis correlatedtunnelings(2.8), whichreducetoproductsof
of the form (2.18), we may now take a different ap- thesevertexoperators. However,dependingonthetun-
proach. Instead of viewing the wire construction as a neling vectors (j), there may be other, less stringent,
s,p
theory, with the Lagrangian (2.9), of scalar fields obey- T
compactifications of these scalar fields that also render
ing the commutation relations (2.1d) with a K-matrix the Lagrangian Lˆ single-valued under z z+L. The
K [Eq. (2.2)] for fermions or K [Eq. (2.3)] for bosons, →
f b parsimoniouscourseofactionistochoosethe“minimal”
we may also view it as a theory, with the Lagrangian
compactification, i.e., the smallest compactification ra-
diusthatstillmaintainsthesingle-valuednessofLˆunder
L˜ˆ..= L˜ˆ0+L˜ˆ{T˜}, (2.20) z →z+L.
9
If the tunneling vectors (j) and (j) correspond to
s p
T T
the tunneling vectors ˜(j) and ˜(j) under the transfor-
s p
T T
mation (2.18) whose only nonzero entries are equal to
1, then there is a clear choice of minimal compactifi-
±
cation. Thischoicecanbeobtainedasfollows. Working
in the tilde basis, we can rewrite the interactions using
the relation [analogous to (2.8)]
e−iT˜TK˜Φ˜ˆ(z) = 2N 2M ψ˜ˆ† (z) T˜j,α, (2.23a)
f,b;j,α
j(cid:89)=1α(cid:89)=1(cid:104) (cid:105)
thereby implicitly defining a new set of fermionic or
bosonic vertex operators,
FIG. 3. (Color online) Pictorial representation of star and
2M plaquette excitations created by the operators (2.31). The
ψ˜ˆ† (z)..= exp i K˜ φ˜ˆ (z) . (2.23b) filledbluecirclerepresentsanapplicationofthevertexoper-
f,b;j,α (cid:32)− α(cid:48)=1 αα(cid:48) j,α(cid:48) (cid:33) ator(2.31a)inthecorrespondingwire,whilethepurpleand
(cid:88)
orange crosses represent defective stars hosting solitons of
The minimal compactification is then obtained by de- opposite signs. Similarly, the filled green square represents
manding that this new set of vertex operators be single an application of the vertex operator (2.31b), and the pur-
valued under z z +L. For any j = 1,...,2N and pleandorangesquaresrepresentdefectiveplaquetteshosting
→ solitons of opposite signs.
α = 1, ,2M, this is achieved by imposing the peri-
···
odic boundary conditions
φ˜ˆ (z+L) φ˜ˆ (z)+2πK˜−1 n , (2.24) Indeed,foranyj,j(cid:48) =1,...,2N andα,α(cid:48) =1, ,2M,
j,α ≡ j,α αα(cid:48) α(cid:48) ···
the equal-time commutator
for n Z2M. Here, there is an important difference
withαre∈spect to Eq. (2.22). Because K˜ is an integer- Q˜ˆj,α,qˆj†(cid:48),α(cid:48)(z) =Q˜αδjj(cid:48)K˜α−α1(cid:48)qˆj†(cid:48),α(cid:48) (2.27)
valued matrix, K˜−1 isgenerically arational-valued ma- (cid:104) (cid:105)
trix. The field φ˜ˆ (z) is thus allowed to advance by indicates that, since K˜−1 is generically a rational ma-
rational (rather thja,αn integer) multiples of 2π when the trix, the quasiparticle operator qˆj†(cid:48),α(cid:48) generically has a
coordinate z is advanced through a full period L. This rationalcharge. Inparticular,ifQ˜ =Qunderthetrans-
crucial distinction is what allows for the existence of formation (2.18), and K˜−1 has at least one rational en-
fractionally-charged operators in the coupled wire ar- try with magnitude smaller than 1, the operator qˆ†
j(cid:48),α(cid:48)
ray, as we now demonstrate.
must then carry a fractional charge.
Fractional quantum numbers appear in the wire con-
struction because the compactification condition (2.24)
allows for the existence of “quasiparticle” vertex opera- 3. Pointlike and linelike excitations
tors
We now outline the relationship between the quasi-
qˆj†,α(z)..= exp −iφ˜ˆj,α(z) (2.25) particlevertexoperatorsdefinedinEq.(2.25)and(pos-
(cid:16) (cid:17) sibly fractionalized) excitations in the array of coupled
for any j = 1,...,2N and α = 1, ,2M that are quantum wires. In the strong-coupling limit U (z)
··· T˜ →
multivalued under the operation z z +L. The fact , the compatibility criteria (2.15) ensure that the
that these vertex operators generica(cid:55)→lly carry fractional ∞quantity
charges can be seen by considering the transformed
charge operator ˜T ˜Φ˜ˆ(z)+α (z), (2.28)
T K T˜
Q˜ˆj,α..= Q2˜πα 2M δαα(cid:48) Ldz ∂zφ˜ˆj,α(cid:48)(z) (2.26) wimhuemre To˜f =theT˜s(cjo)rroerspT˜op(njd),inigs pcionsnineed ptootaenctliaaslsiicnalLˆmin-.
α(cid:88)(cid:48)=1 (cid:90)0 Following Refs. [4, 5] and subsequent works, we id{Te˜n}-
tify excitations in the coupled-wire theory with solitons
for any j = 1,...,2N and α = 1, ,2M. Its normal-
izationisherechosensuchthatthe··fe·rmionicorbosonic that increment the “pinned field” ˜ ˜Φ˜ˆ(z) by an inte-
vertexoperatorsdefinedinEq.(2.23b)havechargeQ˜ . ger multiple of 2π. These excitatiTonKs can therefore be
α
10
(a) (b)
(c) (d)
FIG. 4. (Color online) Deconfinement of star defects within a plane. (a) A single application of the vertex operator (2.31a)
(blue circle) creates a star defect (purple cross) and an anti-star defect (orange cross). (b) Applying a string of vertex
operators (2.31a) moves a star defect by healing one while creating another. Consequently, it costs no extra energy to
separate the two star defects. (c) To turn a corner, it may be necessary to heal a defect with an application of the inverse
of the vertex operator (2.31a) in the appropriate wire (white circle). (d) When two star defects meet, they annihilate one
another. A completely analogous description of plaquette defects also holds starting from the operators Eq. (2.31b).
viewed as living on either the stars or the plaquettes of jacent stars and plaquettes. To see this, we write
the square lattice, rather than within the wires them- out the pinned fields explicitly for all MN tunneling
selves. vectors ˜(j) with j = 1,...,M defined on the stars
s
Wenowdemonstratethatproductsofanappropriate s=1,..T.,N,
number of quasiparticle vertex operators of the form
(2.25) can be used to move the soliton defects to ad-
2M
˜(j)T ˜Φ˜ˆ(z)= v˜(j)K˜ φ˜ˆ (z) φ˜ˆ (z)+φ˜ˆ (z) φ˜ˆ (z) , (2.29a)
Ts K α αα(cid:48) sE,α(cid:48) − sW,α(cid:48) sN,α(cid:48) − sS,α(cid:48)
α,(cid:88)α(cid:48)=1 (cid:104) (cid:105)
and for all MN tunneling vectors ˜(j) with j=1,...,M defined on the plaquettes p=1,...,N,
p
T
2M
T˜p(j)TK˜Φ˜ˆ(z)= w˜α(j)K˜αα(cid:48) φ˜ˆpW,α(cid:48)(z)−φ˜ˆpE,α(cid:48)(z)+φ˜ˆpN,α(cid:48)(z)−φ˜ˆpS,α(cid:48)(z) . (2.29b)
α,(cid:88)α(cid:48)=1 (cid:104) (cid:105)
For any star s=1,...,N or plaquette p=1,...,N from the square lattice and for any α,β =1,...,2M, observe
that, by Eq. (2.21), the equal-time commutators
2M 2M
K˜ φ˜ˆ (z),∂ φ˜ˆ (z(cid:48)) = K˜ φ˜ˆ (z),∂ φ˜ˆ (z(cid:48)) = i2πδ δ(z z(cid:48)) (2.30)
(cid:34) αα(cid:48) sC,α(cid:48) z(cid:48) sC,β (cid:35) (cid:34) αα(cid:48) pC,α(cid:48) z(cid:48) pC,β (cid:35) − αβ −
α(cid:48)=1 α(cid:48)=1
(cid:88) (cid:88)
hold. Here, the uppercase Latin index C =N,W,S,E labels the four cardinal directions. Equation (2.30) indicates
that the pair of fields ∂ φ˜ˆ and ∂ φ˜ˆ can be viewed, up to a multiplicative constant, as canonical conjugates
z s ,α z p ,α
C C
to the pair of fields K˜ φ˜ˆ and K˜ φ˜ˆ (z) that enter the pair of pinned fields ˜(j)T ˜Φ˜ˆ and
α(cid:48) αα(cid:48) sC,α(cid:48) α(cid:48) αα(cid:48) pC,α(cid:48) Ts K
(cid:80) (cid:80)