Assembling Fibonacci Anyons From a $\mathbb{Z}_3$ Parafermion Lattice Model

1.6 MB·English

by  E. M. Stoudenmire

Assembling Fibonacci Anyons From a Z Parafermion Lattice Model
3
E. M. Stoudenmire,1 David J. Clarke,2,3,4 Roger S. K. Mong,2,5,6 and Jason Alicea2,5
1Perimeter Institute of Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada
2Department of Physics and Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, CA 91125, USA
3Microsoft Research, Station Q, University of California, Santa Barbara, CA 93106, USA
4Condensed Matter Theory Center, Department of Physics,
University of Maryland, College Park, Maryland 20742, USA
5Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA
6Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA
(Dated: January 23, 2015)
5 Recentconcreteproposalssuggestitispossibletoengineeratwo-dimensionalbulkphasesupport-
1 ingnon-AbelianFibonaccianyonsoutofAbelianfractionalquantumHallsystems. Thelow-energy
0 degreesoffreedomofsuchsetupscanbemodeledasZ parafermions“hopping”onatwo-dimensional
3
2 lattice. Weusethedensitymatrixrenormalizationgrouptostudyamodelofthistypeinterpolating
n between the decoupled-chain, triangular-lattice, and square-lattice limits. The results show clear
a evidence of the Fibonacci phase over a wide region of the phase diagram, most notably including
J the isotropic triangular lattice point. We also study the broader phase diagram of this model and
1 show that elsewhere it supports an Abelian state with semionic excitations.
2

] I. INTRODUCTION
l
e
-
r The experimental realization of non-Abelian anyons—
t emergent particles with highly exotic exchange
s
. statistics—is anticipated to have widespread impli- t
t 2
a cations. Apart from demonstrating a profound new
m
facet of quantum mechanics, many applications await
- the completion of this ongoing quest. Examples include
d tests of Bell’s inequalities,1 robust quantum memory, t1
n
novel low-temperature circuit elements,2 and most
o t
c importantly intrinsically fault-tolerant ‘topological’ 3
[ quantum computing.3–7 The last of these relies on the

1 remarkable fact that adiabatic exchange (braiding) of FIG. 1. Model of Z3 parafermions with interactions form-
v non-Abelian anyons enacts a unitary rotation within ingananisotropictriangularlatticewithintrachaincouplings
5 the space of locally indistinguishable ground states t3 (horizontal) and interchain couplings t1 (vertical) and t2
0 generated by the anyons. Storage and manipulation of (slanted).
3 qubits encoded in these ground states thus takes place
5 non-locally, so that the quantum information is securely
0
‘hidden’ from local environmental perturbations. defects binding parafermionic zero modes,29 which
.
1 While most of these applications can be carried out comprise natural Majorana generalizations. Al-
0 with any non-Abelian anyon type, topological quan- though such defects require a strongly interacting
5
tum computation carries more stringent demands. Con- host system (unlike Ising anyons), many plausi-
1
sider Ising anyons—or defects that bind Majorana zero ble realizations have been suggested such as lat-
:
v modes—which likely constitute the most experimen- tice quantum anomalous Hall states,30 Abelian quan-
Xi tally accessible non-Abelian anyon. Numerous realis- tum Hall/superconductor heterostructures,31–35 mul-
tic platforms now exist for realizing Ising anyons, most tilayer quantum Hall systems,36,37 and coupled-wire
r
a prominently in quantum Hall systems7,8 and topologi- arrays.38–40 Parafermionic zero modes produce a larger
cal superconductors,9–15 and indeed tantalizing experi- ground-state degeneracy than Majorana modes and thus
mental evidence of these particles has accumulated in enable a denser set of qubit rotations through braid-
both settings.16–28 Braiding Ising anyons, however, only ing. While providing some advantages for quantum
amountsto90◦ qubitrotationsontheBlochsphere. Per- computation,32theirbraidstatisticsneverthelessremains
forming the arbitrary qubit rotations necessary for uni- non-universal.41
versal computing with Ising anyons requires additional Fortunately one can leverage setups supporting
operations that are not topologically protected. parafermionic zero modes to generate non-Abelian
This shortcoming strongly motivates the pursuit anyons allowingbonafide universaltopologicalquantum
of other types of non-Abelian anyons with ‘denser’ computation. To illustrate how this is possible, imagine
braid statistics. One appealing alternative class are nucleatingatwo-dimensional(2D)arrayofparafermionic
2
zero modes in, say, a quantum Hall/superconductor hy- typical 2D models on cylinders can only be obtained for
brid structure. The collection of zero modes encodes circumferences on the order of ten sites along y. Finally,
a macroscopic ground-state degeneracy, similar to a wenotethatmovingawayfromthedecoupled-chainlimit
partially filled Landau level. Hybridizing parafermions lessens the overall entanglement, mitigating the cost of
amongthesitescanliftthisdegeneracyandproducenew, the additional sites needed along the y direction and al-
possibly very exotic 2D phases. Resolving the precise lowing DMRG to even address the behavior of isotropic
state selected by the coupled parafermion modes, how- 2D systems.
ever, poses a highly nontrivial task since the system can In this way we numerically show that the Fibonacci
not be described by free particles (much like the Landau phase extends across a very wide swath of parameter
level problem with Coulomb interactions). spaceinourmodel,persistingfromweaklycoupledchains
Inspired by the important work of Teo and Kane,42 all the way to the isotropic triangular lattice point and
Ref. 43 nevertheless identified an analytically tractable beyond. Evidently the quasi-1D limit pursued earlier is
limit where strongly anisotropic, weakly coupled bynomeansnecessary,butmerelyprovidesaconvenient
parafermion chains could be shown to enter a topolog- entryway into the relevant physics. The broad stability
ically ordered ‘Fibonacci phase’. This phase represents windowforFibonaccianyonsthatweidentifyisoneofthe
a cousin of the so-called Z Read-Rezayi non-Abelian main punchlines of this paper. We also present evidence
3
quantum Hall state,44 yet is built from well-understood for a second topologically ordered state that is Abelian
Abelian states of matter. (The analogy closely parallels and supports a semion as its only nontrivial quasiparti-
the relation between a p + ip superconductor and the cle. This phase naturally emerges from weakly coupled
Moore-Read state.8,9) Like its quantum Hall cousin, the chains upon swapping the sign for the intrachain cou-
Fibonacci phase supports Fibonacci anyons, which are plingconstant, andinoursimulationsalsoappearsquite
theholygrailfortopologicalquantumcomputationsince robust even away from this limit. In fact the Fibonacci
theyallowonetoapproximatearbitraryunitarygatesto and semion phases comprise the only two states that ap-
any desired accuracy solely through braiding.7,45,46 pearedthroughoutthebroad(thoughnotcompletelyex-
haustive) parameter space that we explored numerically.
While this progress is encouraging, the stability of the
For a summary of our results see the octahedral phase
Fibonacciphaseawayfromthesolvablequasi-1Dlimit—
diagram presented as a cutout template in Fig. 16.
as well as the broader phase diagram for coupled 2D
We organize the remainder of the paper as follows.
parafermion arrays—remains largely unknown. Obtain-
Section II motivates the lattice model we study from
ing a more thorough and quantitative understanding of
the viewpoint of physical quantum Hall-based architec-
these physical systems poses a pressing issue given the
tures. Section III analyzes the model, both analytically
importance of finding experimentally accessible realiza-
and numerically, in the two-chain limit—which we argue
tions of Fibonacci anyons for quantum computing. In
already contains precursors of Fibonacci physics. The
this paper we take a major step in this direction by per-
multi-chainsetupofgreatestinterestistackledinSec.IV.
forming extensive density matrix renormalization group
(DMRG)47–49 simulations of lattice parafermions with Wepresentawealthofnumericalevidenceindicatingthe
robustness of the Fibonacci phase; analytically capture
couplings sketched in Fig. 1. A virtue of the model we
thesemionstate; andexplorethebroaderphasediagram
study is that it interpolates between various physically
of the model. Section V discusses future directions and
interesting regimes, including decoupled chains and the
highlights the connection between our study and recent
isotropic triangular- and square-lattice limits as special
relatedworks. Fourappendicescontainadditionaldetails
cases.
of our model and simulation methods.
DMRG naturally complements the weakly-coupled
chain approach to realizing the Fibonacci phase. A
key observation from analytics is that the ‘y’ correla-
II. Z PARAFERMION LATTICE MODEL
tion length for the Fibonacci phase becomes arbitrarily 3
small with weakly coupled chains, albeit at the expense
of extended ‘x’ correlations. This tradeoff is highly fa- To motivate the lattice model we will study, let us re-
vorable for numerics. The short y correlation length al- viewhowanon-AbelianphasesupportingbulkFibonacci
lows us to well-approximate the 2D limit of interest us- anyonscanarisefromcouplingparafermioniczeromodes
ingsystemscomposedofrelativelyfewchains. Moreover, nucleated in a fractional quantum Hall fluid. As a
with sufficient effort DMRG can handle very long chains primer consider some arbitrary Abelian quantum Hall
in the x direction even when each is close to criticality, stateslicedintotwoadjacenthalvesasshowninFig.2(a).
facilitating direct comparisons to analytical predictions. The cut produces a new set of gapless counterpropagat-
Near the decoupled-chain limit we can unambiguously ingedgestatesoppositethetrench. Onecanalwaysfully
pinpoint the Fibonacci phase in our simulations, then gapthesemodes—effectivelyresewingthetwohalves—in
systematically track how it evolves as we gradually en- more than one physically distinct way. The most natu-
hance the interchain coupling—all the while keeping the ral mechanism involves backscattering electrons across
y correlation length manageable. The latter feature is the trench to simply recover the original uninterrupted
important since in practice accurate DMRG results for quantum Hall state as Fig. 2(b) illustrates. Filling the
3
(a) (b) J h
n-1 n n+1
FIG. 3. One-dimensional chain of parafermionic zero modes
arising from inequivalently gapped regions of a trench. Hy-
(c) (d)
bridizationamongnearest-neighborsinthechainisdescribed
by the Hamiltonian in Eq. (3).
byα andα withj thedomain-wallsiteindex. Asa
R,j L,j
specific example, in the ν =2/3 setup α and α al-
L,j R,j
tertheadjacentsuperconductingregionbyaddingcharge
2e/3 to the upper and lower trench edges, respectively.
FIG.2. (a)AtrenchdividingtworegionsofanAbelianquan-
Thisdistinctionismeaningfulsincefractionalchargecan
tum Hall phase supports gapless modes. (b) Gapping these
modes via electron backscattering recovers the original bulk not migrate across the trench. For further discussion on
phase. (c) Introducing either Cooper pairing, or ‘crossed’ this important point see Ref. 43 as well as Appendix A.
tunnelinginabilayersystem,providesaninequivalentwayof Both representations fulfill the conditions
gapping the trench. (d) Domain walls between incompatibly
gapped regions support protected zero modes. α3 =1; α† =α2 (1)
A,j A,j A,j
for A = R or L; moreover, they exhibit ‘anyonic’ com-
mutation relations
trench with a superconductor [blue region in Fig. 2(c)]
providesasecond,intuitivelyquitedifferentmethod: the αR,jαR,j(cid:48) =ei(2π/3)sgn(j(cid:48)−j)αR,j(cid:48)αR,j,
edge modes can then gap out by Cooper pairing elec- (2)
trons from opposite sides of the trench.31–33 Some se- αL,jαL,j(cid:48) =e−i(2π/3)sgn(j(cid:48)−j)αL,j(cid:48)αL,j
tups can support alternative charge-conserving gapping
mechanisms (aside from trivial backscattering).36,37 For inherited from the quantum Hall edge fields from which
instance, in quantum Hall bilayers formed out of Laugh- they derive.
lin states at filling ν = 1/m, one can gap the trench by SupposenowthatthedomainsinFig.3aresufficiently
‘crossed’ tunneling whereby electrons hop between the narrow that nearest-neighbor parafermionic modes hy-
top layer on one side and the bottom layer on the other, bridize appreciably. One can describe the hybridization
sewingthehalvesinyetadifferentmanner. Domainwalls with a Hamiltonian
[Fig. 2(d)] separating regions gapped in such incompati-
H˜ = ω(Jα† α +hα† α )
blewaysbindprotectedzero-energymodesthatformthe 0 − R,2n+1 R,2n R,2n R,2n−1
basic building blocks in our lattice model. (cid:88)n (cid:104) (3)
We are specifically interested in setups that support +H.c. ,
Z parafermionic zero modes, which provide the min-
3 (cid:105)
imal extension of the more familiar Majorana opera- whereω =ei(2π/3),nsumsoverunitcells,andtheJ and
tors. Concrete realizations include (i) domain walls be- h terms couple alternating pairs; see Fig. 3. Physically,
tween pairing- and tunneling-gapped trenches in spin- these terms reflect tunneling of fractional charges from
unpolarized ν = 2/3 quantum Hall fluids43 and (ii) do- onedomainwalltothenextviaapathabove thetrench.
main walls between trivial and ‘crossed’ electron tunnel- (Note that an equivalent form in terms of α is also
L,j
inginν =1/3bilayers.37 Considerforthemomentaone- possible; this form would correspond to tunneling paths
dimensional array of well-separated Z3 zero modes as in below the trench.)
the geometry displayed in Fig. 3. These modes encode a As reviewed in Appendix A the single-chain Hamilto-
ground-state degeneracy that can be understood as fol- nian in Eq. (3) maps to the three-state quantum Potts
lows. In the ν = 2/3 realization each superconducting- model under a non-local ‘Fradkin-Kadanoff’ transforma-
gappedregioncanaccommodatecharge2e/3withouten- tion akin to the Jordan-Wigner mapping in the Ising
ergy penalty, while in the bilayer example each ‘crossed’ model.29,50Manypropertiesimmediatelyfollowfromthis
region can similarly acquire an e/3 dipole for free. identification. Like its Potts analogue, with J,h > 0
Parafermionic zero-mode operators cycle the system the Hamiltonian admits two distinct phases. For h > J
throughthecorrespondingdegenerateground-stateman- the trench enters a trivial state where all parafermion
ifold. Importantly, charges can be added to the trench modes dimerize and gap out pairwise with their neigh-
in two physically distinct ways—and hence there exists bors. On the other hand, for J > h the parafermion
twoinequivalentrepresentationsoftheparafermionoper- operatorsdimerizeinashiftedpatternleaving‘unpaired’
ators. We denote these two parafermion representations modesattheendsofthetrench(assumingopenboundary
4
t t Hamiltonian of this model reads
1 2
y =3 H =H +H
0 ⊥
H = t ωα† (y)α (y)+H.c.
y =2 0 − 3 R,j+1 R,j
(cid:88)y (cid:88)j (cid:104) (cid:105)
(5)
y =1 H = t ωα† (y)α (y+1)
t3 ⊥ − 1 L,j R,j
(cid:88)y (cid:88)j (cid:104)
FIG.4. Aseriesofone-dimensionalparafermionchainsinter- +t2ωαL†,j(y)αR,j−1(y+1)+H.c. .
act by tunneling fractional charge through the quantum Hall
(cid:105)
fluid [Eq. (4)]. For the present work, we consider anisotropic
Our intrachain Hamiltonian H is identical to Eq. (3)
next-neighbor tunneling processes for parafermions arranged 0
inatriangularlattice,allowingustostudy(amongotherlim- but fixed to J =hd=eft3 such that each horizontal
its)theisotropictriangularlattice(t =t =t ),theisotropic parafermion chain remains tuned to its critical point if
1 2 3
square lattice (t = t , t = 0), and weakly coupled chains decoupled from the other chains. The interchain bonds
1 3 2
(t3 (cid:29)t1,t2). then form the triangular lattice pattern of interest, with
vertical couplings t and slanted couplings t in the
1 2
skewed layout of Fig. 1. Throughout we will assume
purely real t . Importantly, the Hamiltonian’s phase
1,2,3
conditions).29Herethesystemexhibitsaprotectedthree- diagram is sensitive to both the magnitude and signs for
fold ground-state degeneracy similar to the topological forthesecouplings, sincetheonlygaugetransformations
phaseofaKitaevchain.10Thesetwophasesareseparated wecanmakecorrespondtoshiftingtheparafermionoper-
by a self-dual critical point at h=J described by a non- ators by factors of ω. Notice that the model interpolates
chiralZ3-parafermionconformalfieldtheory(CFT)with between the square lattice (t2 = 0, t1 = t3), isotropic
central charge c = 4/5.51,52 As our nomenclature hints triangular lattice (t = t = t ), and decoupled chain
1 2 3
theright-andleft-movingparafermionfields atthiscrit- (t = t = 0) limits—allowing us to explore several im-
1 2
ical point are closely related to the lattice parafermion portant cases in one framework.
operators αR/L,j.53 Despite the fact that Eq. (5) describes an inherently
strongly interacting 2D setup, one can make controlled
Now consider a series of N such trenches at vertical
analytic progress in the weakly-coupled-chain limit with
positions y, arranged such that parafermion modes re-
0<t =t t . In this regime the problem reduces to
sideatthesitesofa2Dlattice(Fig.4). Iftrenchesy and 1 2 (cid:28) 3
parafermion CFTs in each chain that hybridize to yield
y+1areseparatedbyafinitedistance,therewillalsobe
a 2D topologically ordered ‘Fibonacci phase’.43 (See Ap-
fractional-charge tunneling processes through the inter-
pendix D for details.) Quite generally, the Fibonacci
vening quantum Hall fluid. Importantly, these processes
phase exhibits the following key properties:
can only couple the ‘left’ parafermion representation in
(i) Within the parent quantum Hall fluid, spatial
trenchytothe‘right’representationiny+1. (Thecorre-
boundaries of the Fibonacci phase support chiral Z
sponding process with right and left interchanged would 3
parafermion modes with central charge c = 4/5. These
pass fractional charge through regions where only whole
residual gapless edge modes represent ‘half’ of the non-
electrons can pass.) A general interchain Hamiltonian
chiral CFT that occurs in each critical chain.
thus reads
(ii)Thebulksupportstwoquasiparticlestypes: trivial
(fermionic/bosonic) excitations and Fibonacci anyons.
N−1 (iii) If the system is defined on a torus or an infinite
H˜⊥ =− t˜j−j(cid:48)αL†,j(y)αR,j(cid:48)(y+1)+H.c. . cylinder, there are two degenerate ground states |1(cid:105) and
(cid:88)y=1 (cid:88)j,j(cid:48) (cid:104) (cid:105) ε —one associated with each quasiparticle type.54
(4) | (cid:105)
(iv) The entanglement entropy of ground state
n=1,ε on an infinite cylinder of large but finite
| (cid:105)
circumference N scales as S = aN γ . Here
y n y n
Because the bulk of each quantum Hall fluid is gapped, −
γ =log( /d ) represents the topological entanglement
the interchain couplings t˜j−j(cid:48) should decay rapidly as a enntropyexDprenssedintermsoftheanyonquantumdimen-
function of separation between sites. Thus taking t˜j−j(cid:48) sions d = 1, d = ϕ and the total quantum dimension
1 ε
non-zero for only the first- or possibly second-nearest-
= 1+ϕ2, with ϕ=(1+√5)/2 the golden ratio.
neighbors in adjacent trenches constitutes a reasonable D
(v)Theentanglementspectrum55—whichcanbecom-
minimal model. (cid:112)
puted from the CFT56—resembles the energy spectrum
Figure 4 depicts the specific pattern of interactions of the physical edge with chiral central charge 4/5.
that we consider. Parafermion modes bound to domain The entanglement spectrum of 1 consists of states de-
walls resideatthe sitesof atriangularlattice andcouple scended from primaries 1,ψ,ψ|†(cid:105) (with scaling dimen-
via anisotropic next-neighbor tunneling processes. The sions 0,2,2 respective{ly), whi}le the spectrum of ε
{ 3 3} | (cid:105)
5
consists of descendents of ε,σ,σ† (with scaling dimen-
sions 2, 1 , 1 respective{ly). The}specificcounting57 of (a)
{5 15 15}
states appears in Fig. 11.
Outside of the highly anisotropic, weakly-coupled-
chain limit, the analytical methods used to establish the
preceding results break down entirely. Addressing the
broader phase diagram of the model—particularly the (b)
extent of the Fibonacci phase—is the central goal of this
paper. For this we turn to density matrix renormaliza-
tion group (DMRG) calculations of the ground states of
Eq. (5), primarily on infinitely long cylinders using the FIG. 6. Pairing of sites in the ground state for the two-chain
ladder system in the limits of (a) dominant t > 0, which
infiniteDMRGalgorithmproposedinRef.58. Toimple- 1
produces a trivial gapped phase, and (b) dominant t > 0,
menttheHamiltonianinEq.(5)wedonotworkdirectly 2
which produces a topological gapped phase with a three-fold
with parafermion operators [Eqs. (1)–(2)]. Instead, fol-
ground-state degeneracy.
lowing Appendix B we map the parafermion degrees of
freedom to Z clock variables, which provides a much
3
moreconvenient(butformallyequivalent)representation
direction (see Fig. 5). As we will see this limit already
for numerics.
contains precursors of the 2D Fibonacci phase that we
Itisworthnotingsometechnicalfeaturesofthemodel
will uncover later on when studying wider geometries.
in Eq. (5) that facilitate our DMRG studies. Retain-
Throughout this section we assume t 0 for simplicity
3
ingallpossiblefirst-andsecond-neighborinterchaincou- ≥
but consider either sign of t .
1,2
plingsinasquare-latticearrangementoftheparafermion
Tounderstandthephasediagramofthetwo-chainsys-
sites would lead to a one-dimensional Hamiltonian (as
tem, first consider the limit where t >0 greatly exceeds
1
seen by the DMRG algorithm) with interactions up to
both t and t . In the extreme case with t = t = 0,
2 3 2 3
a range d = 2N , where N is the circumference of
max y y the ground state is found by pairing each parafermion
the quasi-2D cylinder used for the simulation. In con-
site with the one directly above or below it, yielding the
trast,thepatternoftriangularlatticecouplingswestudy
trivialproductstateillustratedinFig.6(a). (Sayingthat
here (Fig. 1) gives a one-dimensional Hamiltonian with
two parafermions α and α ‘pair’ means that they form
i j
a maximum range scaling only as dmax =Ny. Addition- an eigenstate of 1(ωα†α +H.c.) with maximal eigen-
ally,aswediscussinthenextsection,thetwo-chainlimit 2 i j
value 1.) Because this limit supports a unique ground
(N =2) of the triangular model with open y-boundary
y state protected by a robust gap, this trivial phase per-
conditions is self-dual for a certain choice of parameters.
sists upon restoring sufficiently small t couplings.
Appealing to self-duality allows us to exactly determine 2,3
Considernextthelimitwitht >0muchlargerthant
the location of an important line of critical points that 2 1
andt . Witht =t =0thegroundstateagainarisesby
roughlyrepresentsaremnantoftheFibonacciphasecom- 3 1 3
pairing sites, but now in the skewed pattern depicted in
pressed into a two-chain system. Finally, the properties
Fig. 6(b). Any finite ladder with open boundaries along
of the bona fide 2D Fibonacci phase enumerated above
the horizontal direction thus contains one unpaired site
provide sharp numerical fingerprints that we can use to
at each end—signifying a topologically nontrivial phase.
track the phase diagram in our multi-chain simulations
These two decoupled end sites, taken together, form a
(N >2), which we describe in Sec. IV.
y degenerate three-level system. It follows that there are
threedegenerategroundstatesdistinguishedbytheir‘tri-
ality’, defined as
III. TWO-CHAIN LIMIT
Q= ωα† α , (6)
2n 2n−1
BeforeconsideringthemodelEq.(5)onmulti-legcylin-
n
(cid:89)
ders, it is helpful to understand the limit of only two
coupled chains with open boundary conditions in the y which admits eigenvalues 1, ω, or ω2. Restoring small,
finitet andt onlysplitstheground-statedegeneracyby
1 3
an exponentially small amount in the system size, as the
processes mixing the end states require tunneling across
amacroscopicnumberofsites. (Notethatthisargument
applies only to the ground states. The excited states,
t t
2 1 strictly three-fold degenerate for t1 =t3 =0, generally
t split by an amount decaying only as a power law in the
3 system size.59)
Intuitively, one expects a 1D phase transition between
FIG. 5. Two-chain limit of the Hamiltonian in Eq. (5), with the gapped states sketched in Fig. 6 when t and t be-
1 2
open boundary conditions along the y direction.
come comparable. To be more quantitative, we invoke a
6
2.0
3.0
Topological
0.80006 x + 1.06871
2.5 Trivial 1.8
p2.0 1.6
Ga x)
(
ulk1.5 SL1.4
B
1.0
1.2
0.5
1.0
t =t
1 2
0.0
0.0 0.2 0.4 0.6 0.8 1.0
0.25 0.00 0.25 0.50 0.75
� 1log Lsin πx
✓/⇡ 3 π L
h (cid:16) (cid:17)i
FIG. 7. Bulk gap of the two-chain system (Fig. 5) as a func- FIG.8. EntanglemententropyS (x)foralength-xsubregion
L
tion of θ with t1 =sinθ, t2 =cosθ,√and t3 =1. The vertical of a periodic two-chain system with length L = 60. Data
dashedlinemarksthet1 =t2(=1/ 2)criticalpointseparat- were obtained at the critical point using parameters t1/t3 =
ingtopologicalandtrivialgappedphases. Forthetopological t /t = 1. By fitting to the CFT prediction we extract a
2 3
phase,thefirstexcitedstateisinthequantumnumbersector centralchargec=4/5,consistentwithanalyticalarguments.
with triality 1; for the trivial phase, the first excited states
aretwo-folddegenerateandlieintheω orω2 trialitysectors.
critical line, we use DMRG to study periodic systems of
sizeLalongthehorizontaldirectionandcomputetheen-
weakly-coupled-chainanalysisandfindaninterestingsce-
tanglement entropy S (x) of a subregion of size x. For a
nario. Inthestrictlydecoupled-chainlimit(t =0),the L
1,2 CFTwithcentralchargectheentanglementispredicted
low-energy properties of each chain can be described by
to scale as61
aZ parafermionCFTatc=4/5withapairofcounter-
3
propagating chiral fields.52 Turning on weak interchain c L πx
S (x)= ln sin . (7)
couplingst genericallygapsallofthesefields,butwhen L
1,2 3 π L
one fine-tunes t1 =t2 >0 the top chain’s left-mover and (cid:104) (cid:16) (cid:17)(cid:105)
the bottom chain’s right-mover remain gapless.43 Thus Figure 8 displays our simulation results. For t /t =
1 3
along the line t = t (at least for 0 < t ,t << t ) the t /t = 1 we find c = 4/5 to very high accuracy, con-
1 2 1 2 3 2 3
two-chain system should be critical and described by a firming the prediction that, along the critical t = t
1 2
single non-chiralZ parafermionCFTinwhichtheright- line, the coupled chains host a single pair of gapless Z
3 3
andleft-movershave,inasense,spatiallyseparatedalong parafermion CFT edge modes.
the vertical direction. We can, however, actually reach Figure 9 shows the full t 0 phase diagram of the
3
≥
a much stronger conclusion. Appendix C shows that for two-chain system computed with DMRG. We numeri-
t = t the two-chain Hamiltonian maps to itself un- cally identify the trivial gapped phase occurring in the
1 2
der duality followed by a time-reversal transformation. large t > 0 region by searching for the presence of a
1
Hence there should exist a continuous phase transition finite gap and a unique ground state. To identify the
line precisely along t =t >0 for any t >0. region lying in the topological phase stabilized for large
1 2 3
We confirm all of the above predictions—and ex- t >0,weprimarilyexaminetheentanglementspectrum
2
tend them beyond the analytically tractable regimes— within DMRG. For the entire topological phase each en-
usingDMRG.Unlessotherwisestatedweusetheinfinite tanglement “energy” exhibits a robust three-fold degen-
DMRG algorithm58 to reach the thermodynamic limit eracy. Atvariouspointswithinthisphasewealsochecked
in the horizontal direction while taking open boundary that finite systems with open boundary conditions pos-
conditions along the vertical direction. For both gapped sess three degenerate ground states.
phases we find essentially exact results (truncation er- In addition to the above two gapped phases, we also
rors below 10−12) by keeping only a few hundred states find a large region for either t or t negative where the
1 2
in DMRG. First, we check for the presence of the criti- system is gapless. Numerically computing the central
cal line at t =t by computing bulk gaps using infinite chargeusingthesameapproachasforFig.8,wefindthis
1 2
boundary conditions.60 As shown in Fig. 7 the gap in- entire gapless region has central charge c=1. To under-
deed closes precisely at t = t . (In the horizontal axis standthisphase,considerthelimitt <0andt ,t 0.
1 2 1 2 3
→
θ parameterizes the couplings through t = sinθ and Inthislimittheantiferromagnetict couplingfavorstwo
1 1
t =cosθ with t =1.) of the three states on each rung, effectively projecting
2 3
To additionally extract the central charge along this out the third. This allows a natural mapping to a spin
7
3 Trivial t c = 4/5 tionstocomplementearlieranalyticalstudiesthatapply
1
onlyinthecaseofweaklycoupledchains(t t ). We
3 1,2
(cid:29)
are particularly interested in tracking the extent of the
2 Fibonacciphaseastheinterchaincouplingincreasesaway
from this limit, though we will attempt to address the
nature of nearby states as well. To approach two dimen-
1 sions we study the Hamiltonian of Eq. (5) on cylinders
with N =4,6,8, and 10 sites around the circumference.
y
The cylinders will be taken infinitely long in the hori-
0 zontal direction, in part to avoid complications arising
t
Gapless c=1 2 from possible gapless edge states. For our calculations
we typically retain up to m = 5500 states in DMRG for
truncationerrorsupto10−8 andoftenassmallas10−10.
1
�
A. Fibonacci Phase on an Anisotropic Triangular
2
� Lattice
Topological Until specified otherwise we assume t 0 and fix
3 1,2,3
≥
� t = t d=ef t (Secs. IVB and IVC relax these assump-
3 2 1 0 1 2 3 1 2 ⊥
� � � tions). Ourgoalhereistonumericallyassesshowthesys-
temevolvesaswevaryt /t totunefromthedecoupled-
FIG. 9. Full phase diagram of the two-chain system in units ⊥ 3
where t =1. The trivial phase (green circles) surrounds the chain limit t⊥ = 0 up through the isotropic triangular
3
t1 > 0 axis while the topological phase (blue triangles) sur- latticepointt⊥ =t3 andbeyond. Fromthecoupled-wire
rounds the t >0 axis. The remainder of the phase diagram analysis,oneexpectsthe2Dsystemtoimmediatelyenter
2
shown is gapless with central charge c=1, which we verified the gapped Fibonacci phase upon turning on any small
numericallyforpointsindicatedwithredsquares. Transition but finite t /t 1. In this limit the system should
⊥ 3
linesseparatingthegaplessstatefromthetwogappedphases exhibit a quite lo(cid:28)ng correlation length in the horizontal
are approximate. The critical point along the line t = t is
1 2 directionbut—crucially—anarbitrarilyshortcorrelation
exact, however, and represents a precursor to the Fibonacci
length along y. This feature is extremely attractive for
phase appearing in the 2D limit.
DMRG: it implies that infinite cylinders even with rel-
atively small N possess local properties closely emulat-
y
ing those of the fully 2D system of interest. Our sim-
1/2chain. Perturbativelyreintroducingt andt givesa
2 3 ulation results, even for cylinders as small as N = 4,
y
Hamiltonian of XXZ type. Close to the negative t axis,
1 indeed show strong evidence that the system realizes the
the effective Z coupling is parametrically suppressed rel-
Fibonacci phase once t > 0. In what follows we will
⊥
ative to the XY coupling, leading to a gapless XY phase
numerically recover the characteristics (i) through (v) of
that is known to have c=1.
the Fibonacci phase delineated near the end of Sec. II.
For our purposes the most interesting feature of the
For a wide range of t /t we observe two quasi-
two-chain phase diagram is the c = 4/5 transition line ⊥ 3
degenerategroundstates—bothinthetriality1quantum
occurring at t = t where, remarkably, the low-energy
1 2 number sector—obtained by starting the infinite DMRG
right- and left-movers residing at opposite sides of the
algorithm in randomized initial states. (Which quasi-
ladder do not hybridize even for large t . This criti-
1,2 degenerate ground state appears depends on the pre-
cal point reflects a maximally ‘squeezed’ cousin of the
cise initial state used.) As we will argue later based on
stable 2D Fibonacci phase; in the latter, the right- and
entanglement-entropy scaling, these two states span the
left-movers are separated by macroscopic distances and
entire ground-state subspace. Anticipating Fibonacci-
therefore coexist harmoniously without any fine-tuning.
phase physics, let us call the quasi-ground state with
Inthenextsectionwewillshowthatuponaddingfurther
lower entanglement entropy 1 and the other ε . Fig-
chainsthislinedoesinfactopenintoanextendedregion | (cid:105) | (cid:105)
ure10showstherelativeenergysplitting(E E )/E
that develops into the Fibonacci phase in the 2D limit. ε− 1 | 1|
of these states as a function of t /t , at various system
⊥ 3
sizes. For a fixed circumference N and sufficiently weak
y
t /t , we observe a very small splitting (< 0.1%) with
⊥ 3
IV. TOWARDS TWO DIMENSIONS
weak t dependence. Beyond an N -dependent scale of
⊥ y
t , however, the splitting grows rapidly with the inter-
⊥
Armedwithourinsightsfromthetwo-chainmodelex- chain coupling. We expect that the crossover between
ploredintheprevioussection,wewishtonowdeducethe these behaviors transpires when the correlation length
phase diagram for our parafermion model in the full 2D in the y direction becomes comparable to the circumfer-
limit. For this purpose we use extensive DMRG simula- ence; this interpretation is consistent with the enhanced
8
0.015 22( 5)
N = 4 5 ×
y 61( 10)
4( 3) 15 ×
N = 6 ×
y 11( 4)
E1|0.010 Ny = 8 3 × 157(×4)
/| Ny = 10 3( 2) 4165(×6)
E)1 8(×4)
− 3 × 12( 2)
Eε0.005 351(×4)
( 2 15 ×
5( 2)
3 × 7( 2)
5 ×
0.000 16( 2)
15 ×
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2( 2)
t 3 × 2
⊥ 5
1( 2)
0 15 ×
FIG.10. Relativeenergysplittingbetweenquasi-degenerate
ground states of cylinders with N = 4–10 as a function of
y 1 ε
t⊥ (in units where t3 =1). Notice that for Ny ≥8 the states | (cid:105) | (cid:105)
remain nearly degenerate even up to the isotropic triangular
FIG.11. Entanglement“energies”ofthequasi-groundstates
lattice point t = 1. This is the first evidence suggesting
⊥ |1(cid:105) (left column) and |ε(cid:105) (right column) for the N = 4,
that in the 2D limit the Fibonacci phase enjoys a wide sta- y
t /t =0.2cylinderaftershiftingandrescalingthespectrato
bilitywindowextendingfromweaklycoupledchainspastthe ⊥ 3
match the lowest two levels of the Z parafermion CFT pre-
isotropic triangular lattice limit. 3
diction. The numbers in parentheses indicate the predicted
degeneracies. The same rescaling was used for both spectra,
butwithdifferentshifts;thusthefitrequiresonlythreefitting
robustness of the degeneracy evident in Fig. 10 upon in- parameters.
creasing N .
y
Next we examine the entanglement spectrum of both
cylinders into two semi-infinite halves, the entanglement
quasi-ground states, which for a topological phase is ex-
entropy is predicted to scale as63,64
pected to reveal the gapless low-energy spectrum in the
presence of an open edge.55 Following the approach of
S =aN γ + (8)
n y n
Ref. 62, we plot the entanglement spectrum after apply- − ···
ing a non-universal overall shift and rescaling such that where the first term encodes the boundary law expected
the lowest two levels match the spectrum of the chiral toholdwithinagappedphase. Becausetopologicallyor-
Z3 parafermion CFT. (Each entanglement “energy” (cid:15)i is dered states are locally indistiguishable in the 2D limit,
defined in terms of a reduced-density-matrix eigenvalue the coefficient a should be independent of the ground
pi through (cid:15)i = lnpi.) As shown in Fig. 11 for Ny =4 state n (although it is sensitive to other microscopic de-
−
and t⊥/t3 = 0.2, all the remaining entanglement ener- tails). Furthermore, because DMRG is somewhat biased
giesexhibitthesamepatternofdegeneraciesandrelative to low-entanglement states, it generally favors ground
energy-level spacings as the excitations of the chiral Z3 states of topological phases on infinite cylinders hav-
parafermion CFT on a ring. Specifically, we find that ing well-defined topological flux through the cylinder,
theentanglementspectrumofstate 1 matchestheCFT known as minimum entropy states.65 For such states,
| (cid:105)
level pattern for the superselection sector corresponding γ = log( /d ), with d the quantum dimension of
n n n
to the primary fields 1,ψ,ψ† (and their descendants), the anyonDtype associated with the nth ground state
whilethestate ε correspondstothefieldsε,σ,σ†. From and = d2 is the total quantum dimension of
the fusion algeb|r(cid:105)a of the chiral fields, this suggests that the tDheory. Nonticne that e−2γn = 1 when summed
ε and 1 respectivelycarryFibonacciandidentityflux. over n, pro(cid:112)vid(cid:80)ing a way to chneck whether a complete set
| (cid:105) | (cid:105)
Weobservedsimilaragreementoftheentanglementspec- of ground states has been(cid:80)found.
trum with the CFT prediction for other (small enough) Figure 12 fits our numerically computed entanglement
values of t⊥/t3 and for Ny =6,8,10, but choose to show entropiestotheforminEq.(8). Onesubtletyherearises
the Ny = 4 results to emphasize how quickly the 2D fromthefactthattheparafermionsinoursystemcannot
behavior sets in for these anisotropic cylinders. appear in vacuum, but rather require a host system—
For further evidence that we are observing a topolog- i.e., a ν = 2/3 quantum Hall state or similar ‘parent’
ically ordered phase, we compute the topological entan- topological phase. Consequently, to back out the γ ’s of
n
glement entropy γ for quasi-ground states n = 1,ε . interestwemustshiftourentanglementmeasurementsby
n
| (cid:105)
For a “vertical” entanglement cut dividing our infinite thetopologicalentanglemententropyoftheparentphase,
9
t =0.4,State ε
3 ⊥ | i 0.56 Ny = 4
t =0.6,State ε
⊥ | i N = 6
t =0.4,State 1 y
⊥ | i N = 8
2 t =0.6,State 1 y
⊥ | i 0.52 N = 10
1 y
S
Sn −
1 ε
S
0.48
0
γ = log(ϕ/ )
ε
− D
0.44
γ = log(1/ )
1
1 − D
− 0 2 4 6 8 10 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
N t
y
⊥
FIG. 12. Entanglement entropy fits for N =4,6,8 cylinders FIG.13. Entanglemententropydifferencebetweenthequasi-
y
with two different interchain couplings t (with t =1). All degenerate ground states of N = 4,6,8,10 cylinders as a
⊥ 3 √ y
entanglement-entropyvaluesareshifteddownwardbylog 3 function of interchain coupling t (with t = 1). The hori-
⊥ 3
as explained in the text. The y-intercepts closely match the zontal dashed line denotes the thermodynamic-limit predic-
topological entanglement entropies γ predicted for the Fi- tion logϕ≈0.481. Our N =8 and 10 data further indicate
n y
bonacci phase. that the Fibonacci phase survives even beyond the isotropic
triangularlatticepointinthe2Dlimit,corroboratingtheev-
γ γ e−2γ1 +e−2γε idence presented in Fig. 10.
1 ε
Exact logD≈0.6430 log(D/ϕ)≈0.1617 1
t =0.4a 0.6235 0.1393 1.0442
⊥ The numerical results for S S in Fig. 13 show good
ε 1
t⊥ =0.4b 0.6306 0.1538 1.0186 agreement with this value, e−specially for larger and/or
t =0.6 0.6498 0.1562 1.0043 moreanistropiccylinderswhichareexpectedtohavethe
⊥
weakest finite-size effects.
TABLE I. Intercept values extracted from the fits in Fig. 12
Our numerical evidence regarding the entanglement
fort =0.4,0.6andt =1. Bothagreewiththetheoretically
⊥ 3 entropy, entanglement spectra, and ground-state degen-
predicted topological entanglement entropy shown in the ta-
eracy together strongly indicate the onset of a Fibonacci
ble’sfirstrow,anddemonstratethatthesetofgroundstates
phase over a wide range of parameters.66 These results
iscomplete. (Thelatterconclusionfollowsfromthefactthat
not only corroborate analytical findings for the strongly
the data in the right column are very close to unity.) For
the t⊥ = 0.4 system we give both (a) the intercepts shown anisotropic limit with t⊥/t3 (cid:28) 1; quite remarkably,
in Fig. 12 obtained from fitting all three N = 4,6,8 points Figs.10and13alsorevealthattheFibonacciphaseper-
y
and (b) intercepts obtained from only fitting the N = 4,6 sists into the isotropic-triangular-lattice case t /t =1
y ⊥ 3
entropies that could be computed more accurately. and beyond!
γ = log√3. (The unshifted γ values turn out
− (2/3) − n B. Extent of the Fibonacci Phase
to be negative.) Table I shows that after applying the
shift, the observed γ ’s for two different magnitudes of
n
t agree very well with the theoretical prediction for the The previous subsection reported substantial evidence
⊥
Fibonacci phase quoted in Sec. II: d = 1, d = ϕ and that the model in Eq. (5) realizes the Fibonacci phase
1 ε
= 1+ϕ2, where ϕ=(1+√5)/2 is the golden ratio. along the line t1 =t2 t⊥, for a wide range of t⊥/t3. In
DFurthermore, (e−2γ1 +e−2γε) is very close to unity for light of this finding it≡is interesting to now explore the
both(cid:112)t⊥ values, allowing us to deduce that the system extent of the Fibonacci phase for general t1,t2,t3 ≥ 0.
admits no further ground states beyond 1 and ε . To address this question we fix the ratio t1/t3 and then
The difference Sε S1 of entanglemen|t(cid:105)entro|pi(cid:105)es for vary t2 from 0 to t1, thus scanning a ray in parameter
the ground states h−as been found to converge more space. Alongthislinewecomputethebipartiteentangle-
rapidly as a function of N than linear fits of each in- ment entropy of infinite cylinders, observing clear peaks
y
dividual Sn.62 Within the Fibonacci phase, we expect in the entropy as a function of t2 (Fig. 14) that indi-
cate a transition out of the Fibonacci phase (smoothed
S S = γ +γ =logϕ 0.481 (9) into a crossover due to finite-size effects; we provide sup-
ε 1 ε 1
− − ≈ porting evidence for this interpretation below). These
providedthecylindersizeexceedsthecorrelationlength. peaks are quite broad for systems with t /t 1 but
1 3
≈
10
0.6
S4.0 t1 =0.30 (a)
py t1 =0.40 0.5
o t =0.50
r 1
t
n3.5 t =0.60
E 1 0.4
t
n
e
m
e3.0 t10.3
gl
n
a
t 0.2
n
E2.5 0.60.6 Ny = 8
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1 Ny = 6
t2 0.50.5 Ny = 4
0.0
FIG.14. PeaksinN =8cylinderentan0g.l4e0m.4ententropiesas 0.0 0.1 0.2 0.3 0.4 0.5 0.6
y t
a function of t for various fixed t (t = 1 throughout). To 2
2 1 3
estimate the peak locations, which in the 2D limit coincide
with the transition out of the Fibonta1tc1ci0.p30h.3ase, data points (b) ~t = (1,0,0)
nearthepeakswerefittedtoaquadraticform. Fitsareshown
as thick gray curves while the resulting peak locations are
indicated by vertical dashed lines. The0p.20e.a2k locations give
the four Ny =8 data points in Fig. 15. The t1 =0.4,0.5,0.6 NNy =y =88
data were computed using DMRG with a fixed number of
states m. The t = 0.3 curve was foun0.d10.1by extrapolating NNy =y =66
1
fixed t2 entropies as a power law in m. NNy =y =44
0.00.0
0.00.0 0.10.1 0.20.2 0.30.3 0.40.4 0.50.5 0.60.6 ~t=(0,1,0)
sharpen considerably as t1/t3 approaches zero, presum- t2t2 Fib. phase
ably because for small t /t the y-direction correlation
1,2 3
length remains below the cylinder circumference except
for points very close to the phase boundary. ~t=(0,0,1)
To estimate the locations of the entropy peaks, data
points near each peak were fit to a quadratic as shown
FIG. 15. (a) Estimated extent of the Fibonacci phase near
in Fig. 14 for the case of N =8 cylinders. Figure 15(a)
y
the weakly-coupled-chain limit. Data points correspond to
shows the locations of the entropy peaks thus obtained
the locations of peaks in the entanglement entropy observed
for N =4,6,8 cylinders. For reference the black t =t
y 1 2 when tuning 0<t <t for fixed t in N =4,6,8 cylinders
2 1 1 y
curve represents the critical line for the two-chain limit
(t =1 throughout). The solid curves are fits of the data to
3
discussed in Sec. III; we now see that, as expected, this thefunction(t −t )=C(t +t )8/5withC anN -dependent
2 1 2 1 y
linebroadensintoanextendedphaseonlargercylinders. fittingparameter. Dashedcurvesshowthefitsreflectedacross
As a useful consistency check we note that for a 2D sys- thelinet =t ,sinceinthethermodynamiclimittheHamil-
1 2
tem composed of weakly coupled chains one can analyt- tonian is symmetric about this line. (b) N = 8 fit reflected
y
ically constrain the shape of the phase boundary, which under all t –t –t permutations, estimating the full extent of
1 2 3
should precisely coincide with these entropy peaks ex- theFibonacciphase. Theblacksquaresandtrianglemarkthe
trapolated to the N limit. In particular, scaling locationoftheisotropicsquare-andtriangular-latticemodels.
y
→ ∞
arguments reviewed in Appendix D predict that close to
decoupledchainsthecriticalcouplingst andt satisfy
1c 2c
value of 8/5 to within 10%.
t t =C(t +t )8/5, (10) In the 2D limit, the phase diagram for Eq. (5) must
2c 1c 2c 1c
−
be symmetric under permutations of t , t , and t . We
1 2 3
where C denotes a non-universal constant and we have can exploit this symmetry to roughly estimate the full
employed units where t = 1. Fits of the data to this extent of the Fibonacci phase for all t ,t ,t > 0 based
3 1 2 3
form appear in Fig. 15(a) (solid lines). Despite having on the DMRG data discussed above. As a first step,
only one fitting parameter—and using data for moder- the dashed curves in Fig. 15(a) reflect our DMRG data
ately coupled chains—the fits are remarkably good even about the t =t line. We note, however, that finite-N
1 2 y
forthesmallercylinders. Theagreementmotivatedusto systems break the permutation symmetry; this naive re-
also fit the data to an arbitrary power law; we find that flection thus represents an approximation whose validity
foreachN thefittedexponentagreeswiththepredicted increases with N . Ideally we could have directly mea-
y y