text: t 1t 1 t 1 11 0.6 duce any clear evidence of spontaneous symmetry break- (a) ing or signatures of known topological phases. States appearing when the model contains negative couplings 0.5 also remain mysterious at this point, since throughout this section we have so far focused exclusively on regimes 0.4 with t1,2,3 ≥ 0. If we allow for arbitrary signs of t1,2,3, Fig. 15(b) actually contains only one out of eight faces in the full phase diagram for the Hamiltonian in Eq. (5)! 0.3 (For instance, a second face arises if we take t < 0 but 1 t > 0 and so on; see Fig. 16 for an ‘unfolded’ sketch 2,3 of the eight faces.) In what follows we aim to ﬁll in 0.2 0.60.6 these gaps and deduce the ground states appearing in N = 8 y the broader parameter space. 0.1 Ny = 6 Fortunately, we have not yet exhausted all of the ana- 0.50.5 Ny = 4 lytically soluble windows: with negative couplings a new tractable weakly-coupled-chain limit emerges that lends a 0.0 0.40.4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 great deal of insight into the problem. Consider the case t2 t1 = t2 = 0 with negative intrachain coupling t3 < 0. Un- der a Fradkin-Kadanoﬀ transformation50 each decoupled 0.30.3 (b) ~t = (1, 0, 0) chain here maps to the self-dual point of the antiferro- magnetic three-state Potts model, which is described by (cid:0) (cid:1) a non-chiral U(1) CFT with K-matrix K = 6 − and 0.20.2 central charge c =6 1.67 We respectively denote th6e cor- NNy =y =8 8 responding left- and right-moving chiral boson ﬁelds for 0.10.1 NNy =y =6 6 chain y by φL(y) and φR(y), both 2π-periodic in our nor- malization conventions. NNy =y =4 4 Just as with positive t , we can once again bootstrap 3 0.00.0 oﬀ of these decoupled critical chains to deduce the sys- 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) tem’s response to small but ﬁnite t . To do so we must t t Fib. phase 1,2 2 2 expand the interchain couplings in Eq. (5) in terms of low-energy ﬁelds captured by the CFT. Reference 68 al- ready provided the desired mapping between lattice op- ~t = (0, 0, 1) erators and continuum ﬁelds for the self-dual antiferro- magnetic Potts model; the result reads FIG. 15. captioncolor(a) Estimated extent of the Fibonacci equationcolor phase near the weakly-coupled-chain limit. Data points cor- respond to the locations of peaks in the entanglement entropy αR/L ∼ e2iφR/L + · · · (12) observed when tuning 0 < t < t for ﬁxed t in N = 4, 6, 8 2 1 1 y cylinders (t = 1 throughout). The solid curves are ﬁts of 3 − the data to the function (t t ) = C(t + t )8/5 with C where the ellipsis denotes subdominant contributions. 2 1 2 1 an Ny-dependent ﬁtting parameter. Dashed curves show the Using this mapping, the interchain Hamiltonian H⊥ ﬁts reﬂected across the line t1 = t2, since in the thermody- takes the form namic limit the Hamiltonian is symmetric about this line. (b) equationcolor N = 8 ﬁt reﬂected under all t –t –t permutations, estimat- inyg the full extent of the Fibon1acc2i p3hase. The black squares (cid:88) (cid:90) (cid:104) − and triangle mark the location of the isotropic square- and H⊥ ∼ (t1 + t2)e2iφR(y)e 2iφL(y+1) + H.c. triangular-lattice models. y x (13) (cid:105) + (t − t )X(φ (y), φ (y + 1)) . 1 2 L,R L,R the isotropic square lattice (points in Fig. 15(b) marked by open squares) realizes a state distinct from the The ﬁrst term possesses scaling dimension ∆ = 2 at 3 Fibonacci phase. Pinpointing the precise nature of the decoupled-chain ﬁxed point. In the second line X this state is, however, rather nontrivial. Approaching represents (numerous) subleading terms with dimension the square lattice from the weakly-coupled-chain limit ∆ = 5; these can be enumerated explicitly but we refrain 3 0 < t /t (cid:28) 1 with t = 0 does not provide immediate from doing so here. 1 3 2 insight, as the continuum limit of the t coupling yields Except with ﬁne-tuning, the 2D system’s properties 1 competing relevant interactions whose eﬀect is diﬃcult to are generically dominated by the more relevant (t + t ) 1 2 understand analytically. Similarly, applying DMRG to term above. Its eﬀect is relatively easy to understand for small, anisotropic square-lattice cylinders does not pro- systems composed of a large but ﬁnite number of chains. ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: t

font: default

text: t

text: duce

text: any

text: clear

text: evidence

text: of

text: spontaneous

text: symmetry

text: break-

text: (a)

text: ing

text: or

text: signatures

text: of

text: known

text: topological

text: phases.

text: States

text: appearing

text: when

text: the

text: model

text: contains

text: negative

text: couplings

text: also

text: remain

text: mysterious

text: at

text: this

text: point,

text: since

text: throughout

text: this

text: section

text: we

text: have

text: so

text: far

text: focused

text: exclusively

text: on

text: regimes

text: with

text: If

text: we

text: allow

text: for

text: arbitrary

text: signs

text: of

text: Fig.

text: actually

text: contains

text: only

text: one

text: out

text: of

text: eight

text: faces

text: in

text: the

text: full

text: phase

text: diagram

text: for

text: the

text: Hamiltonian

text: in

text: Eq.

text: (For

text: instance,

text: a

text: second

text: face

text: arises

text: if

text: we

text: take

text: t

text: but

text: t

text: and

text: so

text: on;

text: see

text: Fig.

text: for

text: an

text: ‘unfolded’

text: sketch

text: of

text: the

text: eight

text: faces.)

text: In

text: what

text: follows

text: we

text: aim

text: to

text: ﬁll

text: in

text: these

text: gaps

text: and

text: deduce

text: the

text: ground

text: states

text: appearing

text: in

text: N

text: y

text: the

text: broader

text: parameter

text: space.

text: Ny

text: Fortunately,

text: we

text: have

text: not

text: yet

text: exhausted

text: all

text: of

text: the

text: ana-

text: Ny

text: lytically

text: soluble

text: windows:

text: with

text: negative

text: couplings

text: a

text: new

text: tractable

text: weakly-coupled-chain

text: limit

text: emerges

text: that

text: lends

text: a

text: great

text: deal

text: of

text: insight

text: into

text: the

text: problem.

text: Consider

text: the

text: case

text: with

text: negative

text: intrachain

text: coupling

text: Un-

text: der

text: a

text: Fradkin-Kadanoﬀ

text: each

text: decoupled

text: (b)

text: ~t

text: chain

text: here

text: maps

text: to

text: the

text: self-dual

text: point

text: of

text: the

text: antiferro-

text: magnetic

text: three-state

text: Potts

text: model,

text: which

text: is

text: described

text: by

text: a

text: non-chiral

text: CFT

text: with

text: K-matrix

text: K

text: and

text: central

text: charge

text: c

text: We

text: respectively

text: denote

text: cor-

text: NNy

text: =y

text: responding

text: left-

text: and

text: right-moving

text: chiral

text: boson

text: ﬁelds

text: for

text: NNy

text: =y

text: chain

text: y

text: by

text: φL(y)

text: and

text: φR(y),

text: both

text: in

text: our

text: nor-

text: malization

text: conventions.

text: NNy

text: =y

text: Just

text: as

text: with

text: positive

text: t

text: we

text: can

text: once

text: again

text: bootstrap

text: oﬀ

text: of

text: these

text: decoupled

text: critical

text: chains

text: to

text: deduce

text: the

text: sys-

text: ~t

text: tem’s

text: response

text: to

text: small

text: but

text: ﬁnite

text: t

text: To

text: do

text: so

text: we

text: must

text: t

text: t

text: Fib.

text: phase

text: expand

text: the

text: interchain

text: couplings

text: in

text: Eq.

text: in

text: terms

text: of

text: low-energy

text: ﬁelds

text: captured

text: by

text: the

text: CFT.

text: Reference

text: al-

text: ready

text: provided

text: the

text: desired

text: mapping

text: between

text: lattice

text: op-

text: ~t

text: erators

text: and

text: continuum

text: ﬁelds

text: for

text: the

text: self-dual

text: antiferro-

text: magnetic

text: Potts

text: model;

text: the

text: result

text: reads

text: FIG.

text: captioncolor(a)

text: Estimated

text: extent

text: of

text: the

text: Fibonacci

text: equationcolor

text: phase

text: near

text: the

text: weakly-coupled-chain

text: limit.

text: Data

text: points

text: cor-

text: respond

text: to

text: the

text: locations

text: of

text: peaks

text: in

text: the

text: entanglement

text: entropy

text: αR/L

text: observed

text: when

text: tuning

text: t

text: t

text: for

text: ﬁxed

text: t

text: in

text: N

text: y

text: cylinders

text: (t

text: throughout).

text: The

text: solid

text: curves

text: are

text: ﬁts

text: of

text: the

text: data

text: to

text: the

text: function

text: (t

text: t

text: C(t

text: t

text: with

text: C

text: where

text: the

text: ellipsis

text: denotes

text: subdominant

text: contributions.

text: an

text: Ny-dependent

text: ﬁtting

text: parameter.

text: Dashed

text: curves

text: show

text: the

text: Using

text: this

text: mapping,

text: the

text: interchain

text: Hamiltonian

text: H⊥

text: ﬁts

text: reﬂected

text: across

text: the

text: line

text: since

text: in

text: the

text: thermody-

text: takes

text: the

text: form

text: namic

text: limit

text: the

text: Hamiltonian

text: is

text: symmetric

text: about

text: this

text: line.

text: (b)

text: equationcolor

text: N

text: ﬁt

text: reﬂected

text: under

text: all

text: t

text: –t

text: –t

text: permutations,

text: estimat-

text: inyg

text: the

text: full

text: extent

text: of

text: the

text: The

text: black

text: squares

text: and

text: triangle

text: mark

text: the

text: location

text: of

text: the

text: isotropic

text: square-

text: and

text: H⊥

text: H.c.

text: triangular-lattice

text: models.

text: y

text: x

text: (t

text: t

text: )X(φ

text: (y),

text: φ

text: (y

text: L,R

text: L,R

text: the

text: isotropic

text: square

text: lattice

text: (points

text: in

text: Fig.

text: marked

text: by

text: open

text: squares)

text: realizes

text: a

text: state

text: distinct

text: from

text: the

text: The

text: ﬁrst

text: term

text: possesses

text: scaling

text: dimension

text: at

text: Fibonacci

text: phase.

text: Pinpointing

text: the

text: precise

text: nature

text: of

text: the

text: decoupled-chain

text: ﬁxed

text: point.

text: In

text: the

text: second

text: line

text: X

text: this

text: state

text: is,

text: however,

text: rather

text: nontrivial.

text: Approaching

text: represents

text: (numerous)

text: subleading

text: terms

text: with

text: dimension

text: the

text: square

text: lattice

text: from

text: the

text: weakly-coupled-chain

text: limit

text: these

text: can

text: be

text: enumerated

text: explicitly

text: but

text: we

text: refrain

text: t

text: /t

text: with

text: t

text: does

text: not

text: provide

text: immediate

text: from

text: doing

text: so

text: here.

text: insight,

text: as

text: the

text: continuum

text: limit

text: of

text: the

text: t

text: coupling

text: yields

text: Except

text: with

text: ﬁne-tuning,

text: the

text: system’s

text: properties

text: competing

text: relevant

text: interactions

text: whose

text: eﬀect

text: is

text: diﬃcult

text: to

text: are

text: generically

text: dominated

text: by

text: the

text: more

text: relevant

text: (t

text: t

text: understand

text: analytically.

text: Similarly,

text: applying

text: DMRG

text: to

text: term

text: above.

text: Its

text: eﬀect

text: is

text: relatively

text: easy

text: to

text: understand

text: for

text: small,

text: anisotropic

text: square-lattice

text: cylinders

text: does

text: not

text: pro-

text: systems

text: composed

text: of

text: a

text: large

text: but

text: ﬁnite

text: number

text: of

text: chains.

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 15 couplings yet captures highly exotic topological phases Appendix A: Mapping to Potts chain of matter; this is possible because even with only bilin- ear terms the Hamiltonian is strongly interacting.29 The Here we very brieﬂy review the mapping of a 1D Z model’s simplicity is certainly a boon for the prospect of parafermion chain to the three-state quantum Potts 3 realizing Fibonacci anyons in the underlying quantum- model.29,50 Consider the Hamiltonian in Eq. (3), re- Hall setups. peated here for clarity, Our results also yield indirect implications for systems equationcolor built out of local bosonic degrees of freedom. In an inter- (cid:88)(cid:104) † † esting recent study, Barkeshli et al.72 introduced general- H = − ω (J α α + h α α − ) R,2n+1 R,2n R,2n R,2n 1 ized Kitaev honeycomb models for which the constituent n (cid:105) spin operators could be decomposed in terms of ‘slave + H.c. , (A1) parafermions’. Within that representation the authors identiﬁed a quasi-1D limit that also realizes a phase with Fibonacci anyons, among other quasiparticles. Refer- where again ω = ei(2π/3). We have chosen to express ence 73 conjectured the appearance of Fibonacci anyons (somewhat arbitrarily) the Hamiltonian using ‘right’ in related local bosonic models (though Abelian topo- † parafermion operators that satisfy α3 = 1, α = α2 , logical orders were found in a mean-ﬁeld treatment). R,j R,j R,j and By applying the ‘inverse’ logic from these works one equationcolor can back out local bosonic setups corresponding to the parafermion system that we analyzed. Doing so imme- (cid:48)− α α (cid:48) = ei(2π/3)sgn(j j)α (cid:48)α . (A2) diately allows one to construct bosonic lattice models R,j R,j R,j R,j supporting a phase with Fibonacci anyons—though the Hamiltonians involve somewhat elaborate multi-boson One could equally well employ ‘left’ parafermion oper- interactions. It would be interesting in future numerical ators whose commutator instead reads work to explore the extent of that state when the model equationcolor is deformed towards a simpler limit with only quadratic − (cid:48)− couplings. α α (cid:48) = e i(2π/3)sgn(j j)α (cid:48)α . (A3) L,j L,j L,j L,j One particularly intriguiging wide-open question is whether the Fibonacci phase can appear in a reason- able spatially uniform Hamiltonian describing Abelian In either representation, the mapping to the Potts quantum Hall/superconductor heterostructures or bilay- model proceeds upon nonlocally decomposing the ers (i.e., with the periodic modulations trapping the parafermions via parafermion modes ‘smeared out’). If found, such setups equationcolor might oﬀer relatively simple experimental routes to Fi- (cid:89) (cid:89) bonacci anyons. The fact that we identiﬁed an isotropic αR,2n−1 = σn τm, αR,2n = ωσn τm (A4) ≤ triangular lattice model supporting the Fibonacci phase m<n m n is encouraging and may oﬀer some clues as to how one (cid:89) † (cid:89) † α − = σ τ , α = ω2σ τ .(A5) can construct spatially uniform analogues. We also note L,2n 1 n m L,2n n ≤ m that Barkeshli and Vaezi74 proposed remarkably simple m<n m n ν = 1 + 1 bilayers as uniform Fibonacci anyon hosts 3 3 On the right sides we have introduced local bosonic based on an analysis of the thin-torus limit. A concerted Potts operators σ , τ that satisfy numerical push in this direction seems a very worthwhile n n equationcolor complement to the lattice simulations performed here. † † sectioncolor σ3 = τ3 = 1, σ = σ2, τ = τ2 (A6) n n n n n n ACKNOWLEDGMENTS together with the commutator E. M. S. acknowledges helpful discussions with Lukasz equationcolor Cincio and Juan Carrasquilla. Simulations were per- formed on the Perimeter Institute HPC. We are grateful σ τ = ωτ σ . (A7) n n n n for generous support from the Sherman Fairchild Foun- dation (R. M.), the National Science Foundation through grant DMR-1341822 (J. A.); the Alfred P. Sloan Foun- (In contrast to the parafermion operators, σ and τ m n dation (J. A.); the Caltech Institute for Quantum In- commute for m (cid:54)= n.) Notice that the right and left rep- formation and Matter, an NSF Physics Frontiers Center resentations diﬀer primarily in the form of their non-local with support of the Gordon and Betty Moore Founda- τ strings. One can readily verify that the decomposi- m tion through Grant GBMF1250; and the Walter Burke tions in Eq. (A4) and (A5) preserve the parafermion op- Institute for Theoretical Physics at Caltech. erator algebra. Inserting these expressions into Eq. (A1) sectioncolor yields the desired quantum Potts Hamiltonian, ##LTLine## ##LTLine##

label: paragraph

text: couplings

text: yet

text: captures

text: highly

text: exotic

text: topological

text: phases

text: Appendix

text: A:

text: Mapping

text: to

text: Potts

text: chain

text: of

text: matter;

text: this

text: is

text: possible

text: because

text: even

text: with

text: only

text: bilin-

text: ear

text: terms

text: the

text: Hamiltonian

text: is

text: strongly

text: The

text: Here

text: we

text: very

text: brieﬂy

text: review

text: the

text: mapping

text: of

text: a

text: Z

text: model’s

text: simplicity

text: is

text: certainly

text: a

text: boon

text: for

text: the

text: prospect

text: of

text: parafermion

text: chain

text: to

text: the

text: three-state

text: quantum

text: Potts

text: realizing

text: Fibonacci

text: anyons

text: in

text: the

text: underlying

text: quantum-

text: Consider

text: the

text: Hamiltonian

text: in

text: Eq.

text: re-

text: Hall

text: setups.

text: peated

text: here

text: for

text: clarity,

text: Our

text: results

text: also

text: yield

text: indirect

text: implications

text: for

text: systems

text: equationcolor

text: built

text: out

text: of

text: local

text: bosonic

text: degrees

text: of

text: freedom.

text: In

text: an

text: inter-

text: esting

text: recent

text: study,

text: Barkeshli

text: et

text: introduced

text: general-

text: H

text: ω

text: (J

text: α

text: α

text: h

text: α

text: α

text: ized

text: Kitaev

text: honeycomb

text: models

text: for

text: which

text: the

text: constituent

text: n

text: spin

text: operators

text: could

text: be

text: decomposed

text: in

text: terms

text: of

text: ‘slave

text: H.c.

text: parafermions’.

text: Within

text: that

text: representation

text: the

text: authors

text: identiﬁed

text: a

text: limit

text: that

text: also

text: realizes

text: a

text: phase

text: with

text: Fibonacci

text: anyons,

text: among

text: other

text: quasiparticles.

text: Refer-

text: where

text: again

text: ω

text: We

text: have

text: chosen

text: to

text: express

text: ence

text: conjectured

text: the

text: appearance

text: of

text: Fibonacci

text: anyons

text: (somewhat

text: arbitrarily)

text: the

text: Hamiltonian

text: using

text: ‘right’

text: in

text: related

text: local

text: bosonic

text: models

text: (though

text: Abelian

text: topo-

text: parafermion

text: operators

text: that

text: satisfy

text: α3

text: α

text: α2

text: logical

text: orders

text: were

text: found

text: in

text: a

text: mean-ﬁeld

text: treatment).

text: R,j

text: R,j

text: R,j

text: and

text: By

text: applying

text: the

text: ‘inverse’

text: logic

text: from

text: these

text: works

text: one

text: equationcolor

text: can

text: back

text: out

text: local

text: bosonic

text: setups

text: corresponding

text: to

text: the

text: parafermion

text: system

text: that

text: we

text: analyzed.

text: Doing

text: so

text: imme-

text: α

text: α

text: j)α

text: diately

text: allows

text: one

text: to

text: construct

text: bosonic

text: lattice

text: models

text: R,j

text: R,j

text: R,j

text: R,j

text: supporting

text: a

text: phase

text: with

text: Fibonacci

text: anyons—though

text: the

text: Hamiltonians

text: involve

text: somewhat

text: elaborate

text: multi-boson

text: One

text: could

text: equally

text: well

text: employ

text: ‘left’

text: parafermion

text: oper-

text: interactions.

text: It

text: would

text: be

text: interesting

text: in

text: future

text: numerical

text: ators

text: whose

text: commutator

text: instead

text: reads

text: work

text: to

text: explore

text: the

text: extent

text: of

text: that

text: state

text: when

text: the

text: model

text: equationcolor

text: is

text: deformed

text: towards

text: a

text: simpler

text: limit

text: with

text: only

text: quadratic

text: couplings.

text: α

text: α

text: e

text: j)α

text: L,j

text: L,j

text: L,j

text: L,j

text: One

text: particularly

text: intriguiging

text: wide-open

text: question

text: is

text: whether

text: the

text: Fibonacci

text: phase

text: can

text: appear

text: in

text: a

text: reason-

text: able

text: spatially

text: uniform

text: Hamiltonian

text: describing

text: Abelian

text: In

text: either

text: representation,

text: the

text: mapping

text: to

text: the

text: Potts

text: quantum

text: Hall/superconductor

text: heterostructures

text: or

text: bilay-

text: model

text: proceeds

text: upon

text: nonlocally

text: decomposing

text: the

text: ers

text: (i.e.,

text: with

text: the

text: periodic

text: modulations

text: trapping

text: the

text: parafermions

text: via

text: parafermion

text: modes

text: ‘smeared

text: out’).

text: If

text: found,

text: such

text: setups

text: equationcolor

text: might

text: oﬀer

text: relatively

text: simple

text: experimental

text: routes

text: to

text: Fi-

text: bonacci

text: anyons.

text: The

text: fact

text: that

text: we

text: identiﬁed

text: an

text: isotropic

text: σn

text: τm,

text: ωσn

text: τm

text: triangular

text: lattice

text: model

text: supporting

text: the

text: Fibonacci

text: phase

text: m<n

text: m

text: n

text: is

text: encouraging

text: and

text: may

text: oﬀer

text: some

text: clues

text: as

text: to

text: how

text: one

text: α

text: σ

text: τ

text: α

text: ω2σ

text: τ

text: can

text: construct

text: spatially

text: uniform

text: analogues.

text: We

text: also

text: note

text: n

text: m

text: n

text: m

text: that

text: Barkeshli

text: and

text: proposed

text: remarkably

text: simple

text: m<n

text: m

text: n

text: ν

text: bilayers

text: as

text: uniform

text: Fibonacci

text: anyon

text: hosts

text: On

text: the

text: right

text: sides

text: we

text: have

text: introduced

text: local

text: bosonic

text: based

text: on

text: an

text: analysis

text: of

text: the

text: thin-torus

text: limit.

text: A

text: concerted

text: Potts

text: operators

text: σ

text: τ

text: that

text: satisfy

text: numerical

text: push

text: in

text: this

text: direction

text: seems

text: a

text: very

text: worthwhile

text: n

text: n

text: equationcolor

text: complement

text: to

text: the

text: lattice

text: simulations

text: performed

text: here.

text: sectioncolor

text: σ3

text: τ3

text: σ

text: τ

text: τ2

text: n

text: n

text: n

text: n

text: n

text: n

text: ACKNOWLEDGMENTS

text: together

text: with

text: the

text: commutator

text: E.

text: M.

text: S.

text: acknowledges

text: helpful

text: discussions

text: with

text: Lukasz

text: equationcolor

text: Cincio

text: and

text: Juan

text: Carrasquilla.

text: Simulations

text: were

text: per-

text: formed

text: on

text: the

text: Perimeter

text: Institute

text: HPC.

text: We

text: are

text: grateful

text: σ

text: τ

text: ωτ

text: σ

text: n

text: n

text: n

text: n

text: for

text: generous

text: support

text: from

text: the

text: Sherman

text: Fairchild

text: Foun-

text: dation

text: (R.

text: M.),

text: the

text: National

text: Science

text: Foundation

text: through

text: grant

text: (J.

text: A.);

text: the

text: Alfred

text: P.

text: Sloan

text: Foun-

text: (In

text: contrast

text: to

text: the

text: parafermion

text: operators,

text: σ

text: and

text: τ

text: m

text: n

text: dation

text: (J.

text: A.);

text: the

text: Caltech

text: Institute

text: for

text: Quantum

text: In-

text: commute

text: for

text: m

text: n.)

text: Notice

text: that

text: the

text: right

text: and

text: left

text: rep-

text: formation

text: and

text: Matter,

text: an

text: NSF

text: Physics

text: Frontiers

text: Center

text: resentations

text: diﬀer

text: primarily

text: in

text: the

text: form

text: of

text: their

text: non-local

text: with

text: support

text: of

text: the

text: Gordon

text: and

text: Betty

text: Moore

text: Founda-

text: τ

text: strings.

text: One

text: can

text: readily

text: verify

text: that

text: the

text: decomposi-

text: m

text: tion

text: through

text: Grant

text: and

text: the

text: Walter

text: Burke

text: tions

text: in

text: Eq.

text: and

text: preserve

text: the

text: parafermion

text: op-

text: Institute

text: for

text: Theoretical

text: Physics

text: at

text: Caltech.

text: erator

text: algebra.

text: Inserting

text: these

text: expressions

text: into

text: Eq.

text: sectioncolor

text: yields

text: the

text: desired

text: quantum

text: Potts

text: Hamiltonian,

text: ##LTLine##

text: ##LTLine##


text: S − S 1 ε S yportnE tnemelgnatnE 10 0.56 Ny = 4 4.0 t1 = 0.30 N = 6 t1 = 0.40 y t = 0.50 N = 8 1 y 0.52 N = 10 3.5 t1 = 0.60 y 3.0 0.48 2.5 0.44 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 t 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 2 t ⊥ FIG. 14. captioncolorPeaks in N = 8 cylinder entangle- y FIG. 13. captioncolorEntanglement entropy diﬀerence be- ment entropies as a function of t for various ﬁxed t (t = 1 2 1 3 tween the quasi-degenerate ground states of Ny = 4, 6, 8, 10 throughout). To estimate the peak locations, which in the 2D cylinders as a function of interchain coupling t⊥ (with t3 = 1). limit coincide with the transition out of the Fibonacci phase, The horizontal da≈shed line denotes the thermodynamic-limit data points near the peaks were ﬁtted to a quadratic form. prediction log ϕ 0.481. Our Ny = 8 and 10 data further Fits are shown as thick gray curves while the resulting peak indicate that the Fibonacci phase survives even beyond the locations are indicated by vertical dashed lines. The peak isotropic triangular lattice point in the 2D limit, corroborat- locations give the four N = 8 data points in Fig. 15. The y ing the evidence presented in Fig. 10. t = 0.4, 0.5, 0.6 data were computed using DMRG with a 1 ﬁxed number of states m. The t = 0.3 curve was found by 1 extrapolating ﬁxed t entropies as a power law in m. 2 peaks are quite broad for systems with t /t ≈ 1 but 1 3 sharpen considerably as t /t approaches zero, presum- 1 3 for each N the ﬁtted exponent agrees with the predicted ably because for small t /t the y-direction correlation y 1,2 3 value of 8/5 to within 10%. length remains below the cylinder circumference except In the 2D limit, the phase diagram for Eq. (5) must for points very close to the phase boundary. be symmetric under permutations of t , t , and t . We To estimate the locations of the entropy peaks, data 1 2 3 can exploit this symmetry to roughly estimate the full points near each peak were ﬁt to a quadratic as shown extent of the Fibonacci phase for all t , t , t > 0 based in Fig. 14 for the case of N = 8 cylinders. Figure 15(a) 1 2 3 y on the DMRG data discussed above. As a ﬁrst step, shows the locations of the entropy peaks thus obtained the dashed curves in Fig. 15(a) reﬂect our DMRG data for N = 4, 6, 8 cylinders. For reference the black t = t y 1 2 about the t = t line. We note, however, that ﬁnite-N curve represents the critical line for the two-chain limit 1 2 y systems break the permutation symmetry; this naive re- discussed in Sec. III; we now see that, as expected, this ﬂection thus represents an approximation whose validity line broadens into an extended phase on larger cylinders. increases with N . Ideally we could have directly mea- As a useful consistency check we note that for a 2D sys- y sured entropy peaks near the expected phase transition tem composed of weakly coupled chains one can analyt- curve on the t > t side also. But we found ﬁnite cylin- ically constrain the shape of the phase boundary, which 2 1 ders within this region to be much more highly entangled, should precisely coincide with these entropy peaks ex- preventing an accurate DMRG study. We then further trapolated to the N → ∞ limit. In particular, scaling y permute our N = 8 data to arrive at the phase diagram arguments reviewed in Appendix D predict that close to y shown in Fig. 15(b), parametrized in terms of a vector decoupled chains the critical couplings t and t satisfy 1c 2c equationcolor equationcolor (cid:126)t = (t , t , t ) (11) t − t = C (t + t )8/5, (10) 1 2 3 2c 1c 2c 1c with t + t + t = 1 and all couplings non-negative. 1 2 3 where C denotes a non-universal constant and we have Notably, the isotropic triangular lattice point sits deep employed units where t = 1. Fits of the data to this within the center of the phase—consistent with our re- 3 form appear in Fig. 15(a) (solid lines). Despite having sults from the previous subsection. only one ﬁtting parameter—and using data for moder- sectioncolor ately coupled chains—the ﬁts are remarkably good even C. Global Phase Diagram of the t1–t2–t3 Model for the smaller cylinders. The agreement motivated us to also ﬁt the data to an arbitrary power law; we ﬁnd that Our numerical results presented above indicate that ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S

text: S

text: ε

text: S

text: yportnE

text: tnemelgnatnE

text: Ny

text: N

text: y

text: t

text: N

text: y

text: N

text: y

text: t

text: t

text: FIG.

text: captioncolorPeaks

text: in

text: N

text: cylinder

text: entangle-

text: y

text: FIG.

text: captioncolorEntanglement

text: entropy

text: diﬀerence

text: be-

text: ment

text: entropies

text: as

text: a

text: function

text: of

text: t

text: for

text: various

text: ﬁxed

text: t

text: (t

text: tween

text: the

text: quasi-degenerate

text: ground

text: states

text: of

text: Ny

text: throughout).

text: To

text: estimate

text: the

text: peak

text: locations,

text: which

text: in

text: the

text: cylinders

text: as

text: a

text: function

text: of

text: interchain

text: coupling

text: t⊥

text: (with

text: limit

text: coincide

text: with

text: the

text: transition

text: out

text: of

text: the

text: Fibonacci

text: phase,

text: The

text: horizontal

text: da≈shed

text: line

text: denotes

text: the

text: thermodynamic-limit

text: data

text: points

text: near

text: the

text: peaks

text: were

text: ﬁtted

text: to

text: a

text: quadratic

text: form.

text: prediction

text: log

text: ϕ

text: Our

text: Ny

text: and

text: data

text: further

text: Fits

text: are

text: shown

text: as

text: thick

text: gray

text: curves

text: while

text: the

text: resulting

text: peak

text: indicate

text: that

text: the

text: Fibonacci

text: phase

text: survives

text: even

text: beyond

text: the

text: locations

text: are

text: indicated

text: by

text: vertical

text: dashed

text: lines.

text: The

text: peak

text: isotropic

text: triangular

text: lattice

text: point

text: in

text: the

text: limit,

text: corroborat-

text: locations

text: give

text: the

text: four

text: N

text: data

text: points

text: in

text: Fig.

text: The

text: y

text: ing

text: the

text: evidence

text: presented

text: in

text: Fig.

text: t

text: data

text: were

text: computed

text: using

text: DMRG

text: with

text: a

text: ﬁxed

text: number

text: of

text: states

text: m.

text: The

text: t

text: curve

text: was

text: found

text: by

text: extrapolating

text: ﬁxed

text: t

text: entropies

text: as

text: a

text: power

text: law

text: in

text: m.

text: peaks

text: are

text: quite

text: broad

text: for

text: systems

text: with

text: t

text: /t

text: but

text: sharpen

text: considerably

text: as

text: t

text: /t

text: approaches

text: zero,

text: presum-

text: for

text: each

text: N

text: the

text: ﬁtted

text: exponent

text: agrees

text: with

text: the

text: predicted

text: ably

text: because

text: for

text: small

text: t

text: /t

text: the

text: y-direction

text: correlation

text: y

text: value

text: of

text: to

text: within

text: length

text: remains

text: below

text: the

text: cylinder

text: circumference

text: except

text: In

text: the

text: limit,

text: the

text: phase

text: diagram

text: for

text: Eq.

text: must

text: for

text: points

text: very

text: close

text: to

text: the

text: phase

text: boundary.

text: be

text: symmetric

text: under

text: permutations

text: of

text: t

text: t

text: and

text: t

text: We

text: To

text: estimate

text: the

text: locations

text: of

text: the

text: entropy

text: peaks,

text: data

text: can

text: exploit

text: this

text: symmetry

text: to

text: roughly

text: estimate

text: the

text: full

text: points

text: near

text: each

text: peak

text: were

text: ﬁt

text: to

text: a

text: quadratic

text: as

text: shown

text: extent

text: of

text: the

text: Fibonacci

text: phase

text: for

text: all

text: t

text: t

text: t

text: based

text: in

text: Fig.

text: for

text: the

text: case

text: of

text: N

text: cylinders.

text: Figure

text: y

text: on

text: the

text: DMRG

text: data

text: discussed

text: above.

text: As

text: a

text: ﬁrst

text: step,

text: shows

text: the

text: locations

text: of

text: the

text: entropy

text: peaks

text: thus

text: obtained

text: the

text: dashed

text: curves

text: in

text: Fig.

text: reﬂect

text: our

text: DMRG

text: data

text: for

text: N

text: cylinders.

text: For

text: reference

text: the

text: black

text: t

text: t

text: y

text: about

text: the

text: t

text: t

text: line.

text: We

text: note,

text: however,

text: that

text: ﬁnite-N

text: curve

text: represents

text: the

text: critical

text: line

text: for

text: the

text: two-chain

text: limit

text: y

text: systems

text: break

text: the

text: permutation

text: symmetry;

text: this

text: naive

text: re-

text: discussed

text: in

text: Sec.

text: III;

text: we

text: now

text: see

text: that,

text: as

text: expected,

text: this

text: ﬂection

text: thus

text: represents

text: an

text: approximation

text: whose

text: validity

text: line

text: broadens

text: into

text: an

text: extended

text: phase

text: on

text: larger

text: cylinders.

text: increases

text: with

text: N

text: Ideally

text: we

text: could

text: have

text: directly

text: mea-

text: As

text: a

text: useful

text: consistency

text: check

text: we

text: note

text: that

text: for

text: a

text: sys-

text: y

text: sured

text: entropy

text: peaks

text: near

text: the

text: expected

text: phase

text: transition

text: tem

text: composed

text: of

text: weakly

text: coupled

text: chains

text: one

text: can

text: analyt-

text: curve

text: on

text: the

text: t

text: t

text: side

text: also.

text: But

text: we

text: found

text: ﬁnite

text: cylin-

text: ically

text: constrain

text: the

text: shape

text: of

text: the

text: phase

text: boundary,

text: which

text: ders

text: within

text: this

text: region

text: to

text: be

text: much

text: more

text: highly

text: entangled,

text: should

text: precisely

text: coincide

text: with

text: these

text: entropy

text: peaks

text: ex-

text: preventing

text: an

text: accurate

text: DMRG

text: study.

text: We

text: then

text: further

text: trapolated

text: to

text: the

text: N

text: limit.

text: In

text: particular,

text: scaling

text: y

text: permute

text: our

text: N

text: data

text: to

text: arrive

text: at

text: the

text: phase

text: diagram

text: arguments

text: reviewed

text: in

text: Appendix

text: D

text: predict

text: that

text: close

text: to

text: y

text: shown

text: in

text: Fig.

text: parametrized

text: in

text: terms

text: of

text: a

text: vector

text: decoupled

text: chains

text: the

text: critical

text: couplings

text: t

text: and

text: t

text: satisfy

text: equationcolor

text: equationcolor

text: (t

text: t

text: t

text: t

text: t

text: C

text: (t

text: t

text: with

text: t

text: t

text: t

text: and

text: all

text: couplings

text: non-negative.

text: where

text: C

text: denotes

text: a

text: non-universal

text: constant

text: and

text: we

text: have

text: Notably,

text: the

text: isotropic

text: triangular

text: lattice

text: point

text: sits

text: deep

text: employed

text: units

text: where

text: t

text: Fits

text: of

text: the

text: data

text: to

text: this

text: within

text: the

text: center

text: of

text: the

text: phase—consistent

text: with

text: our

text: re-

text: form

text: appear

text: in

text: Fig.

text: (solid

text: lines).

text: Despite

text: having

text: sults

text: from

text: the

text: previous

text: subsection.

text: only

text: one

text: ﬁtting

text: parameter—and

text: using

text: data

text: for

text: moder-

text: sectioncolor

text: ately

text: coupled

text: chains—the

text: ﬁts

text: are

text: remarkably

text: good

text: even

text: C.

text: Global

text: Phase

text: Diagram

text: of

text: the

text: Model

text: for

text: the

text: smaller

text: cylinders.

text: The

text: agreement

text: motivated

text: us

text: to

text: also

text: ﬁt

text: the

text: data

text: to

text: an

text: arbitrary

text: power

text: law;

text: we

text: ﬁnd

text: that

text: Our

text: numerical

text: results

text: presented

text: above

text: indicate

text: that

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 shocks, then the ratio of optical and gamma-ray ﬂuxes should be relatively constant in time, assuming that the microphysical pa- rameters of the shocks remain constant (and that the conditions τp−p > v4/c and tcool/t < 1 remain satisﬁed in hadronic and lep- tonic scenarios, respectively). Nova Sco is consistent with a tem- porally constant ratio Lγ/Lopt, while nova Del shows evidence for the gamma-ray emission becoming relatively stronger with time (Fig. 3). The apparent increase of Lγ/Lopt in nova Del could result from the reverse shock thermal emission becoming less radiatively ∝ eﬃcient with time, i.e. χ t (eq. [7]). Alternatively, it may reﬂect the early transition in this event to a nebular phase (as occurred on Day 11; Skopal et al. 2014), resulting in a loss of ﬂux out of the measured optical band and into the UV. If nova shocks are indeed ∼ radiative, then we predict that Lγ/Lopt should not exceed 0.06 at any time, since this represents the limit of shock-dominated emis- sion fsh = 1 (eq. [13]) for realistic maximum eﬃciencies (cid:15)γ ∼< 0.3, (cid:15)nth ∼< 0.2. The gamma-ray to optical ﬂux ratio also places a lower limit on the acceleration eﬃciency (cid:15)nth of relativistic non-thermal par- ticles at non-relativistic shocks. For nova Sco and nova Del we constrain (cid:15)nth ∼> 10−2 and 10−3, respectively (Table 1). Unlike tra- ditional studies modeling particle acceleration in supernova rem- nants, nova shocks evolve in real-time, in principle allowing for the study of time-dependent eﬀects. Models used to constrain par- ticle acceleration in supernova remnants also sometimes require as- sumptions about the escape fraction of relativistic particles, e.g. if gamma-rays are produced by the collision of relativistic protons with nearby molecular clouds of irregular geometry. By contrast, nova ejecta serves as a relatively eﬃcient hadronic “calorimeter” since relativistic protons are probably trapped when gamma-rays are observed (eq. [19]). Neither hadronic nor leptonic scenarios for nova gamma-ray − emission can be ruled out by modeling of the γ ray spectrum alone (Ackermann et al. 2014). However, the tension between the value FVi1g3u2r4eS3c.oR(attoiopopfagnaeml)manad-rVay33an9dDoeplt(icbaolttloummipnaonseitli)eas,sLaγf/uLnocptti,ofnorofnotivmaee (cid:15)nth ∼> 10−2 we ﬁnd is required in nova Sco and the much lower elec- since outburst. In Nova Sco, large squares show nights with full NIR+VBR tron acceleration eﬃciency (cid:15)nth ∼< 10−3 inferred from observations coverage, while small triangles show the ﬂux ratio estimated using just the of supernova remnants (e.g. Morlino & Caprioli 2012) and theoret- V-band magnitude adopting the same bolometric correction as measured ical modeling (Kato 2014; Park et al. 2014) appears to disfavor the with the full data color data on Day 15. leptonic scenario. Leptonic models also require a ﬂat injected spec- trum p = 2 to be energetically feasible (eq. [23]). In contrast, ion ∼ acceleration eﬃciencies as high as (cid:15)nth 0.1 are found for non- This shock-powered emission mechanism is similar to that relativistic shocks by hybrid PIC simulations, but only in cases at work in some core collapse supernovae when the ejecta when the upstream magnetic ﬁeld is parallel to the shock normal from the exploding star interacts with dense circumstellar matter (Caprioli & Spitkovsky 2014a); for perpendicular ﬁelds essentially (e.g. Chevalier & Fransson 1994; Larsson et al. 2011), which in ex- no ion acceleration is seen, also inconsistent with our lower lim- treme cases powers the most luminous supernovae yet discovered its on (cid:15)nth. Hadronic scenarios involving ion acceleration at quasi- (Smith et al. 2007). Our ﬁnding that a large fraction of nova opti- parallel shocks thus appear to be the most viable source of relativis- cal emission is powered by shocks requires a revision of standard tic particles responsible for nova gamma-rays. models that instead assume nova thermal emission results from the However, this conclusion is problematic. Most of the power direct outwards transport of thermal energy released in the white dissipated in the system occurs at the reverse shock, making it the dwarf envelope by nuclear burning (e.g. Hillman et al. 2014). The most natural location for particle acceleration. If the fast outﬂow is ultimate energy source (the thermonuclear runaway) is not in ques- ejected over a timescale which is long compared to the dynamical tion, but shock-powered emission taps into the diﬀerential kinetic time near the surface of the hydrostatic white dwarf envelope (i.e., energy of the outﬂow instead of its thermal content advected or a quasi-steady wind), then its magnetic ﬁeld will become domi- diﬀused from small radii. As suggested by Metzger et al. (2014), nated by its azimuthal component at the much larger radii where § shock-powered optical emission might also help explain some of the reverse shock occurs ( ). Such a ﬁeld, being perpendicular to the irregular light curve shapes, including plateaus and secondary the radial shock normal, is not conducive to ion acceleration. maxima, observed in some novae (e.g., Strope et al. 2010). A sim- This mystery could be resolved if particle acceleration oc- ilar mechanism of internal shocks within a time-variable outﬂow curs at localized regions of quasi-parallel shocks, creating spe- ∼ may power an optical transient from the remnant produced by the cial regions of eﬃcient acceleration ((cid:15)nth 0.1) with (cid:15)nth = 0 merger of binary white dwarfs (Beloborodov 2014). across the larger bulk, thereby resulting in an average value of ∼ − − − If the majority of nova optical light is powered by radiative (cid:15)nth 10 3 10 2 consistent with observations. These localized (cid:13) c 2014 RAS, MNRAS 000, 1–?? ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: shocks,

text: then

text: the

text: ratio

text: of

text: optical

text: and

text: gamma-ray

text: ﬂuxes

text: should

text: be

text: relatively

text: constant

text: in

text: time,

text: assuming

text: that

text: the

text: microphysical

text: pa-

text: rameters

text: of

text: the

text: shocks

text: remain

text: constant

text: (and

text: that

text: the

text: conditions

text: τp−p

font: LUMETV+rtxmi

text: and

text: tcool/t

text: remain

text: satisﬁed

text: in

text: hadronic

text: and

text: lep-

text: tonic

text: scenarios,

text: respectively).

text: Nova

text: Sco

text: is

text: consistent

text: with

text: a

text: tem-

text: porally

text: constant

text: ratio

text: Lγ/Lopt,

text: while

text: nova

text: Del

text: shows

text: evidence

text: for

text: the

text: gamma-ray

text: emission

text: becoming

text: relatively

text: stronger

text: with

text: time

text: (Fig.

text: The

text: apparent

text: increase

text: of

text: Lγ/Lopt

text: in

text: nova

text: Del

text: could

text: result

text: from

text: the

text: reverse

text: shock

text: thermal

text: emission

text: becoming

text: less

text: radiatively

font: PFVDVP+txsy

text: eﬃcient

text: with

text: time,

text: i.e.

text: χ

text: t

text: (eq.

text: Alternatively,

text: it

text: may

text: reﬂect

text: the

text: early

text: transition

text: in

text: this

text: event

text: to

text: a

text: nebular

text: phase

text: (as

text: occurred

text: on

text: Day

text: Skopal

text: et

text: al.

text: resulting

text: in

text: a

text: loss

text: of

text: ﬂux

text: out

text: of

text: the

text: measured

text: optical

text: band

text: and

text: into

text: the

text: UV.

text: If

text: nova

text: shocks

text: are

text: indeed

text: radiative,

text: then

text: we

text: predict

text: that

text: Lγ/Lopt

text: should

text: not

text: exceed

text: at

text: any

text: time,

text: since

text: this

text: represents

text: the

text: limit

text: of

text: shock-dominated

text: emis-

text: sion

text: fsh

text: (eq.

text: for

text: realistic

text: maximum

text: eﬃciencies

text: The

text: gamma-ray

text: to

text: optical

text: ﬂux

text: ratio

text: also

text: places

text: a

text: lower

text: limit

text: on

text: the

text: acceleration

text: eﬃciency

text: of

text: relativistic

text: non-thermal

text: par-

text: ticles

text: at

text: non-relativistic

text: shocks.

text: For

text: nova

text: Sco

text: and

text: nova

text: Del

text: we

text: constrain

text: and

text: respectively

text: (Table

text: Unlike

text: tra-

text: ditional

text: studies

text: modeling

text: particle

text: acceleration

text: in

text: supernova

text: rem-

text: nants,

text: nova

text: shocks

text: evolve

text: in

text: real-time,

text: in

text: principle

text: allowing

text: for

text: the

text: study

text: of

text: time-dependent

text: eﬀects.

text: Models

text: used

text: to

text: constrain

text: par-

text: ticle

text: acceleration

text: in

text: supernova

text: remnants

text: also

text: sometimes

text: require

text: as-

text: sumptions

text: about

text: the

text: escape

text: fraction

text: of

text: relativistic

text: particles,

text: e.g.

text: if

text: gamma-rays

text: are

text: produced

text: by

text: the

text: collision

text: of

text: relativistic

text: protons

text: with

text: nearby

text: molecular

text: clouds

text: of

text: irregular

text: geometry.

text: By

text: contrast,

text: nova

text: ejecta

text: serves

text: as

text: a

text: relatively

text: eﬃcient

text: hadronic

text: “calorimeter”

text: since

text: relativistic

text: protons

text: are

text: probably

text: trapped

text: when

text: gamma-rays

text: are

text: observed

text: (eq.

text: Neither

text: hadronic

text: nor

text: leptonic

text: scenarios

text: for

text: nova

text: gamma-ray

text: emission

text: can

text: be

text: ruled

text: out

text: by

text: modeling

text: of

text: the

text: γ

text: ray

text: spectrum

text: alone

text: (Ackermann

text: et

text: al.

text: However,

text: the

text: tension

text: between

text: the

text: value

text: we

text: ﬁnd

text: is

text: required

text: in

text: nova

text: Sco

text: and

text: the

text: much

text: lower

text: elec-

text: since

text: outburst.

text: In

text: Nova

text: Sco,

text: large

text: squares

text: show

text: nights

text: with

text: full

text: NIR+VBR

text: tron

text: acceleration

text: eﬃciency

text: inferred

text: from

text: observations

text: coverage,

text: while

text: small

text: triangles

text: show

text: the

text: ﬂux

text: ratio

text: estimated

text: using

text: just

text: the

text: of

text: supernova

text: remnants

text: (e.g.

text: Morlino

text: Caprioli

text: and

text: theoret-

text: V-band

text: magnitude

text: adopting

text: the

text: same

text: bolometric

text: correction

text: as

text: measured

text: ical

text: modeling

text: (Kato

text: Park

text: et

text: al.

text: appears

text: to

text: disfavor

text: the

text: with

text: the

text: full

text: data

text: color

text: data

text: on

text: Day

text: leptonic

text: scenario.

text: Leptonic

text: models

text: also

text: require

text: a

text: ﬂat

text: injected

text: spec-

text: trum

text: p

text: to

text: be

text: energetically

text: feasible

text: (eq.

text: In

text: contrast,

text: ion

text: acceleration

text: eﬃciencies

text: as

text: high

text: as

text: are

text: found

text: for

text: non-

text: This

text: shock-powered

text: emission

text: mechanism

text: is

text: similar

text: to

text: that

text: relativistic

text: shocks

text: by

text: hybrid

text: PIC

text: simulations,

text: but

text: only

text: in

text: cases

text: at

text: work

text: in

text: some

text: core

text: collapse

text: supernovae

text: when

text: the

text: ejecta

text: when

text: the

text: upstream

text: magnetic

text: ﬁeld

text: is

text: parallel

text: to

text: the

text: shock

text: normal

text: from

text: the

text: exploding

text: star

text: interacts

text: with

text: dense

text: circumstellar

text: matter

text: (Caprioli

text: Spitkovsky

text: for

text: perpendicular

text: ﬁelds

text: essentially

text: (e.g.

text: Chevalier

text: Fransson

text: Larsson

text: et

text: al.

text: which

text: in

text: ex-

text: no

text: ion

text: acceleration

text: is

text: seen,

text: also

text: inconsistent

text: with

text: our

text: lower

text: lim-

text: treme

text: cases

text: powers

text: the

text: most

text: luminous

text: supernovae

text: yet

text: discovered

text: its

text: on

text: Hadronic

text: scenarios

text: involving

text: ion

text: acceleration

text: at

text: quasi-

text: (Smith

text: et

text: al.

text: Our

text: ﬁnding

text: that

text: a

text: large

text: fraction

text: of

text: nova

text: opti-

text: parallel

text: shocks

text: thus

text: appear

text: to

text: be

text: the

text: most

text: viable

text: source

text: of

text: relativis-

text: cal

text: emission

text: is

text: powered

text: by

text: shocks

text: requires

text: a

text: revision

text: of

text: standard

text: tic

text: particles

text: responsible

text: for

text: nova

text: gamma-rays.

text: models

text: that

text: instead

text: assume

text: nova

text: thermal

text: emission

text: results

text: from

text: the

text: However,

text: this

text: conclusion

text: is

text: problematic.

text: Most

text: of

text: the

text: power

text: direct

text: outwards

text: transport

text: of

text: thermal

text: energy

text: released

text: in

text: the

text: white

text: dissipated

text: in

text: the

text: system

text: occurs

text: at

text: the

text: reverse

text: shock,

text: making

text: it

text: the

text: dwarf

text: envelope

text: by

text: nuclear

text: burning

text: (e.g.

text: Hillman

text: et

text: al.

text: The

text: most

text: natural

text: location

text: for

text: particle

text: acceleration.

text: If

text: the

text: fast

text: outﬂow

text: is

text: ultimate

text: energy

text: source

text: (the

text: thermonuclear

text: runaway)

text: is

text: not

text: in

text: ques-

text: ejected

text: over

text: a

text: timescale

text: which

text: is

text: long

text: compared

text: to

text: the

text: dynamical

text: tion,

text: but

text: shock-powered

text: emission

text: taps

text: into

text: the

text: diﬀerential

text: kinetic

text: time

text: near

text: the

text: surface

text: of

text: the

text: hydrostatic

text: white

text: dwarf

text: envelope

text: (i.e.,

text: energy

text: of

text: the

text: outﬂow

text: instead

text: of

text: its

text: thermal

text: content

text: advected

text: or

text: a

text: quasi-steady

text: wind),

text: then

text: its

text: magnetic

text: ﬁeld

text: will

text: become

text: domi-

text: diﬀused

text: from

text: small

text: radii.

text: As

text: suggested

text: by

text: Metzger

text: et

text: al.

text: nated

text: by

text: its

text: azimuthal

text: component

text: at

text: the

text: much

text: larger

text: radii

text: where

text: shock-powered

text: optical

text: emission

text: might

text: also

text: help

text: explain

text: some

text: of

text: the

text: reverse

text: shock

text: occurs

text: Such

text: a

text: ﬁeld,

text: being

text: perpendicular

text: to

text: the

text: irregular

text: light

text: curve

text: shapes,

text: including

text: plateaus

text: and

text: secondary

text: the

text: radial

text: shock

text: normal,

text: is

text: not

text: conducive

text: to

text: ion

text: acceleration.

text: maxima,

text: observed

text: in

text: some

text: novae

text: (e.g.,

text: Strope

text: et

text: al.

text: A

text: sim-

text: This

text: mystery

text: could

text: be

text: resolved

text: if

text: particle

text: acceleration

text: oc-

text: ilar

text: mechanism

text: of

text: internal

text: shocks

text: within

text: a

text: time-variable

text: outﬂow

text: curs

text: at

text: localized

text: regions

text: of

text: quasi-parallel

text: shocks,

text: creating

text: spe-

text: may

text: power

text: an

text: optical

text: transient

text: from

text: the

text: remnant

text: produced

text: by

text: the

text: cial

text: regions

text: of

text: eﬃcient

text: acceleration

text: with

text: merger

text: of

text: binary

text: white

text: dwarfs

text: (Beloborodov

text: across

text: the

text: larger

text: bulk,

text: thereby

text: resulting

text: in

text: an

text: average

text: value

text: of

text: If

text: the

text: majority

text: of

text: nova

text: optical

text: light

text: is

text: powered

text: by

text: radiative

text: consistent

text: with

text: observations.

text: These

text: localized

text: c

text: RAS,

text: MNRAS

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: matrix elements can be easily calculated. The (cid:104)h | state satisﬁes w (cid:104)h |L = (cid:104)h |h and (cid:104)h |h(cid:105) = 1 (3.18) w 0 w w so for most purposes we can use (cid:104)h | in place of (cid:104)h|, and L in place of L . We will use this w n n construction in the next section to create a modiﬁed version of the Virasoro conformal block projector from equation (2.13), utilizing the L in place of the L generators. n n 3.2 Heavy-light Virasoro blocks from a classical background We would like to compute the conformal blocks for the semi-classical heavy-light corre- lator V (z) = (cid:104)O (∞)O (1) [P ] O (z)O (0)(cid:105) (3.19) h H1 H2 h L1 L2 where P projects onto the state |h(cid:105) and all of its Virasoro descendants. We will choose the h − normalization of Vh(z) so that its leading term is zh 2hL in an expansion around z = 0. We now allow h (cid:54)= h and h (cid:54)= h for maximal generality, but we will need to take H1 H2 L1 L2 δ = h − h with |δ | (cid:28) c (3.20) H H1 H2 H and we also deﬁne δ = h − h . This computation is naively much more diﬃcult than L L1 L2 the one in section , because the Virasoro generators act on O to produce factors of h ∝ c. H H This means that the contribution from states containing L− with n > 1 in P no longer n h vanish, since the factors of h in (cid:104)O (∞)O (1)L− |h(cid:105) compensate for factors of c in the H H1 H2 n normalization. However, we will make use of the conformal transformation from the previous section to simplify the computation, basically turning it into a recapitulation of the global conformal case. Let us begin by transforming the light operators in the correlator to w coordinates, so we want to compute the Virasoro blocks for V (w) = (cid:104)O (∞)O (1) [P ] O (w)O (0)(cid:105). (3.21) h H1 H2 h L1 L2 (cid:48) (cid:48) This formula diﬀers from Vh(z) by the Jacobian factor (z (w))hL1(z (0))hL2, which we will restore below. We claim that in the heavy-light semi-classical limit P ≈ (cid:88) Lk−1m1 · · · Lk−nmn|h(cid:105)(cid:104)hw|Lkmnn · · · Lkm11 (3.22) h,w {mi,ki} N{mi,ki} is a projector onto the irreducible representation of the Virasoro algebra with primary |h(cid:105). We have used ≈ because this basis is not orthogonal at higher orders in 1/c, though one can easily write down a formula without approximations by using the matrix of inner products of states instead of just using the normalizations N{ }. We need to show that the normalizations for mi,ki L are identical to those for L , that this is a projection operator satisfying (P )2 = P , n n h,w h,w – 11 – ##LTLine##

text: matrix

text: elements

text: can

text: be

text: easily

text: calculated.

text: The

text: state

text: satisﬁes

text: w

text: |L

text: |h

text: and

text: w

text: w

text: w

text: so

text: for

text: most

text: purposes

text: we

text: can

text: use

text: in

text: place

text: of

text: and

text: L

text: in

text: place

text: of

text: L

text: We

text: will

text: use

text: this

text: w

text: n

text: n

text: construction

text: in

text: the

text: next

text: section

text: to

text: create

text: a

text: modiﬁed

text: version

text: of

text: the

text: Virasoro

text: conformal

text: block

text: projector

text: from

text: equation

text: utilizing

text: the

text: L

text: in

text: place

text: of

text: the

text: L

text: generators.

text: n

text: n

label: section

text: Heavy-light

text: Virasoro

text: blocks

text: from

text: a

text: classical

text: background

text: We

text: would

text: like

text: to

text: compute

text: the

text: conformal

text: blocks

text: for

text: the

text: semi-classical

text: heavy-light

text: corre-

text: lator

text: V

text: (z)

text: (∞)O

text: [P

text: O

text: (z)O

text: h

text: h

text: where

text: P

text: projects

text: onto

text: the

text: state

text: and

text: all

text: of

text: its

text: Virasoro

text: descendants.

text: We

text: will

text: choose

text: the

text: h

text: normalization

text: of

text: Vh(z)

text: so

text: that

text: its

text: leading

text: term

text: is

text: zh

text: in

text: an

text: expansion

text: around

text: z

text: We

text: now

text: allow

text: h

text: h

text: and

text: h

text: h

text: for

text: maximal

text: generality,

text: but

text: we

text: will

text: need

text: to

text: take

text: δ

text: h

text: h

text: with

text: |δ

text: c

text: H

text: H

text: and

text: we

text: also

text: deﬁne

text: δ

text: h

text: h

text: This

text: computation

text: is

text: naively

text: much

text: more

text: diﬃcult

text: than

text: L

text: the

text: one

text: in

text: section

text: because

text: the

text: Virasoro

text: generators

text: act

text: on

text: O

text: to

text: produce

text: factors

text: of

text: h

text: c.

text: H

text: H

text: This

text: means

text: that

text: the

text: contribution

text: from

text: states

text: containing

text: L−

text: with

text: n

text: in

text: P

text: no

text: longer

text: n

text: h

text: vanish,

text: since

text: the

text: factors

text: of

text: h

text: in

text: (∞)O

text: compensate

text: for

text: factors

text: of

text: c

text: in

text: the

text: H

text: n

text: normalization.

text: However,

text: we

text: will

text: make

text: use

text: of

text: the

text: conformal

text: transformation

text: from

text: the

text: previous

text: section

text: to

text: simplify

text: the

text: computation,

text: basically

text: turning

text: it

text: into

text: a

text: recapitulation

text: of

text: the

text: global

text: conformal

text: case.

text: Let

text: us

text: begin

text: by

text: transforming

text: the

text: light

text: operators

text: in

text: the

text: correlator

text: to

text: w

text: coordinates,

text: so

text: we

text: want

text: to

text: compute

text: the

text: Virasoro

text: blocks

text: for

text: V

text: (w)

text: (∞)O

text: [P

text: O

text: (w)O

text: h

text: h

text: This

text: formula

text: diﬀers

text: from

text: Vh(z)

text: by

text: the

text: Jacobian

text: factor

text: (z

text: which

text: we

text: will

text: restore

text: below.

text: We

text: claim

text: that

text: in

text: the

text: heavy-light

text: semi-classical

text: limit

text: P

text: h,w

text: {mi,ki}

text: N{mi,ki}

text: is

text: a

text: projector

text: onto

text: the

text: irreducible

text: representation

text: of

text: the

text: Virasoro

text: algebra

text: with

text: primary

text: We

text: have

text: used

text: because

text: this

text: basis

text: is

text: not

text: orthogonal

text: at

text: higher

text: orders

text: in

text: though

text: one

text: can

text: easily

text: write

text: down

text: a

text: formula

text: without

text: approximations

text: by

text: using

text: the

text: matrix

text: of

text: inner

text: products

text: of

text: states

text: instead

text: of

text: just

text: using

text: the

text: normalizations

text: N{

text: We

text: need

text: to

text: show

text: that

text: the

text: normalizations

text: for

text: mi,ki

text: L

text: are

text: identical

text: to

text: those

text: for

text: L

text: that

text: this

text: is

text: a

text: projection

text: operator

text: satisfying

text: (P

text: P

text: n

text: n

text: h,w

text: h,w

text: ##LTLine##


text: and that it is complete, i.e. that all states in the representation are included with proper normalization. The state (cid:104)h | was deﬁned in equation (3.17). w If P is a projector, then its completeness follows because the L form a basis for h,w n the Virasoro algebra. The fact that P is projector, and the equivalence of normalization h,w factors, follows from the general identiﬁcation of matrix elements (cid:104)h |Lkn · · · Lk1 |h(cid:105) = (cid:104)h|Lkn · · · Lk1 |h(cid:105) (3.23) w mn m1 mn m1 for arbitrary m , k , and n. This identiﬁcation holds because the L and L both satisfy i i m m the Virasoro algebra, so we can use their commutation relations to move m < 0 to the left i and m > 0 to the right. Since both generators annihilate |h(cid:105) when m > 0 and (cid:104)h | and (cid:104)h| i i w when m < 0, the full computation is determined by the Virasoro algebra. Note that despite i † the presence of the adjoint state (cid:104)h | we have not made use of L .2 w n Thus we can compute the heavy-light Virasoro block using V (w) = (cid:104)O (∞)O (1) [P ] O (w)O (0)(cid:105). (3.24) h H1 H2 h,w L1 L2 The matrix elements (cid:104)h |Lkn · · · Lk1 O (0)O (w)(cid:105) (3.25) w mn m1 L1 L2 are identical to the matrix elements of light operators in a ﬂat background, except now they depend on w instead of z. This again follows because the L satisfy the Virasoro algebra, n and act according to equation (2.15) except with L → L and O(z) → O(w). However, the n n heavy matrix elements (cid:104)O (∞)O (1)Lk−1 · · · Lk−n |h(cid:105) (3.26) H1 H2 m1 mn are non-trivial, and have no direct equivalent in a ﬂat background. They can be obtained from equation (3.11), and its generalization to multiple T (w) insertions, simply by expanding in w. In fact, in the case of the global generator L− = ∂ , we need only compute 1 w (cid:104)OH1(∞)OH2(1)Lk−1|h(cid:105) = wli→m0 ∂wk (cid:104)OH1(∞)OH2(1)Oh(w)(cid:105) − − = α hcH1H2h wli→m0 ∂wk (1 − w) h+δH/α (3.27) since L− translates O in the w coordinate system. This diﬀers from the equivalent result 1 h in a ﬂat metric only by the intriguing replacement δ → δ /α. H H We have all the ingredients we need to compute the heavy-light Virasoro block at large (cid:104) | 2One may worry whether the state h is physical and thus whether our construction is meaningful. w P Ultimately, one may view it simply as a convenience for constructing the projection operator , whose h,w action on any state can be deﬁned operationally by using (3.23). The projector thus deﬁned is easily seen P P L L | (cid:105) to satisfy 2 = , and also to project any state onto the basis k−1 . . . k−n h , which is all that is h,w h,w m1 mn required for the construction of the conformal blocks. – 12 – ##LTLine##

text: and

text: that

text: it

text: is

text: complete,

text: i.e.

text: that

text: all

text: states

text: in

text: the

text: representation

text: are

text: included

text: with

text: proper

text: normalization.

text: The

text: state

text: was

text: deﬁned

text: in

text: equation

text: w

text: If

text: P

text: is

text: a

text: projector,

text: then

text: its

text: completeness

text: follows

text: because

text: the

text: L

text: form

text: a

text: basis

text: for

text: h,w

text: n

text: the

text: Virasoro

text: algebra.

text: The

text: fact

text: that

text: P

text: is

text: projector,

text: and

text: the

text: equivalence

text: of

text: normalization

text: h,w

text: factors,

text: follows

text: from

text: the

text: general

text: identiﬁcation

text: of

text: matrix

text: elements

text: |Lkn

text: w

text: mn

text: mn

text: for

text: arbitrary

text: m

text: k

text: and

text: n.

text: This

text: identiﬁcation

text: holds

text: because

text: the

text: L

text: and

text: L

text: both

text: satisfy

text: i

text: i

text: m

text: m

text: the

text: Virasoro

text: algebra,

text: so

text: we

text: can

text: use

text: their

text: commutation

text: relations

text: to

text: move

text: m

text: to

text: the

text: left

text: i

text: and

text: m

text: to

text: the

text: right.

text: Since

text: both

text: generators

text: annihilate

text: when

text: m

text: and

text: and

text: i

text: i

text: w

text: when

text: m

text: the

text: full

text: computation

text: is

text: determined

text: by

text: the

text: Virasoro

text: algebra.

text: Note

text: that

text: despite

text: i

text: the

text: presence

text: of

text: the

text: adjoint

text: state

text: we

text: have

text: not

text: made

text: use

text: of

text: L

text: w

text: n

text: Thus

text: we

text: can

text: compute

text: the

text: heavy-light

text: Virasoro

text: block

text: using

text: V

text: (w)

text: (∞)O

text: [P

text: O

text: (w)O

text: h

text: h,w

text: The

text: matrix

text: elements

text: |Lkn

text: O

text: w

text: mn

text: are

text: identical

text: to

text: the

text: matrix

text: elements

text: of

text: light

text: operators

text: in

text: a

text: ﬂat

text: background,

text: except

text: now

text: they

text: depend

text: on

text: w

text: instead

text: of

text: z.

text: This

text: again

text: follows

text: because

text: the

text: L

text: satisfy

text: the

text: Virasoro

text: algebra,

text: n

text: and

text: act

text: according

text: to

text: equation

text: except

text: with

text: L

text: L

text: and

text: O(z)

text: O(w).

text: However,

text: the

text: n

text: n

text: heavy

text: matrix

text: elements

text: (∞)O

text: Lk−n

text: mn

text: are

text: non-trivial,

text: and

text: have

text: no

text: direct

text: equivalent

text: in

text: a

text: ﬂat

text: background.

text: They

text: can

text: be

text: obtained

text: from

text: equation

text: and

text: its

text: generalization

text: to

text: multiple

text: T

text: (w)

text: insertions,

text: simply

text: by

text: expanding

text: in

text: w.

text: In

text: fact,

text: in

text: the

text: case

text: of

text: the

text: global

text: generator

text: L−

text: we

text: need

text: only

text: compute

text: w

text: ∂wk

text: α

text: ∂wk

text: w)

text: h+δH/α

text: since

text: L−

text: translates

text: O

text: in

text: the

text: w

text: coordinate

text: system.

text: This

text: diﬀers

text: from

text: the

text: equivalent

text: result

text: h

text: in

text: a

text: ﬂat

text: metric

text: only

text: by

text: the

text: intriguing

text: replacement

text: δ

text: δ

text: /α.

text: H

text: H

text: We

text: have

text: all

text: the

text: ingredients

text: we

text: need

text: to

text: compute

text: the

text: heavy-light

text: Virasoro

text: block

text: at

text: large

text: may

text: worry

text: whether

text: the

text: state

text: h

text: is

text: physical

text: and

text: thus

text: whether

text: our

text: construction

text: is

text: meaningful.

text: w

text: P

text: Ultimately,

text: one

text: may

text: view

text: it

text: simply

text: as

text: a

text: convenience

text: for

text: constructing

text: the

text: projection

text: operator

text: whose

text: h,w

text: action

text: on

text: any

text: state

text: can

text: be

text: deﬁned

text: operationally

text: by

text: using

text: The

text: projector

text: thus

text: deﬁned

text: is

text: easily

text: seen

text: P

text: P

text: L

text: L

text: to

text: satisfy

text: and

text: also

text: to

text: project

text: any

text: state

text: onto

text: the

text: basis

text: k−n

text: h

text: which

text: is

text: all

text: that

text: is

text: h,w

text: h,w

text: mn

text: required

text: for

text: the

text: construction

text: of

text: the

text: conformal

text: blocks.

text: ##LTLine##


text: the Sugawara stress tensor, so that by subtracting out this contribution we can factor out the contributions to the extended conformal blocks from the current. More precisely, we deﬁne L(0) ≡ L − Lsug. (3.37) n n n (0) The key point is that the L and J generators provide a basis that factors the algebra into n n separate Virasoro and U(1) sectors: (cid:104) (cid:105) c − 1 L(0), L(0) = (n − m)L(0) + n(n2 − 1)δ , n m n+m 12 n+m,0 (cid:104) (cid:105) L(0), J = 0. (3.38) n m Furthermore, it is straightforward to check that states |φ(cid:105) that are primary with respect to (0) the L ’s and J ’s, with weight h and charge q , are primary under L as well, with weight n n φ n q2 h(0) = h − φ . (3.39) 2k In the conformal block, all descendants of the exchanged state |h (cid:105) can be written in a basis p (0) of J ’s and L s. We restrict our attention to conformal blocks where the exchanged primary n n (0) is neutral, i.e. q = −q ≡ q , q = −q ≡ q . Consider the action of any of the L s H1 H2 H L1 L2 S n inside the three-point function (cid:104)h |φ (z)φ (0)(cid:105): p L1 L2 (cid:18) (cid:19) 1 (cid:104)h |L(0)φ (z)φ (0)(cid:105) = (cid:104)h | L − (J J − + J J − + . . . J − J ) φ (z)φ (0)(cid:105), p n L1 L2 p n 2k 1 n 1 2 n 2 n 1 1 L1 L2 (3.40) where we have used the fact that J with n ≤ 0 annihilates (cid:104)h |. Now, note that n p − 1 z qL2 /k (cid:104)h |φ (z)φ (0)(cid:105) = c − = c . (3.41) p L1 L2 L1L2p zhL1+hL2 hp L1L2hp zh(L01)+h(L02)−hp (0) (0) (0) Thus, we ﬁnd that L acts to the right like L , but with h , h replaced with h , h : n n L1 L2 L1 L2 n − 1 (cid:104)h |L(0)φ (z)φ (0)(cid:105) = (cid:104)h |(h (n + 1)zn + zn+1∂ − q2 zn)φ (z)φ (0)(cid:105) p n L1 L2 p L1 z 2k L L1 L2 − 1 = z qL2 /k(h(L01)(n + 1)zn + zn+1∂z)cL1L2hp zh(L01)+h(L02)−hp − = z qL2 /k [(cid:104)hp|LnφL1(z)φL2(0)(cid:105)]hLi→h(L0i) . (3.42) (0) (0) Descendants with multiple L s follow the same derivation; J ’s commute with L s, so n n n those of them with n ≤ 0 can still be moved to the left where they annihilate (cid:104)h |. Then, one p – 16 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: the

text: Sugawara

text: stress

text: tensor,

text: so

text: that

text: by

text: subtracting

text: out

text: this

text: contribution

text: we

text: can

text: factor

text: out

text: the

text: contributions

text: to

text: the

text: extended

text: conformal

text: blocks

text: from

text: the

text: current.

text: More

text: precisely,

text: we

text: deﬁne

text: L

text: Lsug.

text: n

text: n

text: n

text: The

text: key

text: point

text: is

text: that

text: the

text: L

text: and

text: J

text: generators

text: provide

text: a

text: basis

text: that

text: factors

text: the

text: algebra

text: into

text: n

text: n

text: separate

text: Virasoro

text: and

text: sectors:

text: c

text: (n

text: n

text: m

text: n+m

text: J

text: n

text: m

text: Furthermore,

text: it

text: is

text: straightforward

text: to

text: check

text: that

text: states

text: that

text: are

text: primary

text: with

text: respect

text: to

text: the

text: L

text: ’s

text: and

text: J

text: ’s,

text: with

text: weight

text: h

text: and

text: charge

text: q

text: are

text: primary

text: under

text: L

text: as

text: well,

text: with

text: weight

text: n

text: n

text: φ

text: n

text: h

text: φ

text: In

text: the

text: conformal

text: block,

text: all

text: descendants

text: of

text: the

text: exchanged

text: state

text: |h

text: can

text: be

text: written

text: in

text: a

text: basis

text: p

text: of

text: J

text: ’s

text: and

text: L

text: s.

text: We

text: restrict

text: our

text: attention

text: to

text: conformal

text: blocks

text: where

text: the

text: exchanged

text: primary

text: n

text: n

text: is

text: neutral,

text: i.e.

text: q

text: −q

text: q

text: q

text: −q

text: q

text: Consider

text: the

text: action

text: of

text: any

text: of

text: the

text: L

text: s

text: H

text: S

text: n

text: inside

text: the

text: three-point

text: function

text: |φ

text: (z)φ

text: p

text: (z)φ

text: L

text: (J

text: J

text: J

text: J

text: J

text: J

text: φ

text: (z)φ

text: p

text: n

text: p

text: n

text: n

text: n

text: n

text: where

text: we

text: have

text: used

text: the

text: fact

text: that

text: J

text: with

text: n

text: annihilates

text: Now,

text: note

text: that

text: n

text: p

text: z

text: /k

text: |φ

text: (z)φ

text: c

text: c

text: p

text: hp

text: Thus,

text: we

text: ﬁnd

text: that

text: L

text: acts

text: to

text: the

text: right

text: like

text: L

text: but

text: with

text: h

text: h

text: replaced

text: with

text: h

text: h

text: n

text: n

text: n

text: (z)φ

text: |(h

text: (n

text: zn)φ

text: (z)φ

text: p

text: n

text: p

text: z

text: L

text: z

text: z

text: /k

text: Descendants

text: with

text: multiple

text: L

text: s

text: follow

text: the

text: same

text: derivation;

text: J

text: ’s

text: commute

text: with

text: L

text: s,

text: so

text: n

text: n

text: n

text: those

text: of

text: them

text: with

text: n

text: can

text: still

text: be

text: moved

text: to

text: the

text: left

text: where

text: they

text: annihilate

text: Then,

text: one

text: p

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: O (1) H O (z) L O (1) L O (0) H Figure 1. This ﬁgure suggests an AdS picture for the heavy-light CFT correlator – a light probe interacting with a heavy particle or black hole. We will show that the Virasoro blocks can be computed in this limit by placing the CFT in an appropriately chosen 2d metric, which is related to the ﬂat 2 metric by a conformal transformation. States in CFT can be decomposed into irreducible representations of the Virasoro sym- 2 metry algebra. Correlation functions can be written as a sum over the exchange of these irreducible representations, which are individually called conformal partial waves or Virasoro conformal blocks. The conformal blocks are atomic ingredients in the CFT bootstrap [6– 8], and are particularly crucial for obtaining analytic constraints on CFTs [9–25]. Via the AdS/CFT correspondence, one can interpret the Virasoro blocks as the exchange of a sum of AdS wavefunctions. The states in the Virasoro block correspond with the wavefunctions of some primary object in AdS , plus any number of AdS gravitons. In other words, the Vira- 3 3 soro blocks capture quantum gravitational eﬀects in AdS , providing a way to precisely sum 3 up all graviton exchanges. The Virasoro blocks have also been used to study entanglement entropy [26–35] and scrambling [36] in AdS/CFT, both in the vacuum and in the background of a heavy pure state [37]. There has also been signiﬁcant recent progress [38–40] in using modular invariance to understand the spectrum of CFT at large central charge. 2 We will present a new method for computing the Virasoro conformal blocks in the limit of large central charge 3 c = (cid:29) 1, (1.1) 2G – 2 – ##LTFigure## ##LTLine##

text: O

text: H

text: O

text: (z)

text: L

text: O

text: L

text: O

text: H

text: Figure

text: This

label: caption

text: ﬁgure

text: suggests

text: an

text: AdS

text: picture

text: for

text: the

text: heavy-light

text: CFT

text: correlator

text: a

text: light

text: probe

text: interacting

text: with

text: a

text: heavy

text: particle

text: or

text: black

text: hole.

text: We

text: will

text: show

text: that

text: the

text: Virasoro

text: blocks

text: can

text: be

text: computed

text: in

text: this

text: limit

text: by

text: placing

text: the

text: CFT

text: in

text: an

text: appropriately

text: chosen

text: metric,

text: which

text: is

text: related

text: to

text: the

text: ﬂat

text: metric

text: by

text: a

text: conformal

text: transformation.

text: States

text: in

text: CFT

text: can

text: be

text: decomposed

text: into

text: irreducible

text: representations

text: of

text: the

text: Virasoro

text: sym-

text: metry

text: algebra.

text: Correlation

text: functions

text: can

text: be

text: written

text: as

text: a

text: sum

text: over

text: the

text: exchange

text: of

text: these

text: irreducible

text: representations,

text: which

text: are

text: individually

text: called

text: conformal

text: partial

text: waves

text: or

text: Virasoro

text: conformal

text: blocks.

text: The

text: conformal

text: blocks

text: are

text: atomic

text: ingredients

text: in

text: the

text: CFT

text: bootstrap

text: and

text: are

text: particularly

text: crucial

text: for

text: obtaining

text: analytic

text: constraints

text: on

text: CFTs

text: Via

text: the

text: AdS/CFT

text: correspondence,

text: one

text: can

text: interpret

text: the

text: Virasoro

text: blocks

text: as

text: the

text: exchange

text: of

text: a

text: sum

text: of

text: AdS

text: wavefunctions.

text: The

text: states

text: in

text: the

text: Virasoro

text: block

text: correspond

text: with

text: the

text: wavefunctions

text: of

text: some

text: primary

text: object

text: in

text: AdS

text: plus

text: any

text: number

text: of

text: AdS

text: gravitons.

text: In

text: other

text: words,

text: the

text: Vira-

text: soro

text: blocks

text: capture

text: quantum

text: gravitational

text: eﬀects

text: in

text: AdS

text: providing

text: a

text: way

text: to

text: precisely

text: sum

text: up

text: all

text: graviton

text: exchanges.

text: The

text: Virasoro

text: blocks

text: have

text: also

text: been

text: used

text: to

text: study

text: entanglement

text: entropy

text: and

text: scrambling

text: in

text: AdS/CFT,

text: both

text: in

text: the

text: vacuum

text: and

text: in

text: the

text: background

text: of

text: a

text: heavy

text: pure

text: state

text: There

text: has

text: also

text: been

text: signiﬁcant

text: recent

text: progress

text: in

text: using

text: modular

text: invariance

text: to

text: understand

text: the

text: spectrum

text: of

text: CFT

text: at

text: large

text: central

text: charge.

text: We

text: will

text: present

text: a

text: new

text: method

text: for

text: computing

text: the

text: Virasoro

text: conformal

text: blocks

text: in

text: the

text: limit

text: of

text: large

text: central

text: charge

text: c

text: ##LTFigure##

text: ##LTLine##